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Michael Coey Stuart S. P. Parkin Editors Handbook of Magnetism and Magnetic Materials Handbook of Magnetism and Magnetic Materials J. M. D. Coey • Stuart S. P. Parkin Editors Handbook of Magnetism and Magnetic Materials With 618 Figures and 157 Tables 123 Editors J. M. D. Coey School of Physics Trinity College Dublin Ireland Stuart S. P. Parkin Max Planck Institute of Microstructure Physics Halle (Saale) Germany ISBN 978-3-030-63208-3 ISBN 978-3-030-63210-6 (eBook) ISBN 978-3-030-63209-0 (print and electronic bundle) https://doi.org/10.1007/978-3-030-63210-6 © Springer Nature Switzerland AG 2021 All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface Magnetism is a natural phenomenon that arouses curiosity in people of all ages. Electromagnetism, a mainstay of the industrial revolution, supports urban life and communications everywhere, served by soft magnetic materials that guide and concentrate magnetic flux. Permanent magnets are now ubiquitous flux generators, enabling electric mobility, robotics and energy conversion in a range from μW to MW. A handful of about a dozen optimized bulk functional magnetic materials address well over 90% of practical applications. The ability to pattern magnetic thin films has transformed our subject. Progressively scaled to nanometer dimensions, tiny magnetic regions store binary data, which forms the basis of today’s digital world. Their stray fields are detected using minute and exquisitely sensitive magnetic field sensors formed from atomically engineered multi-layer stacks of magnetic thin films that are the first and, still today, the most important crop product of spin electronics. Spintronics, especially, concerns the generation, manipulation and control of the spin angular momentum, which is the source of the electron’s magnetism. Spin-polarized electrical currents or main pure spin currents with no net charge flow can be used to excite or switch the direction of magnetization of magnetic nano-elements. This has opened the door to a range of magnetic devices with properties that go beyond those of charge-based electronics. There are new prospects for memory, storage and computation that are fundamentally spin based. The emerging field of chiral spintronics combines fundamental aspects of chirality, spin and topology. On a more fundamental level, although the theoretical foundations of magnetism in relativity and quantum mechanics were established a century ago, the behaviour of strongly correlated electrons in solids is an unfailing source of surprises for physicists and chemists, materials scientists and engineers. Model magnetic materials can be created to exhibit an astonishing range of physical properties, and increasingly we are learning how to tailor them to suit a particular practical application or theoretical model. The shift of emphasis from bulk, functional magnets to thin films has transformed the range of elements we can use in our materials. Practically, any stable element in the periodic table can now be pressed into service, because the quantities needed in a device are so minute. A billion thin film devices each needing a few nanograms of some new magnetic material consume just a few grams of an unrecoverable resource. v vi Preface This handbook aims to offer a broad perspective on the state of the art in magnetism and magnetic materials. The discovery and dissemination of reliable knowledge about the natural world is a complex process that depends on interactions of individuals with shared values and presumptions. Information is the primary product of their endeavour. It is contained in in papers, patents, reviews, handbooks monographs and textbooks. This is a perpetual work in progress. Knowledge percolates through this sequence, taking ever-more digestible and definitive forms as it is consolidated or eliminated. Now information technology is facilitating this dynamic. Whereas papers are replaced by more up-to-date papers with new sets of references to trace their pedigree and textbooks may be updated perhaps after 10 years, handbooks are compendia of information that need updating on a shorter timescale. This was impractical within the constraints of traditional publication, but the greater flexibility of electronic publication now opens the possibility for authors to update their contributions as time passes, and perspectives shift. The book’s 34 chapters are organized into four parts. After an introduction to the history and basic concepts in the field, there follow 12 chapters covering the fundamentals of solid state magnetism, and the phenomena related to collective magnetic order. Eight chapters are then devoted to the main classes of magnetic materials – elements, metallic compounds, oxides and other nonmetallic compounds, thin films, nanoparticles and artificially engineered materials. Another six chapters treat the methods for preparing and characterizing magnetic materials, and the final part is devoted to some major applications. No fewer than 85 authors have contributed to this handbook. It has taken longer than we originally anticipated, and the patience of the early responders is sincerely appreciated. The format for subsequent updating of the electronic text is by individual chapter, which will avoid such difficulty in the future. We are grateful to the staff at Springer, Claus Ascheron for initiating the project, Werner Skolaut for his patience and encouragement, and Barbara Wolf for efficiently bringing the handbook to hand. October 2021 J. M. D. Coey Stuart S. P. Parkin Contents Volume 1 Part I Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1 History of Magnetism and Basic Concepts . . . . . . . . . . . . . . . . . . . . . J. M. D. Coey 3 2 Magnetic Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ralph Skomski 53 3 Anisotropy and Crystal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ralph Skomski, Priyanka Manchanda, and Arti Kashyap 103 4 Electronic Structure: Metals and Insulators . . . . . . . . . . . . . . . . . . . . Hubert Ebert, Sergiy Mankovsky, and Sebastian Wimmer 187 5 Quantum Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gabriel Aeppli and Philip Stamp 261 6 Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sergej O. Demokritov and Andrei N. Slavin 281 7 Micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lukas Exl, Dieter Suess, and Thomas Schrefl 347 8 Magnetic Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rudolf Schäfer 391 9 Magnetotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Michael Ziese 435 10 Magneto-optics and Laser-Induced Dynamics of Metallic Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mark L. M. Lalieu and Bert Koopmans 11 Magnetostriction and Magnetoelasticity . . . . . . . . . . . . . . . . . . . . . . . Dirk Sander 477 549 vii viii Contents 12 Magnetoelectrics and Multiferroics . . . . . . . . . . . . . . . . . . . . . . . . . . . Jia-Mian Hu and Long-Qing Chen 595 13 Magnetism and Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ilya M. Eremin, Johannes Knolle, and Roderich Moessner 625 Part II Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657 14 Magnetism of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plamen Stamenov 659 15 Metallic Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Ping Liu, Matthew Willard, Wei Tang, Ekkes Brück, Frank de Boer, Enke Liu, Jian Liu, Claudia Felser, Gerhard Fecher, Lukas Wollmann, Olivier Isnard, Emil Burzo, Sam Liu, J. F. Herbst, Fengxia Hu, Yao Liu, Jirong Sun, Baogen Shen, and Anne de Visser 693 16 Metallic Magnetic Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Wu and X.-F. Jin 809 Volume 2 17 Magnetic Oxides and Other Compounds . . . . . . . . . . . . . . . . . . . . . . . J. M. D. Coey 847 18 Dilute Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alberta Bonanni, Tomasz Dietl, and Hideo Ohno 923 19 Single-Molecule Magnets and Molecular Quantum Spintronics . . . Gheorghe Taran, Edgar Bonet, and Wolfgang Wernsdorfer 979 20 Magnetic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011 Sara A. Majetich 21 Artificially Engineered Magnetic Materials . . . . . . . . . . . . . . . . . . . . 1047 Christopher H. Marrows Part III Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081 22 Magnetic Fields and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083 Oliver Portugall, Steffen Krämer, and Yurii Skourski 23 Material Preparation and Thin Film Growth . . . . . . . . . . . . . . . . . . . 1153 Amilcar Bedoya-Pinto, Kai Chang, Mahesh G. Samant, and Stuart S. P. Parkin 24 Magnetic Imaging and Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203 Robert M. Reeve, Hans-Joachim Elmers, Felix Büttner, and Mathias Kläui Contents ix 25 Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255 Jeffrey W. Lynn and Bernhard Keimer 26 Electron Paramagnetic and Ferromagnetic Resonance . . . . . . . . . . . 1297 David Menard and Robert Barklie 27 Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333 Andrew D. Kent, Hendrik Ohldag, Hermann A. Dürr, and Jonathan Z. Sun Part IV Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367 28 Permanent Magnet Materials and Applications . . . . . . . . . . . . . . . . . 1369 Karl-Hartmut Müller, Simon Sawatzki, Roland Gauß and Oliver Gutfleisch 29 Soft Magnetic Materials and Applications . . . . . . . . . . . . . . . . . . . . . . 1435 Frédéric Mazaleyrat 30 Magnetocaloric Materials and Applications . . . . . . . . . . . . . . . . . . . . 1489 Karl G. Sandeman and So Takei 31 Magnetic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527 Myriam Pannetier-Lecoeur and Claude Fermon 32 Magnetic Memory and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553 Wei Han 33 Magnetochemistry and Magnetic Separation . . . . . . . . . . . . . . . . . . . 1593 Peter Dunne 34 Magnetism and Biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1633 Nora M. Dempsey Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1679 About the Editors Michael Coey was born in Belfast in 1945. He studied physics at Cambridge, and then taught English and physics at the Sainik School, Balachadi (Gujarat). There he read Allan Morrish’s Physical Principles of Magnetism from cover to cover (while recovering from jaundice) before moving to Canada in 1968 to join Morrish’s group at the University of Manitoba for a PhD on Mõssbauer spectroscopy of iron oxides. He has worked on magnetism ever since – a life of paid play. After graduating in 1971, he joined Benoy Chakraverty’s group at the CNRS in Grenoble as a postdoc with a letter of appointment signed by Louis Néel. Entering the CNRS the following year, he worked on the metalinsulator as well as the magnetism of amorphous solids and natural minerals. In France, he built the network of collaborators which sustained much of his career. On a sabbatical with Stefan von Molnar at the IBM Research Center at Yorktown Heights, he learned about magneto-transport and the crystal field. Then, in 1979, he moved to Ireland as a lecturer at Trinity College Dublin and set about establishing a magnetism research group in a venerable but woefully underfunded Physics Department. Luckily, support from the EU substitution programme enabled him to begin research on melt-spun magnetic glasses. Following the discovery of Nd2 Fe14 B permanent magnets in 1984, he and colleagues from Grenoble, Birmingham and Berlin launched the Concerted European Action on Magnets. xi xii About the Editors CEAM blossomed into an informal association of 90 academic and industrial research institutes interested in every aspect of the properties, processing and applications of rare-earth iron permanent magnets. He and his student Sun Hong discovered the interstitial nitride magnet Sm2 Fe17 N3 in 1990. The group investigated other rare-earth intermetallic compounds, as well as magnetic oxide films produced by pulsed-laser deposition. During this period, he and David Hurley started up Magnetic Solutions to develop innovative applications of permanent magnets. The scientific landscape in Ireland was transformed by the establishment of Science Foundation Ireland in 2000, given the mission of developing competitive scientific research in Ireland with a budget to match. His group were able to develop a programme in thin film magnetism and spin electronics, producing Europe’s first magnetic tunnel junctions to exhibit 200 % tunnel magnetoresistance. Later they discovered the first zero-moment ferrimagnetic half-metal and explored the garden of magneto-electrochemistry. Michael coey was a promotor of CRANN, Ireland’s nanoscience research centre, and the Science Gallery, now an international franchise, was his brainchild. Together with Dominique Givord, he launched the Joint European Magnetic Symposia (JEMS) and, while chair of C9, the IUPAP Magnetism Committee, inaugurated the Néel medal that is awarded triennially at the International Conference on Magnetism. The 2015 JEMS meeting in Dublin saw a reunion of many of his 60 PhD students, from all over the world. Together they have published many papers. Books include Magnetic Glasses, 1984 (with Kishin Moorjani): Permanent Magnetism, 1999 (with Ralph Skomski): and Magnetism and Magnetic Materials, 2010. Honours include Fellowship of the Royal Society, International membership of the National Academy of Sciences, a Fulbright fellowship, a Humboldt Prize, the Gold Medal of the Royal Irish Academy and the 2019 Born Medal. He has enjoyed visiting professorships at the University of Strasbourg, the National University of Singapore and Beihang University in Beijing. Michael Coey married Wong May, a writer, in 1973; they have two sons and a grand-daughter. About the Editors xiii Stuart S. P. Parkin is a director of the Max Planck Institute of Microstructure Physics, Halle, Germany, and an Alexander von Humboldt Professor, Martin Luther University, Halle-Wittenberg. His research interests include spintronic materials and devices for advanced sensor, memory and logic applications, oxide thin-film heterostructures, topological metals, exotic superconductors, and cognitive devices. Stuart’s discoveries in spintronics enabled a more than 10,000fold increase in the storage capacity of magnetic disk drives. For his work that, thereby, enabled the ‘big data’ world of today. In 2014, he was awarded the Millennium Technology Award from the Technology Academy Finland and, most recently, the King Faisal Prize for Science 2021 for his research into three distinct classes of spintronic memories. Stuart is a fellow or member of: The Royal Society, the Royal Academy of Engineering, the National Academy of Sciences, the National Academy of Engineering, the German National Academy of Science – Leopoldina, The Royal Society of Edinburgh, The Indian Academy of Sciences, and TWAS – The academy of sciences for the developing world. Stuart is also a fellow of the American Physical Society: the Institute of Electrical and Electronics Engineers (IEEE) the Institute of Physics, London: the American Association for the Advancement of Science (AAAS); and the Materials Research Society. Stuart has published more than 600 papers and has more than 121 issued patents. His h factor is 120. Clarivate Analytics named him a Highly Cited Researcher in 2018, 2019, 2020 and 2021. Stuart’s numerous awards include the American Physical Society International Prize for New Materials (1994); the Europhysics Prize for Outstanding Achievement in Solid State Physics (1997); the 2009 IUPAP Magnetism Prize and Néel Medal; the 2012 von Hippel Award – Materials Research Society; the 2013 Swan Medal – Institute of Physics; an Alexander von Humboldt Professorship – International Award for Research (2014); and ERC Advanced Grant – SORBET (2015). Stuart has been a distinguished visiting professor at several universities worldwide including: National University of Singapore; National Taiwan University; National Yunlin University of Science and Technology, Taiwan; Eindhoven University of Tech- xiv About the Editors nology, The Netherlands; KAIST, Korea; and University College London. Stuart has been awarded four honorary doctorates by: RWTH Aachen University (2007), Eindhoven University of Technology (2008), The University of Regensburg (2011), and Technische Universität Kaiserslautern, Germany (2013). Prior to being appointed to the Max Planck Society, Stuart had spent a large part of his career with IBM Research at the San Jose Research Laboratory, which became the Almaden Research Center when it moved to a new campus. Stuart was appointed an IBM Fellow, IBM’s highest technical honour, by IBM’s chairman, Louis Gerstner in 1999. He received his BA physics and theoretical physics (1977), an MA, and his PhD (1980) from the University of Cambridge. He was a student at Trinity College, Cambridge, where he received an entrance scholarship (1974), a senior scholarship (1975), a research scholarship (1977) and was elected a research fellow (1979). In 2014, he became an honorary fellow. Stuart received a Royal Society European Exchange Fellowship to carry out postdoctoral research at the Laboratoire de Physique des Solides, Université Paris-Sud, France, in 1980–1981 and an IBM World Trade Fellowship to carry out research at IBM in San Jose. Contributors Gabriel Aeppli Physics Department (ETHZ), Institut de Physique (EPFL) and Photon Science Division (PSI), ETHZ, EPFL and PSI, Zürich, Lausanne and Villigen, Switzerland Robert Barklie School of Physics, Trinity College, Dublin, Ireland Amilcar Bedoya-Pinto Max Planck Institute of Microstructure Physics, Halle (Saale), Germany Alberta Bonanni Institut für Halbleiter- und Festkörperphysik, Johannes Kepler University, Linz, Austria Edgar Bonet Néel Institute, CNRS, Grenoble, France Ekkes Brück Delft University of Technology, Delft, The Netherlands Emil Burzo Babes-Bolyai University, Romania, Cluj-Napoca, Romania Felix Büttner Helmholtz-Zentrum Berlin für Materialien und Energie, Berlin, Germany Kai Chang Beijing Academy of Quantum Information Sciences, Beijing, China Long-Qing Chen Materials Research Institute, and Department of Materials Science and Engineering, The Pennsylvania State University, University Park, PA, USA Michael Coey School of Physics, Trinity College, Dublin, Ireland Frank de Boer University of Amsterdam, Amsterdam, The Netherlands Anne de Visser Van der Waals-Zeeman Institute, University of Amsterdam, Amsterdam, The Netherlands Sergej O. Demokritov Institute for Applied Physics and Center for Nanotechnology, University of Muenster, Muenster, Germany Nora M. Dempsey Institut Néel, CNRS & Université Grenoble Alpes, Grenoble, France xv xvi Contributors Tomasz Dietl International Research Centre MagTop, Institute of Physics, Polish Academy of Sciences, Warsaw, Poland WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan Peter Dunne Institut de Physique et de Chimie des Matériaux de Stasbourg, Strasbourg, France Hermann A. Dürr Department of Physics and Astronomy, Uppsala University, Uppsala, Sweden Hubert Ebert München, Department Chemie, Ludwig-Maximilians-Universität, München, Germany Hans-Joachim Elmers Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany Ilya M. Eremin Institut für Theoretische Physik III, Ruhr-Universität Bochum, Bochum, Germany Lukas Exl University of Vienna Research Platform MMM Mathematics – Magnetism – Materials, University of Vienna, and Wolfgang Pauli Institute, Wien, Austria Gerhard Fecher Max-Planck-Institute für Chemische Physik fester Stoffe, Dresden, Germany Claudia Felser Max-Planck-Institute für Chemische Physik fester Stoffe, Dresden, Germany Claude Fermon Service de Physique de l’Etat Condensé, DRF/IRAMIS/SPEC CNRS UMR 3680 CEA Saclay, Gif sur Yvette, France Roland Gauß EIT RawMaterials GmbH, Berlin, Germany Oliver Gutfleisch Technische Universität Darmstadt, Materialwissenschaft, Darmstadt, Germany Wei Han International Center for Quantum Materials, School of Physics, Peking University, Beijing, China J. F. Herbst Research & Development, General Motors R&D Center, Warren, MI, USA Fengxia Hu Institute of Physics, Chinese Academy of Sciences, Beijing, China Jia-Mian Hu Department of Materials Science and Engineering, University of Wisconsin-Madison, Madison, WI, USA Olivier Isnard Institute Néel and Université Grenoble Alpes, Grenoble, France X.-F. Jin Department of Physics and State Key Laboratory of Surface Physics, Fudan University, Shanghai, People’s Republic of China Arti Kashyap IIT Mandi, Mandi, HP, India Contributors xvii Bernhard Keimer Max-Planck Institute for Solid State Research, Germany Stuttgart, Andrew D. Kent Center for Quantum Phenomena, Department of Physics, New York University, New York, NY, USA Mathias Kläui Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany Johannes Knolle Blackett Laboratory, Imperial College London, London, UK Bert Koopmans Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands Steffen Krämer LNCMI-CNRS (UPR3228), EMFL, Univ. Grenoble Alpes, INSA Toulouse, Univ. Toulouse 3, Grenoble, France Mark L. M. Lalieu Department of Applied Physics, Eindhoven University of Technology, Eindhoven, The Netherlands Enke Liu Institute of Physics, Chinese Academy of Sciences, Beijing, China J. Ping Liu University of Texas at Arlington, Arlington, TX, USA Jian Liu Ningbo Institute of Materials Technology and Engineering, Chinese Academy of Sciences, Ningbo, China Sam Liu University of Dayton, Dayton, OH, USA Yao Liu Institute of Physics, Chinese Academy of Sciences, Beijing, China Jeffrey W. Lynn NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg, MD, USA Sara A. Majetich Physics Department, Carnegie Mellon University, Pittsburgh, PA, USA Priyanka Manchanda Howard University, Washington, DC, USA Sergiy Mankovsky München, Universität, München, Germany Department Chemie, Ludwig-Maximilians- Christopher H. Marrows School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom Frédéric Mazaleyrat SATIE, CNRS, École Normale Supérieure Paris-Saclay, Gif-sur-Yvette, France David Menard Department of Engineering Physics, Polytechnique Montreal, Montréal, QC, Canada Roderich Moessner Max-Planck Institut für Physik komplexer Systeme, Dresden, Germany Karl-Hartmut Müller IFW Dresden, Institute for Metallic Materials, Dresden, Germany xviii Contributors Hendrik Ohldag Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, CA, USA Department of Physics, University of California Santa Cruz, Santa Cruz, CA, USA Department of Materials Science, Stanford University, Stanford, CA, USA Hideo Ohno WPI Advanced Institute for Materials Research, Tohoku University, Sendai, Japan Laboratory for Nanoelectronics and Spintronics, Research Institute of Electrical Communication, Tohoku University, Sendai, Japan Center for Spintronics Integrated System, Tohoku University, Sendai, Japan Center for Innovative Integrated Electronic Systems, Tohoku University, Sendai, Japan Center for Science and Innovation in Spintronics (Core Research Cluster), Tohoku University, Sendai, Japan Center for Spintronics Research Network, Tohoku University, Sendai, Japan Myriam Pannetier-Lecoeur Service de Physique de l’Etat Condensé, DRF/ IRAMIS/SPEC CNRS UMR 3680 CEA Saclay, Gif sur Yvette, France Stuart S. P. Parkin Max Planck Institute of Microstructure Physics, Halle (Saale), Germany Oliver Portugall LNCMI-CNRS (UPR3228), EMFL, Univ. Grenoble Alpes, INSA Toulouse, Univ. Toulouse 3, Toulouse, France Robert M. Reeve Institute of Physics, Johannes Gutenberg University Mainz, Mainz, Germany Mahesh G. Samant IBM Research, San Jose, CA, USA Karl G. Sandeman Department of Physics, Brooklyn College of the City University of New York, Brooklyn, NY, USA The Physics Program, The Graduate Center, CUNY, New York, NY, USA Dirk Sander Max Planck Institute of Microstructure Physics, Halle, Germany Simon Sawatzki Technische Darmstadt, Germany Universität Darmstadt, Materialwissenschaft, Vacuumschmelze GmbH & Co.KG, Hanau, Germany Rudolf Schäfer Institute for Metallic Materials, Leibniz Institute for Solid State and Materials Research (IFW) Dresden, Dresden, Germany Institute for Materials Science, Dresden University of Technology, Dresden, Germany Contributors xix Thomas Schrefl Christian Doppler Laboratory for Magnet Design Through Physics Informed Machine Learning, Department of Integrated Sensor Systems, Danube University Krems, Wiener Neustadt, Austria Baogen Shen Institute of Physics, Chinese Academy of Sciences, Beijing, China Ralph Skomski University of Nebraska, Lincoln, NE, USA Yurii Skourski Hochfeld-Magnetlabor Dresden (EMFL-HLD), HelmholtzZentrum Dresden-Rossendorf, Dresden, Germany Andrei N. Slavin Department of Physics, Oakland University, Rochester, MI, USA Plamen Stamenov School of Physics and CRANN, Trinity College, University of Dublin, Dublin, Ireland Philip Stamp Pacific Institute of Theoretical Physics, University of British Columbia, Vancouver, BC, Canada Dieter Suess University of Vienna Research Platform MMM Mathematics – Magnetism – Materials, and Physics of Functional Materials, Faculty of Physics, University of Vienna,Wien, Austria Jirong Sun Institute of Physics, Chinese Academy of Sciences, Beijing, China Jonathan Z. Sun IBM T. J. Watson Research Center, Yorktown Heights, NY, USA So Takei The Physics Program, The Graduate Center, CUNY, New York, NY, USA Department of Physics, Queens College of the City University of New York, Flushing, NY, USA Wei Tang Materials Science and Engineering, Ames Laboratory, Ames, IA, USA Gheorghe Taran Physikalisches Institute, KIT, Karlsruhe, Germany Wolfgang Wernsdorfer Physikalisches Institute, KIT, Karlsruhe, Germany Matthew Willard Materials Science and Engineering, Case Western Reserve University, Cleveland, OH, USA Sebastian Wimmer München, Universität, München, Germany Department Chemie, Ludwig-Maximilians- Lukas Wollmann Max-Planck-Institute für Chemische Physik fester Stoffe, Dresden, Germany D. Wu National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing, People’s Republic of China Michael Ziese Fakultät für Physik und Geowissenschaften, Universität Leipzig, Leipzig, Germany Part I Fundamentals 1 History of Magnetism and Basic Concepts J. M. D. Coey Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Compass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Emergence of Modern Science . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Electromagnetic Revolution [9] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetostatics and Classical Electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Earth’s Magnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Properties of Ferromagnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetism of the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Demise of Classical Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Oxides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermetallic Compounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amorphous Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Fine Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Recording . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods of Investigation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Materials Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 5 5 7 9 9 16 18 20 20 23 25 27 29 32 34 35 36 38 40 40 41 43 43 48 48 50 J. M. D. Coey () School of Physics, Trinity College, Dublin, Ireland e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_1 3 4 J. M. D. Coey Abstract Magnetism is a microcosm of the history of science over more than two millennia. The magnet allows us to manipulate a force field which has catalyzed an understanding of the natural world that launched three revolutions. First came the harnessing of the directional nature of the magnetic force in the compass that led to the exploration of the planet in the fifteenth century. Second was the discovery of the relation between electricity and magnetism that sparked the electromagnetic revolution of the nineteenth century. Third is the big data revolution that is currently redefining human experience while radically transforming social interactions and redistributing knowledge and power. The emergence of magnetic science demanded imagination and observational acuity, which led to the theory of classical electrodynamics. The magnetic field is associated with electric currents and the angular momentum of charged particles in special materials. Our current understanding of the magnetism of electrons in solids is rooted in quantum mechanics and relativity. Yet only since about 1980 has fundamental theory underpinned rational design of new functional magnetic materials and the conception of new spin electronic devices that can be reproduced on ever smaller scales, leading most notably to the disruptive, 60-year exponential growth of magnetic information storage. The development of new magnetic concepts, coupled with novel materials, device and machine designs has become a rich source of technical innovation. Introduction The attraction of ferrous objects to a permanent magnet has been a source of wonder since the Iron Age. Feeble magnets are widespread in nature in the form of rocks known as lodestones, which are rich in magnetite, an oxide mineral with ideal formula Fe3 O4 . Rocky outcrops eventually get magnetized by huge electric currents when lightning strikes, and these natural magnets were known and studied in ancient Greece, Egypt, China, and Mesoamerica. Investigations of magnetic phenomena led to the invention of steel magnets – needles and horseshoes – then electromagnets and eventually the panoply of hard and soft materials that support the modern magnetics industry. Magnetism in a rare example of a science with recorded history goes back well over 2000 years [1, 2]. Theory and practice have been loose partners for most of that time. What people are able to see and rationalize is inevitably conditioned by a priori philosophical beliefs about the world. The scientific method of critically interrogating nature by experimentation and then amassing and exchanging data and ideas among the community of the curious came to be established only gradually. Mathematics emerged as the supporting scaffold of natural philosophy in Europe in the seventeenth century, when precisely formulated natural laws and explanations began to take root. Nevertheless, most of the progress that has been made in magnetism in the past – from the discovery of horseshoe magnets or electromagnetic induction 1 History of Magnetism and Basic Concepts 5 to the development of Alnico – was based on intuition and experience, rather than formal theory. That situation is changing. The discovery of the electron in the closing years of the nineteenth century impelled the great paradigm shift from classical to modern physics. Magnetism, however familiar and practically important it had become, was fundamentally incomprehensible in classical terms. Charged particles were theoretically expected to exhibit no magnetism of any kind. It took 25 years and the insights of quantum mechanics and relativity to resolve that conundrum. Magnetism then went on to play a key role in clarifying basic concepts in condensed matter physics and Earth science over the course of the twentieth century. Now it is a key player in the transformative information technology of the twenty-first century. Early History Aristotle attributed the first reflections on the nature of magnetic attraction to Thales, the early Greek philosopher and mathematician who was born in Miletus in Asia Minor in 624 BC. Thales was an animist who credited the magnet with a soul, on account of its ability to create movement, by attraction. This curious idea was to linger until the seventeenth century. The magnet itself is believed to be named after Magnesia, a city in Lydia in Asia Minor that was a good source of lodestone. In the fifth century BC, when Empedokles postulated the existence of the four elements – Earth, water, air, and fire – magnetism was associated with air. Special effluvia somehow passing through the invisible pores in magnetic material were invoked to explain the phenomenon, a theory echoed much later by Descartes in a mechanistic picture that finally laid the magnet’s soul to rest. The Roman poet Lucretius writing in the first century BC mentions magnetic induction (the ability of a magnet to induce magnetism in pieces of nonmagnetic iron) and for the first time notes the ability of magnets not just to attract but also to repel one another. The Greek approach of developing a philosophical framework into which natural observations were expected to fit was not conducive to open-minded exploration of the natural world. The Compass The Chinese approach to the magnet was more practical. Their magnetism was initially linked to practical concerns of geomancy and divination [3]. The art of adapting the residences of the living and the tombs of the dead to harmonize with local currents of the cosmic breath demanded knowledge of its direction. A south-pointer consisting of a carved lodestone spoon that was free to rotate on a polished baseplate (Fig. 1) was already in use at the time of Lucretius and may have originated hundreds of years earlier. An important discovery, attributed to Zeng Gongliang in 1064, was that iron could acquire a thermoremanent magnetization when rapidly cooled from red heat in the Earth’s magnetic field. A short step 6 J. M. D. Coey Fig. 1 Magnetic direction finders. (a) Baseplate and lodestone spoon of the south-pointer used in China from about the first century BC (Needham, courtesy of Cambridge University Press). (b) A Chinese floating compass from 1044. (c) Fifteenth-century Chinese and (d) Portuguese mariners’ compasses. (Boorstin, courtesy of Editions Robert Laffont) led to the suspended compass needle, which was described by Shen Kuo around 1088, together with declination, the deviation of the needle from a north-south axis. Floating compasses had also been developed by this time, often in the form of an iron fish made to float in a bowl of water. The compass appeared about a century later in Europe, where it was first described by Alexander Neckam in 1190. The direction-finding ability of the magnetic needle or fish was also exploited by Arabs and Persians from the thirteenth century, both for navigation and to determine the sacred direction of Mecca [4]. Compasses (Fig. 1) were the enabling technology for the great voyages of discovery of the fifteenth century, bringing the Ming admiral Cheng Ho to the coasts of Africa in 1433 and Christopher Columbus (who rediscovered declination) to America in 1492, where he landed on the continent where the Olmecs may once have displayed a knowledge of magnetism in their massive stone carvings of human figures and sea turtles dating from the second millennium BC. 1 History of Magnetism and Basic Concepts 7 Before long, the landmasses and oceans of our planet were mapped and explored. According to Francis Bacon, writing in Novum Organum in 1620 [5], the magnetic compass was one of three things, along with printing and gunpowder had “changed the whole face and state of things throughout the world.” All three were originally Chinese inventions. The compass helped to provide us with an image of the planet we inhabit. This was the first of three occasions when magnetism changed the world. The Emergence of Modern Science A landmark in the history of magnetism in Europe was the work of the French crusader monk Petrus Peregrinus. His tract Epistola de Magnete [6] recounts experiments with floating pieces of lodestone and carved lodestone spheres called terella, which he wrote up in Southern Italy during the 1269 siege of Lucera. He describes how to find the poles of a magnet and relates magnetic attraction to the celestial sphere. The same origin had long been associated with the magnet’s directional property in China [3]; we should not forget that before electric light, people were acutely aware of the stars and scrutinized them keenly. Peregrinus’s tract included an ingenious proposal for a magnetic perpetual motion device – a theme that has been embraced by charlatans throughout the ages, right up to the present day. Much credit for the inauguration of the experimental method in a recognizably modern form belongs to William Gilbert. Physician to the English Queen Elizabeth I, Gilbert personally conducted a series of experiments on terellas, which led him to proclaim that the Earth itself was a great magnet. The lodestone or steel magnets aligned themselves not with the celestial sphere, but with the Earth’s poles. He induced magnetism by cooling iron in the Earth’s field and then destroyed it by heating or hammering. Gilbert was at pains to debunk the millennial accretion of superstition that clung to the magnet, confidently advocating in a robust polemical style reliance on the evidence of one’s own eyes. He described his investigations in his masterwork De Magnete, published in 1600 [7]. It is arguably the first modern scientific text. Subsequent developments were associated with improvements in navigation and the prestige of the great voyages of discovery. Gilbert’s theories dominated the seventeenth century up until Edmond Halley’s 1692 shell model for the Earth’s magnetic structure, which strongly influenced compass technology and navigation. Naval interests were the principal drivers of magnetic research during this period, and Halley was sponsored by the British Navy to survey and prepare charts of the Earth’s magnetic field in the North and South Atlantic oceans (Fig. 2), This was in the vain hope of addressing the pressing longitude problem, by pinpointing magnetically the position of a vessel on the Earth’s surface. The following century was marked by the professionalization of natural philosophy (as physical science was then known in Europe) [8]. Accordingly, the natural philosopher with his mantle of theory was rewarded with social status, access to public funding, and credibility beyond that extended to artisans on the one hand and 8 J. M. D. Coey Fig. 2 A section of Halley’s world chart of magnetic variation published in 1700 quacks on the other, such as the colorful Anton Mesmer, who propagated theories of animal magnetism in his salon in Paris or James Graham with his royal Patagonian magnetic bed for nightly rental in a fashionable London townhouse. The English entrepreneur Gowin Knight, representative of a new breed of natural philosopher, greatly improved the quality of bar magnets and compasses, coupling scientific endeavor with manufacturing enterprise and a keen sense of intellectual property. An outstanding technical breakthrough of the eighteenth century was the 1755 discovery by the Swiss blacksmith Johann Dietrich that the horseshoe was an ideal 1 History of Magnetism and Basic Concepts 9 shape for a steel magnet [1]. His invention, a clever practical solution to the age-old problem of self-demagnetization in bar magnets, was enthusiastically promoted by his mentor, the Swiss applied mathematician Daniel Bernoulli, who garnered most of the credit. The Electromagnetic Revolution [9] The late eighteenth century in Europe was a time of great public appetite for lectures and demonstration of the latest scientific discoveries, not least in electricity and magnetism. This effervescent age witnessed rapid developments in the harnessing of electricity, with the 1745 invention of the Leyden jar culminating in Alessandro Volta’s 1800 invention of the voltaic cell. Analogies between electrostatics and magnetism were tantalizing, but the link between them proved elusive. Magnetostatics and Classical Electrodynamics The torsion balance allowed Charles-Augustin de Coulomb to establish in 1785 the quantitative inverse square laws of attraction and repulsion between electric charges, as well as similar laws between analogous magnetic charge or poles that were supposed to be located near the ends of long magnetized steel needles [2]. The current convention is that the north and south magnetic poles are negatively and positively charged, respectively. His image was of pairs of positive and negative electric and magnetic fluids permeating matter, which became charged if one of them dominated or polarized if they were spatially separated. Unlike their electric counterparts, the magnetic fluids were not free to flow and could never be unbalanced in any piece of magnetic material. Coulomb found that the force F between two magnetic poles separated by a distance r fell away as 1/r2 . Siméon Denis Poisson then interpreted Coulomb’s results in terms of a scalar potential ϕm , analogous to the one he used for static electricity, such that the magnetic field could be written as H(r) = −∇ϕm . In modern terms, ϕm is measured in amperes, and H in Am−1 . Magnetic charge qm is measured in Am, and the corresponding potential ϕm = qm /4πr. The magnetic field due to a charge is H(r) = qm r/4πr3 , and Coulomb’s inverse square law for the force between two charges separated by r is F = μ0 qm qm ’r/4πr3 . Here μ0 is the magnetic constant, 4π 10−7 NA−2 , which appears whenever the magnetic field H interacts with matter. (Other equivalent ways of writing the units of μ0 are Hm−1 or TmA−1 .) In Poisson’s opinion, the practice and teaching of mathematics were the purpose of life. He developed his mathematical theory of magnetostatics from 1824, which included the equation that bears his name ∇ 2 ϕm = −ρm , where ρm is the density of magnetic poles. However, the association of H with a scalar potential is only valid in a steady state and when no electric currents are present. The coulombian picture of the origin of magnetic fields was dominant in textbooks until about 1960, and it persists in popular imagery. 10 J. M. D. Coey A revolutionary breakthrough in the history of magnetism came on 21st April 1820, with the discovery of the long-sought link between electricity and magnetism. During a public lecture, the Danish scientist Hans Christian Oersted noticed that a compass needle was deflected as he switched on an electric current in a copper wire. His report, published in Latin a few months later, triggered an experimental frenzy. As soon as the news reached Paris, François Arago (who briefly served as President of France in 1848) immediately performed an experiment that established that a current-carrying conducting coil behaved like a magnet. A week after Arago’s report, André-Marie Ampère presented a paper to the French Academy suggesting that ferromagnetism in a magnetized body was caused by internal currents flowing perpendicular to the axis of magnetization and that it should therefore be possible to magnetize steel needles in a solenoid. Together with Arago, he successfully demonstrated his ideas in November 1820, showing that current loops and coils were functionally equivalent to magnets, and he subsequently established the law of attraction or repulsion between current-carrying wires. Ten days later, the British scientist Humphrey Davy had similar results. The electromagnet was invented by William Sturgeon in 1825; within 5 years Joseph Henry had used a powerful electromagnet in the USA for the first electric telegraph. As early as 1822, Davy’s assistant Michael Faraday produced the first rudimentary electric motor, and Ampère envisaged the possibility that the currents causing magnetism in solids were “molecular” rather than macroscopic in nature. In formal terms, Ampère’s equivalence between a magnet and a current loop of area A carrying a current I is expressed as m = IA (1) where A is in square meters, I is in amperes, and the magnetic moment m is therefore in Am2 . Magnetization, defined in a mesoscopic volume V as M = m/V, has units Am−1 . The direction of m is conventionally related to that of the electric current by the right-hand rule. At the same time as the experimental work of Ampère and Arago, Jean-Baptiste Biot and Félix Savart formulated the law expressing the relation between a current and the field it produces. A current element Iδl generates a field δH = Iδl × r/4πr3 at a distance r. Integrating around a current loop yields an expression for the H-field due to the moment m: H = [3 (m.r) r − m] /4r 3 (2) The form of the field represented by Eq. (2) and illustrated in Fig. 3 is identical to that of an electric dipole, so m is often referred to as a magnetic dipole although we have no evidence for the existence of independent magnetic poles. The dipole moment is best represented by an arrow in the direction of m, although it is still commonplace to see the north-seeking and south-seeking poles of a magnet denoted by the letters N and S. Old habits die hard. Magnetic moments tend to align with magnetic fields in which they are placed. The torque on the dipole m is Γ = μ0 m × H, and the corresponding energy of the 1 History of Magnetism and Basic Concepts 11 Fig. 3 Contours of equal magnetic field produced by a magnetic dipole moment m, represented by the grey arrow dipole is E = − μ0 m . H. These equations are better written in terms of the more fundamental magnetic field B, as discussed below; in free space the two are simply proportional, B = μ0 H, so the torque is =m×B (3) E = −m.B (4) and the corresponding energy is The two rival descriptions of magnetization in solids following from the work of Coulomb or Ampère, based either on magnetic poles or on electric currents, have colored thinking about magnetism ever since (Fig. 4). The poles have no precise, independent physical reality; they are fictitious entities that are a mathematicallyconvenient way to represent the H-field, which is of critical importance in magnetism because it is the local H-field that determines the state of magnetization of a solid. Currents are closer to reality; electric current loops exist, and they do act like magnets. Although it is difficult to attribute the intrinsic spin moment of the electron to a current, the amperian picture of the origin of magnetic fields is generally adopted in modern textbooks. Nineteenth-century electromagnetism owed much to the genius of Michael Faraday. Guided entirely by observation and experiment, with no dependence on formal theory, he was able to perfect the concept of magnetic field, which he 12 +++++ σ m+ jms ----- Fig. 4 Alternative coulombian (left) and amperian (right) descriptions of the magnetization of a uniformly magnetized cylinder, with a magnetic dipole moment m in the direction represented by the black arrow; σm ± is the surface magnetic charge density, jms is the surface electric current density J. M. D. Coey σ m- described by lines of force [10]. Faraday classified substances in three magnetic categories. Ferromagnets like iron were spontaneously magnetized and strongly attracted into a magnetic field; paramagnets were weakly magnetized by a field and feebly drawn into the regions where the field was strongest; diamagnets, on the contrary, were weakly magnetized opposite to the field and repelled by it. Working with an electromagnet, he discovered the law that bears his name and the phenomenon of electromagnetic induction – that a flow of electricity can be induced by a changing magnetic field – in 1831. His conviction that a magnetic field should have some effect on light led to his 1845 discovery of the magneto-optic Faraday effect – that the plane of polarization of light rotates upon passing through a transparent medium in a direction parallel to the magnetization of the medium. The epitome of classical electrodynamics was the set of equations formulated in 1865 by James Clerk Maxwell, the Scottish theoretician, who had “resolved to read no mathematics on the subject till he had first read through Faraday’s ‘Experimental Researches in Electricity’.” Maxwell’s magnificent equations formally defined the relationship between electricity, magnetism, and light [11]. As reformulated by Oliver Heaviside, the equations are a succinct statement of classical electrodynamics. In the opinion of Richard Feynman, Maxwell’s discovery of the laws of electrodynamics was the most significant event of the nineteenth century. The equations in free space are formulated in terms of the fundamental magnetic and electric fields B and E. Using the international system of SI units adopted in this Handbook, the equations read: ∇.B = 0 ε0 ∇.E = ρ (1/μ0 ) ∇ × B = j + ε0 ∂E/∂t (5) ∇ × E = −∂B/∂t The first and third equations express the idea that there are no sources of the magnetic B-field other than time-varying electric fields and electric currents of 1 History of Magnetism and Basic Concepts 13 density j, whereas the second and fourth equations show that the electric field results from electric charge density ρ and time-varying magnetic fields. Maxwell’s equations are invariant in a moving frame of reference, although the relative magnitudes of E and B are altered. The famous wavelike solutions of these equations in the absence of charges and currents are electromagnetic waves, which propagate in free space with velocity c = 1/(ε0 μ0 )1/2 . In SI, the definition of the magnetic constant μ0 is linked to the fine structure constant. To nine significant figures, it is equal to 4π 10−7 NA−2 . ε0 is then related to the definition of the velocity of light. Heinrich Hertz demonstrated Maxwell’s electromagnetic waves experimentally in 1888, and he showed that their behavior was essentially the same as that of light. Hertz could think of no practical application for his work, yet within a few decades, it had become the basis of radio broadcasting and wireless communication! The mechanical effects of electric and magnetic fields were summarized by Hendrik Lorentz in his expression for the force density FL : F L = ρE + j × B (6) The equivalent expression for the force on a particle of charge q moving with velocity v is f = q(E + v × B). Two further fields H and D are introduced in the formulation of Maxwell’s equations in a material medium to circumvent the inaccessibility of the current and charge distributions in the medium. We have no direct way of measuring the atomic charges associated with the polarization of a ferroelectric material or the atomic currents associated with the magnetization of a ferromagnetic material, so we define H and D in terms of fields created by the measurable free charges ρ and free currents j, with dipolar contributions from the magnetization M or polarization P of any magnetic or dielectric material that may be present. The equations now read: ∇.B = 0 ∇.D = ρ ∇ × H = j + ∂D/∂t (7) ∇ × E = −∂B/∂t They are further simplified in a static situation when the time derivatives are zero. The new fields are trivially related to B and E in free space since B = μ0 H and D = ε0 E, but in a material medium, the H-field is defined in terms of the B-field and the magnetization M (the magnetic moment per unit volume) as H = B/μ0 – M or B = μ0 (H + M) (8) 14 J. M. D. Coey H M B +++++ ––––– B = P0(H + M) Fig. 5 B, H, and M for a uniformly magnetized ferromagnetic bar. Eq. (8) is represented by the vector triangle. The H-field can be regarded as originating from a distribution of positive and negative magnetic charge (south and north magnetic poles) on opposite faces Likewise D = ε0 (E + P), where P is the electric polarization. To specify a situation in magnetostatics or electrostatics, any two of the three magnetic or electric fields are needed. (Magnetization M and polarization P are regarded as vector fields.) The defining relation between B, H, and M for a uniformly magnetized ferromagnetic bar is illustrated in Fig. 5. Note that the B-field is solenoidal – the field lines are continuous with no sources or sinks; it is divergenceless and can therefore be expressed as the curl of a vector potential A – whereas the H-field is conservative; it is irrotational provided j is zero and can be expressed as the gradient of a scalar potential. Outside the magnet, the H-field is called the stray field, but within the magnet where it is oppositely oriented to M, the name changes to demagnetizing field. Boundary conditions that B⊥ and H|| are continuous across an interface in a steady state (j = 0) follow from the first and third of Maxwell’s equations 7. B is the fundamental magnetic field, because no elementary magnetic poles exist in nature (∇. B = 0), but it is the local value of H (and perhaps the sample history) that determines the magnetic state of a solid, including its micromagnetic domain structure. The H-field acting in a solid is the sum of the applied field H and the local demagnetizing field Hd created by the solid body itself. When describing the stray field outside a distribution of magnetization M(r) in a solid, the coulombian and amperian descriptions are formally equivalent. The coulombian expression for the magnetic field is obtained by integrating the expression for the field due to a distribution of a magnetic charge qm per unit volume ρm = −∇. M in the bulk, and per unit area σm = M. en at the surface, where en is the unit vector normal to the surface: 1 History of Magnetism and Basic Concepts 1 H (r) = 4π − 15 ∇ · M r − r V |r − r |3 3 M · en r − r d r + S |r − r |3 2 d r (9) This formula gives H(r) both inside and outside the magnetic material. Outside B(r) = μ0 H(r). The amperian expression for the magnetic field produced by a distribution of currents is based on the Biot-Savart expression for the field due to a current element, including contributions from the current density jm = ∇ × M in the bulk, and jms = M × en at the surface: μ0 B (r) = 4π ∇ × M × r − r V |r − r |3 3 (M × en ) × r − r d r + S |r − r |3 2 d r (10) This formula gives B(r) both inside and outside the magnetic material. The same result can be obtained by appropriate integration of Eq. 2 over a magnetization distribution M(r) [12]. For uniformly magnetized ellipsoids, the demagnetizing field Hd is related to the magnetization by H d = −N M (11) where N is a tensor with unit trace [13]. It reduces to a simple scalar demagnetizing factor 0 < N < 1 when the magnetization lies along a principal axis of the ellipsoid. N ≈ 0 for a long needle magnetized along its axis, and N = 1 for a flat plate magnetized perpendicular to the plane. A sphere has N = 1/3. For any shape less symmetric than an ellipsoid, the demagnetizing field is nonuniform. There are useful approximate formulae for square bars and cylinders [14], such as 1/(2n + 1) √ and 1/[(4n/ π) + 1], respectively, but they should not obscure the fact that the demagnetizing field in these shapes really is quite nonuniform. Here n is the ratio of length to diameter. The demagnetizing field is the reason why for centuries magnets were condemned to take awkward shapes of bars or horseshoes to avoid substantial self-demagnetization and why the most successful electromagnetic machines of the nineteenth century were built around electromagnets rather than permanent magnets. The hardened steel magnets of the day showed little coercivity and were easily demagnetized. Demagnetizing fields are also the cause of ferromagnetic domains. The shape constraint on permanent magnets was not lifted until the middle of the twentieth century. Permanent magnets then came to the fore in the design of electric motors and magnetic devices. Fig. 6 illustrates a collection of magnets from the eighteenth, nineteenth, and twentieth centuries. The imaginative world of Maxwell and his followers in the latter part of the nineteenth century when the electromagnetic revolution was in full swing was 16 J. M. D. Coey Fig. 6 Magnets from four centuries; top, seventeenth-century lodestone, nineteenth-century electromagnet; bottom, eighteenth-century horseshoe magnet, twentieth-century alnico and Nd2 Fe14 B magnets (not to scale) actually far removed from our own [15]. They envisaged light and other Hertzian waves as propagating in an all-pervasive aether, which was believed to possess magical mechanical properties – it had to be a massless incompressible fluid, transparent and devoid of viscosity, yet millions of times more rigid than steel! Elaborate mechanical models were envisaged for the waves and fields. In due course it came to be understood that reality was represented by the abstract mathematics, which remained after all the mechanical props had been discarded. The Earth’s Magnetic Field The Earth’s field was the prime focus of attention of magnetism for over a millennium, especially after it was understood that the magnetic field was of terrestrial origin. By the beginning of the nineteenth century, the components of the field were 1 History of Magnetism and Basic Concepts 17 being recorded regularly in laboratories across the world. A comparison of the daily magnetic records at Paris and Kazan, cities lying 4000 km apart, for the same day in 1825, showed astonishingly similar short-term fluctuations. This inspired Carl Friedrich Gauss to establish a worldwide network of 50 magnetic observatories, coordinated from Göttingen, to make meticulous simultaneous measurements of the Earth’s field, in the hope that if enough high-quality data could be collected, the mystery of its origin and its fluctuations might be solved. This heroic pioneering venture in international scientific collaboration amassed stores of data that were enormous for that time. It inspired Gauss to develop spherical harmonic analysis, from which he calculated that the leading, dipolar term accounted for about 90% of the field and that the origin of the stable component was essentially internal. Edward Sabine later spotted that the intensity of the short-term fluctuations tracked the 11-year sunspot cycle, which we now know corresponds to reversals of the solar magnetic field. But in its primary aim, Gauss’s Magnetische Verein must be counted a failure. No amount of data, however copious and precise, could reveal a deterministic origin of a phenomenon that was fundamentally chaotic. Piles of data with no theory or hypothesis through which to view and be tested by them are not very informative. This lesson was learned slowly. The pole picture of the Earth’s magnetic field, albeit with poles that needed to travel tens of kilometers every year to account for the secular variation, yielded eventually in the academy if not in the popular imagination to one based on electric currents driven by convection in the Earth’s liquid core. Joseph Larmor, a dogged believer in the aether, was an early proponent of the geomagnetic dynamo. He demonstrated the precession of a magnet in a magnetic field at a frequency fL = γB/2π that bears his name. The precession is analogous to that of a spinning top in a gravitational field; it is a consequence of the torque on a magnetic moment expressed by Eq. 3. The constant γ, known as the gyromagnetic ratio, is the ratio of the magnetic moment to its associated angular momentum. The proportionality of these two quantities that at first sight appear quite dissimilar, the famous Einstein-de Haas effect, was eventually demonstrated experimentally in 1915 (Fig. 7). Fig. 7 The Einstein-de Haas experiment. The iron rod suspended from a torsion fiber twists when a magnetizing current in the surrounding solenoid is reversed, thereby demonstrating the relationship between magnetism and angular momentum 18 J. M. D. Coey The Properties of Ferromagnets If the luminiferous aether was inaccessible to experimental investigation, as the 1887 Michelson-Morley experiment suggested, the same could not be said for magnetic materials. With its focus on electromagnetism, the nineteenth century brought a flurry of investigations of the magnetic properties of the ferromagnetic metals, iron (discovered in the fourth millennium BC), cobalt (discovered in 1735), and nickel (discovered in 1824) and some of their alloys, which were at the heart of electromagnetic machines. In 1842 James Joule, a brewer and natural philosopher, discovered the elongation of an iron bar when it was magnetized to saturation and demonstrated in a liquid displacement experiment that the net volume was unchanged in the magnetostrictive process, owing to a compensating contraction in the perpendicular directions [16]. Magnetostriction is the reason why transformers hum. Gustav Wiedemann observed that an iron bar twisted slightly when a current was passed through it in the presence of a magnetic field. Anisotropic magnetoresistance (AMR) was discovered by William Thomson in 1856; the resistance of iron or nickel is a few percent higher when measured in the direction parallel to the magnetization than in the perpendicular direction [17]. The Hall effect, the appearance of a transverse voltage when a current was passed through a gold foil subject to a transverse magnetic field was discovered by Edwin Hall in 1879, And the contribution e proportioal to the magnetization of a ferromagnet — tha anomalous Hall effect — was found shortly afterwards, in iron. John Kerr showed in 1877 that the rotation of the plane of polarization of electromagnetic radiation, demonstrated by Faraday for light passing through glass, could also be measured in reflection from polished ferromagnetic metal surfaces [18]. Gauss’s collaborator Wilhelm Weber, who had constructed the first electromagnetic telegraph in 1833, formally presented the idea that molecules of iron were capable of movement around their centers, suggesting that they lay in different directions in an unmagnetized material, but aligned in the same direction in the presence of an applied magnetic field. This was the origin of the explanation of hysteresis by James Alfred Ewing, who coined the name for the central phenomenon of ferromagnetism that he illustrated using a board of small, pivoting magnets [19]. Ewing’s activities as a youthful scottish professor at the University of Tokyo in the 1890s helped to establish the strong Japanese school of research on magnetic materials that thrives to the present day. The hysteresis loop, illustrated in Fig. 8, is the icon of ferromagnetism. Except in very small particles, a magnetized state is always metastable. The saturated magnetic state is higher in energy relative to a multidomain state on account of the demagnetizing field that creates a positive magnetostatic self-energy -½μ0 Ms .Hd dV in the fully magnetized state, where the only contribution to the integral comes from the magnet volume. The hardened steel magnets of the nineteenth century showed little coercivity, Hc Ms , and could only survive as bars and horseshoes where the demagnetizing factor N of Eq. 11 was 1. The principal achievement in technical magnetism in the twentieth century was the mastery of coercivity; this needed new materials having Hc Ms . 1 History of Magnetism and Basic Concepts spontaneous magnetization 19 M remanence coercivity virgin curve initial susceptibility H major loop Fig. 8 The hysteresis loop of magnetization M against magnetic field H for a typical permanent magnet, showing the initial magnetization curve from the equilibrium multidomain state and the major loop. Ms is the saturation magnetization, Mr the remanent magnetization at zero field, and Hc the coercive field required to reduce the magnetization to zero The astonishing transformation of science and society that began in 1820 deserves the name electromagnetic revolution. By the end of the century, electromagnetic engineering was electrifying the planet, changing fundamentally our communications and the conditions of human life and leisure. Huge electric generators, powered by hydro or fossil fuel, connected to complex distribution networks were bringing electric power to masses of homes and factories across the Earth. Electric light banished the tyranny of night. Electric motors of all sorts were becoming commonplace, and public transport was transformed. Telegraph and telephone communication connected people across cities, countries, and continents. Valdemar Poulsen demonstrated magnetic voice recording in 1898. Much of the progress was achieved by engineers who relied on practical knowledge of electrical circuits and magnetic materials, independently of the conceptual framework of electrodynamics that had been developed by the physicists. The electromagnetic revolution and the subsequent electrification of the planet were the second occasion when magnetism changed the world. The century closed with Pierre Curie’s 1895 accurate measurements of the Curie point TC (the critical temperature above which a material abruptly loses its ferromagnetism) and with the all-important discovery of the electron. Yet ferromagnetism was hardly understood at all at a fundamental level at the turn of the century, and it was becoming evident that classical physics was not up to the task. 20 J. M. D. Coey Magnetism of the Electron The discovery of the electron in the closing years of the nineteenth century was a huge step toward the modern understanding of magnetism. The elementary charged particle with mass me = 9.109 10−31 kg and charge e = −1.602 10−19 C had been named by the Irish scientist George Johnstone Stoney in 1891, several years before Jean Perrin in France actually identified negatively charged particles in a cathode ray tube and J. J. Thompson in England measured their charge to mass ratio e/me , by deflecting the electrons in a magnetic field and making use of Eq. 6. Another Irish scientist, George Francis FitzGerald, suggested in 1900 that magnetism might be due to rotational motion of these electrons. They turned out to be not only the carriers of electric current but also the essential magnetic constituent of atoms and solids. The Demise of Classical Physics At the beginning of the twentieth century, the contradictions inherent in contemporary physics could no longer be ignored, but 25 years were to elapse before they could be resolved. In that heroic period, classical physics and the lingering wisps of aether were blown away, and a new paradigm was established, based on the principles of quantum mechanics and relativity. Magnetism in particular posed some serious puzzles. In order to account for the abrupt disappearance of ferromagnetism at the Curie point, Pierre Weiss, who had developed Ewing’s concept of magnetic domains, postulated in 1907 the existence of an internal molecular field. H i = nW M (12) proportional to magnetization in order to explain the spontaneous magnetization within them. His theory of ferromagnetism was based on Paul Langevin’s 1905 explanation of the Curie law susceptibility of an array of disordered classical magnetic moments. χ = C/T (13) Susceptibility χ can be conveniently defined as the dimensionless ratio M/H, where H is the applied magnetic field. The expression is modified for a ferromagnet above its Curie point where it becomes the Curie-Weiss law χ = C/(T – θp ) with θp ≈ TC . With Eq. (12) and Langevin’s theory of paramagnetism, Weiss invented the first mean-field theory of a phase transition. For iron, where M = 1.71 MAm−1 , the Weiss constant nW is roughly 1000. According to Maxwell’s equation ∇. B = 0, the component of B normal to the surface of a magnet is continuous, so there should 1 History of Magnetism and Basic Concepts 21 be a stray field of order μ0 Hs ∼ 1000 T in the vicinity of a magnetized iron bar. In fact, the observed stray fields are a thousand times smaller. Furthermore if, as Ampère believed, all magnetism was traceable to circulating electric currents, the magnetization of an iron bar requires an incredible surface current of 17,100 A for every centimeter of its length. How could such a current be sustained indefinitely? Why does the iron not melt? What did the sobriquet molecular really mean? The anomalous Zeeman splitting of spectral lines in a magnetic field was another mystery. In retrospect, the most startling result was a theorem proved independently in their theses by Niels Bohr in 1911 and Hendrika van Leeuwen in 1919. They showed that at any finite temperature and in any magnetic or electric field, the net magnetization of a collection of classical electrons vanishes identically. So, in stark contrast with experiment, classical electron physics was fundamentally incompatible with any kind of magnetism! By 1930, quantum mechanics and relativity had ridden to the rescue, and a new understanding of magnetism emerged in terms of the physics of Einstein, Bohr, Pauli, Dirac, Schrödinger, and Heisenberg. The source of magnetism in matter was identified with the angular momentum of elementary particles, especially the electron [20]. The connection between angular momentum and magnetism had been demonstrated directly on a macroscopic scale in 1915 by the Einsteinde Haas experiment (Fig. 7), where angular recoil of a suspended iron rod was observed when its magnetization was reversed by an applied field. It turned out that the perpetual currents in atoms were quantized in stationary states that did not decay and that the angular momentum of the orbiting electrons was a multiple of Planck’s constant = 1.055 10−34 Js. Furthermore, the electron itself possessed an intrinsic angular momentum or spin [20] with eigenvalues of ±½ along the axis of quantization defined by an external field. Weiss’s molecular field was no magnetic field at all, but a manifestation of electrostatic coulomb interactions constrained by Wolfgang Pauli’s exclusion principle, which forbade the occupancy of a quantum state by two electrons with the same spin. The intrinsic angular momentum of an electron with two eigenvalues had been proposed by Pauli in 1924; Samuel Goudsmit and George Uhlenbeck demonstrated a year later that the spin angular momentum had a value of ½. The Pauli spin matrices representing the three components of spin angular momentum are s= 01 0 −i 1 0 , , 10 i 0 0 −1 /2 (14) The corresponding electronic magnetic moment was the Bohr magneton, μB = e/2me or 9.274 × 10−24 Am2 , twice as large as the moment associated with a unit of orbital angular momentum in Bohr’s model of the atom. The gyromagnetic ratio of magnetic moment to angular momentum for the electron spin is γ ≈ e/me , so the Larmor precession frequency eB/2πme for the electron is 28 GHzT−1 . 22 J. M. D. Coey The problem of the electron’s magnetism was finally resolved by Paul Dirac in 1928 when he succeeded in writing Schrödinger’s equation in relativistically invariant form, obtaining the non-relativistic electron spin in terms of the 2 × 2 Pauli matrices. Together with Dirac, Werner Heisenberg formulated the exchange interaction represented by the famous Heisenberg Hamiltonian H = –2J S i .S j (15) to describe the coupling between the vector spins Si and Sj of two nearby manyelectron atoms i and j. The spin vectors S are the spin angular momenta in units of . The value of the exchange integral J was closely related to Weiss’s molecular field coefficient nW and depends strongly on interatomic distance. It can be positive, if it tends to align the two spins parallel (ferromagnetic exchange), or negative if it tends to align the pair antiparallel (antiferromagnetic exchange). The value of S is obtained from the first of the three rules, discussed below, that were formulated by Friedrich Hund around 1927 for finding the ground state of a multi-electron atom. The exchange interactions among the electrons of the same atom are much stronger than those between the electrons of adjacent atoms given by Eq. (15). The fundamental insight that magnetic coupling of electronic spins is governed by electrostatic coulomb interactions, subject to the symmetry constraints of quantum mechanics, was the key needed to unlock the mysteries of ferromagnetism. Exchange is discussed in Chap. 2, “Magnetic Exchange Interactions.” The magnetic moment of an atom or ion is the sum of two contributions. One arises from the intrinsic spin angular momentum of the atomic electrons. The other comes from their quantized orbital angular momentum. The moments associated with each type of angular momentum have to be summed according to the rules of quantum mechanics. The moment associated with ½ of spin angular momentum is practically identical to that associated with of orbital angular momentum, namely, one Bohr magneton in each case. The quantum theory of magnetism is therefore the quantum theory of angular momentum. Hund’s rules were an empirical prescription for determining the total angular momentum of the many-electron ground state of electrons belonging to the same atom or ion. Firstly, the rule is to maximize the spin angular momentum S while respecting the Pauli principle that no two electrons can be in the same quantum state. Secondly, the orbital angular momentum L is maximized, consistent with the value of S, and thirdly the spin and orbital momenta are coupled together to form the total angular momentum J = L ± S, according to whether the electronic shell is more or less than half full. The total magnetic moment (in units of μB ) is then related to the total angular momentum (in units of ) by a numerical Landé g-factor, which is 1 for a purely orbital moment and 2 for pure spin. The spin-orbit coupling, which arises in the atom from motion of the electron in the electrostatic potential of the charged nucleus and gives rise to Hund’s third rule, is another key interaction. Of fundamentally relativistic character, it emerges naturally from Dirac’s relativistic quantum theory of the electron, and it turns out to be at the root of many of the most interesting phenomena in magnetism, including 1 History of Magnetism and Basic Concepts 23 magneto-optics, magnetocrystalline anisotropy, and the spin Hall effect. The spinorbit interaction for a magnetic ion is represented by the Hamiltonian L.S, where L is the orbital angular momentum of the many-electron atom in units of and is the atomic spin-orbit coupling constant. Like the exchange constant J , has dimensions of energy. Felix Bloch in 1930 described the spin waves that are the quantized elementary excitations of a ferromagnetic array of atoms whose spins are coupled by Heisenberg exchange. These excitations have an angular frequency ω and a wavevector k that are related by the dispersion relation ω = Dk2 , where D is the spin wave stiffness constant. It is proportional to J . The first quantum theories of magnetism regarded the electrons as localized on the atoms or ions, but an alternative magnetic band theory of ferromagnetic metals was developed by John Slater and Edmund Stoner in the 1930s. It accounted for the non-integral, delocalized spin moments found in Fe, Co, and Ni and their alloys, although the theory in its original form greatly overestimated the Curie temperatures. The delocalized, band electron model of Slater and the localized, atomic electron model of Heisenberg were two distinct paradigms for the theory of magnetism that persisted until sophisticated computational methods for treating the many-body interelectronic correlations in the ground state of multi-electron atoms were devised toward the end of the twentieth century. The differences between the two approaches are epitomized in the calculation of the paramagnetic susceptibility. Pauli found a small temperature-independent susceptibility resulting from Fermi-Dirac statistics for delocalized electrons, whereas Léon Brillouin had used Boltzmann statistics and the Bohr model to derive the Curie law susceptibility of an array of atoms with localized electrons. The sixth Solvay Conference, held in Brussels in October 1930 (Fig. 9), was devoted to magnetism [21]. It followed four years of brilliant discoveries in theoretical physics, which set out the modern electronic theory of condensed matter. Yet the immediate impact on the practical development of functional magnetic materials was surprisingly slight. Dirac there made the perceptive remark “The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” Magnetic Phenomenology In view of the immense computational challenge posed by many-body electron physics in 1930, a less fundamental theoretical approach was needed. Louis Néel pursued a phenomenological approach to magnetism with notable success, oblivious to the triumphs of quantum mechanics. His extension of the Weiss theory to two equal but oppositely aligned magnetic sublattices led him to the idea of antiferromagnetism in his 1932 doctoral thesis. This hidden magnetic order 24 J. M. D. Coey Fig. 9 The 1930 Solvay Conference on Magnetism Back row: Herzen, Henriot, Verschaffelt, Manneback, Cotton, Errera, Stern, Piccard, Gerlach, Darwin, Dirac, Bauer, Kapitza, Brioullin, Kramers, Debye, Pauli, Dorfman, van Vleck, Fermi, Heisenberg. Front row: de Donder, Zeeman, Weiss, Sommerfeld, Curie, Langevin, Einstein, Richardson, Cabrera, Bohr, de Haas awaited the development of neutron scattering in the 1950s before it could be directly revealed, initially for MnO. Néel went on to explain the ferrimagnetism of oxides such as magnetite, Fe3 O4 , the main constituent of lodestone, in terms of two unequal, antiferromagnetically coupled sublattices. The three most common types of magnetic order, and their temperature dependences, are illustrated in Fig. 10. The spinel (MgAl2 O4 ) structure of magnetite has an A sublattice of 8a sites with fourfold tetrahedral oxygen coordination and twice as many 16d sites with sixfold octahedral coordination forming a B sublattice. The spinel structure is illustrated in Fig. 14 where the 8a sites are at the centers of the blue tetrahedra, which have oxygen ions at the four corners, and the 16d sites are at the centers of the brown octahedra, which have six oxygen ions at the corners. The numbers of each type of site in the unit cell are indicated by the labels. The 16d sites in magnetite are occupied by a mixture of ferrous Fe2+ and ferric Fe3+ ions with electronic configurations 3d5 and 3d6 and spin moments of 5 μB and 4 μB , respectively, whereas the 8a sites are occupied by oppositely aligned Fe3+ ions. This yields a net spin moment of 4 μB per formula (0.48 MAm−1 ) – a quantitative explanation of the magnetism of the archetypical magnet in terms of lattice geometry and the simple rule that each unpaired electron contributes a spin moment of one Bohr magneton. Néel added two new categories of magnetic substances – antiferromagnets and ferrimagnets – to Faraday’s original three. Their magnetic ordering temperatures are known as antiferromagnetic or ferrimagnetic Néel temperatures. The ferrimagnetic one is also called a Curie point. 1 History of Magnetism and Basic Concepts 25 1/c 1/c T T C ,qp 1/c qp T TN M M qp M B B B A TC T T T fN B TN A T A T fN T A Fig. 10 Schematic temperature dependences of the inverse susceptibility (top) and (sub)lattice magnetization (bottom) of a ferromagnet (left), an antiferromagnet (center), and a ferrimagnet (tight) Micromagnetism For many practical purposes, it is possible to follow in the footsteps of Néel, sidestepping the complications engendered by the atomic and electronic basis of magnetism, and regard magnetization as a continuous vector in a solid continuum [13], as people have for about 200 years. The iconic hysteresis loop M(H) (Fig. 8) is the outcome of a metastable structure of domains of uniformly magnetized ferromagnetic Weiss domains separated by narrow domain walls between domains magnetized in different directions. The structure depends on the thermal and magnetic history of a particular sample. Aural evidence for discontinuous jumps in the size of the domains as the magnetization was saturated was first heard by Heinrich Barkhausen in 1919 with the help of a pickup coil wound around some ferromagnetic wires, a rudimentary amplifier, and a loudspeaker. Then in 1931 the domains were directly visualized by Francis Bitter using a microscope focused on a polished sample surface and a colloidal suspension of magnetite particles that were drawn by the stray field to the domain walls. These colloids, known as ferrofluids, behave like ferromagnetic liquids. The idea of a domain wall as a region where the magnetization rotates progressively from one direction to the opposite one in planes parallel to the wall was introduced by Felix Bloch in 1932. His walls create no bulk demagnetizing 26 J. M. D. Coey Fig. 11 Two types of 180◦ domain walls: a) the Bloch wall and b) the Néel wall field and cost little magnetostatic energy because ∇. M = 0; the magnetization in each plane is uniform, and there is no component perpendicular to the planes (see Eq. 9). The exchange energy cost, written in the continuum approximation as A(∇M)2 where A ∝ J , is balanced by the anisotropy energy cost associated with the magnetization in the wall that is misaligned with respect to a magnetic easy axis of the crystal. Magnetic anisotropy is introduced below, and it is discussed in detail in Chap. 3, “Anisotropy and Crystal Field.” A Néel domain wall, where the magnetization rotates in a plane perpendicular to the wall so that ∇. M = 0 in the bulk, but there is no surface magnetic charge, is higher in energy except in thin films. The two types of wall are illustrated in Fig. 11. In principle, the sum of free energy terms associated with exchange, anisotropy, and magnetostatic interactions, together with the Zeeman energy in an external field, could be minimized to yield the M(H) loop and the overall domain structure of any solid. Further terms can be added to take into account the effects of imposed strain and spontaneous magnetostriction. In practice, however, crystal defects such as grain boundaries spoil the continuum picture and can exert a crucial influence on the walls. It is then necessary to resort to models to develop an understanding of hysteresis. The basic theory of micromagnetism was developed by William Fuller Brown in 1940 [13]. The magnetostatic interaction between the magnetic dipoles that constitute the magnetization is a dominant factor. The dipole fields fall off as 1/r3 (Eq. 2), providing a long-range interaction unlike exchange, which is short-range because it depends on an overlap between electronic wavefunctions that decays exponentially with interatomic spacing. This is why weak magnetostatic interactions that are of order 1 K for a pair of ions are able to compete on a mesoscopic length scale with the much stronger exchange interactions of electrostatic origin that can be of order 100 K to control the domain structure of a given ferromagnetic sample. Magnetocrystalline anisotropy is represented phenomenologically in the theory by terms in the energy that depend on the orientation of M with respect to the local crystal axes. The electrostatic interaction of localized atomic electrons with the potential created by all the other atoms in the crystal is known as the crystal field 1 History of Magnetism and Basic Concepts 27 interaction; the effect of chemical bonding with the ligands of an atom is the ligand field interaction. The two effects are comparable in magnitude for 3d ions [22]. Magnetocrystalline anisotropy arises from the interplay of the crystal/ligand field and spin-orbit coupling. The simplest case is for uniaxial (tetragonal, hexagonal, rhombohedral) crystals, where the leading term in the energy density is of the form Ea = K1 sin2 θ + . . . .. (16) where θ is the angle between M and the symmetry axis. Two opposite easy directions lie along the crystal axis if the anisotropy constant K1 is positive, but there are many easy directions lying in an easy plane perpendicular to the crystal axis (θ = π/2) when K1 is negative. Anisotropy arises also from overall sample shape, due to the demagnetizing energy ½MHd , which gives another contribution in sin2 θ that depends on the demagnetizing factor N with K1 sh = 1 μ0 Ms 2 (1 − 3N ) 4 (17) where Ms is the spontaneous magnetization. There is obviously no shape anisotropy for a sphere, which has N = 1/3. An expression equivalent to (16) at the atomic scale is εa = Da sin2 θ , where Da /kB ∼ 1 K. The magnitude of the crystal field energy is comparable to the magnetostatic energy, but it is much smaller than the exchange energy in practical magnetic materials. It remains challenging to calculate K1 or Da precisely in metals. An instructive paradox arising from Brown’s micromagnetic theory is his result that the coercivity Hc of a perfect, defect-free ferromagnetic crystal lattice must exceed the anisotropy field Ha = 2 K1 /μ0 Ms . In practice Hc is rarely as much as a fifth of Ha . The explanation is that no real lattice is ever free of defects, which act as sites for the nucleation of reverse domains or as pinning centers for domain walls. The sequence of metastable states represented on the hysteresis loop is generally dominated by asperities and lattice defects that are very challenging to characterize in any real macroscopic sample. Control of these defects in modern permanent magnets having Hc Ms has been as much a triumph of metallurgical art as physical theory. Micromagnetism is the subject of Chap. 7, “Micromagnetism.” Magnetic Materials The traditional magnetic materials were alloys of the ferromagnetic metals, Fe, Co, and Ni. The metallurgy and magnetic properties of these alloy systems were the focus of investigations of technical magnetism in the first half of the twentieth century, when useful compositions were developed such as Permendur, Fe50 Co50 , the alloy with the highest magnetization (1.95 MAm−1 ); Permalloy Fe20 Ni80 , which has near-zero anisotropy and magnetostriction, together with very high relative permeability (μr = (1 + χ) ≈ 105 ); and Invar Fe64 Ni36 a composition with near- 28 J. M. D. Coey Fig. 12 Unit cells of the ferromagnetic elements Fe (body-centered cubic, left), Ni (face-centered cubic, center), and Co, Gd (hexagonal close-packed, right) [29], with kind permission from Cambridge University Press zero thermal expansion around room temperature. The early investigations are well summarized in Bozorth’s 1950 monograph [23]. The fourth ferromagnetic element at room temperature is the rare earth gadolinium. The crystal structures of these elemental ferromagnets are illustrated in Fig. 12. An important practical advance in the story of permanent magnet development was the thermal processing of a series of Al-Ni-Co-Fe alloys, the Alnico magnets, that was initiated in Japan in 1932 by Tokushichi Mishima. Their coercivity relied on achieving a nanostructure of aligned acicular (needle-like) regions of Co-Fe in a matrix of nonmagnetic Ni-Al. It was the shape of the ferromagnetic regions that gave the alloys some built-in magnetic anisotropy (Eq. 17), but it still had to be supplemented with global shape anisotropy by fabricating the Alnico into a bar or horseshoe in order to avoid self-demagnetization. The mastery of coercivity that was acquired over the course of the twentieth century (Fig. 13) was spectacular, and burgeoning applications in technical magnetism of soft and hard magnetic materials were the direct consequence. The terms “soft” and “hard” were derived originally from the magnetic steels that were used in the nineteenth century. The most useful figure of merit for the hard, permanent magnets is the maximum energy product |BH|max , equal to twice the energy in the stray field produced by a unit volume of magnet. The SI unit is kJm−3 . Energy product doubled every 12 years for most of the twentieth century, thanks to the discovery in the 1960s of rare earth cobalt intermetallic compounds and the discovery of new rare earth ironbased materials in the 1980s. Comparable progress with decreasing hysteresis losses in soft, electrical steels continued to the point where they became a negligible fraction of the resistive losses in the copper windings of electromagnetic energy converters. Ultrasoft amorphous magnetic glasses were developed in the 1970s. Applications of soft and hard magnetic materials are discussed in Chaps. 29, “Soft Magnetic Materials and Applications,” and 28, “Permanent Magnet Materials and Applications” respectively. A good working knowledge of the quantum mechanics of multi-electron atoms and ions had been developed by the middle of the twentieth century, mainly from 1 History of Magnetism and Basic Concepts 29 Nd-Fe-B Fig. 13 The development of coercivity over the ages and in the twentieth century observations of optical spectra and the empirical rules formulated by Hund to specify the ground state L, S, and J multiplet, which is the one of interest for magnetism. All this led naturally to a focus on the localized electron magnetism found in the 3d and 4f series of the periodic table. For 3d ions in solids, the ionic moment is essentially that arising from the unpaired electron spins left after filling the orbitals according to the Pauli principle and Hund’s first rule. The orbital moment expected from the second rule is quenched by the crystal field, which impedes the orbital motion so that it barely contributes to the ionic magnetism. But the crystal field is weaker for the 4f elements in solids, whether insulating or metallic, and the magnetism is more atomic-like with spin and orbital contributions coupled by the spin-orbit interaction according to Hund’s third rule to yield the total angular momentum J. Microscopic quantum theory began to play a more important part in magnetic materials development after the 1970s with the advent of rare earth permanent magnets SmCo5 and especially Nd2 Fe14 B, when an understanding of the intrinsic, magnetocrystalline anisotropy in terms of crystal field theory and spin-orbit coupling began at last to make a contribution to the design of new permanent magnet materials. Magnetic Oxides The focus on localized electron magnetism in the 1950s and 1960s led to systematic investigations of exchange interactions in insulating compounds where the spin 30 J. M. D. Coey moments of magnetic 3d ions are coupled by indirect overlap of their wavefunctions via an intervening nonmagnetic anion, usually O2− . A systematic empirical understanding of the dependence of these superexchange interactions on electron occupancy and bond angle emerged in the work of Junjiro Kanamori and John Goodenough [24], based on the many new magnetic compounds that were being fabricated at that time. There is a multitude of solid solutions between end-members, with extensive opportunities to tune magnetic properties by varying the chemical compositions of oxide families such as ferrites [25]. Superexchange, like direct exchange in the ferromagnetic 3d elements, depends on the overlap of wavefunctions of adjacent atoms and decays exponentially with interatomic distance. The magnetite family of cubic spinel ferrites M2+ Fe3+ 2 O4 was the first to be thoroughly investigated, with M = Mg, Zn, Mn, Fe, 2/3Fe3+ (γFe2 O3 ), Co, or Ni. Ferrimagnetic Neél temperatures of these ferrites range from 700 to 950 K, although spinel itself (MgAl2 O4 ) is nonmagnetic. Several of the insulating compounds with Mn, Ni, and Zn are suitable as soft magnetic materials for audio- or radiofrequency applications. Other important families investigated at that time were 3+ garnets, perovskites, and hexagonal ferrites. The garnet ferrites R3+ 3 Fe5 O12 have a large cubic unit cell containing 160 ions, with ferrimagnetically aligned ferric iron in both tetrahedral 24d and octahedral 16a sites, and large R3+ ions in eightfold oxygen coordination in deformed cubal 24c sites. R may be any rare earth element, including Y, which forms yttrium iron garnet (YIG), Y3 Fe5 O12 , a superlative microwave material that exhibits ultra-low magnetic losses on account of its insulating character. The net magnetic moment of YIG is 5μB per formula unit. Substituting magnetic rare earths in the structure provides an opportunity to study superexchange between 3d and 4f ions. That interaction is weak, and the 4f ions couple antiparallel to the 24d site iron, but their sublattice magnetization decays much faster with temperature, giving rise to the possibility of a compensation temperature, where the net magnetization of the two ferrimagnetic sublattices crosses zero at a temperature below the ferrimagnetic Neél point. The compensation temperature of Gd3 Fe5 O12 , for example, is 290 K, whereas its ferrimagnetic Néel point is at 560 K, a typical value for the whole rare earth iron garnet series. Another important oxide family, the hexagonal ferrites especially M2 Fe12 O19 , where M = Ba2+ or Sr2+ , have uniaxial anisotropy and crystallize in the magnetoplumbite structure. There are four Fe3+ sites in the structure, including a fivefold 2b site with trigonal symmetry where the threefold axis is parallel to the c-axis of the hexagonal unit cell. The net ferrimagnetic moment is 20 μB per formula unit, since eight iron ions belong to one sublattice and four to the other. The large nonmagnetic M cations occupy sites that would otherwise belong to a hexagonal close-packed oxygen lattice. The 2b site contributes rather strong uniaxial anisotropy, and the anisotropy field of 1.4 MAm−1 is more than three times the magnetization (0.38 MAm−1 ), making it possible in the early 1950s to achieve coercivity comparable to the magnetization and manufacture cheap ceramic magnets in any desired shape, thereby overcoming the shape barrier that had impeded the development permanent magnets for a millennium. A million tonnes of these ferrite magnets is sold every year. 1 History of Magnetism and Basic Concepts 31 The drawback of any oxide magnetic material is that its magnetization is never more than a third of that of metallic iron. This is unavoidable because most of the unit cell volume is occupied by large, nonmagnetic O2− anions, with the high-spin ferric iron Fe3+ or other magnetic ions confined to the interstices in the oxygen lattice. To make matters worse, a ferrimagnetic structure reduces the magnetization further. There are relatively few ferromagnetic oxides; CrO2 is one example. It is not an insulator, but a half metal, with a gap in the minority-spin conduction band. A search for insulating ferromagnetic oxides in the 1950s led to the investigation of ABO3 compounds with the perovskite structure. Here the magnetic B cations occupy the 1a octahedral sites, and the nonmagnetic A cations occupy the 12-coordinated 1b sites in the ideal cubic structure. It proved to be possible to obtain ferromagnetism provided the A cations are present in two different valence states. This works best in mixed-valence manganites [26], with composition 2+ 3+ (3d4 ) (La3+ 0.7 M0.3 )MnO3 where M = Ba, Ca, or Sr. The resulting mixture of Mn 4+ 3 and Mn (3d ) on B sites leads to electron hopping with spin memory from one 3d3 core to another. This is the ferromagnetic double exchange interaction, envisaged by Clarence Zener in 1951. Similar electron hopping occurs for Fe2+ and Fe3+ in the octahedral sites of magnetite. A consequence is that the oxides, though ferromagnetic, are no longer insulating, and the Curie temperatures are not particularly high – they do not exceed 400 K. A notable feature of the mixed-valence manganites, related to their hopping conduction, is the “colossal magnetoresistance” observed near the Curie point, where there is a broad maximum in the resistance that can be suppressed by applying a magnetic field of several tesla. All four oxide structures are presented in Fig. 14. They illustrate the importance of crystal chemistry for determining magnetic properties. Fig. 14 Crystal structures of magnetic oxides: perovskite (top left), spinel (bottom left), garnet (center), magnetoplumbite (right). The oxygen coordination polyhedral around the magnetic cations (tetrahedrons, blue, or octahedrons, brown) is illustrated. The spheres are large nonmagnetic cations. Unit cells are outlined in black. Magnetoplumbite is hexagonal, and the others are cubic [31], with kind permission from APS 32 J. M. D. Coey Research on localized electron magnetism in oxides and related compounds has passed through three phases. Beginning with studies of polycrystalline ceramics from about 1950, single crystals were grown for specific physical investigations after about 1970, and then in the late 1980s, following the high-temperature superconductivity boom, came the growth and characterization of ferromagnetic and ferrimagnetic oxide thin films and first steps toward all-oxide spin electronics. A similar pattern was followed by sulfides, fluorides, and other magnetic compounds. All are discussed further in Chap. 17, “Magnetic Oxides and Other Compounds.” Intermetallic Compounds A rich class of functional magnetic materials is the intermetallic compounds of rare earth elements and transition metals. The atomic volume ratio of a 4f to a 3d atom is about three, so the alloys tend to be stoichiometric line compounds rather than solid solutions. The first of these was SmCo5 , developed for permanent magnet applications in the USA in the mid-1960s by Karl Strnat. It was followed by Sm2 Co17 in the early 1970s, and then in 1983 came the announcement of the independent discovery of the first iron-based rare earth magnet, the ternary Nd2 Fe14 B, by Masato Sagawa in Japan and John Croat in the USA. This was a breakthrough because iron is cheaper and more strongly magnetic than cobalt. Nd2 Fe14 B has since come to dominate the global high-performance magnet market, with an annual production in excess of 100,000 tonnes. The coercivity needed in these optimized rare earth permanent magnets is comparable to their magnetization, and the optimization of the microstructure of a new hard magnetic material to attain the highest possible energy product, which scales as Ms 2 but can never exceed ¼μ0 Ms 2 , is a long empirical process. It generally takes many years to achieve a coercivity as high as 20–30% of the anisotropy field [28]. The battle to create the metastable hysteretic state that permits a permanent magnet to energize the surrounding space with a large stray field is never easy to win, and each material requires a different strategy. The fundamental significance of these intermetallics and related interstitial compounds such as Sm2 Fe17 N3 that were discovered in the 1990s is that crystal field theory and quantum mechanics were involved in their design. All have a uniaxial crystal structure with a single easy axis and strong magnetocrystalline anisotropy. Such anisotropy is a prerequisite for the substantial coercivity, Hc Ms needed to overcome the shape barrier and create a magnet with any desired form. The practical significance of the rare earth permanent magnets has been the appearance of a wide range of compact, energy-efficient electromagnetic energy converters that are being used in consumer products, electric vehicles, aeronautics, robotics, and wind generators. Besides magnetocrystalline anisotropy, another potentially useful consequence of the spin-orbit interaction in rare earth intermetallics is their strong magnetostriction. The rare earth elements order magnetically at or below room temperature so, just as for the permanent magnets, it was necessary to form an intermetallic 1 History of Magnetism and Basic Concepts 33 Fig. 15 Crystal structures of ferromagnetic intermetallic compounds: YFe2 (cubic, left) SmCo5 (hexagonal, top centre), Co2 MnSi (cubic, bottom centre), Nd2 Fe14 B (tetragonal, right). Fe and Co Mn are the small brown/red, blue, and scarlet spheres. Rare earths are the large spheres. Si and B are grey and black compound with iron or cobalt to obtain a functional material with a useful Curie temperature that should be substantially greater than room temperature to ensure adequate magnetic stability. A functional magnetostrictive material has to be magnetically soft, and this was achieved in the RFe2 rare earth Laves phase compounds by Arthur Clark in 1984, who combined Dy and Tb, which have the same sign of magnetostriction, but compensating anisotropy of opposite sign, in the cubic alloy (Tb0.3 Dy0.7 )Fe2 , known as Terfenol-D. Single crystals exhibited Joulian magnetostriction of up to 2000 parts per million (ppm), a hundred times greater than Joule had measured 150 years earlier in pure iron [16] (see Chaps. 28, “Permanent Magnet Materials and Applications,” and 11, “Magnetostriction and Magnetoelasticity”). Magnetically soft rare earth intermetallics are also of interest as magnetocaloric materials for solid-state refrigeration when their Curie point is close to room temperature (see Chap. 30, “Magnetocaloric Materials and Applications”). Some crystal structures of rare earth intermetallics are shown in Fig. 15. Among the other intermetallic families, the ordered body-centered cubic Heusler families of X2 YZ or XYZ alloys are notable in that they include a wide variety of magnetically ordered compounds, such as the magnetic shape-memory alloy NiMnSb or the half-metallic ferromagnet Co2 MnSi, which, like CrO2 , has a gap at the Fermi level for minority-spin electrons. Information on a great many metallic magnetic materials is collected in Chap. 4, “Electronic Structure: Metals and Insulators.” 34 J. M. D. Coey Model Systems Magnetism has proved to be a fertile proving ground for condensed matter theory. The first mean-field theory was Weiss’s molecular field of magnetism, later generalized by Lev Landau in the USSR in 1937. There followed more sophisticated theories of phase transitions, with magnetism providing much of the data to support them. The single-ion anisotropy of rare earth ions due to the local crystal field reduces the effective dimensionality of the magnetic order parameter from three to two for easy-plane (xy) anisotropy or from three to one for easy-z-axis (Ising) anisotropy. Magnetically ordered compounds can be synthesized with an effective spatial dimension of one (chains of magnetic atoms), two (planes of magnetic atoms), or three (networks of magnetic atoms), as well as ladders and isolated motifs. Magnetism has provided a treasury of materials that show continuous phase transitions as a function of temperature or quantum phase transitions at zero temperature as a function of pressure or magnetic field, as well as topological phases such as the two-dimensional xy model, investigated by David Thouless, Michael Kosterlitz, and Duncan Haldane. It is frequently possible to realize magnetic materials that embody the essential electronic or structural features of the theoretical models. An early theoretical milestone was Lars Onsager’s 1944 solution of the twodimensional Ising model, where spins are regarded as one-dimensional scalars that can take only values of ±1. The behavior of more complex and realistic systems such as the three-dimensional Heisenberg model near its Curie temperature was solved numerically using the renormalization group technique developed by Kenneth Wilson in the 1970s. The ability to tailor model magnetic systems, with an effective spatial dimension of 1 or 2 due to their structures of chains or planes of magnetic ions and an effective spin dimension of 1, 2, or 3 determined by magnetocrystalline anisotropy due to the combination of the crystal/ligand field and the spin-orbit interaction, was instrumental in laying the foundation of the modern theory of phase transitions. The theory is based on universality classes where power-law temperature variations of the order parameter and its thermodynamic derivatives with respect to temperature or magnetic field in the vicinity of the phase transition are characterized by numerical critical exponents that depend only by the dimensionality of the space and the magnetic order parameter. Another fecund line of enquiry was “Does a single impurity in a metal bear a magnetic moment?” This was related to Jun Kondo’s formulation of a problem concerning the scattering of electrons by magnetic impurities in metals and its eventual solution in 1980. In the presence of antiferromagnetic coupling between an impurity and the conduction electrons of a metallic host, the combination enters a nonmagnetic ground state below the Kondo temperature TK . The Kondo effect is characterized by a minimum in the electrical resistivity. The study of magnetic impurities in metals focused attention on the relation between magnetism and electronic transport, which has proved extremely fruitful, leading to several Nobel Prizes and the emergence in the 1990s of spin electronics. 1 History of Magnetism and Basic Concepts 35 The exchange interaction between two dilute magnetic impurities in a metal is long-range, decaying as 1/r3 while oscillating in sign between ferrromagnetic and antiferromagnetic, where r is their separation. The following is the RudermanKittel-Kasuya-Yosida (RKKY) exchange interaction J (r) = aJsd 2 (sinξ − ξcosξ) /ξ4 (18) where a is a constant, Jsd is the exchange coupling between the localized impurity and the conduction electrons, and ξ is twice the product of r and the Fermi wavevector. It was studied intensively in the 1970s in dilute alloys such as AuFe or CuMn, known as spin glasses (the host is in bold type, and the impurity in italics). The impurity in these hosts retains its moment at low temperatures, and the RKKY exchange coupling J (∇) between a pair of spins is as likely to be ferromagnetic (positive) as antiferromagnetic (negative). The impurity spins freeze progressively in random orientations around a temperature Tf that is proportional to the magnetic concentration. The nature of this transition to the frozen spin glass state was exhaustively debated. A related issue, the long-range exchange interactions associated with the ripples of spin polarization created by a magnetic impurity in a metal, led to an understanding of complex magnetic order in the rare earth metals ( Chap. 14, “Magnetism of the Elements”). The magnetism of electronic model systems such as a chain of 1s atoms with an on-site coulomb repulsion U when two electrons occupy the same site, formulated by John Hubbard in 1963, has proved to be remarkably complex. Control parameters in the Hubbard model are the band filling and the ratio of U to the bandwidth, and they lead to insulating and metallic, ferromagnetic, and antiferromagnetic solutions. Amorphous Magnets An important question, related to the dilute spin glass problem, was what effect does atomic disorder have on magnetic order and the magnetic phase transition in magnetically concentrated systems? Here a dichotomy emerges between ferromagnetic and antiferromagnetic interactions. The answer for materials with ferromagnetic exchange and a weak local electrostatic (crystal field) interaction is that the atomic disorder has little effect. Techniques for rapidly cooling eutectic melts at rates of order 106 Ks−1 developed around 1970 produced a family of useful amorphous ferromagnetic alloys based on Fe, Co, and Ni, with a minor amount of metalloid such as B, P, or Si. These metallic glasses, frequently in the form of thin ribbons obtained by melt spinning, were magnetically soft and proved that ferromagnetic order could exist without a crystal lattice. There are no crystal axes, and weak local anisotropy due to the local electrostatic interactions averages out. The magnetic metallic glasses are mechanically strong and have found applications in transformer cores and security tags. 36 J. M. D. Coey Amorphous materials with antiferromagnetic interactions are qualitatively different. Whenever the superexchange neighbors in oxides or other insulating compounds form odd-membered rings, these interactions are frustrated. No collinear magnetic configuration is able to satisfy them all. In crystalline antiferromagnets like rocksalt-structure NiO, the partial frustration leads to a reduced Néel temperature, but in fully frustrated pyrochlore-structure compounds, for example, the Néel point is completely suppressed. In the amorphous state, however, frustration has a spatially random aspect, and it leads to random spin freezing with a tendency to antiferromagnetic nearest-neighbor correlations, known as speromagnetism. The situation for amorphous rare earth intermetallic alloys, which are best prepared by prepared by rapid sputtering, is different. There the local anisotropy at rare earth sites is strong, and does not average out, but it tends to pin the rare earth moments to randomly oriented easy axes in directions that are roughly parallel to that of the local magnetization of the 3d ferromagnetic sublattice for the light rare earths and roughly antiparallel to it for the heavy rare earths. The sign of the 3d-4f coupling changes in the middle of the series, so that amorphous Gd-Fe alloys, for example, are ferrimagnetic. (Gd is the case where there are no orbital moment and no magnetocrystalline anisotropy on account of its half-filled, 4f7 shell.) Rapid quenching can also be used to produce nanocrystalline material with isotropic crystallite orientations of nanocrystals embedded in an amorphous matrix. Certain soft magnetic materials have such a two-phase structure. Nanocrystalline Nd-Fe-B produced by rapid quenching shows useful coercivity due to domain wall pinning at the Nd2 Fe14 B nanocrystallite boundaries, but the remanence is only about half the saturation magnetization on account of the randomly directed easy axes of the tetragonal crystallites. The magnitude of the anisotropy and the nanoscale dimension are critical for the averaging that determines the magnetic properties. Magnetic Fine Particles An early approach to the difficult problem of calculating hysteresis was to focus on magnetization reversal in single-domain particles that were too small to benefit from any reduction in their energy by forming a domain wall. Edmund Stoner and Peter Wohlfarth proposed an influential model in 1948. The particles were assumed each to have a single anisotropy axis, and the reverse field parallel to the axis necessary for magnetic reversal was the anisotropy field Ha = 2Ku /μ0 Ms , potentially a very large value. There was no coercivity when the field was applied perpendicular to the axis. Insights arose from the substantial deviation of real systems from the idealized Stoner-Wohlfarth model. Meanwhile, the following year Néel, seeking to understand the remanent magnetism and hysteresis of baked clay and igneous rocks, proposed a model of thermally driven fluctuations of the magnetization of nanometer-sized ferromagnetic particles of volume V, a phenomenon known as superparamagnetism. The fluctuation time depended exponentially on the ratio of the energy barrier to magnetic 1 History of Magnetism and Basic Concepts 37 reversal reversal ≈ Ku V to the thermal energy kB T. Here Ku is the uniaxial anisotropy (Eq. 16) of shape or magnetocrystalline origin. The expression for the time τ that elapses before a magnetic reversal is τ = τ0 exp (/kB T ) (19) where the attempt frequency 1/τ0 was taken to be the natural resonance frequency, ∼109 Hz. When the particles are superparamagnetic, the magnetization of particles smaller than a critical size fluctuates rapidly above a critical blocking temperature. The magnetization at lower temperatures, or for larger particles, does not fluctuate on the measurement timescale, and the particles are then said to be blocked. The blocking criterion for magnetic measurements at room temperature is defined, somewhat arbitrarily, as /kB T ≈ 25, corresponding to τ ≈ 100 s and ≈ 1 eV (see Chap. 20, “Magnetic Nanoparticles”). The 10-year stability criterion is /kB T ≈ 40. Cooling an ensemble of particles through the blocking temperature Tb = Ku V/25kB in a magnetic field leads to a relatively stable thermoremanent magnetization. The typical size of iron oxide particles that are superparamagnetic at room temperature is 10 nm. The magnetization of baked clay becomes blocked on cooling through Tb in the Earth’s magnetic field. From the direction of the thermoremanent magnetization of appropriately dated hearths of pottery kilns, records of the historical secular variation of the Earth’s field could be established, a topic known as archeomagnetism. Application of the same idea of thermoremanent magnetization to cooling of igneous rocks in the Earth’s field provided a direct and convincing argument for geomagnetic reversals and continental drift; rocks cooling at different periods experienced fields of different polarities (Fig. 16), which followed an irregular sequence on a much longer timescale than the secular variation. The reversals could be dated using radioisotope methods on successive lava flows. This gave birth to the subfield of paleomagnetism and in turn allowed dating of the patterns of remanent magnetization picked up in oceanographic surveys conducted in the 1960s that established the reality of seafloor spreading. The theory of global plate tectonics has had far-reaching consequences for Earth science [29]. Superparamagnetic particles have found other practical uses. Ferrofluids, the colloidal suspensions of nanoparticles in oil or water with surfactants to inhibit agglomeration, are just one. They behave like anhysteretic ferromagnetic liquids. Individual particles or micron-sized polymer beads loaded with many of them may be functionalized with streptavidin and used as magnetic labels for specific biotin-tagged biochemical species, enabling them to be detected magnetically and separated by high-gradient magnetic separation based on the Kelvin force on a particle with moment m, fK = (m.∇)B. Medical applications of magnetic fine particles include hyperthermia (targeted heating by exposure to a high-frequency magnetic field) and use as contrast agents in magnetic resonance imaging. However the most far-reaching application of magnetic nanoparticles so far has been in magnetic recording. 38 J. M. D. Coey Fig. 16 Polarity of the thermoremanent magnetization measured across the floor of the Atlantic ocean (left). Current polarity is dark; reversed polarity is light. The pattern is symmetrical about the mid-ocean ridge, where new oceanic crust is being created. Random reversals of the Earth’s field over the past 5 My, which are dated from other igneous lava flows, determine the chronological pattern (right) that is used to determine the rate of continental drift, of order centimeters per year. (McElhinney, Palaeomagnetism and Plate Tectonics [29], courtesy of Cambridge University Press) Magnetic Recording Particulate magnetic recording enjoyed a heyday that lasted over half a century, beginning with analog recording on magnetic tapes in Germany in the 1930s through digital recording on the hard and floppy discs that were introduced in the 1950s and 1960s, before eventually being superseded by thin-film recording in the late 1980 [27]. Particulate magnetic recording [30] was largely based on acicular particles of γFe2 O3 often doped with 1–2% Co. Elongated iron particles were also used, and acicular CrO2 was useful for rapid thermoremanent reproduction of videotapes on account of its low Curie temperature. Magnetic digital tape recording with hard ferrite particulate media continues to be used for archival storage. The trend with magnetic media has always been to cram ever more digital data onto ever smaller areas. This has been possible because magnetic recording technology is inherently scaleable since reading is done by sensing the stray field of a patch of magnetized particles. It follows from Eq. 2 that since the dipole field decays as 1/r3 and the moment m ∼ Mr3 , the magnitude of B is unchanged when everything else shrinks by the same scale factor – at least until the superparamagnetic limit KV/kB T ≈ 40 is reached, at which point the magnetic records become thermally unstable. To continue the scaling to bit sizes below 1 History of Magnetism and Basic Concepts 39 Fig. 17 Exponential growth of magnetic recording density over 50 years. The lower panel shows the magnetized magnetic medium with successive generations of read heads based on anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR), and tunnel magnetoresistance (TMR) 100 nm, granular films of a highly anisotropic tetragonal Fe-Pt alloy are used to maintain stability of the magnetic records on ever-smaller oriented crystalline grains. The individual grains are less than 8 nm in diameter. Over the 65-year history of hard disc magnetic recording, the bit density has increased by eight orders of magnitude, at ever-decreasing cost (Fig. 17). Copies cost virtually nothing, and the volume of data stored on hard discs in computers and data centers doubles every year, so that as much new data is recorded each year as was ever recorded in all previous years of human history. This data explosion is unprecedented, and the third magnetic revolution, the big data revolution, is sure to have profound social and economic consequences. Although flash memory has displaced the magnetic hard discs from personal computers. The huge data centres, which are the physical embodiment of the ‘cloud’ where everything we download from the interenet is stored continue to use hard disc drives. 40 J. M. D. Coey Methods of Investigation Magnetism is an experimental science, and progress in understanding and applications is generally contingent on advances in fabrication and measurement technology, whether it was fourteenth-century technology to fabricate a lodestone sphere or twenty-first-century technology to prepare and pattern a 16-layer thin-film stack for a magnetic sensor. The current phase of information technology relies largely on semiconductors to process digital data and on magnets for long-term storage. For many physical investigations, magnetic materials are needed in special forms such as single crystals or thin films. Crystal growers have always been assiduously cultivated by neutron scatterers and other condensed matter physicists. Only with single crystals can tensor properties such as susceptibility, magnetostriction, and magnetotransport be measured properly. Nanoscale magnetic composites have extended the range of magnetic properties available in both hard and soft magnets. After 1970, thin-film growth facilities (sputtering, electron beam evaporation, pulsed laser deposition, molecular beam epitaxy) began to appear in magnetism laboratories worldwide. Ultra-high vacuum has facilitated the study of surface magnetism at the atomic level, while some of the motivation to investigate magneto-optics or magnetoresistance of metallic thin films, especially in thin-film heterostructures, arose from the prospect of massively improved magnetic data storage. Experimental methods are discussed in the chapters in Part 3 of this Handbook. Materials Preparation Silicon steel has been produced for electromagnetic applications by hot rolling since the beginning of the twentieth century. Annual production is now about 15 million tonnes, half of it in China. Permanent magnets, soft ferrites, and specialized magnetic alloys are produced in annual quantities ranging from upward of a hundred to a million tonnes. All such bulk applications of magnetism are highly sensitive to the cost of raw materials. This effectively disqualifies about a third of the elements in the periodic table and half of the heavy transition elements from consideration as alloy additives in bulk material. Newer methods such as mechanical alloying of elemental powders and rapid quenching from the melt by strip casting or melt spinning have joined the traditional methods of high-temperature furnace synthesis of bulk magnetic materials. The transformation of magnetic materials science that has gathered pace since 1970 has been triggered by the ability to prepare new materials for magnetic devices in thin-film form. The minute quantity of material needed for a magnetic sensor or memory element, where the layers are tens of nanometers thick, means that any useful stable element can be considered. Platinum, for example, may sell for $30,000 per kilogram, yet it is an indispensable constituent of the magnetic medium in the 400 million hard disc drives shipped each year that sell for about $60 each. 1 History of Magnetism and Basic Concepts 41 Uniform magnetic thin films down to atomic-scale thicknesses are produced in many laboratories by e-beam evaporation, sputtering, pulsed laser deposition, or molecular beam epitaxy, and the more complex tools needed to make patterned multilayer nanometer-scale thin-film stacks are quite widely available in research centers, as well as in the fabs of the electronics industry, which deliver the hardware on which the technology for modern life depends. Experimental Methods Advances in experimental observation underpin progress in conceptual understanding and technology. The discovery of magnetic resonance, the sharp absorption of microwave or radiofrequency radiation by Zeeman split levels of the magnetic moment of an atom or a nucleus in a magnetic field, or the collective precession of the entire magnetic moment of a solid was a landmark in modern magnetism. Significant mainly for the insight provided into solids and liquids at an atomic scale, electron paramagnetic resonance (EPR) was discovered by Yevgeny Zavoisky in 1944, and Felix Bloch and Edward Purcell established the existence of nuclear magnetic resonance (NMR) 2 years later. In 1958, Rudolf Mössbauer discovered a spectroscopic variant making use of low-energy gamma rays emitted by transitions from the excited states of some stable isotopes of iron (Fe57 ) and certain rare earths (Eu151 , Dy161 , etc.). All except Zavoisky received a Nobel Prize. The hyperfine interactions of the multipole moments of the nuclei (electric monopole, magnetic dipole, nuclear quadrupole) offered a point probe of electric and magnetic fields at the heart of the atom. Larmor precession of the total magnetization of a ferromagnet in its internal field, usually in a resonant microwave cavity, was discussed theoretically by Landau and Evgeny Lifshitz in 1935, and ferromagnetic resonance (FMR) was confirmed experimentally 10 years later. Of the non-resonant experimental probes, magnetic neutron scattering has probably been the most influential and generally useful. A beam of thermal neutrons from a nuclear reactor was first exploited for elastic diffraction in the USA in 1951 by Clifford Shull and Ernest Wohlan, who used the magnetic Bragg scattering to reveal the antiferromagnetic order in MnO. Countless magnetic structures have been determined since, using the research reactors at Chalk River, Harwell, Brookhaven, Grenoble, and elsewhere. Magnetic excitations can be characterized by inelastic scattering of thermal neutrons, with the help of the triple-axis spectrometer developed in Canada by Bertram Brockhouse at Chalk River in 1956. Complete spin-wave dispersion relations provide a wealth of information on anisotropy and exchange. Newer accelerator-based neutron spallation sources at ISIS, Oak Ridge, and Lund provide intense pulses of neutrons by collision of highly energetic protons with a target of a heavy metal such as tungsten or mercury. They are most useful for studying magnetization dynamics. The low neutron scattering and absorption cross sections of most stable isotopes mean that neutrons can penetrate deeply into condensed matter. 42 J. M. D. Coey Besides neutrons, other intense beams of particles or electromagnetic radiation available at large-scale facilities have proved invaluable for probing magnetism. The intense, tunable ultraviolet and X-ray radiation from synchrotron sources allows the measurement of magnetic dichroism from deep atomic levels and permits the separate determination of spin and orbital contributions to the magnetic moment. The spectroscopy is element-specific and distinguishes different charge states of the same element. Spin-sensitive angular-resolved photoelectron spectroscopy makes it possible to map the spin-resolved electronic band structure. Muon methods are more specialized; they depend on the Larmor precession of short-lived (2.20 μs) positive muons when they are implanted into interstitial sites in a solid. Magnetic scattering methods are discussed in Chap. 25, “Magnetic Scattering.” The specialized instruments accessible at large-scale facilities supplement the traditional benchtop measurement capabilities of research laboratories. Perhaps the most versatile and convenient of these, used to measure the magnetization and susceptibility of small samples, is the vibrating sample magnetometer invented by Simon Foner in 1956 and now a workhorse in magnetism laboratories across the world. The sample is vibrated in a uniform magnetic field, produced by an electromagnet or a superconducting coil, about the center of a set of quadrupole pickup coils, which provide a signal proportional to the magnetic moment. Since sample mass rather than sample volume is usually known, it is generally the mass susceptibility χ m = χ /ρ that is determined. Superconducting magnets now provide fields of up to 20 tesla or more for NMR and general laboratory use. The 5–10 T magnets are common, and they are usually cooled by closed-cycle cryocoolers to avoid wasting helium. Coupled with superconducting SQUID sensors, ultrasensitive magnetometers capable of measuring magnetic moments of 10−10 Am2 or less are widely available. (The moment of a 5 × mm2 ferromagnetic monolayer is of order 10−8 Am2 .) High magnetic fields, up to 35 T, require expensive special installations with water-cooled Bitter magnets consuming many megawatts of electrical power. Resistive/superconducting hybrids in Tallahassee, Grenoble and Tsukuba, and Nijmegen can generate steady fields in excess of 40 T. Higher fields imply short pulses; the higher the field, the shorter the pulse. Reusable coils generate pulsed fields approaching 100 T in Los Alamos, Tokyo, Dresden, Wuhan, and Toulouse. Magnetic domain structures are usually imaged by magneto-optic Kerr microscopy, magnetic force microscopy, or scanning electron microscopy, although scanning SQUID and scanning Hall probe methods have also been developed. The Bitter method with a magnetite colloid continues to be used. All these methods image the surface or the stray field near the surface. Ultra-fast, picosecond magnetization dynamics are studied by optical pulse-probe methods based on the magneto-optic Kerr effect (MOKE). Transmission electron microscopy reveals the atomic structures of thin films and interfaces with atomic-scale resolution, while Lorentz microscopy offers magnetic contrast and holographic methods are able to image domains in three dimensions. Atomic-scale resolution can be achieved by point-probe methods with magnetic force microscopy or spin-polarized scanning tunnelling microscopy. The shift of focus in magnetism toward thin films and 1 History of Magnetism and Basic Concepts 43 thin-film devices has been matched by the development of the sensitive analytical methods needed to characterize them. Hysteresis in thin films is conveniently measured by MOKE or by anomalous Hall effect (AHE) when the films are magnetized perpendicular to their plane. Magnetic fields and measurements are discussed in Chap. 22, “Magnetic Fields and Measurements” and other chapters in Part 3. An important consequence of the increasing availability of commercial superconducting magnets from the late 1960s was the development of medical diagnostic imaging of tissue based on proton relaxation times measured by NMR. Thousands of these scanners in hospitals across the world provide doctors with images of the hearts, brains, bones, and every sort of tumor. Computational Methods After about 1980, computer simulation began to emerge as a third force, besides experiment and theory, to gain insight into the physics of correlated electrons in magnetic systems. Contributions are mainly in two areas. One is calculation of the electronic structure, magnetic structure, magnetization, Curie temperature, and crystal structure of metallic alloys and compounds by using the density functional method. Magnetotransport in thin-film device structures can also be calculated. Here there is potential to seek and evaluate new magnetic phases in silico, before trying to make them in the laboratory. This magnetic genome program is in its infancy; success with magnetic materials to date has been limited, but the prospects are enticing. The other area where computation has become a significant source of new insight is micromagnetic simulation. The domain structure and magnetization dynamics of magnetic thin-film structures and model heterostructures are intensely studied, both in industrial and academic laboratories. Simulation overcomes the surface limitation of experimental domain imaging. Software is generally based on finite element methods or the Landau-Lifshitz-Gilbert equation for magnetization dynamics. Spin Electronics As technology became available in the 1960s and 1970s to prepare high-quality metallic films with thicknesses in the nanometer range, interest in their magnetostansport properties grew. The terrain was being prepared for the emergence of a new phase of research that has grown to become the dominant theme in magnetism today – spin electronics. Spin electronics is the science of electron spin transport in solids. Many chapters in the Handbook deal with its various aspects. For a long time, conventional electronics treated electrons simply as elementary Fermi-Dirac particles carrying a charge e, but it ignored their spin angular momentum ½. At first this was entirely justified; charge is conserved – the electron has no tendency to flip between states with charge ± e, no matter how strongly 44 J. M. D. Coey it is scattered. But angular momentum is not conserved, and spin flip scattering is common in metals. Perhaps one scattering event in 100 changes the electron spin state, so the spin diffusion length ls should be about ten times the mean free path λ of the electron in a solid. When electronic device dimensions were many microns, there was no chance of an electron retaining the memory of any initial spin polarization it may have had, unless the device itself was ferromagnetic. Anisotropic magnetoresistance, where the scattering depends slightly on the relative orientation of the current and magnetization because of spin-orbit coupling, can be regarded as the archetypical spin electronic process. The relative magnitude of effect in permalloy, for instance, is only ∼2%, but the alloy is extremely soft, on account of simultaneously vanishing anisotropy and magnetostriction, so a permalloy strip with current flowing at 45◦ to the magnetic easy axis along the strip for maximum sensitivity – which can be achieved by a superposed “barber pole” pattern of highly conducting gold – makes a simple, miniature sensor for low magnetic fields, with a reasonable signal-to-noise ratio. AMR sensors replaced inductive sensors in the heads used to read data from hard discs in 1990, and the annual rate of increase of storage density improved sharply as a result. Meanwhile, research activity on thin-film heterostructures where the layer thickness was comparable to the spin diffusion length began to pick up as more sophisticated thin-film vacuum deposition tools were developed. Spin diffusion lengths are 200 nm in Cu, or about ten times the mean free path, as expected, but they are shorter in the ferromagnetic elements and sharply different for majorityand minority-spin electrons. The mean free path for minority-spin electrons in Co is only 1 nm. Particularly influential and significant was the work carried out in 1988 in the groups of Peter Grunberg in Germany and Albert Fert in France on multilayer stacks of ferromagnetic and nonferromagnetic elements that led to the discovery of giant magnetoresistance (GMR). The effect depended on electrons retaining some of their spin polarization as they emerged from a ferromagnetic layer and crossed a nonmagnetic layer before reaching another ferromagnetic layer. Big changes of resistance were found when the relative alignment of the adjacent ferromagnetic iron layers in an Fe-Cr multilayer stack was altered from antiparallel to parallel by applying a magnetic field (Fig. 18). At first, large magnetic fields and low temperatures were needed to see the resistance changes, but the structure was soon simplified to a sandwich of just two ferromagnetic layers with a copper spacer that became known as a spin valve. Spin valves worked at room temperature, and they were sensitive to the small stray fields produced by recorded magnetic tape or disc media. In order to make a useful sensor, it was necessary to pin the direction of magnetization of one of the ferromagnetic layers while leaving the other free to respond to an in-plane field (Fig. 19). It was here that the phenomenon of exchange bias came to the rescue. First discovered in Co/CoO core shell particles by Meiklejohn and Bean in 1956, it was extended to antiferromagnetic/ferromagnetic thin-film pairs in Néel’s laboratory in Grenoble in the 1960s. By pinning one ferromagnetic layer with an adjacent antiferromagnet (initially NiO), a useful GMR sensor could be produced with a magnetoresistance change of order 10%. Exchange-biased GMR read heads 1 History of Magnetism and Basic Concepts 45 Fig. 18 Original measurement of giant magnetoresistance of a FeCr multilayer stack, where the iron layers naturally adopt an antiparallel conduction, which can be converted to a parallel configuration in an applied field [31] developed by Stuart Parkin and colleagues went into production at IBM in 1998 – a remarkably rapid transfer from a laboratory discovery to mass production. Exchange bias was the first practical use of an antiferromagnet. The Nobel Physics Prize was awarded to Fert and Grunberg for their work in 2007. Subsequent developments succeeded in eliminating the influence of the stray field of the pinned layer on the free layer by means of a synthetic antiferromagnet. This was another sandwich stack, like the slimmed-down spin valve, except the spacer was not copper, but an element that transferred exchange coupling from one ferromagnetic layer to the other. Ruthenium proved to be ideal, and a layer just 0.7 nm thick was found to be ideal for antiferromagnetic coupling [32]. GMR’s tenure as read-head technology was to prove as short-lived as that of AMR. A new pretender with a much larger resistance change was based on the magnetic tunnel junction (MTJ), a modified spin valve where the nonmagnetic metal spacer is replaced by a thin layer of nonmagnetic insulator. Electron tunneling across an atomically thin vacuum barrier had been a striking prediction of quantum mechanics implicit in the idea of the wavefunction. The thin barrier was at first made of amorphous alumina, but it was replaced by crystalline MgO after it was found in 2004 that junctions where the MgO barrier acts as a spin filter exhibit tunneling magnetoresistance (TMR) in excess of 200% [33, 34] (Fig. 19). The adoption of 46 J. M. D. Coey B B I free pinned free pinned af I ΔR af ΔR MR = (R↑↓−R↑↑)/R↑↑ B Spin valve sensor B Magnetic tunnel junction (MTJ) Fig. 19 Magnetic bilayer spin-valve stacks used as sensor (left) or as a memory element (right). In each case, the magnetization lies in-plane, and the lower ferromagnetic reference layer is pinned by exchange bias with the purple underlying antiferromagnetic layer, while the upper ferromagnetic free layer changes its orientation in response to the applied magnetic field. The change in stack resistance is plotted as a function of applied field. The magnetoresistance ratio MR is defined as the normalized resistance change between parallel and antiparallel orientation of the two ferromagnetic layers TMR sensors in read heads in 2005 was accompanied by a change from in-plane to perpendicular recording on the magnetic medium. Despite the changing generations of readers, the hard disc writer remained what is always had been, a miniature electromagnet that delivers sufficient flux to a patch of magnetic medium to overcome its coercivity and write the record. The extreme demands of magnetic recording have driven contactless magnetic sensing to new heights of sensitivity and miniaturization requiring increasingly hard magnetic media and new ways of writing them. Thin-film GMR and TMR structures have also taken a new life as magnetic switches for nonvolatile memory and logic. Most prominent is magnetic random access memory (MRAM), where huge arrays of memory cells are based on magnetic tunnel junctions. Magnetic sensing is discussed in Chaps. 31, “Magnetic Sensors,” and 22, “Magnetic Fields and Measurements.” Magnetic thin-film technology has now advanced to the point where uniform layers in synthetic antiferromagnets and magnetic tunnel junctions only a few atoms thick are routinely deposited on entire 200 or 300 mm silicon wafers. A corollary of the short spin diffusion length of electrons in metals is the short distance – a few atomic monolayers – necessary for an electron to acquire spin polarization on transiting a ferromagnetic layer. Spin-polarized electron currents are central to spin electronics. 1 History of Magnetism and Basic Concepts 47 The relation between magnetism and the angular momentum of electrons was unveiled in Larmor precession and the Einstein-de Haas experiment over a hundred years ago, but only in the present century has it become commonplace to associate electric currents with short-range flows of angular momentum. A spin-polarized current carrying its angular momentum into a ferromagnetic thin-film element can exert torque in two ways. It can create an effective magnetic field, causing Larmor precession of the magnetization of the element, and it can exert spin transfer torque, described by John Slonczewski in 1996 that counteracts damping of the precession and can be used to stabilize high-frequency oscillations or switch the magnetization without the need for an external magnetic field. Spin torque switching is effective for elements smaller than 100 nm in size, and unlike switching by current-induced “Oersted” fields, it is scalable – an essential requirement for electronic devices. Luc Berger showed that spin torque can also be used to manipulate domain walls. A recurrent theme in the recent development of magnetism is the role of the spinorbit interaction. It is critically important in thin films [35], being responsible not only for the Kerr effect, magnetocrystalline anisotropy, and anisotropic magnetoresistance but also for the anomalous Hall effect and the spin Hall effect, whereby spin-orbit scattering of a current passing through a heavy metal or semiconductor produces a buildup of electrons with opposite spin on opposite sides of the conductor. This transverse spin current created by spin-orbit scattering enables the injection of angular momentum into an adjacent ferromagnetic layer and the change of its magnetization direction, an effect known as spin-orbit torque. Conversely, the inverse spin Hall effect is the appearance of a voltage across the heavy metal on pumping spin-polarized electrons into it from an adjacent ferromagnet, for example, by exciting ferromagnetic resonance. The origin of the intrinsic anomalous Hall effect was an open question in magnetism, for well over a hundred years. A consensus is now building that it is due to the geometric Berry phase acquired by electrons moving adiabatically through a magnetic medium. The phase can be acquired from a non-collinear spin structure in real space or from topological singularities in the band sturcture in reciprocal space. Circular micromagnetic defects, known as skyrmions are also topologically protected. Another manifestation of spin-orbit interaction is the Rashba effect; when an electric current is confined at an interface or surface, it tends to create a spin polarization normal to the direction of current flow. One of the most remarkable surface phenomena, arising from work by Haldane in 1988, is the possibility of topologically protected spin currents. A special feature of the band structure ensures that electrons at the surface or edges of some insulators or semiconductors are in gapless states. Electrons in these states can propagate around the surface without scattering, and they exhibit a spin order that winds around the surface as the direction of electron spin is usually locked at right angles to their linear momentum. Electrons at surfaces and interfaces can behave quite differently from electrons in the bulk, and interfaces are at the heart of electronic devices. The introduction of topological concepts into the discussion of spin-polarized electronic transport 48 J. M. D. Coey and magnetic defects is providing new insight into magnetism at the atomic and mesoscopic scales. Conclusion Magnetism since 1945 has been an area rich in discovery and useful applications, not least because of the tremendous increase in numbers of scientists and engineers working in the field. Magnet ownership for citizens of the developed world has skyrocketed from 1 or 2 magnets in 1945 to 100–200 60 years later or something of order a trillion if we count the individual magnetic bits on a hard disc in a desktop computer. Countless citizens throughout the world during this period already experienced magnetism’s bounty at first hand in the form of a cassette tape recorder, and nowadays they can access the vast stores of magnetically recorded information in huge data centers via the Internet using a handheld device. Magnetism is therefore playing a crucial role in the big data revolution that is engulfing us, by enabling the permanent data storage, from which we can make instant copies at practically no cost. It may deliver more nonvolatile computer memory if MRAM proves to a winning technology and possibly facilitate data transfer at rates up to the terahertz regime with the help of spin torque oscillators. There are potential magnetic solutions to the problems of ballooning energy consumption and the data rate bottleneck. There is potential to implement new paradigms for computation magnetically. While there is no certainty regarding the future form of information technology, improved existing solutions often have an inside track. Magnetism and magnetic materials may be a good bet. There have been half a dozen paradigm shifts – radical changes in the ways of seeing and understanding the magnet and its magnetic field – during its 2000-year encounter with human curiosity. Implications of the big data revolution for human society are only beginning to come into focus, but they are likely to be as profound as on the previous two occasions when magnetism changed the world. This Handbook is a guide to what is going on. Acknowledgments The author is grateful to Science Foundation Ireland for continued support, including contracts 10/IN.1/I3006, 13/ERC/I2561 and 16/IA/4534. Appendix: Units By the middle of the nineteenth century, it was becoming urgent to devise a standard set of units for electrical and magnetic quantities in order to exchange precise quantitative information. The burgeoning telegraph industry, for example, needed a standard of electrical resistance to control the quality of electrical cables. Separate electrostatic and electromagnetic unit systems based on the centimeter, the gram and the second had sprung into existence, and Maxwell and Jenkin proposed combining them in a coherent set of units in 1863. Their Gaussian cgs system was adopted 1 History of Magnetism and Basic Concepts 49 internationally in 1881. Written in this unit system, Maxwell’s equations relating electric and magnetic fields contain explicit factors of c, the velocity of light. Maxwell also introduced the idea of dimensional analysis in terms of the three basic quantities of mass, length, and time. The magnetic field H and the induction B are measured, respectively, in the numerically identical but dimensionally different units of oersted (Oe) and gauss (G). Another basic unit, this time of electric current, was adopted in the Système International d’Unités (SI) in 1948. The number of basic units and dimensions in any system is an arbitrary choice; the SI (International System of Units) uses four insofar as we are concerned, the meter, kilogram, second, and ampere (or five if we include the mole). The system has been adopted worldwide for the teaching of science and engineering at school and universities; it embodies the familiar electrical units of volt, ampere, and ohm for electrical potential, current, and resistance. Maxwell’s equations written in terms of two electric and two magnetic fields contain no factors of c or 4π in this system (Eq. 7), but they inevitably crop up elsewhere. B and H are obviously different quantities. The magnetic field strength H, like the magnetization M, has units of Am−1 . The magnetic induction B is measured in tesla (1 T ≡ 1 kgs2 A−2 ). Magnetic moments have units of Am2 , clearly indicating the origin of magnetism in electric currents and the absence of magnetic poles as real physical entities. The velocity of light is defined to be exactly 299,792,458 ms−1 . The two constants μ0 and ε0 , the permeability and permittivity of free space, are related by μ0 ε0 = c2 , where μ0 was 4π 10−7 kgs−2 A−2 according to the original definition of the ampere. However, in the new version of SI, which avoids the need for a physical standard kilogram, the equality of μ0 and 4π 10−7 is not absolute, but it is valid to ten significant figures. Only two of the three fields B, H, and M are independent (Fig. 4). The relation between them is Eq. 8, B = μ0 (H + M). This is the Sommerfeld convention for SI. The alternative Kenelly convention, often favored by electrical engineers, defines magnetic polarization as J = μ0 M, so that the relation becomes B = μ0 H + J. We Table 1 Numerical conversion factors between SI and cgs units Physical quantity B-field (magnetic flux density) H-field (magnetic field intensity) Magnetic moment Magnetization Specific magnetization Magnetic energy density Dimensionless susceptibility M/H Symbol B SI to cgs conversion 1 tesla = 10 kilogauss H 1 kAm−1 = 12.57 oersted m M σ 1 Am2 = 1000 emu 1 Am−1 = 12.57 gauss† 1 Am2 kg−1 = 1 emu g−1 (BH) χ 1 kJm−3 = 0.1257 MGOe 1 (SI) = 1/4π (cgs) *symbol G; § symbol Oe; † 4πM; Note: 12.57 = 4π; 79.58 = 1000/4π cgs to SI conversion 1 gauss* = 0.1 millitesla 1 oersted§ = 79.58 Am−1 1 emu = 1 mAm2 1 gauss† = 79.58 Am−1 1 emu g−1 = 1 Am2 kg−1 1 MGOe = 7.96 kJm−3 1 (cgs) = 4π (SI) 50 J. M. D. Coey follow the Sommerfeld convention in this Handbook. The magnetic field strength H is not measured in units of Tesla in any generally accepted convention, but it can be so expressed by multiplying by μ0 . At the present time, Gaussian cgs units remain in widespread use in research publications, despite the obvious advantages of SI. The use of the cgs system in magnetism runs into the difficulty that units of B and H, G and Oe, are dimensionally different but numerically the same; μ0 = 1, but it normally gets left out of the equations, which makes it impossible to check whether the dimensions balance. Table 1 lists the conversion factors and units in the two systems. The cgs equivalent of Eq. 8 is B = H + 4πM. The cgs unit of charge is defined in such a way that ε0 = 1/4πc and μ0 = 4π/c so factors of c appear in Maxwell’s equations in place of the electric and magnetic constants. Convenient numerical conversion factors between the two systems of units are provided in Table 1. Theoretical work in magnetism is sometimes presented in a set of units where c = = kB = 1. This simplifies the equations, but does nothing to facilitate quantitative comparison with experimental measurements. References 1. Kloss, A.: Geschichte des Magnetismus. VDE-Verlag, Berlin (1994) 2. Matthis, D.C.: Theory of Magnetism, ch. 1. Harper and Row, New York (1965) 3. Needham, J.: Science and Civilization in China, vol. 4, part 1. Cambridge University Press, Cambridge (1962) 4. Schmid, P.A.: Two early Arabic sources of the magnetic compass. J. Arabic Islamic Studies. 1, 81–132 (1997) 5. Fowler, T.: Bacon’s Novum Organum. Clarendon Press, Oxford (1878) 6. Pierre Pèlerin de Maricourt: The Letter of Petrus Peregrinus on the Magnet, AD 1292, Trans. Br. Arnold. McGraw-Hill, New York (1904) 7. Gilbert, W: De Magnete, Trans. P F Mottelay. Dover Publications, New York (1958) 8. Fara, P.: Sympathetic Attractions: Magnetic Practices, Beliefs and Symbolism in EighteenthCentury England. Princeton University Press, Princeton (1996) 9. Mottelay, P.F.: Bibliographical History of Electricity and Magnetism. Arno Press, New York (1975) 10. Faraday, M.: Experimental Researches in Electricity, volume III. Bernard Quartrich, London (1855) 11. Maxwell, J.C.: A Treatise on Electricity and Magnetism, two volumes. Clarendon Press, Oxford (1873) (Reprinted Cambridge University Press, 2010) 12. Bertotti, G.: Hysteresis in Magnetism. Academic Press, New York (1998) 13. Brown, W.F.: Micromagnetics. Interscience, New York (1963) 14. Sato, M., Ishii, Y.: Simple and approximate expressions of demagnetizing factors of uniformly magnetized rectangular rod and cylinder. J. Appl. Phys. 66, 983–988 (1989) 15. Hunt, B.J.: The Maxwellians. Cornell University Press, New York (1994) 16. Joule, J.P.: On the Effects of Magnetism upon the Dimensions of Iron and Steel Bars. Philosoph. Mag. Third Series. 76–87, 225–241 (1847) 17. Thomson, W.: On the electrodynamic qualities of metals. Effects of magnetization on the electric conductivity of nickel and iron. Proc. Roy. Soc. 8, 546–550 (1856) 18. Kerr, J.: On rotation of the plane of the polarization by reflection from the pole of a magnet. Philosoph. Mag. 3, 321 (1877) 1 History of Magnetism and Basic Concepts 51 19. Ewing, J.A.: Magnetic Induction in Iron and Other Metals, 3rd edn. The Electrician Publishing Company, London (1900) 20. Tomonaga, S.: The Story of Spin. University of Chicago Press, Chicago (1974) 21. Marage, P., Wallenborn, G. (eds.): Les Conseils Solvay et Les Débuts de la Physique Moderne. Université Libre de Bruxelles (1995) 22. Ballhausen, C.J.: Introduction to Ligand Field Theory. McGraw Hill, New York (1962) 23. Bozorth, R.M.: Ferromagnetism. McGraw Hill, New York (1950) (reprinted Wiley – IEEE Press, 1993) 24. Goodenough, J.B.: Magnetism and the Chemical Bond. Interscience, New York (1963) 25. Smit, J., Wijn, H.P.J.: Ferrites; Physical Properties of Ferrrimagnetic Oxides. Philips Technical Library, Eindhoven (1959) 26. Coey, J.M.D., Viret, M., von Molnar, S.: Mixed valence manganites. Adv. Phys. 48, 167 (1999) 27. Wang, S.X., Taratorin, A.M.: Magnetic Information Storage Technology. Academic Press, San Diego (1999) 28. Coey, J.M.D. (ed.): Rare-Earth Iron Permanent Magnets. Clarendon Press, Oxford (1996) 29. McElhinney, M.W.: Palaeomagnetism and Plate Tectonics. Cambridge University Press (1973) 30. Daniel, E.D., Mee, C.D., Clark, M.H. (eds.): Magnetic Recording, the First Hundred Years. IEEE Press, New York (1999) 31. Baibich, M.N., Broto, J.M., Fert, A., Nguyen Van Dau, F., et al.: Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Phys. Rev. Lettters. 61, 2472 (1988) 32. Parkin, S.S.P.: Systematic variation on the strength and oscillation period of indirect magnetic exchange coupling through the 3d, 4d and 5d transition metals. Phys. Rev. B. 67, 3598 (1991) 33. Parkin, S.S.P., Kaiser, C., Panchula, A., Rice, P.M., Hughes, B., et al.: Giant tunneling magnetoresistance with MgO (100) tunnel barriers. Nat. Mater. 3, 862–867 (2004) 34. Yuasa, S., Nagahama, T., Fukushima, A., Suzuki, Y., Ando, K.: Giant room-temperature magnetoresistance in single-crystal Fe/MgO/Fe magnetic tunnel junctions. Nat. Mater. 3, 868–871 (2004) 35. Sinova, J., Valenzuela, S.O., Wunderlich, J., Bach, C.H., Wunderlich, J.: Spin hall effects. Rev. Mod. Phys. 87, 1213 (2015) Michael Coey received his PhD from the University of Manitoba in 1971; he has worked at the CNRS, Grenoble, IBM, Yorktown Heights, and, since 1979, Trinity College Dublin. Author of several books and many papers, his interests include amorphous and disordered magnetic materials, permanent magnetism, oxides and minerals, d0 magnetism, spin electronics, magnetoelectrochemistry, magnetofluidics, and the history of ideas. 2 Magnetic Exchange Interactions Ralph Skomski Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum-Mechanical Origin of Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Electron Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Electron-Electron Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stoner Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heisenberg Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hubbard Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specific Exchange Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intra-Atomic Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indirect Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Itinerant Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bethe-Slater Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Metallic Correlations and Kondo Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange and Spin Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Curie Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Order and Noncollinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Waves and Anisotropic Exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiferromagnetic Spin Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensionality Dependence of Quantum Antiferromagnetism . . . . . . . . . . . . . . . . . . . . . . . Frustration, Spin Liquids, and Spin Ice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 57 57 59 60 61 63 65 66 67 69 72 75 78 83 83 84 89 92 93 94 95 99 R. Skomski () University of Nebraska, Lincoln, NE, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_2 53 54 R. Skomski Abstract The electrostatic repulsion between electrons, combined with quantum mechanics and the Pauli principle, yields the atomic-scale exchange interaction. Intraatomic exchange determines the size of the atomic magnetic moments. Interatomic exchange ensures long-range magnetic order and determines the ordering (Curie or Néel) temperature. It also yields spin waves and the exchange stiffness responsible for the finite extension of magnetic domains and domain walls. Intra-atomic exchange determines the size of the atomic magnetic moments. Positive and negative exchange constants mean parallel (ferromagnetic) and antiparallel (antiferromagnetic) spin alignments. As a rule, direct exchange and Coulomb interaction favor ferromagnetic spin structures, whereas interatomic hopping tends to be ferromagnetic and is often the main consideration. The basic interatomic exchange mechanisms include superexchange, double exchange, Ruderman-Kittel exchange, and itinerant exchange. Exchange interactions may also be classified according to specific models or phenomena. Examples are Heisenberg exchange, Stoner exchange, Hubbard interactions, anisotropic exchange, Dzyaloshinski-Moriya exchange, and antiferromagnetic spin fluctuations responsible for high-temperature superconductivity. From the viewpoint of fundamental physics, exchange interactions differ by the role of electron correlations, the strongly correlated Heisenberg exchange and weakly correlated itinerant exchange at the opposite ends of the spectrum. Correlations are also important for the understanding of some exotic exchange phenomena, such as frustration and quantum spin liquid behavior. Introduction Solid-state magnetism is caused by interacting atomic moments or “spins” (Fig. 1). In the absence of such interactions, the spins would point in random directions, and the net magnetization would be zero. Ferromagnetic (FM) order requires positive interactions (a), which favor parallel spin alignment, ↑↑, and yields a nonzero net magnetization. Antiferromagnetic (AFM) order (b) is caused by negative interactions and corresponds to antiparallel spin alignment ↑↓ between neighboring atoms. For reasons discussed below, these interactions are referred to as exchange interactions. Aside from the interatomic exchange illustrated in Fig. 1, there are intra-atomic exchange interactions. For example, Fe2+ ions in oxides have six 3d electrons and the spin structure ↑↑↑↑↑↓, which yields a net atomic moment of 2 μB . Magnetic moments in transition-metal elements and alloys tend to be noninteger, as exemplified by Ni, which has a moment of 0.61 μB per atom. Such non-integer moments reflect the itinerant Stoner exchange, which contains both inter- and intra-atomic contributions. By about 1920, it had become clear that magnetostatic interactions cannot explain ferromagnetism at and above room temperature. Weiss’ mean-field theory assumes that a molecular field stabilizes ferromagnetic order. However, the 2 Magnetic Exchange Interactions 55 Fig. 1 Interatomic exchange and magnetic order: (a) antiferromagnetism (AFM) and (b) ferromagnetism (FM) molecular fields required in the theory (several 100 teslas) are much higher than typical magnetostatic interaction fields, which are only of the order of 1 tesla. Equating the thermal energy kB T with the Zeeman energy μo μB H yields the conversion μB /kB = 0.672 K/T, meaning that low-temperature thermal excitations of about 1 kelvin destroy any magnetic order caused by magnetic fields of about 1 T. The weakness of Zeeman and other magnetic interactions reflects the relativistic character of magnetism: The ratio of magnetic and electrostatic interactions is of the order of 1/α 2 , where α = 1/137 is Sommerfeld’s fine structure constant. Aware of the smallness of purely magnetic interactions, Werner Heisenberg concluded in 1928 that ferromagnetic order must be of electrostatic origin, realized on a quantum-mechanical level [1]. He found that the Coulomb repulsion between electrons U (r 1 , r 2 ) = e2 1 4πεo | r 1 − r 2 | (1) in combination with the Pauli principle yields a strong effective field consistent with experiment. The Pauli principle forbids the occupancy of an orbital by two electrons of parallel spin. In real space, it yields an exchange hole, that is, electrons with parallel spins (↑↑) stay away from each other, while electrons with antiparallel spin (↑↓) can come arbitrarily close, which carries a Coulomb-energy penalty. In a nutshell, this is the origin of ferromagnetic exchange. A different consideration is that even electrons of antiparallel spin (↑↓) avoid each other to some extent due to their Coulomb repulsion, which is known as the correlation hole. The correlation hole weakens the trend toward ferromagnetism. Consider two electrons 1 and 2 in two atomic orbitals L (for left) and R (for right). The real-space part of the wave function is 56 R. Skomski 1 ± (r 1 , r 2 ) = √ (φL (r 1 )φR (r2 ) ± φR (r 1 )φL (r 2 )) 2 (2) where the upper and lower signs correspond to ↑↓ and ↑↑, respectively. Using Eq. (2) to evaluate the Coulomb interaction EC = Ψ ± ∗ U Ψ ± dr1 dr2 yields an energy splitting of ±JD . The integral ±JD = φL∗ (r 1 )φR∗ (r 2 )U (r 1 , r 2 )φR (r 1 )φL (r 2 )dV1 dV2 (3) is referred to as the exchange integral or direct exchange. If JD is positive, then the energy of the FM state is lower than that of the AFM state, favoring ↑↑ alignment. Since direct exchange JD is of electrostatic origin, it has the right order of magnitude to explain ferromagnetism. However, equating the net exchange J with JD has a number of flaws. First, JD is the electrostatic self-interaction energy of a fictitious charge distribution ρ F (r) = – e φ ∗ L (r) φ R (r) and therefore always positive. This is at odds with experiment, because antiferromagnetism is well established in many materials. Second, Eq. (2) means that the left (L) and right (R) atoms harbor exactly one electron each [1, 2]. This approximation, known as the Heitler-London approximation in chemistry, amounts to ignoring the ionic configurations φ L (r1 ) φ L (r2 ) and φ R (r1 ) φ R (r2 ). In fact, some iconicity is expected on physical grounds, because electrons hop between atoms and therefore temporarily create ionic configurations. Third, the wave functions φ L (r) and φ R (r) exhibit some overlap So = φ L (r) φ R (r) dr. This overlap is responsible for interatomic hopping, affecting the one-electron levels and reducing the net exchange. Furthermore, overlap corrections diverge with increasing number N of electrons involved, which is known as non-orthogonality catastrophe. Sections “One-Electron Wave Functions” and “Electron-Electron Interactions” solve the overlap problem by using Wannier-type orthogonalized orbitals. It is nontrivial to predict magnetism from the chemical composition. For example, MnBi, ZrZn2 , and CrBr3 are all ferromagnetic but do not contain any ferromagnetic element. Section “Specific Exchange Mechanisms” describes a number of important exchange mechanisms in metals and insulators. It is important to distinguish between intra-atomic exchange, which is responsible for the formation of atomic magnetic moments, and interatomic exchange, which determines the type of magnetic order and the ordering temperature. Examples of magnetic order are the FM and AFM structures of Fig. 1, but there also exist noncollinear spin structures, caused, for example, by competing exchange or Dzyaloshinski-Moriya interactions (Sections “Curie Temperature” and “Magnetic Order and Noncollinearity”). Beyond determining magnetic order, exchange is important in micromagnetism, where the exchange stiffness affects the sizes of magnetic domains and domain walls (Section “Spin Waves and Anisotropic Exchange”). Exchange is also involved in various “exotic” magnetic systems, such as frustrated spin structures and quantum-spin liquids (Sections “Curie Temperature” and “Magnetic Order and Noncollinearity”). 2 Magnetic Exchange Interactions 57 Quantum-Mechanical Origin of Exchange Exchange reflects the interplay between independent-electron level splittings (T ), Coulomb repulsion (U ), and the direct exchange integral (JD ). This section elaborates the fundamentals of this relationship, starting from one-electron wave functions (Section “One-Electron Wave Functions”), introducing wave functions of interacting electrons (Section “Electron-Electron Interactions”), and discussing a number of fundamental limits and models (Sections “Stoner Limit”, “Correlations”, “Heisenberg Model”, and “Hubbard Model”). One-Electron Wave Functions Well-separated atoms are described by atomic wave functions of on-site energy Eat , but in molecules and solids, the wave functions of neighboring atoms overlap and yield interatomic hybridization. This section focuses on two s electrons in atomic dimers, such as H2 and hypothetical Li2 (Fig. 2). Most features of this model can be generalized to solids, although solids exhibit additional many-electron effects. The one-electron Hamiltonian corresponding to Fig. 2 is H.1 (r) = 2 2 ∇ + VL (r) + VR (r) 2me (4) where L and R stand for right and left, respectively, and VL/R (r) = Vo (| r – RL/R | ) are the atomic potentials. In the case of hydrogen-like atoms of radius Rat = ao /Z, the atomic ground state has the eigenfunction φ(r)∼ exp (–r/Rat ) and the energy Eat = Z2 e2 /8πεo ao . The vicinity of the second atom means that the atomic wave functions φ L (r) = |Lo > and φ R (r) = |Ro > overlap, which is described by the overlap Fig. 2 Symmetric and antisymmetric wave functions: (a) atomic wave function, (b) Wannier function, (c) antibonding state |σ *>, and (d) bonding state |σ > 58 R. Skomski integral So =< Lo VR Ro > (5) The overlap causes interatomic hopping and yields a level splitting into bonding and antibonding states. The respective energies are Eo ± T , where T is the hopping integral. The hybridized eigenfunctions (Fig. 2(c-d)) are |σ > ∼ |Lo > + |Ro > (bonding) and |σ ∗ > ∼ |Lo > – |Ro > (antibonding), both having So -dependent normalizations. The label σ refers to the ss σ -bond between the two s orbitals and leads to T < 0 in this specific model. The on-site energy Eo differs from the atomic energy Eat by the crystal-field energy ECF ≈ <Lo |VR |Lo >. Traditional and Modern Analyses The determination and interpretation of the hopping integral T require care, because hopping affects the net exchange J and may change its sign. T is often approximated by To = <Lo |VR |Ro > = φL∗ (r) VR (r) φR (r) dr (6) but this interpretation is qualitative only, because the overlap correction to T is of the same order of magnitude as To itself. Equation (6) therefore conflates the related phenomena of hopping and wave-function overlap. This distinction is related to the abovementioned non-orthogonality catastrophe. Orthogonality problems are avoided by the use of orthogonalized atomic wave functions or Wannier functions [3]. These functions are similar to atomic wave functions but contain some admixture of neighboring orbitals to ensure orthogonality. In the present model [4] 1 1 |L> = √ |σ > + |σ ∗ > and |R> = √ |σ > − |σ ∗ > 2 2 (7) Figure 2(b) shows one of these Wannier functions. The two wave functions (c-d) correspond to rudimentary wave vectors k = 0 (bonding) and k = π/a (antibonding). Solids are very similar in this regard, except that k varies continuously (band structure). It is instructive to discuss the parameters involved for large interatomic distances R [2]. In this extreme-tight-binding limit, So = ½ (R/Rat )2 exp(–R/Rat ), ECF = 2(Rat /R)Eat , and T = So ECF . Since ECF decreases only slowly, scaling as 1/R, the asymptotic behavior of T is governed by the exponential decay of So . In terms of the Wannier functions |L > and |R>, the one-electron Hamiltonian of Eq. (4 ) assumes the very simple matrix structure H1 = Eo T T Eo (8) 2 Magnetic Exchange Interactions 59 The diagonalization of this Hamiltonian is trivial, reproducing E± = Eo ± T and yielding 1 1 |σ > = √ (|L> + |R>) and |σ ∗ > = √ (|L> − |R>) 2 2 (9) Equations (8, 9) remove the overlap integral from explicit consideration and constitute a great scientific and practical simplification. Electron-Electron Interactions Ferromagnetism is caused by electron-electron interactions. Addition of the Coulomb energy H12 = U (r 1 , r 2 ) to Eq. (4) yields H (r 1 , r 2 ) = H1 (r 1 ) + H1 (r 2 ) + U (r 1 , r 2 ) (10) To diagonalize this Hamiltonian, it is convenient to use two-electron wave functions Ψ i constructed from Wannier functions, namely, Ψ 1 = |LL>, Ψ 2 = |LR>, Ψ 3 = |RL>, and Ψ 4 = |RR>. Since <L|R > = 0, these functions are all orthogonal, and Eq. (10) becomes ⎛ U ⎜T H= ⎜ ⎝T JD T 0 JD T T JD 0 T ⎞ JD T ⎟ ⎟ T ⎠ U (11) where a physically unimportant zero-point energy has been ignored. The Coulomb parameter U is the extra energy required to put a second electron onto a given atom (R or L), essentially U= e2 4πεo n(r 1 ) n(r 2 ) dr 1 dr 2 | r1 − r2 | (12) where n(r) = nL/R (r). Unlike JD , which decrease exponentially with interatomic distance, U is an atomic parameter and more or less independent of crystal structure. Both U and T tend to be large, several eV, whereas JD is rather small, typically of the order of 0.1 eV. This indicates that JD is not the only or even the most important contribution to interatomic exchange. For example, the exchange in the H2 molecule is antiferromagnetic, in spite of JD being positive. Equation (11) can be diagonalized analytically. There are two low-lying states 1 1 |↑↑> = √ |LR> − √ |RL> 2 2 (13) 60 R. Skomski cos χ sin χ |↑↓> = √ (|LR> + |RL>) + √ (|LL> + |RR>) 2 2 (14) where tan (2χ ) = –4T /U [4]. Equation (14) is a superposition of two Slater determinants, described by the mixing angle χ . The corresponding energy levels are E↑↑ = –JD E↑↓ U = +D − 2 (15) U2 4 4T 2 + (16) Defining an effective exchange as J = E↑↑ –E↑↑ /2 yields U J = JD + − 4 T2+ U2 16 (17) This equation shows that interatomic hopping (T ) reduces the net exchange interaction. The effect comes from the admixture of |LL> and |RR> to |LR> + |RL> (Eq. (14)) which is ignored in Eq. (2). Stoner Limit In the metallic limit of strong interatomic hopping (T J = U + JD − |T | 4 U ), Eq. (17) becomes (18) This equation predicts ferromagnetism for sufficiently small hopping T and roughly corresponds to the Stoner theory [5] of itinerant transition-metal magnets (Section “Itinerant Exchange”). Since U JD , the driving force behind Stoner ferromagnetism is the Coulomb integral U , not the direct exchange JD [6]. The interatomic hopping competes against the electron-electron interactions described by the Stoner parameter I = U /4 + JD , whereas a refined calculation for transition metals yields I = U /5 + 1.2 Jat [7]. Here Jat is the intra-atomic exchange, which merges with the interatomic exchange in the itinerant limit. Equation (18) yields a very simple and scientifically successful explanation, namely, that ferromagnetism occurs when the one-electron level splitting, ±|T | in the model of Section “Electron-Electron Interactions”, is sufficiently small compared to the nearly crystal-independent Coulomb parameter U . Figure 3 illustrates the physics behind this mechanism. The Coulomb repulsion U favors the FM configuration, but the FM alignment carries a one-electron energy penalty. More precisely, 2 Magnetic Exchange Interactions 61 Fig. 3 Origin of magnetism in the independent-electron picture. The one-electron level splitting into bonding (σ ) and antibonding (σ *) states favors ↑↓ spin pairs, whereas the Coulomb repulsion between the two |σ > electrons yields ↑↑ coupling so long as the Coulomb energy is larger than the one-electron level splitting. The independent-electron nature of this picture is seen from two features. First, the electrons occupy one-electron levels (σ and σ *). Second, the Coulomb interaction can be interpreted as an effective field (Stoner exchange field) the “one-electron” contributions of this section are actually independent-electron contributions treated on a quantum-mechanical mean-field level [8], because level splittings such as ±|T | depend on all other electrons in the system. An alternative view on the Stoner limit is that antisymmetrized wave functions |Ψ > diagonalize the leading one-electron part (T -part) of Eq. (11) and can therefore be used to evaluate electron-electron interactions (U and JD ) in lowest-order perturbation theory. The antisymmetric wave functions have the character of Slater determinants if the spin is included. For example, Fig. 3 corresponds to |FM > = σ (r1 ) σ ∗ (r2 ) − σ ∗ (r1 ) σ (r2 ) ↑ (1) ↑(2) (19) |AFM > = σ (r1 ) σ (r2 ) (↑(1) ↑(2) −↑ (1) ↑(2)) (20) and In this method, known as the independent-electron or quantum-mechanical meanfield approximation in solid-state physics and the molecular-orbital (MO) method in chemistry, individual electrons move in an effective potential or “mean field” Veff (r) created by all electrons in the system (Section “Itinerant Exchange”). Correlations The quantum-mechanical mean-field approach, which is the rationale behind the local-density approximation to density-functional theory (LSDA DFT), has been highly successful in magnetism, but some red flags indicate the need for a more thorough analysis. For example, the mean-field result of Eq. (18) leads to the prediction of positive (FM) exchange for JD = 0 so long as |T | < 14 U . In fact, putting JD = 0 in the exact result of Eq. (17) shows that the exchange is always 62 R. Skomski negative for JD = 0. Ferromagnetic coupling requires |T | < JD U + JD 2 (21) which is qualitatively different from |T | < 14 U . The limitations of the quantum-mechanical mean-field approach are defined by the treatment of the correlation hole. The correlation energy is defined [9] as the difference between the correct many-electron energy and the corresponding oneelectron (independent-electron) energy obtained from a single Slater determinant (Hartree-Fock determinant). Consider |Ψ AFM > of Eq. (20), whose real-space part has the structure |σ σ > = 1 (|LL> + |LR> + |RL> + |RR>) 2 (22) This wave function has an ionic character of 50%, that is, the electrostatically unfavorable configurations |LL> and |RR> provide half the weight. Since the electrons equally occupy all two-electron states, Eq. (22) lacks a correlation hole. The Coulomb penalty associated with the unfavorable ionic contribution leads to an overestimation of the AFM energy and therefore to an overestimation of the trend toward ferromagnetism. In reality, electron correlations lead to a partial suppression of the |LL> and |RR > occupancies, described by the mixing angle χ in Eq. (14). The Heisenberg limit, Ψ + in Eq. (2) and χ = 0 in Eq. (14), has |Ψ AFM > ∼ |LR> + |RL>, which corresponds to an ionic character of 0% and to a fully developed correlation hole. The Heisenberg model is said to be overcorrelated, as opposed to the undercorrelated independent-electron approach. An interesting approach is the use of Coulson-Fischer wave functions, that is, of Slater determinants constructed not from |L> and |R> but from combinations such as |L> + λ |R>, where λ ≈ |T |/U for small hopping [10]. This unrestricted Hartree-Fock approximation contains a part of the correlations at the expense of symmetry breaking in the Hamiltonian [9]. The approximation is sufficient to reproduce the correct AFM wave function, Eq. (14), for the H2 model of Eq. (11), but this finding cannot be generalized to arbitrary many-electron systems. Near the equilibrium H-H bond length of about 0.74 Å, the electrons are delocalized, described by Eq. (22) and λ = 1, but above 1.20 Å, the electrons localize very rapidly and λ approaches zero. Correlations primarily affect AFM spin configurations [6]. For example, the FM wave function |σ σ ∗ > – |σ ∗ σ > = |LR> – |RL>, Eq. (13), is independent of T and U and therefore unaffected by correlations. The reason for the absence of ionic configurations in Eq. (13) is the Pauli principle, which creates the exchange hole and forbids |LL > and |RR > occupancies with parallel spin. Correlations effects are most important in half-filled bands, where ferromagnetism means that all bonding and antibonding real-space orbitals are occupied by ↑ electrons and the net energy 2 Magnetic Exchange Interactions 63 gain due to interatomic hybridization is zero. Electrons (or holes) added to halffilled bands do not suffer from this constraint and make ferromagnetism easier to achieve. Solid-state correlations are multifaceted and yield many more or less closely related magnetic phenomena, such as spin-charge separation (Section “Antiferromagnetic Spin Chains”), wave-function entanglement, and the fractional quantumHall effect (FQHE). The determination of correlations is demanding even- or medium-sized molecules or clusters, because the number of configurations to be considered increases exponentially with system size. For example, the complete description of a single CH4 molecule (10 electrons) requires the diagonalization of a matrix containing 43,758 × 43,758 determinants [9]. Some methods to describe correlations [9–13] are microstate approaches, such as those in this chapter, selfenergy methods, the evaluation of matrix elements between Slater determinants (known as the configuration interactions, CI), dynamical mean-field theory (DMFT) [14, 15], and the Bethe ansatz [16, 17]. Unlike LSDA+U, the DMFT is a true correlation approach, because the electrons keep their individuality and the mean-field character refers to the spatial aspect of the correlations only. Some other correlation approaches, such as the Hubbard model, are briefly discussed in Section “Hubbard Model”. Heisenberg Model In the strongly correlated Heisenberg limit (U J = JD − T ), Eq. (17) becomes 2T2 U (24) Putting U = ∞ yields J = JD and reproduces the naïve Heisenberg result of Eq. (3). Expressions very similar to Eq. (24) can be derived for solids [3], but the method is cumbersome and the resulting picture not very transparent. It is often better to eliminate hopping terms and to consider spin Hamiltonians. To replace the real-space wave functions (R and L) by spins (↑ and ↓), one considers the full wave functions in the Heisenberg limit, namely, the AFM singlet |AFM> = 1 (|LR> + |RL>) (|↑↓> − |↓↑>) 2 (25a) and a FM triplet 1 |FM ↑↑> = √ (|LR> − |RL>) |↑↑> 2 |FM0> = 1 (|LR> − |RL>) (|↑↓> + |↓↑>) 2 (25b) (25c) 64 R. Skomski 1 |FM ↓↓> = √ (|LR> − |RL>) |↓↓> 2 (25d) The triplet (25b-d) reflects Sz = (−1, 0, +1) for S = 2 · ½ = 1 and is split by an external magnetic field (Zeeman interaction). In the Heisenberg model, one considers the spin part and implicitly understands that the spins are located on neighboring atoms. The model involves spin operators S = 12 σ, where σ is the vector formed by the Pauli matrices. The mathematical direct-product identity ⎛ 1 ⎜0 σx ⊗ σx + σy ⊗ σy + σz ⊗ σz = ⎜ ⎝0 0 0 1 1 0 0 1 −1 0 ⎞ 0 0⎟ ⎟ 0⎠ 1 (26) reproduces the eigenfunctions and the singlet-tripletsplitting of Eq. (25), so that the Heisenberg Hamiltonian can be written as H=–2 J S x ⊗ S x +S y ⊗ S y +S z ⊗ S z , in vector notation, H = –2 J S 1 · S 2 . An alternative approach is to apply angularmomentum algebra to S = S 1 + S 2 , using S 2 = S 1 2 + 2 S 1 · S 2 + S 2 2 and S 1 2 = S 1 2 = 3/4, and exploiting that S 2 = S(S + 1) is equal to 2 (S = 1, ↑↑), and S 2 = 0 (S = 0, ↑↓). Considering atomic spins of arbitrary size S ≥ 1/2, performing a lattice summation over all spin pairs (compare Fig. 1), and including an external magnetic field, the Heisenberg Hamiltonian becomes H = −2 i>j Jij S i · S j − g μo μB i S1 · H i (27) where the Jij are often treated as parameters. Solutions of the Heisenberg model will be discussed in Sections “Spin Waves and Anisotropic Exchange”, “Antiferromagnetic Spin Chains”, and “Dimensionality Dependence of Quantum Antiferromagnetism”. Some definitions of J involve a factor of 2, depending on whether the summation is over all atoms (subscript ij) or only over pairs of atoms (subscript i > j). Even opposite signs are sometimes chosen, using J > 0 and J < 0 for AFM and FM interactions, respectively. The most common definition of J , used in Eq. (27), is actually an exchange per electron, not per atom. The AFM-FM energy difference per pair of atoms, E(Sz = 0) – E(Sz = 2S), is equal to 4 S (S + 1/2) J and diverges in the classical limit (S = ∞). The divergence is removed by introducing renormalized atomic exchange constants Jat = 2S 2 J or Jat = 2S (S + 1). In the classical limit (S = ∞), Jat = Jat and H = −Jat s 1 · s 2 , where the unit vector s = S/S = M/Ms describes the local magnetization direction. The classical energy splitting between the ↑↑ and ↑↓ states, namely, ±Jat , is formally the same as that for S = 1/2, ±J . 2 Biquadratic exchange, H = –B S · S , as well as other higher-order terms, may arise for several reasons, for example, in T /U expansions of the full Hamiltonian [3]. In the case of spin 1/2 interactions, they do not yield new physics, 2 Magnetic Exchange Interactions 2 because S · S = 3/16 − nonzero for S ≥ 1. 65 1 2 S · S , but biquadratic exchange effects are Hubbard Model Completely ignoring the small direct exchange JD in equations such as (11, 12, 13, 14, 15, 16, 17, 18) leads to the Hubbard model. Generalized to solids, the Hubbard Hamiltonian is + + H = i,j Tij ĉi↑ ĉj↑ + ĉi↓ ĉ↓ + U i n̂i↑ n̂i↓ (28) where n̂ = ĉ+ ĉ [18, 19]. In the Hubbard model, correlation effects are described by the Coulomb interaction U . Equation (21) indicates that the Hubbard model does not predict ferromagnetism in half-filled bands, but this argument cannot be generalized to arbitrary bands and band fillings. The bare Coulomb interaction is very high, about 20 eV for the iron-series elements, but this value is reduced to about 4 eV due to intra-atomic correlations and screening by conduction electrons. The screening (Sections “Itinerant Exchange” and “Metallic Correlations and Kondo Effect”) depends on the crystal structure, and eg orbitals tend to have slightly higher U values than t2g orbitals, so that U varies somewhat for a given element. Table I shows typical U values for the three transition-metal series [20]. Note that the effects of U are complemented by the moderately strong intra-atomic exchange Jat , also listed in Table I. Approximate values for U in some main-group elements are 8.0 eV (C), 3.1 eV (Ga), and 4.2 eV (As). In rare earths, U is equal to and best obtained from the spectroscopic SlaterCondon parameter F0 . It is of the order 10 eV and somewhat increases with number of 4f electrons. The Hubbard U yields a number of correlation effects. One of them is the suppression of metallic conductivity for large values of U (Mott localization), which reflects the splitting of metallic bands into upper and lower Hubbard bands with opposite spin directions. The effect is very similar to the Coulson-Fischer Table 1 Typical values of screened Coulomb integrals (U ) and screened intra-atomic exchange (Jat )(Jat ) nv 3 4 5 6 7 8 9 10 11 Sc V Ti Cr Mn Fe Co Ni Cu U Jat 2.4 3.1 3.3 4.5 4.5 3.9 4.4 4.0 5.7 0.4 0.5 0.6 0.7 0.7 0.7 0.8 0.8 0.8 Y Zr Nb Mo Tc Ru Rh Pd Ag U Jat 1.7 2.4 2.7 3.7 3.9 4.2 4.0 3.8 4.8 0.3 0.4 0.5 0.5 0.6 0.6 0.6 0.6 0.6 Lu Hf Ta W Re Os Ir Pt Au U Jat 1.5 2.0 2.4 3.5 3.7 4.1 3.8 3.6 4.0 0.3 0.3 0.4 0.5 0.5 0.5 0.5 0.5 0.6 66 R. Skomski Fig. 4 Hubbard interpretation of band gaps: (a) Mott-Hubbard insulator, (b) charge-transfer insulator, (c) simple interpretation of Hubbard-Mott transition, and (d) refined Hubbard transition involving a correlated metal phase known as the Brinkman-Rice (BR) phase electron localization in the H2 molecule (Section “Correlations”). Some oxides are antiferromagnetic Mott-Hubbard insulators, Fig. 4a, but many are charge-transfer insulators [21], where the 2p-3d gap Δ is smaller than the Hubbard gap U (Fig. 4b) and the transition to metallic behavior involves hopping between cation 3d and anion 2p states. The trend toward charge transfer behavior increases from early to late transition metals and from oxides to halides. In spite of the simplicity of Eq. (28), there have been no exact solutions for the Hubbard model so far, except for a few special cases. Even the well-known Hubbard band splitting (Fig. 4c) is a simplification. A detailed analysis, using Gutzwiller wave functions [19] and dynamical mean-field theory (DMFT) [14], yields a correlated-metal or Brinkman-Rice phase [22] with metallic quasiparticles in the middle of the Hubbard gap (Fig. 4d). This quasiparticle peak is analogous to impurity peaks near band edges, for example, in the gaps of semiconductors. The difference is that the disorder responsible for the peak is not caused by impurity atoms but by correlated electrons (and holes) randomly occupying lattice sites. Specific Exchange Mechanisms The involvement of Coulomb integral (U ), and exchange integral (JD ), and oneelectron level splitting (T ) is a common feature of exchange interactions, but the interplay between these quantities varies greatly among magnetic solids. 2 Magnetic Exchange Interactions 67 Intra-Atomic Exchange Atomic wave functions inside a given atom are orthogonal, so that the ferromagnetic exchange is not weakened by one-electron level splittings involving hopping between different orbitals (T = 0). On the other hand, one-electron energy differences between shells and subshells are typically large, several eV. In terms of Fig. 3, these energy differences provide a forbiddingly one-electron level splitting. Ferromagnetic intra-atomic exchange is therefore almost exclusively limited to the nearly degenerate electrons in the partially filled inner subshells of transitionmetal atoms, namely, 3d, 4d, and 5d electrons in the iron, palladium, and platinum series, respectively, 4f electrons in rare-earth (lanthanide) atoms, and 5f electrons in actinides. Hund’s Rules Intra-atomic exchange and spin-orbit coupling give rise to the hierarchy of three Hund’s rules [23]. The rules, which are empirical but have a sound physical basis, determine the magnetic ground state of atoms or ions. Hund’s first rule reflects intra-atomic exchange and states that the total spin S is maximized so long as the Pauli principle is not violated. The number of one-electron orbitals per subshell is 2 l + 1, which yields 5 orbitals per d-shell and 7 orbitals per f-shell. In the first half of each series, all spins are ↑, and for half-filled shells, the total spin moment is therefore 5 μB (d-shells) and 7 μB (f -shells). Additional electrons are ↓ due to the Pauli principle. For example, Co2+ has a 3d7 electron configuration and the spin structure 3d (↑↑↑↑↑↓↓). Quantum states characterized by quantum numbers L and S form terms denoted by 2S + 1 L. For example, the term symbol 2 F means L = 3 and S = ½. The next consideration is Hund’s second rule, which states that the orbital angular moment L is maximized, subject to the value of S. The vector model usually employed in magnetism assumes L-S (Russell-Saunders) coupling, where the total orbital moment L = i Li and the total spin moment S = i S i combine to yield the total moment J = L + S. The operators obey angular-momentum quantum mechanics, for example, S 2 = S (S + 1), L2 = L (L + 1), and J2 = J (J + 1). The opposite limit of j-j coupling, where the spin-orbit interaction dismantles the total ionic spin and orbital moments, is important only for the ground state of very heavy elements (Z > 75) and for excited states of most elements [24], which are usually of no concern in magnetism. Spin-orbit coupling causes the terms to split into multiplets, which are denoted by 2S + 1 LJ, and obey |L – S| ≤ J ≤ |L + S|. Hund’s third rule describes how spin (S) and orbital moment (L) couple to yield the total angular momentum (J): Less than half-filled subshells have J = |S – L|, whereas more than half-filled shells exhibit J = |S + L|. This rule explains, for example, the large atomic magnetic moments of the heavy rare earths, such as 10 μB per atom in Dy3+ and Ho3+ . 68 R. Skomski Consider, for simplicity, the Hund’s-rules ground state of the p2 configuration, realized, for example, in free carbon atoms. There are six one-electron states (px↑ , py↑ , pz↑ , px↓ , py↓ , pz↓ ), but the Pauli principle reduces the 6 × 6 = 36 two-electron microstates to 15 Slater determinants. For example, |↑ ◦ ↓> ∼ |x↑ (r1 )y↓ (r2 ) – y↓ (r1 )x↑ (r2 )> means that the px (Lz = +1) and py (Lz = −1) orbitals are both occupied by ↑ electrons, while the pz orbital (Lz = 0) is empty. The 15 microstates of the p2 configuration form three terms: 1 S (L = 0, S = 0), 1 D (L = 2, S = 0), and 3 P (L = 1, S = 1). Hund’s first rule uniquely establishes the ground-state term 3 P, because the other two terms have zero spin. The term contains (2 L + 1) (2S + 1) = 9 Slater determinants, for example, |↑ ↑ ◦>, where Lz = 1 and Sz = 1. Hund’s third rule predicts J = L – S = 0, corresponding to a nonmagnetic ground state. The p2 ground-state wave function is a superposition of three Slater determinants described by Clebsch-Gordan coefficients C(L, Lz , S, Sz |J, Jz ) [2, 25]. Explicitly 1 1 1 |ψ> = √ |↑ ◦ ↓> + √ |↓ ◦ ↑> − √ |◦ ↑↓ ◦> 3 3 3 (29) The involvement of two or more Slater determinants indicates that correlations are not necessarily be important even in seemingly simple systems. Hund’s rules are obeyed fairly accurately by rare-earth ions in metallic and nonmetallic environments. For example, the ground-state multiplets of rare-earth ions obey J = |L ± S|, whereas excited multiplets have relatively high energies, with notable exceptions of Eu3+ and Sm3+ , where the splitting is only about 0.1 eV [26]. One reason for the applicability of the rules is that the 4f -shell radii of about 0.5 Å are much smaller than the atomic radii of about 1.8 Å. This enhances the spin-orbit coupling and reduces the interaction with surrounding atoms. By contrast, Hund’s rules are often violated in 3d, 4d, and 5d transition metals, where orbital moments are quenched. Moment Projections and Quenching Exchange interactions are between spins S, not between total moments J=L+S, which makes it necessary to project the total moment onto the spin moment. Similarly, the Zeeman interaction with an external field involves L + 2 S, not J = L + S and S. In the Zeeman case, projection of L + 2 S onto J yields the symbolic replacement L + 2 S → g and the moment m = g J. The g-factor is obtained by using (L + 2 S) · J = g J2 and J (J + 1) = L (L + 1) + 2L · S + S (S + 1) (30) The result of the calculation is g= 3 1 S (S + 1) − L (L + 1) + 2 2 J (J + 1) (31) which yields g = 1 – S/(J + 1) and g = 1 + S/J for the first and second halves of the lanthanide series, respectively. To account for spin-only character of 2 Magnetic Exchange Interactions 69 interatomic exchange, the atomic projection S · J = (g − 1) J2 must be used. The corresponding de Gennes factor G = (g – 1)2 J(J + 1) is important for the finite-temperature behavior of rare-earth magnets, where it controls the Curie temperature. One implication of Eq. (31) is that the vectors L, S, and J are not necessarily (anti)parallel but described by the vector model of angular momenta [24]. A good example is Sm3+ , which has antiparallel spin and orbital moments L = 5 and S = 5/2, respectively, so that L – 2S could naïvely be expected to yield a zero magnetic moment. In fact, g = 2/7 and m = 0.71 μB , which corresponds to angles of 22◦ between L and J and of 44◦ between S and J. Hund’s rules are often violated in metallic and nonmetallic transition-metal magnets. The d orbitals of iron-, palladium-, and platinum-series atoms are fairly extended, so that interactions with neighboring atoms outweigh Hund’s rules considerations. The rules regarding L are affected most, because the orbital moment is normally quenched, L ≈ 0. For example, bcc iron has a magnetization of about 2.2 μB , but only about 5% is of orbital origin. The reason is that orbital moments require an orbital motion of the electrons, but this motion is disrupted by the crystal field introduced in Section “One-Electron Wave Functions”. Note that L = 0 means J = S and, according to Eq. (31), g = 2. High-spin Low-spin Transitions Crystal-field interactions cause the five 3d levels of transition-metal ions to split. In magnets with cubic crystal structure, this splitting is of the eg -t2g type: The |z2 > and |x2 – y2 > orbitals form the eg dublet, whereas the |xy>, |xz>, and |yz> orbitals form the t2g triplet. The crystal-field interaction yields a moment reduction if the splitting is larger than the combined effect of U and JD . Such transitions are known as high-spin low-spin transitions. For example, in octahedral environments, the energy of the t2g triplet is lower than that of the eg dublet. Co2+ has seven 3d electrons, which translate into the spin configuration t2g (↑↑↑↓↓) eg (↑↑) and a moment of 3 μB . However, in the limit of large crystalfield splitting, one of the two eg↑ electrons “falls down” in the sense of Fig. 3 and occupies the empty t2g↓ orbital, yielding the spin configuration t2g (↑↑↑↓↓↓) eg (↑) and a moment of 1 μB . Examples are the Co2+ complexes [Co(H2 O)6 ]2+ (high spin) and [Co(CN)6 ]4− (low spin). Indirect Exchange The model of Section “Electron-Electron Interactions” describes the so-called direct exchange between nearest neighbors, where the hopping integral T competes against U and JD . Exchange in solids is often indirect, mediated by conduction electrons or by intermediate atoms, such as oxygen. Superexchange Transition-metal oxides frequently exhibit exchange bonds of type Mm+ -O2− -Mm+ , where Mm+ is a transition-metal cation. This type of exchange is known as superexchange and also realized in magnetic halides such as MnF2 . 70 R. Skomski The net exchange is tedious to calculate [27], but a transparent physical picture emerges if one assumes that U and JD compete against T and that the outcome of this competition is largely determined by T , similar to Eq. (24). For one 3d level per transition-metal atom (M) and one oxygen 2p level (O), Eq. (8) becomes ⎛ ⎞ EM Tpd(L) 0 H = ⎝ Tpd(L) EO Tpd(R) ⎠ 0 Tpd(R) EM (32) Here, EM and EO are the atomic on-site energies, and Tpd(R/L) describes the hopping between M and O atoms. When Tpd(R) = Tpd(L) , a unitary transformation using ⎛ √1 2 0 √1 2 ⎞ ⎟ ⎜ =⎝ 0 1 0 ⎠ √1 0 √1 2 (33) 2 partially diagonalizes the Hamiltonian and yields ⎛ √EM Q+ HQ = ⎝ 2Tpd 0 √ ⎞ 2Tpd 0 EO 0 ⎠ 0 EM (34) The transformation couples the wave functions of the two transition-metal atoms. One of the coupled M levels is nonbonding (bottom-right matrix element), whereas the other one (top left) hybridizes with the oxygen, thereby creating a level splitting between the two coupled M orbitals. In the Heisenberg limit, Tpd is small, and the diagonalization of Eq. (34) yields the transition-metal level splitting ±Teff , where Teff = Tpd 2 /|EM –EO |. Substitution of Teff into Eq. (24) yields the effective transition-metal exchange Jeff = JD − 2 Tpd 4 U (EM − EO )2 (35) Since JD is small, the hopping normally wins, and the exchange in most oxides is therefore antiferromagnetic. However, the non-s character of the 2p and 3d orbitals causes Tpd to depend on the type of d orbital (eg or t2g ) and on the bond angle. Figure 5 compares a 180◦ bond (a) with a 90◦ bond (b). In (a), the two p-d bonds differ by the sign of the involved 2p wave-function lobe, but Tpd(R) = −Tpd(L) leaves Eq. (35) unchanged. In (b), Tpd(R) = 0 by symmetry, because the hopping contributions of the two oxygen lobes (+ and –) cancel each other. This implies Tpd(R) = 0 in Eq. (32), and the two transition-metal orbitals are no longer coupled (Teff = 0). 2 Magnetic Exchange Interactions 71 Fig. 5 Overlap and exchange: (a) nonzero overlap (180◦ bond) and zero overlap (90◦ bond). In (a), the hopping integral is nonzero, corresponding to antiferromagnetic indirect exchange, but in (b), the hopping integral is zero by symmetry The above analysis is the basis for the Goodenough-Kanamori-Anderson rule [27, 28], which states that exchange in oxides is antiferromagnetic for bond angles θ B > 90◦ Teff 2 > 0 but ferromagnetic for bond angles of θ B = 90◦ (Teff = 0). Examples of the former are rock salt, spinel, and wurtzite oxides, where the predominant bond angles are 180◦ , 125◦ , and 109◦ , respectively [27]. Ferromagnetic exchange dominates in CrO2 [27], where the Cr4+ ions yield a net moment of 2 μB per formula unit. Ruderman-kittel Exchange The exchange interaction of localized magnetic moments in metals is mediated by conduction electrons, which is known as the Ruderman-Kittel-Kasuya-Yosida or RKKY mechanism. Electrons localized at Ri and conduction electrons of wave vector k undergo a strong intra-atomic s-d exchange –Jsd S k · S i δ (r–R i ), so that the localized electrons perturb the sea of conduction electrons. The perturbed wave functions are ψ k (r) = k ck exp (i k · r) dk, where the integration is limited to wave vectors |k| < kF . The wavevector cutoff affects the real-space resolution of the response ψ(r) and means that details smaller than about 1/kF , such as δ(r – Ri ), cannot be resolved. As a consequence, the electron density n(r)∼ ψ k (r) ψ k (r) dk contains a wavelike oscillatory contribution. The oscillations are spin-dependent and yield the oscillatory RKKY exchange 72 R. Skomski J (R) = Jo 2kF R cos (2kF R) − sin (2kF R) (2kF R)4 (36) between localized moments at Ri and Rj = Ri + R. In metals, kF ∼ n1/3 is large, and the oscillation period does not exceed a few Å. In dilute magnetic semiconductors (DMS), n can be made small by adjusting doping level and/or temperature, and the RKKY interaction is then a nanoscale effect. Equation (36) describes exchange interactions mediated by free electrons, but the underlying perturbation theory can also be used to treat arbitrary independentelectron systems, such as tight-binding electrons in metals [29] and DMS exchange mediated by shallow nonmagnetic donors (or acceptors) [30]. At finite temperatures, the thermal smearing of the Fermi surface yields an exponential decay of the oscillations, with a decay length proportional to kF /T. Double Exchange Intra-atomic exchange favors parallel spin alignment, and electrons retain a “spin memory” while hopping between atoms. This process translates into a ferromagnetic exchange contribution first recognized by Zener [28, 31]. Double exchange occurs in mixed valence oxides, such as Fe3 O4 . This oxide contains Fe3+ and Fe2+ ions on B-sites. The latter can be considered as Fe3+ ions plus an extra electron that can hop more or less freely between the d5 ion cores. The double-exchange mechanism is important in magnetoresistive perovskites (manganites). The parent compound, LaMnO3 , contains Mn3+ ions only and is an antiferromagnetic insulator. Partially replacing La3+ by Sr2+ creates a charge imbalance that is compensated by the formation of Mn4+ ions. In both Mn3+ and Mn4+ , the low-lying t2g triplets are occupied by three well-localized 3d electrons, but in Mn3+ , there is an additional eg electron that yields ferromagnetic double exchange and metallic conductivity. Itinerant Exchange The magnetism of 3d, 4d, and 5d elements and alloys is fairly well described by the independent-electron approximation, which corresponds to the use of a single big Slater determinant. The electrons move in the solid, and the corresponding hopping competes against the electrostatic electron-electron interaction. The simplest approach is to replace the crystal potential V(r) by a chargeneutralizing homogeneous background V(r) = const. (jellium model). The only free parameter describing the corresponding homogeneous but not necessarily free electron gas is the electron density n. It is convenient to parameterize n in terms of the average inter-electronic distance rs = (3/4πn)1/3 and to relate rs to the freeelectron Fermi wave vector kF = (9π/4)1/3 /rs . Typical values of kF ao are 0.34 (Cs), 0.72 (Cu), and 1.03 (Be) [8]. The inverse magnetic susceptibility of the jellium is [32] 2 Magnetic Exchange Interactions 73 χp π 1 =1− + 2 (0.507 ln (kF ao ) − 0.162) χ kF ao kF ao 2 (37) where χ p = (α/2π)2 ao kF is the susceptibility of the non-interacting electron gas (Pauli susceptibility). The onset of ferromagnetism corresponds to χ = ∞, that is, to 1/χ = 0. Equation (37) includes the key distinction between kinetic energy (hopping), scaling as 1/rs 2 ∼ kF 2 , and Coulomb interaction, scaling as 1/rs ∼ kF . The Pauli susceptibility reflects the kinetic energy, whereas –π/kF ao is the independentelectron Coulomb correction, which corresponds to Bloch’s early theory of itinerant exchange [8, 33]. As the electron gas gets less dense and kF becomes smaller, the π/kF ao term in Eq. (37) predicts ferromagnetism for kF ao < 1/π, which is close to the value for alkali metals such as Cs. Experimentally, the alkali metals are not particularly close to ferromagnetism, which is caused by d and f electrons, not by a homogeneous electron gas. In fact, the last term in Eq. (37), which scales as 1/kF 2 and reflects the so-called random-phase approximation (RPA), negates the Bloch prediction of ferromagnetism – χ (kF ao ) never reaches zero in Eq. (37). The physics behind the RPA is that the charge of any individual electron is screened by the other electrons in the metal, which amounts to a reduction of the net Coulomb repulsion from 1/rs to an exponentially decaying interaction. In other words, the screening electrons form a quasi-particle cloud around the electron and renormalize the Coulomb interaction. The Stoner theory replaces Eq. (37) by the semiphenomenological expression χp = 1 − I D(EF ) χ (38) where the Stoner parameter I ∼ 1 eV [34] describes the electron-electron interaction (Section “Stoner Limit”). Equation (38) predicts ferromagnetism for high densities of states (DOS), when the paramagnetic state becomes unstable and the magnets satisfy the Stoner criterion (EF ) > 1/I. The DOS of d electrons is much higher than that of the jellium electrons implied in Eq. (37), which explains the occurrence of ferromagnetism in transition metals. Alternatively, since the DOS (density of states) is inversely proportional to the bandwidth W ∼ |T |, ferromagnetism occurs in narrow bands. This finding is in agreement with the general analysis of Section “Antiferromagnetic Spin Chains”. Band Structure and Magnetism The hopping aspect of magnetism is determined by the band structure and by the metallic density of states (DOS). Both are obtained from the eigenvalues and eigenfunctions of Hamiltonians of the type H=− 2 2 ∇ + 2me j Vo (r − r j ) (39) 74 R. Skomski where the lattice-periodic potential depends, in general, on the electron distribution. The eigenfunctions of Eq. (39) are Bloch states ψ(r) = exp (ik · r) u(r) and electron densities n(r) = u*(r)u(r). Equation (39) describes delocalized electrons whose electrical conductivity is infinite due to the absence of scattering matrix elements. This includes the tight-binding limit of well-separated atoms, where the hopping integrals decrease exponentially with interatomic distance, T ∼ exp (–R/Ro ), but the conductivity remains infinite even for large R [8]. At zero temperature, the magnets are well described by these Bloch-periodic wave functions. This includes the explanation of non-integer moments, which are caused by the smearing of oneelectron wave functions and spin densities over many lattice sites. Inhomogeneous Magnetization States Wave-function and magnetization inhomogeneities may have several reasons. Wave-function localization requires the breaking of structural periodicity due to disorder (Anderson localization) or finite temperature. Near Tc , atomic-scale itinerant moments behave like Heisenberg spin vectors (“spin fluctuations”) of random orientation but well-conserved magnitude, the latter involving some short-range order. Experimentally, this localization manifests itself as a characteristic specific-heat contribution [9]. This spin-fluctuation picture is realized both in strong ferromagnets (e.g., Co), where the ↑ band is filled, and in weak ferromagnets such as Fe, where the ↑ band is only partially filled. Deviations from wave-function periodicity also occur due to electron correlations (Mott localization, Section “Hubbard Model”), competing exchange in perfectly periodic lattices (Section “Magnetic Order and Noncollinearity”), and surface effects. Very weak itinerant ferromagnets (VWIFs), such as ZrZn2 (Tc = 17 K), barely satisfy the Stoner criterion, and their behavior is qualitatively different from that of strong and weak ferromagnets [35, 36]. Thermal excitations act as local spin perturbations that can be described by the wave-vector-dependent susceptibility χ (k) [3]. For VWIFs, a good approximation is χ= χo |I D (EF ) − 1 + f (k)| (40) and f (k) = a2 k2 . Here χ o is the interaction-free susceptibility, approximately equal to the Pauli susceptibility χ p of Eq. (37), and a is an effective interatomic distance. Inverse Fourier transform of Eq. (40) yields |M(r)| ∼ exp.(−r/ξ ), where r is the distance from the perturbation and ξ = a/|1–I D (EF )|1/2 . In VWIFs, I D ≈ 1, so that ξ is large by atomic standards and blurs the distinction between intra- and interatomic exchange. The Stoner transition, I D = 1, yields ξ = ∞ and corresponds to Bloch-periodic wave functions. A rough Curie temperature approximate is [37]. Tc 2 TS 2 + Tc =1 TJ (41) 2 Magnetic Exchange Interactions 75 This equation interpolates between the Heisenberg limit TJ (spin rotations) and the Stoner limit Ts (moment reduction). Strongly exchange-enhanced Pauli paramagnets, such as Pt, are close to the onset of ferromagnetism and have I – 1/(EF ) 0. Magnetic impurities create spinpolarized clouds of radius ξ in these materials. The corresponding radial dependence M(r) of the magnetization combines a pre-asymptotic exponential decay (r ξ ) with RKKY oscillations for large distances (r ξ ). The exponential decay length ξ is described by Eq. (40), in close analogy to VWIFs. For example, magnetic surfaces of Co2 Si nanoparticles spin-polarize the interior of the particles with a penetration depth ξ [38]. Spin polarized clouds in strongly exchange-enhanced Pauli paramagnets are also known as a paramagnons [3]. Left to themselves, these quasiparticles slowly decay, and by considering the time dependence of the fluctuations, f (k) → f (k, ω) in Eq. (4), one can show that the relaxation time diverges at the phase transition (critical slowing down). Bethe-Slater Curve It is of practical importance to have some guidance concerning the strength and sign of the exchange in a given metallic magnet. An early attempt was the semiphenomenological Bethe-Slater-Néelcurve [39], which plots the net exchange or the ordering temperature as a function of the interatomic distance or number of electrons. There are many versions of this curve, and Fig. 6 shows one of them. The curve predicts antiferromagnetism for small interatomic distances, ferromagnetism for intermediate distances, and the absence of magnetic order in the limit of very large distances. Experiment, the results of Section “Stoner Limit”, and detailed calculations [40] grant some credibility to the approach, but the curve has nevertheless severe flaws [27, 41]. Equation (38) shows that the onset of ferromagnetism is predominantly determined the density of states (EF ) at the Fermi level. This density somewhat increases Fig. 6 Early version of the Bethe-Slater-Néel curve [27, 39] 76 R. Skomski with interatomic distance, but a more important consideration is the position of the Fermi level relative to the big peaks in the DOS. These peaks tend to vary substantially among materials with similar chemical composition but different crystal structures. For example, many transition-metal-rich intermetallic alloys have interatomic distances of about 2.5 Å but show big differences in spin structures and magnetic ordering temperatures. A specific example is the distinction between fcc and bcc Fe structures. First, the interatomic distance R = 2Rat in fcc iron, 2.53 Å, is actually a little bit larger than that in bcc Fe, 1.48 Å, so that Fig. 6 cannot explain the ferromagnetism of bcc Fe. Second, the plot ignores that bcc and fcc iron have very different crystal structures. One difference is the number of nearest neighbors, namely, 8 in the bcc structure and 12 in the fcc structure. The bandwidth increases with the number z of neighbors, so the ferromagnetism tends to be more difficult to create in dense-packed structures (z = 12 . . . 14) and easier to create at surfaces (z = 4 . . . 6). However, the number of neighbors is not the main consideration, because fcc Ni and fcc Co have 12 nearest neighbors but are both ferromagnetic. More important is the location of the big peaks in the density of states. For nearly half-filled d-shells (Cr, Mn), one wants to have the peaks somewhere in the middle of the band, whereas for nearly filled d-shells (Co, Ni), main peaks near the upper band edge are preferred. An accurate determination of the peak positions can only be done numerically, but the moments theorem [42], dealing with µm = Em (E) dE, provides some guidance [41]. The respective zeroth, first, and second moments describe the total number of states, the band’s center of gravity, and the bandwidth, all unimportant in the present context. The third moment, μ3 , parameterizes the asymmetry of the DOS, that is, whether the main peaks of the DOS are in the middle of the band (μ3 = 0) or close to the upper band edge (μ3 < 0). It can be shown [42] that μ3 reflects the absence or presence of equilateral nearest-neighbor triangles in the structure, the former yielding centered main peaks and the latter creating main peaks near the upper band edge. Figure 7 provides a very simple example of this relationship. Equilateral nearest-neighbor triangles are present in the fcc structure but not in the bcc structure, which corresponds to bcc ferromagnetism in the middle of the series and fcc (or hcp) ferromagnetism for Co and Ni. Fe is intermediate, but bcc Fe becomes ferromagnetic more easily than fcc Fe. Manganese Isolated manganese atoms have half-filled 3d shells and a magnetic moment of 5 μB per atom, which corresponds to a magnetization of approximately 5 T in dense-packed Mn structures. If this magnetization could be realized in a ferromagnetic material, it would revolutionize technology, particularly since Mn is a relatively inexpensive element. However, most Mn-based permanent magnets, such as MnAl, MnBi, and Mn2 Ga, exhibit rather modest magnetizations of the order of 0.5 T [43]. The main reason for the low magnetization of Mn magnets is the halffilled character of the Mn bands. 2 Magnetic Exchange Interactions 77 Fig. 7 Crystallographic motifs and density of states: (a) square and (b) equilateral triangle. The density of states is largest in the middle (a) and near the top of the level distribution (b). The atomic orbitals (red) are of the s-type, but in [42], it can be seen that 3d electrons behave similarly, and the present figure can be generalized to three-dimensional lattices Fig. 8 Exchange interactions in hypothetical simple-cubic Mn [45] Magnetizations as high as μo Ms = 3.2 T (3.25 μB per atom) have been reported in thin-film Fe9 Co62 Mn29 deposited on MgO [44], where DFT calculations predict 2.90 μB per atom [45]. A traditional interpretation in terms of Fig. 6 is that dilution by Fe and Co atoms enhances the average distance between Mn atoms. The tetragonal structure of the Fe-Co-Mn alloy is loosely related to that of L21 -ordered Mn2 YZ Heusler alloys, where the Mn atoms occupy a simple-cubic sublattice and exhibit ferromagnetic exchange [46]. DFT calculations (Fig. 8) actually indicate that the Mn-Mn exchange never becomes ferromagnetic. Furthermore, the example of L10 -ordered MnAl shows that large Mn-Mn distances are not necessary for 78 R. Skomski ferromagnetic exchange: The dense-packed Mn sheets in the (001) planes of MnAl, which form a square lattice, exhibit a strong FM intra-layer exchange J [47]. This underlines the crucial role of atomic neighborhoods. Metallic Correlations and Kondo Effect The situation in 3d metals is intermediate between the uncorrelated itinerant limit (U /W = 0) and the strongly correlated Heisenberg limit, with U /W ratios of the order of 0.5 [9]. For example, electron-electron interactions cause a bare electron to become surrounded or “dressed” by other electrons, forming a quasiparticle of finite lifetime, because electrons constantly enter and leave the dressing cloud. The corresponding relaxation time τ ≈ / Im (Σ), where Σ is the self-energy, decreases with increasing interaction strength. For metallic electrons of energy Ek , the lifetime is approximately EF 3 /V2 (Ek – EF )2 [48], meaning that weak interactions and vicinity to the Fermi surface yield well-defined and slowly decaying quasiparticles which constitute a Fermi liquid. As pointed out in Section “Correlations”, the independent-electron approximation involves a single Stater determinant and does not account for correlation effects. The treatment of correlations requires several Slater determinants, such as the two determinants of the model of Section “Electron-Electron Interactions” and the three determinants forming the ground state of the p2 configuration (Section “Intra-Atomic Exchange”). An example of correlated manyelectron states is the Gutzwiller wave function |> = exp –η i n̂i↑ n̂i↓ |o >, where the parameter η depends on U /W and the exponential term has the effect of creating new Slater determinants from |Ψ o > [9, 19]. The Gutzwiller method can be interpreted as a many-electron extension of the Coulson-Fischer approach. It is sometimes claimed or implied that density-functional theory becomes exact if one goes beyond the local-density approximation and that LSDA+U approaches account for correlations. This argumentation is questionable for several reasons. First, density-functional theory provides the correct ground-state energy [12, 49] if the density functional is known, but the exchange interaction is an energy difference between the ferromagnetic and other spin configurations (AFM, PM) and therefore involves excited states. Second, the density functional is not known very well. The local-density approximation uses a potential inspired by and well adapted to nearly homogeneous dense electron gases. The eigenfunctions used in LSDA, known as Kohn-Sham (KS) orbitals, are pseudo-wave functions without a well-defined quantum-mechanical meaning. They serve to determine the density functional [49] and lack, for example, Gutzwiller-type projection features. The local character of the LSDA, which can be improved by gradient corrections [50], is not essential in this regard: Hartree-Fock theory involves a single Slater determinant but is highly nonlocal [8]. Other density functionals, such as the Runge-Zwicknagel functional for highly correlated electrons in dimers [51] and the density functional 2 Magnetic Exchange Interactions 79 for Bethe-type crystal-field interactions of rare-earth 4f electrons [52], bear little or no resemblance to the LSDA. The underlying problem is that the density functional is a generating functional very similar to partition function Z and free energy F= – kB T ln Z in equilibrium thermodynamics [52, 53]. The generating functionals correspond to Legendre transformations, and in thermodynamics, the transformations are realized through the term – T S, where S is the entropy. Once Z is determined by the summation or integration over all microstates, such as the atomic positions ri in a liquid, temperature-dependent physical properties are obtained in a straightforward way from F(T). The theory is exact in principle, but the predictions depend on the accuracy of the partition function. One example is that low- and high-temperature expansions have different domains of applicability. Another example is the statistical mean-field approximation, including Oguchi-type nonlocal corrections [54], which are unable to describe critical fluctuations. In density-functional theory, the Legendre transformation is realized through the integral – V(r) n(r) dr [53]. The density functional is obtained by eliminating the microstate information in Ψ ( . . . , ri , . . . , rj , . . . ) and yields the ground state for each lattice potential V(r). This lattice potential is the DFT equivalent of the temperature in thermodynamics, and the accuracy of the predictions depends on the quality of the generating functional. The density functionals used in LSDA are not calculated but obtained through intelligent and experimentally supported guesswork. An exception is the weakly correlated limit (U ≈ 0), where the KS orbitals become quantum-mechanical wave functions with well-defined physical meaning. There are two reasons for the great success of the LSDA, and its extensions have two main reasons. First, transition metals are only weakly correlated and therefore amenable to ad hoc improvements using “second-principle” approaches (materials-specific choices of methods and parameters). Second, the KS Slater determinants used in LSDA are of the unrestricted Hartree-Fock type (Section “Correlations”), which are constructed from wave functions having symmetries lower than that of the Hamiltonian [9, 10]. Unrestricted HF determinants can be expanded in terms of “regular” Slater determinants and therefore contain some correlations [9]. The “Plus U” Method The LSDA+U method modifies the KS one-electron potential by a potential that depends on the electron’s atomic orbital i, essentially [55]. 1 Vi (r) = VLSDA (r) + U − ni 2 (42) A crude approximation is U ∼ U . The presence of U suppresses ↑↓ occupancies in highly correlated 3d and 4f orbitals. The LSDA+U can be used, for example, to adjust the charge state of magnetic ions (configurations) to their experimental values. Such adjustments are sometimes necessary, because there is only one 80 R. Skomski Fig. 9 LSDA+U for bcc Fe: (a) magnetic moment, (b) weak ferromagnetism and (c) strong ferromagnetism. The direct exchange and double-counting corrections are ignored in this figure KS determinant available to account for the uncorrelated subsystem (one Slater determinant) and for the ion’s intra-atomic couplings (several Slater determinants). Strictly speaking, U is a well-defined first-principle quantity [55], not a fitting parameter that can be chosen to obtain a desired computational result. Figure 9 illustrates this point for the magnetic moment of bcc Fe, calculated using the VASP code for with U varying from 0 to 6. The moment m per Fe atom (a) exhibits an increase from 2.21 μB to 3.07 μB , the experimental value being about 2.22 μB . Near U = 0.9 eV (dashed vertical line), the slope dm/dU changes from about 0.4 μB /eV to 0.1 μB /eV, caused by the unphysical transition from weak to strong ferromagnetism (b-c). 2 Magnetic Exchange Interactions 81 Fig. 10 Model describing quantum-spin-liquid corrections in solids. The quantum-mechanical mean-field (MF) approximation self-consistently treats an independent electron in a sea of surrounding electrons (gray) and corresponds to one Slater determinant. Atomic Heisenberg spins having S = 1/2 yield 2z + 1 Slater determinants Noncollinear Density-functional Theory The Heisenberg model is based on quantum rotations of atomic spins of fixed magnitude S 2 = S (S + 1). This is a rough approximation for transition metals, where electrons are delocalized (itinerant) and atomic moments are often non-integer. However, rotations of electron spins (S = 1/2), which are realized through Pauli matrices and yield spin-wave functions such as ψ(θ ) = (cos½θ , sin½θ ), can be implemented in the LSDA and used to describe noncollinear spin states, including antiferromagnets [56]. This approach corrects, for example, much of the great overestimation of the Curie temperature in the Stoner theory. The spin-wave functions ψ(θ ) are of the quantum-mechanical mean-field type, weakly correlated, and not eigenstates of the Heisenberg Hamiltonian. Figure 10 illustrates the many-electron aspect of the approximation. The model treats one ↓ electron in a sea of ↑ electrons. The left part of the figure corresponds to the quantum-mechanical mean-field approximation, where electrons interact with an effective medium. In a slightly more realistic picture, the interaction with z nearest neighbors is individualized through exchange bonds, as shown for z = 3 and z = 5. The model, which assumes S = 1/2 Heisenberg spins and nearest-neighbor exchange J< 0, is exactly solvable. The ground state has one ↑ electron and z ↓ electrons, which leads to the involvement of (z + 1) Slater determinants. The admixture of these determinants describes whether the ↓ electron stays in its original central place (Néel state) or “leaks” into the crystalline environment [57]. The calculation shows that the reversed spin occupies neighboring atoms (dark gray) with a combined weight of 50%, thereby affecting net exchange and ordering temperature. Sections “Antiferromagnetic Spin Chains” and “Dimensionality Dependence of Quantum Antiferromagnetism” considers the lattice aspect of this spin leakage. Exchange in the Kondo Model The Kondo effect, characterized by a resistance minimum, is a correlation effect caused by the exchange interaction of localized 82 R. Skomski Table 2 Kondo temperatures (in kelvin) for some transition-metal impurities in nonmagnetic hosts (gray column) [59, 65] Rh Pd Pt Cu Ag Au Zn Al Cr – 100 200 1.0 0.2 0.01 3 1200 Mn 10 0.01 0.1 0.01 0.04 0.01 1.0 530 Fe 50 0.02 0.3 22 3 0.3 90 5000 Co 1000 0.1 1 2000 – 200 – – impurity spins with conduction electrons [9, 58]. Below the Kondo temperature TK , the interaction couples the conduction electrons to the impurity spins, which enhances the electrical resistivity. Some Kondo temperatures for Cr, Mn, Fe and Co in various matrices are shown in Table 2. The simplest Kondo model is of the Anderson-impurity type, where a single conduction electron, described by a delocalized orbital |c>, interacts with a localized state |f > [9]. The Coulomb U is negligibly small for the delocalized orbital |c> but large for the localized orbital |f >. Furthermore, the on-site energy of the localized electron (bound state) is lower than that of the delocalized electron by E. In terms of the wave functions |ff >, |fc>, |cf>, and |cc>, the Hamiltonian is ⎛ U − E ⎜ T H=⎜ ⎝ T 0 T 0 0 T ⎞ T 0 0 T ⎟ ⎟ 0 T ⎠ T E (43) Since U T , the |ff > state (energy U − E) does not play any role in the groundstate determination. In the absence of hybridization (T = 0), the ground state would be degenerate, |f c> ± |c f >, both states having the energy E = 0 and containing one localized and one delocalized electron. The first excited antiferromagnetic state, |cc> = |c↑ c↓ >, has the energy E = E, meaning that the localized electron becomes a conduction electron. The hopping integral T does not affect the ferromagnetic state |f c>−|c f >, because a localized ↑ electron cannot hop into a delocalized orbital that already contains a ↑ electron. However, the localized ↑ electron can hop into a delocalized orbital containing an electron of opposite spin, which lowers the energy of the antiferromagnetic state. The corresponding singlet (↑↓) ground state has an energy of –2T 2 /E = –2JK , roughly translating into a Kondo-temperature of TK = 2T 2 /kB E. Above TK , the |↑↓> and |↑↑> states are populated with approximately equal probability, the two electrons effectively decouple, and the resistivity drops. In reality, there are many conduction electrons, so an integration over all k-states is necessary [58]. The main contribution comes from electrons near the Fermi level, 2 Magnetic Exchange Interactions 83 which form a Kondo screening cloud of size ξ proportional to 1/TK and yield a Kondo temperature TK = (W/kB ) exp (−1/ [2 JK D(EF )]). Due to its exponential dependence on JK and (EF ), TK varies greatly among systems [59]. Table II shows some examples. TK is lowest for impurities in the middle of the 3d series and largest for nearly filled or nearly empty 3d shells, as exemplified by TK = 5000 K for Ni in Cu. The dependence JK ∼ 1/S indicates that Kondo exchange become less effective in the classical limit. Heavy-fermion compounds, such as UPt3 and CeAl2 , can be considered as Kondo lattices where conduction electrons interact with localized 4f or 5f electrons [9]. Exchange and Spin Structure Exchange affects spin structure and magnetic order in many ways. It determines the ordering temperature, gives rise to a variety of collinear and noncollinear spin structures, and influences micromagnetism through the exchange stiffness A. Exchange phenomena include quantum-spin-liquid behavior, high-temperature superconductivity, and Dzyaloshinski-Moriya interactions. Curie Temperature In spite of its simplicity, the Heisenberg model (27) is very difficult to solve, especially in two and three dimensions. A great simplification is obtained by using the identity S i · S J = S i · <S j > + <S i > · S j + Cij + co (44) and neglecting the thermodynamic correlation term Cij = (S i − <S i >) · S j − <S j > and the constant co = <S i > · <S j >. The latter is physically unimportant, because it does not affect the thermodynamic averaging. The former is important only in the immediate vicinity of the Curie temperature, where it describes critical fluctuations [60, 61]. Substituting Eq. (44) into Eq. (27) and assuming z nearest neighbors of spin moments S yield the factorized single-spin Hamiltonian H = −2 z J S · <S> − 2μo μB S · H (45) This equation amounts to the introduction of a mean field μo μB H =2 z J <S> and maps the complicated Curie-temperature problem onto the much simpler problem of a spin S in a magnetic field. This approximation (45) is the thermodynamic meanfield approximation, which must be distinguished from the quantum-mechanical mean-field approximation used to treat electron-electron interactions. The partition function belonging to Eq. (45) is a sum over the 2 S + 1 Zeeman levels Sz . The field dependence of <S> has the form of a Brillouin function (BS ), 84 R. Skomski and self-consistently evaluating <S> yields the Curie temperature Tc = 2 S(S + 1) zJ 3 kB (46) A generalization of this equation to two or more sublattices will be discussed in Section “Dimensionality Dependence of Quantum Antiferromagnetism”. This generalization includes the Néel temperature of antiferromagnets. The spin excitations leading to Eq. (46) consist in the switching of individual spins S i . The corresponding energies are rather high, with temperature equivalents close to Tc . At low temperatures, the mean-field approximation predicts exponentially small deviations from the zero-temperature magnetization M(0), which is at odds with experiment. In fact, the low-temperature behavior M(0) – M(T) of Heisenberg magnets is governed by low-lying excitations (spin waves) (Section “Spin Waves and Anisotropic Exchange”) and described by Bloch’s law, M(0) – M(T) ∼ T3/2 , in three dimensions. Magnetic Order and Noncollinearity Depending on the sign of the interatomic exchange, there are several types of magnetic order. Figure 11 shows some examples. Often there are two or more sublattices [4, 54], and the division into sublattices can be of structural or magnetic origin. Ferrimagnetism (FiM) normally reflects chemically different sublattices, such as Fe and Dy sublattices in Dy2 Fe14 B. Antiferromagnetism (AFM) is also caused by negative interatomic exchange constants, but the different sublattices are chemically and crystallographically equivalent. For example, CoO crystallizes in the rock-salt structure, but the Co forms two sublattices of equal and opposite magnetization. Ferromagnetism is frequently encountered in metals (Fe, Co, Ni) and alloys (PtCo, SmCo5 , Nd2 Fe14 B), the latter having different ferromagnetic sublattices. CrO2 is a ferromagnet, but most oxides and halides are antiferromagnetic (MnO, NiO, MnF2 ) or ferrimagnetic (Fe3 O4 , BaFe12 O19 ). Many oxides of chemical composition MFe2 O4 crystallize in the spinel structure, which contains one cation per formula unit on tetrahedral sites [...] (M2+ , sublattice A) and two cations per formula unit on octahedral sites {...} (Fe3+ , sublattice B). The exchange between the A and B sublattices is negative, which yields a ferrimagnetic spin structure. The cation distribution over the A and B sites depends on both chemical composition and magnet processing. For example, Fe3 O4 crystallizes in the so-called “inverse” spinel structure [Fe3+ ] {Fe2+ Fe3+ }(O2− )4 [65]. The total magnetization, measured in μB per formula unit, is therefore [−5] + {5 + 4} = 4. In the classical limit, the mean-field Curie temperature is given by the lowest eigenvalue of the N × N matrix in the equation kB T <si > = j Jij <sj > (47) 2 Magnetic Exchange Interactions 85 Fig. 11 Spin structures (schematic): ferromagnets (FM), antiferromagnet (AFM), ferrimagnet (FI), Pauli paramagnet (PM), and noncollinear spin structure (NC) This matrix equation is easily generalized to quantum-mechanical case, by carefully counting neighbors and using the appropriate de Gennes factors and Brillouin functions [54, 62]. The number N of sublattices is equal to the number of nonequivalent atoms. In disordered solids, all atom are nonequivalent and N → ∞. For two sublattices A and B, Eq. (47) becomes 3 kB T <sA > = j JAA <sB > + JAB <sB > (48a) 3 kB T <sB > = j JBA <sB > + JBB <sB > (48b) Here JAA/BB and JAB/BA are the classical intra- and intersublattice exchange constants, respectively, and the factor 3 reflects the classical limit of the Brillouin functions. The solution of Eq. (48) is Tc = 1 6 kB (JAA + JBB ) ± (JAA − JBB )2 + 4 JAB JBA (49) The two sublattices often have different numbers of atoms, so that JAB = JAB in general, but the two intersublattice exchange constants enter Eq. (49) in the form of the product JAB JBA , and it is sufficient to consider J ∗ = (JAB JBA )1/2 . For one-sublattice ferromagnets, where JBB = J ∗ = 0, Tc is equal to JAA /3kB . 86 R. Skomski Various scenarios exist for two-sublattice magnets. In the simplest AFM case, the two intrasublattice exchange interactions JAA = JBB = 0 and J ∗ <0, yielding the Néel temperature TN = –Tc = |J ∗ |/3kB . Metallic Sublattices Sublattice effect also occur in metals. In rare-earth transitionmetal (RE-TM) magnets, the RE-TM exchange (spin-spin) coupling is AFM for the light rare earths and FM for the heavy rare earths. The orbital moment of the TM sublattice is negligible, but inside each rare-earth atom, L and S are antiparallel for light RE and parallel for heavy RE. This yields the spin structures [S↑ ]TM [S↓ L↑ ]RE for the light and [S↑ ]TM [S↓ L↓ ]RE for the heavy rare earths. The large but opposite moment of the heavy rare-earth atoms yields a zero net magnetization in some transition-metal-rich alloys. This spin state is referred to as compensated ferrimagnetism (CFiM) and normally occurs at some compensation temperature T0 , because different sublattices tend to have different temperature dependences of magnetization [54, 63]. This is one of the features that distinguish CFiM from AFM. Compensation occurs quite frequently in ferrimagnets, including oxides such as rare-earth garnets R3 Fe5 O12 . A rule of thumb for the exchange in transition-metal alloys is the switch rule: The exchange is negative for interactions between late and early transition-metal atoms but positive otherwise. For example, Co and Pt are both late transition-metal elements, so the Co and Pt moments in CoPt are parallel. While the switch rule includes alloys containing heavy transition-metal atoms (3d-4d and 3d-5d alloys), it is not very reliable for elements in the middle of the series [64]. It also describes impurities in host lattices and RE-TM intermetallic compounds, because rare earths count as early transition metals, with one 5d electron contributing to the exchange [65]. With the exception of very weak itinerant ferromagnets, intersublattice interactions in metals are well described by the Heisenberg model, and equations like (49) provide good estimates of the ordering temperature [62]. For example, transition-metal-rich rare-earth permanent magnets have JTT J ∗ ≈ JRT and JRR ≈ 0, so that the rare-earth contribution to the Curie temperature is given by Tc ≈ (JTT /3kB ) 1 + J ∗2 /JTT 2 . As a function of the number of 4f electrons, it peaks in the middle of the lanthanide series, because J ∗2 involves the de Gennes factor (Section “Intra-Atomic Exchange”) Noncollinear Spin Structures There is a rich variety of noncollinear spin structures. Spin glasses are disordered materials whose local magnetization is frozen below some spin-glass transition temperature Tf [66, 67]. The definition of Tf , the spin state below and above Tf , the nature of the transition, and the microscopic description are nontrivial, but there is normally a distribution of exchange interactions Jij , caused, for example, by RKKY interactions between localized moments in a metallic host. In the simplest case, Jij = ±Jo for the interaction with z neighbors. For z → ∞, the eigenvalue distribution of the random matrix Jij obeys Wigner’s semicircle law √ [66], and the corresponding mean-field estimate is Tf = z Jo /kB . The situation is 2 Magnetic Exchange Interactions 87 further complicated by different types of disorder that can occur. Chemical disorder means atomic substitutions with little or no changes in atomic positions. Bond and topological disorders involve substantial changes in atomic positions and in Jij = |r i − r j | , but in the latter case, there is no continuous transformation connecting the ordered and disordered lattices [67]. Helimagnetism arises when competing exchange interactions between nearest and next-nearest neighbors yield spin spirals of wave vector k. Such structures are realized, for example, in the heavy rare-earth elements, where k || ez [68]. Consideration of a-b planes labeled by magnetization angles θ n yields the classical Heisenberg energy E = –J n cos (θn+1 –θn ) –J n cos (θn+2 –θn ) (50) where J and J are the exchange interaction between neighboring layers, respectively. The energy (50) is minimized by the ansatz θ n + 1 = θ n + δ, where δ ∼ 1/k is the magnetization rotation between subsequent layers: J + 4J cos δ sin δ = 0 (51) Aside from including FM (δ = 0) and AFM (δ = π) states, has this equation noncollinear or helimagnetic (0 < δ < π) solutions, δ = arccos −J /4J . Noncollinear spins structures are very common for elements in the middle of the iron series, notably Cr and Mn, where the exchange contains competing antiferromagnetic interactions. Elemental Cr forms a spin-density wave where the AFM sublattice magnetization exhibits a real-space oscillation with a periodicity of about 6 nm [59]. Dzyaloshinski-Moriya Interactions Noncollinearity may also arise from relativistic Dzyaloshinski-Moriya (DM) interactions [69–71], which occur in structures with violated or “broken” inversion symmetry. Examples are MnSi [72], α-Fe2 O3 (hematite) [65], and structurally disordered magnets such as spin glasses [66], as well as in artificial magnetic nanostructures [73]. DM interactions are described by the Hamiltonian HDM = – i>j Dij · Si × Sj , wherethe direction of the DM vector Dij = −Dj i is given by Dij ∼ n (r i − r n ) × r j − r n . In this expression, i and j denote the two DM-interacting spins, and rn is the position of a magnetic or nonmagnetic neighbor (Fig. 12). Physically, d electrons hop from atom i to atom n and then to atom j. Unless rn is located on the line connecting ri and rj (and the cross product determining D is zero), the hopping sequence involves a change of direction at rn , which creates a partial orbit around rn and some spin-orbit coupling that affects the spins i and j. The DM interaction changes the spin projections onto the plane created by the vectors ri – rn and rj – rn : It tries to make Si and Sj parallel to ri – rn and rj – rn , respectively. 88 R. Skomski Fig. 12 Dzyaloshinski-Moriya Interactions in (a–b) crystals and (b–c) thin films. The red atoms are magnetic, whereas the blue and white atoms are nonmagnetic but have weak (white) and strong (blue) spin-orbit coupling Since DM interactions are caused by spin-orbit coupling, they are a weak relativistic effect, comparable to micromagnetic dipolar interactions and to magnetocrystalline anisotropy. They compete against the dominant Heisenberg exchange and create canting angles of the order of 1◦ in typical magnetic materials [74]. By contrast, noncollinearities due to competing ferromagnetic or antiferromagnetic exchange (Eq. (51)) can assume any value between 0◦ and 180◦ . However, DM canting angles substantially larger than 1◦ are possible in materials with weak Heisenberg exchange (low Tc ). DM effects are strongly point-group-dependent, and the absence of inversion symmetry is a necessary but not sufficient condition [75]. For example, inverse cubic Heusler alloys have zero net DM interactions in spite of their noncentrosymmetric point group Td . Figure 12(a–b) illustrates DM interactions in an orthorhombic bulk 2 Magnetic Exchange Interactions 89 crystal without inversion symmetry (point group C2v ). The fictitious crystal has an equiatomic MT composition, where M is a magnetic or nonmagnetic metallic element and T is a transition-metal element. The structure yields a spin spiral in the x-z plane, that is, perpendicular to the net DM vector. B20-ordered cubic crystals such as MnSi (point group T) are unique in the sense that their space group (P21 3) is achiral due to the 180◦ character of the 21 screw axis but becomes chiral through the incorporation of a chiral MnSi motifs. Figure 12(c–d) shows the effect of Dzyaloshinski-Moriya interactions in thin films with perpendicular anisotropy and fourfold (C4v ) or sixfold (C6v ) symmetry (side view). When a patch of magnetic material is deposited on a material with strong spin-orbit coupling, for example, Co on Pt, the modified spin structure is reminiscent of a hedgehog. Such DM interactions are of interest in the context of magnetic skyrmions. For example, bubble domains in thin films have a nonzero skyrmion number and therefore yield a topological Hall effect (THE) [76], but DM interactions change the spin structure of the bubble and the THE, thereby adding new physics. Spin Waves and Anisotropic Exchange The low-lying excitations in Heisenberg magnets are of the spin-wave or magnon type. Spin waves are of interest in experimental and theoretical physics and also important in applied physics (microwave resonance, exchange stiffness). Chapter SPW is devoted to spin waves, and in this chapter, the focus is on exchange in spin1/2 Heisenberg magnets, where quantum effects are most pronounced. To solve the ferromagnetic Heisenberg model, it is convenient to rewrite the exchange term in Eq. (27) as S i · S i+1 = 1 + − + Si Si+1 + Si− Si+1 + Sz,i Sz,i+1 2 (52) The Sz operators measure the spin projections, Sz |↑>= + 12 |↑> and Sz |↓>= − 12 |↓>, but leave the wave function unchanged. The spin-flip operators S + and S − rise and lower the spin by one unit, respectively: S + |↓>=|↑> and S − |↑>=|↓>. Since the S = 1/2 Heisenberg model has only two spin states, S + |↑>= 0 and S − |↓>= 0, or symbolically S + S + = 0 and S − S − = 0. The products of S + and S − in Eq. (52) have the effect of interchanging spins of opposite sign: |↑↓ > becomes |↓↑ > and vice versa. The ferromagnetic state, symbolically |0 > = |↑↑↑↑↑↑↑↑...>, is an eigenstate of the Hamiltonian, because each of the spin-flip terms contains an S + operator that creates a zero. One might naively expect that a single switched spin creates an excited eigenstate, for example, |i > = |↑↑↑↓↑↑↑↑...>, where Ri is the position of the flipped spin. However, the spin-flip operators move the flipped spin and thereby create wave functions |i + 1 > and |i–1>. The low-lying eigenstates of the ferromagnetic chain are actually plane-wave superpositions of single-spin 90 R. Skomski Fig. 13 Spin wave (schematic) flips, |ψ k > = exp.(ik·Rj ) |j>. These wave-like excited states are the spin waves or magnons. Each magnon corresponds to one switched spin, but the reversal is delocalized rather than confined to a single atom (Fig. 13). The corresponding excitation energy is E = 4(1 – cos(k a)). For arbitrary crystals and spins [63] E(k) = 2S jJ R j 1– cos k · R j (53) where Rj is the distance between the exchange-interacting atoms. Of particular interest is the long-wavelength limit, where the dispersion relation (53) becomes quadratic. The three monatomic cubic lattices (sc, bcc, fcc) have [63] E(k) = 2SJ a 2 k 2 (54) This equation cannot be generalized to more complicated cubic crystals, because E(k) is governed by the interatomic distance Rj , not by the lattice constant a, which can be very large due to superlattice formation. Application of Eq. (53) to crystals without second-order structural anisotropy and z nearest neighbors (distance R) yields E(k) = 2zSJ (1– sin(kR)/kR), which has the long-wavelength limit E(k) = z SJ R 2 k 2 3 (55) For sc, bcc, and fcc lattices, Eq. (55) is equivalent to Eq. (54). In good approximation, it can also be applied to slightly noncubic structures. For example, elemental cobalt has R = 2RCo , where RCo = 1.25 Å nm is the atomic radius of fcc and hcp Co. Strongly anisotropic structures, such as multilayers, require an explicit evaluation of Eq. (53). It is common to write this relation as E = D k2 , where D is the spin-wave stiffness. In micromagnetism, it is convenient to write the exchange energy as E = A [∇s]2 dV where A is the exchange stiffness. Comparison of Eqs. (54) and (56) yields (56) 2 Magnetic Exchange Interactions 91 Table 3 Spin-wave stiffness D and exchange stiffness A for some materials [78] Material Fe Co Ni Ni80 Fe20 Co2 MnSn Fe3 O4 A = 2 c S2 J a D meV/nm2 2.8 4.5 4.0 2.5 2.0 5.0 A pJ/m 20 28 8 10 6 7 (57) where c is the number of atoms per unit cell (c = 1 for sc, c = 2 for bcc, c = 4 for fcc). Similar to Eq. (54), Eq. (57) cannot be used for arbitrary crystals, whose lattice constants can be very large, and for dilute magnets [90]. In terms of the interatomic distance, the rule of thumb is A ≈ zS 2 J /5R. Values of spin wave stiffness D and exchange stiffness A for some common magnets are given in Table 3. Anisotropic Exchange Anisotropic exchange is a vague term, used for a variety of physically very different phenomena, sometimes even for the Dzyaloshinski-Moriya interactions. Spin waves are affected by magnetocrystalline anisotropy, especially in noncubic magnets. The anisotropy adds a spin-wave gap Eg = E(k = 0) to Eq. (54) and also affects the exchange stiffness. For example, in uniaxial (tetragonal, hexagonal, trigonal) magnets, one needs to distinguish A|| (along the c-axis) and A⊥ (in the ab-plane). The difference is particularly large in multilayers, where the intra-layer exchange (A⊥ ) is often much stronger than the interlayer exchange (A|| ). The Heisenberg interaction behind this type of anisotropic exchange remains isotropic, as in Eq. (27), and the difference between A|| and A⊥ is caused by the nonrelativistic bond anisotropy, Jij = J R i − R j [79]. Very different physics are involved in the so-called anisotropic Heisenberg model, which derives from the (isotropic) Heisenberg model by the replacement J ŝ · ŝ → J ŝx · ŝx + ŝy · ŝy + Jz ŝz · ŝ z (58) The exchange anisotropy J = Jz –J /Jz , which has the same relativistic origin as magnetocrystalline anisotropy and the Dzyaloshinski-Moriya interaction (Section “Magnetic Order and Noncollinearity”), is normally very small compared to the average or “isotropic” exchange Jo = 2J + Jz /3. However, J becomes non-negligible when Jo is very small, for example, in some compounds with low Curie temperature [80]. 92 R. Skomski The XY and Ising models are obtained by putting Jz = 0 and J = 0 in Eq. (58), respectively. In classical statistical mechanics, these models have the spin dimensions n = 2 (XY) and n = 1 (Ising), as compared to n = 3 (Heisenberg model), n = ∞ (spherical model), and n = 0 [4, 60]. The spin dimension has a profound effect on the onset of ferromagnetism in D-dimensional crystals. Ising ferromagnets have Tc = 0 in one real-space dimension (D = 1) but Tc > 0 for D ≥ 2. Heisenberg magnets have Tc = 0 in one and two real-space dimensions but Tc > 0 for D ≥ 3. For all ferromagnetic spin dimensionalities, statistical mean-field theory is qualitatively correct in D > 4 real-space dimensions, with logarithmic corrections in D = 4. For the geometrical meaning of D = 4, see Figs. 7.9, 7.10 in Ref. [4]. Two-dimensional magnets (D = 2) are particularly intriguing. The Heisenberg model (n = 2) predicts Tc = 0, but adding an arbitrarily small amount of uniaxial anisotropy to the Heisenberg model yields Tc > 0 [81]. This feature has recently received renewed attention in the context of the two-dimensional van der Waals (VdW) magnetism. The two-dimensional XY model (n = 2, D = 2) yields a Thouless-Kosterlitz transition with a power-law decay of spin-spin correlations but no long-range magnetic order. There are actually two types of Ising models, characterized by similar Hamiltonians, J = 0 in Eq. (58), but different Hilbert spaces. Ising’s original model is de facto a classical Heisenberg model with infinite magnetic anisotropy [82, 83], which leads to two spin orientations, ↑ and ↓. The quantum-mechanical Ising model, also known as the “spin-1/2 Ising model in a transverse field” [84–86], is physically very different. For example, it allows states with <sx > = 0 and <sy > = 0, whereas the idea of the (original) Ising model is to suppress such states, <sx > = <sy > = 0. Exchange anisotropy or exchange bias in thin films means that a pinning layer yields a horizontal hysteresis-loop shift in a free layer. The bias is realized through FM or AFM exchange at the interface between the pinning and soft layers, but its ultimate origin is the magnetocrystalline anisotropy of the pinning layers, which is often an antiferromagnet. The situation is physically similar to the horizontal and vertical hysteresis-loop shifts sometimes observed in hard-soft composites, which are inner-loop effects. Micromagnetically, the exchange-energy density is not confined to the atomic-scale interface but extends into the pinning and free layers, so that the net interlayer exchange energy per film area is generally very different from the atomic-scale interlayer exchange [73]. Experimental Methods There are many methods to investigate exchange, directly and indirectly, some of which are briefly mentioned here. Magnetic measurements are used to determine Curie temperatures Tc ∼ J , from which exchange constants can be deduced. The low-field magnetization of antiferromagnets is zero, but high fields tilt the AFM sublattices and yield a small magnetization M(H ) ∼ H /J . 2 Magnetic Exchange Interactions 93 The exchange may also be deduced from the low-temperature M(T) curves, because the Bloch law involves the exchange stiffness. Magnetic force and, to a much lesser extent, anomalous magneto-optic microscopies used to investigate magnetic domain structures, which contain implicit information about the exchange stiffness. A direct method to probe exchange is magnetic resonance. Neutron diffraction and, to a much lesser extent, X-ray diffraction (XRD) are important methods to probe spin structure. The magnetic XRD signal is much weaker than the neutron-diffraction signal. Interatomic exchange can be probed by a variety of methods, such as X-ray magnetic dichroism, which also allows a distinction of L and S contributions to the atomic moments. Electron-transport measurements are frequently used to gauge and confirm exchange effects. Antiferromagnetic Spin Chains Spin waves are particularly intriguing in antiferromagnets, whose low-lying states correspond to the highest excited states in the ferromagnetic case [8, 16, 17]. By analogy with FM ground state, |↑↑↑↑↑↑↑↑...>, one could intuitively assume that the AFM ground state is a superposition of the two quasi-classical Néel states |AFM (1)>=| ↑↓↑↓↑↓↑↓ · · · > (59) |AFM (2)>=| ↑↓↑↓↑↓↑↓ · · · > (60) and However, the spin-flip terms in Eq. (52) do not transform Eqs. (59) and (60) into each other but create pairs of parallel spins (spinons), for example S4 + S5− |↑↓↑↓↑↓↑↓ · · · >=|↑↓↑↑↓↓↑↓ · · · > (61) Using the Néel states to evaluate Eq. (27) yields an AFM ground-state energy of −0.5 J per atom, compared to the exact Bethe result of −2 J (ln 2 − 1/4) = −0.886 J [8]. Systems with such complicated ground states are also known as quantum spin liquids (QSL). The underlying physics is very similar to the spin mixing discussed below Fig. 10 but now involves an infinite number of spins. The derivation of the Heisenberg model, Section “Hubbard Model”, was based on the neglect of interatomic hopping. Strictly speaking, this is meaningful only when U is large and the band is half-filled. In more- or less-than-half-filled bands, a fraction of the electrons can move almost freely. Such magnets have both charge and spin degrees of freedom, and the corresponding extension of the Heisenberg chain is known as the Tomonaga-Luttinger model or Luttinger liquid [87, 88]. A typical wave function is |↑↓↑↑↓◦↑↓>, which contains one hole. The model has a number of interesting features. For example, spin and charge excitations move with different velocities, the former being slower, because spin excitations have lower 94 R. Skomski energies δE = ω than charge excitations. This is an example of a correlation effect known as spin-charge separation [88]. By contrast, in the itinerant limit, charge and spin degrees are closely linked. Spin-charge separation is important in the Kondo mechanism (where low-energy spin flips determine the resistivity) and in hightemperature superconductivity (Section “Dimensionality Dependence of Quantum Antiferromagnetism”). The electron distribution n(E) of a Luttinger liquid is very different from a Fermi liquid [88]. Weak correlations in metallic magnets create particle-hole quasiparticles but leave the Fermi surface otherwise intact. Strong correlations, as in the Luttinger liquid, completely destroy the Fermi surface, and n(E) becomes a smooth function. Dimensionality Dependence of Quantum Antiferromagnetism The Luttinger liquid is a typical one-dimensional effect: Arbitrarily small perturbations of structural, thermal, or quantum-mechanical origin destroy long-range magnetic order. Quantum-spin-liquid effect in higher dimensions are generally less pervasive but not necessarily unimportant. In antiferromagnets, it is possible to redefine the operators Si ± in Eq. (52) by reversing the spin in each second atom, which assimilates the AFM problem to the FM problem and allows the consideration of spin waves. However, this procedure creates terms of the type Si + Sj + and Si − Sj − , where the atoms i and j belong to different sublattices. These terms go beyond straightforward spin-wave theory, which exclusively involves Si + Sj − and Si − Sj + . The additional terms yield a quantum-mechanical mixture of the two sublattices and require an additional diagonalization procedure known as Bogoliubov transformation [3]. The sublattice admixture reduces both the energy of the AFM state and the sublattice magnetizations, the latter meaning that the ↑ sublattice acquires some ↓ character and vice versa. The ground-state energy is discussed most conveniently by starting from two interacting spins having S = 1/2, as described by Eq. (26). For a given AFM exchange J < 0, the energies of the FM and AFM states scale as S2 = 1/4 and –S(S + 1) = −3/4, respectively. More generally, the AFM energy is proportional to –S(S + δ), where δ describes the intersublattice admixture and 0 < δ < 1. Spinwave theory yields δ = 0.363 for the linear chain, δ = 0.158 for the square lattice, and δ = 0.097 for the simple-cubic lattice. The one-dimensional value is close to the exact result δ = 0.386 for S = 1/2. A rough estimate for the relative reduction of the sublattice magnetization in hypercubic magnets (z = 2d) is 0.15/S(d − 1), corresponding to sublattice magnetizations of 0%, 70%, and 85% in one-, two-, and three-dimensional magnets having S = 1/2. The complete magnetization collapse in one dimension is caused by the involve ment of the integral k−1 dk ∼ kd – 1 dk, which exhibits an infrared (small-k) divergence in one dimension. The same integral is behind Bloch’s law and the Wagner-Mermin theorem, and in all cases, the divergence indicates that fluctuations destroy long-range magnetic order in one dimension. However, the underlying physics is different: The fluctuations considered in Bloch’s law and in the Wagner- 2 Magnetic Exchange Interactions 95 Mermin theorem are of thermodynamic origin, whereas the present ones are zero-temperature quantum fluctuations. These fluctuations, which are largest for S = 1/2, are a correlation effect and therefore difficult to treat in density-functional calculations. One example is high-temperature superconductivity (HTSC) in La2-x Srx CuO4 , which involves 3d9 states in the Cu-O planes of the oxides [9] and where spin-1/2 quantum fluctuations trigger the formation of Cooper pairs. The parent compound La2 CuO4 is a strongly correlated antiferromagnetic semiconductor, but Sr doping drives the system toward a phase transition. The denominator in Eq. (40) becomes small, meaning that spin fluctuations (antiferromagnetic paramagnons) are strongly enhanced by Sr doping. Furthermore, both spin-charge separation [9] and critical slowing down cause the spin fluctuations to evolve very sluggishly, so that they can play the role of phonons in BCS superconductors. Frustration, Spin Liquids, and Spin Ice A number of exotic topics in physics are more or less closely related to exchange interactions. This subsection discusses both classical and quantum-mechanical implications of frustration, as well as some related micromagnetic questions. Frustration Ring configurations with antiferromagnetic interactions and odd numbers N of atoms offer intriguing physics. Let us start with theclassical exchange energy between atoms at Ri and Rj , which is equal to −Jij cos φi − φj . Consider an equilateral triangle (N = 3) with antiferromagnetic nearest-neighbor interactions (Fig. 14). If the exchange was ferromagnetic, then φ i = 0 (or φ i = const.) would simultaneously minimize the energy of all bonds and yield a ground-state energy of −3J . The corresponding antiferromagnetic solution, of energy −3 |J |, does not exist, because three antiferromagnetic bonds cannot be simultaneously realized in a triangle (Fig. 14a). This is referred to as magnetic frustration. Fig. 14b shows that bond angles of 120◦ , rather than 180◦ , may be realized for all spins, and the corresponding ground-state energy is −1.5 |J |, somewhat lower than the energy − |J | of (a). The classical frustration problem of Fig. 14 is elegantly summarized by a construction known as Frost’s circle. The approach was developed to describe the hopping of p electrons in cyclic molecules [89] but can also be applied to s-state electrons such as those in Fig. 7 and to interatomic exchange, because the involvement of τ ij and Jij is mathematically equivalent. For a ring of N atoms with nearest-neighbor exchange J , the energy eigenvalues per atom are En = –2 J cos (2 π n/N ) (62) where n = 0, ..., N–1, N. These energies can be arranged on a circle, as exemplified by the example N = 5 and ferromagnetic coupling (J > 0) in Fig. 14(c). The 96 R. Skomski Fig. 14 Classical frustration in rings of N atoms: (a) frustrated state (N = 3), (b) ground state (N = 3), and (c) graphical solution (Frost’s cycle) for N = 5 FM ground state has n = 0 and the energy −2J . However, the figure shows that energy eigenvalues are not necessarily symmetric with respect to changing the sign of J . For odd values of N, the AFM ground state is double degenerate and characterized by nearest-neighbor spin angles (1–1/N) 180◦ , as opposed to the 180◦ expected for ideal antiferromagnetism. The incomplete antiparallelity leads to a ground-state energy 2J cos (π/N), higher than that of an ideal antiferromagnet (2J ). This analysis shows exchange in antiferromagnets is different from exchange in ferromagnets, even in the classical limit. The quantum-mechanical ground state of the AFM spin-1/2 Heisenberg triangle is obtained by using Eq. (52) to evaluate the matrix elements between the states |↑↑↓>, |↑↓↑>, and |↓↑↑>. It is sufficient to consider Sz = 1/2, since none of the terms in Eq. (52) changes the total spin projection Sz . For example, spin configurations such as |↑↑↑ > (Sz = 3/2) and |↑↑↓ > (Sz = 1/2) do not mix. Furthermore, Sz = − 1/2 is equivalent to Sz = +1/2 and does not need separate consideration. For the three states with Sz = 1/2, the Hamiltonian is ⎛ ⎞ ⎛ ⎞ 100 111 3 H = J ⎝0 1 0⎠ − J ⎝1 1 1⎠ 2 001 111 (63) The diagonalization of this matrix is trivial and yields one FM eigenstate (1, 1, 1) of energy – 3J /2 > 0 and two AFM eigenstates of energy 3J /2 < 0, for example, |Ψ 1 > = (2, −1, −1) and |Ψ 2 > = (0, 1, −1). Explicitly 1 |1 >= √ (2| ↑↑↓ >−| ↑↓↑ >−| ↓↑↑ >) 6 and (64a) 2 Magnetic Exchange Interactions 1 |2 > = √ (| ↑↓↑ >−| ↓↑↑ >) 2 97 (64b) Since the two AFM states are degenerate, |Ψ > = c1 |Ψ 1 > + c2 |Ψ 2 > is also an eigenstate; complex numbers c1/2 mean a spin component in the y-direction. Alternatively, Eq. (64b) is a product of the type (|↑↓>− |↓↑>)⊗|↑> and contains a maximally entangled AFM singlet |↑↓>− |↓↑>. According to Eq. (63), the corners of the triangle are equivalent, so that the singlet may be placed on any of the three bonds. The spin-1/2 Heisenberg square has the spin projections Sz = (0, ±1, ±2). In the antiferromagnetic ground state (Sz = 0), there are six spin configurations, namely, the two Néel states |↑↓↑↓> and |↓↑↓↑>, as well as six non-Néel states with pairs of parallel spins, such as |↑↑↓↓>. Classically, non-Néel states are not expected to appear in the ground state, but the diagonalization of the corresponding 6 × 6 matrix yields an AFM ground-state singlet with a strong admixture of non-Néel character, namely 1 1 1 |>= | ↑↓↑↓ >+ | ↓↑↓↑ >− √ (| ↑↑↓↓ >+ . . . ) 2 2 8 (65) The total weight of the non-Néel configurations in the ground state is 50%. This ground state is rather complicated but, unlike Eq. (64), not frustrated. Quantum-spin Liquids The quantum-mechanical behavior of the structures of Fig. 14 is liquid-like, similar to the Luttinger liquid of Sect. “Antiferromagnetic Spin Chains”. Two-dimensional magnets are particularly interesting. Figure 15 shows 2D lattices with the triangular structural elements required for AFM frustration. The wave functions are complicated but contain AFM singlets, and there are many ways of arranging these singlets on a lattice (a). Some materials investigated as QSL materials, such as ZnCu3 (OH)6 Cl2 (herbertsmithite), form Kagome lattices (b). Triangular and Kagome lattices exhibit similar frustration behaviors, but Kagome lattices differ by having low-lying excitations [91]. Frustration is also important in spin-ice materials, such as pyrochlore-ordered Dy2 Ti2 O7 . The pyrochlore structure consists of tetrahedra whose corner atoms are magnetic. The strong crystal field forces the moments to lie on lines from the corners to the centers of the tetrahedra, but to fix the spin direction (inward or outward), one needs an additional criterion known as the “two-in, two-out” rule. There are many two-in two-out configurations, which creates a spin-ice situation reminiscent of Fig. 15a. There are excited spin-ice states where all spins point inward (four in) or outward (four out), which yields an accumulation of magnetic charge (south poles or north poles) in the middle of the tetrahedron. Such accumulations are also known as magnetic monopoles, but this characterization is misleading. Magnetic monopoles are high-energy elementary particles having B·dA = 0. Their existence cannot 98 R. Skomski Fig. 15 Frustrated lattices in two dimensions: (a) triangular lattice and (b) Kagome lattice Fig. 16 Static magnetic field sources: (a) magnetic dipole and (b) “magnetic monopole” be ruled out, but they have never yet been observed in the universe. Solid-state “monopoles” are no monopoles, because they are formed from dipoles and do not violate B·dA = 0. For illustrative purposes, the magnetic charges near the ends of a long bar magnet may be regarded monopoles, but these are not real monopoles but merely the ends of long dipoles. Figure 16 compares a magnetic dipole (a) with a putative magnetic monopole (b). The configuration (b) may be created in the form of a magnetic-dipole layer or by some radial magnetization distribution in a magnetic material. In any case, it requires compensating south-pole charges inside the sphere, so that B · dA = 0. As a consequence, the magnetic field is actually zero outside the sphere, so that the finite-length arrows in (b) are without physical basis. In the context of new materials, it is important to keep in mind that exchange interactions are described in terms of Hamiltonians. The equation of motion of any system is more fundamentally governed by the Lagrangian and its time integral, the action. The difference can be ignored in flat spaces but is important in curved and periodic spaces, where it corresponds to the Berry phase. The contributions of the phase are basically of a ‘zero-Hamiltonian’ type and ignored in this chapter. 2 Magnetic Exchange Interactions 99 Acknowledgments This chapter has benefited from help in details by P. Manchanda and R. Pathak and from discussions with B. Balamurugan, C. Binek, X. Hong, Y. Idzerda, A. Kashyap, P. S. Kumar, D. Paudyal, T. Schrefl, D. J. 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Balasubramanian, B., Manchanda, P., Pahari, R., Chen, Z., Zhang, W., Valloppilly, S.R., Li, X., Sarella, A., Yue, L., Ullah, A., Dev, P., Muller, D.A., Skomski, R., Hadjipanayis, G.C., Sellmyer, D.J.: Chiral Magnetism and High-Temperature Skyrmions in B20-Ordered Co-Si. Phys. Rev. Lett. 124, 057201-1-6 (2020) 91. Mendels, P., Bert, F.: Quantum kagome frustrated antiferromagnets: One route to quantum spin liquids. C. R. Phys. 17, 455–470 (2016) Ralph Skomski received his PhD from Technische Universität Dresden in 1991. He worked as a postdoc at Trinity College, Dublin, and at the Max-Planck-Institute in Halle, before moving to the University of Nebraska, Lincoln, where he is presently a Full Research Professor. He is an analytical theorist with primary research interests in magnetism, nanomaterials, and quantum mechanics. 3 Anisotropy and Crystal Field Ralph Skomski, Priyanka Manchanda, and Arti Kashyap Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Phenomenology of Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lowest-Order Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisotropy and Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tetragonal, Hexagonal, and Trigonal Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Higher-Order Anisotropy Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anisotropy Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal-Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . One-Electron Crystal-Field Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal-Field Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Many-Electron Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Orbit Coupling and Quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rare-Earth Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rare-Earth Ions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operator Equivalents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Single-Ion Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Temperature Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Transition-Metal Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perturbation Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Orbit Matrix Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal Fields and Band Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Itinerant Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 107 108 109 112 114 115 116 118 122 123 127 132 133 136 139 141 145 147 149 150 151 R. Skomski () University of Nebraska, Lincoln, NE, USA e-mail: [email protected] P. Manchanda Howard University, Washington, DC, USA A. Kashyap IIT Mandi, Mandi, HP, India e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_3 103 104 R. Skomski et al. First-Principle Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other Anisotropy Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetostatic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Néel’s Pair-Interaction Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two-Ion Anisotropies of Electronic Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dzyaloshinski-Moriya Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Antiferromagnetic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetoelastic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Low-Dimensional and Nanoscale Anisotropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Surface Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Random Anisotropy in Nanoparticles, Amorphous, and Granular Magnets . . . . . . . . . . . . Giant Anisotropy in Low-Dimensional Magnets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A: Spherical Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Point Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C: Hydrogen-Like Atomic 3d Wave Functions . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 156 159 159 161 162 164 165 166 167 167 169 171 174 174 176 177 179 Abstract Magnetic anisotropy, imposed through crystal-field and magnetostatic interactions, is one of the most iconic, scientifically interesting, and practically important properties of condensed matter. This article starts with the phenomenology of anisotropy, distinguishing between crystals of cubic, tetragonal, hexagonal, trigonal, and lower symmetries and between anisotropy contributions of second and higher orders. The atomic origin of magnetocrystalline anisotropy is discussed for several classes of materials, ranging from insulating oxides and rare-earth compounds to iron-series itinerant magnets. A key consideration is the crystal-field interaction of magnetic atoms, which determines, for example, the rare-earth single-ion anisotropy of today’s top-performing permanent magnets. The transmission between crystal field and anisotropy is realized by spin-orbit coupling. An important crystal-field effect is the suppression of the orbital moment by the crystal-field, which is known as quenching and has a Janus-head effect on anisotropy: the crystal field is necessary to create magnetocrystalline anisotropy, but it also limits the anisotropy in many systems. Finally, we discuss some other anisotropy mechanisms, such as shape, magnetoelastic, and exchange anisotropies, and outline how anisotropy is realized in some exemplary compounds and nanostructures. Introduction Magnetic anisotropy means that the energy of a magnetic body depends on the direction of the magnetization with respect to its shape or crystal axes. It is a quantity of great importance in technology. For example, it is crucial for a material’s ability to serve as a soft or hard magnet; governs many aspects of data storage and processing, 3 Anisotropy and Crystal Field 105 such as the areal density in the magnetic recording; and affects the behavior of microwave and magnetic-cooling materials. In the simplest case of uniaxial anisotropy, the energy depends on the polar angle θ but not on the azimuthal angle φ of the magnetization direction: Ea = V K1 sin2 θ + K2 sin4 θ + K3 sin6 θ (1) Here the Kn are the n-th anisotropy constants and V is the crystal volume. The first anisotropy constant K1 is often the leading consideration. Ignoring K2 and K3 , the anisotropy energy is equal to K1 V sin2 θ , and two cases need to be distinguished. K1 > 0 yields energy minima at θ = 0 and θ = 180◦ , that is, the preferential magnetization direction is along the z-axis (easy-axis anisotropy). When K1 < 0, the energy is minimized for θ = 90◦ (easy-plane anisotropy). The magnitudes of the room-temperature anisotropy constants K1 vary from less than 5 kJ/m3 in very soft magnets to more than 17 MJ/m3 in SmCo5 . A variety of rare-earth-free transition-metal alloys have anisotropies between 0.5 and 2.0 MJ/m3 . YCo5 , where the Y is magnetically inert, has K1 = 5.0 MJ/m3 . This chapter deals with the phenomenological description and physical origin of anisotropy. A key question is how magnetic anisotropy depends on crystal structure and chemical composition. The main contribution to the anisotropy energy of most materials is magnetocrystalline anisotropy (MCA), which involves spin-orbit coupling, a relativistic interaction [1]. This mechanism involves two steps. First, the electrons that carry the magnetic moment interact with the lattice, via electrostatic crystal field and exchange interactions. Second, the spin-orbit coupling (SOC) ensures that the spin magnetization actually takes its orientation from the lattice. In the absence of spin-orbit coupling, anisotropic arrangements of atoms do not introduce magnetocrystalline anisotropy. A good example is the Heisenberg model, H = –ij J R i –R j S i · S j , which is magnetically isotropic even if the exchangebond distribution (Ri – Rj ) is highly anisotropic, for example, in thin films and nanowires. Magnetocrystalline anisotropy is not the only contribution. Magnetostatic dipolar interactions are important in some nanostructured materials and also in materials where the magnetocrystalline anisotropy is zero by coincidence. Shape anisotropy (Sect. “Néel’s Pair-Interaction Model”) is a dipole contribution of importance in some permanent magnets (alnicos), and in magnets with a noncubic crystal structure, there is also a small dipole contribution to the MCA. The latter is particularly important in Gd-containing magnets, because Gd3+ ions do not exhibit anisotropic crystal-field interactions (Sect. “Crystal-Field Theory”) but has a large dipole moment (S = 7/2). Magnetic anisotropy is most widely encountered in ferro- and ferrimagnets, but it is also present in antiferromagnets (Sect. “Magnetoelastic Anisotropy”), disordered magnets (Sect. “Random Anisotropy in Nanoparticles, Amorphous, and Granular Magnets”), paramagnets, and diamagnets. An example of an anisotropic diamagnet is graphite, where the magnitude of the susceptibility is 40 times higher along the hexagonal c-axis than in the basal plane [2], due to the high mobility of the electrons in the graphene-like carbon sheets that make up the graphite structure. 106 R. Skomski et al. Anisotropies in bulk materials and this films are closely related to the orbital moment L (see below) and therefore to the Bohr-van Leeuwen theorem, which suggests that a magnetic field acting on electrons does not change the magnetization of solids. In fact, the field leaves the energy of the electrons unchanged, because the Lorentz force is perpendicular to the velocity, but nevertheless changes the orbital moment. Note that the definition and physical interpretation of orbital moments in solids is a rather recent development, associated with the discoveries of Berry phase and bulk-boundary correspondence. Berry-phase effects are importantant in curved and periodic spaces and can be considered as geometrical constantenergy phenomena. Furthermore, magnetization processes reflect the rotation rather than creation of atomic moments. Neither the concept of spin roatation nor the Berry phase where known in the early 20th century, when the theorem was formulated. Depending on the relative strength of the spin-orbit coupling compared to the interatomic interactions (crystal field, exchange, hopping), there are two important limits. The magnetocrystalline anisotropy of high-performance permanent magnets, such as SmCo5 and Nd2 Fe14 B, is largely provided by the rare-earth 4f electrons [3, 4]. These electrons are close to the nucleus (R ≈ 0.5 Å), which means that they exhibit a strong spin-orbit coupling (of the order of 200 meV) but do not exhibit much interaction with the crystalline environment (of the order of 10 meV). The orbits of the 4f electrons, as well as the charge distribution n4f (r), are determined by Hund’s rules. Specifically, Hund’s first rule, which has its origin in the Pauli principle, states that the total spin S is maximized. The remaining degeneracy with respect to total orbital moment L is removed by Hund’s second rule, which means that the orbital moment L is maximized due to intra-atomic exchange. Finally, Hund’s third rule describes how the spin-orbit interaction rigidly couples orbitalmoment vector L to the spin vector S. Due to Hund’s third rule, a change in the magnetization angles θ and φ yields a rigid rotation of the charge distribution n4f (r). Figure 1 explains how this Fig. 1 Basic mechanism of magnetocrystalline anisotropy, illustrated by an Sm3+ ion (blue) in a tetragonal environment (yellow). The Sm spin (arrow) is rigidly coupled to the prolate Hund’s-rules 4f charge cloud of the Sm, and the anisotropy energy is equal to the electrostatic interaction energy between the Sm3+ electrons and the electrostatic crystal field. Due to electrostatic repulsion, the energy of (a) is lower than (b), and the anisotropy is easy-axis (K1 > 0). If the prolate Sm3+ ion was replaced by an oblate Nd3+ ion, the situation would be reversed to easy-plane anisotropy (K1 < 0) 3 Anisotropy and Crystal Field 107 rotation translates into magnetocrystalline anisotropy. The electric field created by neighboring atoms in the crystal (yellow) is weaker than the spin-orbit coupling and has little effect on the 4f electronic structure, but the electrostatic interaction of the 4f shell with the crystal causes the energy to depend on θ and φ. Atomic crystal-field charges are normally negative, which amounts to a repulsive interaction between 4f electrons and neighboring atoms. Figure 1 shows an Sm3+ ion with a prolate charge distribution. For the tetragonal crystal environment shown in the figure, the electrostatic energy of (a) is lower than (b), corresponding to easy-axis anisotropy (K1 > 0). In the opposite limit of the Fe-series transition-metal magnets, the spin-orbit interaction is much weaker than the interatomic interactions involving 3d electrons. The 3d orbits are therefore determined by the crystal field and interatomic hopping, and the spin-orbit coupling is a small perturbation unless the unperturbed energy levels are accidentally degenerate. In other words, crystal-field interactions determine rare-earth anisotropy but are merely an important starting consideration for the determination of iron-series transition-metal anisotropies. This chapter starts with the phenomenology of magnetic anisotropy (Sect. “Phenomenology of Anisotropy”), followed by analyses of crystal-field interactions (Sect. “Crystal-Field Theory”), rare-earth anisotropy (Sect. “Rare-Earth Anisotropy”), and transition-metal anisotropy (Sect. “Other Anisotropy Mechanisms”). The last section deals with some special topics, Phenomenology of Anisotropy Magnetocrystalline anisotropy is usually parameterized in terms of anisotropy constants, as in Eq. (1). The definition of these constants is somewhat arbitrary, tailored towards experimental and micromagnetic convenience, and there is some variation in notation. Another approach is to use anisotropy coefficients of order 2n, obtained by expanding the magnetic energy in spherical harmonics (Appendix A). For example, the anisotropy coefficients κ 2 0 , κ 4 0 , κ 6 0 roughly correspond to the uniaxial anisotropy constants K1 , K2 , and K3 , respectively. Anisotropy coefficients are difficult to access by direct magnetic measurements, but they form an orthonormal system of functions that do not mix crystal-field contributions of different orders. Anisotropy energies per atom, typically 1 meV or less, are much smaller than the energies responsible for moment formation (about 1000 meV). This means that the spontaneous magnetization Ms = |M| is essentially fixed and that anisotropy energies can be expressed in terms of the magnetization angles φ and θ . It is convenient to choose a coordinate frame where M = Ms sin θ cos φ ex + sin θ sin φ ey + cos θ ez (2) The respective directions x, y, and z with unit vectors ex , ey , and ez often correspond to the crystallographic a-, b-, and c-axes in crystals of high symmetry. 108 R. Skomski et al. Lowest-Order Anisotropies The first- (or second-)order anisotropy constant K1 often provides a good description of the anisotropy of magnetic substances where higher-order anisotropy constants are negligible. There are, however, many exceptions to this rule. For example, the relative magnitudes of the different Kn depend on crystal structure and temperature, and the φ-dependence of the anisotropy cannot be neglected in many cases. On the other hand, simplifications arise because many anisotropy constants are zero by crystal symmetry. Figure 2 shows the corresponding hierarchy: the number of anisotropy constants decreases in the direction of the arrows. Fig. 2 Relations between crystal systems; the arrows point in directions of increasing symmetry and decreasing numbers of anisotropy constants. The arrows also suggest how degeneracies may arise. For instance, stretching a cubic crystal so that c > a = b creates a tetragonal crystal, whereas stretching a hexagonal crystal so that a > b creates an orthorhombic crystal 3 Anisotropy and Crystal Field 109 Let us start with the lowest-order anisotropy. Expansion of the magnetic energy in spherical harmonics shows that there are five second-order terms Y2 m , corresponding to five anisotropy coefficients κ 2 m , namely, κ 2 −2 , κ 2 −1 , κ 2 0 , κ 2 1 , and κ 2 2 . However, three Euler angles are necessary to fix the anisotropy axes ex , ey , and ez relative to the crystal axes, so there remain only two independent anisotropy constants. Explicitly Ea (θ, φ) 2 = K1 sin2 θ + K1 sin θ cos 2φ V (3) where K1 is the lowest-order (or second-order) in-plane anisotropy constant; K1 and K1 are generally of comparable magnitude. Equation (3) can be used for any crystal, but by symmetry, K1 = 0 for trigonal (rhombohedral), hexagonal, and tetragonal crystals. Only triclinic monoclinic and orthorhombic crystals (red unit cells in Fig. 2) have K1 = 0. Examples of such low-symmetry compounds are the monoclinic 3:29 intermetallics [5, 6], such as Nd3 (Fe1-x Tix )29 . Cubic symmetry is not compatible with these second-order anisotropy contributions, which means that K1 = 0 and K1 = 0. For example, the replacement of the tetragonal environment in Fig. 1 by a cubic environment would mean that the x-, y-, and z-directions are all equivalent, which can only be achieved if K1 = K1 = 0 in Eq. (3). However, there exists a differently defined fourth-order “cubic” K1 c that reproduces Eq. (3) for small angles θ . To describe cubic anisotropy, one must add spherical harmonics so that the sum does not violate cubic symmetry. Based on Table 16, there are only two independent terms that satisfy this condition: K c Ea K1 c 2 = 4 x 2 y 2 + y 2 z2 + z2 x 2 + 6 x 2 y 2 z2 V r r (4) where x/r = cosα x , y/r = cosα y , and z/r = cosα z are the direction cosines of the magnetization. From the power-law behavior of r in Eq. (4), we see that K1 c and K1 c are fourth- and sixth-order anisotropy constants, respectively. K1 c > 0 favors the alignment of the magnetization along the [001] cube edges, which is called irontype anisotropy, while K1 c < 0 corresponds to an alignment along the [98] cube diagonals and is referred to as nickel-type anisotropy . The subscript “c” is often omitted, but when uniaxial and cubic anisotropies need to be distinguished, then it is better to use Ku and/or K1 c to distinguish the respective anisotropy constants. Anisotropy and Crystal Structure The number of anisotropy constants rapidly increases with increasing order and decreasing symmetry . By definition, Ea (−M) = Ea (M), so that we need evenorder spherical harmonics only (gray rows in Table 16). The maximum number of anisotropy constants is therefore 5 (up to second order), 14 (up to the fourth order), and 27 (up to the sixth order). Since the anisotropy axes do not necessarily corre- 110 R. Skomski et al. spond to the crystallographic axes, three of these anisotropy constants effectively function as Euler angles to fix the orientation of the anisotropy axes. The anisotropy axes are known for most crystals of interest in magnetism, which reduces the number of anisotropy constants to 2 (up to second order), 11 (up to the fourth order), and 24 (up to the sixth order). Anisotropy constants of the eighth- and higher-order occur in itinerant magnets, for example, (Sect. “Transition-Metal Anisotropy”), but they are usually very small and rarely considered. The Euler angles must be considered in crystals with low symmetry. Table 1 lists the crystal systems, point groups, and space groups for some magnetic substances. Appendix B gives a complete list of all 32 points and 230 space groups. Triclinic crystals always need three Euler angles to relate the crystallographic a-, b-, and c-axes to the magnetic x-, y-, and z-axes. In all other noncubic crystals, the caxis is parallel to the z-axis, and one needs at most one Euler angle φ o . This angle corresponds to a rotation of the crystal around the c-axis, and the identity cos(β – β o ) = cos (β) cos (β o ) + sin (β) sin (β o ) can be used to get rid of one in-plane anisotropy constant at the expense of introducing a generally unknown rotation angle. As a macroscopic property, magnetic anisotropy is determined by the point group of the crystal. Most magnetic substances with orthorhombic, tetragonal, rhombohedral (trigonal), or hexagonal structures belong to the cyclic (C) or dihedral (D) Schönflies groups. The groups Cn have a single n-fold rotation axis (c-axis), whereas Cnh and Cnv also have one horizontal and n/2 vertical mirror planes, respectively. The dihedral groups Dn have an n-fold rotation axis and n/2 additional twofold rotation axes perpendicular to the c-axis. The cubic crystal system contains tetrahedral (T) and octahedral (O) Schönflies groups. Most noncubic crystal structures of interest in magnetism belong to the highly symmetric Schönflies point groups Cnv , Dn , Dnh , and Dnd , which have φ o = 0. This includes hexagonal (6 mm, 622, 6/mmm, 62m), trigonal (3 m, 32, 3m), tetragonal (4 mm, 422, 4/mmm, 42m), and orthorhombic (2 mm, 222, 2/mmm) crystals. Nonzero values of φ o need to be considered in crystals with point groups Cn , Cnh , and Sn . This is the case for all triclinic and monoclinic crystals and for some hexagonal (6, 6/m, 6), trigonal (3, 3), and tetragonal (4, 4/m, 4) point groups. An example is the monoclinic 3:29 structure [5], which has the point group C2h . To elaborate on the role of the point groups, it is instructive to compare Cn and Cnh with Cnv . Figure 3 shows a top view of a fictitious tetragonal crystal. The fourfold symmetry axis is clearly visible, and since the horizontal mirror plane is in the plane of the paper, the figure describes both Cn and Cnh . Some of the nonmagnetic atoms act as “ligands” (red crosses) and create a crystal field that acts on the rare-earth ions (blue) and establishes local easy axes (dashed line). The local 3 Anisotropy and Crystal Field 111 Table 1 Crystal systems, point groups, space groups, Strukturbericht notation, and prototype structures for some compounds of interest in magnetism. Not all the examples are ferromagnetic Crystal system Monoclinic Monoclinic Orthorhombic Point group C2h (2/m) C2h (2/m) D2h (mmm) Space group C2/m C2/c Pnma Tetragonal Tetragonal Tetragonal D4h (4/mmm) D4h (4/mmm) D4h (4/mmm) P4/mmm P42 /mnm I4/mmm Trigonal Trigonal D3 (32) D3d (3m) P32 12 R3m Trigonal Hexagonal Hexagonal D3d (3m) C6v (6 mm) D6h (6/mmm) R3c P63 mc P63 /mmc Hexagonal Cubic Cubic Cubic Cubic Cubic D6h (6/mmm) T (23) Th (m3) Td (43m) Td (43m) Oh (m3m) P6/mmm P21 3 Pa3 F43m I43m Fm3m Cubic Cubic Oh (m3m) Oh (m3m) Im3m Pm3m Cubic Cubic Oh (m3m) Oh (m3m) Pn3m Fd3m Cubic Oh (m3m) Ia3d Examples D015 (AlCl3 ): DyCl3 B26 (tenorite): CuO C37 (Co2 Si): Co2 Si; D011 (cementite): Fe3 C; goethite: α-FeO(OH); orthorhombic perovskite: SrRuO3 L10 (CuAu): PtCo, FePd, FePt, MnAl, FeNi Rutile (C4): CrO2 , MnF2 , TiO2 ; Nd2 Fe14 B ThMn12 (D2b ): Sm(Fe11 Ti); Al3 Ti (D022 ): Al3 Dy, GaMn3 D04 (CrCl3 ): CrCl3 (P31 12) C19 (α-Sm): Sm, NbS2 ; Th2 Zn17 : Sm2 Co17 , Sm2 Fe17 N3 D51 (corundum): α-Fe2 O3 B4 (wurtzite): MnSe A3 (hcp): Co, Gd, Dy; B81 (NiAs): MnBi, FeS; C7 (MoS2 ): TaFe2 ; C14 (MnZn2 hexagonal laves phase): TaFe2 , Fe2 Mo; C36 (MgNi2 hexagonal laves): ScFe2 ; D019 (Ni3 Sn): Co3 Pt*; PbFe12 O19 (magnetoplumbite): BaFe12 O19 , SrFe12 O19 ; Th2 Ni17 : Y2 Fe17 D2d (CaCu5 ): SmCo5 FeSi (B20): MnSi, CoSi, CoGe Pyrite (C2): FeS2 C1b (half-Heusler): MnNiSb A12 (α-Mn): Mn A1 (fcc): Ni; B1 (NaCl): CoO, NiO, EuO, US; D03 (AlFe3 ): Fe3 Si; L21 (cubic Heusler): AlCu2 Mn; D8a (Th6 Mn23 ): Dy6 Fe23 A2 (bcc): Fe, Cr B2 (CsCl): NiAl, FeCo, AlCo, B3 (zincblende): CuCl, MnS, GaAs; E21 (cubic perovskite): BaTiO3 ; L12 (AuCu3 ): Fe3 Pt C3 (cuprite): CuO2 C15 (cubic Laves phase): SmFe2 , TbFe2 , UFe2 , ZrZn2 ; H11 (spinel): Fe3 O4 Fe3 Al2 Si3 O12 (garnet): Y3 Fe5 O12 , Gd3 Fe5 O19 112 R. Skomski et al. Fig. 3 Top view on a unit cell of a tetragonal crystal with C4 or C4h symmetry. Nonmagnetic ligands (red crosses) create a crystal field of low symmetry and local easy axes (dashed lines) that are unrelated to the crystal axes (gray lines). For clarity, the figure shows only some of the atoms in the unit cell crystal field may have very low site symmetry, with local easy axes unrelated to the a- and b-axes (gray). However, since the local easy axes obey the fourfold rotation symmetry, the sum of all local anisotropy contributions is fourth-order, and there is no second-order in-plane contribution. The in-plane anisotropy is of the type cos(4φ – 4φ o ), and in the present example, the angle φ o ≈ 40◦ is equal to the angle between the dashed local easy axes and the crystal axes. Going from C4(h) to C4v introduces vertical mirror planes. These two planes ensure that each local easy axis of angle φ o has a counterpart with –φ o , so that the net anisotropy directions are now parallel to the crystal axes. The picture outlined in Fig. 2 and Table 1 focuses on crystallographic point groups. A more general approach would be to consider magnetic point groups, as exemplified by the noncubic (tetragonal) electronic structure of magnets having a cubic crystal structure and a layered antiferromagnetic spin structure. However, the layers can lie in any of the equivalent cubic lattice planes, so the magnetic anisotropy remains cubic. Tetragonal, Hexagonal, and Trigonal Anisotropies The anisotropy constants belonging to a given point group can be derived by applying the symmetry elements of the group to the expansion of the magnetic energy in terms of spherical harmonics. For example, the fourfold rotation symmetry of tetragonal magnets, Fig. 3, is compatible with cos4φ terms but not with cos2φ 3 Anisotropy and Crystal Field 113 or cos6φ terms. Up to the sixth order, the anisotropy of magnets with a tetragonal crystal structure is described by Ea = K1 sin2 θ + K2 sin4 θ + K2 sin4 θ cos 4φ + K3 sin6 θ + K3 sin6 θ cos 4φ (5) V Without further modification, this equation can be used for the space groups C4v , D4 , D4h , and D2d . In particular, all tetragonal compounds listed in Table 1 belong to the highly symmetric point group D4h . The point groups C4 , C4h , and S4 have a lower symmetry and require consideration of fourth-order angular shifts φ o . The anisotropy of orthorhombic crystals differs from Eq. (5) by additional second-order terms, similar to the K2 term in Eq. (3). The corresponding anisotropy energy expression for trigonal symmetry is Ea 2 4 3 V = K1 sin θ + K2 sin θ + K2 sin θ cos θ cos (3 φ) 6 6 + K3 sin φ + K3 sin θ cos (6φ) + K3 sin3 θ cos3 θ cos (3 φ) (6) Without modification, this equation can be used for the trigonal point groups C3v , D3 , and D3d as well as for hexagonal crystals, which can be considered as degenerate trigonal crystals (Fig. 2). Hexagonal point symmetry is ensured by putting K2 = K3 = 0, and no further modification is necessary for the point groups C6v , D6 , D6h , and D3h . The lower-symmetry point groups C3 , C3h , and S3 (trigonal) and C6 , C6h , C3h , and S6 (hexagonal) require the consideration of an angular shift φ o in each φ-dependent term. Note that the relationship between trigonal, rhombohedral, and hexagonal crystals is complicated. The term rhombohedral denotes the translational symmetry (Bravais lattice), whereas the closely related term trigonal refers to the point symmetry. Some trigonal crystals have hexagonal rather than rhombohedral translation symmetry. The trigonal space groups whose names begin with P (for primitive) are hexagonal, whereas those starting with R are rhombohedral. For example, Table 1 shows that α-Sm and Sm2 Co17 belong to the space group R3m and are both trigonal and rhombohedral. Translationally, the difference between hexagonal (P) and rhombohedral (R) is similar to the difference between primitive (P) and body-centered (I) cubic crystals, the rhombohedral cell having two extra lattice points. Equations (5) and (6) also describe cubic crystals, which can be considered as degenerate tetragonal or trigonal crystals (Fig. 2). Stretching a cubic crystal along the [001]-axis yields a tetragonal crystal, whereas stretching it along the [111] cube diagonal yields a rhombohedral crystal. The tetragonal symmetry axis (θ = 0) is therefore parallel to the cubic [001] direction, and the anisotropy constants obey K1 = K1 c , K2 = − 7K1 c /8 + K2 c /8, K2 = – K1 c /8 – K2 c /8, K3 = – K2 c /8, and K3 = K2 c /8. In the trigonal case, θ = 0 refers to the [111] direction, and the √ in lowestorder anisotropy constants are K1 = − 3K1 c /2, K2 = 7K1 c /12, and K2 = 2K1 c /3. 114 R. Skomski et al. Higher-Order Anisotropy Effects Equations (5) and (6) are relatively easy to use in experimental magnetism and theoretical micromagnetism. However, unlike spherical harmonics, the energy terms in these equations are nonorthogonal and mix anisotropy contributions of different orders 2n. For example, uniaxial anisotropy, Eq. (1), has the following presentation in terms of spherical harmonics: Ea V 0 0 = κ22 3cos2 θ − 1 + κ84 35 cos4 θ − 30cos2 θ + 3 0 6 231 cos6 θ − 315 cos4 θ − 105 cos2 θ − 5 + κ16 (7) Comparison of Eqs. (1) and (7) shows that K1 contains not only second-order (κ 2 0 ) but also fourth-order (κ 4 0 ) and sixth-order (κ 6 0 ) contributions. In more detail, K1 = – 3κ 2 0 /2 – 5κ 4 0 – 21κ 6 0 /2, K2 = 35κ 4 0 /8 + 189κ 6 0 /8, and K3 = −231κ 6 0 /16. In many cases, the only important anisotropy contribution is K1 = −3κ 2 0 /2, but in some cases κ 2 0 = 0 and K1 are dominated by fourth-order terms. An important example is Nd2 Fe14 B (tetragonal) in a narrow temperature range below room temperature, where κ 4 0 causes the sign of K1 to change (Fig. 14(d) in Sect. “Temperature Dependence” and Ref. 7). Anisotropy contributions of the same order tend to have similar magnitudes, which is important for understanding experimental data. For example, the two uniaxial anisotropy constants K1 and K2 provide a consistent fourth-order description of hexagonal crystals but not of tetragonal crystals, because the non-uniaxial K2 term in Eq. (5) is also of the fourth order. For second-order uniaxial anisotropies, see Sect. “Lowest-Order Anisotropies”. Higher-order anisotropy constants may have drastic effects if K1 ≈ 0 by coincidence, for example, due to competing sublattice contributions. For example, uniaxial anisotropy with K1 < 0 and K2 > − K1 /2 yields easy-cone magnetism, where the negative K1 makes the c-axis an unstable magnetization direction but the positive K2 prevents the magnetization from reaching the basal plane (a-b-plane). In this regime, the preferred magnetization direction lies on a cone around the c-axis, described by the angle θ c = arcsin (|K1 |/2K2 ). The temperature dependences of K1 and K2 are generally very different; K2 usually negligible at high temperatures. As a consequence, the preferential magnetization direction may change as a function of temperature, which is known as a spin-reorientation transition. A similar film thickness-dependent transition is observed in films where surface and bulk anisotropy contributions compete. The ratio K1 /μo Ms has the dimension of a magnetic field, which makes it possible to compare anisotropies with applied magnetic fields and coercivities. It is customary to define the corresponding anisotropy field of K1 -only uniaxial magnets as Ha = 2K1 μo Ms (8) 3 Anisotropy and Crystal Field Table 2 First- and second-order anisotropy constants at room temperature [9–11] 115 Substance Fe Ni Co Fe3 O4 Nd2 Fe14 B Sm2 Fe17 N3 Sm2 Fe17 C3 YCo5 Y2 Co17 Tm2 Co17 Sm2 Fe14 B K1 (MJ/m3 ) 0.048 −0.005 0.53 −0.011 4.9 8.6 7.4 5.8 4.0 1.6 −12.0 K2 (MJ/m3 ) 0.015 0.005 0 0.028 0.65 1.46 0.74 −0.3 0.3 0.2 −0.29 Structure Cubic Cubic Hexagonal Cubic Tetragonal Rhombohedral Rhombohedral Hexagonal Hexagonal Hexagonal Tetragonal The anisotropy field is defined in a formal way and does not actually exist inside a magnet; it is equal to the external field that creates a certain effect on the magnet. Subject to shape anisotropy corrections (Sect. “Magnetostatic Anisotropy”), the anisotropy field establishes an upper limit to the coercivity Hc . In practice, Hc Ha , which is known as Brown’s paradox. An approximate relation is Hc = α Ha , where α 1 is the Kronmüller factor [8, 9]. The inclusion of higher-order anisotropies gives rise to different nonequivalent anisotropy field definitions. For example, using Eq. (1) and comparing the energies for θ = 0 and θ = 90◦ lead to Ha = 2(K1 + K2 + K3 )/µo Ms . The initial slope of the perpendicular magnetization curves yields the same Ha , whereas the nucleation field of uniaxial magnets is not affected by K2 and K3 , so that Eq. (8) remains valid for uniaxial magnets of arbitrary order. In cubic magnets, the anisotropy fields for irontype anisotropy (K1 > 0) are described by Eq. (8), whereas nickel-type anisotropy (K1 < 0) yields Ha = − 4 K1 /3μo Ms (Table 2). Anisotropy Measurements Sucksmith-Thompson method. The experimental determination of magnetic anisotropy is easiest if single crystals or c-axis-aligned single-crystalline powders or thin films are available. The Sucksmith-Thompson method uses a magnetic field H perpendicular to the c-axis and measures the magnetization M in the field direction [12]. Starting from Eq. (1), ignoring K3 and adding the Zeeman energy yield the energy density η(M/Ms ): η = K1 M2 Ms 2 + K2 M4 Ms 4 − μo MH (9) where M = Ms sinθ . Minimizing the energy, ∂η/∂M = 0, and dividing the result by μo M yields 116 R. Skomski et al. 2 K1 H 4 K2 = + M2 M μo Ms 2 μo Ms 4 (10) Plotting H/M as a function of M2 yields K1 and K2 from the intercept and slope of the straight line, respectively. Approach to saturation. Samples are often polycrystalline. In the ideal case of noninteracting grains with second-order uniaxial anisotropy, the corresponding random-anisotropy problem can be solved explicitly. The approach to saturation obeys M(H ) = Ms Ha 2 1− 15H 2 (11) In practice, this method requires the fitting of the three parameters: Ms , the sought-for Ha = 2 K1 /μo Ms , and a high-field susceptibility that must be used to ensure that ∂M/∂H = 0 for H = ∞. Note that Eq. (11) does not predict the sign of K1 , because both easy-axis and easy-plane ensembles yield the same asymptotic behavior. Note that Eq. (11) is essentially a random-anisotropy relation (Sect. “Random Anisotropy in Nanoparticles, Amorphous, and Granular Magnets”). Torque magnetometry. A single-crystalline magnetic sample experiences a mechanical torque –∂Ea /∂α, where α is a magnetization angle relative to the crystal axes. The angle α is varied with the help of a rotating magnetic field, and the torque is monitored as a function of the field direction, for example, by measuring the twisting angle of a filament to which the sample is attached. The interpretation of the torque depends on the crystalline orientation of the sample, but if the torque axis is parallel to a magnetocrystalline symmetry axis, the corresponding anisotropy constants are readily obtained as Fourier components of the torque curves [13]. Magnetic circular dichroism. Single-ion anisotropy is closely related to the orbital moment and approximately proportional to the latter in iron-series transitionmetal magnets (Sect. “Perturbation Theory”). A direct way to probe orbital (and spin) moments on an atomic scale is X-ray magnetic circular dichroism (XMCD). Circular dichroism means that circularly polarized photons pass through the sample and that the absorption is different for left- and right-polarized light [14–16]. This is because the orbital moment reflects atomic-scale circular currents that interact with light. Furthermore, due to spin-orbit coupling, the light also interacts with spin, so that XMCD can also be used to simultaneously measure the spin moment. Crystal-Field Theory Electrons in solids occupy states reminiscent of atomic orbitals, even in metals. This applies, in particular, to the partially filled inner shells of transition-metal elements, such as the iron-series 3d shells and rare-earth 4f shells. The electrons in 3 Anisotropy and Crystal Field 117 Fig. 4 Angular dependence of 3d wave orbitals: (a) real eigenfunctions and (b) top view on a mixture of states constructed from m > ∼ exp (ιmφ) with m = ±2. Red and yellow areas in (a) indicate regions of positive and negative wave functions ψ, respectively, and the darkness in (b) indicates the electron density ψ*ψ. The wave functions shown in this figure are all eigenfunctions of the free atoms, but in solids (b), the crystal field, symbolized by ligands (black dots), favors real wave functions (top), whereas spin-orbit coupling favors complex wave functions | ± m > (bottom). Details of this “quenching” behavior will be discussed in Sect. “Spin-Orbit Coupling and Quenching” the inner shells, which often carry a magnetic moment, interact with the crystalline environment. The crystal-field (CF) interaction of the Sm3+ ion in Fig. 4 is one example, but a similar picture is realized in 3d ions, especially in oxides. Itinerant magnets, such as 3d metals, require additional considerations, because their electronic structure is largely determined by interatomic hopping (band formation). Crystal-field theory had its origin in the study of transition-metal complexes in the last decade of the nineteenth century [17]. An example was the distinction between violet and green [Co(NH3 )6 ]3+ Cl3 3− , which indicates energy-level differences of stereochemical origin. The quantitative crystal-field theory dates back to Bethe [18], who treated the atoms as electrostatic point charges. Since then, the crystal-field theory has been extended to include quantum mechanical bonding effects in a generalization are known as ligand-field (LF) theory [19]. As emphasized by Ballhausen [17], the latter is quantitatively superior to Bethe’s CF theory but leaves the main conclusions of the latter unchanged. In practice, the terms are often used interchangeably: the atoms surrounding a magnetic ion are called ligands in both complexes and solids, and the term ligand field is sometimes used. Physically, both electrostatic and hybridization effects contribute to the crystal field (ligand field), even in oxides. The focus of this section is on the traditional electrostatic crystal-field theory, but some interatomic 118 R. Skomski et al. hybridization effects will be discussed in the context of itinerant anisotropy (Sect. “Transition-Metal Anisotropy”). One-Electron Crystal-Field Splitting The wave functions and charge distributions of the electrons are obtained from the Schrödinger equation. Hydrogen-like 3d wave functions are listed in Appendix C. The angular parts of the wave functions follow from the spherical character of the intra-atomic potential and are the same for Fe-series 3d, Pd-series 4d, and Pt-series 5d electrons. However, the radial parts differ for the three series, and they also depend on non-hydrogen-like details of the atomic potentials. Figure 4 shows the angular distribution of the five 3d orbitals ψ μ (r). In a free atom, the five orbitals are degenerate, but in solids and molecules, they undergo crystal-field interactions described by the Hamiltonian: HCF = V (r) n(r) dV (12) where V (r) is the crystal or ligand-field potential and n(r) = ψ ∗ (r)ψ(r) refers to the d or f orbital(s) in question. To understand crystal-field effects, it is necessary to consider the shape of the orbitals. Atomic wave functions and charge distributions such as those shown in Figs. 4 and 1, respectively, have characteristic prolate, spherical, or oblate shapes. The larger the magnitude of the quantum number m = lz , the more oblate or flatter the orbitals, as we can in Fig. 4. This is because large orbital moments, m = ± 2 in Fig. 4 , correspond to a pronounced circular electron motion in the plane perpendicular to the quantization axis (z-axis). By contrast, the prolate |z2 > orbital, which has m = 0 has its electron cloud close to the z-axis. In a crystalline environment, the different orbital shapes correspond to different electrostatic interactions. Crystalfield charges are negative [20], so that the interaction between the 3d or 4f electronic charge clouds and those of the surrounding atoms is repulsive. As a consequence, the prolate |z2 > orbital prefers to point in interstitial directions between the atomic neighbors, rather than towards them. The opposite is true for the oblate orbitals with m = ± 2. The electrostatic repulsion between the 3d electrons and those of the neighboring atoms removes the degeneracy of the five 3d levels and yields the famous eg -t2g splitting in an environment with cubic symmetry. Figure 5(a) shows the |z2 > orbitals in a cubal environment where the central atom is coordinated by 8 neighbors. The |z2 > orbital points in an electrostatically favorable direction and has a very low energy. The charge distribution of the |x2 -y2 > orbital also points in directions away from the neighboring atoms or ligands, and it can be shown that the |x2 -y2 > and |z2 > have the same energy, forming a so-called eg doublet. The |xy>, |xz>, |yz> orbitals are equivalent by symmetry and form a t2g triplet. The charge 3 Anisotropy and Crystal Field 119 Fig. 5 A 3d orbital (z2 ) in some crystalline environments: (a) cubal, (b) octahedral, (c) tetrahedral, and (d) tetragonally distorted cubal. Note that (a), (b), and (c) have cubic symmetry, whereas (d) is tetragonal distributions of the triplet orbitals are closer to the ligands, so that the triplet energy is higher than the doublet energy. The opposite splitting is realized in an octahedral environment, Fig. 5(b), where the central atoms are coordinated by six ligand atoms. In this environment, the |x2 y2 > and |z2 > orbitals point directly towards the neighboring atoms, whereas the |xy>, |xz>, |yz> orbitals point in interstitial directions. The tetrahedral environment (c) has no inversion symmetry but is otherwise very similar to the cubal environment. Basically, the cubic e-t2 crystal-field splitting is reduced by a factor 2, because there are only four neighbors. Symmetries lower than cubic partially or completely remove the eg and t2g degeneracies. Figure 5(d) illustrates this for a tetragonally distorted cubal environment. Compared to (a), the ligands move towards the basal plane, which lowers the energy of the |z2 > orbital but raises that of the |x2 y2 > orbital. As a consequence, these states no longer form a doublet. Similarly, the |xz> and |yz> orbitals become somewhat more favorable compared to the |xy > orbital, because their charge distribution has a substantial out-of-plane component. This splits the t2g triplet, but |xz> and |yz> remain degenerate, because the x and y directions are equivalent in a tetragonal crystal. Figure 6 summarizes the eg -t2g splitting and the evolution of the levels due to a symmetry-breaking tetragonal distortion. It is important to note that halffilled (and full) 3d shells have spherical charge distributions and do not interact with anisotropic crystal fields. Equivalently, the CF interaction leaves the center of gravity of the 3d levels unchanged. This can be used to gain some quantitative information about the level splitting. For example, the eg -t2g splitting, also known 120 R. Skomski et al. Fig. 6 Crystal-field splitting of 3delectrons in cubic and tetragonal environments as 10Dq, consists of an energy shift of +6Dq for the doublet and a –4Dq shift for the triplet. Table 3 lists the crystal-field splittings for the most symmetric point groups in each crystal system and for axial symmetry. The levels are described by Mullikan symmetry labels, using t and e for triplets and doublets, respectively [24]. Singlets are denoted by a or b, depending on whether the reference axis is an n-fold rotation axis (a) or not (b). The subscripts 1, 2, and 3 indicate C2 symmetries around crystal axes, and primes ( ) and double-primes ( ) refer to horizontal mirror symmetry and antisymmetry, respectively. The subscript g (German gerade “even”) denotes inversion symmetry, which exists for the cubal coordination, Fig. 5(a), but not for the otherwise very similar tetrahedral coordination, Fig. 5(c). However, the inversion symmetry of the 3d wave functions means that there are no levels with subscript u (German ungerade “odd”), so that no confusion arises by dropping the subscript g [25]. In each crystal system, the complexity of the symmetry labels decreases with decreasing symmetry. For example, the eg -t2g splitting is limited to the highly symmetric point group Oh : the respective cubic compounds FeS2 (space group Th ), MnNiSb (space group Td ), and FeSi (space group T) have eg -tg , e-t2 , and e-t splittings. 3 Anisotropy and Crystal Field 121 Table 3 Crystal-field splittings of 3d electrons. The colors indicate the crystal-field multiplet structure: one doublet and one triplet (red), one singlet and two doublets (yellow), three singlets and one doublet (green), and five singlets (blue). The listed point groups are the most symmetric ones in each crystal system – less symmetric point groups yield modified symbols, such as missing subscripts g. In linear molecules (point groups D∞h and C∞v ), the multiplets a1 , e1 , and e2 are also known as + , , and , respectively 122 R. Skomski et al. Crystal-Field Expansion It is convenient to expand the crystal-field potential V (r) into spherical harmonics Yl m (θ , φ). The corresponding expansion coefficients Al m are known as crystalfield parameters and play an important role in crystal-field theory and magnetism. Treating the ligands (i = 1 ... N) as electrostatic point charges [18] located at Ri yields the crystal-field potential energy: V (r) = − e 4πεo N i=1 qi | Ri − r | (13) This sum is easily converted into a sum of spherical harmonics by exploiting the identity: 4 π rl l (2 l + 1) R l+1 1 = |R−r | |m|<l Y1 m∗ (Θ, Φ) Y1 m (θ, φ) (14) so long as R > r. Strictly speaking, the l-summation extends from zero to infinity, but the symmetry of n(r) in Eq. (12) means that the only relevant terms are l = 2, 4 (d-orbitals) and l = 2, 4, 6 (f -orbitals). Inserting Eq. (14) into Eq. (13) and summing over all ligands leads to the cancellation of Yl m (θ , φ) terms that are incompatible with the symmetry of the crystal. For example, cubic crystals have V (r) = 20A4 0 x 4 + y 4 + z4 − 3r 4 /5 (15a) where the dimensionless crystal-field parameter 4πεo R5 A4 0 /qe is equal to −7/16, 7/18, and 7/36 for the octahedral, cubal, and tetrahedral ligands of Fig. 5, respectively. The r4 term in Eq. (15a) is isotropic and not necessary for the description of magnetic anisotropy, but it ensures that the center of gravity of the energy is conserved during crystal-field splitting. Since x2 + y2 + z2 = r2 , Eq. (15a) is equivalent to V (r) = −40A4 0 x 2 y 2 + y 2 z2 + z2 x 2 − r 4 /5 (15b) and to any linear combination of Eqs. (15a) and (15b). The structure of this equation mirrors that of Eq. (4) for the anisotropy of cubic magnets. A third version of Eq. (15) will be discussed in the context of operator equivalents. Uniaxial crystal fields are described by V (r) = A2 0 3 z2 − r 2 + A4 0 35 z4 − 35 z2r 2 + 3 r 4 +A6 0 231 z6 − 315 z4 r 2 + 105 z2 r 4 − 5 r 4 (16) 3 Anisotropy and Crystal Field Table 4 Crystal-field parameters for some noncubic rare-earth transition-metal intermetallics [10] 123 Compound R2 Fe14 B R2 Fe17 R2 Fe17 N3 A2 0 K/ao 2 300 34 −358 A4 0 K/ao 4 −13 −3 −39 From Eq. (14) we see that the small parameter in the ligand-field expansion is r/R, that is, the ratio of d-shell radius to interatomic distance. For this reason, A4 0 is typically smaller than A2 0 by a factor of order (r/R)2 , or about one order of magnitude. Exceptions are, for example, weakly distorted cubic structures. Another way of interpreting crystal fields is to expand V (r) into a Taylor series with respect to x, y, and z. The nonzero expansion coefficients are the crystal-field parameters Al m , where l denotes the l-th spatial derivative of V (r). In particular, A2 0 ∼ ∂ 2 V (r) /∂z2 or, in terms of the electric field, A2 0 ∼ ∂Ez /∂z. This means that A2 0 is essentially an electric field gradient . The point-charge model accurately describes the symmetry of the crystal field [20] and yields semiquantitatively correct numerical predictions for a variety of systems. It was originally developed for insulators but also approximates rare-earth ions in metals where the electrostatic interaction is screened by conduction electrons [21]. This surprisingly broad applicability has its origin in the superposition principle of crystal-field interactions, which states that the effects of different ligand atoms are additive in very good approximation [20]. Experimentally, crystal-field effects are measured most directly by spectroscopy, for example, optical spectroscopy or inelastic neutron scattering, but there are also indirect measurements, such as rareearth anisotropy measurements (Table 4). Many-Electron Ions A fixed number n of inner-shell electrons of an ion is called a configuration, such as 3dn and 4fn . In practice, the configuration corresponds to the ions’ charge state. All rare-earth elements form tripositive ions, R3+ , as exemplified by Sm3+ (4f5 ) and Dy3+ (4f9 ). Some form R2+ shells such as europium in EuO or R4+ in mixedvalence and heavy-fermion compounds such as CeAl3 [22, 23]. Transition-metal ions show a greater variety, most commonly T2+ , T3+ , and T4+ , where the ionic charge is determined by chemical considerations. For example, Fe3 O4 contains both Fe2+ (3d6 ) and Fe3+ (3d5 ) ions to charge-compensate the O2− anions. The n electrons are distributed over the available 2 × (2 l + 1) one-electron states and labeled by sz = ±1/2 and lz = −l, ..., l – 1, l. The relationship between these electrons is largely governed by the Pauli principle, by Hund’s-rules for electronelectron interactions, and by spin-orbit coupling. The Pauli principle means that each real-space d or f orbital can accommodate at most one↑ and one↓ electron. Subject to the Pauli principle, there are several ways to place n electrons onto the 10 124 R. Skomski et al. one-electron 3d levels, each combination corresponding to a many-electron state. These can be divided into terms characterized by well-defined total spin S = i si and orbital quantum numbers L = i li (i = 1 ... n), with each term containing (2 S + 1) (2 L + 1) states. The terms are usually denoted by 2S + 1 L, where 2S + 1 is the spin multiplicity and L is denoted by as S (L = 0), P (L = 1), D (L = 2), F (L = 3), G (L = 4), H (L = 5), and I (L = 6). More generally, it is common to use capital letters for ionic properties, and S, P, D, F are analogous to one-electron states s, p, d, and f. An example is the 3d2 configuration, realized, for example, in Ti2+ . The first electron can occupy any of the 2 × 5 states, leaving nine states for the second electron. This yields 90/2 = 45 permutations, each corresponding to a two-electron state. The highest L is achieved by placing two electrons in the lz = 2 state, (↑↓, −, −, −, −). This yields S = 0 and L = 4, that is, a1 G term containing 9 states. The wave function (↑, ↑, −, −, −) has S = 1 and L = 3 and therefore belongs to a3 F term, which contains 21 states. The other 3d2 terms are 1 D (5 states) and 3 P (9 states), and 1 S (1 state). Similar term analyses can be made for all configurations [17, 24, 26] but will not be discussed here, because in magnetism our main interest is the ground-state term. A trivial case is 3d1 , which corresponds to a single term 2 D. As far as symmetry is concerned, the crystal-field splittings of ions are equal to those of the one-electron states [17]. For example, the octahedral splitting eg -t2g for a single d electron corresponds to Eg -T2g in D ions. Table 5 shows basic the CF splittings of many-electron terms in cubic, tetragonal, and trigonal environments. The subscript-free symmetry labels A (singlet), B (singlet), E (doublet), and T (triplet) are of the lowest-symmetry type, and the numbers indicate two or more distinct levels. Note that most point groups have subscripts (1, 2, g, u) that are important in spectroscopy but not for the explanation of magnetic anisotropy. Without interactions, the terms of a configuration would be degenerate. In reality, the degeneracy is removed by the electron-electron interaction: 1 U= 4πεo ρ (r) ρ r dV dV | r − r | (17) Table 5 Basic branching table for crystal-field splittings of many-electron ions. Both groundstate and excited terms are included, and the table is not restricted to d electrons. For example, the free-ion triplet of a single p electron (P) remains unaffected by a cubic crystal field but exhibits a singlet-doublet splitting in tetragonal and trigonal crystals Term S P D F G H Cubic CF A T E+T A+2T A+E+2T E+3T Tetragonal CF A A+E A + 2B + E A + 2B + 2E 3A + 2B + 2E 3A + 2B + 3E Trigonal CF A A+E A + 2E 3A + 2E 3A + 3E 3A + 4E 3 Anisotropy and Crystal Field 125 where ρ(r) is the electron charge density. The corresponding term splittings are large, 1.8 eV for Co2+ , and often dominate the behavior of the ion. The term energies E(L, S) can be calculated in a straightforward way, by applying the lowestorder perturbation theory to Eq. (17), E(L, S)=< (L,S) | U | (L, S) >[17, 27]. However, the ground-state term is more easily obtained from Hund’s rules. The first rule states that the total spin S= i si is maximized. In the above 3d2 example, there are two terms with maximum S, namely, 3 F and 3 P, both having S=1. Hund’s second rule acts as a tiebreaker, by favoring large L= i li . Since F and P mean L=3 and L=1, respectively, 3 F is the ground-state term of the 3d2 configuration. Table 6 shows some basic properties of 3d ions; 4f ions will be discussed in the context of rare-earth anisotropy (Sect. Crystal-Field Theory). Hund’s first rule yields another simplification: in the ground-state term of the 3d5 configuration, there are five ↑ electrons which occupy the five available orbitals, lz = −2, ... +1, +2. This yields L = lz = 0, meaning that empty, half-filled, and completely filled 3d shells are all S-type ions. This principle carries over to magnetic anisotropy: from Table 5 we see that S states do not undergo crystal-field splitting but remain in their highly symmetric degenerate A state. The corresponding charge distribution is spherical, and the ion does not contribute to the magnetocrystalline anisotropy (except via admixture with a higher excited state). Figure 7 shows the level splittings of the ground-state terms of the 3d ions in an octahedral crystal field. Note the half-shell symmetry of the splittings: aside from the sign, there are only two nontrivial cases, namely, one electron or hole (d1 , d4 , d6 , d9 ) and two electrons or holes (d2 , d3 , d7 , d8 ). The crystal-field interaction is normally weaker than the intra-atomic exchange. However, very strong crystal fields may negate Hund’s rules and cause a transition to a low-spin state. For example, octahedrally coordinated Fe2+ has the configuration 3d6 , and, according to Fig. 7, a T2g ground state, that is, t2g (↑↑↑↓)-eg (↑↑). The two ↑ electrons in the eg -doublet experience a competition between electronelectron interaction, which favors parallel spin alignment, and the CF, which favors t2g occupancy. In very strong crystal fields, the electronic structure becomes t2g (↑↑↑↓↓↓)-eg (empty), and the ion loses its magnetic moment. This is an example of a high-spin low-spin transition. Aside from d6 , the three ions d4 , d5 , and d7 Table 6 Electronic configurations of 3d ions. The listed terms are the ground-state terms Ion 3d1 3d2 3d3 3d4 3d5 3d6 3d7 3d8 3d9 Example Ti3+ , V4+ Ti2+ , V3+ V2+ , Cr3+ Cr2+ , Mn3+ Mn2+ , Fe3+ Fe2+ , Co3+ Co2+ , Ni3+ Ni2+ , Pd2+ Cu2+ Term 2D 3F 4F 5D 6S 5D 4F 3F 2D L 2 3 3 2 0 2 3 3 2 S 1/2 1 3/2 2 5/2 2 3/2 1 1/2 126 R. Skomski et al. Fig. 7 Crystal-field splittings of the ground-state terms of 3d ions in a weak octahedral crystal field. The energy unit Dq is one tenth of the eg -t2g splitting undergo a high spin low spin in strong octahedral crystal fields, leading to spin moments of 2 μB (d4 ) and 1 μB (d5 , d7 ). It is instructive to plot the term energies as a function of the crystal field, using the eg -t2g splitting 10Dq to quantify the crystal field in an Orgel diagram. An extension of the Orgel diagram is the Tanabe-Sugano diagram , where both the crystal field (Dq) and the term energies are divided by the Racah parameter B [24]. This parameter links Hund’s second rule, namely, the maximization of L, to the underlying intra-atomic electron-electron interaction and satisfies E(3 P) – E(3 F) = 15B. The ground-state energy is used as the energy zero, which helps to visualize transitions. Figure 8 shows a Tanabe-Sugano diagram where the ground-state term changes from high spin 5 T2g (blue line) to low spin 1 A1g (red line). Any splitting ± E of a degenerate state lowers the energy by about E if the level is only partially occupied. For example, the tetragonal lattice distortion of Fig. 6 means that the eg doublet splits into a low-lying a1g state and a b1g state and a single electron in the eg doublet moves to the a1g level. The resulting crystalfield energy gain competes against the mechanical energy necessary to tetragonally distort the crystal. However, the former is linear in strain ε, whereas the latter is quadratic, so that the CF should always create a small distortion. This is known as the Jahn-Teller effect. 3 Anisotropy and Crystal Field 127 Fig. 8 Tanabe-Sugano diagram for a 3d6 ion [24]. The energy unit Dq is one tenth of the eg -t2g splitting and B is the Racah parameter [28]. The vertical line indicates a transition from a high-spin state (blue) to a low-spin state (red) Spin-Orbit Coupling and Quenching The interatomic interactions (U ) remove the degeneracy between different terms and create ions with well-defined L and S. However, L and S do not interact and can point in any direction. In reality, they are subject to relativistic spinorbit coupling, which causes the terms to split into multiplets of well-defined total angular momentum J, denoted by 2S + 1 LJ . Figure 9 illustrates the origin of spinorbit coupling: the orbital motion of the electron (L) creates a magnetic field that couples to the electron’s own spin (S). This coupling is important for both isotropic magnetism (moment formation, and exchange) and magnetic anisotropy. The key role of spin-orbit coupling in the explanation of magnetic anisotropy was first recognized and exploited by Bloch and Gentile in 1931 [1]. The quasiclassical model of Fig. 9 correctly reproduces the order of magnitude of the spin-orbit coupling, aside from a factor 1/2 (Thomas correction). The spinorbit coupling may be derived directly from the relativistic Dirac wave equation. The coupling is a fourth-order term in the Pauli expansion of the relativistic energy, similar to the v4 term in the equation: me c 2 1+ v2 1 1 = me c 2 + me v 2 − me v 4 2 2 8 c (18) 128 R. Skomski et al. Fig. 9 Spin-orbit coupling in a free ion (schematic). The orbiting spin acts like a current loop and creates a magnetic field that acts on the spin. The nucleus does not actively participate in the spinorbit coupling but merely serves to curve the trajectory of the electron: a circular racetrack would do equally well Electromagnetic effects are added by including scalar and vector potentials [10, 29]. The result of the calculation is the SOC energy [29]: Hso = 3 s · ∇V × k 2 2 2me c (19) This equation shows that the spin-orbit coupling favors a spin direction perpendicular to both potential gradient and direction of motion. For example, electrons in thin films experience a Rashba effect [30], and there is a small interstitial contribution to the magnetocrystalline anisotropy [31]. The Rashba effect means that electrons of wave vector k move in the film plane and experience a potential gradient perpendicular to the film, which naturally occurs due to broken inversion symmetry at thin-film surfaces and interfaces. According to Eq. (19), the spin then prefers to lie in the plane, in one direction perpendicular to k. In the opposite inplane spin direction, the energy is enhanced, which is referred to as Rashba splitting of the electron levels. The potential gradient is most pronounced near the atomic nuclei, and for hydrogen-like 1/r potentials Hso = Ze2 2 l·s 2 2 2me c 4πεo r 3 (20) Using Appendix C, we can evaluate the average <1/r3 > and obtain Hso = ξ l · s. Here ξ is the spin-orbit coupling constant: ξ= 2 Z 4 e2 1 2 3 2 2me c ao 4πεo n3 l l + 1/ (l + 1) 2 (21) 3 Anisotropy and Crystal Field 129 It is instructive to discuss relativistic phenomena in terms of Sommerfeld’s finestructure constant, α = e2 /4πεo c ≈ 1/137. Electrons in atoms and solids have velocities of the order of v = αc, so that from Eq. (18): me c 2 1 + v2 1 1 = me c 2 + me α 2 c 2 − me α 4 c 4 2 8 c2 (22) Similarly, Eq. (21) becomes ξ= 1 me 4 4 2 Z α c + 1 2 3 n l l + /2 (l + 1) (23) This equation captures the relativistic nature of spin-orbit coupling and magnetic anisotropy. In terms of powers of α, ξ is a small relativistic correction, similar to the v4 term in Eqs. (18) and (22), but Z, which is largest for inner-shell electrons in heavy elements, greatly enhances the effect in partially filled shells. Tables 7 and 8 show values of spin-orbit coupling constants ξ for 3d, 4d, 5d, 4f, and 5f elements, obtained from Hartree-Fock calculations [32, 33]. A comparison of experimental data and theoretical predictions indicates that these tables have an accuracy of the order of 10% [32–34]. Furthermore, ξ somewhat increases with ionicity [34]: going from T2+ to T3+ and T+ , respectively, changes the SOC constant of late 3d elements by about ±10%. There are two limits for many-electron spin-orbit coupling. Russell-Saunders coupling means that the ion has well-defined values of L = i li and S = i si . They are good quantum numbers, and the SOC is a weak perturbation. This limit is realized when the intra-atomic interactions are stronger than ξ . In the opposite limit of j-j coupling, the i-th electron first experiences a one-electron SOC so that J = i (li + si ). Most solid-state magnetism involves Russell-Saunders coupling, but j-j coupling is important in two limits: (a) low-lying levels of very heavy elements, such actinides, and (b) excited levels of most elements, except very light ones. In (a), the j-j coupling is imposed by the large λ in heavy atoms, whereas in (b), it reflects the increased electron separation in excited states. In many cases, Russell-Saunders Table 7 Spin-orbit coupling constants for electrons in the partially filled dipositive 3d, 4d, and 5d transition-metal ions. ξ of the inner 1s, 2s, and 2p electrons in heavy elements is much stronger than the values in this table, but closed shells do not exhibit a net spin-orbit coupling d1 d2 d3 d4 d5 d6 d7 d8 d9 Sc Ti V Cr Mn Fe Co Ni Cu ξ (meV) 10 15 22 31 41 53 68 86 106 Y Zr Nb Mo Tc Ru Rh Pd Ag ξ (meV) 32 48 65 84 106 129 156 186 221 La Hf Ta W Re Os Ir Pt Au ξ (meV) 69 196 244 302 360 419 485 556 633 130 Table 8 Spin-orbit coupling constants ξ for tripositive 4f and 5f transition-metal ions [32, 33] R. Skomski et al. f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 f11 f12 f13 f14 Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu ξ (meV) 85 102 120 139 159 182 205 230 257 286 318 352 422 143 Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lw ξ (meV) 197 234 271 308 348 390 432 478 525 576 629 686 – – coupling yields the correct multiplet structure but j-j coupling causes quantitative deviations in the level spacing. For example, the j-j coupling effect on the 3 P2 -3 P0 splitting is negligible in C, about 20% in Si, and dominates in Ge, Sn, and Pb [26]. Magnetic anisotropy reflects low-lying excitations, and Russell-Saunders coupling therefore applies to both transition metals and rare earths. The Russell-Saunders coupling establishes the vector model, where J = L + S. Using the Hund’s-rules ground-state terms to evaluate i ξ i li · si yields the ionic spin-orbit coupling Λ L·S, where Λ = ± ξ /2S for less and more than half-filled shells, respectively [17]. The change of sign at half filling yields Hund’s third rule: for the early elements in each series, J = L – S, and for the late elements, J = L + S. Each multiplet has 2 J + 1 Zeeman-like intramultiplet levels, Jz = − J, ..., (J – 1), J, and the degeneracy of these levels is removed by a magnetic field or by the crystal field. Due to the g-factor of the electron, a magnetic field couples to (L + 2S) rather than to (L + S). This makes it necessary to project L + 2S onto J, so that (L + 2S) · J = g J2 . The Landé g-factor of the ion g = 1 for pure orbital magnetism (L = J) and g = 2 for pure spin magnetism (L = 0). For arbitrary L and S, less and more than half-filled shells exhibit g = 1 – S/(J + 1) and g = 1 + S/J, respectively. The exchange between magnetic ions involves spin only, which mandates the use of the projection S · J = (g − 1) J2 and yields de Gennes factor G = (g – 1)2 J(J + 1). This makes it possible to write the exchange interaction as J = G Jo , where Jo is a J-independent Heisenberg exchange constant. Spectroscopic and magnetic measurements indicate that Hund’s rules are well satisfied in rare-earth ions (Sect. “Crystal-Field Theory”), but iron-series transitionmetal ions systematically violate them, especially the third rule. For example, g ≈ 2 for iron-series atoms in almost all metallic and nonmetallic crystalline environments. In other words, the magnetic moments of Fe, Co, and Ni originate 3 Anisotropy and Crystal Field 131 Fig. 10 Quenching of the 3d orbital moment (schematic). The crystal field creates an energy landscape that inhibits the circular orbital motion of the electron and leads to the charge density of Fig. 4(b) nearly exclusively from the spin of the 3d electrons, and the atoms look as if L = 0. For example, iron has a magnetization of about 2.2 μB , but only about 5% of this moment is of orbital origin. This effect is known as orbital-moment quenching. Quenching was first recognized explicitly by van Vleck in 1937 [35]. Figure 10 illustrates the physics behind this effect, namely, the disruption of the electron’s orbital motion by the crystal field. Mathematically, the difference between quenched and unquenched wave functions is that between real and complex spherical harmonics (Appendix A). Consider the two states |x2 –y2 >∼cos(4φ) and |xy>∼ sin(4φ), which are shown in the top row of Fig. 7. Using lz = −i∂/∂φ to calculate <lz > = − i ψ* lz ψ dφ yields <lz > = 0; it is completely quenched. Pictorially, the electron “oscillates” in the valleys between the CF potential mountains, as indicated by the dashed line in Fig. 10, and these oscillations yield no net orbital motion. The respective electron densities <ψ|ψ> for |x2 –y2 > and |xy>, namely, ρ = 1 + cos(8φ) and ρ = 1–2cos(8φ), exhibit complementary minima and maxima, and the positioning of mountains decide which of the two densities yields the lower energy. Rather than asking why the orbital is quenched, we should therefore ask how an orbital moment arises in a solid. Unquenched orbitals are described by wave functions of the type exp (±imφ) = cos(mφ) ± i sin(mφ), or |±2>= |x2 –y2 >± |xy>. These functions describe an uninhibited orbital motion and yield <lz > = ± 2 in units of . However, the corresponding electron charge cloud is ringlike, ρ = const., so that the electron occupies both valley and energetically costly hill regions, rather than being confined to valleys. The competition between spin-orbit coupling (SOC) and crystal field (CF) decides whether the orbital moment is quenched. In the 4f case, the SOC is large, and the orbital motion of the electrons remains essentially unquenched by the CF, as in Fig. 9. The opposite is true for 3d electrons, where the SOC is a small perturbation to the CF, leading to nearly complete quenching. 132 R. Skomski et al. Rare-Earth Anisotropy The magnetocrystalline anisotropy of permanent-magnet materials, such as Nd2 Fe14 B and SmCo5 , largely originates from the rare-earth sublattice. K1 values are 4.9 MJ/m3 and 17.0 MJ/m3 , respectively. By comparison, bcc iron has K1 = 0.05 MJ/m3 [10]. The 4f wave functions are nearly unquenched, so that the magnetocrystalline anisotropy energy is equal to the crystal-field energy, as in Fig. 1. The basic physical picture of this single-ion anisotropy is clear, but a few questions remain; it is necessary to determine the shape of the rare-earth 4f shells or ions and to quantify the relationship between crystal-field interaction and anisotropy. Another question is the temperature dependence. Anisotropy energies per ion correspond to very low temperatures, at most a few kelvins, so the observation of anisotropy at and above room temperature must be explained (Table 9). Table 9 Anisotropy, magnetization, and Curie temperature of some rare-earth transition-metal intermetallics [9, 10, 37] Substance YCo5 SmCo5 NdCo5 Y2 Fe14 B Pr2 Fe14 B Nd2 Fe14 B Sm2 Fe14 B Dy2 Fe14 B Er2 Fe14 B Y(Co11 Ti) Sm(Fe11 Ti) Y(Fe11 Ti) Y2 Co17 Nd2 Co17 Sm2 Co17 Dy2 Co17 Er2 Co17 Y2 Fe17 Y2 Fe17 N3 Sm2 Fe17 Sm2 Fe17 N3 TbFe2 aK c 1 for TbFe2 K1 (RT)a MJ/m3 5.2 17.2 0.7 1.06 5.6 4.9 −12.0 4.5 −0.03 −0.47 4.9 0.89 −0.34 −1.1 3.3 −2.6 0.72 −0.4 −1.1 −0.8 8.9 0.013 μo Ms (RT) T 1.06 1.07 1.23 1.36 1.41 1.61 1.49 0.67 0.95 0.93 1.14 1.12 1.25 1.39 1.20 0.68 0.91 0.84 1.46 1.17 1.54 0.84 Tc K 987 1003 910 571 565 585 618 593 557 940 584 524 1167 1150 1190 1152 1186 320 694 389 749 730 Structure Hexagonal (CaCu5 ) Hexagonal (CaCu5 ) Hexagonal (CaCu5 ) Tetragonal (Nd2 Fe14 B) Tetragonal (Nd2 Fe14 B) Tetragonal (Nd2 Fe14 B) Tetragonal (Nd2 Fe14 B) Tetragonal (Nd2 Fe14 B) Tetragonal (Nd2 Fe14 B) Tetragonal (ThMn12 ) Tetragonal (ThMn12 ) Tetragonal (ThMn12 ) Hexagonal (Th2 Ni17 ) Rhombohedral (Th2 Zn17 ) Rhombohedral (Th2 Zn17 ) Hexagonal (Th2 Ni17 ) Hexagonal (Th2 Ni17 ) Hexagonal (Th2 Ni17 ) Hexagonal (Th2 Ni17 ) Rhombohedral (Th2 Zn17 ) Rhombohedral (Th2 Zn17 ) Cubic (laves) 3 Anisotropy and Crystal Field 133 Rare-Earth Ions Rare-earth atoms tend to form tripositve ions in both metals and insulators. Since spin-orbit coupling is very strong for inner-shell electrons in heavy elements, the 4f electrons experience a rigid coupling of their spin and orbital moments, with unquenched orbitals and Hund’s-rules spin-orbit coupling. Magnetic anisotropy is an intramultiplet effect, involving the 2 J + 1 magnetic quantum states Jz of the ground-state multiplet. Excited multiplets have relatively high energies, with the notable exceptions of Eu3+ and Sm3+ [33]. In the former, this energy is only about 40 meV, but the ground-state moment of Eu2+ is zero, and the element often adopts a Eu2+ configuration with half-filled shell and zero anisotropy. Otherwise, the Eu3+ ion shows strong van Vleck susceptibility: the Eu3+ moment is zero in its J = 0 ground-state multiplet, where the contributions from S = 3 and L=3 cancel, but the first-excited multiplet (7 F1 , J = 1) is only 330 K above the 7 F0 ground-state multiplet. In the case of Sm3+ , the splitting between the ground-state multiplet (6 H5/2 ) and the first-excited multiplet (6 H7/2 ) is about 100 meV (∼1000 K) [33], so that interatomic interactions and thermal excitations yield some admixture of 6 H7/2 character (J-mixing). The focus of this section is on ground-state multiplets, with a brief discussion of the excited Sm multiplet. To determine the crystal-field energy, it is first necessary to specify the shape of the 4f shells. Why is the Sm3+ ion in Fig. 1 prolate rather than oblate? Interchanging oblate and prolate shapes changes the sign of K1 and has far-reaching implications. A tentative answer is provided by the angular dependence of the (real) one-electron 4f wave functions, which are shown in Fig. 11. States with m = ±3, favored by Hund’s second rule, are prolate, whereas the m = 0 state is oblate. The strong spinorbit coupling then creates axially symmetric superpositions exp.(±mφ) from states with equal |m|, and Hund’s rules determine how the one-electron orbitals combine to yield many-electron orbitals. Like any electric charge distribution, the many-electron 4f shell can be expanded in spherical harmonics. This multipole expansion provides a successively improved description of angular features. In the zeroth order, the 4f shell is approximated by a sphere of charge Q = Qo and does not support any anisotropy. The first-order corresponds to an electric dipole moment Q1 , which is absent by wave-function symmetry. The lowest-order electric moment is the second-order quadrupole moment Q2 , which describes the prolaticity of a charge distribution. Table 10 lists some Hund’s-rules ground-state properties of the tripositive rare-earth ions, including Q2 . There is a systematic dependence of the ground-state ionic shape on the number of 4f electrons. Gd3+ has a half-filled shell and a spherical charge distribution because Hund’s rules mandate seven ↑ electrons having l = 3, 2, 1, 0, −1, −2, −3, so that L = i li = 0 (S-state ion) and Q2 = 0. The other elements follow a quarter-shell rule: the first and third quarters of the series have oblate ions, and the second and fourth quarters have prolate ions. This rule is a consequence of particle- 134 R. Skomski et al. Fig. 11 Angular dependence of 4f wave functions. Red and yellow areas indicate regions of positive and negative wave functions, respectively. As in Fig. 4, the wave functions shown here are the real ones, and m refers to the wave functions |m > ∼ exp.(imφ) from which these wave functions are constructed hole symmetry in each half shell: 6 electrons are equivalent to a half-filled shell (7 electrons) with one hole. By Hund’s rules, the first electron(s) in a shell have a large |m| and are oblate (Fig. 11), corresponding to a negative Q2 . Removing an electron with a large |m| from a half-filled shell yields one oblate hole, which is the same as a prolate electron distribution with a positive Q2 . Table 10 is limited to the quadrupole moment Q2 . Higher-order multipole moments provide a refined description of the angular dependence of the rare-earth 4f electron cloud. The third-order octupole and fifth-order triakontadipole moments are zero by symmetry, but the fourth-order hexadecapole moment (16-pole, Q4 ) and the sixth-order hexacontatetrapole (64-pole, Q6 ) are generally nonzero. Figure 12 shows the zoology of the angular dependence of the 4f charge distributions up to the fourth order. For Hund’s-rules ions, the number of animals is limited by the symmetry of the wave functions (Figs. 4 and 11), namely, n ≤ 4 for 3d ions and to n ≤ 6 for 4f ions [38]. Furthermore, since the rare-earth 4f electrons are unquenched, the 4f charge distribution shows axial symmetry, and there are no multipole contributions Ql m with m = 0. The anisotropy corresponding to the unquenched quadupole 3 Anisotropy and Crystal Field 135 Table 10 Hund’s-rules ground states of 4f ions. The orbitals listed from left to right, lz = 3, 2, 1, 0, −1,–2, −3. moment of rare-earh ions can be very high, up to a few K per atom in temperature units [36]. This temperature scale needs to be distinguished from that governing the temperature dependence of anisotropy constants, which involves interatomic exchange (Sect. 4.4.4). 136 R. Skomski et al. Fig. 12 Cartoon illustrating the electrostatic R3+ multipole moments up to the fourth order (Q0 , Q2 , and Q4 ). The 4f charge distributions n(r) derive from Figs. 4 and 11 and are both axially and inversion symmetric Operator Equivalents The next step is to quantitatively determine the interaction V (r) n (r) dV (Sect. “One-Electron Crystal-Field Splitting”) between the crystal field and the 4f charge distribution. This can be done explicitly, in a straightforward but cumbersome way, but a more elegant method is to use operator equivalents. Both approaches assume that the crystal field, V (r) or Al m , and the 4f charge distribution, n(r) or Qn , are known. The straightforward method is best explained by considering the lowest-order uniaxial limit, where Eq. (16) reduces to V (r) = A2 0 3z2 –r 2 . Substituting this expression into Eq. (12) yields HCF = A02 3z2 − r 2 n(r)dV (24) 3 Anisotropy and Crystal Field 137 By definition, the integral in this equation is equal to Q2 , so that HCF = A2 0 Q2 . Equation (24) is exact and easily generalized to other Al m , but the problem remains to determine Q2 as a function of the ion’s electronic properties and magnetization angles. For example, the rare-earth crystal field is normally far too weak to affect the term and multiplet structures, but it usually affects the intramultiplet structure. These energy values can all be obtained by specifying n(r), but this is a very tedious method. A much more elegant approach is the use of Stevens operator equivalents Ol m . The idea is to replace the real-space coordinates (x/r, y/r, z/r) in expressions such as Eqs. (24) by the vector operator (Jx , Jy , Jz ), using J± = Jx ± iJy and identities such as J2 = J(J + 1). The lowest-order noncubic operator equivalents are O2 0 (J ) = 3 Jz 2 –J (J + 1) (25) corresponding to 3z2 – r2 and O2 2 (J ) = 1 2 J+ + J− 2 2 (26) corresponding to x2 – y2 = ½(x + iy)2 + ½(x – iy)2 . The derivation of higher-order operator equivalents [33, 38] is straightforward but tedious. For example, the fourthorder cubic crystal-field expression Eq. (15a) consists of the term 1 1 20 x 4 + y 4 + z4 − 3r 4 /5 = 35z4 − 30 z2 r 2 + 3r 4 + 5 (x + iy)4 + (x–iy)4 2 2 (27) 2 and corresponds to O4 0 + 5 O4 4 . Here O4 0 = 35 Jz 4 − 30J (J + 1) Jz + 1 4 2 2 4 4 2 25Jz − 6J (J + 1) + 3J (J + 1) and O4 = 2 J+ + J– . The operators have been tabulated in Refs. 33 and especially 38. It is also possible to define operator equivalents Ol m (L, Lz ) and related spin Hamiltonians Hspin (S, Sz ) for 3d ions (Sect. “Transition-Metal Anisotropy”), but the underlying physics is different from the presently considered rare-earth limit, because L and S are only weakly coupled (quenching). The occurrence of Jz and of the ladder operators J± greatly simplifies the calculation of matrix elements of magnetic ions in a crystal field or exchange field. For Sm3+ , J = 5/2 yields Jz = ±5/2, ±3/2, and ± 1/2, corresponding to O2 0 = 10, O2 0 = −2, and O2 0 = −8. The magnitude of the splitting is determined by A2 0 and by the radial part of n(r), but the evaluation of the Ol m is sufficient to determine the relative energies, namely, 5:–1:–4 in the present example. The multipole moments are straightforward linear functions of the operator equivalents: Ql = θl < r l >4f Ol 0 (28) 138 R. Skomski et al. Here the Stevens coefficients θ 2 = α J , θ 4 = β J , and θ 6 = γ J are rare-earth specific constants that describe how Hund’s rules affect the shape of the R3+ ions [38]. For example, Sm3+ has α J = 13/32 ·5·7, β J = 2·13/33 ·5·7·11, and γ J = 0. There is no sixth-order crystal-field interaction for Sm3+ (γ J = 0), because the ground-state multiplet has J = 5/2 < n/2. However, as mentioned in Sect. “Rare-Earth Ions”, Sm3+ exhibits a rather unusual low-lying excited multiplet, which has J = 7/2 and may give a small nonzero γ J contribution due to thermal or quantum mechanical admixture. Rare-earth ions in magnetically ordered compounds experience an interatomic exchange field HJ , so that the rare-earth Hamiltonian becomes [39] H = l,m Bl m Ol m (J, Jz ) + 2 μo (g–1) J · H J + g μo J · H (29) Here Bm n = θ n < rn >4f Al m and g J·H describes the comparatively weak Zeeman interaction and HJ is the exchange field. The quantities L, S, and λ enter this equation only indirectly, via Hund’s rules and J = L ± S. However, O l m contains intramultiplet excitations (−J < Jz < J), and the raising and lowering operators J± in Eq. (26) indicate that off-diagonal crystal fields, such as A2 2 , can change Jz . To exactly diagonalize Eq. (29), it is necessary to include matrix elements <Jz | Ol m (J ) | Jz >, where Jz = J’z . These matrix elements are known [38] but complicate the calculations and the evaluation of the results. Major simplifications arise if the term involving the exchange energy is much larger than the CF interaction. This is approximately the case in rare-earth transitionmetal (RE-TM) intermetallics such as Nd2 Fe14 B [39, 40], where the exchange field is roughly proportional to the RE-TM intersublattice exchange JRT . This strong exchange field stabilizes states with Jz = ±J, where the sign determines the net magnetization but does not affect the anisotropy. Intramultiplet excitations, caused by the operators J± , are effectively suppressed, and only the Ql = θl < r l >4f Ol 0 terms remain to be considered. Furthermore, putting Jz = ±J drastically simplifies the operator equivalents: O2 0 = 2 J · (J − 1/2) (30) O4 0 = 8 J · (J − 1/2) · (J –1) · (J –3/2) (31) O6 0 = 16 J (J − 1/2) · (J –1) · (J –3/2) · (J − 2) · (J − 5/2) (32) The corresponding 4f charge distributions are axially symmetric around the quantization axis (z-axis), and their multiple moments are given by Eq. (28). Table 11 lists multipole moments derived from Eqs. (30)–(32). 3 Anisotropy and Crystal Field Table 11 Rare-earth multipole moments Ql = θl < r l > Ol 0 for Jz = J, measured in ml . ao = 0.529 Å is the Bohr radius 139 Element 4f1 Ce3+ 4f2 Pr3+ 3 4f Nd3+ 5 4f Sm3+ 7 4f Gd3+ 8 4f Tb3+ 9 4f Dy3+ 10 4f Ho3+ 11 4f Er3+ 12 4f Tm3+ 13 4f Yb3+ Q2 /ao 2 −0.748 −0.713 −0.258 0.398 0 −0.548 −0.521 −0.199 0.190 0.454 0.435 Q4 /ao 4 1.51 −2.12 −1.28 0.34 0 1.20 −1.46 −1.00 0.92 1.14 −0.79 Q6 /ao 4 0 5.89 −8.63 0 0 −1.28 5.64 −10.0 8.98 −4.50 0.73 Single-Ion Anisotropy The anisotropy constants are extracted by rotating the magnetization, that is, by rotating the 4f charge distribution and calculating the energy. It is convenient to choose a coordinate frame where J is fixed, that is, to actually rotate the crystal field around the rare-earth ions. This can be done for each ligand separately, because crystal fields obey the superposition principle. It starts conveniently from an axial coordination, R || ez , and the corresponding crystal fields A 2 , A 4 , and A 6 are referred to as intrinsic crystal fields [20]. In the point-charge model, A2 (R) = –eq/4πεo R 3 . Due to the axial symmetry of the 4f charge distribution, the rotation of R into the correct direction relative to the 4f moment involves a polar angle Θ. For example A2 0 = A2 1 3cos2 − 1 2 (33) describes the rotation of a single ligand. By adding the contributions from all ligands, one can create any crystal field and any relative orientation between crystal and magnetic moment. This approach is not limited to uniaxial anisotropy. Equation (16) is uniaxial, but it contains a z4 term, and by rotating different charges onto the x- and y-axes, one can create crystal fields of the type x4 + y4 + z4 , which are cubic. Figure 13 illustrates the rotation of the crystal around the rare-earth ion for a fourth-order anisotropy contribution. Note that none of the rare-earth ions in Fig. 12 has the ghost shape, but quadrupole moments (Q2 ) do not interact with crystal fields having fourfold symmetry, so that Fig. 13 actually applies to the UFOs (Ce, Tb) and to the digesting snakes (Sm, Er, Tb). Since crystal rotations, for example, Θ = 45◦ in Fig. 13(c), and magnetization rotations are equivalent, Eq. (33) also describes the energy as a function of the magnetization angle, that is, the anisotropy energy per rare-earth atom. Explicitly 140 R. Skomski et al. Fig. 13 Cartoon-like “shaking-ghost” interpretation of fourth-order rare-earth anisotropies. Since the head, feet, and hands of the ghost are made from negatively charged 4f electrons, electrostatics favors (a) over (b) and (c). The latter two have the same crystal-field energy, but (c) is easier to calculate, because it leaves the axis of quantization (arrow) unchanged Ea = 1 Q2 A2 0 3 cos2 θ − 1 2 (34) Comparison with Eq. (1) yields K1 = − 3 A2 0 Q2 2VR (35) where VR is the crystal volume per rare-earth atom. This equation resolves the rareearth anisotropy problem by separating the properties of the 4f shell, described by Q2 , from the crystal environment, described by A2 0 . Crystal-field parameters such as A2 0 describe the surroundings of the rare-earth ion and therefore change little across an isotructural series of compounds with different rare earths. Examples are A2 0 values of 300 K/ao 2 for R2 Fe14 B, 34 K/ao 2 for R2 Fe17 , and – 358 K/ao 2 for R2 Fe17 N3 . In a given crystalline environment, the sign of the rare-earth anisotropy depends on whether the ion is prolate or oblate. A positive K1 is obtained by using oblate ions, such as Nd3+ , on sites where the crystal-field parameter A2 0 is positive, and prolate ions, such as Sm3+ , in crystalline environments where A2 0 is negative. This explains the use of neodymium in hard R2 Fe14 B and RT12 N alloys, whereas samarium is preferred in RCo5 , R2 Fe17 N3 , and RT12 intermetallics. The rare-earth ions responsible for the anisotropy must be magnetic, whereas both magnetic and nonmagnetic ligand atoms contribute to the crystal field. An interesting example is interstitial nitrogen in Sm2 Fe17 , which changes the anisotropy from easy-plane to easy-axis [41]. Using volume VR per rare-earth ion as a unit volume, the uniaxial anisotropy constants are 3 21 K1 = − A2 0 Q2 − 5 A4 0 Q4 − A6 0 Q6 2 2 (36) 3 Anisotropy and Crystal Field K2 = 141 35 0 189 A4 Q4 + A6 0 Q6 8 8 K3 = − 231 0 A6 Q6 16 (37) (38) Tetragonal magnets also have K2 = 1 5 A4 4 Q4 + A6 4 Q6 8 8 K3 = − 11 A6 4 Q6 16 (39) (40) whereas hexagonal magnets exhibit only one in-plane term K3 = − 1 A6 6 Q6 16 (41) Cubic anisotropy can be considered as a special limiting case of tetragonal anisotropy. Using Eqs. (36)–(40) and dropping terms absent incompatible with cubic symmetry yields K1 c = −5 A4 0 Q4 − K2 c = 21 0 A6 Q6 2 231 0 A6 Q6 2 (42) (43) A striking feature in the last two equations is the absence of independent inplane crystal-field parameters, such as A4 4 . While a separate consideration of O4 4 , as contrasted to O4 4 ∼ Q4 , is not necessary for rare earths due to the axial symmetry of the 4f charge clouds, the non-uniaxial CF parameters are not independent but obey A4 4 = 5A4 0 and A6 4 = − 21A6 0 in cubic symmetry. Temperature Dependence Magnetic anisotropy exhibits a temperature dependence that is usually much more pronounced than that of the spontaneous magnetization. It vanishes at the Curie point. Figure 14 shows schematic temperature dependences of the anisotropy constants for some classes of magnetic materials. Anisotropy energies per atom intrinsically correspond to rather low temperatures, of order 1 K for. Magnetic anisotropy at or above room temperature therefore requires the help of an interatomic exchange field Hex , which stabilizes the directions of the atomic moments against thermal fluctuations. 142 R. Skomski et al. Fig. 14 Temperature dependence of anisotropy (schematic): (a) basic dependence in elemental magnets, (b) bcc Fe, (c) RCo5 alloys, and (d) Nd2 Fe14 B. The curves in (a) are schematic and less smooth in practice [70], which reflects subtleties in the electronic structure Typical rare-earth transition-metal (RE-TM) intermetallics exhibit a strong rareearth anisotropy contribution, and for TM-rich intermetallics, this contribution dominates below and somewhat above room temperature. For example, the lowtemperature anisotropy constants K1 are 26 MJ/m3 for SmCo5 and 6.5 MJ/m3 for Sm2 Co17 , as compared to room-temperature values of 17 MJ/m3 and 4.2 MJ/m3 . The exchange field necessary to realize the RE anisotropy contribution is largely provided by the rare-earth transition-metal (RE-TM) intersublattice exchange JRT , rather than the weaker rare-earth rare-earth (RE-RE) exchange [42]. The RE-TM interaction is proportional to J·Hex , that is, the rare-earth ions behave like paramagnetic ions in an exchange field Hex ∼ JRT MT created by 3 Anisotropy and Crystal Field 143 and proportional to the transition-metal sublattice magnetization MT . Depending on the sign of Hex , the RE-TM exchange favors Jz = ±J, and the corresponding low-temperature anisotropy is described by Ol m (J, Jz ) =Ol m (J, ±J ), as in Eqs. 30–32. However, thermal excitation leads to the population of intermediate intramultiplet levels with |Jz | < J. The randomization becomes important above some temperature T ∗ ∼ JRT /kB , which is typically of order 100–200 K, Fig. 14(c–d). Below T*, |Jz | ≈ J, and the anisotropy is only slightly reduced. Above T*, the rare-earth anisotropy contribution is strongly reduced. In the hightemperature limit, kB T JRT , all Jz levels are equally populated and the rare-earth anisotropy vanishes, because m Ol m (J, m) = 0. The orientations of the 4f charge clouds are thermally randomized and the net shape of the charge clouds becomes spherical. To quantify the temperature dependence, one must evaluate the thermal averages < Ol m >th . At low temperatures, the quantization of Jz plays a role. The exchange splitting between Jz = ±J and ± (J – 1) is of order JRT , so that the anisotropy remains constant or “plateau-like” for T T*, Fig. 14(c). Above T*, the discrete level splitting is less important and Jz can be considered as a continuous quantity. This means that Jz = J cosθ and HRT = –JRT cos (θ ), and the operator equivalents entering the anisotropy expression simplify to Legendre polynomials, for example, O2 0 ∼ 12 3cos2 θ − 1 = P2 (cos θ ). The thermal averages π m < cos θ >= N exp 0 JRT kB T cos θ cosm θ sin θ dθ (44) are readily evaluated by a high-temperature expansion of the exponential function and yield the rare-earth anisotropy [43]. K1 (T ) = K1 (0) JRT 2 15 kB T 2 (45) For anisotropies of arbitrary order m, it can be shown that Km ∼ (JRT /T )2m . Equation (44) can also be used as a classical estimation for iron-series elements and for the TM anisotropy contribution in RE-TM intermetallics. However, in this case, J is not an independent interaction parameter (JRT ) but determined by the Curie temperature, JTT ≈ kB Tc , and the high-temperature limit of Eq. (45) is no longer meaningful. For small θ , Eq. (44) leads to <cosm θ > = 1−mk B T /JTT . The exponent m = 1 yields the magnetization, whereas values m>1 are necessary to determine the anisotropy, which is proportional to <Pm > = 1–m (m + 1) kB T /2JTT . These relations correspond to the famous Akulov-Callen m(m + 1)/2 power laws [44–46]: Km/2 (T ) = Km/2 (0) Ms (T ) Ms (0) m(m+1)/2 (46) 144 R. Skomski et al. Table 12 First and second-order anisotropy constants at low temperatures (LT) and at room temperature Element Fe Co Ni Nd2 Fe14 B Pr2 Fe14 B Sm2 Fe17 N3 LT K1 (MJ/m3 ) 0.052 0.7 −0.012 −18 24 12 Table 13 Transition-metal and rare-earth contributions to the room-temperature magnetocrystalline anisotropy [10]. All values are in MJ/m3 RT K2 (MJ/m3 ) −0.018 0.18 0.03 48 −7 3 Structure Refs. K1 (MJ/m3 ) 0.048 0.41 −0.005 4.3 5.6 8.6 Compound Nd2 Fe14 B Sm(Fe11 Ti) Sm2 Fe17 N3 Sm2 Co17 SmCo5 K2 (MJ/m3 ) −0.015 0.15 −0.002 0.65 ≈0 1.9 K1 4.9 4.8 8.6 3.3 17.0 K1T 1.1 0.9 −1.3 −0.4 6.5 bcc fcc hcp tetr. tetr. rhomb. K1R 3.8 3.9 9.9 3.7 10.5 [47] [47] [47] [48] [48] [48] Symmetry Tetragonal Tetragonal Rhombohedral Rhombohedral Hexagonal In other words, 2nd-, 4th-, and sixth-order anisotropy contributions are proportional to the third, tenth and 21st powers of the magnetization, respectively. Equation (46), which is valid up to about 0.65 Tc for Fe, means that higher-order anisotropy contributions rapidly decrease with increasing temperature. A crude approximation, based on Ms ∼ (1 – T/Tc )1/3 and used in Fig. 14(a), yields the linear dependence K1 (T) ≈ K1 (0) (1 – T/Tc ) for the first anisotropy constant K1 of uniaxial magnets (Table 12). In summary, the temperature dependence of the anisotropy is a very complex phenomenon. Each crystallographically nonequivalent site generally yields a different anisotropy contribution with a different temperature dependence, and the distinction is most pronounced between rare-earth (4f ) and transition-metal (3d) sites. As a rule of thumb, the RE or TM contributions dominate at low or high temperatures [40, 49], and their respective temperature dependences are approximately given by Eqs. (44 and 45) and Eq. (46). In the latter case, K1 ∼ Ms 3 (uniaxial magnets) and K1 ∼ Ms 10 (cubic magnets). Actinide (5f ) anisotropy is limited by the interatomic exchange, although the spin-orbit coupling is very large, and its temperature dependence follows that of the magnetization, K1 ∼ Ms [50]. The anisotropy of 3d–5d (and 3d–4d) intermetallics, such as tetragonal PtCo, largely originates from the heavy transition-metal atoms, but this anisotropy is realized via spin polarization by the 3d sublattice, roughly corresponding to K1 ∼ Ms 2 [51, 52]. The same dependence is obtained for the two-ion (magnetostatic) contribution to the magnetocrystalline anisotropy, Sect. “Two-Ion Anisotropies of Electronic Origin”, because the magnetostatic energy scales as Ms 2 (Table 13). 3 Anisotropy and Crystal Field 145 Transition-Metal Anisotropy Typical second- and fourth-order iron-series transition-metal anisotropies are 1 MJ/m3 and 0.01 MJ/m3 , respectively, with large variations across individual alloys and oxides (Tables 14 and 15). The anisotropy constants are often quoted in meV or μeV per atom, especially in the computational literature dealing with metallic magnets. A rule-of-thumb conversion for dense-packed iron-series transition-metal magnets is 1 meV = 14.4 MJ/m3 . In alloys, the anisotropy must be multiplied by the volume fraction f of the transition metals. For example, the transition-metal contribution to the anisotropy of transition-metal-rich rare-earth intermetallics corresponds to f ≈ 0.7, because about 30% of the crystal volume is occupied by the rare-earth atoms. The magnetic anisotropy 3d magnets is largely dominated by the degree of quenching (Sect. “Spin-Orbit Coupling and Quenching”). For oxides, the degree of quenching was implicitly considered by Bloch and Gentile [1], whereas Brooks (1940) explicitly considered quenching in itinerant iron-series magnets [53]. An explanation of quenching in itinerant magnets is provided by the model Hamiltonian: Table 14 Anisotropy, magnetization, and Curie temperature of some oxides [9–11, 37, 63] K1 (RT) MJ/m3 α-Fe2 O3 −0.007 γ-Fe2 O3 −0.0046 Fe3 O4 −0.011 MnFe2 O4 −0.003 CoFe2 O4 0.270 NiFe2 O4 −0.007 CuFe2 O4 −0.0060 MgFe2 O4 −0.0039 BaFe12 O19 0.330 SrFe12 O19 0.35 PbFe12 O19 0.22 BaZnFe17 O27 0.021 Y3 Fe5 O12 −0.0007 Sm3 Fe5 O12 −0.0025 Dy3 Fe5 O12 −0.0005 CrO2 0.025 NiMnO3 −0.26 (La0.7 Sr0.3 )MnO3 −0.002 Sr2 FeMoO6 0.028 Substance μo Ms (RT) T 0.003 0.47 0.60 0.52 0.50 0.34 0.17 0.14 0.48 0.46 0.40 0.48 0.16 0.17 0.05 0.56 0.13 0.55 0.25 Tc K 960 863 858 573 793 858 728 713 723 733 724 703 560 578 563 390 437 370 425 Structure Rhombohedral (Al2 O3 ) Cubic (disordered spinel) Cubic (ferrite) Cubic (ferrite) Cubic (ferrite) Cubic (ferrite) Cubic (ferrite) Cubic (ferrite) Hexagonal (M ferrite) Hexagonal (M ferrite) Hexagonal (M ferrite) Hexagonal (W ferrite) Cubic (garnet) Cubic (garnet) Cubic (garnet) Tetragonal (rutile) Hexagonal (FeTiO3 ) Rhombohedral (perovskite) Orthorhombic 146 R. Skomski et al. Table 15 Anisotropy, magnetization, and Curie temperature of some transition-metal structures. PT indicates a structural change near or below the Curie temperature Fe Co (α) K1 (RT) MJ/m3 0.048 0.53 μo Ms (RT) T 2.15 1.76 Tc K 1043 1360 Co (β) Ni Fe0.96 C0.04 Fe4 N −0.05 −0.0048 −0.2 −0.029 1.8 0.62 2.0 1.8 1388 631 (PT) 767 Fe16 N2 Fe3 B Fe23 B6 1.6 −0.32 0.01 2.7 1.61 1.70 (PT) 791 698 Fe0.65 Co0.35 FeNi 0.018 1.3 2.43 1.60 1210 (PT) Fe0.20 Ni0.80 FePd −0.002 1.8 1.02 1.37 843 760 FePt 6.6 1.43 750 CoPt 4.9 1.00 840 Co3 Pta MnAl 2.1 1.7 1.38 0.62 1000 650 MnBi 1.2 0.78 630 Mn2 Ga 2.35 0.59 (PT) Mn3 Ga 1.0 0.23 (PT) Mn3 Ge 0.91 0.09 (PT) NiMnSb −6.3 1.10 698 Fe7 S8 0.320 0.19 598 Substance a Extrapolation Structure Refs. Cubic (bcc) Hexagonal (hcp) Cubic (fcc) Cubic (fcc) Tetragonal Cubic (modified fcc) Tetragonal Tetragonal Cubic (C6 Cr23 ) Cubic (bcc) Tetragonal (L10 ) Cubic (fcc) Tetragonal (L10 ) Tetragonal (L10 ) Tetragonal (L10 ) Hexagonal Tetragonal (L10 ) Hexagonal (NiAs) Tetragonal (D022 ) Tetragonal (D022 ) Tetragonal (D022 ) Cubic (half-Heusler) Monoclinic [68] [68] to fully ordered Co3 Pt has been suggested to yield 3.1 H= E1 (k) 0 0 E2 (k) +λ 0 i −i 0 MJ/m3 [69] [68] [70] [37] [71] [72] [73] [37] [68] [37] [74] [74] [74] [75] [74] [9] [76] [76] [76] [37] [37] [67] (47) where E1 (k) and E2 (k) are two 3d subbands connected by a spin-orbit matrix elements ±iλ. The spin-orbit term favors a nonzero net orbital moment, as required 3 Anisotropy and Crystal Field 147 for magnetic anisotropy, but λ ≈ 50 meV is usually much smaller than |E1 (k) – E2 (k)|, the latter being comparable to the bandwidth W of several eV. However, even for |E1 (k) – E2 (k)| = W, perturbation theory leads to a small orbital moment and some residual anisotropy. Furthermore, accidental degeneracies E1 (k) = E2 (k) yield the eigenvalues ±λ and completely unquenched orbitals. The corresponding anisotropy energy, about 50 meV per atom, is then huge compared to typical ironseries anisotropies of 0.1 meV, or about 1 MJ/m3 . The practical challenge is to add the spin-orbit couplings of all atoms (index i): Hso = i λi l i · s i (48) to the isotropic Hamiltonian and to determine the anisotropy contributions from all bands and k-vectors. To quantitatively determine the anisotropy, this procedure must be performed for different spin direction s, s || ez and s || ex . Perturbation Theory The simplest approach to 3d anisotropy is the perturbation theory as originally developed by Bloch and Gentile [1] and later popularized by van Vleck [35] and Bruno [54]. The idea is to consider the Hamiltonian H = Ho + Hso , where Ho (l i ) is the nonrelativistic isotropic part and to consider Hso as a small perturbation. In the independent-electron approximation, the lowest-order correction proportional to ξ i = λ is obtained by using the perturbed wave functions |μ k σ >, where μ is a 3d subband index and the index σ = {↑, ↓} labels the spin direction. Lowest-order perturbation theory, linear in λ, uses completely quenched orbitals, <li > = 0, and therefore <li ·si > = < li > ·si = 0. The next term is quadratic in λ. For a single electron of wave function |μ k σ >, the corresponding anisotropy energy is Ek = λ2 μ,σ k <μk σ |l · s|μ k σ ><μ k σ |l · s|μkσ > Eμ k σ −Eμkσ (49) The total second-order anisotropy energy is obtained by summation over all electrons. Since the SOC leaves the centers of gravity of the one-electron energies unchanged, there is no net anisotropy contribution from level pairs |μ k σ > and |μ k σ > when both levels are occupied (o) or unoccupied (u). The summation (or integration) is therefore limited to |μkσ >= |o>and |μ k σ > = |u>: E = −λ o,u of <o | l · s | u> <u | l · s | o> Eu −Eo (50) The numerical determination of the anisotropy constants requires the evaluation E for several spin directions s = ½(σ x ex + σ y ey + σ z ez ), where σ x , σ y , 148 R. Skomski et al. and σ z are Pauli’s spin matrices. Equation (50) is sometimes reformulated in form of a statement that the anisotropy energy is proportional to the quantum average of angular orbital moment. However, this equivalence is limited to small orbital moments [55] – rare-earth orbital moments are fixed by Hund’s rules (Fig. 13) and do not change as a function of magnetization direction. The spin summation is greatly simplified by the factorization of the unperturbed wave functions, |μ k σ >= |μ k>|σ >, but the k-space summations can only be performed numerically for most systems. The factorization into |μk> and |σ > makes it possible to formally perform a summation over |μ k>, |μ k >, and |σ > only, leaving the spin s unaveraged. This leads to a spin Hamiltonian of the general many-electron type: Hspin = −λ2 S · K · S (51) where K is a 3 × 3 real-space anisotropy matrix [56, 57]. For uniaxial anisotropy, Eq. (51) reduces to the anisotropy term: Hspin = D Sz 2 –S (S + 1) /3 (52) This expression, which mirrors other second-order anisotropy expressions, is not restricted to magnetocrystalline anisotropy but can also be used for dipolar anisotropy (see Sect. “Magnetostatic Anisotropy”). It is most useful for 3d ions, where D is often considered an adjustable parameter. As a rough approximation, Eq. (52) can also be used for metallic Fe and Co (S ≈ 1). It cannot be used to describe the anisotropy of independent conduction electrons (S = ½) nor for Ni (S ≈ ½), because S = ½ yields Sz 2 – S(S + 1)/3 = 0 for Sz = ±½. It is, however, possible to consider classical averages over a number of electrons, which yields Hspin = D cos2 θ –1/3 and K1 = −D. Generalizing the perturbation expansion to arbitrary orders n yields anisotropy constants of the order: Kn/2 ∼ λn Vo (Eo − Eu )n−1 (53) where Vo is the crystal volume per transition-metal atom. This important relation, known as spin-orbit scaling, was first deduced for lowest-order cubic anisotropy, where n = 4 and K1 c ∼ K2 ) [1]. In this case, the anisotropy constant scales as λ4 /A3 , where A is the energy-level splitting in the absence of spin-orbit coupling (crystal-field splitting or bandwidth). This scaling behavior explains the low cubic anisotropy of bcc iron (0.05 MJ/m3 ) and Ni (−0.005 MJ/m3 ), as compared to that of hexagonal Co (0.5 MJ/m3 ) and YCo5 (5 MJ/m3 ). Equation (53) provides a semiquantitative understanding of transition-metal anisotropies. In metallic systems, Eu – Eo ∼ W, where the bandwidth W is about 5 eV for iron-series (3d) magnets and somewhat larger for palladium -series (4d), 3 Anisotropy and Crystal Field 149 platinum-series (5d), and actinide (5f ) magnets. The spin-orbit coupling rapidly increases as the atoms get heavier (Tables 7 and 8), so that heavy transitionmetal elements are able to support very high anisotropies so long as the induced magnetic moments on the heavy atoms are appreciable. In particular, FePt magnets are important in magnetic recording [58], but both the low Curie temperature and the low intrinsic magnetic moment per heavy transition-metal atom make it very difficult to exploit the high anisotropy of very heavy atoms, up to 1000 MJ/m3 for actinide compounds such as uranium sulfide [59]. As outlined in Eqs. (49 and 50), quantitative anisotropy calculations require a summation of all occupied and unoccupied states. This summation involves matrix elements <o|l·s|u>, which couple wave functions of equal |Lz |, namely, Lz = ±1 and Lz = ±2, where the quantization axis (z-axis) is parallel to the spin direction (see below). These matrix elements affect the sign and magnitude of the anisotropy but do not change its order of magnitude, because they are of order unity. The order of magnitude of the anisotropy is given by the spin-orbit coupling, which is essentially fixed for a given element (Tables 7 and 8) and by the denominator Eo – Eu , which requires a detailed discussion. Spin-Orbit Matrix Elements In Eq. (50), the itinerant wave functions |o > and |u > are of the Bloch type and can therefore be expanded into atomic wave functions. Including spin, there are 10 3d orbitals per atom, which yield 100 matrix elements <l·s > for each spin direction. However, the number of independent matrix elements is drastically reduced by symmetry. First, for the highly symmetric point groups Cnv , Dn , Dnh , and Dnd (Sect. “Anisotropy and Crystal Structure”), only three spin and orbital-moment directions need to be considered, namely, x, y, and z. Second, the matrix elements between ↑↑ and ↓↓ pairs are the same, whereas those for ↑↓ and ↓↑ are equal and opposite in sign. Third, many of the remaining matrix elements are zero by symmetry [60]. Explicit matrix elements are obtained by applying equations such as lˆ z = i(y∂/∂x – x∂/∂y) or lˆ z = −i∂/∂φ to the real or quenched 3d wave functions of Fig. 4. For example, |xy>∼ sin(4φ) and |x2 –y2 >∼ cos(4φ) yield: <xy | l̂z |x 2 –y 2 >=2i (54) This matrix element is imaginary and creates an imaginary (unquenched) admixture to the wave function, as required for magnetocrystalline anisotropy. For degenerate |xy> and |x2 –y2 > levels, this matrix element yields the eigenfunctions exp.(±2iφ) = cos(2φ) ± i sin(2φ), the orbital momentum <lz > = ±2, and the orbital moment ±2μB . In terms of Fig. 10, the spin-orbit coupling acts as a perturbation that promotes hopping from one valley into the next and thereby creates a small net orbital motion. As outlined above (Sect. “Spin-Orbit Coupling and Quenching”), this motion is 150 R. Skomski et al. 2 2 ˆ Fig. 15 The three “canonical” d electron orbital-momentum √ matrix elements: (a) <x –y |l z |xy > = 2i, (b) <xz|lˆ z |yz>= i, and (c) <z2 | l̂x | yz>= 3 i. The dashed lines are out of the paper plane and visualize the direction of l̂ , but the actual length of the lines is zero, because all orbitals belong to the same atom responsible for the small orbital contribution to the magnetic moment of itinerant magnets, such as Fe, and for the corresponding magnetic anisotropy. The five 3d orbitals yield a fairly large number of matrix elements such as that in Eq. (54), but due to symmetry, many of them are zero, and only three are nonequivalent. Figure 15 illustrates these three “canonical” matrix elements. Figure 15(a) corresponds to Eq. (54) and is encountered only once, aside from the conjugate complex value –2i created by interchanging xy and x2 –y2 . The matrix element of Fig. 15(b) occurs five times, namely, in form of <xz|lˆ z |yz>, <xy|lˆ x |xz>, <xy|lˆ y |yz>, <x2 –y2 |lˆ x |yz>, and <x2 –y2 |lˆ y |xz>, whereas that of Fig. 15(c) has two realizations, namely, <z2 |lˆ x |yz> and <z2 |lˆ y |xz>. A physical interpretation of matrix elements <ψ 1 |l̂ |ψ 2 > is that <ψ 1 |ψ 2 > = 0 but the angular momentum operator rotates ψ 2 and thereby creates overlap with ψ 1 . The rotation angle is equal to π/m, where m is the magnetic quantum number of the orbitals, so that π/4 in Fig. 15(a, c) and π/2 in Fig. 15(b). Crystal Fields and Band Structure An important question is the relation between electrostatic crystal-field interaction and the interatomic hopping that leads to band formation. In the Mott insulator limit of negligible interatomic hopping, the energy differences Eo – Eu correspond to the ionic CF level splittings outlined in Sect. “Crystal-Field Theory”. However, many oxides are Bloch-Wilson insulators, whose insulating character is a bandfilling effect. This means that band effects are not negligible in many or most oxides. Hybridization-type ligand fields, which include band formation, do not alter the qualitative physics of crystal-field theory [17] but are often stronger than the electrostatic crystal fields and strongly affect quantitative anisotropy predictions. 3 Anisotropy and Crystal Field 151 For example, the eg -t2g crystal-field splitting in transition-metal monoxides is of the order of 1 eV, as compared to 3d bandwidths of about 3 eV [62]. It is important to note that properly set up band structure calculations, from firstprinciple (Sect. “First-Principle Calculations”) or based on tight-binding approximations, automatically include crystal-field effects. This is easily seen by considering a tight-binding model that is nonperturbative as regards spin-orbit coupling. The Hamiltonian is H=− 2 2 ∇ + j Vo r − Rj + j Hso Rj 2me (55) where the matrix elements of Hso are those of Fig. 15. The lattice periodicity is accounted for by the ansatz ψkμ (r) = N j exp ik · Rj φμ r − Rj (56) where the index μ labels the orbitals, such as |xy↑>. Putting Eq. (56) into Eq. (55) yields, in matrix notation Eμμ (k) = Eo δμμ + Aμ δμμ + m exp (ik · Rm ) tμμ (Rm ) + Eso,μμ (57) Here Eo is the on-site energy, Aμ is the subband-specific crystal-field energy, and tμμ (Rm ) is the matrix containing the interatomic hopping integrals. The crystal-field term is easily derived by splitting the potential energy j Vo (r – Rj ) into an on-site term Vo (r – Ri ), which enters Eo , and a crystal-field term j = i Vo (r – Rj ). Itinerant Anisotropy Figure 16 shows an explicit example, namely, a monatomic tight-binding spin chain with two partially occupied ↓ subbands near the Fermi level, namely, |xy > and |x2 –y2 >, whereas Fig. 17 illustrates the corresponding band structure and anisotropy. In terms of the fundamental Slater-Koster hopping integrals [64], txy, xy = Vddπ and tx2–y2 ,x2–y2 = ¾Vddσ + ¼Vddδ , whereas txy ,x2–y2 is zero by symmetry. The ratio Vddσ :Vddπ :Vddδ is about +6:-4:+1 [65], so that the model creates two cos(ka) bands of nearly equal widths W ≈ 2Vddπ but opposite slope. The two bands, shown as dashed curves in Fig. 17, exhibit a crossing at k = π/a. The solid curves in Fig. 17 differ from the dashed ones by including crystal-field and spin-orbit interactions. First, the charge distributions of the |x2 –y2 > orbitals (bottom row in Fig. 16) point towards each other, so that the crystal-field charges felt by the |x2 –y2 > orbitals are more negative than those felt by the |xy> orbitals. This yields an equal CF shift of the two bands and shifts the crossing to slightly lower k-vectors. Second, for s || ez , which is perpendicular to the plane of the paper in Fig. 16, Eq. (54) yields an off-diagonal spin-orbit matrix element which mixes 152 R. Skomski et al. the bands and creates an avoided crossing near k = π/a. The gap at this degenerate Fermi-surface crossing (DFSC), 4λ, and the derived anisotropy energy K1 (k), shown in Fig. 17(b), are finite, in contrast to the perturbative result of Eq. (50), where the anisotropy contribution diverges at Eo (k) = Eu (k). To appreciate this peak, is useful to recall that typical noncubic 3d anisotropies are of the order of 0.1 meV per atom, as compared to SOC constants λ of about 50 meV and bandwidths W in excess of 1000 meV. In other words, the avoided crossings in Fig. 17(a) may look tiny on the scale of the bandwidth but they are huge compared to anisotropies actually realized in solids. The bottom panel in Fig. 17(b) shows the k-space integrated density of states as a function of the occupancy n of the spin-down |xy> and |x2 –y2 > bands. In analogy to Eq. (50), it is sufficient to restrict the integration to the matrix elements between occupied (o) and unoccupied (u) states, as schematically shown in Fig. 17(a). The anisotropy, which favors a magnetization perpendicular to the chain, also exhibits a DFSC peak for half filling, near k = π/a, although this peak is much less pronounced than the k-space peak. The simple model of Fig. 16 elucidates a major aspect of itinerant anisotropy, namely, that different pairs of 3d subbands yield positive or negative anisotropy contributions, depending on which of the three canonical matrix elements are realized in each magnetization direction. Including spin, this creates 10 × 10 = 100 different contributions. Each of these contributions depends on the band filling and may further split due to the involvement of different neighbors. As a consequence, the anisotropies exhibit a complicated oscillatory dependence on d-band filling. Figure 18 illustrates this point for a nanoparticle with a completely filled ↑ band. Fig. 16 Monatomic spin-chain model (top) with two orbitals per site, |xy > (center) and |x2 –y2 > (bottom) 3 Anisotropy and Crystal Field 153 Fig. 17 Magnetic anisotropy of the spin chain of Fig. 16: (a) band structure without CF and SOC interactions (dashed lines) and with CF and SOC interaction (solid lines) and (b) anisotropy as a function of the electron wave vector in units of 1/a (top) and band filling of the |xy> and |x2 – y2 > orbitals (bottom). The gray area in (a) shows the occupied states used to define the electron count 0 ≤ n ≤ 2 in the bottom part of (b). The peaks in (b) are caused by degenerate Fermi-surface crossing near k = π/a Fig. 18 First-order anisotropy constant of a hexagonal nanoparticle: (a) structure and (b) tightbinding anisotropy as a function of the number of d electrons (after Ref. 66) By comparison, the anisotropy of rare-earth atoms in a given atomic environment yields only two minima and two maxima, given by the quarter-shell rule of Sect. “Rare-Earth Ions”. This simplicity originates from Hund’s rules, which yield electron clouds of well-defined shape as a function of the number of f electrons. In the itinerant case, each k-state corresponds to a different shape of the electron cloud. This complicated picture starts to emerge in the simplest itinerant picture, 154 R. Skomski et al. namely, in the diatomic pair model [60, 61], where the situation is reminiscent of a quarter-shell rule. It is instructive to compare the contributions of the nonperturbative DFSC anisotropy peaks with the perturbative volume anisotropy due to Eu – Eo ∼ W. The latter corresponds to the nearly homogeneous background in the top of Fig. 17(b) and to the constant slopes near n = 0 and n = 2 in the bottom of Fig. 17(b). In systems where the peak contribution is strong, a very dense k-point mesh is necessary, or else the numerical error gets very big. The relative contribution of the peaks depends on both the order of the anisotropy and the dimensionality of the magnet. In one-dimensional magnets, the bulk and peak contributions to K1 are comparable, as one may guess from the bottom of Fig. 17(b). More generally, Eq. (53) means that perturbative anisotropy contributions scale as Km = W(λ/W)2m . The peak contributions have a strength of λ but are restricted to a small kspace volume of (l/W)d , so the corresponding anisotropy contribution scales as λ(λ/W)d = W(λ/W)d + 1 . The peak contributions are therefore strongest in lowdimensional magnets. They are of equal importance for d = 2 m – 1, that is for K1 in one-dimensional magnets and K2 (K1 c ) in three-dimensional magnets. The latter is fundamentally important, because it includes the anisotropy of cubic magnets such as Fe and Ni. The former is important from a practical viewpoint, because quasi-onedimensional reflection from lattice planes creates pronounced peaks in the density of states [77, 78]. First-Principle Calculations The explanation of magnetocrystalline anisotropy by Bloch and Gentile [1] led to the first attempt by Brooks in 1940 to describe itinerant anisotropy numerically [53]. Early attempts to compute the anisotropy of itinerant magnets [53, 79–82] led to substantial errors, such as wrong signs of K1 in cubic magnets. The errors are partially due to the DFSC peaks discussed above, but they also reflect the limitations of approximations such as tight binding. The use of self-consistent first-principle density functional theory (DFT) has improved the situation in recent decades [83–85], although reliable anisotropy calculations have remained a challenge, especially for cubic magnets. Second-order anisotropy calculations for noncubic transition-metals alloys, transition-metal contributions in rare-earth intermetallics, and ultrathin films [86–91] are better described by DFT and have typical errors of the order of 20–50%. However, in uniaxial magnets having nearly cubic atomic environments, such as hcp Co, the situation is comparable to cubic magnets. The Kohn-Sham equations, which form the basis of density functional theory, are nonrelativistic. Spin-orbit coupling needs to be added in form of Eq. (50), which is a second-order relativistic approximation, or a fully relativistic form, starting from the Dirac equation. The latter is implemented in many modern codes, for example, in the Vienna Ab Initio Simulation Package (VASP). The simplest method to compute second-order anisotropies uses the so-called magnetic force theorem 3 Anisotropy and Crystal Field 155 [92, 93]. In this approach, the energy differences between two magnetization directions are approximated by the difference of band-energy sums along different magnetization directions, which can be achieved by a one-step diagonalization of the full Hamiltonian. A better approach is to use total energy calculations, where the energy is self-consistently calculated for each spin direction. A specific problem is Hund’s second rule, which states that intra-atomic electron-electron exchange favors states with large orbital momentum. The effect is parameterized by the Racah parameter B and, in itinerant magnets, is known as orbital polarization [89, 94]. The relative importance of this intra-atomic exchange effect is reduced by band formation, but anisotropy calculations require a very high accuracy, so that the corresponding orbital polarization effect cannot be ignored in general. A simple but fairly accurate approach is to add an orbital polarization term –½BL2 to the Hamiltonian, where B is of the order of 100 meV [94]. This term lowers the energies of |xy> and |x2 – y2 > orbitals and enhances those of |z2 > orbitals. The example of orbital polarization shows that correlation effects are important in the determination of the anisotropy. In a strict sense, correlation effects involve two or more Slater determinants [17], but sometimes their definition includes Hund’s rule correlations. The latter are of the one-electron or independent-electron type in the sense of a single Hartree-Fock-type Slater determinant [23]. Density functional theory is, in principle, able to describe anisotropy, because anisotropy is a ground-state property for any given spin direction. However, very little is known about the density functional beyond the comfort zone of the free electron-inspired local spin density approximation [95], including gradient corrections. For example, rare-earth anisotropy, which is largely determined by the crystal-field interaction of 4f charge distribution, can be cast in form of a density functional [96], but this functional looks very different from the LSDA functional and its gradient extensions. One approach to approximately treat correlations is LSDA+U, where a Coulomb repulsion parameter is added to the density functional [97]. The parameter U or, in a somewhat more accurate interpretation, U ∗ = U –J is well-defined in the sense that it should not be used to adjust theoretical results to achieve an agreement with the experiment. Treating U as an adjustable parameter yields substantial errors, of the order of 1 MJ/m3 for Ni [98]. However, similar to Hund’s-rules correlations and LSDA, the LSDA+U approximation does not go beyond a single Stater determinant. For example, it does not specifically address many-electron phenomena such as spin-charge separation. The merit of the approach consists in replacing local or quasilocal LSDA-type density functionals by density functionals that are somewhat less inadequate for highly correlated systems. In particular, U suppresses charge fluctuations and thereby improves the accuracy of the energy levels connected by spin-orbit matrix elements [84]. Calculations going beyond a single Slater determinant are still in their infancy. An analytic model calculation has yielded Kondo-like corrections to the anisotropy [96], and dynamical mean-field theory (DMFT) is being used to investigate the effect of charge fluctuations beyond one-electron LSDA+U [99]. 156 R. Skomski et al. Case Studies The magnetic anisotropies of a number of cubic and hexagonal 3d compounds are only partially understood, both quantitatively and qualitatively. In cubic crystals, the smallness of the anisotropy constants makes numerically calculations susceptible to errors, for example, due to electron-electron correlations. Anisotropies in hexagonal (and trigonal) magnets are higher, but their theoretical determination is complicated by the fact that hexagonal crystal fields (sixfold symmetry) do not quench 3d states (two- or fourfold symmetry). This quenching behavior is one reason for the relatively high anisotropy of hexagonal magnets like BaFe12 O19 , SrFe12 O19 , and YCo5 , as contrasted to tetragonal 3d magnets, such as steel. Hexagonal Co also belongs to this high-anisotropy category, given that the atomic environment of the Co atoms is nearly cubic. Hexagonal ferrites. The anisotropy of Ba and Sr ferrites, which are widely used as moderate-performance permanent magnets, is poorly understood in terms of quantitative density-functional theory, partially due to the very narrow energy levels. Nevertheless, early research by Fuchikami [57] traces the anisotropy to Fe atoms on sites with a trigonal environment. An intriguing aspect of the system is that all iron atoms in MFe12 O19 = (MO)·(Fe2 O3 )6 are ferric, Fe3+ , characterized by half-filled 3d shells and zero anisotropy in the ground state. In more detail, the crystal-field splitting yields an S = 5/2 ground state where two ↑ electrons occupy a low-lying |xz> and |yz> doublet (e ), two ↑ electrons occupy an excited |xy> and |x2 –y2 > doublet (e ), and the fifth ↑ electron occupies a |z2 > singlet (a 1 ) of intermediate energy. The first-excited spin configuration is of the low-spin type (S = 3/2), realized by one ↑ electron from the excited e level becoming an e ↓ electron. This spin configuration supports substantial anisotropy, because it has odd numbers of electrons in two unquenched doublets. The splitting between the S = 3/2 and S = 5/2 levels is fairly large (about 1 eV), but the admixiture of S = 3/2 character due to spin-orbit coupling is sufficient to create an anisotropy of the order of 0.3 MJ/m3 . Nickel. The anisotropies of the cubic transition metals (bcc Fe, fcc Co, fcc Ni) have remained a moderate challenge to computational physics. Calculated anisotropy constants are often wrong by several hundred percent and may even have the wrong sign, that is, they predict the wrong easy axis. The choice of methods, for example, with respect to the inapplicability of the force theorem to fourth-order anisotropies, is one question [92, 93]. For instance, when a generalized gradient approximation is used instead of the LSDA, the results are improved for bcc Fe but not for Ni and Co [100]. In fact, the available choice of methods and density functionals adds a “second-principle” component to first-principle calculations, whose only input should be the atomic positions. Another problem is numerical accuracy, depending on the number of k-points used. A particularly well-investigated system is nickel [80, 82–84], where problems are exacerbated by the smallness of the magnetic anisotropy (Table 12). The anisotropy of Ni is determined by several contributions that largely cancel each other: DFSC effects (Sect. “Itinerant Anisotropy”) are important, and the sum of the 3 Anisotropy and Crystal Field 157 anisotropy contributions from different orbitals and k-space regions is nearly zero. It is also known that LSDA+U-type one-electron correlations are important in Ni. An LSDA+U or “static DMFT” calculation was performed for Fe and Ni [84]. Values of U* = 0.4 eV and U* = 0.7 eV have been advocated for Fe and Ni, respectively, leading to anisotropy constants of 0.02 MJ/m3 for Fe (experiment: 0.05 MJ/m3 ) and − 0.04 MJ/m3 for Ni (experiment −0.005 MJ/m3 ). The Ni anisotropy is overestimated, but the sign is correct, and a major reason for the correct sign is the absence of a pocket near the X point of the fcc Brillouin zone. Without U, the Fermi level cuts the pocket and spin-orbit matrix elements between occupied and unoccupied states, similar to Fig. 17(a), creating an unphysical positive anisotropy contribution. YCo5 . The intermetallic compound YCo5 , which crystallizes in the hexagonal CaCu5 structure, has the largest anisotropy among all know iron-series transitionmetal intermetallics, about 8 MJ/m3 at low temperature and 5 MJ/m3 at room temperature [101]. Nearly all this anisotropy arises from the Co sublattices, in spite of Y being a relatively heavy atom. According to Table 7, the spin-orbit coupling of Y (32 meV) is not much smaller than that of Co (68 meV), but according to Eq. (50), the effect of atomic SOC on the anisotropy scales is λ2 s2 , and the magnitude of the Y spin is only about 0.3 μB , as compared to about 1.4 μB for Co [89]. In other words, the anisotropy of YCo5 is about ten times greater than that of hcp Co, in spite of the magnetically largely inert Y. There are two reasons for the high anisotropy of YCo5 . First, the structure of the YCo5 consists of alternating Co and Y-Co layers, in contrast to the nearly cubic atomic environment in hcp Co. In this framework, the Y acts as a nonmagnetic crystal-field source with a contribution similar to a vacuum. This has been shown in a computer experiment [101] where the Y atoms were replaced by fictitious empty interstices without any changes to the Co positions. The replacement reduces the anisotropy by only 13%, which confirms that the anisotropy of YCo5 is largely due to the anisotropic distribution of the Co atoms. A secondary reason for the high anisotropy is that the electronic structure of YCo5 supports a fairly strong orbital moment, about 0.2 μB per Co atom [93], as compared to about 0.1 μB per atom in hcp Co [82]. The less quenched orbital moment in YCo5 , which translates into enhanced anisotropy, partially reflects the presence of degenerate |xy> and |x2 – y2 > states near the Fermi level [89]. According to Eq. (54), the mixing of these states yields an orbital moment of up to 2 μB per atom and a disproportionally strong anisotropy contribution (Fig. 17). More importantly, the bands are very narrow near the Fermi level, which reduces the denominator Eo – Eu in Eq. (50). Iron, steel, and Fe nitride. Purified iron is magnetically very soft, but steel formation due to the addition of carbon (Fe100–x Cx , x ≈ 4) drastically enhances the coercivity [70, 102, 103]. The underlying physics is that carbon causes a martensitic phase transition in bcc Fe, leading to a tetragonally distorted phase [70]. Figure 19 illustrates this mechanism, which is responsible for both the mechanical and magnetic hardnesses of steel. The carbon occupies the octahedral interstitial sites in the middle of the faces of the bcc unit cell (a). These octahedra are strongly 158 R. Skomski et al. Fig. 19 Martensitic distortion of bcc Fe: (a) undistorted unit cell and (b) unit cell distorted along the c-axis (dashed line). The martensitic distortion involves spontaneous symmetry breaking along the a-, b-, or c-axis and extends over many interatomic distances, typically over several micrometers √ distorted: perpendicular to the faces, the Fe-Fe distance is smaller by a factor 2 than along the face diagonals. In a hard-sphere model based on an Fe radius of 1.24 Å, the radius of the interstitial site is 0.78 Å along the face diagonals but only 0.19 Å perpendicular to the face. The atomic radius of C is about 0.77 Å [103], so that the interstitial occupancy requires a strong tetragonal distortion. This distortion breaks the cubic symmetry locally and, via elastic interactions between C atoms on different interstitial sites, macroscopically. For example, 4 at% C yields an enhancement of the c/a ratio by 3.5% [103]. Figure 19(b) shows the C occupancy for a tetragonal distortion along the c-axis. The martensitic lattice strain and the chemical effect due to the presence of the carbon atoms yield almost equal uniaxial anisotropy contributions [102], and K1 is negative for Fe1-x Cx , of the order of −0.2 MJ/m3 . Cobalt addition changes the sign of the volume magnetoelastic constant (Sect. “Magnetoelastic Anisotropy”) and therefore the sign of the strain effect [70]. The magnetization is as high as 2.43 T in Fe65 Co35 , and the corresponding Honda steel [104] has a coercivity of μo Hc = 0.020 T, as compared to 0.004 T for ordinary carbon steel. Such steels dominated permanent-magnet technology in the early twentieth century and have recently attracted renewed attention. Substantial anisotropy, K1 = 9.5 MJ/m3 , and a magnetization of μo Ms = 1.9 T have been predicted for tetragonally distorted FeCo with c/a = 1.23 [105], although such a strong distortion is virtually impossible to sustain metallurgically. Experimental room-temperature anisotropies reach about 2.1 MJ/m3 [106] and require a large amount of Pt (about 75 vol.%). The behavior of interstitial N in Fe is similar to that of C [107], but nitrogen has the additional advantage of improving the magnetization in tetragonal superlattices of Fe8 N, or Fe16 N2 [108]. It is well-established that α -Fe16 N2 has a very high magnetization [109, 110], about 2.8 ± 0.4 T, but the precise value has been a subject of debate. An LSDA+U prediction of the magnetization is 2.6 T, which 3 Anisotropy and Crystal Field 159 includes a U contribution of 0.3 T [98]. Using U as an adjustable second-principle parameter enhances the magnetization at a rate of 0.1 T/eV [111], but very large values of U correspond to a heavy Fermion-like behavior that is contradictory to the band structure of Fe8 N and to explicit first-principle calculations [98]. The roomtemperature K1 of the material is about 1.6 MJ/m3 [71]. LSDA and GGA reproduce the correct order of magnitude [112]. Other Anisotropy Mechanisms The magnetocrystalline anisotropy of Sects. Rare-Earth Anisotropy and 5 dominates the behavior of most magnetic materials. Less commonly considered or more exotic anisotropy mechanisms provide the leading contributions in a few systems and substantial corrections in others. For example, two-ion anisotropies of magnetostatic or electronic origin are usually much smaller than single-ion anisotropies, but they dominate if the latter are zero by symmetry or by chance. An exotic mechanism is the anisotropy of superconducting permanent magnets [113], which is not an anisotropy in a narrow sense but reflects the interaction of local currents with the real-structure features after field-cooling. Magnetostatic Anisotropy Magnetostatic dipole-dipole interaction between atomic spins yields a magnetostatic contribution to the magnetocrystalline anisotropy (MCA). Relativistically, both spin-obit coupling and magnetostatic interactions are of the same order in the small parameter v/c [29], but the similarities end here, and it is customary to treat magnetostatic anisotropy contributions separately from MCA involving spin-orbit coupling. The magnetostatic interaction energy between two dipole moments m and m , located at r and r , respectively, has the form EMS = μo 3m · R m · R − m · m R 2 4π R5 (58) where R = r – r . The total magnetostatic energy is obtained by summation over all spin pairs. In continuum theory, the summation must be replaced integration, i ... mi = ... M(r) dV, and it can be shown that EMS = ½μo H2 (r) dV or, equivalently EMS 1 = − μo 2 M (r) · H (r) dV (59) where H is the self-interaction field. In a homogeneously magnetized body, the energy EMS depends on the direction of m = m . Figure 20 shows the “compass-needle” interpretation of this anisotropy contribution. Neighboring spins lower their magnetostatic energy by aligning 160 R. Skomski et al. Fig. 20 Magnetostatic contribution to the magnetocrystalline anisotropy of a layered magnet with tetragonal symmetry. The energy of the spin configuration (a) is higher than that of (b), because the former creates a relatively large magnetic field between the layers parallel to the nearest-neighbor bond direction R/R, and in noncubic compounds, this amounts to magnetic anisotropy. The corresponding contribution to K1 , which can exceed 0.1 MJ/m3 , is especially important in some noncubic Gd-containing magnets, because Gd combines a large atomic moment (7 μB ) with zero crystalfield anisotropy due to its half-filled 4f shell. In cubic magnets, the anisotropy arising from Eq. (58) is exactly zero [1], because it is a second-order anisotropy. The anisotropy of Fig. 20 is closely related to the phenomenon of shape anisotropy. If a homogeneously magnetized magnet has the shape of an ellipsoid, then H(r) in Eq. (59) is also homogeneous inside the magnet (demagnetizing field). For ellipsoids of revolution magnetized along the axis of revolution, H = –N M, where N is the demagnetizing factor, that is, N ≈ 0 for long needles, N = 1/3 for spheres, and N ≈ 1 for plate-like ellipsoids [10, 114]. Turning the magnetization in a direction perpendicular to the axis of revolution yields H ⊥ = – 1–2N M. Putting H|| and H⊥ into Eq. (59) and comparing the energies EMS yields the shape anisotropy constant: Ksh = μo 1 − 3 N M2 4 (60) This constant adds to the magnetocrystalline anisotropy constant, Keff = K1 + Ksh . However, some precautions are necessary when using this equation. Consider a slightly elongated magnet with N = 1/4 and zero magnetocrystalline anisotropy. Equation (60) then predicts a positive net anisotropy constant Keff = μo M2 /16, corresponding to a preferred magnetization direction parallel to the axis of revolution. This is contradictory to the experiment. In fact, the “shape anisotropy” of macroscopic magnets is merely a demagnetizing field energy. The demagnetizing factor N is defined for uniform magnetization, corresponding to the Stoner-Wohlfarth model in micromagnetism, and this nanoscale uniformity is also exploited to evaluate Eq. (59). However, in 3 Anisotropy and Crystal Field 161 Fig. 21 Micromagnetic nature of shape anisotropy in a slightly prolate but defect-free ellipsoid. Imperfections, including nonellipsoidal edges, cause reduced nucleation fields (coercivities), which is known as Brown’s paradox macroscopic magnets, the magnetization state becomes nonuniform (incoherent) due to magnetization curling [9]. The curling leads to vortex-like magnetization states for which a shape anisotropy can no longer be meaningfully defined. Curling reflects the strength of the magnetostatic interaction relative to the interatomic exchange and occurs when the radius of the ellipsoid exceeds the coherence radius Rcoh ≈ 5(A/μo Ms 2 )1/2 , or about 10 nm for a broad range of ferromagnetic materials. Figure 21 elaborates the micromagnetic character of shape anisotropy by showing the external nucleation field (coercivity) as a function of the particle radius. Atomic-scale magnetism, as in Fig. 20, is realized on a sub nm length scale. On this scale, the interatomic exchange is sufficient to ensure a parallel spin alignment, and the magnetic anisotropy is a well-defined quantity. Elongated nanoparticles, for example, fine-particle magnets such as Fe amalgam [13], have radii of the order of 10 nm and are well-described by Eq. (60). Shape anisotropy is also exploited in alnico magnets [115–118], which contain needles of high-magnetization FeCo embedded in a nonmagnetic NiAl matrix. The radius R of the needles is smaller than about 50 nm but substantially larger than Rcoh , which reduces the shape anisotropy by a factor Rcoh 2 /R2 [9]. Néel’s Pair-Interaction Model The magnetocrystalline anisotropies of Sects. “Rare-Earth Anisotropy” and 5 are single-ion anisotropies, that is, they can be expressed in terms of atomic spin operators such as ŝz 2 . The underlying physical phenomenon is the spin-orbit 162 R. Skomski et al. coupling, which is separately realized in each atom and described by Eq. (21). The single-ion mechanism does not exclude interactions between spins, such as exchange, but the net anisotropy of a magnet is obtained by adding all single-ion contributions. Examples of two-ion anisotropies are the magnetostatic anisotropy, just discussed and epitomized by Eq. (58), and Néel’s phenomenological pairinteraction model [119]. The latter uses an expansion of the anisotropy energy in direction cosines. In the lowest order, the pair energy is equal to L (cos2 ψ – 1/3), where L is a phenomenological parameter and ψ is the angle between bond axis and magnetization direction. Néel’s expression is reproduced by putting m = m’ in Eq. (58), that is, by assuming a uniform magnetization direction. Single-ion and Néel two-ion anisotropies yield anisotropy-energy expressions of the correct symmetry, but this does not mean that they are physically equivalent. For example, both magnetic and nonmagnetic atoms contribute to the crystal field acting on rare-earth ions, but the latter make no contribution in the Néel model, because is based on pairs of magnetic atoms. Nonmagnetic ligands yield big anisotropy effects in some materials, such as Sm2 Fe17 interstitially modified by N or C [41, 120]. The alloy crystallizes in the rhombohedral Th2 Zn17 structure, where each Sm atom is coordinated by three 9e interstitial sites, as shown in Fig. 22(a). The anisotropy of Sm2 Fe17 is easy plane, that is, the Sm moment lies in the x-y-plane plane, which also contains the 9e triangle. Heating powders of Sm2 Fe17 in N2 gas (gas-phase interstitial modification) causes the nitrogen atoms to occupy the 9e interstices, yielding the approximate composition Sm2 Fe17 N3 . The nitrogen addition changes the anisotropy from easy-plane (K1 = −0.8 MJ/m3 ) to easy-axis (K1 = 8.6 MJ/m3 ), because the virtually nonmagnetic N atoms act as strongly negative crystal-field charges and repel the tips of the 4f charge distribution, Fig. 22(b). One- and two-ion anisotropies are difficult to distinguish experimentally, partly because interatomic exchange keeps neighboring spins parallel. The temperature dependence of the anisotropy is sometimes used as a criterion, scaling as K1 (T) ∼ Ms (T)2 for magnetostatic anisotropy. However, a very similar behavior is observed in L10 magnets such as FePt and CoPt, where the anisotropy is of the single-ion type but requires proximity spin polarization of the 5d electrons by the 3d electrons [51, 52]. Two-Ion Anisotropies of Electronic Origin Two-ion anisotropy is sometimes equated with magnetostatic anisotropy, but there are also quantum-mechanical two-ion mechanisms [121]. The simplest example is the two-ion anisotropy model described by the S = 1/2 Hamiltonian: H = –Jxx Ŝx · Ŝx –Jyy Ŝy · Ŝy –Jzz Ŝz · Ŝz (61) In the isotropic Heisenberg model, J xx = J yy = J zz = J , but generally J xx = J J zz due to spin-orbit coupling. There is no single-ion anisotropy in the model, = because the operator equivalent O 2 0 (S) = 3 Sz 2 – S(S + 1) is zero for S = 1/2 and yy 3 Anisotropy and Crystal Field 163 Fig. 22 Anisotropy of Sm2 Fe17 N3 : (a) interstitial sites surrounding the Sm3+ ion in Sm2 Fe17 (blue) and (b) change of the easy-axis direction due to interstitial nitrogen (yellow). Since this anisotropy mechanism involves one magnetic atom only, it cannot be cast in form of a Néel interaction Sz = ± 1/2, but the “combined” spin S = 1, with Sz = 0 and Sz = ±1, supports second-order anisotropy. In the uniaxial limit, J xx = J yy = J o + J and J zz = J o – 2 J , where J o is the isotropic Heisenberg exchange and J is relativistically small. Diagonalization of Eq. (61) yields a singlet (S = 0) with wave function |↓↑ – ↑↓ > and energy 3J o /4, as well as triplet (S = 1). The triplet contains the Sz = ± 1 states |↑↑ > and |↓↓>, both of energy – J o /4 – J /2, and the Sz = 0 state |↓↑ + ↑↓>, which has the energy – J o /4 + J . Figure 23 shows the corresponding energy levels for J o > 0 and an anisotropy splitting 3 J /2 > 0. The anisotropic part of the triplet energy can be written as Ea = − J 2 3Sz 2 − S (S + 1) (62) Formally, this expression is a spin Hamiltonian in form of an operator equivalent, but here the spin S is the combined spin of the two atoms. J reflects spin-orbit coupling, very similar to singleThe parameter ion anisotropy and Dzyaloshinski-Moriya interactions. As emphasized in the introduction, the Heisenberg model is isotropic, even if the bond distribution 164 R. Skomski et al. Fig. 23 Level splitting for the two-ion model of Eq. (61). The anisotropic triplet is very similar to an L = 1 or J = 1 term in ionic crystal-field theory, except that the two coupled spins reside on different ions is anisotropic, for example, in a thin film. For example, ignoring spin-orbit coupling and trying to explain electronic two-ion anisotropy in terms of SlaterPauling-Néel distance dependences yields lattice-anisotropic exchange constants Jo (z – z ) = Jo (x – x ) but does not reproduce the spin-anisotropic exchange constants Jzz (r – r ) = Jxx (r – r ) required in Eq. (61). Anisotropic exchange is usually mixed with single-ion anisotropy and relatively small, as exemplified by hexagonal Co, whose saturation magnetization decreases by about 0.5% on turning the magnetization from the easy magnetization direction into the basal plane [122]. The small parameter involved is essentially K1 Vat /Jo , so that the effect can be enhanced by reducing Jo . However, since Tc ∼ Jo , this strategy is limited to lowtemperature magnets [123, 124]. Starting from the isotropic Heisenberg model (J ), the addition of an anisotropy term Ea ≈ –K1 Sz 2 and putting K1 = ∞ yields the classical single-ion Ising model [125–129]. The model, which has greatly advanced the understanding of thermodynamic phase transitions, is characterized by Sz = ±S, whereas Sx = Sy = 0 reflects the “squeezing” of quantum-mechanical degrees of freedom due to the high anisotropy. The model requires S ≥ 1, because Eq. (52) yields zero anisotropy for S = 1/2. However, the underlying quantum-fluctuations are ignored in classical models in the first place, and it is common to interpret the Ising model as a classical spin1/2 model. Quantum-mechanical Ising models are obtained by putting J xx = J yy = 0 in Eq. (61) while allowing nonzero values of Sx and Sy , for example, in a transverse magnetic field [130, 131]. Such two-ion models are important in the context of quantum-phase transitions. Dzyaloshinski-Moriya Interactions An interaction phenomenon closely related to single-ion anisotropy, electronic pair anisotropy, and anisotropic exchange is the Dzyaloshinski-Moriya (DM) interaction HDM = − ½ ij Dij · Si × Sj [132–135], where i and j refer to neighboring atoms. The DM vector Dij = − Dji reflects the local environment of the magnetic atoms and is nonzero only in the absence of inversion symmetry. Like the spin- 3 Anisotropy and Crystal Field 165 orbit coupling, the DM interaction is derived from the Dirac equation and is of the same order relativistically. Phenomenologically, the interaction favors noncollinear spin states, because HDM = 0 if the spins Si and Sj are parallel. Micromagnetically, the DM interactions can be expressed in terms of magnetization gradients ∇S and then assume the form of Lifshitz invariants. The corresponding energy contributions depend on the point group of the crystal or film and are zero even for some crystals without inversion symmetry [136]. DM interactions occur in some crystalline materials, such as α-Fe2 O3 (haematite), in amorphous magnets, in spin glasses, and in magnetic nanostructures [37, 135, 137]. The resulting canting is small, because D competes against the dominant Heisenberg exchange J, but the canting is easily observed in hematite and other canted antiferromagnets where there is no ferromagnetic background. The micromagnetism of the DM interactions [138] and its competition with singleion anisotropy is important in the context of magnetic vortices, for example, in MnSi [139]. The spin angles between neighboring atoms are comparable to angles encountered in domain walls, of the order of 1◦ for material ordered at room temperature, which reflects the common relativistic origin of both phenomena (D in the DM interactions and K1 determining the domain-wall width). DM noncollinearities are not be confused with noncollinearities caused by competing Heisenberg exchange interactions. Antiferromagnetic Anisotropy Magnetic anisotropy is not restricted to ferromagnets, because the single-ion mechanism is operative in each magnetic sublattice. As in ferromagnets, the net anisotropy is obtained by adding all sublattice anisotropy contributions. The resultant is usually nonzero; single-ion anisotropy requires a magnetic moment on each atom, but it does not require a nonzero net magnetization. An example is CoO, where K1 ≈ 1 MJ/m3 [78]. Antiferromagnetic anisotropy can, in principle, be extracted from the spin-flop field. When the antiferromagnet is subjected to a sufficiently strong magnetic field parallel to easy axis, the net magnetization jumps from zero to a finite value [129]. The corresponding spin-flop field Hsf μo μB Hsf = 2 K1 Vat (J ∗ − K1 Vat ) (63) reflects the competition between intersublattice exchange J * and anisotropy K1 . Snce J * Vat K1 in most materials, Hsf Ha , and high fields are needed to produce the spin-flop, even in fairly soft materials. The anisotropy energy remains unchanged on reversing the magnetization direction, Ea (M) = Ea (−M). This means that there should be no odd-order anisotropy contributions. However, exchange bias caused by the exchange coupling of a ferromagnetic and an antiferromagnetic phase yields an apparent unidirectional 166 R. Skomski et al. anisotropy on cooling through a blocking temperature that was first observed as an asymmetric shift of the hysteresis loop by Meiklejohn and Bean 1956 [140], in their study Co nanoparticles coated with an antiferromagnetic CoO layer. Exchange bias may be best characterized as an inner-loop effect, caused by the external field’s inability to overcome the high anisotropy field of the antiferromagnetic phase. Magnetoelastic Anisotropy Straining a magnet with a cubic crystal structure yields a noncubic structure with nonzero second-order magnetic anisotropy. This mechanism contributes, for example, to the magnetic anisotropy of steel (Sect. “Case Studies”). The same consideration applies to isotropic magnetic materials, such as amorphous and polycrystalline magnets, if they are rolled and extruded. However, the change in K1 is usually very small for metallurgically sustainable strain. Magnetoelastic anisotropy is also important in soft magnets, especially in permalloy-type materials (Fe20 Ni80 ), where the cubic anisotropy is small and the magnetoelastic contribution, caused by magnet processing or a substrate, often dominates the total anisotropy. Magnetoelastic anisotropy is physically equivalent to magnetocrystalline anisotropy, because a strained lattice is merely an unstrained lattice with modified atomic positions. For example, the atomic environment in Fig. 1 can be considered as a tetragonally strained cubic environment. In many cases it is sufficient to describe a uniaxially strained isotropic medium by the magnetoelastic energy: HME V =− λs E E 3 cos2 θ − 1 ε + ε2 − ε σ 2 2 (64) where σ is the uniaxial stress, ε = l/l denotes the elongation along the stress axis, E is Young’s modulus, and θ is the angle between the magnetization and strain axis. The strength of the magnetoelastic coupling is described by the saturation magnetostriction λs . Putting σ = 0 and θ = 0 and minimizing the magnetoelastic energy with respect to ε yields the elongation ε = λs . A magnet which has a spherical shape in the paramagnetic state becomes a prolate ferromagnet when λs > 0 but an oblate ferromagnet when λs < 0. Physically, the spin alignment creates, via spin-orbit coupling, an alignment of the atomic electron distributions and a change in lattice parameters. Since λs is only 10–100 ppm in most ferromagnetic compounds, a moderate stress σ = Eε can outweigh the spontaneous magnetostriction. This then produces a magnetoelastic anisotropy energy density: HME V =− λs σ 3 cos2 θ − 1 2 (65) 3 Anisotropy and Crystal Field 167 and the magnetoelastic contribution to K1 KME = 3λs σ /2, which may be fairly large. For cubic crystallites, there are two independent magnetostriction coefficients in the lowest order, and the polycrystalline average over all possible orientations is [141] λs = 2 3 λ100 + λ111 5 5 (66) where the quantities λ100 and λ111 are the spontaneous magnetostriction along the cube edge and the cube diagonal, respectively. Experimental room-temperature values of λs , measured in parts per million (10−6 ), are −7 for Fe, −33 for Ni, +75 for FeCo, +40 for Fe3 O4 , −1560 for SmFe2 , and + 1800 for TbFe2 , and practically zero for Py (permalloy, Fe20 Ni80 ) [11, 70, 115]. For example, highcarbon steel (Fe94 C6 ) has E = 200 GPa and is strained by about 5% [103], so that KME ≈ −0.1 MJ/m3 (see the discussion of steel magnets in Sect. “Case Studies”). Low-Dimensional and Nanoscale Anisotropies Nanostructuring opens a new dimension to anisotropy research and practical applications. Surface and interface anisotropies become important on the nanoscale, and it is possible to realize atomic environments not encountered in the bulk [9, 142]. Examples are thin films and multilayers, nanowires, single atoms, molecules, and nanodots on surfaces, nanogranular thin-film, and bulk magnets [142]. Figure 24 shows some of these nanostructures, whose dimensions range from less than 1 nm, for adatoms and monatomic nanowires, to 100 nm or more in nanostructured composites. Most structures can be produced freestanding or deposited on substrates, and advanced techniques are available for their fabrication and characterization (see the other chapters of this book and Refs. [15, 143, 144]). From a theoretical viewpoint, arbitrarily small anisotropies are important in the theory of two-dimensional phase transitions, because they can change the universality class from Heisenberg-like to Ising-like and even create a nonzero Curie temperature [145, 146]. Surface Anisotropy Surface and interface anisotropies, which are closely related, play an important role in magnetic thin films and nanostructures. Surface anisotropies easily dominate the bulk anisotropy in nanostructures with cubic or amorphous crystal structures, but surface and interface contributions are also of interest in noncubic systems. For example, L10 -ordered magnets such as FePt and CoPt can be considered as naturally occurring multilayers. In line with other 3d anisotropies, the sign and magnitude of surface anisotropies are difficult to predict, but some crude rules of thumb exist for 168 R. Skomski et al. Fig. 24 Anisotropic nanostructures: (a) thin films (L10 -FePt/MgO), (b) free-standing Pd zigzag nanowire, (c) monatomic Fe nanowire on Pt(001), and (d) Co adatom on an insulating substrate. First-principle calculations often use periodic arrays of supercells with sufficiently big airgaps (a) the anisotropy as a function of band filling [60, 61]. For example, anisotropy often changes sign between Fe and Co, the Fe preferring an easy axis perpendicular to the Fe-Fe bonds (perpendicular to the plane). Surface anisotropy tends to dominate when the thin-film thickness is in the range of a few atomic layers. Phenomenologically [88, 147] K1 = KS /t + KV (67) where t is the film thickness, KS is the surface anisotropy, and KV includes the bulk magnetocrystalline and shape anisotropies. Typical iron-series surface anisotropies are of the order of 0.1–1 mJ/m2 [147], or 0.03–0.3 meV per surface transitionmetal atom, which corresponds to bulk equivalents of 0.5–5 MJ/m3 . When KV and KS favor in-plane and perpendicular anisotropy, respectively, then there is a spinreorientation transition from perpendicular to in-plane as the thickness exceeds KS /|KV |. Note that Eq. (67) does not mean that the anisotropy is limited to the surface: the equation is asymptotic, with small contributions from subsurface atoms and from atoms deeper in the bulk. Thin-film, multilayer, surface, and interface anisotropies have the same physical origin as bulk anisotropies, mostly single-ion anisotropy with magnetostatic correc- 3 Anisotropy and Crystal Field 169 Fig. 25 Effect of surface index on the surface of bcc Fe: (a) fourth-order in-plane anisotropy for a (001) surface and (b) second-order in-plane anisotropy for a (011) surface. Gray atoms are subsurface atoms [148] tions. The anisotropic distribution of exchange bonds at the surface does not create magnetic anisotropy. The Heisenberg Hamiltonian is isotropic, even if the exchange bonds Jij = J(ri – rj ) are anisotropic. Only relative angles between neighboring spins matter, and the Heisenberg model is silent about the easy magnetization directions. Both the easy axes and the strength of the anisotropy depend on the index of the surface, and there is no reason to expect that the anisotropy axis should necessarily be normal to the surface. For example, the (100) surface of bcc Fe, Fig. 25(a), has fourfold in-plane symmetry and yields a fourth-order anisotropy contribution. By comparison, the (011) surface, Fig. 25(b), has a twofold in-plane symmetry, which yields two nonzero lowest-order anisotropy constants , K1 and K1 [148]. Surface defects often yield substantial anisotropy contributions [88, 144]. Stepped surfaces are an example, which can also be considered as high-index surfaces [144, 149]. Random Anisotropy in Nanoparticles, Amorphous, and Granular Magnets Many magnetic materials are characterized by random easy axes n(r), so that the uniaxial anisotropy-energy expression K1 sin2 θ must be replaced by Ha = – K1 (n · s)2 dV (68) where s = M(r)/Ms . Atomically, K1 in Eq. (68) is the same as the K1 in Sect. “Lowest-Order Anisotropies”, the only difference being the randomness of the local c-axis. Random anisotropy is important in a variety of materials, including hard and soft-magnetic polycrystalline solids [150–155], amorphous magnets [124, 137, 156], spin glasses [135], and nanoparticles [143, 157]. One example is the approach to saturation in polycrystalline materials (Sect. “Anisotropy Measurements”). Nanoparticles and nanoclusters are defined very similarly, but in a strict sense, the former are random objects, whereas the latter are characterized by well-defined atomic positions. Typical nanoparticles contain surface patches with many different indexes, and the corresponding anisotropy contributions add. The net anisotropy of nanoparticles is generally biaxial, involving both K1 and K1 , and there is generally no physical justification for considering nanoparticles as 170 R. Skomski et al. uniaxial magnets. This can be seen from Eq. (3): aside from accidental degeneracies, there is always one axis of lowest energy. However, uniaxiality goes beyond having an axis of lowest energy (easy axis), because it also requires the absence of “secondary” anisotropy axes perpendicular to the easy axis. The secondary anisotropy is important, because it causes hysteresis loops to deviate from the uniaxial predictions. Consider a nanoparticle with a highly disordered surface, so that each of the NS surface atoms yields an anisotropy contribution ±Ko , where Ko is 0.03–0.3 meV (Sect. “Surface Anisotropy”) and ± refers to orthogonal easy axes. For NS = ∞, the surface anisotropy would average to zero, but in patches of finite NS , the averaging is incomplete. The addition of NS random contributions √ ±Ko creates a Gaussian distribution√of net anisotropies of the order of ±Ko / NS per surface atom [9], or Keff = Ko NS /N averaged over all N atoms in the particle. Here the negative sign means that the easiest axis switches into a direction perpendicular to the reference axis (z-axis). Since Ns ∼ R2 and N ∼ R3 , Keff scales as 1/R. Atomic-scale random-anisotropy effects in bulk solids were first discussed in the context of amorphous magnets, which exhibit random-field [158], randomanisotropy [159], and random-exchange spin glasses [135, 137]. In less than four dimensions, the ground state of random-anisotropy magnets does not exhibit longrange ferromagnetic order [135]. However, this does not preclude the use of random-anisotropy magnets as nanostructured magnetic materials, where hysteretic properties are important [155, 160] and true equilibrium is rarely reached. The coercivity and remanence of atomic-scale random-anisotropy magnets were first investigated in the late 1970s [151, 156], but a very similar situation is encountered in nanocrystalline magnets [161, 162]. The random anisotropy in Eq. (68) creates magnetic hysteresis. In the case of noninteracting random-anisotropy grains, which also includes noninteracting nanoparticles, the M(H) loops are obtained by adding the Zeeman interaction –μo Ms H·s dV to Eq. (68), finding the M(H) loop for each direction n, and then averaging over all n. In terms of Ha = 2 K1 /μo Ms , the behavior near remanence is M(H) = Ms (1/2 + 2H/3Ha ). In particular, the remanence ratio Mr /Ms = M(0)/Ms is equal to 1/2. Performing the same analysis for cubic magnets with iron-type (K1 > 0) and nickel-type (K1 < 0) anisotropy yields the remanence ratios 0.832 and 0.866, respectively. Replacing the easy-axis anisotropy by easy-plane anisotropy yields a very similar curve for H > 0, namely, M(H) = Ms (π/4 + H/3Ha ), and the same asymptotic behavior (Sect. “Anisotropy Measurements”). However, the coercive behavior is very different: random easy-axis anisotropy yields Hc = 0.479 Ha , whereas easy-plane anisotropy leads to Hc = 0. Intergranular exchange modifies the hysteresis loops, creating some coercivity in the easy-plane ensembles but reducing the coercivity in the case of easyaxis anisotropy. The exchange energy density, A(∇σ m )2 , is largest for rapidly varying magnetization directions σ m , so that exchange effects are most pronounced grain with small radius R. In the weak-coupling limit, A/R2 K1 , there are quantitative corrections to the hysteresis loop [9], but the strong-coupling behavior is qualitatively different. 3 Anisotropy and Crystal Field 171 In the limit of infinite exchange, all grain magnetizations would be parallel, σ m (r) = σ mo , and the average anisotropy of Eq. (68) would be zero by symmetry for isotropic magnets. Large but finite exchange means that N grains are coupled ferromagnetically, where N increases with A. Each grain yields an anisotropy contribution ±K1 , but as in the above nanoparticle √ case, the anisotropy does not average to zero but exhibits a distribution ±K1 / N. This yields the total energy density: η= 1 A − K1 √ 2 L N (69) where L is the magnetic correlation length, that is, the radius of the correlated regions. In d dimensions, it is given by N = (L/R)d . Putting this expression into Eq. (69) and minimization with respect to L yields the scaling relation L ∼ R (δo /R)4/(4–d) (70) where δ o = (A/K)1/2 is the domain-wall-width parameter. Equation (70) shows that d = 4 is a marginal dimension below which small grains (R < δ o ) yield intergranular correlations (L√> R). In three dimensions, L ∼ 1/R3 . Since K1 / N can be considered as an effective anisotropy, the formation of correlated regions reduces the coercivity: Hc ∼ Ha (R/δo )2d/(4–d) (71) In three dimensions, this means that the coercivity of random-anisotropy magnets scales as R6 [156]. This dependence helps to reduce the coercivity of soft-magnetic materials [155]. For example, K1 is virtually zero for amorphous alloys Fe40 Ni40 B20 and Gd25 Co75 [37]. Random anisotropy magnets having large grain sizes are in a weak-coupling regime and exhibit high coercivities of the order of 2K1 /μo Ms , and there is a fairly sharp transition between the strong-coupling (small R) and weakcoupling (large R) regimes. Giant Anisotropy in Low-Dimensional Magnets Very high anisotropies per atom are possible in small-scale nanostructures such as adatoms on surfaces or monatomic wires. These high anisotropies indicate unquenched orbital moments , due to either high spin-orbit coupling or high crystalfield symmetry. The former is realized for Co atoms on Pt(111) [163], where a giant magnetic anisotropy of about 9 meV per Co atom has been measured. Platinum is predisposed toward strong anisotropy, because it is close to the onset of ferromagnetism and possesses a spin-orbit coupling of about 550 meV. A single atom of Fe or Co easily spin-polarizes several Pt atoms, which then make large contributions to the anisotropy. Atomically thin nanowires, such as the zigzag wire 172 R. Skomski et al. in Fig. 24(b), may support very high anisotropy, partly due to pronounced van-Hove peaks in the density of states. In terms of Eq. (53), van-Hove singularities near the Fermi level correspond to small energy differences Eu – Eo . For example, an anisotropy of 5.36 meV per atom has been predicted for free-standing ladders of Pd atoms [164]. An upper limit to the anisotropy per atom is given by the spin-orbit coupling constant, λ ≈ 50 meV for the late iron-series transition metals. This huge value corresponds to 140 MJ/m3 for dense-packed atoms. It is unlikely that this anisotropy could be exploited in nanotechnology, because anisotropy is defined as anisotropy energy per unit volume and the requirement of isolated or freestanding wires leads to a dilution of the anisotropy. Densification is incompatible with such huge anisotropy, because crystal formation involves interactions of the order 1000 meV, which tend to quench the orbital moment. Quenching is ineffective in free-standing monatomic nanowire however, and anisotropy energies of 20–60 meV have been predicted or experimentally inferred for these structures. In 3d systems, anisotropies as high 6–20 meV/atom have been calculated for free-standing linear monatomic Co wires [165]. Some monatomic 4d and 5d wires exhibit larger anisotropies, up to 60 meV per atom in stretched Rh and Pd, respectively [166]. The high anisotropy of freestanding monatomic nanowires indicates that some levels undergo little or no quenching. The wires have C∞ symmetry, which leaves the states with nonzero Lz , namely, |xz> and |yz> (Lz = ± 1) and |xy> and |x– y2 > (Lz = ± 2), completely unquenched so long as the spin is parallel to the symmetry axis of the wire (z-axis). Figure 26 compares the corresponding level splitting with the tetragonal one in Fig. 6. Physically, the electrons freely orbit around the wires, because there are no in-plane crystal-field charges that could perturb this motion. The corresponding wave functions, |xz > ± i |yz > and |xy > ± i |x– y2 >, yield anisotropy energies of up to λ and 2λ, respectively, depending on the number of electrons in the system. Configurations similar to Fig. 6 also exist in a few crystalline environments. Recent experiments have indicated that a Co ad-atom deposited on MgO shows the giant magnetic anisotropy of 58 meV [167]. This huge anisotropy requires a degeneracy between two levels of equal |Lz |. Co adatoms on MgO(001) have C4 symmetry. Due to Hund’s rules, the Co2+ ion (3d7 ) has one electron in the xy-xz doublet, and this degeneracy yields a large orbital moment, <Lz > ≈ 1, and a huge anisotropy. The example of Co on MgO shows high anisotropy energies can also be obtained in some crystalline environments. The C4 argument can be extended to vertically embedded but laterally isolated wires. Such configurations might conceivably be used for magnetic recording. In terms of thermal stability, 50 meV corresponds to 580 K, or about 2kB T per atom. For magnetic recording, one would need about 50 kB T, or chain lengths of 25 strongly exchange-coupled 3d atoms. Heavier elements have stronger spin-orbit couplings but cannot be used for this purpose, because 3 Anisotropy and Crystal Field 173 Fig. 26 Crystal-field splitting in an insulating free-standing nanowire. Since states with the same quantum number |Lz | > 0 form degenerate states (doublets), there is no quenching, and large magnetocrystalline anisotropies K1 Vo ∼ λ are possible their interatomic exchange is too small to ensure ferromagnetic alignment at room temperature. It is uncertain whether any of the approaches outlined in this section could be used to improve areal recording densities. Multiferroic aspects of magnetic anisotropy are an important aspect of current research in solid-state physics and nanoscience. Electric-field control of magnetic anisotropy in magnetic nanostructures could enable entirely new device concepts, such as energy-efficient electric field-assisted magnetic data storage. Due to screening by conduction electrons in metals, there is no electric-field dependent bulk anisotropy, but the surface anisotropy changes via the filling of the 3d orbitals, which is modified by the electric field. This was demonstrated L10 -FePd and FePt thin films immersed in a liquid electrolyte [168], where the coercivity can be modified by an applied electric field. A common scenario is that an electric field yields a modest change in K1 , which modifies the coercivity of the films and could be exploited for magnetization switching [169, 170]. Similar mechanisms are realized in nanowires on substrates, Fig. 24 [171]. For example, the application of an electric field has been predicted to change the sign of K1 of organometallic vanadium-benzene wires [172]. Mechanical strain and adsorbate atoms on thin films may have a similar effect [16]. Acknowledgments This chapter has benefited from discussions with B. Balamurugan, C. Binek, R. Choudhary, J. Cui, P. A. Dowben, A. Enders, O. Gutfleisch, G. C. Hadjipanayis, H. Herper, X. Hong, S. S. Jaswal, P. Kharel, M. J. Kramer, P. Kumar, A. Laraoui, L. H. Lewis, S.-H. Liou, J.-P. Liu, R. W. McCallum, O. N. Mryasov, D. Paudyal, R. Sabirianov, S. S. Sankar, T. Schrefl, D. J. Sellmyer, J. E. Shield, A. K. Solanki, and A. Ullah. The underlying work was or has been supported by ARO (W911NF-10-2-0099), DOE (DE-FG02-04ER46152), NSF EQUATE (OIA2044049), partially NSF-DMREF (1729288), HCC, and NCMN. 174 R. Skomski et al. Appendices Appendix A: Spherical Harmonics Separating radial (r) and angular (θ , φ) degrees of freedom, any function f (θ , φ) can be expanded into spherical harmonics Yl m (θ , φ). The present chapter uses this expansion to describe (i) atomic wave functions ψ(r), as in Figs. 4 and 11, (ii) atomic charge densities n(r), (iii) crystal-field potentials V(r) and operator equivalents O m l , and (iv) magnetic anisotropy energies Ea (θ , φ). These quantities differ by radial part and physical meaning, but their angular dependences are all described by m Yl m (θ, φ) = Nl exp (imφ) Pl m (cos θ ) (72) where the Pl m are the the associated Legendre polynomials. Concerning sign and magnitude of the normalization factor N l m , we use the convention (2l + 1) (l − m)! 4π (l + m)! Nl m = (73) It is sometimes useful to express Eq. (1) in terms of Cartesian coordinates or “direction cosines” x, y, and z. Last but not least, the complex functions exp.(imφ) may be replaced by real functions, using exp.(±imφ) = cos (mφ) ± i sin(mφ). These real spherical harmonics, also known as tesseral harmonics, are often convenient, because charge densities, crystal-field potentials, and anisotropy energies are real by definition. However, the distinction remains important in quantum mechanics, because complex and real spherical harmonics correspond to unquenched and quenched wave functions, respectively. A very frequently occurring function is Y2 0 = 1 2 5 3 cos2 θ –1 4π (74a) or Y2 0 1 = 2 5 3z2 − r 2 4π r2 (74b) Note that the Cartesian coordinates require a factor 1/rl , which ensures that the Yl are dimensionless and that the expansion is in terms of direction cosines x/r, y/r, and z/r. Up to the sixth order, there are Table 16 lists real and complex spherical harmonics up to the sixth order. m 3 Anisotropy and Crystal Field 175 Table A.16 Spherical harmonics √ in several representations. For m = 0, the real representation requires an additional factor 1/ 2, because the normalization behavior of cos(mφ) ± sin(mφ) differs from that of exp.(imφ). Furthermore, the minus sign in N l m is not used for m = 0. The following formulae can be used to extract the full spherical harmonics from the table: Yl m = π-1/2 fN fP exp.(imφ), Yl m = π-1/2 fN fR /rl (m = 0), and Yl m = (2π)-1/2 fN fR /rl (m = 0). Anisotropy energies involve even-order spherical harmonics only (gray rows) Function 0 Y0 Y1 1 Y1 0 Y1 – 1 Y2 2 Y2 1 Y2 0 Y2 – 1 Y2 – 2 Y3 3 Y3 2 Y3 1 Y3 0 Y3 – 1 Y3 – 2 Y3 – 3 Y4 4 Y4 3 Y4 2 Y4 1 Y4 0 Y4 – 1 Y4 – 2 Y4 – 3 Y4 – 4 Y5 5 Y5 4 Y5 3 Y5 2 Y5 1 Y5 0 Y5 – 1 Y5 – 2 Y5 – 3 Y5 – 4 Y5 – 5 Y6 6 Y6 5 Y6 4 Y6 3 Y6 2 Y6 1 Y6 0 Y6 – 1 Y6 – 2 Y6 – 3 Y6 – 4 Y6 – 5 Y6 – 6 fN = S ࣨm fP = Plm fR = rlYlm/ࣨ m 1/2 1 1 x sinT – 1/2 · 3/2 z cosT 1/2 · 3 y sinT 1/2 · 3/2 x2 – y2 sin2T 1/4 · 15/2 xz sinT cosT – 1/2 · 15/2 3 z2 – r2 3 cos2T – 1 1/4 · 5 yz sinT cosT 1/2 · 15/2 2 xy sin2T 1/4 · 15/2 x3 – 3 xy2 sin3T – 1/8 · 35 2 z (x2 – y2) sin T cosT 1/4 · 105/2 x (5 z2 – r2) sinT cos2T – cosT – 1/8 · 21 z (5 z2 – 3r2) cos3T – 3 1/4 · 7 2 y (5 z2 – r2) sin cos – cos T T T 1/8 · 21 2 2 xyz sin T cosT 1/4 · 105/2 3 x2y – y3 sin3T 1/8 · 35 x4 – 6 x2y2 + y4 sin4T 3/16 · 35/2 3 xz (x2 – 3 y3) sin T cosT – 3/8 · 35 2 2 2 (x – y2) (7 z2 – r2) sin cos – 1 T T 3/8 · 5/2 3 xz (7 z2 – 3 r2) sinT cos T – 3 cosT – 3/8 · 5 4 2 3/16 35 z4 – 30 z2r2 + 3 r4 35 cos T – 30 cos T + 3 yz (7 z2 – 3r2) sinT cos3T – 3 cosT 3/8 · 5 2 2 2 xy (7 z2 – r2) sin T cos T – 1 3/8 · 5/2 yz (3 x2 – y2) sin3T cosT 3/8 · 35 4 xy (x2 – y2) sin4T 3/16 · 35/2 x (x4 – 10 x2y2 + 5 y4) sin5T – 3/32 · 77 z (x4 – 6 x2y2 + y4) sin4T cosT 3/16 · 385/2 2 3 2 – 3 y2)·(9 z2 – r2) x (x cos – 1 sin T T – 1/32 · 385 z (x2 – y2) (3 z2 – r2) sin2T cos3T – cosT 1/8 · 1155/2 z (21 z4 – 12 z2r2 + r4) sinT 21 cos4T – 14 cos2T + 1) – 1/16 · 165/2 z (63 z4 – 70 z2r2 + 15 r4) 63 cos5T – 70 cos3T + 15 cosT 1/16 · 11 y (21 z4 – 12 z2r2 + r4) sinT 21 cos4T – 14 cos2T + 1) 1/16 · 165/2 2 3 2 xyz (3z2 – r2) sin cos – cos T T T 1/8 · 1155/2 y (3x2 – y2) (9z2 – r2) sin3T cos2T – 1 1/32 · 385 4 xyz (x2 – y2) sin4T cosT 3/16 · 385/2 y (5 x4 – 10 x2y2 + y4) sin5T 3/32 · 77 x6 – 15 x4y2 + 15 x2y4 – y6 sin6T 1/64 · 3003 5 x (x4 – 10 x2y2 + 5 y4) sin cos T T – 3/32 · 1001 4 4 2 (x – 6 x2y2 + y4)(11 z2 – r2) sin T cos T – 1 3/32 · 91/2 3 3 xz (x2 – 3 y2)(11 z2 – 3 r2) sin T cos T – 3 cosT – 1/32 · 1365 (x2 – y2)(33 z4 – 18 z2r2 + r4) sin2T 33 cos4T – 18 cos2T + 1) 1/64 · 1365 5 3 xz (33 z4 – 30 z2r2 + 5 r4) – 1/16 · 273/2 sinT (33 cos T – 70 cos T + 5 cosT ) 6 4 2 231 z6 – 315 z4r2 + 105 z2r2 – 5 r6 231 cos – 315 cos + 105 cos – 5 T T T 1/32 · 13 5 3 yz (33z4 – 30z2r2 + 5r4) sinT (33 cos T – 70 cos T + 5 cosT ) 1/16 · 273/2 2 4 2 2 xy (33 z4 – 18 z2r2 + r4) sin T 33 cos T – 18 cos T + 1) 1/64 · 1365 3 yz (3 x2 – y2)(11 z2 – 3r2) sin3T cos3T – 3 cosT 1/32 · 1365 4 xy (x2 – y2)(11 z2 – r2) sin4T cos2T – 1 3/32 · 91/2 zy (5 x4 – 10 x2y2 + y4) sin5T cosT 3/32 · 1001 6 xy (6 x4 – 20 x2y2 – 6 y4) sin T 1/64 · 3003 176 R. Skomski et al. Appendix B: Point Groups Table A.17 Less common space and point groups. The space groups in bold characters are frequently encountered in magnetism and separately considered in the main text of the chapter (Table 1) Crystal system Triclinic Triclinic Monoclinic Monoclinic Monoclinic Orthorhombic Point group C1 (1) Ci (1) C2 (2) Cs (m) C2h (2/m) D2 (222) Orthorhombic C2v (mm2) Orthorhombic D2h (mmm) Tetragonal Tetragonal Tetragonal Tetragonal C4 (4) S4 (4) C4h (4/m) D4 (422) Tetragonal C4v (4 mm) Tetragonal D2d (42m) Tetragonal D4h (4/mmm) Trigonal Trigonal Trigonal Trigonal Trigonal Hexagonal Hexagonal Hexagonal Hexagonal Hexagonal Hexagonal Hexagonal Cubic C3 (3) S6 (3) D3 (32) C3v (3 m) D3d (3m) C6 (6) C3h (6) C6h (6/m) D6 (622) C6v (6 mm) D3h (6m2) D6h (6/mmm) T (23) Space group P1 P1 P2, P21 , C2 Pm, Pc, Cm, Cc C2/m, C2/c, P2/m, P21 /m, P2/c, P21 /c P222, P2221 , P21 21 2, P21 21 21 , C2221 , C222, F222, I222, I21 21 21 Pmm2, Pmc21 , Pcc2, Pma2, Pca21 , Pnc2, Pmn21 , Pba2, Pna21 , Pnn2, Cmm2, Cmc21 , Ccc2, Amm2, Aem2, Ama2, Aea2, Fmm2, Fdd2, Imm2, Iba2, Ima2 Pnma, Pmmm, Pnnn, Pccm, Pban, Pmma, Pnna, Pmna, Pcca, Pbam, Pccn, Pbcm, Pnnm, Pmmn, Pbcn, Pbca, Cmcm, Cmce, Cmmm, Cccm, Cmme, Ccce, Fmmm, Fddd, Immm, Ibam, Ibca, Imma P4, P41 , P42 , P43 , I4, I41 P4, I4 P4/m, P42 /m, P4/n, P42 /n, I4/m, I41 /a P422, P421 2, P41 22, P41 21 2, P42 22, P42 21 2, P43 22, P43 21 2, I422, I41 22 P4mm, P4bm, P42 cm, P42 nm, P4cc, P4nc, P42 mc, P42 bc, I4mm, I4cm, I41 md, I41 cd P42m, P42c, P421 m, P421 c, P4m2, P4c2, P4b2, P4n2, I4m2, I4c2, I42m, I42d P4/mmm, P42 /mnm, I4/mmm, P4/mcc, P4/nbm, P4/nnc, P4/mbm, P4/mnc, P4/nmm, P4/ncc, P42 /mmc, P42 /mcm, P42 /nbc, P42 /nnm, P42 /mbc, P42 /nmc, P42 /ncm, I4/mcm, I41 /amd, I41 /acd P3, P31 , P32 , R3 P3, R3 P32 12, P312, P321, P31 12, P31 21, P32 21, R32 P3m1, P31m, P3c1, P31c, R3m, R3c R3m, R3c, P31m, P31c, P3m1, P3c1 P6, P61 , P65 , P62 , P64 , P63 P6 P63 /mmc, P6/m, P63 /m P622, P61 22, P65 22, P62 22, P64 22, P63 22 P63 mc, P6mm, P6cc, P63 cm P6m2, P6c2, P62m, P62c P6/mcc, P63 /mcm P21 3, P23, F23, I23, I21 3 (continued) 3 Anisotropy and Crystal Field 177 Table A.17 (Continued) Crystal system Cubic Cubic Cubic Cubic Point group Td (43m) Th (m3) O (432) Oh (m3m) Space group F43m, I43m, P43m, P43n, F43c, I43d Pa3, Pm3, Pn3, Fm3, Fd3, Im3, Ia3 P432, P42 32, F432, F41 32, I432, P43 32, P41 32, I41 32 Fm3m, Im3m, Pm3m, Pn3m, Fd3m, Ia3d, Ia3d, Pn3n, Pm3n, Fm3c, Fd3c Appendix C: Hydrogen-Like Atomic 3d Wave Functions Hydrogen-like 3d wave functions are obtained by solving the Schrödinger equation for n = 3 (third shell) and l = 2 (d electrons). There are 2 l + 1 = 5 different orbitals, and each can be occupied by up to two electrons. Explicitly, where N = √ |xy> = R3d (r)sin2 θ sin 2φ (75) |x 2 − y 2 > = N R3d (r) sin2 θ cos 2φ (76) |xz> = 2N R3d (r) sin θ cos θ cos φ (77) |z2 > = R3d (r) 3 sin2 θ − 1 (78) |yz > = 2N R3d (r) sin θ cos θ sin φ (79) 15/16π, ao = 0.529 Å, and R3d (r) = 4Z 5/2 r 2 Zr exp − ao 81ao2 30ao3 (80) Aside from the real set of wave functions, there exist complex wave functions of the type exp.(±imφ). The two sets of wave functions are linear combinations of each other, and both are solutions of the Schrödinger equation. However, they are nonequivalent with respect to orbital moment and magnetic anisotropy. More generally, Ψ (r, φ, θ ) = Rn l (r) Yl m (φ, θ ), where it is convenient to express the radial wave functions in terms of the parameter ro = ao /Z: 2 r R1s = exp − ro ro 3 178 R. Skomski et al. R2s 1 = 2 2ro 3 R2p R3s r 1 r = exp − 2ro 2 6ro 3 ro 2 = 81 3ro 3 R3p r r 2− exp − ro 2ro r r r2 27 − 18 + 2 2 exp − ro 3ro ro r r r2 6 − 2 exp − ro 3ro ro 4 = 81 6ro 3 R3d = r2 4 r exp − 3ro 81 30ro 3 ro 2 R4f = r3 1 r exp − 4ro 768 35ro 3 ro 3 From the radial wave functions, the following averages are obtained: <r 2 > = n2 ro 2 2 5n + 1 − 3l (l + 1) 2 <r> = ro 2 3 n − l (l + 1) 2 <1/r> = <1/r 2 > = <1/r 3 > = 1 n2 r o 2 n3 ro 2 (2l + 1) 2 n3 ro 3 l (l + 1) (l + 2) 3 Anisotropy and Crystal Field 179 These formulae have numerous applications. For example, <r> and the square root of <r2 > are used to estimate shell radii, <1/r> gives the electronic energy, and <1/r3 > determines the strength of the spin-orbit coupling on which magnetocrystalline anisotropy relies. References 1. Bloch, F., Gentile, G.: Zur Anisotropie der Magnetisierung ferromagnetischer Einkristalle. Z. Phys. 70, 395–408 (1931) 2. Jäger, E., Perthel, R.: Magnetische Eigenschaften von Festkörpern. 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Spin Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Band Structure Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relativistic Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adiabatic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Itinerant Magnetism of Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stoner Model of Itinerant Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Slater-Pauling Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heusler Alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Electronic Energy and Magnetic Configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Total Electronic Energy and Magnetic Ground State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange Coupling Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magneto-Crystalline Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnon Dispersion Relations Based on the Rigid Spin Approximation . . . . . . . . . . . . . . . Spin Spiral Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Excitation Spectra Based on the Dynamical Susceptibility . . . . . . . . . . . . . . . . . . . . . . . . . Finite-Temperature Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods Relying on the Rigid Spin Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Methods Accounting for Longitudinal Spin Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . Coherent Treatment of Electronic Structure and Spin Statistics . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 190 190 196 201 204 208 208 211 213 217 218 220 224 230 231 233 237 240 240 242 244 249 H. Ebert () · S. Mankovsky · S. Wimmer München, Department Chemie, Ludwig-Maximilians-Universität, München, Germany e-mail: [email protected]; [email protected]; [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_4 187 188 H. Ebert et al. Abstract This chapter gives an overview on the various methods used to deal with the electronic properties of magnetic solids. This covers the treatment of noncollinear magnetism, structural and spin disorder, as well as relativistic and many-body effects. An introduction to the Stoner theory for itinerant or band magnetism is followed by a number of examples with an emphasis on transition metal-based systems. The direct connection of the total electronic energy in the ground state and its magnetic configuration is considered next. This includes mapping the dependence of the energy on the spin configuration on a simplified spin Hamiltonian as provided, for example, by the Heisenberg model. Another important issue in this context is magnetic anisotropy. As it is shown, considering excitations from a suitable reference state provides a powerful tool to search for stable phases, while calculating the wave vector- and frequency-dependent susceptibility gives a sound basis to understand the dynamical properties of magnetic solids. Finally, magnetism at finite temperature is dealt with starting from a pure classical treatment of the problem and ending with schemes that deal with quantum mechanics and statistics in a coherent way. Introduction Theory and modeling always played an important role for the understanding and development of magnetism [1, 2, 3, 4]. An early example for this is the presence of ring currents suggested by Ampère to explain the properties of permanent magnetic materials. Another example is the introduction of the molecular field by Weiss when discussing magnetism at finite temperature. Another important milestone in the theory of magnetism is the Bohr-van Leeuwen theorem [3] that unambiguously made clear that magnetism is a quantum mechanical phenomenon and for that reason requires a corresponding description. In line with this, Heisenberg’s investigation on the relation between the energy and the spin configuration clearly demonstrated that spontaneous spin-magnetic ordering is connected with the exchange interaction and is not due to the much weaker dipole-dipole interaction. Interestingly, the existence of the electronic spin follows directly from the Dirac equation that is the proper relativistic counterpart of Schrödinger’s equation [5]. Another important direct consequence of Dirac’s equation is the presence of spin-orbit coupling [5] that gives rise to many technologically important phenomena as the magnetocrystalline anisotropy, magnetostriction [6, 7], anomalous Hall [8], and magneto-optical Kerreffect [9]. Although these phenomena were discovered already in the nineteenth century, a proper explanation could be given only much later on the basis of quantum mechanics. Spin-orbit coupling is also the origin of many other important new effects that are exploited in the field of spintronics as, for example, the spin Hall effect [10] and spin-orbit torque [11]. In addition, one has to mention the spin-orbitinduced anisotropy in the magnetic exchange interaction. Apart from adding to the magnetocrystalline anisotropy, this includes the so-called Dzyaloshinskii-Moriya 4 Electronic Structure: Metals and Insulators 189 interaction that is responsible for chiral spin configurations leading in particular to skyrmionic spin structures [12]. Starting from a description of electron-electron interaction in the framework of quantum electrodynamics, one is led to the Breit interaction that can be seen as a current-current interaction [13]. This correction to the isotropic electron-electron Coulomb interaction is anisotropic and leads to the magnetic shape anisotropy [14]. Apart from explaining magnetic phenomena on a fundamental level in detail, theory nowadays provides in most cases a quantitative description of these. Corresponding numerical studies are in general based on a treatment of the underlying electronic structure within the framework of spin-density functional theory [15] to deal with electronic exchange and correlation. In spite of the many successes of this ab initio approach to magnetism, there are several limitations. This concerns first of all the impact of strong correlations or many-body effects that are often discussed on the basis of simplified models or hybrid schemes as the LDA+DMFT [16] (see section “Spin Density Functional Theory”). In addition, there are still open questions. For example, a coherent definition of orbital magnetism [17] and with this a prescription for its calculation were suggested only recently. Another important field to mention in this context is magnetism at finite temperature. Although the necessary quantum-statistical formalism [4] is available, one is in practice most often forced to use approximate schemes. These include in particular statistical or dynamical simulations on the basis of quasi-classical spin Hamiltonians. Ab initio theory is still extremely helpful in this case as it provides realistic parameters. This holds true, for example, for the exchange coupling tensor [18] and the anisotropy constant entering the extended Heisenberg Hamiltonian or for the Gilbert damping parameter occurring within the Landau-Lifshitz-Gilbert equation [19]. Starting from a basic knowledge in solid state theory [20], this chapter gives an overview on the methods used to deal with the electronic properties of magnetic solids (section “Electronic Structure Theory”). This covers in particular the treatment of noncollinear magnetism, structural and spin disorder, as well as relativistic and many-body effects. An introduction to the Stoner theory for itinerant or band magnetism is followed by a number of examples with an emphasis on transition metal-based systems (section “Itinerant Magnetism of Solids”). The direct connection of the total electronic energy of the ground state and its magnetic configuration is considered next (section “Total Electronic Energy and Magnetic Configuration”). This includes the mapping of the complex energy landscape representing the dependence on the spin configuration on a simplified spin Hamiltonian as provided, for example, by the Heisenberg model. Another important issue of this section will be magnetic anisotropy. As it will be shown, considering excitations from a suitable reference state provides a very powerful tool to search for stable phases. Calculation of the wave vector- and frequency-dependent susceptibility provides a sound basis for understanding the dynamical properties of magnetic solids (section “Excitations”). Finally, magnetism at finite temperature is dealt with in the last section that presents a series of methods starting from a pure classical treatment of the problem and leading to schemes that deal with quantum mechanics and statistics in a coherent way (section “Finite-Temperature Magnetism”). 190 H. Ebert et al. Electronic Structure Theory Dealing with the electronic structure of magnetic solids requires to account for exchange and correlation and to solve the resulting electronic structure problem. Spin density functional theory provides a powerful and flexible platform to tackle the first issue while still allowing for extensions aiming at an improved treatment of correlation effects. Concerning the second issue, there are many band structure schemes now available that provide the necessary accuracy. In particular, they allow dealing with two important aspects that are often of crucial importance for magnetic properties, namely, disorder and relativistic effects. Spin Density Functional Theory Most computational investigations on the electronic structure of magnetic materials are nowadays based on density functional theory (DFT) [15] or extensions to it. The major goal of this approach is to reduce the complicated many-body problem connected with the electron system of an atom, molecule, or solid effectively to a single-particle problem. The formal basis for this tremendous simplification is laid by the theorems of Kohn and Hohenberg [21] that introduce the electron density n(r) as the basic system variable: 1. The total ground state energy E of any many-body system is a functional of the density n(r) E[n] = F [n] + d 3 r n(r) Vext (r), (1) where Vext (r) is an arbitrary external potential, in general the Coulomb potential of the nuclei, and F [n] itself is a functional of the density n(r) but does not depend on Vext (r). 2. For any many-electron system, the functional E[n] for the total energy has a minimum equal to the ground-state energy E0 = E[n0 ] at the ground-state density n0 (r). Applying the variational principle to the minimal property of the energy functional Kohn and Sham [22] derived Schrödinger-like single-particle equations whose solution allows calculating any property of the system. For this purpose, the functional F [n] is split into three parts: F [n] = T [n] + 3 d r d 3r n(r) n(r ) + Exc [n], |r − r | (2) with the two first terms representing the kinetic and Coulomb or Hartree energy of the electrons. The last term is a universal functional Exc [n] that represents all 4 Electronic Structure: Metals and Insulators 191 exchange and correlation effects. Introducing an auxiliary system of noninteracting particles with the same density n(r) as the real one, the corresponding kinetic energy can easily be expressed leading to the formal definition of the corresponding exchange and correlation energy functional Exc [n] as containing all remaining many-body effects. As it is common in electronic structure theory of solids in Eq. (2) and the following atomic Rydberg units, (h̄ = 1, me = 1/2, e2 = 2, and c = 2/α with α the fine-structure constant) are used. As the functional Exc [n] is universal, the resulting scheme can in principle be applied without modification to spin-magnetic, i.e., spin-polarized, systems. However, as the available formulations for the functional are far too complicated to be applied to real systems, Exc [n] has to be represented in practice by a suitable approximation. For this, it is advantageous to replace DFT by the corresponding spin-density functional theory (SDFT) that was introduced by von Barth and Hedin [23] and Rajagopal and Callaway [24]. Restricting to the situation of collinear magnetism with the spin-quantization axis along the global magnetization, this leads to the following Schrödinger-like single-particle equations: −∇ 2 + Vσeff (r) φiσ (r) = iσ φiσ (r). (3) Here φiσ (r) and iσ are the wave function and energy of the single-particle state i with spin character σ (up or down). While these quantities have a priori no physical meaning, they can nevertheless be used to determine the central properties of the system. This applies in particular for the spin densities nσ (r) = Nσ |φiσ (r)|2 , (4) i=1 as the basic variables of the system. In Eq. (4), the summation runs over all Nσ states with their energy iσ below the Fermi energy EF that in turn is determined by the requirement N= d 3 r n↑ (r) + n↓ (r) , (5) where N = N↑ + N↓ is the total number of electrons. Obviously, the corresponding particle density n(r) and spin magnetization m(r), given by the relations: n(r) = n↑ (r) + n↓ (r) (6) m(r) = n↑ (r) − n↓ (r), (7) may also be chosen as alternative basic variables of a spin-polarized system. The spin-dependent effective potential Vσeff (r) entering Eq. (3) is determined by the requirement that the total energy E[n↑ , n↓ ] that now has to be seen as 192 H. Ebert et al. a functional of the spin densities takes a minimum. This leads finally to the expression: Vσeff (r) = Vσext (r) + 2 d 3r n(r ) + Vσxc (r), |r − r | (8) with Vσxc (r) = δExc [n↑ , n↓ ] . δnσ (r) (9) Equations (3), (8), and (9) constitute the coupled Kohn-Sham equations that obviously have to be solved self-consistently. With this accomplished, the total energy of the system can be obtained from: E[n↑ , n↓ ] = Nσ iσ − d 3r d 3r σ =↑,↓ i=1 − n(r) n(r ) |r − r | d 3 r Vσxc (r) nσ (r) + Exc [n↑ , n↓ ]. (10) σ =↑,↓ When the impact of spin-orbit coupling is included into the formalism (see section “Relativistic Effects”), the corresponding expression for E[n↑ , n↓ ] is very convenient for investigations on the magnetic anisotropy. In this case, one can in general neglect the change of nσ (r) with the orientation of the magnetization, and one has to consider only the first term in Eq. (10) representing the single-particle energies of the system. The collinear formulation given above is adequate for most situations. In case of noncollinear magnetic configurations with the orientation of the spin magnetization changing with position, one has in principle to use the spin-matrix formulation of SDFT [23]. This implies that the wave functions φiσ (r) in Eq. (3) have to be replaced by spinors φi (r), i.e., two component wave functions, that in general will have no pure spin character. However, very often one can still assume a uniform orientation of the magnetization within an atomic cell. In this case the potential is also spin-diagonal in a local frame of reference with the spin-quantization axis along the orientation m̂ of the local spin moment m. Accordingly, one can represent the spin-dependent part of the effective 2 × 2 potential matrix function by: V spin (r) = σ · m̂ B(r), (11) where σ is the vector of 2 × 2 Pauli spin matrices [5] and the effective field B(r) = (V ↑ (r) − V ↓ (r))/2 is given by the difference of the spin-up and spin-down potentials in the local frame. Alternatively, the spin-diagonal potential in the local frame may be related to the 2 × 2 potential matrix function in the global frame by 4 Electronic Structure: Metals and Insulators 193 means of a transformation matrix U (θ, φ) such that: σ · m̂ = U † (θ, φ) σz U (θ, φ). This matrix can be obtained from Eq. (80) (see below) by setting θ and φ according to the orientation m̂ and q = 0. In either case, the form of the spin-dependent potential again implies spinor or two-component wave functions. As indicated, the major problem of SDFT is that the functional Exc [n↑ , n↓ ] for the exchange-correlation is not known. A useful expression for this, that can be justified for slowly varying densities, is supplied by the local spin-density approximation (LSDA): LSDA Exc [n↑ , n↓ ] = d 3 r n(r) (n↑ (r), n↓ (r)). (12) 2 2 0 0 -2 -2 E (eV) Ejkσ (eV) Here (n↑ (r), n↓ (r)) is the exchange-correlation energy per electron for the homogeneous free electron gas with uniform spin densities n↑ (r) and n↓ (r) that can be determined with high accuracy [25]. A fit to numerical results for (n↑ (r), n↓ (r)) allows giving explicit expressions for the corresponding spin-dependent exchangecorrelation potential Vσxc (r) = Vσxc [n↑ (r), n↓ (r)]. As an example for a spin-polarized solid, Fig. 1 gives the band structure or dispersion relation Ej kσ and the density of states (DOS) nσ (E) of bcc-Fe as calculated within LSDA (see section “Band Structure Methods”). These curves clearly show the exchange splitting due to the spin dependency of the exchange-correlation potential when compared with results for the corresponding paramagnetic state. Integrating nσ (E) up to the Fermi energy EF obviously gives the number of electrons Nσ with spin character σ . The corresponding spin-magnetic moment M = N↑ − N↓ is given in Table 1 for bcc-Fe, fcc-Ni, and hcp-Co [27]. Obviously, -4 -4 -6 -6 -8 -8 Γ Δ wave vector k X 3 2 1 n↓(E) (sts./eV) 0 1 2 n↑(E) (sts./eV) 3 Fig. 1 Dispersion relation Ej kσ (left) and density of states nσ (E) (right) of ferromagnetic bcc-Fe as calculated within LSDA. Results for the majority (↑) and minority (↓) spin states are given in red and blue, respectively. In addition, results for the corresponding paramagnetic state are given in black [26] 194 H. Ebert et al. Table 1 Spin-magnetic moment of bcc-Fe, fcc-Ni, and hcp-Co calculated on the basis of the LSDA and the GGA, compared with experimental values. The calculated magnetic moments given in the last two columns have been obtained for the theoretical equilibrium and experimental lattice parameter, respectively. (All data taken from [27]) Magnetic moment (μB ) Fe Co Ni Wigner-Seitzradius (a.u.) LSDA 2.59 GGA 2.68 Expt. 2.67 LSDA 2.54 GGA 2.63 Expt. 2.62 LSDA 2.53 GGA 2.63 Expt. 2.60 Bulk modulus (Mbar) 2.64 1.74 1.68 2.68 2.14 1.91 2.50 2.08 2.86 Cohesive energy (eV) 7.32 6.31 4.28 5.98 4.52 4.39 5.45 4.18 4.44 (atheo ) 2.08 2.20 – 1.50 1.63 – 0.59 0.65 – (aexpt ) 2.14 2.17 2.22 1.62 1.63 1.72 0.61 0.63 0.61 the results depend to some extent on the chosen functional for Vσxc (r) (LSDA or GGA) and the lattice parameter. Nevertheless, the calculated moments are in fairly good agreement with experiment. Although LSDA turned out to be astonishingly successful for many situations, it nevertheless shows severe limitations. For example, LSDA leads in general to an over-binding as can be seen from the Wigner-Seitz radius given in Table 1 that is too small when compared to experiment. Furthermore, calculations on ferromagnetic Fe led to a lower total energy for the fcc instead of the bcc structure (see section “Total Electronic Energy and Magnetic Ground State”). These deficits could be removed when the generalized gradient approximation (GGA) was introduced that expresses GGA [n , n ] not only in terms of the spin densities n (r) but also of their Exc ↑ ↓ σ gradients ∇nσ (r). A more systematic route to derive accurate exchange-correlation energies and corresponding potentials is supplied by the optimized potential method that leads to a functional expressed in terms of the Kohn-Sham orbitals [15]. Unfortunately, this approach is numerically much more demanding than the very efficient LSDA or GGA schemes in particular when a reliable representation of correlation is incorporated. Accordingly, the development of parametrizations for the exchange-correlation potential that are at the same time efficient and sufficiently accurate is a field of ongoing research. A major problem of LSDA and comparable SDFT schemes is the accurate treatment of correlation effects in case of moderate or strong correlations as they occur in systems with narrow energy bands. A way to cure this problem is the use of the GW method [28] that is applicable to moderately correlated systems. Application to Ni, for example, showed in particular a narrowing of the d-band when compared to LSDA-based results [29] as expected from photoemission [30]. A scheme that is applicable also to strongly correlated materials at a much lower 4 Electronic Structure: Metals and Insulators 195 numerical cost is the LDA+U [31] that accounts for static correlations by adding corrections to the LDA or LSDA Hamiltonian that depend on the Hubbard Coulomb parameter U . In case that U is much larger than the band width W , a situation typically met in oxides, the correction term leads in particular to a splitting into an upper and lower Hubbard band. Dynamical correlations, on the other hand, are accounted for by the dynamical mean field theory (DMFT) that when merged with the LSDA leads to the combined Hamiltonian [32, 33]: H = HLSDA + 1 σσ Umm n̂ilmσ n̂ilm σ 2 il − mσ m σ 1 † † Jmm ĉilmσ ĉilm σ̄ ĉilm σ̄ ĉilmσ 2 il − mσ m il mσ Δl n̂ilmσ . (13) i=id ,l=d Here, ĉ, ĉ† , and n̂ are creation, annihilation, and particle density operators that refer to atomic orbitals labeled by the quantum numbers l, m, and σ and site index i, with σ σ and J Umm mm corresponding to Coulomb and exchange integrals, respectively. The quantity Δl represents the so-called double counting term that takes care that static correlations are not accounted for twice – by the LSDA Hamiltonian HLSDA and by its complementary DMFT counterpart HDMFT . As Eq. (13) indicates, the DMFT correction is usually restricted to the correlated subsystem of the system as, for example, d-states in case of transition metals (i = id , l = d). Furthermore, the Coulomb and exchange integrals are assumed to be site diagonal (singlesite approximation) leading for the many-body problem to the same situation as for the Anderson impurity model (AIM). Accordingly, all the various many-body techniques available to deal with the AIM can also be used when dealing with the combined LSDA+DMFT Hamiltonian. In most cases, Eq. (13) is dealt with by calculating in a first step the one-electron Green function (see section “Band Structure Methods”) associated with HLSDA . In a next step, the single-site problem is solved by a so-called impurity solver that allows representing the impact of HDMFT in terms of a corresponding complex and energy-dependent self-energy ΣDMFT (E). Finally, making use of the Dyson equation (see Eq. (21) below), the one-electron Green function of the system is updated. Figure 2 shows typical results for the spin-dependent self-energy ΣDMFT (E) of ferromagnetic Ni. The characteristics of these curves lead to the various correlationinduced features expected from photoemission [30]: the real part gives rise to a renormalization of the energy bands leading to band narrowing, reduction of the exchange splitting, and the occurrence of a satellite structure at 6 eV binding energy. The imaginary part, on the other hand, implies a finite lifetime of the electronic state that increases for the d-states with distance from the Fermi energy. This is reflected in the DOS curves by a smearing-out of its structure when compared to the LSDA result. 196 H. Ebert et al. Fig. 2 Left: real (red) and imaginary (blue) parts of the spin-resolved self-energy ΣDMFT (E) of ferromagnetic Ni. Right: corresponding DOS obtained on the basis of plain LSDA and LSDA+DMFT [26] Band Structure Methods Most methods for calculating the electronic structure of solids on the basis of the Kohn-Sham equation (3) assume three-dimensional periodicity for the potential Vσeff (r). This implies that the corresponding solutions ψj k (r) are Bloch states that transform under a lattice translation as: ψj k (r + R n ) = eik·R n ψj k (r) (14) and accordingly can be labeled by the wave vector k and an additional band index j . A direct consequence of this is that Bloch states for different wave vectors k are orthogonal. This property simplifies the solution of the band structure problem tremendously when using the variational principle to solve Eq. (3) and transforming that way the problem to solving an algebraic eigenvalue problem. Constructing in this case the basis functions such that they obey Eq. (14), one is led to a secular equation with finite dimension for each k-vector H k − Ej k S k α j k = 0 (15) with the Hamilton and overlap matrices, H k and S k , referring to the basis functions and Ej k and α j k the associated eigenvalues and eigenvectors, respectively. Obviously one still has great freedom to construct a suitable basis set, and accordingly there is a large number of methods and corresponding computer codes available [34]. One route to set up an electronic structure method is to take the tightly bound core states as frozen. This allows introducing effective pseudo-potentials that do not show the singularity of the Coulomb potential when 4 Electronic Structure: Metals and Insulators 197 approaching the atomic nucleus. As a consequence, one may use even plane waves as basis functions. Higher accuracy and flexibility, however, can be achieved by using a technique called projector-augmented wave method (PAW) [35] that mediates between pseudo-potential and so-called all-electron methods, with the latter aiming to solve the Kohn-Sham equations directly. Again one may distinguish between methods using analytical or numerical basis functions. Within the LMTO method [36], for example, the Kohn-Sham equation is solved numerically at a fixed energy Eν and angular momentum l for a spherical potential inside an atomic cell that is approximated by a sphere. A linear combination of these solutions φlν (r) and their energy derivatives φ̇lν (r) are augmented outside the atomic cell in a way that leads to a decaying muffin-tin orbital that solves the Kohn-Sham equation also within all neighboring atomic cells. The Bloch sum of such muffin-tin orbitals is obviously a suitable basis function. By construction, it is energy independent, but with an appropriate choice of Eν it will account, within the energy regime of interest, for the energy dependence of the exact solution up to first order. This is a common feature of all so-called linear methods [36] like the LAPW [36, 37]. The special choice of the basis function of the LMTO method has the additional feature that it is minimal: this means that one can restrict the angular momentum expansion of the basis function in line with chemical intuition, i.e., for transition metals, one should go at least up to d-states with l = 2. Solving the resulting secular equation or algebraic eigenvalue problem, one gets finally the energy eigenvalue Ej k of the Bloch states together with a corresponding representation of their wave functions ψj k (r) in terms of the radial functions φlν (r) and their energy derivatives φ̇lν (r): ψj k (r) = jk jk Alm φlν (r) Ylm (r̂) + Blm φ̇lν (r) Ylm (r̂), (16) lm jk jk where the expansion coefficients Alm and Blm are given by the eigenvectors α j k and Ylm (r̂) are spherical harmonics. A similar representation of the Bloch wave functions is obtained for most other all-electron band structure methods. Although most band structure methods are formulated as k-space methods assuming three-dimensional periodicity, they can nevertheless be applied to situations with lower dimensionality or symmetry by means of the super-cell technique. This is illustrated for the case of a random substitutionally disordered binary alloy Ax B1−x in Fig. 3. Instead of dealing with an, in principle, infinitely large unit cell that represents the random distribution of the A- and B-atoms on the geometric lattice, periodic boundary conditions are imposed implying the use of a finite size super-cell that is enlarged when compared to the unit cell of the underlying lattice. To achieve a reliable representation of the configurational average of a disordered alloy, obviously the unit cell has to be large enough and a representative average has to be taken concerning the atomic configuration within the super-cell [38] using, for example, the concept of a special quasi-random structure [39]. As indicated by the lower panel of Fig. 3, the same type of reasoning can be applied when dealing with the problem of a disordered spin configuration of a solid that may occur due to thermal spin fluctuations [40]. The super-cell technique is also frequently applied 198 H. Ebert et al. Fig. 3 Top row: representation of a random substitutionally disordered binary alloy Ax B1−x (left) by means of the super-cell technique (middle) and by means of an effective medium theory (right). Bottom row: corresponding application of these schemes to the problem of a disordered spin configuration in case of reduced dimensionality of the system as, for example, when dealing with surfaces or impurities in a host system. In the latter case obviously the size of the super-cell has to be chosen large enough to avoid an interaction of impurities in neighboring super-cells. Instead of representing the electronic structure of a solid in terms of Bloch states with associated wave functions ψj k (r) and energy eigenvalues Ej k , this can be done by means of the corresponding retarded single-particle Green function G+ (r, r , E). With the solutions to the band structure problem available, the Green function G+ (r, r , E) can be given via the so-called Lehmann spectral representation [41] + ψj k (r) ψj†k (r ) G (r, r , E) = lim →0 jk E − Ej k + i , (17) that allows straightforwardly to derive convenient expressions, for example, for the density of states n(E) and electron density n(r), respectively: 1 n(E) = − π d 3 r G+ (r, r, E) (18) 4 Electronic Structure: Metals and Insulators 1 n(r) = − π 199 EF dE G+ (r, r, E). (19) Although Eq. (17) is frequently used, it is not very convenient as one needs in principle the whole spectrum connected with the underlying electronic Hamiltonian to get the Green function for a given energy E. An alternative to this is offered by the multiple scattering theory-based KKR (Korringa-Kohn-Rostoker) formalism that leads to the following expression for G+ (r, r , E) [42]: G+ (r, r , E) = L,L i× ii ZLi (r, E) τLL (E) ZL (r , E) −δii ZLi (r < , E) JLi× (r > , E) (20) L that in particular does not require Bloch translational symmetry for the system considered. The functions ZLi (r, E) and JLi (r, E) in Eq. (20) are regular and irregular solutions, respectively, to the Kohn-Sham equation with angular character L = (l, m) for r in the atomic cell at site i and a specific normalization [42]. The ii (E) is the so-called scattering path operator that transfers a wave with quantity τLL character L coming in at site i into a wave going out from site i with character L with all intermediate scattering events accounted for. From Eq. (18), it is obvious ii (E) determines in particular the variation of the that the site-diagonal quantity τLL density of states ni (E) at site i with energy E. The use of the Green function offers many advantages when dealing with embedded subsystems, response functions, spectroscopy, disorder, or the manybody problem. To a large extent, this is due to the Dyson equation that allows to express the Green function G+ (r, r , E) of a complex system on the basis of that of a simpler reference system (G+ 0 (r, r , E)) and the arbitrary perturbing Hamiltonian Hpert (r) that connects the two systems: G+ (r, r , E) = G+ 0 (r, r , E) + Ω d3 r G+ 0 (r, r , E) Hpert (r ) G+ (r , r , E), (21) with Ω the region for which Hpert (r) has to be accounted for. For a substitutional impurity, this would include the atomic cell of the impurity and the region of the neighboring host atoms that are distorted by the impurity. The Green function formalism is particularly useful when dealing with the electronic structure of disordered systems. By using the concept of the molecular field, Soven [43] introduced the Coherent Potential Approximation (CPA) approach when dealing with disordered substitutional alloys. The corresponding hypothetical effective CPA medium plays the role of the molecular field and is constructed such that it represents the configurational average for the alloy as accurate as possible 200 H. Ebert et al. (see Fig. 3). The standard CPA is a so-called single-site theory implying that the occupation of neighboring lattice sites is uncorrelated, i.e., short-range order is excluded. Within the KKR approach, the CPA medium is therefore determined by requiring that for an Ax B1−x alloy, the embedding of an A- or B-atom into the CPA medium should on the average lead to no additional scattering: ii ii x τ ii A + (1 − x) τ B = τ CPA . (22) 2 2 0 0 0 -2 -2 -2 -4 -4 -6 -6 -6 -8 -8 Δ Pd -8 2 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 n↑(E) (sts./eV) n↑(E) (sts./eV) 2 2 0 0 0 -2 -2 -2 -4 -4 -6 -6 -6 -8 -8 wave vector k X E (eV) E (eV) Γ Ni -4 Γ Δ wave vector k X Ni Pd E (eV) -4 E (eV) 2 E (eV) E (eV) Here the component-projected scattering path operators τ ii α represent the single-site embedding of the component α into the CPA medium according to Eq. (21). When using these quantities together with the corresponding component-related wave functions in Eq. (20), one gets obviously access to component-specific properties as the partial DOS of an alloy. Corresponding results for the disordered ferromagnetic alloy fcc-Ni0.8 Pd0.2 are shown in Fig. 4. The left column gives the spin-resolved band structure in terms of the Bloch spectral function AB σ (k, E) that can be seen as the Fourier transform of the real space Green function G+ σ (r, r , E), while the -8 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 n↓(E) (sts./eV) n↓(E) (sts./eV) Fig. 4 Left: spin-resolved Bloch spectral function AB σ (k, E) of the disordered ferromagnetic alloy fcc-Ni0.8 Pd0.2 calculated on the basis of the CPA. Middle and right column: corresponding spinresolved partial density of states nασ (E) for α = Ni or Pd, respectively. The top and bottom row give results for spin up and down, respectively. As a reference, the dispersion relation Ej kσ of pure Ni Ni is superimposed as a black line to AB σ (k, E) (left). In addition, nσ (E) for pure ferromagnetic Pd Ni (middle) and of nσ (E) for pure paramagnetic Pd (right) are included in the figures as dashed lines [26] 4 Electronic Structure: Metals and Insulators 201 middle and right columns give the spin-resolved partial density of states nασ (E) for α = Ni or Pd, respectively. Comparison of AB σ (k, E) with the dispersion relation Ej kσ of fcc-Ni and of nNi σ (E) for the alloy with that for pure Ni clearly shows the smearing-out of these curves for the alloy in particular in the regime of the d-states. This reflects the fact that for the alloy the wave vector k is not a good quantum number. In fact the width of the AB σ (k, E) functions can be interpreted as a measure for the electronic lifetime to be used in the calculation of the residual resistivity on the basis of the Boltzmann formalism [44]. The right column of Fig. 4 gives results for the partial DOS nPd σ (E) of Pd in fcc-Ni0.8 Pd0.2 and for pure paramagnetic Pd that clearly shows that Pd gets spin-polarized due to hybridization leading to an induced spin-magnetic moment of about 0.21 μB . It is important to note that the concept of the CPA is not restricted to alloys but can be applied to any type of disorder. Making use of the alloy analogy, the CPA is used, for example, within the disordered local moment (DLM) model [45] to perform an average over spin configurations connected with thermal spin fluctuations (see section “Coherent Treatment of Electronic Structure and Spin Statistics”). Relativistic Effects A most coherent way to account for the influence of relativistic effects is to work on the basis of the Dirac Hamiltonian [5]: ĤD = −icα · ∇ + 1 2 c (β − 1) + V̄ (r) + β σ · B(r) + eα · A(r). 2 (23) Here c is the speed of light, αi and β are the standard 4 × 4 Dirac matrices, cα is the electronic velocity operator, σi are the 4 × 4 spin matrices, and the local potential may involve a spin-independent part V̄ (r), an effective magnetic field (B(r)) coupling only to the spin [46] and the vector potential (A(r)) coupling to the electronic current, where the effective fields B(r) and A(r) combine exchange-correlation and possible external contributions. This approach implies a four-component electronic wave function ψ(r, E) or bi-spinor, respectively, with a large and small component [5]. Corresponding versions based on this framework have been worked out for several band structure methods [47, 48] assuming in general a spin-dependent potential only (A(r) = 0). The appealing feature of these schemes is that they treat all relativistic effects and magnetic ordering on the same footing. Alternatively, one may apply a Foldy-Wouthuysen transformation [5] to the Dirac equation. Considering only a spherical scalar potential V (r) in Eq. (23) and keeping only terms up to the order of 1/c2 , this leads to three relativistic corrections when compared to the Schrödinger Hamiltonian [49]: Hmass = −β 1 4 p c2 (24) 202 H. Ebert et al. 1 2 ∇ V (r) 2c2 1 1 ∂V (r) = 2 σ · l. c r ∂r HDarwin = (25) HSOC (26) The first two corrections, mass enhancement (Hmass ) and Darwin (HDarwin ) terms, do not involve the spin operator σ , and for that reason, they are called scalar relativistic. In general, these corrections give rise to a downward shift in energy for s- and p-states and, in response to this, to an upward shift in energy for d-states [50]. For a paramagnetic metal, this influences the density of states at the Fermi level and this way the tendency toward spontaneous formation of spin magnetism via the Stoner mechanism (see section “Stoner Model of Itinerant Magnetism”). The third correction term is the spin-orbit coupling (HSOC ), which is given in its commonly simplified form, that holds in case of a spherically symmetric scalar potential V (r). As HSOC couples the electronic orbital and spin degrees of freedom, this term will lead to the removal of degeneracies for any material. For a spinpolarized material, it leads furthermore to a reduction in symmetry as reflected by many important properties as, for example, the magnetocrystalline anisotropy [51,6] or galvanomagnetic [8] and magneto-optical [9] properties. In practice, most electronic structure calculations are based on relativistic correction schemes that first of all aim to eliminate the small component from the Dirac equation leading to a two-component formalism [52, 53]. Within socalled scalar relativistic calculations, spin-orbit coupling is ignored leading to no technical changes when compared to a nonrelativistic calculation on the basis of the LSDA that treat spin-up and spin-down states separately. However, including HSOC does not allow this simplification any more requiring higher computational effort. While HSOC can still be accounted for when setting up the basis functions of a band structure scheme [52, 54], it is in most cases seen as a correction to a scalar relativistic Hamiltonian and therefore accounted for only in the variational step of a standard band structure scheme [36, 37]. As an illustration, Fig. 5 gives the dispersion relation Ej k of ferromagnetic Ni calculated in a non-, scalar, and fully relativistic way. The left panel of the figure shows the impact of the scalar relativistic corrections Hmass and HDarwin . The middle panel shows results that account in addition for the spin-diagonal part of HSOC proportional to lz σz [55]. Although this term leaves spin as a good quantum number, it obviously removes many degeneracies and band crossings giving rise, for example, to spin-orbit-induced orbital magnetic moments [55]. Accounting finally for the full spin-orbit coupling HSOC (right panel), one finds further removals of band crossings and a coupling of the two spin systems. This spin mixing gives rise to so-called hot spots in the band structure [56] that are important, for example, for spin-flip relaxation processes. The relativistic corrections mentioned so far concern only the kinetic part of the electronic Hamiltonian, but not the effective potential. Starting from a fully relativistic framework in fact corresponding corrections have to be expected when 4 Electronic Structure: Metals and Insulators 203 -0.5 -0.5 -1 Ejkσ (eV) -1 -1.5 -1.5 -2 -2 NREL -2.5 Γ Δ FREL SOC zz X Γ Δ wave vector k X Γ Δ -2.5 X Fig. 5 Dispersion relation Ej k of ferromagnetic Ni. Left: spin-resolved curves Ej kσ for non(black lines) and scalar (up and down triangles) relativistic mode. Middle: relativistic mode that accounts only of the spin-diagonal part of HSOC proportional to lz σz (black lines) and scalar relativistic mode (up and down triangles). Right: fully (black lines) and scalar (up and down triangles) relativistic mode [26] considering the free electron gas as a reference system [15]. The impact of such corrections to the exchange-correlation potential has been monitored in particular for paramagnetic materials [57]. For magnetic materials, a coherent derivation of nonrelativistic spin-density functional theory could be given by starting from a relativistic formalism [24]. Later on, a relativistic formulation of spin-density functional theory assuming a spin-dependent exchange-correlation potential that couples only to the spin degree of freedom was worked out [58, 46] leading to Eq. (23) with A(r) = 0. This simplification seems to be acceptable for many magnetically ordered transition metal systems. On the other hand, spin-orbit coupling gives for spin-polarized materials automatically rise to orbital magnetism [59] that can be associated with a corresponding orbital current. Accordingly, relativistic spin-density functional theory should in principle be replaced by a current density formalism with the four current as a basic system variable [15], i.e., one has B(r) = 0 and A(r) = 0 in Eq. (23). While various developments have been made in this direction [60], there are no functionals for the corresponding exchange-correlation available at the moment. As a consequence, the feedback of orbital magnetism or polarization on the electronic Hamiltonian has been accounted for primarily by means of hybrid schemes like the OP- [61], the LSDA+U- [62], or the LSDA+DMFT-formalism [63]. Another consequence of a fully relativistic formalism is a modification of the Coulomb potential and the occurrence of the Breit interaction [13]. The latter one can be seen as a current-current interaction and accordingly can be represented by a corresponding vector potential A(r) in Eq. (23). For magnetic materials, the Breit interaction contributes not only to the total energy but also to its magnetic anisotropy. In fact it has been pointed out that the Breit interaction is the quantum 204 H. Ebert et al. mechanical source for the classical dipole-dipole interaction giving rise to the magnetic shape anisotropy [14]. Adiabatic Dynamics The features of the electronic band structure determine not only the ground-state properties of a material but also the dynamical properties of electrons in the presence of external time-dependent perturbations. When a perturbation varies slowly in time, the dynamics of the electronic subsystem can be efficiently described using the concept of the Berry phase [64] arising during the adiabatic evolution of electronic quantum states. According to its definition, the Berry phase is connected to a parameter-dependent Hamiltonian [65,66,67]. For systems with a periodic effective potential Veff (r + R n ) = Veff (r) with R n a lattice translation vector, leading to a Bloch-like solution of the eigenvalue problem (see Eq. (14)): ψj k (r) = eik·r uj k (r), (27) 2 p̂ the unitary transformation of the Hamiltonian H = 2m + Veff (r) leads to a momentum-dependent Hamiltonian (for details see [67]) H(k) = e−ik·r Heik·r = (p̂ + h̄k)2 + Veff (r). 2m (28) This allows treating the Brillouin zone as the parameter space of the transformed Hamiltonian H(k) with cell-periodic eigenfunctions uj k (r). The Berry phase is expressed as an integral over the path C in the parameter space of the electronic momentum k, γj (k) = dk · Aj (k). (29) C The integral is gauge-invariant for a closed path, while the Berry vector potential, or the Berry connection Aj (k) = i uj k ∂ uj k , ∂k (30) is a gauge-dependent quantity. This value, treated in analogy to electrodynamics as a vector potential, gives access, using Stokes’ theorem, to a gauge-invariant quantity called Berry curvature playing the role of a magnetic field in the parameter space: Ω j (k) = ∇ k × Aj (k), or, alternatively, (31) 4 Electronic Structure: Metals and Insulators Ωj,μν (k) = ∂μ Aj,ν − ∂ν Aj,μ = −2 ∂μ uj k | ∂ν uj k , γj = dS · Ω j (k), S 205 (32) (33) with the integration over an arbitrary surface enclosed by the path C. In case of a translation-invariant crystal, a closed path in Eq. (29) occurs due to a torus topology of the Brillouin zone as any two points k and k + G are fully equivalent for any reciprocal lattice vector G, leading to an integration over a Brillouin zone in Eq. (29). Concerning applications of the Berry phase concept, we focus here on the electron dynamics in the presence of an electric field E entering the Hamiltonian trough a time-dependent uniform vector potential A(t), preserving the translation symmetry of the system. The group velocity of a state (j, k) is given to first order by an expression [65, 68] consisting of two terms, the usual band dispersion contribution as well as a so-called anomalous velocity proportional to the Berry curvature of the bands, v j (k) = ∂j (k) e − E × Ω j (k), h̄∂k h̄ (34) with the Berry curvature Eq. (32) given in the form [69, 67]: Ω j (k) = i ∇ k uj k | × |∇ k uj k . (35) The Berry curvature is nonzero either in non-centrosymmetric systems or in systems with broken time-reversal symmetry and vanishes in the case when both symmetries are present, leading to elimination of the anomalous velocity in Eq. (34) [67]. Thus, due to these symmetry properties of the Berry curvature, the anomalous Hall effect can be observed in ferromagnetic systems. One has to stress the crucial role of the spin-orbit interaction required for a nonzero Berry curvature in FM systems, leading to avoided crossings of the energy bands which give in turn the most pronounced contributions in the vicinity of the Fermi surface that can be seen in Fig. 6a [70]. As one can see in Eq. (34), the anomalous velocity is always transverse to the electric field giving rise to a Hall current. The corresponding contribution to the Hall conductivity associated with the anomalous velocity term was demonstrated first by Karplus and Luttinger [72] and is called Karplus-Luttinger mechanism. In terms of the Berry curvature, it is given by the expression [65, 68, 69, 73]: KL σαβ = e2 h̄ BZ dDk γ f (j (k))αβγ Ωj (k), (2π )D n (36) with D the dimensionality of the system and αβγ the Levi-Civita tensor. As one can see, this contribution is determined by all occupied states, and it is the only term 206 H. Ebert et al. Fig. 6 (a) Fermi surface in the (010) plane (solid lines) and integrated Berry curvature −Ω z (k) in atomic units (color map) of fcc Fe. (From Yao et al. [70]); (b) Berry curvature projected onto the (k3 , k1 ) plane for Mn3 Ge, where k3 and k1 are aligned with the kz and kx axes, respectively. It is calculated by integrating along the k2 direction. (From Ref. [71]); (c) Energy dispersion of Mn3 Ge along k2 with (k3 , k1 ) fixed at the point with the largest Berry curvature, indicated by a black dashed circle in (b). (From Ref. [71]). (Figure are printed with permission from [70] by the American Physical Society; Figure (b) and (c) reprinted with permission from [71] by the American Physical Society.) contributing to the AHC for insulating systems. It is independent on the nature of the impurities and their concentration and therefore is called intrinsic contribution to the AHE. Note, however, that this contribution is not the only intrinsic contribution in metals. The so-called side-jump mechanism suggested by Berger [74] is contributed by the electrons at the Fermi surface and also yields a Hall conductivity that is independent of the impurity concentration [69]. Note that the anomalous thermoelectric transport driven by statistical forces due to a temperature gradient ∇T (the same concerns also the transport driven by chemical potential gradients ∇μ) cannot rely on the anomalous velocity term as it vanishes in the absence of an electric field. A corresponding theory giving the intrinsic contribution to transverse thermoelectric transport has been reported by Xiao et al. [75]. It is based on the generalization to finite temperatures of a theory giving a Berry-phase correction to the orbital magnetization [76, 77, 17]: 4 Electronic Structure: Metals and Insulators M(r) = BZ j + 207 dDk f (j (k))mn (k) (2π )D 1 dDk e Ω j (k)log 1 + e−β(j (k)−μ) , D β BZ (2π ) h̄ (37) j where mn (k) = (e/2h̄)i ∇ k uj,k |[j (k) − H(k)] × |∇ k uj,k is the orbital moment of state n. By doing some transformations, this expression can be written also in the form [67]: Mz (r) = BZ j dDk 1 f (j (k))mn,z (k) + D (2π ) e df ()σxy (). (38) It has two different contributions associated with the self-rotation of the wave packet representing an electron and with the center-of-mass motion, respectively. The first term, obviously, occurs by treating the carrier as a wave packet having finite spread in the phase space. The second term comes from the Berry-phase correction to the electron density of states [67]. A crucial point for thermoelectric transport is that the conventional expression used for the current density is incomplete as it is derived for the carrier treated as a point particle. Having a corresponding expression for the local current J obtained by treating the carrier as a wave packet, introducing the concept of a transport current j = J − ∇ × M(r), (39) and using the expression Eq. (37) for the orbital magnetization density M(r), one obtains the transport current as given by: dDk g(r, k)ṙ D BZ (2π ) dDk e 1 Ω(k)log(1 + e−β((k)−μ) ). −∇ × β BZ (2π )D h̄ j = −e (40) (41) With this expression, it is straightforward to calculate various thermoelectric responses to statistical forces [67]. Thus, the expression for an anomalous Nernst conductivity αxy is given in terms of the intrinsic anomalous Hall conductivity σxy : αxy 1 =− e d −μ ∂f σxy () . ∂μ T (42) Talking about the AHE in antiferromagnets (AFM), one has to distinguish the systems according to their symmetry. The AHE does not occur in collinear AFMs symmetric with respect to time-reversal symmetry T combined with a half-magnetic 208 H. Ebert et al. unit-cell translation Ta/2 or spatial inversion I, i.e., either Ta/2 T symmetry or IT (see [78]). However, in AFM materials having a symmetry violating these conditions, the AHE can be observed. Accordingly, a large intrinsic AHE was predicted in Mn3 Ir [79] and Mn3 Ge [80] by performing first-principles electronic structure calculations. As already indicated, the avoided band crossings near the Fermi surface give the dominant contribution to the AHE. An example of a calculated k-resolved Berry curvature for Mn3 Ge [71] is plotted in Fig. 6b. It is dominating in the area highlighted in red giving rise to the large AHC. In order to investigate the origin of the hot spot at (0.127, 0.428), the band structure is plotted in Fig. 6c along k2 varying from 0 to 1. As one can see, the Fermi level crosses two small gaps around k2 = 0 and 0.5. This implies that the entanglement between occupied and unoccupied states must be very strong around these two points, giving a large contribution to the Berry curvature and in turn to the AHE. Finally, it should be noted that also the real-space topology of a material as, for example, the presence of a noncoplanar, chiral spin texture can give rise to an intrinsic AHC. This phenomenon, that requires neither a finite external magnetic field, nor a finite net magnetic moment, nor even spin-orbit coupling, is commonly termed topological or chirality-induced Hall effect. The newly emerging field of topological antiferromagnetic spintronics [81] deals with this and related response phenomena at the intersection of antiferromagnetic spintronics with topology. Itinerant Magnetism of Solids Historically there was for a long time a heated discussion whether the model of itinerant or band magnetism is a suitable platform to discuss the magnetic properties of a specific solid or whether the assumption of local magnetic moments is more appropriate. With the advances in electronic structure theory to provide schemes that allow treating the electronic structure of solids from wide-band metallic solids up to narrow-band oxides, including localized systems showing the formation of electronic multiplets, the competition between these extreme models gets more or less obsolete. Accordingly, this section gives only a brief introduction to the theory of itinerant or band magnetism on the basis of the Stoner model followed by discussing the main electronic features of two prototype class of materials: disordered alloys of transition metals and the Heusler alloys. Stoner Model of Itinerant Magnetism A rather simple criterion for the spontaneous formation of ferromagnetic order is provided by the Stoner model for itinerant magnetism. Starting point is the spinprojected DOS for a paramagnetic solid as sketched in Fig. 7. Application of an external magnetic field Bext gives rise to the Zeeman splitting ΔEZ = 2μB Bext for spin-up and spin-down states with μB the Bohr magneton. The flip of the spin for some states in the vicinity of the Fermi energy EF reestablishes a common Fermi 4 Electronic Structure: Metals and Insulators 209 E E E EF EF EF 2 μBBext n↓(E) n↑(E) n↓(E) n↑(E) n↓(E) n↑(E) Fig. 7 Left: spin-dependent DOS n(EF ) at the Fermi energy EF for a paramagnetic metal. Middle: Zeeman splitting ΔEZ = 2 μB Bext due to an external magnetic field Bext . Right: spin flip leads to a common Fermi energy EF and a finite spin-magnetic moment M = (N↑ − N↓ ) energy for both spin systems leading to a net spin-magnetic moment M = (N↑ − N↓ ). For small magnetic fields Bext , the resulting Pauli spin susceptibility χ = M μB /Bext is determined by the density of states n(EF ) at the Fermi energy EF : χ0 = 2μB n(EF ). (43) An imbalance of N↑ and N↓ also changes the total energy due to the exchange interaction. The corresponding spin-dependent correction for the electron energies Ej kσ for spin σ (with σ = ±1/2) may be written as [82]: Ej kσ = Ej k + sign (σ ) M I (44) with the Stoner exchange integral I originally seen as a parameter. Accounting for this correction, in addition one is led to the enhanced spin susceptibility χ = S χ0 , (45) with S the Stoner enhancement factor: S= 1 . 1 − I n(EF ) (46) The paramagnetic state remains stable as long as I n(EF ) < 1 holds. However, when I n(EF ) approaches the value 1, the enhancement factor S and with this the induced magnetic moment diverge indicating an instability. In fact, I n(EF ) > 1 for the paramagnetic reference state implies that the increase of kinetic energy associated with the flip of the spin for electrons at the Fermi energy is more than compensated by the resulting change in the exchange-correlation energy even without an external field leading to a stabilization of the ferromagnetic state [83]. Accordingly, the Stoner criterion I n(EF ) > 1 indicates the spontaneous formation of ferromagnetic spin order for a solid. 210 H. Ebert et al. Linear response theory allows deriving explicit expressions for the static [84, 85, 86] and dynamic [87, 88] spin susceptibility of arbitrary systems that confirm Eqs. (43) and (45) even for inhomogeneous, i.e., non-bulk, systems. Working within the framework of SDFT provides in particular a clear prescription for the calculation of the Stoner exchange integral [89, 84]: I= d 3 r γ (r)2 K(r), (47) with the induced spin polarization γ (r) and the exchange-correlation kernel K(r): γ (r) = |ψj k (r)|2 δ(EF − Ej k )/n(EF ) (48) jk K(r) = − 1 2 δ 2 Exc δm2 (49) . m(r)=0 Corresponding numerical results for the Stoner exchange-correlation integral I (left) and density of states n(EF ) (right) are given in Fig. 8 for the 3d, 4d, and 5d transition metal rows. As one notes, I varies smoothly within a transition metal row. Accordingly, the product I n(EF ) primarily reflects the variation of n(EF ) with atomic number. In line with experiment, the Stoner criterion for ferromagnetic ordering is met only for the late 3d transition metals Fe, Co, and Ni. Because of the increase of the d-band width when going from a 3d metal to the corresponding isoelectronic 4d or 5d metal, the Stoner product decreases and with this the tendency toward ferromagnetic ordering. This trend is best seen for the sequence of fcc-metals Ni-Pd-Pt that leads from a ferromagnet to strongly enhanced Pauli paramagnets with a Stoner enhancement factor of 5.96 and 2.16, respectively. 1 3d 4d 5d 2 n(EF) (1/eV) I (eV) 0.9 0.8 0.7 0.6 3d 4d 5d 1.5 1 0.5 0.5 0.4 0 Ti Zr V Cr Mn Fe Nb Mo Tc Ru Co Rh Ni Pd Cu Ag Ti V Cr Mn Fe Zr Nb Mo Tc Ru Co Ni Rh Pd Cu Ag Hf Ta Ir Pt Au Hf Ta W Ir Au W Re Os Re Os Pt Fig. 8 Stoner exchange-correlation integral I (left) and density of states n(EF ) (right) at the Fermi energy EF for the 3d, 4d, and 5d transition metal rows [26] 4 Electronic Structure: Metals and Insulators 211 As the Stoner integral I in general does not change much with the atomic environment, the Stoner factor is primarily determined by the DOS n(EF ) at the Fermi energy EF . As the d-band width W of the transition metals decreases with coordination number, this leads usually to an increase of n(EF ). This implies a corresponding increase of the tendency toward ferromagnetic ordering with reduced dimensionality. Accordingly, magnetically ordered surface layers have been predicted by theory for the paramagnetic metals V and Pd while the experimental situation is unclear. For free and deposited transition metal clusters, many SDFTbased calculations led to finite spin-magnetic moments as, for example, for free Ru13 , Rh13 , and Pd13 clusters [90]. Additional calculations for the paramagnetic state gave a large peak for the DOS near the top of the valence band with the Fermi energy located at its maximum, i.e., the magnetic ordering could be explained on the basis of the Stoner criterion. These results are fully in line with experimental data for RuN and PdN clusters; for example, mRh ≈ 0.8 μB /atom in Rh9 , mPd < 0.4 μB /atom in Pd13 and mRu < 0.32 μB /atom in Ru10 clusters [91]. In accordance with the Stoner criterion, an increase of cluster size led to a decrease of magnetic moments as, for example, mRh ≈ 0.16 μB /atom in Rh34 and mRu < 0.09 μB /atom in Ru1115 clusters. The Stoner criterion was also applied successfully to deposited atoms. For example, in line with experiment, self-consistent LSDA-based calculations predicted a finite and vanishing spin-magnetic moment for Fe and Ni atoms, respectively, on a chalcogenide topological insulator surface [92]. Slater-Pauling Curve The substitutional magnetic alloys of 3d transition metals are often seen as prototype materials for itinerant metallic magnetism. The experimental data on the average magnetic moment M per atom of these systems is summarized by the well-known Slater-Pauling curve shown in Fig. 9 (top). There are two main branches to be seen: one leading from Fe to Cu with a slope of −45◦ and another one from Fe to Cr with a slope of +45◦ . M is obviously given by M = N↑ − N↓ , where Nσ is the average number of valence electrons with spin character σ . With the total number of valence electrons Z = N↑ + N↓ , one gets the simple relation M = 2 N↑ − Z. On the basis of the outdated rigid-band model that postulates a common electronic band structure for both components of a binary alloy, all Ni and Co alloys are considered to be strong ferromagnets with their spin-up band filled. Accordingly, N↑ is constant and M decreases with Z explaining the right main branch of the Slater-Pauling curve. For Z = 8.25, one may assume that the Fermi energy is at the top of the spin-up band. Decreasing Z, one can now expect that N↓ stays constant leading to M = Z − 2 N↓ . This obviously gives a simple explanation for the second branch including in particular the Fe-Cr and Fe-V alloys. Using the tight-binding version of the CPA combined with the Hartree-Fock approximation, Hasegawa and Kanamori [94] could already give an alternative qualitative explanation for the Slater-Pauling curve avoiding the unrealistic assumptions 212 H. Ebert et al. 2.5 FeCo magnetic moment (µ B) Experiment FeTi FeNi 2 CoFe 1.5 FeMn NiCo NiFe CoCr 1 FeCr 0.5 CoMn NiMn NiCu NiCr 0 NiV 24 25 2.5 2 FeSc 1.5 27 26 electron number / atom 28 FeCo Theory magnetic moment (µ B) NiTi FeV FeNi FeCu CoFe NiFe CoMn(2) CoCr NiCo 1 CoMn (1) NiMn FeTi 0.5 FeV NiCu NiTi FeCr NiCr 0 NiV 24 25 27 26 electron number / atom 28 Fig. 9 Top: experimental Slater-Pauling curve, i.e., the average magnetic moment corresponding to the saturation magnetization of Fe-, Ni-, and Co-based alloys vs. average number of electrons per atom. Bottom: corresponding theoretical results obtained by means of the KKR-CPA. The fcc, instead of hcp, structure is assumed for Co-based alloys. For Co-Mn, two solutions, CoMn(1) with a Mn local moment parallel to the bulk magnetization and CoMn(2) with an antiparallel moment, are obtained. (All data taken from [93]) of the rigid-band model. This approach could be improved a lot by Akai using the KKR-CPA within the framework of SDFT leading for most alloy systems to a quantitative agreement between theory and experiment [93] (see bottom panel of Fig. 9). Most importantly, this implies that the average moments are well described by the effective CPA medium. As found by experiment, the curves on the left-hand side, connected with Febased bcc alloys, have a slope of about +45◦ although there is some spread. The common feature of the alloys belonging to these subbranches is that the solute atoms (Sc, Ti, V, and Cr) have negative local magnetic moments; i.e., they are aligned antiparallel to the moments of the host (Fe, Co, and Ni). These negative 4 Electronic Structure: Metals and Insulators 213 moments can be associated with the appearance of hole states above or at the top of the majority-spin d-bands. These hole states have a large amplitude at the solute atoms causing a negative local moment there, as well as the rapid decrease of the average moment with the solute concentration. The existence of the hole states in the majority-spin band also affects the concentration dependence of the average moment. In the case that no holes exist in the majority-spin band, the main origin of the concentration dependence of the average moment is the reduced number of available d-electrons as determined by the average number of valence electrons. In fact, the straight line with a slope of −45◦ in the right half of the Slater-Pauling curve, that is mainly associated with Ni-based fcc alloys, is explained this way. Contrarily, in the case that holes exist in the majority-spin band, it is mainly the missing number of majority d-states that causes the concentration dependence. Thus, the slope of the average moment against the total number of electrons varies depending on the solute atoms; the smaller the difference in the number of the valence electrons, the steeper the slope. The above discussion, however, is useful only for simple cases where the magnetic state is rather stable, typically in the region of the strong ferromagnetism of Ni. More delicate situations as, for example, the Ni-Fe, Ni-Mn, and Fe-Mn alloys in the vicinity where ferromagnetism becomes instable, however, need a more detailed and specific discussion [93]. An important feature of the CPA calculations is that the alloy components essentially keep their intrinsic properties. Fe, Co, and Ni, for example, have in general in the various alloys a spin moment close to that of the pure elements (2.2, 1.7, and 0.6 μB , respectively; see Table 1). This is fully in line with results of neutron scattering experiments or XMCD (X-ray magnetic circular dichroism) experiments that give access to the spin and orbital moment in an element-specific way via the XMCD sum rules [9]. Figure 10 shows corresponding results for the average spinand orbital magnetic moment per atom in fcc-Fex Ni1−x together with componentspecific data as calculated via the relativistic KKR-CPA on the basis of the LSDA and LSDA+DMFT, respectively, in comparison with experiment [95]. As one notes, the individual spin-magnetic moments of Fe and Ni in fcc-Fex Ni1−x show only a rather weak concentration dependence. This also applies for the spin-orbit-induced orbital moments that are defined by the expectation value of the angular momentum operator lz (see, e.g., the discussion in [17]). These findings are in full agreement with the individual moments determined via XMCD at the L2,3 -edges of Fe and Ni [95]. Here, it is interesting to note that the theoretical results for the spin moment hardly depend on the computational mode, i.e., whether the calculations are based on the LSDA or the LSDA+DMFT. The orbital moment, on the other hand, depends strongly on the computational mode. In particular it is found that inclusion of correlation effects via the DMFT improves agreement with experiment. Heusler Alloys There are plenty of experimental and theoretical investigations on Heusler alloys in the literature because of their very rich variety of magnetic properties including 214 H. Ebert et al. 1.5 0.08 1 LSDA LSDA+DMFT Expt 0.5 0 0 0.2 0.4 xFe 0.6 0.04 0 0 0.8 3 mspin (μB) morb (μB) LSDA LSDA+DMFT Expt 1 0 0.8 0.2 0.4 xFe 0.1 0.05 0.6 0 0 0.8 0.2 0.4 xFe 0.6 0.8 0.25 1 Ni 0.2 morb (μB) 0.8 LSDA LSDA+DMFT Expt Ni 0.15 0.6 0.4 0 0 0.6 0.15 2 0.2 0.4 xFe LSDA LSDA+DMFT Expt Fe 2.5 1.5 0.2 0.2 Fe mspin (μB) LSDA LSDA+DMFT Expt 0.12 morb (μB) mspin (μB) 2 LSDA LSDA+DMFT Expt 0.2 0.4 xFe 0.1 0.05 0.6 0.8 0 0 0.2 0.4 xFe 0.6 0.8 Fig. 10 Top row: average spin- (left) and orbital (right) magnetic moment per atom in fccFex Ni1−x as calculated on the basis of the LSDA and LSDA+DMFT, respectively, in comparison with experiment. In addition, the individual moments calculated for Fe (middle row) and Ni (bottom row) are given. The experimental component-specific data stem from XMCD measurements. (All data taken from [95]) ferromagnetism, antiferromagnetism, helimagnetism, and Pauli paramagnetism. Depending on their crystal structure, one can distinguish two families of Heusler alloys: semi-Heuslers of the type XYZ with C1b structure and full-Heuslers of the type X2 YZ with L21 structure. Most of the Heusler alloys are metals; however, for some systems also, half-metallic and semiconducting behavior has been observed. The half-metallicity found in certain Heusler magnets [62, 87, 96, 97, 98] attracted especially strong interest over the last 30 years because of its possible 4 Electronic Structure: Metals and Insulators 215 Fig. 11 Left: spin-resolved DOS for Co2 MnSi showing a bandgap for the minority states. Right: comparison of the spin polarization obtained by in situ SRUPS on a Co2 MnSi thin film with the calculated DOS-derived spin polarization, the calculated UPS spin polarization including broadening effects and considering only bulk states, and the calculated total UPS spin polarization including broadening effects with additional surface state contributions [99] use in spintronics and magneto-electronics. Half-metallic materials exhibit metallic behavior only for one spin direction, while for the other spin direction the Fermi level is located in a bandgap. Figure 11 shows the spin-resolved DOS for Co2 MnSi as a representative example. Defining the spin polarization p of a material in terms of the spin-dependent density of states nσ (E) according to p= n↑ (E) − n↓ (E) n↑ (E) + n↓ (E) (50) one has for half-metallic materials 100% spin polarization at the Fermi energy that should lead to a fully spin-polarized electric current. For real materials, however, the spin polarization may be influenced in various ways. Mavropoulos et al. [100], for example, demonstrated the impact of spin mixing caused by spin-orbit coupling for the Heusler alloys of the type XMnSb. As expected, this influence increased with increasing atomic number of the element X, as reflected by the ratio n↓ (EF )/n↑ (EF ) = 0.25, 0.30, 0.35, 0.75, and 2.70 found for the series X = Co, Fe, Ni, Pd, and Pt. As a consequence, one can expect a spin polarization well below 100% for compounds with heavier elements. The influence of the surface on the spin polarization has been studied by Galanakis [101] investigating the (001) surfaces of the semi-Heusler alloys NiMnSb, CoMnSb, and PtMnSb and for the full-Heusler alloys Co2 MnGe, Co2 MnSi, and Co2 CrAl. In general, a rather strong modification of electronic and magnetic properties has been found for the surface region. In the case of semiHeuslers, the Ni-, Co-, or Pt-terminated surface has a rather large DOS at the Fermi level for minority-spin states, while for the MnSb-terminated surfaces the calculated properties are close to those of the bulk. Nevertheless, half-metallicity disappears also in this case due to surface states, resulting in a spin polarization at the Fermi 216 H. Ebert et al. level of 38%, 46%, and 46% for NiMnSb, CoMnSb, and PtMnSb, respectively. A similar behavior was found for the Co-terminated surfaces of the full-Heusler alloys Co2 MnGe, Co2 MnSi, and Co2 CrAl. In the case of the MnGe-terminated (001)surface of Co2 MnGe, the spin polarization vanishes also due to the surface states, although the CrAl termination of Co2 CrAl leads to a very high spin polarization of around 84%. In experiment, the spin polarization of a material can be determined among others by tunneling experiments or by spin-resolved photoemission [102, 103, 104]. Figure 11 shows corresponding experimental and theoretical spin-polarization curves of Co2 MnSi [99] for the UPS-regime (hν = 21.2 eV) that showed for the surface regime a value of 93% at the Fermi level, the largest observed so far. NiMnSb was among the first semi-Heusler compounds predicted to be halfmetallic [105] and was intensively investigated since then by experiment as well as theory [106, 107, 108]. The Sb atoms with the atomic configuration 5s2 5p3 lead to the formation of a deep lying narrow s- and a p-band at around 12 and 3 − 5 eV, respectively, below the Fermi energy EF . Accordingly, these states are not involved in the formation of the bandgap near EF . The Ni and Mn d-states hybridize with each other as well as with the sp-states of Sb leading to the formation of bonding and antibonding bands. For the paramagnetic state of NiMnSb, the Fermi level lies in the middle of an antibonding band mainly associated with the d-states of Mn. Accordingly, exchange splitting of the antibonding band leads to a gain in energy accompanied by formation of a strong magnetic moment for the Mn atom. As a result, the Fermi energy moves to the energy gap separating bonding and antibonding minority-spin states. Due to this, there are nine minority-spin states per unit cell below EF , one due to the Sb-like s-band, three due to the Sb-like p-bands, and five due to the Ni-like d-bands, that are all occupied. As atoms forming the alloy contribute 22 electrons per unit cell, the majority-spin band contains 22 − 9 = 13 electrons, resulting in a moment of 4 μB per unit cell. Plotting the total spin-magnetic moment per unit cell, M, of NiMnSb together with that of other semi-Heusler half-metallic compounds as a function of the total number of valence electrons Z, one can see that M – in analogy to the Slater-Pauling curve for the binary transition metal alloys – follows the relation: M = Z −18 [107] (see Fig. 12 (left)). This relation is a consequence of the complete occupation of the nine minority-spin bands and follows directly from the definitions Z = N↑ + N↓ and M = N↑ − N↓ that lead to mt = Z − N↓ [98]. The occurrence of half-metallicity in the case of the prototype full-Heusler compounds Co2 MnSi and Co2 MnGe was also predicted by electronic structure calculations [109, 110]. Similar to the case of semi-Heusler alloys, the sp-bands of Si and Ge are located well below EF . For that reason, they do not participate in the formation of the energy gap that is caused by the hybridization of Mn and Co dstates. As a result, for the spin-polarized state of the material, the Fermi level is again located within the minority-spin energy gap, so that the minority band contains 1 sband and 3 p-bands derived from the sp-element and 8 Co-related d-bands, which are fully occupied by 12 electrons [98]. As shown by Galanakis et al. [111], the spin-magnetic moment of the full-Heusler alloys accordingly follows the relation M = Z − 24. Corresponding theoretical and experimental data are summarized in 4 Electronic Structure: Metals and Insulators 217 6 NiMnSe NiMnTe 0 18 -2 20 22 Z 24 20 M CoTiSb Co2MnSb Co2FeSi Co2FeAl Ni2MnAl Rh2MnGe Rh2MnSn Rh2MnPb Co2VAl Fe2MnAl 2 = Z18 = M 16 CoVSb FeMnSb CoCrSb NiVSb Co2MnAs Rh2MnIn Rh2MnTl Co2TiSn Fe2CrAl Co2TiAl Fe2VAl Mn2VGe 24 2 CoMnSb IrMnSb NiCrSb M M RhMnSb 4 CoFeSb NiFeSb Co2CrAl Fe2MnSi Ru2MnSi Ru2MnGe Ru2MnSn Z- NiMnSb PdMnSb PtMnSb 4 0 Co2MnSi Co2MnGe Co2MnSn 6 Co2MnAl Co2MnGa Rh2MnAl Rh2MnGa Ru2MnSb Mn2VAl 22 24 26 Z 28 30 32 Fig. 12 Left: calculated total spin moments per unit cell for several semi-Heusler alloys. Experimental values are given for NiMnSb (3.85 μB ), PdMnSb (3.95 μB ), PtMnSb (4.14 μB ), and CoTiSb (nonmagnetic) [107]. Right: calculated total spin moments for several full-Heusler alloys. Experimental values are given for Co2 MnAl (4.01 μB ), Co2 MnSi (5.07 μB ), Co2 MnGa (4.05 μB ), Co2 MnGe (5.11 μB ), Co2 MnSn (5.08 μB ), Co2 FeSi (5.9 μB ), Mn2 VAl (1.82 μB ), and Fe2 VAl (nonmagnetic) [111]. In both figures the dashed line represents the corresponding Slater-Pauling curve. The open circles represent the compounds deviating from this curve the Slater-Pauling plot in Fig. 12 (right). The difference to the Slater-Pauling curve of the binary transition metal alloys in Fig. 9 is due to the fixed number of minorityspin electrons in the half-metallic Heusler compounds. In this case, increasing Z leads to a filling of the majority band, while for the ferromagnetic transition metal alloys, the relation M = 10 − Z for the right branch is a result of the full occupation of five majority-spin d-states and charge neutrality achieved by filling the minorityspin d-states. Within the family of Heusler compounds, there are furthermore the so-called inverse full-Heusler compounds that have a similar chemical formula X2 YZ but crystallize in the so-called Xα structure [98]. The prototype of the inverse fullHeusler compounds is Hg2 TiCu [112]. As Fig. 13 demonstrates, the magnetic moments per unit cell as a function of valence electrons also follow in this case corresponding Slater-Pauling rules. Total Electronic Energy and Magnetic Configuration The calculation of the electronic total energy allows to seek for the magnetic ground state of a solid. This applies to the crystal structure as well as to the specific magnetic ordering or spin configuration, respectively. For many purposes, it is helpful to represent the neighborhood of the ground state or a suitable magnetic reference state by mapping the complex configuration dependence of the total energy of the system on an approximate spin Hamiltonian. An issue in this context is spin-orbit coupling that removes energetic degeneracies of competing spin configurations, may 218 H. Ebert et al. M 2 0 -2 -4 M = Z-18 Ti2NiAl Ti2CuAl Ti2CoAl Ti2CoSi Ti2FeAs Ti2FeSi Sc2NiAl V2MnAl Sc2CoSi V2CrSi Ti2FeAl Sc2NiSi V2CrAl Ti2MnAs Ti2MnAl Ti2CrAl Sc2MnAl Ti2VAs Sc2FeAl Sc2CrSi Ti TiSi V2VAl 2 Sc2VAs Sc2MnSi Sc CrAs Ti2VAl 2 Ti2VSi Sc2VSi Sc2VAl Sc2CrAl 13 15 17 19 21 23 M = Z-24 4 2 M 4 0 -2 V2CoSi V2FeAs Cr2CrAs Cr2FeAl Mn2CrAl Cr2CoAs Mn2FeSi Mn2MnAs Mn2CoAl Cr2CrSi V2MnSi V2FeAl -4 Cr2CrAl 20 22 Mn2NiAs Mn2CoAs Mn2NiSi Mn2FeAs Mn2CoSi Cr2NiSi V2NiAs Cr2CoAl Cr2MnAs V2NiAl Mn MnAl 2 Cr2MnSi Mn2CrSi V2CoAl V2FeSi V2MnAs Cr2MnAl 24 Z 26 28 Cr2NiAl Cr2CoSi Cr2FeAs Mn2FeAl Mn2MnSi Mn2CrAs 30 Z Fig. 13 Total spin-magnetic moments per unit cell (in μB ) as a function of the total number of valence electrons Z in the unit cell for several compounds. The lines represent the two of the various forms of the Slater-Pauling rule [98]. The compounds within the frames follow one of these rules and are perfect half-metals, while the rest of the alloys slightly deviate. For this reason, their total spin-magnetic moment is represented by an open red circle. The sign of the spin-magnetic moments has been chosen so that the half-metallic gap is in the spin-down band lead to the anisotropic Dzyaloshinsky-Moriya exchange interaction, and gives rise to magnetocrystalline anisotropy. Total Electronic Energy and Magnetic Ground State Access to the electronic total energy Etot (see Eq. (10)) in principle allows to determine the magnetic ground state of any solid. As a corresponding example for this, results of LSDA-based calculations for Fe in the para- (PM), ferro- (FM), as well as antiferromagnetic (AFM) state with bcc and fcc structure are shown in Fig. 14 [27]. Obviously, the use of the LSDA led to an over-binding, i.e., to a lattice parameter that is too small and a bulk modulus that is too large when compared with experiment (see Table 1). Most importantly, however, the paramagnetic fccphase was found as the ground state instead of the ferromagnetic bcc-phase. Use of the GGA on the other hand improved the situation very much giving in particular the ferromagnetic bcc-phase as the ground state. Another example for a search for the magnetic ground state configuration is given by Fig. 15. In this case, the so-called fixed spin moment method was used to explore the dependency of the total energy Etot on the lattice parameter and average atomic moment of disordered fcc-Fex Ni1−x alloys. As one notes, a double minimum occurs in the vicinity of the concentration x = 0.65 indicating the competition between a low-volume, low-spin moment phase and a high-volume, high-spin moment phase. In fact, this has been seen as a possible explanation for the occurrence of the invar effect for that composition. Another study on the invar effect, however, stressed the importance of a noncollinear spin configuration [114]. 4 Electronic Structure: Metals and Insulators 219 400 600 200 E (meV) E (meV) PM bcc FM bcc 0 PM bcc 400 200 PM fcc AFM fcc PM fcc FM bcc 0 -200 60 65 70 75 Volume (a.u.) 70 80 80 75 85 Volume (a.u.) 90 Fig. 14 Total energy Etot of paramagnetic (PM) bcc and fcc, ferromagnetic (FM) bcc, and antiferromagnetic (AFM) fcc-Fe as a function of the volume as obtained within the LSDA (left) and GGA (right), respectively. The curves are shifted in energy so that the minima of the FM-bcc curves, corresponding to Etot = 0, coincide. (All data taken from [27]) 2.0 2.0 M avg ( μ Β/atom ) 2.5 M avg ( μ Β/atom ) 2.5 1.5 1.0 1.0 0.5 0.0 6.3 1.5 Fe 60 Ni 40 6.4 6.5 6.6 a (a.u.) 6.7 6.8 6.9 0.5 0.0 6.3 Fe 65 Ni 35 6.4 6.5 6.6 6.7 6.8 6.9 a (a.u.) Fig. 15 Total energy or binding surfaces for the disordered ferromagnetic alloys Fe60 Ni40 and Fe65 Ni35 . (All data taken from [113]) The occurrence of a noncollinear spin configuration is quite common for actinide compounds but also for a number of transition metal-based systems [115]. A prominent example for this is Mn3 Sn with its hexagonal unit cell given in Fig. 16. Scalar relativistic calculations gave for all shown noncollinear spin configurations a total energy well below that of competing collinear spin configurations. Ignoring spin-orbit coupling, only the angle between the spin moments is relevant leading to a degeneracy for all four spin configurations. However, if spin-orbit coupling is taken into account, the degeneracy is removed and the resulting moment may deviate to some extent from the orientation found for the scalar relativistic calculations (see thin arrows for configuration (c) and (d) in Fig. 16). The presence of spinorbit coupling implies that the spin-magnetic moment is accompanied by an orbital one. It is interesting to note that the orientation of the spin-orbit-induced orbital magnetic moment may and will deviate from that of the spin moment if symmetry of the system allows. Another interesting property of spin-compensated noncollinear 220 H. Ebert et al. Mn Sn z=1/4 z=3/4 (a) (b) 1 2 1 3 2 3 2 (c) (d) 1 1 3 2 3 Fig. 16 Crystal and magnetic structure of Mn3 Sn. Rotations of the magnetic moments leading to weak ferromagnetism in the structure in (c) and (d) are shown only for atoms in the z = 0.25 plane (thin arrows). Moments of the atoms in the z = 0.75 plane are parallel to the moments of the corresponding atoms of the z = 0.25 plane. (All data taken from [115]) antiferromagnets having certain magnetic space groups is the occurrence of the anomalous Hall effect that is usually not expected for an antiferromagnet [79]. However, group theoretical considerations unambiguously show that the Hall effect may even show up for spin-compensated solids [116]. The first observations of the AHE [117] as well as of its thermoelectric analog, the anomalous Nernst effect [118], in a non-collinear antiferromagnet could in fact both be made in Mn3 Sn. The occurrence of spin-polarized currents in this and related materials currently attracts a lot of attention as well [119,120]. Finally, a criterion for the instability of a collinear spin structure with respect to a transition to a noncollinear one was formulated by Sandratskii and Kübler: If the collinear magnetic structure under consideration is not distinguished by symmetry compared with the noncollinear structures obtained with infinitesimal deviations of the magnetic moments from collinear directions, this structure is unstable [121]. Exchange Coupling Parameters When dealing with competing magnetic configurations, it is not always possible or not necessary to perform full ab-initio calculations. In these cases, one may adopt a multi-scale approach that uses the classical Heisenberg Hamiltonian: 4 Electronic Structure: Metals and Insulators H=− 221 Jij m̂i · m̂j , (51) i=j with m̂i(j ) the orientation of the magnetic moment on the lattice site i(j ), for corresponding simulations. The isotropic exchange coupling parameters Jij , on the other hand, are calculated in an ab initio way. This can be done by applying a corresponding version of the so-called Connolly-Williams method [122]. This implies to calculate the total energy for many different magnetic configurations within a super-cell and to determine the exchange coupling parameters by fitting the energy on the basis of Eq. (51). This way one obviously achieves a mapping of the complicated energy landscape E({m̂i }) on the rather simple expression in Eq. (51) involving only pair interactions that is easy to handle. Accordingly, a more accurate representation of the energy as a function of the magnetic configuration can therefore be expected by a cluster expansion as suggested by various authors [123, 124]. Another approach to determine the exchange coupling parameters Jij in Eq. (51) is to consider the change of the single-particle energy ΔEij if two magnetic moments on sites i and j change their relative orientation. The necessary formal developments started with the work of Oguchi et al. who expressed the difference in energy between the ferro- and antiferromagnetic state of a solid making use of multiple scattering theory and Lloyd’s formula [125]. Lichtenstein et al. [126] extended this approach dealing with the coupling energy ΔEij associated with an individual pair (i, j ) of atoms. If ΔEij is expressed to lowest order with respect to the orientation angle of the moments m̂i and m̂j , one gets a one-to-one mapping of the exchange coupling energy ΔEij to the Heisenberg Hamiltonian in Eq. (51), with the exchange coupling constants Jij given by [126]: Jij = 1 4π EF ij ji −1 −1 −1 τ↑ tj−1 dE Trace ti↑ − ti↓ ↑ − tj ↓ τ↓ , (52) ij where ti↑(↓) is the spin-dependent single site scattering matrix for site i and τ↑(↓) is the spin-dependent scattering path operator matrix connecting sites i and j . Results for the isotropic exchange coupling parameter Jij of Fe and Co as a function of the distance Rij between sites i and j that have been obtained using an analogous expression derived within the LMTO-GF formalism [127] are shown in Fig. 17. The big advantage of this approach is that it can be applied with comparable effort to more complex systems like disordered substitutional alloys [128], Heusler alloys [129, 130, 131], diluted magnetic semiconductors [132, 133, 134, 135, 136, 137], magnetic surface films [127,138], or finite deposited clusters [139,140]. In addition, it should be mentioned that an approach similar to that leading to Eq. (52) was worked out by Katsnelson and Lichtenstein [141] that allows for an improved treatment of correlated systems. If spin-orbit coupling is accounted for, the exchange coupling parameter in the Heisenberg Hamiltonian has to be replaced by a corresponding tensor: 222 H. Ebert et al. 20 16 20 16 Fe 12 Jij (meV) Jij (meV) 12 Co 8 8 4 4 0 0 -4 -4 1 2 1.5 2.5 Rij (units of lattice parameter) 1 2 1.5 2.5 Rij (units of lattice parameter) Fig. 17 Isotropic exchange coupling constants Jij of Fe and Co as a function of the distance Rij between sites i and j . (All data taken from [127]) H=− i=j =− i=j − m̂i J m̂j + ij K(m̂i ) (53) i Jij m̂i · m̂j − m̂i J S m̂j ij i=j D ij · m̂i × m̂j + Ki (m̂i ), i=j (54) i with the accompanying single-site magnetic anisotropy represented by the term K(m̂i ). In Eq. (53), the coupling tensor J has been decomposed into its isotropic ij part Jij , its traceless symmetric part J S , and its antisymmetric part. The latter ij one, that may occur in case of systems without inversion symmetry, is often represented in terms of the so-called Dzyaloshinsky-Moriya (DM) vector D ij , with βγ γβ Dijα = 12 (Jij − Jij ) and a cyclic sequence of the Cartesian indices α, β, and γ . A corresponding generalization of the nonrelativistic expression for Jij given in Eq. (52) to its relativistic tensor form was worked out by various authors [18, 142] and applied in particular to cluster systems [124, 143] with the interest focusing on the impact of the DM interaction. It should be emphasized once more that Eq. (51) and extensions to it supply an approximate mapping of the complicated energy landscape E({m̂i }) of a system calculated in an ab initio way onto a simplified analytical expression. This implies corresponding limitations [124] in particular due to the use of the rigid spin approximation (RSA) [144]. It is interesting to note that a coupling tensor of the same shape as in Eq. (53) occurs for the indirect coupling of nuclear spins mediated by conduction electrons. In this case the mentioned restrictions do not apply. As a consequence, the linear response formalism on the basis of the Dyson equation can 4 Electronic Structure: Metals and Insulators 223 be used without restrictions to determine the corresponding nuclear spin-nuclear spin coupling tensor [145]. Another approach using the same idea as the Connolly-Williams method is based on the total energy ΔE(q, θ ) = E(q, θ ) − E(0, θ ) calculated for noncollinear spin spirals (see section “Spin Spiral Calculations”) that are characterized by the wave vector q and tilt angles (θ, φ(R i ) = q · R i ) for the moment mi on the atomic position R i . In the case of a small tilt angle θ , ΔE(q, θ ) can be represented in terms of the Fourier transform J (q) of the real space exchange coupling parameters [146]: E(q, θ ) = E0 (θ ) − θ2 J (q). 2 (55) Accordingly, performing an inverse Fourier transformation, one can determine the real-space interatomic exchange coupling parameters J0j . In the case of a simple Bravais lattice, this is given by the expression J0j = 1 −iqR 0j e J (q). N q (56) Uhl et al. [147] applied this scheme among others for a study of the invar system Fe3 Pt. From their numerical results for ΔE(q, θ ), they evaluated the spin-wave stiffness constant A and the exchange parameter J0 that allows to give an estimate for the Curie temperature on the basis of the mean field approximation (MFA) (see section “Finite-Temperature Magnetism”). Similar work was also done for twodimensional systems as for example magnetic surface films. Within their study on an Fe film on W(110), Heide et al. also accounted for the spin-orbit coupling [148]. This way they could determine not only the spin-wave stiffness constant A but also the Dzyaloshinsky-Moriya interaction vectors D. As discussed in section “Spin Density Functional Theory”, dealing with systems with narrow electronic energy bands, in order to go beyond the local spin-density approximation, the LDA+U or +DMFT methods can be used to properly account for strong electronic correlation in these materials. In order to adapt the method for the treatment of exchange interactions formulated within the LSDA [126, 18, 142], a corresponding theory was developed by Katsnelson and Lichtenstein [141] that employs an analog of the local force theorem to derive expressions for effective exchange parameters, Dzyaloshinsky-Moriya interaction, and magnetic anisotropy in highly correlated systems. The authors demonstrated for the particular case of ferromagnetic Fe that treating correlation effects beyond the LSDA (within the LDA+Σ approach) in the exchange interactions results in a spin-wave spectrum and spin-wave stiffness which are in better agreement with experiment than those obtained within plain LSDA. The important role of additional contributions to the exchange coupling of the correlation interactions has been demonstrated, e.g., for half-metallic ferromagnetism in CrO2 [149], magnetic properties of CaMnO3 [150], and magnetic properties of transition metal oxides [151]. Another important feature of the exchange interactions observed in various materials are their strong 224 H. Ebert et al. orientation dependence that would require to go beyond the Heisenberg model when considering the finite-temperature or spin-wave properties of these materials. This problem was recently discussed by different groups [151, 152, 153, 154]. Magneto-Crystalline Anisotropy Magnetocrystalline anisotropy denotes the dependence of the total energy of a system on the orientation m̂ of its magnetization with the anisotropic part EA (m̂) of the energy taking a minimum for m̂ along a so-called easy direction of the magnetization. Usually, EA (m̂) is split into the intrinsic material-specific magnetocrystalline anisotropy (MCA) energy EMCA (m̂) and the extrinsic shape anisotropy energy Eshape (m̂) determined by the shape of the sample: EA (m̂) = EMCA (m̂) + Eshape (m̂). (57) Considering the difference in energy ΔEX (m̂, m̂ ), with X=A, MCA or shape, respectively, for the magnetization oriented along directions m̂ and m̂ , respectively, one has accordingly ΔEX (m̂, m̂ ) = EX (m̂) − EX (m̂ ). A convenient phenomenological representation of the magnetocrystalline anisotropy energy can be given by an expansion in terms of spherical harmonics Ylm (m̂) EMCA (m̂) = m=l κlm Ylm (m̂) (58) l even m=−l or alternatively by an expansion in powers of the direction cosines (α1 , α2 , α3 ) = (m̂ · x̂, m̂ · ŷ, m̂ · ẑ): EMCA (m̂) = b0 + i,j bij αi αj + bij kl αi αj αk αl + . . . (59) i,j,k,l Assuming degeneracy of the energy upon time-reversal, i.e., flip of the magnetization, only terms that are even with respect to the orientation m̂ of the magnetization can occur in these equations. Further restrictions on the expansions are imposed by the crystal symmetry of the investigated material [7]. Considering, for example, a hexagonal system with the expansion up to sixth order, the corresponding hex (m̂) are given by: expressions for EMCA hex EMCA (m̂) = K̃0 + K̃1 Y20 (θ, φ) + K̃2 Y40 (θ, φ) +K̃3 Y60 (θ, φ) + K̃4 Y64 (θ, φ) (60) = K0 + K1 (α12 + α22 ) + K2 (α12 + α22 )2 + K3 (α12 + α22 )3 +K4 (α12 − α22 ) (α14 − 14α12 α22 + α24 ), (61) 4 Electronic Structure: Metals and Insulators 225 2 where the coefficients are interconnected by the relations K̃1 = 21 (7K1 + 8K2 + 8 16 1 8K3 ), K̃2 = 385 (11K2 + 18K3 ), K̃3 = 231 K3 and K̃4 = 10,395 K4 . The shape anisotropy energy Eshape (m̂) is usually associated with the classical dipole-dipole interaction of the individual magnetic moments mν on the lattice sites [6, 155]. Accordingly, it can be determined straightforwardly by a corresponding lattice summation: Edip (m̂) = mν mν c2 νν Rn [R nνν · m̂]2 1 − 3 , |R nνν |3 |R nνν |2 1 (62) with R nνν = R n + ρ ν − ρ ν , where a periodic system has been considered with lattice vectors R n and ρ ν the basis vectors within the unit cell. For transition metal systems, the intrinsic part of the magnetic anisotropy energy EMCA (m̂) has to be ascribed to the spin-orbit coupling. Accordingly, the corresponding energy difference ΔESOC (m̂, m̂ ) can be determined by total energy calculations with the magnetization oriented along directions m̂ and m̂ , respectively, and taking the difference. Obviously, this implies a full SCF calculation for both orientations and taking the difference of large numbers to get ΔESOC (m̂, m̂ ). The problems connected with this approach can be avoided by making use of the so-called magnetic force theorem that allows to approximate ΔESOC (m̂, m̂ ) by the difference of the single particle or band energies (see Eq. (10)) for the two orientations obtained using a frozen spin-dependent potential [6, 156]: ΔESOC (m̂, m̂ ) = − EFm̂ dE N m̂ (E) − N m̂ (E) 1 − nm̂ (EFm̂ ) (EFm̂ − EFm̂ )2 + O(EFm̂ − EFm̂ )3 . 2 (63) Here EFm̂ is the Fermi energy for the magnetization along m̂ while nm̂ (E) and E N m̂ (E) = dE nm̂ (E ) are the corresponding DOS and integrated DOS, respectively. This approach is used extensively for compounds and layered systems and leads typically to anisotropy energies that deviate less than 10% from results obtained from full SCF calculations. By using, in the case of layered systems, layer-resolved data for the DOS in Eq. (63), a corresponding layer decomposition of the anisotropy energy ΔESOC (m̂, m̂ ) could be achieved [157]. Application of this scheme for the spatial decomposition of ΔESOC (m̂, m̂ ) shows that the dominating contributions originate in general from the interface or surface layers, respectively. Equation (63) implies that spin-orbit coupling is accounted for within the underlying electronic structure calculations. Instead one can start from a scalar relativistic calculation and treat HSOC as a perturbation. Solovyev et al. [158] used this approach on the basis of the Green function method in combination with the Dyson equation Eq. (21). This allowed to write the spin-orbit-induced correction 226 H. Ebert et al. ESOC (m̂) to the single-particle energies as a sum of two-site interactions: ESOC (m̂) = EF dE δN(E) = Eij (m̂) (64) ij with Eij (m̂) = − 1 2π EF ij j ji i dE Trace G0 (m̂) HSOC G0 (m̂) HSOC , (65) ij where G0 (m̂) are real space structural Green function matrices corresponding to the scattering path operator in Eq. (20) [42]. In contrast to Eq. (63), Eq. (65) provides a unique spatial or component-wise decomposition of the magnetocrystalline energy. Solovyev et al. [158] used this approach for a detailed study of the ordered compounds TX with T = Fe, Co and X = Pd, Pt having CuAu structure. This way they could in particular show that the hybridization between the T and X sublattices essentially determines their magnetocrystalline anisotropy. In addition, an expression analogous to Eq. (65) allowed to demonstrate and discuss the interconnection between the energy correction E(m̂) and the spin-orbit-induced orbital magnetic moment μorb represented by the expectation value of the angular momentum operator l. Using a similar approach as sketched here, this relation was already investigated before by Bruno [159] and also by van der Laan [160]. Assuming a strong ferromagnet with the majority band filled, the relation: 1 ESOC (m̂) = − C ζ σ · l , 4 (66) was derived, where C is a constant and ζ represents the strength of the spin-orbit coupling. This equation was used in numerous experimental studies that exploited the XMCD (X-ray magnetic circular dichroism) and the associated sum rules [9] to determine in an element specific way the change of the angular momentum Δl when changing the orientation of the magnetization from m̂ to m̂ to get a component resolved estimate for the corresponding anisotropy energy ΔESOC (m̂, m̂ ). A further approach to calculate the spin-orbit-induced anisotropy energy is to consider the torque T (θ ) exerted on a magnetic moment m when the magnetization is tilted by the angle θ away from its equilibrium orientation (easy axis). The corresponding expression for T (θ ), T (θ ) = j k occ ψj k ∂HSOC ψj k , ∂θ (67) was given first by Wang et al. [161] for the case that the electronic structure is represented in terms of Bloch states. A more general expression was obtained on the basis of multiple scattering theory [162]: 4 Electronic Structure: Metals and Insulators Tαm̂û = − 1 π EF dE ∂ −1 0 ln det t( , m̂) − G ∂α û 227 (68) where the torque component with respect to a rotation of the magnetization around an axis û is considered and where t(m̂) is the single-site t-matrix for an orientation of the moments along m̂ and G0 is the corresponding free electron Green function matrix. On the basis of Eq. (67) or (68), respectively, the anisotropy energy ΔESOC (m̂, m̂ ) is obtained from the torque by integrating along a path connecting m̂ and m̂ . This approach is especially suited when dealing with systems with uniaxial anisotropy. Neglecting in this case the dependence on φ, the anisotropy energy can be represented as ESOC (θ ) = K0 + K1 sin2 (θ ) + K2 sin4 (θ ) with the torque given by: T (θ ) = dESOC (θ ) = K1 sin(2θ ) + 2K2 sin(2θ ) sin2 θ. dθ (69) For the special setting θ = π/4 and φ = 0, one has therefore: ESOC (π/2) − ESOC (0) = K1 + K2 = T (π/4). (70) This implies that if the contribution K1 sin2 (θ ) to ESOC (θ ) dominates, a situation often met, K1 and with this ESOC (θ ) can be obtained from a single calculation for the special settings. Otherwise, K1 and K2 can be obtained by a fit to a sequence of calculations for varying angles θ . The contribution ΔEdip (m̂, m̂ ) to the total anisotropy energy that is associated with the dipole-dipole interaction of the individual magnetic moments is usually treated classically by evaluating a corresponding Madelung sum (see Eq. (62)) [155, 163, 164]. While for most cases ΔESOC (m̂, m̂ ) is much larger than ΔEdip (m̂, m̂ ), both contributions are often found for layered systems to be in the same order of magnitude. As ΔEdip (m̂, m̂ ) always favors an in-plane orientation of the magnetization while ΔESOC (m̂, m̂ ) in general favors an out-of-plane orientation, one may have a flip of the easy axis from out-of-plane to in-plane with increasing thickness of the magnetic layers. Such a behavior has been found, for example, for Con Pdm multilayers as shown in Fig. 18 [163]. Similar results were obtained for the magnetic surface layer system Fen /Au(001) that shows a change from out-of-plane to in-plane anisotropy if the number n of Fe layers is larger than 3 [155]. In particular in cases for which ΔESOC (m̂, m̂ ) and ΔEdip (m̂, m̂ ) are of the same order of magnitude, it seems questionable to treat the first contribution quantum mechanically and the second one in a classical way. Although it was pointed out already nearly 30 years ago that ΔEdip (m̂, m̂ ) is caused by the Breit interaction [14], there is only little numerical work done in this direction [165,166]. Including a vector potential in the Dirac equation Eq. (23) that represents the corresponding current-current interaction such numerical work has been done on magnetic surface films and multilayer systems. It turned out in all investigated 228 H. Ebert et al. Co1Pd2 0.34 Co4Pd2 0 0 0.4 ΔE (meV) Co2Pd4 1 ΔE (meV/unit cell) 0.68 Co1Pd5 2 Kt (mJ/m ) 2 0 -0.4 -0.8 Co3Pd3 -1 2 4 6 8 10 12 14 t (Å) -0.34 -1.2 1 2 3 4 5 6 Number of Fe layers Fig. 18 Left: calculated total anisotropy energy ΔE of Con Pdm multilayers with (111)-oriented fcc structure as a function of the thickness t of the magnetic Co layers. Corresponding experimental data for the product of the anisotropy energy density K and Co thickness t are shown for polycrystalline films deposited at two different temperatures (triangles up and down). (All data taken from [163]) Right: SOC-induced (ΔESOC ; circles) and dipole-dipole (ΔEdip ; triangles) contributions to the total anisotropy energy (ΔE; squares) for the magnetic surface layer system Fen /Au(001) as a function of the number n of Fe layers. (All data taken from [155]) cases that the classical treatment on the basis of the dipole-dipole interaction leads to results for ΔEdip (m̂, m̂ ) that are very close to those of a coherent quantummechanical calculation that accounts also for the Breit interaction. Starting from the 1950s, compounds of rare-earth (RE) with 3d transition metal (TM) elements, as, for example SmCo5 , or Nd2 Fe14 B, attracted much attention because of their strong magnetic anisotropy. In these materials, the MCA is primarily determined by the RE sublattice, while the TM sublattice is responsible for the magnetic ordering [167]. For that reason, the simplified two-sublattice Hamiltonian f d-f H = Hd + HCEF + Hex (71) f is often used to discuss their properties, where Hd and HCEF characterize the TM d-f and RE, respectively, sublattices while Hex describes the exchange interactions f between the two. Within the single-ion model, HCEF accounts for the interaction of the aspherical 4f-charge with the crystalline electric field (CEF). Due to strong spin-orbit interaction for the 4f-electrons, rotation of the magnetization leads to a rotation of their aspherical charge cloud. This in turn results in a dependency of the electrostatic energy on the orientation of the 4f-magnetic moment as described by the Hamiltonian [168, 169, 170] f HCEF = n,m n Am n θJ n r 4f Onm 4f . (72) 4 Electronic Structure: Metals and Insulators 229 Here Am n are crystal field parameters for the angular momentum quantum numbers n and m determined by the charge contribution in the system excluding the 4felectrons, θJ n are Stevens’ factors depending on the total angular momentum quantum number J , Onm 4f are the expectation values of the Stevens’ operators, and r n 4f are the expectation values of r n calculated for the 4f-states of the RE atom. As the quantities θJ n and Onm 4f are all tabulated, calculation of the crystal field f n parameters Am n together with r 4f allows to fix HCEF and with this to determine the corresponding phenomenological anisotropy constants Ki [168, 169]. While the first calculations in the field have been done adopting a spherical approximation for the potential [171,172], later work clearly demonstrated the need to use a nonspherical potential. Such calculations have been performed, for example, by Richter et al. [173] on SmCo5 representing the itinerant s-, p-, and d-electrons via band states, while the localized Sm 4f-states are treated within the atomic like socalled open shell scheme. Hummler and Fähnle report on corresponding calculations on the CEF parameters for the whole RECo5 series with RE=Ce . . . Yb [170]. Their results for A02 r 2 4f and A04 r 4 4f are plotted in Fig. 19 in comparison with experiment. This type of calculations on bulk materials led in general to satisfying agreement with experiment and clearly showed that the naive point charge model is completely inadequate for an estimate of the CEF parameters: point charges chosen according to the chemical valency of the elements are much too high when compared to ionic charges obtained from self-consistent calculations. In addition, it turned out that the parameters Am n are determined by about 80% by the charge distribution on the RE site while the point charge model assumes a lattice of ionic point charges surrounding the RE site. Corresponding work has also been performed in order to investigate the magnetic anisotropy at the surface or interface of RE-based compounds. Calculations of 0 0 -100 A4 < r >4f (K) A2 < r >4f (K) -10 -20 4 -300 0 0 2 -200 -400 -30 -40 -500 Ce Pr Nd Er Sm Gd Dy Yb Pm Eu Tb Ho Tm Ce Pr Nd Er Sm Gd Dy Yb Pm Eu Tb Ho Tm Fig. 19 Comparison of theoretical and experimental values for A02 r 2 4f (left) and A04 r 4 4f (right) parameters for the series of RECo5 compounds with RE = Ce . . . Yb. The full circles (full squares) are theoretical results for the experimental lattice parameters (for the lattice parameters fixed to those of GdCo5 ). Experimental values are shown as open squares and crosses. (All data taken from [174]) 230 H. Ebert et al. A02 r 2 have been done, for example, for the Nd sites of the (001)-surface of Nd2 Fe14 B [175]. It turned out that the sign of A02 r 2 depends on the positions of the Nd atoms in the unit cell supplying this way an explanation for the different coercivity of crystalline and sintered Nd2 Fe14 B. Similar calculations have been performed also to investigate the impact of Dy impurities on the coercivity of Nd2 Fe14 B [176]. From these, it was found that the parameter A02 r 2 for Dy atoms in the surface region of Nd2 Fe14 B also may have a positive or negative sign depending on its position, leading finally to a decrease of the coercivity of sintered samples. P. Novák et al. introduced a scheme for the calculation of the crystal field parameters that avoids the assumption of an inert 4f-charge cloud and allows for the hybridization of the 4f-states with the surrounding electronic states [177]. The approach is based on a local Hamiltonian represented in the basis of Wannier functions and expanded in a series of spherical tensor operators. Applications to RE impurities in yttrium aluminate showed that the calculated crystal field decreases continuously as the number of 4f-electrons increases and that the hybridization of 4f-states with the states of the oxygen ligands is important. This method has been successfully applied also to calculate crystal field parameters for RE impurities in LaF3 [178]. Dealing with ferrimagnetic materials composed of several, inequivalent magnetic sublattices, calculating the magnetic anisotropy may become more complicated as an additional canting between the sublattices introduced by an external field may play a significant role and should be taken into account to get a reasonable agreement with experiment. This was demonstrated for the RE-TM ferrimagnet GdCo5 [179], where the authors report a first-principles magnetizationversus-field (FPMVB) approach giving temperature-dependent magnetization as a function of an externally applied magnetic field in excellent agreement with experiment. Excitations Many dynamical as well as finite-temperature properties of the magnetization of a solid can be understood and described on the basis of magnetic excitations. In the low-energy, small-wave vector regime, one has to deal with the collective magnon excitations that can be investigated by various techniques. An approximate approach builds on the use of calculated exchange coupling parameters in combination with the so-called rigid spin approximation. More accurate results can be expected from self-consistent spin-spiral or frozen-magnon calculations that also allow exploring the magnetic phase space in an efficient way. Both approaches, however, do not give access to single-particle or Stoner excitations. On the other hand, using the concept of the dynamical susceptibility depending on frequency and wave vector, a coherent description of magnon and Stoner excitations is achieved. 4 Electronic Structure: Metals and Insulators 231 Magnon Dispersion Relations Based on the Rigid Spin Approximation When considering the magnetization dynamics of solids, one usually assumes the magnetization to be collinear inside an atomic cell i oriented along the common direction m̂i implying a coherent rotation of the magnetization within the cell during progress of time (rigid spin approximation (RSA)) [144]. As a consequence, the equation of motion for the magnetization can be replaced by the equation of motion for the local magnetic moments mi = mi m̂i that can be written as [144, 180, 181]: 2μB 1 ∂E d m̂i = − × m̂i , h̄ mi ∂ m̂i dt (73) where the right-hand side represents the torque acting on the magnetic moment mi . Making use of the harmonic approximation for the energy, Eq. (73) yields for the spin waves uλν (q) = uλν eiqR n with wave vector q the following eigenvalue problem [181]: h̄ ωλ (q) uλν = 2μB νν J (q) uλν , mν (74) ν for solids with translational symmetry. Here the eigenvectors uλν numbered by the index λ represent small deviations of magnetic moments from the direction of the ground state and the J νν (q) are the Fourier transforms of the interatomic exchange coupling parameters with ν labeling the basis atoms within a unit cell. Solution of the eigenvalue problem in Eq. (74) obviously yields the frequencies ωλ (q) of the various collective spin-wave eigenmodes that can be compared with magnon excitation energies as deduced, for example, from neutron scattering. Corresponding results obtained for Fe and Ni are given in Fig. 20 in comparison with experiment. Although good agreement between theory and experiment is achieved, one has to stress that the theoretical results depend on the method used to calculate the J νν (q) parameters. The data shown by a solid line were obtained using the exchange coupling parameters Jij calculated on the basis of the Lichtenstein formula Eq. (52) [182]. In the case of a lattice with one atom per unit cell, the magnon energy spectra E(q) possess only a single branch. Therefore, the bccFe and fcc-Ni magnon spectra in Fig. 20 could be obtained by a simple Fourier transformation [182] E(q) = 4μB J0j (1 − eiq·R j ). m (75) j The minima of E(q) for bcc-Fe along the Γ − H and H − N directions to be seen in Fig. 20 are so-called Kohn anomalies which occur due to long-range RKKY interactions. It turned out that these minima appear only if the summation in Eq. (75) 232 H. Ebert et al. 600 Expt. 1 Expt. 2 Halilov et al. Pajda et al. Fe 400 300 200 300 200 Expt Halilov et al. Pajda et al. 100 100 0 Γ Ni 400 E(q) (meV) E(q) (meV) 500 500 N Γ P H 0 L N Γ X W K Γ Fig. 20 Magnon dispersion relations for bcc-Fe (left) and fcc-Ni (right) along high-symmetry directions in the Brillouin zone, in comparison with experiment (open symbols [182]). Solid lines represent the results by Pajda et al. [182], while full circles show the results by Halilov et al. [180] 700 Co E(q) (meV) E(q) (meV) Expt Theory 25 500 400 300 20 15 200 10 100 5 0 Γ Gd 30 600 M K Γ A L 0 Γ M K Γ A Fig. 21 Left: magnon dispersion relation for hcp-Co along high-symmetry lines in the Brillouin zone [180]. Right: magnon dispersion relation for hcp-Gd (full lines) in comparison with experimental data. (All data taken from [183]) is performed over a sufficiently large number of atom shells around the central atomic site with index 0. As an example for a lattice with a multiatom basis, Fig. 21 (left) displays the magnon spectrum calculated for hcp-Co [180]. As there are two atomic sites in the unit cell, solving the eigenvalue problem Eq. (74) leads to two magnon branches. A similar approach was applied to hcp-Gd [183]. The corresponding experimental data shown in Fig. 21 (right) were obtained at T = 78 K. This was accounted for in the calculations within the RPA (see section “Methods Relying on the Rigid Spin Approximation”) leading to a simple rescaling of the magnon energies proportional to the temperature-dependent average magnetization. 4 Electronic Structure: Metals and Insulators 233 Spin Spiral Calculations Usually calculations of the electronic structure for magnetic systems are performed assuming a collinear spin-magnetic structure and using the smallest unit cell corresponding to the space group of the system. However, this configuration does not have to correspond to the ground state of the system. A possible way to search for the proper magnetic ground state is to consider incommensurate spinspiral configurations. Furthermore, within the adiabatic approximation, spin spirals can be seen as a representation of transverse spin fluctuations. Therefore, selfconsistent calculations on static spin spirals or so-called frozen magnons give access to the energies of spin-wave excitations that can be used in particular to investigate the finite-temperature magnetism. Considering a corresponding spin spiral characterized by the wave vector q and the tilt angles θν and φν , the variation of the spin-magnetic moment mnν from site to site may be expressed via: mnν = mν [cos(q · R n + φν ) sin θν , sin(q · R n + φν ) sin θν , cos θν ], (76) where ν labels the atomic site in the unit cell located at lattice vector R n . Because of broken translational and rotational symmetry, the presence of a spin spiral in principle implies an increased unit cell compared to a collinear spin configuration. However, as shown by Brinkman and Elliot [184, 185] as well as Herring [186], one can make use of the fact that a spin-spiral structure characterized by the wave vector q is invariant with respect to a so-called generalized translation: Tn = {α(φ)|αR |t n }, (77) if spin-orbit coupling is neglected. Here, the vector t n specifies a spatial translation combined with a spatial rotation αR and a spin rotation about the ẑ axis by the angle α(φ) = α(q · t n ). This property allows to formulate the generalized Bloch theorem [187]: Tn ψj k (r) = e−ik·t n ψj k (r), (78) that specifies the behavior under a generalized translation for the two-component eigenfunctions ψj k of a Hamiltonian with a noncollinear spin-dependent potential of the form (see also Eq. (11)): V (r) = nν q† Unν (θν , φν ) ↑ Vnν (r) 0 ↓ 0 Vnν (r) q Unν (θν , φν ). (79) Here n specifies the Bravais lattice vector R n , ν gives the position ρ ν of an atom q in the unit cell, and Unν is a spin-transformation matrix that connects the global frame of reference of the crystal to the local frame of the atom site at R n + ρ ν that has its magnetic moment mnν tilted away from the global z-direction. The 234 H. Ebert et al. x ϕ=qR m θ z y q Fig. 22 Geometry of a spin spiral with the wave vector q along the z-direction q transformation Unν is characterized by the Euler angles θnν and φnν as it is shown in Fig. 22. Assuming a collinear alignment of the spin density within the atomic cell at (n, ν), it is natural to use a local frame of reference with its z-axis oriented along q mnν . The corresponding transformation matrices Unν occurring in Eq. (79) can be q written as a product of two independent rotation matrices Unν = Un (θν , φν , q) = Uν (θν , φν ) UqR n , where the matrix UqR n depends only on the translation vector R n [187]: q Unν = cos θ2ν sin θ2ν − sin θ2ν cos θ2ν i 0 e 2 φν − 2i φν 0e i e 2 q·Rn 0 − 2i q·Rn 0e . (80) Results for the wave vector-dependent energy and spin-magnetic moments per atom obtained by Uhl et al. [188] from corresponding spin-spiral calculations for γ -Fe are shown in Fig. 23. This work was based on the LSDA and used the augmented spherical wave (ASW) band structure method in combination with the atomic sphere approximation (ASA). In addition, noncollinearity within an atomic cell was neglected. The investigations on γ -Fe by Kurz et al. [189], on the other hand, avoided these simplifications by the use of the LAPW band structure method. The noncollinear magnetic structure was imposed by a constraining magnetic field applied to the magnetic moments of the atoms. Furthermore, the GGA was used for the exchange-correlation potential. In spite of the various technical differences between the two studies, the results shown in Fig. 23 agree fairly well and justify the approach used by Uhl et al. as well as many others. The implementation of the spin-spiral method within the KKR band structure method allows dealing not only with ordered materials but also with random alloys [190]. Figure 24 gives as an example the energy (left) and individual spin moments (right) for a spin-spiral magnetic structure in Fe0.5 Mn0.5 as a function of the wave vector q directed along the [001] direction. One can see a transition from the antiparallel alignment of the Fe and Mn magnetic moments at small q to a parallel alignment when q approaches the boundary of the Brillouin zone at |q| = 2π/a. As it is seen from the left part of Fig. 24, the latter magnetic configuration is energetically more stable. Apart from exploring the magnetic phase space by performing self-consistent spin-spiral calculations, the technique can also be used to get access to magnon 4 Electronic Structure: Metals and Insulators 235 0 2 Kurz et al. Uhl et al. 1.5 -20 mspin (μB) E(q) (meV) -10 -30 1 -40 Kurz et al. Uhl et al. 0.5 -50 -60 Γ X 0 Γ W X W Fig. 23 The energies of the spin-spiral structure with respect to the energy of the FM state (left) and spin-magnetic moments per atom (right) calculated for γ -Fe for the wave vector q varying along Γ − X − W in the Brillouin zone. Open diamonds represent the results obtained by Uhl et al. [188] while full circles represent the results obtained by Kurz et al. [189]. (All data taken from [188] and [189]) 3 0.2 MFe MMn Mtotal 2.5 Mn and Fe parallel Mspin (μB) Espin spiral (eV) 2 0.1 Mn and Fe antiparallel 1.5 1 0.5 0 0 -0.5 0 0.2 0.4 qz 0.6 0.8 1 -1 0 0.2 0.4 qz 0.6 0.8 1 Fig. 24 Left: energy of a spin spiral in a Fe0.5 Mn0.5 alloy calculated for the wave vectors q = 2π a (0, 0, qz ) along the [001] direction. Right: local magnetic moments on Fe and Mn atoms as a function of the wave vector q. (All data taken from [190]) excitation energies h̄ω(q). In the case of simple lattices, the energy ΔE(q, θ ) of a spin spiral with wave vector q and tilt angle θ can be used directly to get h̄ω(q) from the expression [191]: 4 ΔE(q, θ ) . θ→0 m sin2 θ h̄ω(q) = lim (81) This approach was used, for example, by Halilov et al. [180] to calculate the magnon energy spectra for Fe and Ni represented in Fig. 20. Obviously, the results are in a reasonably good agreement with those of Pajda et al. [182] that are based on a 236 H. Ebert et al. calculation of the real space exchange coupling parameters Jij via the Lichtenstein formula Eq. (52). As one notes, the minima for Fe along the Γ − H and H − N directions are given by both approaches. However, the magnons obtained by the spin-spiral calculations are softer, because of the self-consistent relaxation within the electronic structure calculations. As another example, Fig. 25 represents spin-wave dispersion curves obtained for the full-Heusler alloys Cu2 MnAl, Pd2 MnSn, Ni2 MnSn, and the L12 -type ferromagnet MnPt3 [192]. The simplified approach used in this work did not fully account for the magnetic sublattices of the investigated systems providing for that reason only the first magnon branch. Nevertheless, this already led to values for the Curie temperature calculated within the RPA approach (see section “Methods Relying on the Rigid Spin Approximation”) in reasonable agreement with experiment. As a general trend, one can see in Fig. 25 that the calculated magnon energies h̄ω(q) are too high when compared with experiment. The authors attribute this to the treatment of electronic correlations on the basis of the GGA. In fact, previous work on the series of Heusler alloys Co2 Mn1−x Fex Si [62] clearly demonstrated the impact of correlation effects by comparison of results based on the LSDA, LSDA+U, and LSDA+DMFT. 200 Expt (4.2 K) Theory Cu2MnAl 60 Pd2MnSn Expt (50 K) Theory E(q) (meV) E(q) (meV) 150 100 20 50 0 Γ 80 [100] [110] Γ X Ni2MnSn [111] L 0 Γ 200 Theory Expt (50 K) [100] [110] [111] Γ X MnPt3 L Expt (80 K) Theory 150 E(q) (meV) E(q) (meV) 60 40 20 0 Γ 40 100 50 [100] [111] [110] X Γ L 0 Γ [100] [110] X M [111] Γ R Fig. 25 Calculated (solid lines) spin-wave dispersion curves h̄ω(q) in the first Brillouin zone along high-symmetry directions for the L21 -type full-Heusler and L12 -type ferromagnets. As indicated, the experimental data stem from neutron diffraction measurements at various temperatures. (All data taken from [192]) 4 Electronic Structure: Metals and Insulators 237 The use of the spin-spiral technique for the calculation of the full magnon energy spectrum in case of complex compounds was demonstrated by Şaşıoğlu et al. [193]. In this case, a set of spin-spiral calculations is required to obtain all exchange coupling parameters J νν (q) that enter the eigenvalue problem Eq. (74). Excitation Spectra Based on the Dynamical Susceptibility Despite the many successful applications of the adiabatic approach for the investigation of spin-wave excitations, one has to stress that it has severe limitations. In particular it can be applied only to systems for which single-particle or the so-called Stoner excitations can be neglected [194, 195]. Figure 26 gives a simplified picture of the exchange-split band structure of an itinerant ferromagnet in the vicinity of its Fermi level. Excitation of an electron from an occupied majority-spin state below the Fermi level to an empty minority state may not only be associated with a spin flip but also with a change of the electronic wave vector. As it is visualized in the right panel of Fig. 26, these Stoner excitations lead to a broad continuum that in general overlaps and hybridizes with the discrete magnon dispersion spectrum. For this and other reasons, a more sophisticated description of spin-wave excitations was worked out making use of the linear response formalism within the framework of time-dependent density functional theory [196,197,87,198]. This allows expressing the magnetization Δmi (r, q, ω) induced by a magnetic field B(r, q, ω), with its time and spatial dependency expressed by the wave vector q and frequency ω, respectively, in terms of a corresponding susceptibility tensor [2]: Δmi (r, q, ω) = j (82) Ω Minority-spin δ ΔE = ε U Majority-spin Electron wave vector, k EF Excitation energy, ε Electron energy, E Δk = q d 3 r χ ij (r, r , q, ω) B j (r , q, ω). Stoner continuum U Magnons 0 δ Excitation wave vector, q Fig. 26 Left: schematic representation of Stoner excitations in an itinerant ferromagnet. A majority electron is excited from an occupied state below the Fermi level to an unoccupied minority state above the Fermi level. Right: continuum of Stoner excitations for a metallic ferromagnet 238 H. Ebert et al. Within spin-density functional theory and making use of circular coordinates, one may write in particular for the transverse susceptibility tensor element χ ± a Dysonlike equation [87]: χ ± (r, r , q, ω) = χ0± (r, r , q, ω) d 3 r χ0± (r, r , q, ω) Kxc (r ) χ ± (r , r , q, ω), (83) + Ω xc (r) where χ0± is the unenhanced susceptibility, Kxc (r) = Bm(r) is the exchangecorrelation kernel function, and Bxc (r) and m(r) are the local exchange-correlation field and magnetization, respectively. As discussed, for example, by Bruno [199] as well as by Katsnelson and Lichtenstein [200], it is the second term in Eq. (83) that gives rise to the enhancement of the transverse susceptibility. The dynamical susceptibility gives not only access to the energetics of magnetic excitations but also to their lifetime characterizing this way the dissipation of the energy. Outside the Stoner continuum, the loss tensor associated with χ + shows peaks at frequencies corresponding to the excitation of spin waves. Inside the Stoner continuum, these show a finite width due to the hybridization with the Stoner excitations. In this case, one has approximately: χλ+ (q, ω) ≈ Aλ (q) , (ω − ω0λ (q))2 + βλ (q)2 (84) with the amplitude Aλ (q), the spin-wave energy ω0λ (q), and inverse lifetime βλ (q). As an example, Fig. 27 shows the magnon dispersion curves ω0 (q) together with the corresponding broadening for the magnon states as deduced from the dynamical susceptibility as calculated for bcc-Fe and fcc-Co [203]. As one can clearly see from the given width, Stoner excitations have only a very small influence on the long-period magnons. Results for ω0λ (q) and βλ (q) for the Heusler alloy Co2 MnSi that has three magnetic sublattices are shown in Fig. 28. The corresponding eigenvectors (EV) Fig. 27 Spin waves of bcc-Fe (left) and fcc-Co (right). Solid circles correspond to ω0 (q), while the error bars denote full width at half maximum of the peak. Solid line denote spin-wave energies obtained using the magnetic force theorem [203] 4 Electronic Structure: Metals and Insulators 600 EV 1 EV 2 EV 3 400 200 0 Γ EV 2 EV 3 150 β(q) (meV) ω(q) (meV) 800 239 100 50 [ξ00] X K [ξξ0] Γ [ξξξ] L Γ [ξ00] X K [ξξ0] Γ [ξξξ] L Fig. 28 Energies ω0λ (left) and inverse lifetimes βλ (middle) of three spin-wave modes in Co2 MnSi together with the corresponding eigenvectors (right); arrows indicate the orientations of the magnetic moments. The basis atoms are Co at (1/4, 1/4, 1/4)a and (3/4, 3/4, 3/4)a and Mn at (1/2, 1/2, 1/2)a. The parameter β1 of EV 1 does not exceed 5 meV and is not shown. (All data taken from [87]) of the resulting spin-wave modes are given on the right-hand side of the figure. The acoustic mode that is lowest in energy has a vanishing energy for q = 0, and its value for βλ is very small (therefore not shown in Fig. 28). The optical modes, on the other hand, appear at higher energies where the continuum density is appreciable. Accordingly, their inverse lifetime βλ is quite large and depends strongly and non-monotonously on the wave vector q. For the Heusler alloy Cu2 MnAl, only one magnetic sublattice has to be considered, and accordingly, there is only one acoustic spin-wave mode. In contrast to Co2 MnSi, a more pronounced damping is found in this case. The influence of the Stoner excitations can also be seen for the spin-wave energies. Calculating these by use of the so-called adiabatic approach [202] that neglects the hybridization, the spin-wave energies are higher and in less good agreement with experiment [202]. Comparable studies based on the dynamical susceptibility were done (i) to investigate the Landau damping in Fe(100) and Fe(110) films and the effect of the substrate on this [201], (ii) to study the Landau damping of spin waves and large Rh moments induced by the AFM magnons in FeRh [203], and (iii) on acoustic magnons in the long-wavelength limit in order to analyze the Goldstone violation in many-body perturbation theory [204]. The concept of the dynamic spin susceptibility has been applied also to paramagnetic systems at finite temperatures by Staunton et al. [205, 206]. Due to the use of the multiple scattering formalism, investigations on alloys could be made, for example, on paramagnetic Cr0.95 V0.05 and antiferromagnetic Cr0.95 Re0.05 above the Néel temperature TN . While the work sketched here was primarily based on the linear response formalism applied within the framework of time-dependent density functional theory, similar work on quasiparticle and collective electronic excitations in solids was done using techniques from many-body perturbation theory [207]. 240 H. Ebert et al. Finite-Temperature Magnetism Dealing with the impact of finite temperatures in a quantitative way is a big challenge for theory. Accordingly, many different techniques on various levels of sophistication are in use for that purpose. Most of these employ the adiabatic approximation that decouples the electronic and magnetic degrees of freedom. One type of such approaches, that proved to be astonishingly successful for many situations, starts from the properties of low-energy magnetic excitations by calculating real-space exchange coupling parameters or the energies of spin spirals. In a second step, this information is used in combination with classical statistical methods including in particular the Monte Carlo method to deduce temperaturedependent magnetic properties. More advanced schemes, however, are based on a coherent description of the electronic structure and statistics. While the disordered local moment (DLM) method still relies on the adiabatic approximation, this does not apply, for example, to the functional-integral method or various many-body approaches used within the dynamical mean field theory (DMFT) that account for finite temperature in a coherent way. Methods Relying on the Rigid Spin Approximation Within standard Stoner theory, a spin-dependent but collinear electronic structure is assumed, and finite temperatures are accounted for only via the Fermi distribution function. Accordingly, the resulting critical temperatures are much too high. More successful approaches to deal with magnetism at finite temperatures, on the other hand, allow for transverse spin excitations. A simple model accounting for this was suggested already by P. Weiss who considered a magnet as a system of localized magnetic moments that order spontaneously due to an effective molecular or Weiss field hW (T ) = w m(T ) n̂ that depends on the average magnetic moment m(T ) on an atomic site. The factor w is the molecular or Weiss field constant: w= 3kB TC , m20 (85) which is determined by the magnetic moment m0 = m(0) at T = 0 K and the critical temperature TC . Quite general, the temperature-dependent magnetic moment m(T ) along n̂ is determined by the statistical average over all possible orientations ê with the probability distribution for the local magnetic moments given by: P n̂ (ê) = e−β hW n̂·ê d ê e−β hW n̂·ê , (86) where hW = w m(T ) and β = 1/(kB T ). The various techniques discussed in section “Exchange Coupling Parameters” allow to deduce the Weiss or mean field 4 Electronic Structure: Metals and Insulators 241 constant from the calculated exchange coupling parameters Jij . Within this mean field approach (MFA), application of classical spin statistics leads to: TCMFA = 2 2 J0j = J0 , 3kB 3kB (87) j =0 MFA in case of an elemental ferromagnet, with J = for 0 the Curie temperature TC j =0 J0j [126]. A more accurate approach to deal with finite-temperature magnetism is provided by the random-phase approximation Green function (RPA-GF) method which also can be based on a combination of the Heisenberg model and SDFT calculations (see section “Exchange Coupling Parameters”). The decoupling scheme suggested by Tyablikov leads to an approximate expression for the one-particle Green magnon function [208, 209]: Gm (z) = 1 1 , N q z − E(q) (88) with E(q) the magnon energies that allows expressing the critical, i.e., Curie or Néel, respectively, temperature TcRPA as [182]: 6 1 = − lim RPA z→0 m k B Tc Gm (z), (89) in case of a ferro- or antiferromagnet, respectively. The MFA approach accounts for all spin-wave excitations with the same weight leading in general to an overestimation of the Curie temperature. The RPA-GF approach, on the other hand, accounts in particular for the low-energy excitations in a much more adequate manner. As a consequence, more accurate results for the critical temperature are normally obtained this way when compared with experiment. This behavior has been found within numerical work on the elemental ferromagnets Fe, Co, and Ni [182] but also for other systems, as, for example, the Heusler alloys (Ni,Cu)2 MnSn, (Ni,Pd)2 MnSn [210], NiMnSb, CoMnSb, Co2 MnSi, and Co2 CrAl [193], for Gdbased intermetallic compounds GdX (X = Mg, Rh, Ni, Pd) [211] as well as for zincblende half-metallic ferromagnets GaX X = N, P, As, Sb [212]. The exchange coupling parameters in these works have been obtained either from spin-spiral calculations [193,212] or in real space, on the basis of the force theorem [182,211]. In all cases, reasonable agreement with experiment could be achieved, except for fcc-Ni, a system for which the application of a Heisenberg Hamiltonian with fixed spin moments seems questionable. Corresponding investigations on the spin-wave spectra and Curie temperatures have also been performed for L21 -type full-Heusler FM alloys and L12 -type XPt3 alloys [192]. It was found that the Curie temperatures are in good agreement with experiment when the Stoner gap was large enough so that the magnon regime is well separated from the Stoner excitations. In this case, single-particle spin-flip 242 H. Ebert et al. Stoner excitations make only a small contribution to the excitation spectra at low energies, so that magnon excitations make the dominating contribution to thermodynamics. It is interesting to note that the RPA-GF formalism fulfills the Mermin-Wagner theorem [213], i.e., in the absence of magnetic anisotropy it leads for two-dimensional systems to TC = 0 K [214]. Corresponding studies by Pajda et al. on the Curie temperature in Fe and Co films that included a finite magnetic anisotropy led to an oscillating behavior as a function of film thickness [127]. These oscillations could be ascribed to the oscillating behavior of the exchange coupling parameters due to thickness-dependent quantum well states. An extension of the RPA-GF method (RPA-CPA) to deal within the CPA also with disordered alloys was worked out by Bouzerar and Bruno [215]. Starting from the Stratonovich-Hubbard functional integral method [83], Kübler [191] derived an expression for the critical, i.e., Curie or Néel, respectively, temperature: ⎡ ⎤−1 2 2⎣1 1 ⎦ k B Tc = L (90) 3 ν ν N q,n jn (q) that is applicable to systems with several atoms per unit cell. Here Lν is the socalled local moment of atom ν that should be found in a self-consistent way [191]. However, in the case of well-localized magnetic moments, values of mν at T = 0 K can be used as a reasonable approximation [216]. Finally, jn (q) in Eq. (90) is the exchange function after diagonalization of jνν (q) [191]. This expression accounts in particular for the fact that different atomic types have in general different moments, while the expressions in Eqs. (88) and (89) that are often applied to multicomponent systems are based on the assumption that Lν = m0 with m0 , the average saturation moment. As is demonstrated by the results in Table 2, Eq. (90) gives in general results for the critical temperature that are in good agreement with experiment. An important alternative to the RPA-GF scheme for dealing with magnetic properties at finite temperatures is provided by the Monte Carlo method that is used to deal with the statistical aspect of the problem. Corresponding work again is in general based on the Heisenberg Hamiltonian (see Eq. (51)) with the exchange parameters calculated in an ab initio way. Numerous successful applications have been done, both for ordered compounds [218] and for disordered alloys such as diluted magnetic semiconductors like Ga1−x Mnx As [135, 219, 220] or Heusler alloys [129, 221]. In most cases, good agreement with RPA-based results as well as with corresponding experimental data for the critical temperature could be achieved. Methods Accounting for Longitudinal Spin Fluctuations Despite the good results often obtained via the RPA and MC methods, one has to stress that they are based on a classical spin Hamiltonian and therefore are 4 Electronic Structure: Metals and Insulators 243 Table 2 Structure, magnetic moments on M1 and M2 sublattices, as well as critical Curie or Néel temperature calculated via the MC (TcMC ), RPA (TcRPA ), or MFA (TcMFA ) approaches in comparison with experiment. (All data taken from [191] and [217]) System Fe Co Ni FeNi CoNi FeNi3 CoNi3 NiMnSb Mn2 VAl Co2 FeSi Mn3 Al Mn3 Ga Mn3 Ga RhMn3 Structure bcc fcc fcc CuAu CuAu AuCu3 AuCu3 C1b L21 L21 L21 L21 DO22 AuCu3 M1 (μB ) 2.330 1.410 0.630 2.551 1.643 2.822 1.640 3.697 −0.769 2.698 −2.258 −2.744 −2.829 3.066 M2 (μB ) – – – 0.600 0.673 0.588 0.629 0.303 1.374 1.149 1.128 1.363 2.273 – TcMC/RPA (K) 1060MC 1080MC 510MC 972RPA 1149RPA 986RPA 733RPA 968RPA 580RPA 1058RPA 196RPA 314RPA 762RPA 1059RPA TcMFA (K) 1460 1770 660 1130 1538 1290 925 1281 663 1267 342 482 1176 – Exp Tc (K) 1043 1388 633 790 1140 870 920 730 760 1100 – – – 855 suitable only for systems with their magnetic moments depending only weakly on the temperature. As the latter assumption is not always fulfilled, an extension of the Heisenberg Hamiltonian was suggested that is meant to account for longitudinal fluctuations [222, 217] that express the total energy by an expansion in even powers of the magnetic moments per atom m: E(M, q, θ ) = n An m2n + Jn (q, θ ) m2n . (91) n Here the functions Jn (q, θ ) are proportional to the energy difference between the ferromagnetic and the spin-spiral states specified by wave vector q and tilt angle θ . Monte Carlo simulations for bcc-Fe, fcc-Co and fcc-Ni performed on this basis [217] led to a rather good agreement with experiment as one can see from Table 2. Another model to account for temperature-induced longitudinal spin fluctuations was suggested by Ruban et al. [223] that also led for Fe and Ni to a rather realistic description of the magnetic properties at finite temperatures. An important class of materials, for which longitudinal spin fluctuations are of great importance, are alloys and compounds composed of magnetic and otherwise nonmagnetic elements. Such systems exhibit so-called covalent magnetism [224, 225], i.e., the magnetization of the nonmagnetic component is caused by the spontaneously magnetized atoms via a spin-dependent hybridization of the electronic states. Ležaić et al. [129, 221], for example, emphasized the need to account for longitudinal fluctuations of the magnetic moment induced on the Ni atoms for a proper description of the temperature dependence of the spin polarization at the Fermi energy EF when performing Monte Carlo investigations 244 H. Ebert et al. on the half-metallic Heusler alloy NiMnSb. For that purpose, they used an extended Heisenberg Hamiltonian: Hext = 1 Jij mi · mi + (a m2i + b m4i ), 2 ij (92) i∈N i that allows accounting for transverse and longitudinal magnetic fluctuations connected with the temperature-dependent induced magnetic moment on the Ni atom. The Heusler alloy NiMnSb was also investigated by Sandratskii et al. [212] using the spin-spiral approach and treating the magnetic moment of Ni atom as being induced. Mryasov et al. [226] found that the induced magnetic moment of Pt plays a crucial role for the magnetic anisotropy of FePt at finite temperatures. To account for this, a renomalization of the Fe-Fe exchange interactions according to: J˜ij = Jij + I Pt χPt m0Pt 2 Jiν Jνj (93) ν∈Pt was suggested. Here I Pt characterizes the local exchange interaction of the Pt atoms, m0Pt is the Pt magnetic moment in the ordered ferromagnetic state, and χPt is the partial spin susceptibility of Pt. The impact of a renormalization of the Fe-Fe interactions due to the induced Rh magnetic moment has also been demonstrated for the stabilization of the ferromagnetic state and for the control of the antiferromagnet-ferromagnet phase transition of FeRh [227]. Another scheme to account within Monte Carlo simulations for the impact of the induced magnetic moments on nonmagnetic alloy components leading to a renomalization of the exchange interactions between the magnetic components was worked out by Polesya et al. [128]. Figure 29 (left) shows corresponding results for the Curie temperatures of the ferromagnetic alloy Fex Pd1−x that are in very good agreement even for the Pd-rich side of the alloy system. Corresponding calculations have been done for FeRh to investigate its antiferromagnet-ferromagnet phase transition [229]. The right panel of Fig. 29 shows results of Kudrnovský et al. [228] for fcc-Ni1−x Pdx . Investigating the finite-temperature magnetism of various Ni-based transition metal alloys, these authors concluded that Bruno’s formulation [199] for the renormalized RPA gives the most satisfying agreement with experiment. Coherent Treatment of Electronic Structure and Spin Statistics The methods sketched above that are based on the Heisenberg model obviously have problems to account for longitudinal spin fluctuations. In addition, they consist in an incoherent combination of electronic structure calculations and classical 4 Electronic Structure: Metals and Insulators T (K) 400 Theory Expt. 1 Expt. 2 Expt. 3 Expt. 4 RPA MFA rRPA Expt. 600 T (K) 500 245 300 400 200 200 100 0 0 0.05 0.1 xFe 0.15 0.2 0 0 0.2 0.4 0.6 0.8 1 xNi Fig. 29 Calculated Curie temperature TC of disordered alloys for fcc-Fex Pd1−x (left) using the Monte Carlo method and fcc-Nix Pd1−x (right) using the MFA, RPA, and renormalized RPA (rRPA) approaches, in comparison with experiment. (All data taken from [128] and [228]) statistics. These problems can be avoided by the use of the disordered local moment (DLM) method [230] that deals with the temperature-dependent magnetization within self-consistent electronic structure calculations. The slow dynamics of the magnetic moments spatially localized on the atoms when compared to the fast electron propagation and relaxation time scale allows to make use of the adiabatic approximation. Assuming ergodicity for the system of local magnetic moments, the time average required to calculate the average magnetization at a given temperature can be replaced by the average over the ensemble of all orientational configurations characterized by a set of unit vectors {êi }. Within the DLM theory, this is determined by the corresponding single-site probabilities P n̂ (êi ) obtained from Eq. (86) for the Weiss field hW = hn̂ . Following Györffy et al. [230], hn̂ is determined by approximating the free energy F corresponding to the microscopic Hamiltonian H of the considered system by the free energy F0 based on the trial Hamiltonian H0n̂ = j hn̂ · êj with hn̂ = hn̂ n̂ and using the Feynman-Peierls inequality [231]: F ≤ F0 + H − H 0 , (94) where the canonical distribution Eq. (86) is used to calculate the average. Using the Weiss field hn̂ as a variational parameter to minimize the right-hand side of Eq. (94) one is led to [162]: 3 h = 4π n̂ d êi êi Hn̂ êi , (95) where Hn̂ êi denotes the average of Hn̂ with the restriction that the orientation of the moment on site i is fixed to êi [230]. Self-consistent electronic structure calculations for a given temperature result in a temperature-dependent magnetic moment that automatically accounts for longitudinal fluctuations. The resulting 246 H. Ebert et al. magnetization M(T ) = M(T )n̂ can be obtained from M(T ) = mloc (T ) d êi P n̂ (êi ) n̂ · êi , (96) with the local moment mloc to be determined self-consistently. Based on a relativistic formulation of the DLM method (RDLM), Staunton et al. [162] worked out a scheme to investigate the magnetocrystalline anisotropy at finite temperature. This implies in particular a corresponding extension of Eq. (68) that allows calculating the magnetic torque together with the Weiss field. Results for the temperature-dependent magnetization (M(T )) and uniaxial magnetocrystalline anisotropy (K(T )) that have been obtained by an application of this approach to L10 -ordered FePt are shown in Fig. 30 [232]. In line with experiment, the orientation of the easy axis was found to be perpendicular to the Fe- and Pt-layers for all temperatures. Also the temperature dependence of the anisotropy constant K(T ) is in good agreement with experiment. The single-ion anisotropy model, on the other hand, fails to give a correct qualitative description for K(T ). Similar RDLM-based investigations have been performed also for the L10 -ordered FePd [162]. As in the case of FePt, the easy axis in FePd is perpendicular to the Fe- and Pd-layers, with the uniaxial magnetocrystalline anisotropy also showing the scaling behavior K(T ) ∼ [M(T )/M(0)]2 . Buruzs et al. [233] used the RDLM method to investigate the temperaturedependent magnetic properties of Co films deposited on Cu(100) (Con /Cu(100)) with the thickness n of the Co film varying from 1 to 6. The resulting Curie temperatures are given in the Table 3. In agreement with experiment, it was found that the magnetization is oriented parallel to the surface for almost all temperatures below the Curie temperature, except for the system with n = 2. Based on the calculation of the anisotropy constant, a temperature-induced reversal ΔESOC (meV) M(T) 0.8 -0.5 0.6 0.4 -1.5 0.2 0 -1 -2 200 400 600 800 Temperature T (K) 0.2 0.4 0.6 0.8 2 (M(T)) Fig. 30 RDLM calculations on FePt. Left: the magnetization M(T ) versus T for the magnetization along the easy [001] axis (filled squares). The full line shows the mean field approximation to a classical Heisenberg model for comparison. Right: the magnetic anisotropy energy ΔESOC as a function of the square of the magnetization M(T ). The filled circles show the RDLM-based results, the full line give K(T ) ∼ [M(T )/M(0)]2 , and the dashed line is based on the single-ion model function. (All data taken from [232]) 4 Electronic Structure: Metals and Insulators Table 3 Calculated Curie temperatures (K) for Con /Cu(100) [233] 247 n TC 1 1330 2 933 3 897 4 960 5 945 6 960 of the anisotropy direction from in-plane to out-of-plane was predicted. A more detailed investigation led to the conclusion that the spin reorientation is driven by a competition of exchange and single-site anisotropies [18]. Zhuravlev et al. [234] used the RDLM method to investigate the origin of the anomalous temperature dependence of the magnetocrystalline anisotropy in (Fe1−x Cox )2 B alloys. In contrast to a conventional monotonous variation of the MCA energy with increasing temperature, the anisotropy in (Fe1−x Cox )2 B shows a non-monotonous temperature dependence due to increasing magnetic disorder. This behavior of the experimental data was quantitatively reproduced by the RDLMbased calculations. It turned out that the observed temperature dependence of the anisotropy is caused by a modification of the electronic structure induced by spin fluctuations which can result in a selective suppression of spin-orbit-induced hot spots (see section “Relativistic Effects”). In contrast to the expectations based on other models, this peculiar electronic mechanism may lead to an increase, rather than decrease, of the anisotropy with decreasing magnetization. Another approach to account for temperature-induced charge and spin fluctuations when dealing with the magnetic properties of itinerant-electron magnets at finite temperature is based on the functional integral method [4]. Within this approach, the corresponding auxiliary exchange field is introduced using the Hubbard-Stratonovich transformation [235]. This allows to describe the spin fluctuation contribution to the free energy in a rather simple way. Based on the functional-integral method, Kakehashi has proposed to use the dynamical coherentpotential approximation (CPA) theory to go beyond the adiabatic theories in metallic magnetism [236], which was first formulated for a model Hamiltonian. In order to apply this approach to realistic systems, the dynamical CPA theory has been extended by a combination with the LMTO band structure method (see section “Band Structure Methods”) [237]. Using this approach, the hightemperature susceptibility is expected to follow the Curie-Weiss law: χCW (T ) = m2eff . 3(T − TC ) (97) Using the atomic exchange coupling parameter J , the values J = 0.9, 0.94, and 0.9 eV for Fe, Co, and Ni and the Curie temperatures TC = 1930, 2250, and 620 K, respectively, are found. The corresponding effective magnetic moments meff = 3.0, expt 3.0, and 1.93 μB are in reasonable agreement with experiment (meff = 3.2 , 3.15, and 1.6 μB , respectively) The importance of correlation effects for finite-temperature magnetism has been investigated within the framework of the dynamical mean field theory (DMFT) (see section “Spin Density Functional Theory”) [16,32]. As demonstrated by Kakehashi 248 H. Ebert et al. 1 2 χ meff 3TC M(T)/M(0) Fe 0 0 0.5 -1 Ni 1 T/TC 1.5 2 0 1000 TCW (K) 1 500 0.2 CPA+DMFT Peschard 1925 Chechernikov 1962 0.4 0.6 xNi 0.8 1 Fig. 31 Left: temperature dependence of the magnetization M(T )/M(0) and the inverse ferromagnetic susceptibility for Fe (open squares) and Ni (open circles) compared with experimental results for Fe (squares) and Ni (circles). Right: CPA+DMFT-based and experimental values for the Curie-Weiss temperature of Fe1−x Nix alloys as a function of Ni concentration. (All data taken from [16, 32] (left) and [239] (right)) [238], concerning the treatment of finite temperatures within electronic structure calculations, this approach is essentially equivalent to the dynamical CPA theory mentioned above. Figure 31 (left) shows results for the calculated temperature dependence of the magnetic moment and the inverse ferromagnetic susceptibility of Fe and Ni. The effective magnetic moments are found to be meff = 3.09 μB for Fe and meff = 1.5 μB for Ni. The corresponding estimated Curie temperatures are 1900 and 700 K for Fe and Ni, respectively. A combination of the DMFT with the CPA alloy theory that treats substitutional disorder and electronic correlations on equal footing has been used by Poteryaev et al. [239] to investigate the magnetic properties of Fe1−x Nix alloys. The calculated Curie temperatures shown in Fig. 31 (right) are obviously in good agreement with experiment. Also in line with experiment, a linear variation with temperature has been found for the calculated inverse magnetic susceptibilities at high temperatures. Recently, Patrick and Staunton have put forward a computational scheme for the description of finite-temperature magnetic properties of RE-TM compounds [240]. 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Kakehashi, Y.: Many-body coherent potential approximation, dynamical coherent potential approximation, and dynamical mean-field theory. Phys. Rev. B 66, 104428 (2002) 239. Poteryaev, A.I., Skorikov, N.A., Anisimov, V.I., Korotin, M.A.: Magnetic properties of Fe1−x Nix alloy from CPA+DMFT perspectives. Phys. Rev. B 93, 205135 (2016) 4 Electronic Structure: Metals and Insulators 259 240. Patrick, C.E., Staunton, J.B.: Rare-earth/transition-metal magnets at finite temperature: selfinteraction-corrected relativistic density functional theory in the disordered local moment picture Phys. Rev. B 97, 224415 (2018) Hubert Ebert studied physics and received his Ph.D. from the Ludwig-Maximilians-University Munich in 1986. After a postdoc stay at the University of Bristol (UK), he worked for several years at the central laboratory for research and development of Siemens Company in Erlangen. Since 1993 he is professor for theoretical physical chemistry at the university of Munich. Sergiy Mankovsky studied physics in Moscow institute of Physics and Technology. In 1992 he received the degree of candidate of physico-mathematical sciences (an analogue of the Ph.D.) from the Kurdyumov Institute for Metal Physics of the N.A.S. of Ukraine, where he worked as a research scientist until 2001. Since 2001 he works at the Ludwig-Maximilians-University Munich. Sebastian Wimmer received his Ph.D. from the LudwigMaximilians-Universität München in 2018. Until 2019 he worked in the group of Prof. Dr. Hubert Ebert, focusing on the firstprinciples description of spintronic and spincaloritronic linear response properties of metals and alloys. 5 Quantum Magnetism Gabriel Aeppli and Philip Stamp Contents Spin Paths and Spin Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Importance of Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quantum Relaxation in Dipolar Nets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large-Scale Coherence and Entanglement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Future Directions and Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 265 268 271 276 277 Abstract Macroscopic quantum effects have been familiar since the discovery of superfluids and superconductors over 100 years ago. In the last few decades, it has been understood how large-scale quantum effects can also show up in “spin space.” The collective tunneling of many spins was observed in magnetic nanomolecules and in insulating dipolar-coupled spin arrays, and the tunneling of ferromagnetic domain walls has also been cleanly identified. To see largescale coherence or entanglement effects, the decoherence caused by interactions with the environment (particularly with nuclear spins) must be controlled. Theory indicates ways of doing this, and systems ranging from classic magnetic compounds to deterministically doped silicon will make the job easier. Coherent G. Aeppli () Physics Department (ETHZ), Institut de Physique (EPFL) and Photon Science Division (PSI), ETHZ, EPFL and PSI, Zürich, Lausanne and Villigen, Switzerland e-mail: [email protected] P. Stamp () Pacific Institute of Theoretical Physics, University of British Columbia, Vancouver, BC, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_5 261 262 G. Aeppli and P. Stamp control of quantum spin arrays, and large-scale quantum spin superpositions, is a likely prospect for the future. Magnetism has its microscopic origin in quantum mechanics which determines basic parameters such as single-site anisotropy and intersite couplings. However, once the parameters have been fixed, classical reasoning is usually all that is required to model and engineer real materials and devices. In this sense, magnetism is not different than any other branch of condensed matter physics: quantum mechanics is essential to describe very small objects such as individual nuclei, atoms, and molecules, but the emergent behavior of macroscopic ensembles of such objects can almost always be explained in classical terms. There are a few striking exceptions, most notably for fractional quantum Hall systems and Bose condensates. These depend for their existence on “macroscopic wave functions” [1], which correlate simultaneously the dynamics of all the particles into a singlewave function Ψ ∼ |Ψ (r, t)|eiϕ(r,t) , with a phase ϕ(r, t) varying in space and time. So, can any other kinds of large-scale quantum behavior be expected in physics, not entailing a Laughlin state, Bose condensation, or supercurrents? We report here on some of the remarkable ways this can happen, not in real space but in spin space. These developments are relatively recent and constitute a new class of quantum phenomena – and in the last three decades or so, experiments have begun to confront theory in a rigorous way. Spin Paths and Spin Phase There is a remarkable way of formulating quantum mechanics, found by Feynman during his doctoral studies [2], which allows a neat appreciation of these developments. One writes the usual amplitude Gba , for the transition between two quantum states |ψa and |ψb , as a sum of “amplitudes” over all possible paths between them – the system simultaneously explores all these paths. The probability that the transition will then occur is the usual “amplitude squared,” i.e., |Gba |2 . The amplitude or weighting factor for the μ-th path is ei Aμ /h̄ , exponentiating a phase ϕ = Aμ /h̄, where Aμ is the “action” of this path. Crucially, the action is just that for the corresponding classical system (e.g., a particle) moving along the same path, i.e., Aμ = dtL, where L is the classical Lagrangian. This sum over paths, or “path integral,” brings out vividly the essential role of phase interference between different paths. It has been enormously fruitful, both as a pedagogical guide [3] and in advancing our understanding of basic quantum physics and quantum field theory [4]. An obvious question, which notoriously worried Feynman, is how to then deal with quantum spins, which have no classical limit (if the spin quantum number S is finite, then when h̄ → 0, the spin moment h̄S disappears!). The answer turned out to be interesting [5, 6, 7]. Imagine a unit sphere in spin space, with a “particle” of unit charge moving on its surface – its coordinate n̂(t) = S(t)/S defines the spin direction (see Fig. 1). The fake charge 5 Quantum Magnetism 263 accumulates a conventional “potential phase” by coupling to a potential Ho (S) (the spin Hamiltonian) on the sphere. Crucially, however, the charge also couples to a fake monopole of strength q = h̄S, situated at the center of the sphere. This adds a “kinetic” or Berry phase [6], having the same form as the Aharonov-Bohm phase [8] for a charge moving in real space (see Fig. 1a). One then sums over paths on the spin sphere, to reproduce exactly the dynamics of a quantum spin. Tunneling spins: As an example, suppose Ho (S) has local potential minima – at low energy, the spin must then quantum tunnel between these minima to move at all. For simplicity, we imagine two paths, connecting the lowest states | ↑, | ↓ in these energy minima (Fig. 1b). This is just like the famous “two-slit” problem – the tunneling amplitude o will be controlled by the interference due to the different phases ϕ1 , ϕ2 accumulated along the two paths [3]. Two points are interesting here. First, as discussed by Bogachek and Krive [10], and later others [11, 12], one can manipulate the two paths by applying a transverse magnetic field H⊥ o , thereby giving oscillations in the tunneling amplitude (Fig. 1c). Second, by considering only the lowest state in each potential well, we have in effect truncated the Hamiltonian Ho (S) to a two-level system (justified if the temperature is low enough that higher spin states are inactive). We can write Ho (S) → Ho (τ̂ ) = o τ̂x , where τ̂ is the Pauli vector. The eigenstates are just the symmetric and antisymmetric combinations of | ↑ and | ↓. If the paths are oppositely directed but otherwise symmetric, then ϕ1 = −ϕ2 = π S, and the transition amplitude (and hence the tunneling splitting) between the two eigenstates is then ∼ (eiϕ +e−iϕ ) ∝ cos(π S), giving a gap between the lowest and first excited states if S is integer-valued, but no gap if S is half-integer. This last result was of course already noted by Kramers [13] in 1930; spin paths are not required to understand it! Interest in spin phases was really set off by Haldane’s remarkable prediction [9] that a large gap existed in the spectrum of an integer spin chain (but not a half-integer one). If we add an extra “longitudinal” field Hoz to lift the degeneracy of the minima (i.e., of the two states | ↑ and | ↓), by an “energy bias,” o = gμB Sz Hoz . Then our two-level system (or qubit, in the language of quantum information) has Hamiltonian: Ho (τ̂ ) = o τ̂ x + o τ̂ z (1) We see the tunneling term o tries to drive quantum transitions between | ↑ and | ↓, but is hindered by the longitudinal field bias o , which pushes the states | ↑, | ↓ out of resonance. Notice that we can manipulate both o and o with external fields. Quantum Ising Spin Networks: Now imagine an interacting “spin net” of twolevel systems τ̂i , with i = 1, 2, ..N, and each with its own tunneling amplitude i and bias i . Adding interspin interactions between them generalizes (1) to: HoQI = i τ̂ix + i τ̂iz + Vij τ̂iz τ̂jz i ij (2) 264 G. Aeppli and P. Stamp a b flux ω n(t) q c 1E+1 1E+0 1E-1 1E-2 90 1E-3 o 1E-4 50 Δ(K) 1E-5 o o 20 1E-6 o 1E-7 7 1E-8 0 o 1E-9 1E-10 0 Fig. 1 (Continued) 1 2 H (T) 3 4 5 5 Quantum Magnetism 265 We only include longitudinal couplings (coupling the z-components of the spins) in the interaction Vij ; in many interesting magnetic systems, the other interaction terms are strongly suppressed relative to these. Notice that the total longitudinal field bias blocking quantum transitions on the i-th site is now ξi = i + j Vij τjz , instead of just i . This “Quantum Ising” model describes a large variety of physical systems which, at low temperature, truncate to a set of interacting two-level systems. The best known examples in quantum magnetism are dipolar-coupled magnetic molecules and ions and quantum spin glasses. However, H0QI also describes a set of N interacting qubits – and if we can manipulate the couplings in it, we have a toy model for a quantum computer. In quantum computation (reviewed in [14, 15, 16]), one creates and manipulates N-qubit states which are “entangled,” i.e., which cannot in general be written as products over separate spins. Such states are fundamentally nonclassical, as first discussed by Schrödinger and Einstein in 1935. The “quantum information” in them is encoded in the 2N −1 relative “spin phases” between the different qubits. To make and use such states is one of the great goals in this field – but it will be hard. To see why, we must first understand the main obstacle in the way. The Importance of Decoherence Decoherence arises when a quantum system interacts with its environment, and their Feynman paths and quantum phases become entangled. Even if they later decouple, averaging over the unknown environmental states then “smears” over states of the system, destroying phase coherence over some timescale τφ (the “decoherence time”). Decoherence in many-particle systems is usually much larger than expected, Fig. 1 Path integrals on the spin sphere. In (a), we show a spin S as a unit charge, moving on the unit sphere around a magnetic monopole of strength q = h̄S, along a path which has coordinate n̂(t) = S(t)/S, and encloses a solid angle . The kinetic phase φB = q/h̄ A · dn, where A is the monopole vector potential (compare the “Aharonov-Bohm” phase [8] accumulated by a charged particle moving through an ordinary magnetic vector potential A(r) in real space). For a closed path, Gauss’s theorem then shows that the enclosed flux from the fake monopole is just φB = S. In (b), a biaxial (easy ẑ-axis, hard x̂-axis) potential field Ho (S) is added (dark areas are regions of higher potential). The spin moves preferentially between the two minimum energy states at the poles by tunneling along the pair of minimum action paths (shown as dashed lines), with amplitudes 12 μ eiϕμ respectively, where μ = 1, 2 labels the paths, and the μ are real. An external field H⊥ o , applied along the hard x̂-axis, pulls the two states, and the paths between them, toward x̂, thereby reducing the enclosed area on the unit sphere. In this “symmetric” ˜ case, ϕ1 = −ϕ2 = ϕ(H⊥ o ) and 1 = 2 = o . The total tunneling amplitude is then just the ⊥ ) cos ϕ(H⊥ ). (c) shows the resulting oscillations in ˜ o (H⊥ sum 12 o (eiϕ + e−iϕ ), i.e., ) = (H o o o o ˜ o (H⊥ o ), for a typical biaxial potential (easy axis ẑ, hard axis x̂), as a function of transverse field ⊥ H⊥ o . If Ho is rotated away from x̂ by an angle φ (shown here in degrees), then |ϕ1 | = |ϕ2 |, and 1 = 2 , i.e., one path is favored over the other, and the oscillations are lost 266 G. Aeppli and P. Stamp a b + E(t) ε(t) t=0 t γk - E(t) c Vo Dipolar-Dominated Regime Li Ho x Y1-x F4 0.2 K Mn-12 0.1 K Quantum ξ o Relaxation Regime Fe-8 1K Fig. 2 (Continued) Δo Quantum Coherence Regime γk 5 Quantum Magnetism 267 and still somewhat mysterious (witness the debate over the saturation of dephasing times in mesoscopic conductors [17, 18]). Moreover, superpositions and entangled states are extraordinarily sensitive to even very small environmental interactions. To get a feeling for decoherence, let us go back to our toy tunneling spin, and now couple it to a bath of “satellite” spins. In the real world, these satellite spins describe localized modes (defects, nuclear and paramagnetic spins, etc.) which couple to the central spin [19]. There are also delocalized environmental modes like electrons, photons, phonons, etc., which also cause decoherence [21, 22], and which can be described as a bath of oscillators [20]). How do the satellite spins and oscillators dephase the “central” tunneling spin? We explain this again using path integral language (Fig. 2). There are three main decoherence mechanisms: (i) Typically the environmental modes have their own dynamics, creating an extra fluctuating “noise” field on the central system. This adds random phases to each path, eventually destroying phase coherence between them (“noise decoherence”). The noise can even push the central spin in and out of resonance (Fig. 2a). (ii) When the central spin tunnels, it causes a sudden “kick” perturbation on the satellite spins, giving them an extra phase which is entangled with the central spin phase – thence dephasing the central spin dynamics (“topological decoherence” [23]). (iii) The field on the k-th satellite, from the central spin, flips with the central spin between two (in general noncollinear) orientations (Fig. 2b). Between flips, the satellite precesses in these fields. Summing over all central spin paths, each involving a different accumulated satellite precessional phase, Fig. 2 The role of decoherence: In (a), we see the effect of a randomly fluctuating environmental noise bias ε(t) (black curve) on a tunneling two-level qubit with tunneling matrix element o . The two levels having adiabatic energies ±E(t), with E 2 (t) = 2o + ε2 (t), are shown as red and blue curves. The system can only make transitions when near “resonance” (i.e., when |ε(t)| is ∼ o or less, the regions shown in green). In (b), we show schematically the motion of a satellite spin, in the presence of a qubit which is flipping between two different states | ↑ and | ↓. When the qubit flips, the qubit field acting on the k-th satellite spin rapidly changes, from ↑ ↓ γk to γk (or vice versa). Between flips, the spin precesses around the qubit field, accumulating an extra “precessional” phase. Averaging over this phase gives precessional decoherence. The sudden change of qubit field also perturbs the satellite spin phase, giving further decoherence (the “topological decoherence” mechanism [23]). (c) shows the important parameters for a spin network – the typical tunneling splitting o , the characteristic energy Vo of interspin interactions, and the energy scale ξo governing interactions with the environment (which in insulating magnetic systems at very low T comes from the coupling to nuclear spins). For definiteness, we show the parameter range covered in this space by experiments in crystals of Mn-12 and F e-8 molecular magnets, and in LiH ox Y1−x F4 ; energy scales are in temperature units. In these systems, o is controlled by varying the transverse field, and Vo is varied by changing x (in LiH ox Y1−x F4 ), or by diluting the molecules in solution (in F e-8 and Mn-12) 268 G. Aeppli and P. Stamp gives “precessional” decoherence. In an oscillator bath, the central spin flip slightly shifts the oscillator wave functions – for metallic environments, this gives Anderson’s “orthogonality catastrophe” [24], a very strong decoherence mechanism. Notice that decoherence may involve very little energy transfer – it is not necessarily a dissipative process. In magnetic systems, the worst low-T decoherence will come from very low-energy localized modes, particularly nuclear spins, which cause very little dissipation, but lots of precessional decoherence [19, 25]. Delocalized modes like phonons, photons, and electrons cause strong decoherence (and strong dissipation) at higher energy, where they have a high density of states. Thus, at intermediate energies (typically around 0.01 − 0.5 K), decoherence is at a minimum. This “window” of low decoherence is of great practical importance – it also exists for many other solid-state systems [26]. The basic problem with our toy model (2) for a quantum computer is now clear – it ignores decoherence. If we couple each of the spins in the spin net to an environment, there are now three main energy scales (Fig. 2c). A “quantum” parameter o (the typical value of i ) drives the dynamics, along with interspin interactions of typical strength Vo ; but a coupling of each spin to the environment, having some effective energy scale ξo , destroys phase coherence. If we could switch off ξo (i.e., stay in the Vo − o plane in Fig. 2c), we would have perfect quantum behavior, with Vij correlating the entangled dynamics of vast numbers of qubits – this would be true macroscopic quantum spin entanglement. But how close are we to this goal? Quantum Relaxation in Dipolar Nets In fact most work has been done on systems with dipolar interspin couplings, having non-negligible environmental interactions – i.e., near the Vo − ξo plane in Fig. 2c. These systems are very complex – but a simple theoretical picture can be given. Consider first a single spin qubit τi . The net bias i on τiz now includes a dynamic contribution from the nuclear spin environment. This typically fluctuates over an energy range ξo ∼ Eo , where Eo defines the energy width of the multiplet of nuclear spin states coupled to S; this width is easy to calculate if the hyperfine couplings are known. Then if i is within a “tunneling window” of width ξo around zero bias, the fluctuating field can actually bring the qubit to resonance (recall Fig. 2a), where it can make inelastic (i.e., incoherent) transitions [25]. Now consider an interacting network – assuming here for definiteness that go = 1 (the “dipolar-dominated” regime in Fig. 2c). As the resonant spins o /Vo tunnel, a “hole” of width ξo should appear in the distribution of longitudinal fields in the system, around zero (see Fig. 3a). The interaction contribution j Vij τj to i then plays a key role – it slowly changes as the τj relax, bringing more spins into the tunneling window (hole “refilling”). The total spin distribution is then predicted 5 Quantum Magnetism a 269 [P (ξ,t) - P ( ξ,t)] εo Vo 2ξo 0 Fig. 3 (Continued) ξ 270 G. Aeppli and P. Stamp Fig. 3 Collective tunneling dynamics of dipolar nets, when o ξo , Vo (incoherent tunneling relaxation regime). In (a), we show the short-time evolution of the distribution function M(ξ, t) = P↑ (ξ, t) − −P↓ (ξ, t), where Pσ (ξ, t) is the normalized probability that a spin in a state |σ = | ↑ or | ↓ finds itself in a bias field ξ at time t. Different colors show the distribution at different times. At short times, a “tunneling hole” of width ξo appears, driven by inelastic tunneling transitions involving nuclear spins. The dipolar interactions gradually modify the shape and width of the hole at later times. This figure was produced by Monte Carlo simulations for a sample starting in a strongly annealed state. (b) shows measurements of the function M(ξ = o , t = 0) on a strongly annealed F e-8 crystal (from Ref. [30]), obtained by extracting the square root relaxation rate −1 Γsqrt ≡ τQ ∼ (2o /Vo )ξo N (o ), where N (ξ ) is the “density of states” of spins in a bias energy ξ (see text). If one lets M(ξ, t) relax for a time tW before examining it, the tunneling hole is revealed. Closer examination (lower graph) shows that for small initial magnetization Min (i.e., strong annealing), the hole has an intrinsic linewidth, revealed at short waiting times tW . This linewidth ξo is caused by the nuclear spins (see text). (c) shows the tunneling matrix element extracted from measurements of relaxation in a transverse field H⊥ o (from ref. [31]). These experiments found the −1 oscillatory dependence of τQ , and thence |o |, on H⊥ o (compare text, and Fig. 1c) to relax, with a characteristic “square root” relaxation [27] ∼ (t/τQ )1/2 . One gets −1 (o ) ∼ (2o /Vo )ξo N(o ), where N(ξ ) is the “density of states” of spins in a bias τQ energy ξ , i.e., ξo N (H ) is the number of spins in the tunneling window, centered at the external field bias energy o = gμB SHoz . Many experiments, using ensembles of magnetic nanomolecules such as F e-8 and Mn-12 (which truncate to two-level systems at low T ), have now tested this theory. Square root relaxation was found at short times [28, 29, 30]. Wernsdorfer et −1 al. [30] found the time-evolving hole, of width ξo , by measuring τQ (ξ = o ) for many different values of o (Fig. 3b). In strongly annealed samples (where M(ξ ) is a known Gaussian independent of sample shape), they also extracted 2o from measurements of τQ , and showed how it oscillated in a transverse field (Fig. 3c), and then confirmed this in independent “Landau-Zener” relaxation measurements. We emphasize these oscillations are not evidence for coherent tunneling, quite the 5 Quantum Magnetism 271 contrary – the experiments observe incoherent relaxation rates! In a very striking result [32], the nuclear isotopes were varied (substituting 2 H for 1 H , or 57 F e for 56 F e). This changed the hole width and the relaxation rate, giving independent measurements of ξo which were consistent with the calculated value. This is fairly direct evidence for the nuclear spin-mediated tunneling mechanism. The Leiden group [33] has also done low-T NMR on Mn-12, seeing not only how nuclear spins control the tunneling dynamics but also how the nuclear dynamics in turn is controlled by the molecular tunneling dynamics. Notice these are all results for short-time quantum relaxation. At longer times, multi-spin correlations intervene, causing a breakdown of the square root – the system moves into the quantum spin glass regime, of fundamental interest [35, 36]. Only quantum tunneling, simultaneously involving many spins, allows the system to escape local potential minima. Experiments on the insulator LiH o0.44 Y0.56 F4 , in which the lowest magnetic doublets (i.e., two-level systems) of J = 8 H o3+ ions interact primarily via dipolar interactions, have looked at this tunneling relaxation (Fig. 4a). Remarkably, much of the relaxation (here to a ferromagnetic ground state) goes via collective dissipative tunneling of domain walls (see Fig. 4a); this purely quantum effect has been definitively confirmed by observing its dependence on an applied transverse field [37]. For these dense H o arrays, we can also reinterpret the long-time relaxation as a quantum optimization process. One relaxes the system toward the ground state not by reducing the temperature (as in thermal annealing optimization protocols [38]), but by “quantum annealing” – exposing the system at very low T to a large transverse field H⊥ o , allowing it to quantum relax, and then reducing H⊥ o to zero, thence freezing the dynamics [39]. One then reads the final state – which is the “solution” to the problem of energy optimization. Finally, one can also explore the regime where ξo Vo , i.e., where dipolar interspin interactions are unimportant, and the nuclear environment dominates completely. Here, a “toothcomb” structure was expected in the quantum relaxation rate, reflecting the level structure of nuclear spins [25]. This was recently found (Fig. 4b) in experiments [40] on dilute concentrations of H o ions in LiH ox Y1−x F4 (there was also an interesting catch – residual inter-H o interactions can cause pairs of spins to co-flip, giving a doubling of the teeth). We see that the study of the quantum relaxation of a spin net reveals the essential physics governing the incoherent spin dynamics. Now we can turn to the coherent spin dynamics. Large-Scale Coherence and Entanglement The acid test of our understanding comes with large-scale quantum effects – where decoherence must be rigorously suppressed. Of course the traditional view has been that too many microstates are involved in any macrostate for this to be possible [41]. However, the modern picture is different. Macroscopic tunneling: In pioneering work, predictions of macroscopic tunneling between different flux states in superconducting SQUID rings [20] were 272 Fig. 4 (Continued) G. Aeppli and P. Stamp 5 Quantum Magnetism c 273 70 0.11 K 0.11 K with pump 0.15 K χ'' (emu/mole Ho) 60 50 40 30 20 10 0 1 10 100 1000 f (Hz) Fig. 4 Tunneling dynamics of the LiH ox Y1−x F4 system. In (a), we show typical behavior for the rate of microscopic domain wall depinning in LiH o0.44 Y0.56 F4 , as a function of inverse temperature (from Ref. [37]); the crossover from thermal activation to T -independent tunneling −1 relaxation occurs when T ∼ 50 mK. (b) shows the relaxation rate τQ (H ) of the H o spins in a −3 very dilute system (x = 2 × 10 ), from Ref. [40]. The main peaks in the “toothcomb” pattern, each separated by the H o hyperfine energy, come from nuclear spin-mediated tunneling of single H o ions between the lowest doublet states. The n = 8, 9 peaks shown in the inset come from coflip tunneling of pairs of H o ions, mediated by residual inter-H o interactions. (c) shows spectral hole burning for x=0.045, from Ref. [57]. The absorptive part of the magnetic susceptibility is measured as a function of frequency in the linear response regime using a probe amplitude of 0.04 Oe, giving a broad maximum centered at a frequency which depends strongly on temperature. The same spectroscopy in the presence of a 0.2 Oe pump at 5 Hz shifts the spectrum, cuts off its tails, and most important, inserts a sharp hole at 5 Hz. quantitatively verified in the 1980s [42, 43]. In magnets, similar tunneling was predicted for large domain walls pinned to defects [44], and also found in experiments [45,46]. In experiments on large domain walls in mesoscopic Ni wires (of thickness ∼20 − 80 nm), one sees a crossover to a T -independent escape rate of the walls from a pinning center, in an applied field; the dependence of the rate on field can be compared with theory [46]. This tunneling involved roughly 107 spins – not far short of the number of Cooper pairs involved in SQUID tunneling. A revealing set of experiments [46] also looked at microwave absorption between different levels – these represented the quantized dynamics of the wall center of mass, trapped in the pinning potential well. A fairly detailed picture can be given of these experiments [47, 48]. Such tunneling phenomena involve a collective degree of freedom (SQUID flux, magnetic domain wall position) which does indeed involve a huge number of 274 G. Aeppli and P. Stamp microstates. So how can it happen? One reason is that all electronic microstates (Bogoliubov quasiparticles, magnons) are strongly gapped, by energies EG ∼ 10 K. To such high-energy excitations, the collective coordinate tunneling, over a timescale τo , seems very slow. The amplitude to excite them is then exponentially small, ∼ O(e−EG τo ), by elementary time-dependent perturbation theory. Of course there are also many very low-energy excitations (defects, paramagnetic impurities, nuclear spins, etc. – the “satellite spins” discussed in section “The Importance of Decoherence”), which can entangle with the collective tunneling coordinate. However, they cause little dissipation, because their energy is so low – their direct effect on “single-shot” tunneling is then rather weak. Large-scale coherence: Coherent superpositions, on the other hand, require phase coherence between many successive tunneling events [21,22]. Now, the low-energy environmental microstates are indeed very dangerous [19]. So is macroscopic state superposition feasible? In superconductors, the answer to this question came a considerable time ago, including early experimental evidence for macroscopic flux state superpositions [49, 50, 51]. The decoherence times τφ for superconducting qubits have undergone a spectacular rise since the year 2000 so that today, they are viewed as leading candidates for the fundamental building blocks of quantum computers. Analogous macroscopic superpositions in magnets - e.g., of “giant qubit” superpositions of two different magnetization states in a large magnetic particle – have not yet been confirmed. Some years ago, absorption experiments in very large ferritin molecules (with Neél vector ∼ 23, 000 μB ) showed sharp resonances, attributed to coherent tunneling of individual ferritin molecules [53]. However, these results were hard to understand theoretically, and no other group has confirmed them; and experiments on much smaller molecules like Mn-12 or F e-8 have never seen coherence. The basic difficulty is that low-energy decoherence from the hyperfine coupling to nuclear spins is expected to be large (in ferritin, the hyperfine coupling to a single 57 F e nucleus is much larger than the tunnel splitting!), and at higher energies, phonon contributions are not negligible [25, 19]. However, experiments may have simply been looking in the wrong place. The “window of opportunity” between nuclear spin and phonon energy scales is actually very wide; by applying strong transverse fields, one can increase o to values much higher than hyperfine couplings (compare Fig. 1c), but still much lower than most phonon energies, and optimize the decoherence rate. By combining isotopic purification with choice of material, one can also remove almost all nuclear spins. Experimentalists like to define a decoherence Q-factor Qφ = o τφ , which tells us how many coherent oscillations the system can show before decoherence sets in. Elementary theory [25, 52] then indicates that for an spin S in an insulator, we have 2 /SK E , where K is the anisotropy energy per electronic spin an optimal Qφ ∼ θD o o o of the magnet, θD the Debye energy, and Eo is again the spreading of the nuclear multiplet. For example, if θD = 300 K, Ko = 1 K, and S = 106 , then by reducing 0.1 K, we should get “mesoscopic” coherent dynamics (i.e., Qφ > 1). Eo to Large-scale entanglement: We next turn to multi-qubit entangled states, but now involving microscopic spins. It is estimated that quantum information processing 5 Quantum Magnetism 275 Fig. 5 Design for a nuclear spin-based quantum computer, from Kane [54]. Two cells in a one-dimensional array, containing 31P donors and electrons in a Si host, are separated by a barrier from metal gates on the surface. The “A gates” control the resonance frequency of the nuclear spin qubits, while the “J gates” control the electron-mediated coupling between adjacent nuclear spins (QUIP) will be possible, using error correction [14,16], if the single qubit coherence Q-factor Qφ > 104 − 105 . This should easily be possible with microscopic spins – note from above that the optimal Qφ ∼ O(1/S). Thus, theory clearly indicates that QUIP is feasible with microscopic magnetic qubits, provided electronic decoherence is absent (e.g., magnetic ions in insulators or semiconductors, or perhaps insulating molecular crystals). Many proposals have appeared along these lines – as an example, consider that due to Kane [54] in Fig. 5, using networks of nuclear spins in semiconductors to do the computation. Reasonable estimates of decoherence rates [54] give very small numbers here – problems should only arise, as before, from very low-energy excitations (e.g., 1/f noise from charge defects). Again, applying strong fields should help [55], and measurements of the decoherence rates will be crucial, in this and other designs [56]. Experiments over the last years have given very long spin relaxation times for the spins associated with isolated impurities and quantum dots in silicon. While this is very interesting for quantum information science, the finding of sharp resonances in a strongly interacting many-body system using spectral hole-burning [57, 64, 69] in LiH o0.045 Y0.995 F4 (see Fig. 4c) is important for the science of disordered magnetism. The decoherence was remarkably small, in spite of the long-range interH o dipole interactions. The data were explained as a collective effect, involving tunneling of large clusters of H o spins. These results are both surprising and exciting – they indicate that we may be close to manipulating entangled mesoscopic spin states. To do fully fledged quantum computations will require controlling individual spins or groups of spins, i.e., control of the parameters i , i , and Vij . Control of i and i can obviously be done by varying transverse and longitudinal external fields – in fact, all quantum logic operations can be implemented by varying only one of these three parameters, and one can also use timed pulses in creative ways (which also help with decoherence [58,59]), so control of Vij is not crucial. One possibility 276 G. Aeppli and P. Stamp is to use engineered heterostructures to control the local fields [54]; another would be to use magnetic STM tips, although the practicality of this for anything other than demonstration experiments involving a very small number of qubits is questionable. A more difficult problem will be to measure the quantum state of the spin qubits, without affecting their operation. One can imagine many possibilities – for example, bringing in superconducting or magnetic devices, whose tunneling into the qubit depends on its polarization [60], or using optical methods. Over the last two decades, great progress has been made on both adiabatic and gated quantum computation. For solid-state implementations, the leading contenders have been superconducting qubits, whose decoherence times have dramatically improved, and for which very complex circuits can be constructed. It is beyond the scope of this chapter on magnetism to review these developments on magnetism, except to mention that the quantum annealer manufactured by D-wave systems [62] is built to simulate the transverse field Ising model, and can therefore be viewed as a programmable version of LiH o1−x Yx F4 . Equally relevant here and for the future of magnetism are experiments showing control of the magnetic interactions between the very simple S=1/2 spins associated with either donors [34] or quantum dots [67] in silicon. Future Directions and Open Problems When many of the authors of this volume began graduate work in the 1980s, the idea of large-scale quantum phenomena in magnetic systems was hardly a topic for discussion – for ∼70 years, quantum mechanics was only used in magnetism to discuss atomic and nuclear spins and the microscopic interactions (exchange, spinorbit, etc.) operating on them. Now we are discussing and even seeing quantum phenomena at much larger scales, where magnetic variables were previously treated as classical – it is in this sense that the “quantum” is being put back into magnetism. We are on the threshold of a very different era – in which coherent spin states, having no classical analogue, may come to play a role as important as the macroscopic wave function in superconductors. As always, it is difficult to make predictions about a fast-moving field. The preparation and readout of coherent multi-spin states may well require techniques from spintronics [61], implemented on submicron scales. The key challenges here will be (i) to marry spintronics with the science of collective quantum spin states, under progressively less extreme experimental conditions of magnetic field and temperature, and (ii) to understand how to suppress electronic decoherence in conducting magnets. This latter is a hard problem because spin current is not a conserved quantity, which has made it difficult to find a rigorous theory of spin dynamics in conducting magnets, although moving to antiferromagnets where mesoscopic quantum tunneling of domain walls has been identified (in the common metal chromium) [68] may be a promising route. The development of new materials having the correct mix of optical, electronic, and magnetic properties will be crucial, and theory will need to be developed to model candidate materials and devices. It is sobering that even for an insulator as simple as LiH ox Y1−x F4 , many key 5 Quantum Magnetism 277 discoveries – e.g., the coherent hole burning for x=0.045 – were unexpected, and dictated by such factors as the availability of samples with particular compositions at sale prices. We also note that there are many interesting spin systems apart from electronic magnets. 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Commun. 10, 4001 (2019). https://doi.org/10.1038/s41467019-11985-1 280 G. Aeppli and P. Stamp Gabriel Aeppli is professor at ETHZ and EPFL, and head of Photon Science at PSI. After working at NEC, AT&T, IBM and MIT on problems from liquid crystals to magnetic data storage, he co-founded the London Centre for Nanotechnology and BioNano Consulting. His focus is on implications and development of photon science and nanotechnology for information processing and health care. Philip Stamp received his PhD from the Univ of Sussex. After postdoctoral work in Massachusetts, Grenoble, and Santa Barbara, he held positions in the Univ of British Columbia, and as a Spinoza Professor in Utrecht. He is currently director of the Pacific Institute of Theoretical Physics in Vancouver. He works on theoretical quantum gravity and condensed matter theory. 6 Spin Waves Sergej O. Demokritov and Andrei N. Slavin Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Waves in 3D and 2D Systems: Theory and Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . Theory of Spin Waves in 3D and 2D Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brillouin Light Scattering a Powerful Tool for Investigation of Spin Waves . . . . . . . . . . . . Spin Waves in 1D Magnetic Elements: Standing and Propagating Waves . . . . . . . . . . . . . . . . BLS in Laterally Confined Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lateral Quantization of Spin Waves in Magnetic Stripes . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Wave Wells and Edge Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implementation of Micro-Focus BLS for Laterally Patterned Magnetic Systems . . . . . . . . Propagating Waves in 1D Magnetic Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Control and Conversion of the Propagating Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inductive Excitation of Spin Waves in 1D Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Torque Transfer Effect and Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin Waves in 0D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin-Torque Nano-Oscillator (STNO) and Emitted Spin Waves . . . . . . . . . . . . . . . . . . . . . Spin-Hall Nano-Oscillator (SHNO) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nature of Spin Wave Modes Excited in 0D Magnetic Nanocontacts . . . . . . . . . . . . . . . . . . Coupling of a STNO and 1D Spin-Wave Waveguide to Each Other . . . . . . . . . . . . . . . . . . Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 284 284 288 291 291 291 296 300 302 307 311 315 320 321 324 329 336 339 341 S. O. Demokritov () Institute for Applied Physics and Center for Nanotechnology, University of Muenster, Muenster, Germany e-mail: [email protected] A. N. Slavin Department of Physics, Oakland University, Rochester, MI, USA e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_6 281 282 S. O. Demokritov and A. N. Slavin Abstract Spin waves are the dynamic eigen-excitations of a magnetic system. They provide the basis for the description of spatial and temporal evolution of the magnetization distribution of a magnetic object. The unique features of spin waves such as the possibility to carry spin information over relatively long distances, the possibility to achieve submicrometer wavelength at microwave frequencies, and controllability by electronic signal via magnetic fields make these waves uniquely suited for implementation of novel integrated electronic devices characterized by high speed, low power consumption, and extended functionalities. The history of spin waves clearly shows a progressively increasing interest for the spin waves in magnetic samples of reduced dimensionality. Since 1950s the focus of the researchers has moved from 3D to 2D objects – thin films and magnetic multilayers, resulting in the discovery of the surface Damon-Eshbach spin-wave mode. Later in 1990s 1D stripes became the most actively studied magnetic systems, bringing about the discovery of lateral quantization and edge modes. Finally, theoretical prediction of the spin-torque effect and development of novel techniques for nanofabrication allowed for the investigation of magnetic 0D objects such as spin-torque nano-oscillator. In this chapter we follow this historical trend and describe the recent development of spin-wave studies. Introduction Felix Bloch introduced the concept of spin waves (SPW), as the lowest-energy magnetic states above the ground state of a magnetic medium [1]. Bloch theoretically considered quantum states of magnetic systems with spins slightly deviating from their equilibrium orientations, and found that these disturbances were dynamic: they propagate as waves through the medium. It is interesting to note that the concept of dynamic spin waves was introduced to explain experimental data obtained in static measurements. From spin-wave theory Bloch was able to predict that the magnetization of a three-dimensional (3D) ferromagnet at low temperatures should deviate from its zero-temperature value with a T3/2 dependence (the famous Bloch law), instead of the exponential dependence given by the mean field theory. Albeit the Bloch theory has restricted itself by assuming the dominating exchange interaction, now we know that the relativistic magnetic dipole interaction plays a decisive role in the properties of spin waves having the wavelengths that are much smaller than the interatomic distance in the magnetic medium. Moreover, phenomenologically, spin waves in a wide interval of wavevectors (30 < k < 106 cm−1 ) that is most important for the practical applications are, on one hand, almost entirely determined by the magnetic dipole-dipole interaction, and, on the other hand, can be correctly described when the retardation effects are neglected. Such spin waves are usually called dipolar magnetostatic waves or magnetostatic modes. Due to the anisotropic properties of the magnetic dipole interaction, the frequency of a spin 6 Spin Waves 283 wave depends on the orientation of its wavevector relative to the orientation of the static magnetization. For higher values of the wavevectors, when the exchange interaction cannot be neglected, one speaks about dipole-exchange spin waves. Spin waves are the dynamic eigen-excitations of a magnetic system. They provide the basis for the description of spatial and temporal evolution of the magnetization distribution of a magnetic object under the general assumption that locally the length of the magnetization vector is constant. This is correct, if, first, the temperature is far below the Curie temperature of the medium, as is assumed throughout this chapter, and, second, if no topological anomalies such as vortices or domain walls are present. The latter is fulfilled for samples in a single-domain state, i.e., magnetized to saturation by an external bias magnetic field. The unique features of spin waves such as the possibility to carry spin information over relatively long distances, the possibility to achieve submicrometer wavelength at microwave frequencies, and controllability by electronic signal via magnetic fields make these waves uniquely suited for implementation of novel integrated electronic devices characterized by high speed, low power consumption, and extended functionalities. The utilization of spin waves for integrated electronic applications is addressed within the emerging field of magnonics [2–5]. Although the application of spin waves for microwave signal processing has been intensively explored since many decades, recent advances in spintronics and nanomagnetism, as well as the development of novel techniques for nanofabrication and measurements of high-frequency magnetization dynamics created essentially new possibilities for magnonics and brought it onto a new development stage. Of particular importance here is the recent discovery of the spin-transfer torque (STT) [6–8] and the spin-Hall effect (SHE) [9–11], both of which have already been demonstrated to enable novel device geometries and functionalities [12–16]. The history of spin waves clearly shows a progressively increasing interest to the spin waves in magnetic samples of reduced dimensionality. Although the original theory of Bloch was developed for 3D magnets, since 1950s the focus of the researchers has moved to two-dimensional (2D) objects – thin films and magnetic multilayers, resulting in the discovery of quantized standing spin-wave resonances [17, 18], surface Damon-Eshbach spin-wave mode [19, 20], and the interlayer coupling [21]. Later in 1990s quasi-one-dimensional (1D) stripes became the most actively studied magnetic systems, bringing about the discovery of lateral quantization [22] and edge modes [23, 24]. Finally, as mentioned above, theoretical prediction of the spin-torque effect (STE) [6, 7] and development of novel techniques for nanofabrication allowed for investigation of magnetic zerodimensional (0D) objects such as spin-torque nano-oscillator (STNO) [25–31]. In this chapter we follow this historical trend. In the Sect. “Spin Waves in 3D and 2D Systems: Theory and Experiment” we describe the theory of spin waves in 3D and 2D magnetic systems, as well as the main experimental techniques for spin-wave studies in such systems. In Sect. “Spin Waves in 1D Magnetic Elements: Standing and Propagating Waves” we focus on spin waves in quasi-1D systems: laterally quantized and localized spin-wave edge modes in magnetic stripes. We also discuss the propagating wave modes in magnetic waveguides. The experimental data are 284 S. O. Demokritov and A. N. Slavin complemented by a short theoretical description of the spin-torque effect and its role in the damping compensation for propagating spin waves. In Sect. “Spin Waves in 0D” we describe the spin-wave dynamics of 0D systems on the example of an inplane magnetized STNO structure supporting a self-localized solitonic spin-wave “bullet” mode. Conclusions are given in Sect. “Conclusion and Outlook”. Spin Waves in 3D and 2D Systems: Theory and Experiment This section comprises two subsections. Subsection “Theory of Spin Waves in 3D and 2D Systems” is devoted to the general theory of spin waves in both 3D and 2D systems. It also demonstrates the modification of the spin-wave properties due to the dimensionality reduction. Subsection “Brillouin Light Scattering a Powerful Tool for Investigation of Spin Waves” describes the Brillouin light scattering – the main experimental techniques for the investigation of spin waves. Since spin waves in 3D and 2D systems have been intensively studied in the past, the sections devoted to their description are short, and the main attention is paid to the spin waves in the quasi-1D and 0D geometries. Theory of Spin Waves in 3D and 2D Systems The dynamics of the magnetization vector is described by the Landau-Lifshitz torque equation [32]: 1 dM = M × H eff − (1) γ dt where M = MS + m(R, t) is the total magnetization, MS and m(R , t) are the vectors of the saturation and the variable magnetization correspondingly, γ is the modulus of the gyromagnetic ratio for the electron spin (γ/2π = 2.8 MHz/Oe), and H eff = − δW δM (1a) is the effective magnetic field calculated as a variational derivative of the energy function W, where all the relevant interactions in the magnetic substance have been taken into account (see, e.g., [33–35]). For the case of an unbounded 3D ferromagnetic medium the variable magnetization m(R, t) depends on time t and on the three-dimensional radius-vector R. In the spin-wave analysis it is usually assumed that the variable magnetization m(R, t) is small compared to the saturation magnetization MS , i.e., the angle of magnetization precession is small. In this case the variable magnetization can be expanded in a series of plane spin waves (having a 3D wavevector q): m (R, t) = q mq exp (iqR) . (2) 6 Spin Waves 285 The spectrum of dipole-exchange spin waves is in an unbounded ferromagnetic medium which is given by the Herrings-Kittel formula [36]. ω = 2πf = γ 1/2 2A 2 2A 2 H+ H+ q q + 4π Ms sin2 θq Ms Ms (3) where A is the exchange stiffness constant, H is the applied magnetic field, and θ q is the angle between the directions of the wavevector q and the static magnetization MS with sin2 θ q being the matrix element of the dipole-dipole (magnetostatic) interaction. Analyzing Eq. (3), one concludes that if the exchange can be neglected q2 < < HMS /2A and q 2 << 2π MS2 /A, the spin-wave frequency is independent of q. It depends solely on θ q demonstrating that the nonexchange (magnetostatic) spinwave spectrum is anisotropic and nondispersive. In contrast, for q2 > > HMS /2A and q 2 >> 2π MS2 /A the spectrum of purely exchange spin waves is isotropic since the spin wave frequency solely depends on q = |q|. The transition from 3D to 2D can be made if one considers a magnetic film with a finite thickness d. In the following, we assume a Cartesian coordinate system oriented in such a way that the film normal is along the x-axis, and axes y and z are in the film plane, with the external field H and the static magnetization MS being aligned along the z-axis. Correspondingly, because the translational invariance along the direction normal to the film surfaces (axis x) is broken, the three-dimensional spin wave wavevector is represented as a sum of a two-dimensional continuous inplane wavevector q and quantized wavevector κ p ex (p = 0,1,2 . . . ) along the film thickness: q = q + κ p ex , while the three-dimensional radius vector is represented as R = R + x ex . Then, the distribution of the variable magnetization along the film thickness (axis x) can be represented as a Fourier series in a complete set of orthogonal functions mp (x) [37]: m (R, t) = mp (x) exp iq R . (2a) q ,p These functions mp (x) are chosen in such a way that they satisfy the exchange differential operator of the second order and the exchange boundary conditions at the film boundaries [38]: ∂m ∂x + D m|x=±d/2 = 0, (4) x=±d/2 where D is the so-called “pinning” parameter determined by the ratio of the effective surface anisotropy ks and the exchange stiffness constant A: D = ks /A. The modes mp (x) can be interpreted as the modes of the spin-wave resonance [17] in a particular geometry, and are sometimes called perpendicular standing spin waves (PSSW). The discrete transverse wavenumbers κ p for these modes are determined from the eigenvalue problem for the exchange differential operator 286 S. O. Demokritov and A. N. Slavin with the boundary conditions Eq. (4). Note that in an finite-in-plane nonellipsoidal magnetic film samples, the “pinning” of the dynamic magnetization at the lateral edges of the sample could be determined by the local inhomogeneity of internal dynamic magnetic field, and a different set of the “in-plane” eigenfunctions for the expansion of the in-plane components of the variable magnetization can be obtained in that case [39, 40]. Assuming that the thickness spin-wave modes mp (x) do not hybridize, it is possible to obtain an approximate “diagonal” dispersion equation for the spin-wave modes in a magnetic film of a finite thickness that is similar to the classical Kittel equation (3) [37]: 2A 2 2 ωp = 2πfp = γ H + q + κp Ms 1/2 2A 2 2 H+ q + κp + 4π Ms Fpp κp , q , d , Ms (5) where Fpp (κ p , q , d) is the matrix element of the dipole-dipole interaction in a film defined by Eq. (46) in [37]. In the case of “unpinned” spins at the film surfaces ∂m ∂x = 0, (6) x=±d/2 it is possible to obtain a simple explicit expression for the transverse spin-wave wavenumber κ p = pπ /d, p = 0, 1, 2, . . . , and the expression for the matrix element Fpp (κ p , q , d) for an arbitrary angle between q|| and MS can be written in the form: Fpp = 1 + Ppp (q) 1 − Ppp (q) 4π MS H + (2A/MS ) q 2 qy2 q2 q2 − Ppp (q) z2 q , (7) where q 2 = qy2 + qz2 + pπ d 2 = q2 + pπ d 2 , (8) and the function Ppp (q , p) is defined in [37]. We present here only the expression for this function for the lowest (quasiuniform) thickness mode (p = 0) [37]: P00 1 − exp −q d . = P00 q d = 1 + q d (9) 6 Spin Waves 287 If the lowest thickness spin wave mode (p = 0) is propagating perpendicular to the bias magnetic field, q = (qy , 0), its dispersion equation obtained from (5) using (6), (7), and (8) in the nonexchange limit (A = 0) has the form: ω0 qy d = 2πf0 qy d 1/2 (10) , = γ H (H + 4π MS ) + (4π MS )2 P00 qy d 1 − P00 qy d which for qy d < < 1 is similar to the dispersion equation obtained by Damon and Eshbach for the dipolar surface mode [19]: ωDE qy d = 2πfDE qy d 1/2 . = γ H (H + 4π MS ) + (2π MS )2 1 − exp −2qy d (11) Thus, the spectrum of spin wave modes propagating perpendicular to the direction of the bias magnetic field in an in-plane magnetized magnetic film obtained from (5), (7), and (8) contains the lowest dipole-dominated mode (p = 0) with a quasi-uniform thickness profile, which is analogous to the spin waves in 3D bulk samples, and higher exchange-dominated spin-wave modes (p > 0), whose thickness profiles are approximately described by the PSSWs. The higher spin-wave modes are created because of the broken translational invariance along the x-direction. The frequencies of these modes (p > 0) in the long wave limit qy d 1 can be calculated from the following expression: 2A pπ 2 2 qy + ωp = 2πfp = γ H + Ms d 2 1/2 (12) 2 /H pπ 4π M 2A s qy2 + + 4π Ms + H qy2 , H+ Ms d pπ/d which is obtained from Eq. (5) using the expressions for the dipole-dipole matrix elements Fpp (qy d) calculated in [37]. Figure 1 illustrates the typology of the lowest quasi-uniform (p = 0) dipoledominated spin-wave modes in the quasi-2D case of a magnetic film for different mutual orientations between the in-plane wavevector, q , and the static magnetization, MS . Three different geometries are shown. If MS is in the film plane, and if q is perpendicular to MS , the surface or Damon-Eshbach (DE) mode exists (see Eqs. (10) and (11)). If q and MS are collinear in the film plane, a mode with a negative dispersion, or, so-called, the backward volume magnetostatic mode (BV) exists and has the group velocity that is antiparallel to the wavevector. Finally, if the magnetization MS is perpendicular to the film plane, the existing mode is the, so-called, forward volume magnetostatic mode. (FV) In the latter case the dipoledipole matrix element F00 (q) in Eq. (5) can be expressed as: 288 S. O. Demokritov and A. N. Slavin Fig. 1 Typology of spin wave modes in a magnetic film for different orientations of the magnetization, MS , and the in-plane wavevector, q F00 q d = P00 q d , (13) with P00 (q d) given by Eq. (9). Brillouin Light Scattering a Powerful Tool for Investigation of Spin Waves The spin-wave spectrum of a magnetic system can be investigated by various techniques: ferromagnetic resonance [41], time-resolved Kerr magnetometry [24, 42–44], and Brillouin light scattering spectroscopy (BLS) [45–47]. The BLS experimental technique has a number of advantages for the investigations of magnetic structures. It combines the possibility to study the dynamics of magnetic systems in the frequency range of up to 500 GHz (corresponding time resolution is 2 ps) with a high lateral resolution of 20–40 μm for the regular setup and down to 220 nm for the, so-called, micro-focus BLS. In both cases the spatial resolution is defined by the size of the laser beam focus. Another important advantage of BLS is its very high sensitivity, which allows us to register thermally excited spin wave modes, so the coherent excitation of the magnetic element by an external signal is not necessary. This property of the 6 Spin Waves Fig. 2 Scattering process of a photon from a spin wave (magnon). θ is the scattering angle 289 hqI hω I θ hqS hωS hq, hω BLS is especially useful for the experimental investigations of complicated, strongly confined spin-wave modes in patterned magnetic elements, which will be considered in the next sections. The BLS process can be considered as follows (see Fig. 2): monoenergetic photons (visible light, usually green line 532 nm or blue line 473 nm) with the wave vector qI and frequency ωI = cqI interact with the elementary quanta of spin waves (magnons), characterized by the magnon wave vector q and frequency ω. Due to the conservation laws resulting from the time- and space-translation invariance of a 3D system the scattered photon increases or decreases its energy and momentum if a magnon is annihilated or created: ωS = (ωI ± ω) (14) q S = q I ± q (15) Measuring the frequency shift of the scattered light one obtains the frequency of the spin wave participating in the BLS process. From Eq. (15) it is evident that the wave vector qS − qI , transferred in the scattering process, is equal to the wave vector q of the spin wave. Changing the scattering geometry one can sweep the value of q and measure the corresponding ω(q). Thus, the spin-wave dispersion ω(q) can be studied. Note here that for the 3D scattering process the maximum accessible wavevector q = 2qI , the double value of the light wave vector, corresponds to the backscattering geometry with the scattering angle θ = 180◦ . The electromagnetic field of the scattered wave is proportional to the product of the Faraday/Kerr and other magneto-optical constants of the medium and the amplitude of the dynamic magnetization, corresponding to the spin wave [45]. Thus, the BLS intensity, determined by the squared field, is directly proportional to the dynamic magnetization squared. Magneto-optical effects relate the dielectric tensor of the medium with Cartesian components of its magnetization. Usually the nondiagonal elements of the tensor, the magneto-optical effects (magnetic birefringence and the Faraday effect and the corresponding dichroisms) are responsible for the scattering. Therefore, the plane of polarization of the scattered light is rotated by 90◦ with respect to that of the incident light. The conservation laws, given by Eqs. (14) and (15), follow from the time invariance of the problem and the translation invariance of an infinite medium, 290 S. O. Demokritov and A. N. Slavin correspondingly. However, if the scattering volume is finite, the selection rule for the momentum is broken. For the scattering volume with a size less or comparable with the wavelength of the light, any spin wave with a wavevector comparable with that of light contributes to the scattering process. The confinement of the scattering volume can come from the finite size of the element under consideration and/or from a small scattering volume of the laser beam. For a thin film or for nontransparent bulk materials, the thickness of the scattering volume is strongly confined along the direction normal to the surfaces. Therefore, only the in-plane wave vector is conserved in the light scattering experiments. As shown in Fig. 3, in the backscattering geometry the transferred in-plane wavevector q|| is determined by the angle of incidence q|| = 2qsinα, with q being the absolute value of the wave vector of the incident light. For green laser light q|| varies in the typical range of (0–2.5) × 105 cm−1 . This approach is illustrated in Fig. 4 showing the spin-wave dispersion of a permalloy (Ni80 Fe20 ) film with a thickness of 20 nm, measured in the DE geometry at the applied magnetic field H = 500 Oe. The experimental data presented in Fig. 4 were obtained by varying α . The solid line in the figure is the result of calculation based on Eq. (11) with the value of the permalloy magnetization 4πMS = 9.8 kG. Fig. 3 Backscattering process from a thin film. qi is the wavevector of the incident light; qs is the wavevector of the incident light; α is the angle between the wavevectors and the film normal Spin-wave frequency (GHz) Fig. 4 The spin-wave dispersion of a permalloy (Ni80 Fe20 ) film with parameters listed in the text measured using BLS. The solid line is the result of calculation based on Eq. (11) qs α qi 14 12 10 8 6 0.0 0.5 1.0 1.5 5 -1 q|| (10 cm ) 2.0 2.5 6 Spin Waves 291 Spin Waves in 1D Magnetic Elements: Standing and Propagating Waves In this section, we will consider spin-wave modes in arrays of micron-size quasi-1D magnetic elements (stripes and waveguides). We will discuss lateral quantization of DE modes in a longitudinally magnetized stripe due to its finite width as well as localization of spin-wave modes near the edges of the stripes. After that, we review recent experimental investigations of spin-wave propagation, excitation, and control in microscopic waveguides. BLS in Laterally Confined Systems Before we consider new effects resulting from the lateral confinement in stripes, let us first focus on BLS from laterally confined excitations. It has been already mentioned in the previous sections that the form of the conservation laws, which determine the BLS process, depends on the dimensionality of the studied system: due to lack of translational invariance of a thin magnetic film along the normal to the film surface, the corresponding component of the wavevector is not conserved. Instead, the scattering angle (see Fig. 3) determines the 2D inplane vector, q|| only, whereas all the thickness modes possessing this q|| contribute to BLS, albeit with different intensity according to their thickness profiles. If now the in-plane translational invariance of the magnetic film is broken by patterning, the in-plane wavevector is no longer fully conserved in the BLS process. In the case of a spin-wave mode localized in a long stripe, the only conserved component is the component of along the stripe axis. In analogy with the films, all the laterally confined modes contribute to BLS, whereas the mode profile along the stripe width (more specific: the corresponding Fourier component of the dynamic magnetization) defines the contribution of the mode to BLS. Finally, if the confinement takes place in all three dimensions, no conservation laws for wavevectors can be applied. One should perform a Fourier analysis of the 3D distribution of the dynamic magnetization of a particular mode to calculate its contribution to the BLS intensity. Lateral Quantization of Spin Waves in Magnetic Stripes Mathieu et al. [22] and Jorzick et al. [48] investigated spin-wave excitations by BLS in arrays of permalloy stripes. They have found the effect of lateral quantization of spin waves due to a finite width of the stripes and observed several dispersionless spin-wave modes. Since these experiments provide the first account for spin-wave modes in 1D magnetic systems and heavily contribute to quantitative understanding of spin wave quantization effects in systems with reduced dimensions, we consider them in detail. The samples were made of 20–40 nm thick permalloy films deposited in UHV onto a Si(111) substrate by means of e-beam evaporation using using 292 S. O. Demokritov and A. N. Slavin Fig. 5 Scanning electron micrographs of permalloy stripes with a width of 1.8 μm and a separation of 0.7 μm. (Reprinted with permission from [45], © 2001 by Elsevier) X-ray lithography with a following lift-off process. Several types of periodic arrays of stripes with stripe widths w = 1.7 and 1.8 μm and distances between the stripes above 0.5 μm were prepared. Thus, the interaction between the stripes were negligible. The length L of the stripes was 500 μm, ensuring 1D-properties of the stripes. The patterned area was 500 × 500 μm2 , allowing BLS investigation with a large beam diameter, providing a good wavevector resolution. One of the studied arrays is shown in Fig. 5. In agreement with the shape-anisotropy arguments, the magnetic easy axis of the array was along of the stripe axis. Let us consider a magnetic stripe magnetized in plane along the z-direction and having a finite width w along the y-direction as shown in Fig. 5. A boundary condition similar to Eq. (6) at the lateral edges of the stripe should be imposed: m|x=±d/2 = 0 (16) One should emphasize that the internal field in the stripe is strongly nonhomogeneous due to the nonellipsoidal shape. This nonhomogeneity is of particular importance close to the edges. In the considered geometry, only the dynamic internal field is nonhomogeneous, since the static magnetization is along the edge and does not contribute to the demagnetizing effects. Nevertheless, this nonhomogeneous dynamic internal field results in specific mechanism for pinning of the magnetization [39]. Therefore, the Eq. (16) differs from Eq. (6) written for 6 Spin Waves 293 an unconfined film possessing a homogeneous internal field. The corresponding quantization of qy is then obtained as: qyn = nπ w (17) where n = 1,2, . . . . Using Eqs. (4), (11), and (12) and the quantization expression (17) one can calculate the frequencies of these so-called width (or laterally quantized) modes. The profile of the dynamic part of the magnetization m in the nth mode can be written as follows: mn (y) = An sin w nπ y+ w 2 (18) Equation (18) describes a standing mode consisting of two counter-propagating waves with quantized wavenumbers, nπ /w. Note here that due to the truncation of the sin-function at the stripe boundaries the modes are no more infinite plane waves and the quantized values are not true wavevectors. In BLS experiments with backscattering geometry the in-plane wavevector q|| , transferred in the light scattering process, was oriented perpendicular to the stripes, and its value was varied by changing the angle of light incidence, α, measured from the surface normal: as illustrated in Fig. 3. Figure 6 shows a typical BLS spectrum for the sample with a stripe width of 1.8 μm and as transferred wavevector q|| = 0.3 × 105 cm−1 , while an external field of 500 Oe was applied along the stripe axis. As it is seen in Fig. 6, the spectrum contains four distinct modes near 7.8, 9.3, 10.4, and 14.0 GHz. By varying the applied field, the spin wave frequency for each mode was measured as a function of the field, as displayed in Fig. 7. The observed dependence of all frequencies on the field confirms that all detected modes are magnetic excitations. The dispersion law of the modes was obtained by varying the angle of light incidence, α, and, thus the magnitude of the transferred wavevector, qy . The obtained results are displayed in Fig. 8 for two with the same stripe thickness of 40 nm and width of 1.8 pm, but with different stripe separations of 0.7 pm (open symbols) and 2.2 pm (solid symbols). The dispersion measured on the arrays with the same lateral layout but with a stripe thickness of 20 nm is presented in Fig. 9. It is clear from Fig. 8 that one of the detected modes presented by circles (near 14 GHz) is the PSSW mode, corresponding to p = 1 in Eq. (12). This mode is not seen in Fig. 9 due to its much higher frequency caused by the smaller stripe thickness. In the region of low wavevectors the spin wave modes show a disintegration of the continuous dispersion of the DE mode of an infinite film into several discrete, resonance-like modes with a frequency spacing between the lowest lying modes of approximately 0.9 GHz for d = 20 nm and 1.5 GHz for d = 40 nm. As it is clear from Figs. 8 and 9, there is no significant difference between the data for the stripes with a separation of 0.7 and 2 μm. This fact indicates that the mode splitting is purely caused by the quantization of the spin waves in a single stripe due to its finite width. In other words, the studied stripes can be considered as independent 1D elements. 294 S. O. Demokritov and A. N. Slavin PSSW BLS Intensity (a.u.) 3000 2000 1000 0 6 10 14 Frequency Shift (GHz) Fig. 6 Experimental BLS spectrum obtained from the stripe array with a stripe thickness of 40 nm, a width of 1.8 μm, and a separation of 0.7 μm. The applied field is 500 Oe orientated along the stripe axis. The transferred wavevector of q|| = 0.3 × 105 cm−1 is oriented perpendicular to the wires. The discrete spin-wave modes are indicated by arrows. PSSW stands for perpendicular standing spin-wave mode. (Reprinted with permission from [45], © 2001 by Elsevier) Spin Wave Frequency [GHz] 24 n=3 n=2 n=1 PSSW 20 16 12 Q-DE 8 4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Magnetic Field [kOe] Fig. 7 Frequencies of the in-plane quantized Damon-Eshbach modes (Q-DE) as a function of the applied field, obtained from the BLS spectra similar to that shown in Fig. 6. The lines are calculated using Eq. (11) with quantized wavevectors, determined by the quantization numbers n = 1,2,3. The line labeled PSSW shows the frequency of the first perpendicular standing spin-wave mode. (Reprinted with permission from [45], © 2001 by Elsevier) 6 Spin Waves 295 Spin Wave Frequency (GHz) 17 16 d = 40 nm 15 14 13 n=5 12 n=4 11 n=3 10 9 n=2 8 n=1 7 6 0.0 0.5 1.0 5 1.5 -1 2.0 2.5 q|| (10 cm ) Spin Wave Frequency (GHz) Fig. 8 Obtained spin-wave dispersion curves for an array of stripes with a stripe thickness of 40 nm, with a width of 1.8 μm and a separation of 0.7 μm (open symbols) and 2.2 μm (solid symbols). The external field applied along the stripe axis is 500 Oe. The solid horizontal lines indicate the results of a calculation using Eq. (11) with the quantized wavevectors, determined by the quantization numbers n = 1,2,3,4,5. The dashed lines showing the hybridized dispersion of the Damon-Eshbach mode and the first PSSW mode were calculated numerically for a continuous film with a thickness of 40 nm. (Reprinted with permission from [45], © 2001 by Elsevier) 14 13 d = 20 nm 12 11 n=5 10 n=4 9 n=3 8 n=2 n=1 7 6 0.0 0.5 1.0 5 1.5 -1 2.0 2.5 q|| (10 cm ) Fig. 9 Obtained spin-wave dispersion curves for an array of stripes for the same conditions as in Fig. 8, but with a stripe thickness of 20 nm. (Reprinted with permission from [45], © 2001 by Elsevier) 296 S. O. Demokritov and A. N. Slavin The main features of the observed spin wave modes in magnetic stripes can be summarized as follows: (i) For low wavevector values (0–0.8 × 105 cm−1 ) the discrete modes do not show any noticeable dispersion, behaving like standing wave resonances. (ii) Each discrete mode is observed over a continuous range of the transferred wavevector qy . (iii) The lowest two modes appear very close to zero wavevector, the higher modes appear at higher values. (iv) The frequency splitting between two neighbored modes is decreasing for increasing mode number. (v) There is a transition regime (qy = 0.8–1.0 × 105 cm−1 ) where the well-resolved dispersionless modes converge towards the dispersion of the continuous film (see dashed lines in Figs. 8 and 9). All these features can be explained if one implies that the observed discrete, dispersionless spin wave modes result from a confinement of the DE modes in the stripes. The confinement causes width-dependent quantization of the in-plane wavevector of the mode, as discussed above. Considering the corresponding Fourier component of the dynamic magnetization one can reproduce the measured BLS intensity of each modes [48]. The frequency of the observed modes can be derived by substituting the obtained quantized values of the wavevector, qyn , determined by Eq. (17) into the dispersion equation of the DE mode, Eq. (11). The results of these calculations are shown in Figs. 8 and 9 by the solid horizontal lines. For the calculation the geometrical parameters (stripe thickness d = 20 or 40 nm, stripe width w = 1.8 pm) and the independently measured material parameters 4πMs = 10.2 kG and γ = 2.95 GHz/kOe were used. Without any fit parameters the calculation reproduces all mode frequencies very well. Spin Wave Wells and Edge Modes In the previous subsection, the simplest geometry with the applied magnetic field aligned along the stripe axis was considered. In this case, the static field is homogeneous, while the dynamic field is not homogeneous resulting in the pinned boundary conditions for the dynamic magnetization. If, however, the applied field is directed along the width of a thin stripe, both the static and the dynamic internal field are strongly inhomogeneous. Spin waves propagating along the field are affected in this case not only by the confinement effects but also by the above inhomogeneities. Since in unconfined media a wave with q||MS is called the backward volume (BV) magnetostatic wave, we refer to this experimental geometry as the BV geometry, contrary to the geometry described in the previous subsection, which we call the DE geometry. Figure 10 shows two typical BLS spectra obtained from an array of stripes for the external in-plane magnetic field He = 500 Oe for different orientations of the field. The spectrum (a) is recorded with both transferred wavevector and He aligned along the width of the stripe (the BV geometry), whereas the spectrum (b) is obtained for He oriented along the stripe axis, thus presenting the DE geometry 6 Spin Waves 297 LM BLS Intensity (a.u.) Band (a) Quantized DE - Modes PSSW (b) 0 5 10 15 Frequency shift (GHz) 20 Fig. 10 BLS spectra obtained on the stripe array described in the text, q = 0.3 × 105 cm−1 and He = 500 Oe for (a) the DE geometry and (b) the BV geometry. LM indicates the localized mode. (Reprinted with permission from [23], © 2002 by the American Physical Society) discussed in detail above. As it is seen in Fig. 10, both spectra contain several distinct peaks corresponding to spin-wave modes. The high-frequency peaks can easily be identified as exchange-dominated PSSW modes. Thus, a narrow PSSW peak in spectrum (b) confirms the homogeneity of the static internal field in the DE geometry. On the contrary, a broad PSSW peak in spectrum (a) clearly indicates a strong inhomogeneous distribution of the internal field across the stripe in the BV geometry. For a large He the magnetization is parallel to the applied field within almost the entire stripe. Therefore, poles are created at the edges of the stripe, which decrease the internal magnetic field in those regions. A detailed analysis shows that the internal static field, Hi , has a broad maximum in the center of the stripe while it is vanishing completely near the edges of the stripe [49, 50]. To get further inside into the physics of the observed spin-wave modes, their dispersion was measured by varying q. It is displayed in Fig. 11 for both orientations of He . Figure 11a representing the DE geometry is very similar to Figs. 8 and 9: it clearly demonstrates the lateral quantization of the DE spin waves, resembling a typical “staircase” dispersion. The frequency of the PSSW mode coincides with that of the PSSW mode for the unpatterned film and corresponds to an internal field of Hi = He = 500 Oe, thus corroborating a negligible static demagnetizing field in the stripes magnetized along their long axes. The dispersion presented in Fig. 11b and representing the BV geometry differs completely from that shown in Fig. 11a. First, the PSSW mode is split into two modes, with frequencies 298 S. O. Demokritov and A. N. Slavin Fig. 11 Spin wave dispersion of the stripe array measured at He = 500 Oe for (a) the DE geometry with the quantization numbers of the quantized modes as indicated and (b) the BV geometry. In the latter case, the shadowed region represents the band of non-localized spin modes, whereas LM indicates the localized mode. The solid lines represent the results of calculation. (Reprinted with permission from [23], © 2002 by the American Physical Society) corresponding to internal fields of Hi = 300 Oe and Hi = 0 Oe, respectively, in agreement with the above qualitative discussion. Second, a broad peak is seen in the spectra in the frequency range 5.5–7.5 GHz over the entire accessible interval of q. The shape of the peak varies with q, thus indicating different contributions of unresolved modes to the scattering cross-section at different q. Third, a separate, low-frequency, dispersionless mode with a frequency near 4.6 GHz (indicated as “LM” in Figs. 10 and 11b) is observed over the entire accessible wave vector range (qmax = 2.5 × 105 cm−1 ) with almost constant intensity. This is a direct confirmation of a strong lateral localization of the mode within a region with the width z = 2π /qmax = 250 nm. From the low frequency of the mode, one can conclude that it is localized near the edges of the elements, where the internal field vanishes. A quantitative analytical description of the spin-wave modes observed in the BV geometry is made as follows. The frequency of the spin wave as a function of q and H is given by Eq. (4). Contrary to the DE geometry, where H = He = Hi , the demagnetizing effects in the BV geometry are very large: Hi is strongly inhomogeneous and differs from He . To evaluate Hi : 6 Spin Waves 299 Hi (x, y, z) = He − N (x, y, z) · 4π MS (19) where Nzz (x, y, z) is the demagnetizing factor. Here we assume a Cartesian coordinate system, in which the x-axis is perpendicular to the plane of the elements, the y-axis is along the long axes of the stripes, and the z-axis is along the width of the stripe: He ||ez . The field profile of the internal field H(z) obtained from Eq. (19) for He = 500 Oe is shown in the inset of Fig. 12. For Hi > 0 the magnetization is parallel to He . Near the edges, however, regions with Hi = 0 and with continuously rotating magnetization are formed [49, 50], which reflect spin waves propagating from the middle of the stripe towards these regions. Moreover, a spin wave propagating in an inhomogeneous field might encounter the second turning point even if the magnetization is uniform. In fact, for large enough values of the internal field there are no allowed real values of q, consistent with the spin-wave dispersion (Eq. 4) – a potential well for propagating spin waves is created. Similar to the potential well in quantum mechanics, the conditions determining the frequencies fr of possible spinwave states in the well created by the inhomogeneous internal field are as follows: 2 q (H (z), f ) dz + ψ1 + ψ2 = 2rπ (20) where r = 1,2,3, . . . and q(H(z),f ) is found from the spin-wave dispersion Eqs. (4) and (5), and ψ l , ψ 2 are the phase jumps at the left and right turning points, between which Eq. (4) has a real solution q(z) for a fixed frequency f. We will illustrate these ideas in the following. The dispersion curves for spin waves with q||He and p = 0 calculated using Eqs. (4) and (5) for different constant values of the field are presented in Fig. 12. A dashed horizontal line shows the frequency of the lowest spin-wave mode f1 = 4.5 GHz obtained from Eq. (20) for the lowest value r = 1 in good agreement with the experiment. It can be seen from Fig. 12 that for H > 237 Oe there are no spin waves with the frequency f1 = 4.5 GHz. Therefore, the lowest mode can exist only in the spatial regions in the magnetic stripe where 0 Oe < H < 237 Oe. The corresponding turning points are indicated in the inset of Fig. 12 by the vertical dashed lines. Thus, the lowest mode is localized in the narrow region z near the lateral edges of the stripe where 0.26 < |z/w| < 0.39. The mode is composed of exchange-dominated plane waves with qmin < q < qmax , as indicated in Fig. 12. The higher-order spin-wave modes with r > 1 having their frequencies above 5.3 GHz are not strongly localized under the used experimental conditions and exist everywhere in the stripe where the internal field is positive (0 < |z/w| < 0.39). In the experiment, they show a band, since the frequency difference between the fr and fr + 1 modes is below the frequency resolution of the BLS technique. Note here that several localized modes can be observed at higher values of He [51]. 300 S. O. Demokritov and A. N. Slavin Fig. 12 Dispersion of plane spin waves in the BV geometry at constant internal fields as indicated. Inset: the profile of the internal field in a stripe. z shows the region of the lowest mode localization. (Reprinted with permission from [23], © 2002 by the American Physical Society) Implementation of Micro-Focus BLS for Laterally Patterned Magnetic Systems All the above BLS data were obtained in the so-called Fourier microscope mode [48]. In this case, the diameter of the laser beam was kept quite large (typically 30–50 μm), allowing fulfilling the wavevector conservation law Eq. (15), and the frequencies and the intensities of the studied spin-wave modes were investigated as a function of transferred wavevector. A complimentary approach is the micro-focus BLS [46]. Here the coherent laser light focused onto the surface of the magnetic system into a diffraction-limited spot by using a high-quality microscope objective lens with a large numerical aperture. While the frequency shift of the scattered light is equal to the frequency of the magnetization oscillations, its intensity (referred to as BLS intensity) is proportional to the intensity of magnetization oscillations at the position of the probing spot. The latter fact enables direct spatial imaging of the spin-wave intensity by twodimensional rastering of the probing spot over the sample surface (Fig. 13a). The acquired intensity maps, such as that shown in Fig. 13b, allow one to obtain information about the spatial characteristics of spin waves. In the case of spin-wave 6 Spin Waves 301 a) Microwave current Probing light Antenna H0 Magnonic waveguide Substrate b) 1 -1 10 BLS intensity, a.u. c) -2 10 1 cos (ϕ) 0 500 nm -1 Fig. 13 (a) Sketch of a typical micro-focus BLS experiment on spin-wave propagation in a microscopic waveguide. (b) and (c) Representative examples of the two-dimensional maps of the spin-wave intensity (b) and phase (c) recorded by micro-focus BLS. (© 2015 IEEE. Reprinted, with permission, from [47]) beams propagating in waveguides, they also provide important information about the damping and the spatial characteristics of the spin-wave beam. For reliable two-dimensional imaging of spin waves, the spatial resolution of the micro-focus BLS apparatus is of crucial importance, which is found to be about 250 nm for the wavelength of 532 nm in agreement with classical optics. This resolution can be further improved to about 50 nm [52] by utilizing the principles of near-field optical microscopy. However, the use of this approach inevitably leads to a reduction of the sensitivity, which noticeably increases the time needed for BLS measurements. Therefore, the experiments described here were performed by using free-space-optics micro-focus BLS apparatus with the resolution of 250 nm, which is sufficient to address most of the spin-wave propagation phenomena. In agreement with the discussion in Sect. “BLS in Laterally Confined Systems”, high-spatial resolution of the micro-focus BLS technique is incompatible with the wavevector resolution due to the large uncertainty of the light-scattering angle associated with the tight focusing of the probing light. Thus, the information about the wavevector of the wavelength of the studied spin waves is lost, and the spinwave dispersion cannot be obtained on the usual way. However, this drawback can be eliminated by making use of the time invariance of the BLS-process, which results in frequency/phase conservation in the light-scattering process. It means 302 S. O. Demokritov and A. N. Slavin that the phase of the scattered light is directly correlated to the phase of the magnetization oscillations in the spin-wave mode. This phase information can be acquired by utilizing interference of the scattered light with the light modulated by the signal used to excite the magnetization oscillations. This approach enables direct measurements of the phase difference between the excitation signal and the phase of a propagating spin wave at the given spatial location, providing, for example, direct information about the spin-wave wavelength. The phase-resolution technique was first demonstrated for standard macro-BLS apparatus [53] and was subsequently adapted for micro-focus BLS measurements [54, 55]. Figure 13c shows a representative example of the measured spatial phase map for spin waves propagating in a submicrometer-width magnonic waveguide, excited by microwave current in the antenna. The plotted value is cos(ϕ), where ϕ is the phase difference between the microwave current and the magnetization oscillations in the spin wave. The spatial period of cos(ϕ)-function is equal to the spin-wave wavelength at a given excitation frequency. Therefore, by varying the latter and measuring the spatial period, one can obtain the complete information about the spin-wave dispersion characteristics of the studied waveguide. Propagating Waves in 1D Magnetic Structures By analyzing the lateral quantization of spin waves in stripes in the previous subsections, we have considered standing waves, i.e., we implied that the component of the wavevector along the stripe axis is zero. To extend this approach to propagating waves in such a waveguide we should consider two components of the wavevector: one component quantized due to finite width of the stripe and another component continuously varying as illustrated in Fig. 14. In fact, if the quantized components qzn are known, the spectrum of normal waveguide modes can be obtained from the two-dimensional dispersion surface described by Eq. (4) and by cutting it along qy at the fixed qzn as illustrated in Fig. 14a by the curves labeled as DE1 − DE3 . For the sake of clearness we project these curves onto the frequency- qy plane, as shown in Fig. 14b, keeping in mind that the different curves correspond to different qzn . As seen from Fig. 14b, the considered modes propagate perpendicular to He , i.e., they are analogues of the DE mode in an extended film, their dispersion curves are shifted to lower frequencies with respect to that of the unconfined DE mode, and this shift increases with the increase of the mode number. This is not surprising, since all these modes are characterized by a nonzero component of the wavevector qz parallel to He . Since the dipolar magnetic energy is known to decrease with the increase of this component causing the backward dispersion of the BV modes (Fig. 14a), the dispersion curves of the waveguide modes shift to lower frequencies with the increase of the mode number, which corresponds to the increase in qz . We emphasize that the above-described approach to calculation of the dispersion curves of the normal waveguide modes is a rough approximation, since they do not take into account the reduction of the static magnetic field inside the waveguide 303 a) z y DE3 DE DE1 f0 DE2 DE1 DE2 BV qy DE3 H0 qz b) Frequency, GHz Fig. 14 (a) Two-dimensional dispersion spectrum of spin waves in an extended in-plane magnetized ferromagnetic film. Inset shows the geometry of the stripe waveguide and transverse profiles of the dynamic magnetization for normal waveguide modes. (b) Calculated (solid lines) and measured (symbols) dispersion curves for a waveguide with the width w = 800 nm and the thickness d = 20 nm magnetized by the static field He = 900 Oe. Dashed line shows the dispersion curve for Damon-Eshbach mode in an extended film. (© 2015 IEEE. Reprinted, with permission, from [47]) Frequency 6 Spin Waves 10 DE1 DE DE2 f1 9 DE3 8 q2 q1 0 1 2 3 -1 4 q3 5 6 qy, μm caused by the demagnetization effects, which can be further taken in account using Eq. (19). The data of Fig. 14b show that the dispersion spectrum of waveguide modes supports multimode propagation of spin waves at all frequencies above f0 . For example, by exciting spin waves at the frequency f1 (see Fig. 14b), one simultaneously excites a number of modes with different longitudinal wavevectors qy . Neglecting attenuation of spin waves, the spatial distribution of the intensity of the dynamic magnetization in these patterns can be described as (cf. Eq. 18): I (y, z) = n w nπ z+ An sin w 2 exp −iq n y 2 (21) where An are the amplitudes of the modes and qn are their longitudinal wavevectors at the given excitation frequency (see Fig. 14b). Figure 15 shows the results of calculations based on Eq. (21) performed for different ratios between the amplitudes An and the dispersion data taken from Fig. 14b. In the simplest case, where the only present mode is the fundamental mode with n = 1 (Fig. 15a), the intensity distribution is uniform in the longitudinal direction and shows a half-sine profile in the transverse direction. The co-propagation of the fundamental mode and the mode with n = 2 (Fig. 15b) results in an appearance of a “snake”-like pattern, which becomes more pronounced with the increase of A2 . We note that, because of the symmetry reasons, the mode n = 2 possessing an antisymmetric distribution of the dynamic magnetization across the waveguide width (inset in Fig. 14a) is 304 z, μm a) A1=1, A2=0, A3=0 0.4 0 -0.4 z, μm b) A1=1, A2=0.15, A3=0 0.4 0 -0.4 z, μm Fig. 15 Interference patterns for the three lowest-order waveguide modes calculated for different ratios between their amplitudes, as labeled. Calculations were performed for the waveguide with the width of 800 nm and the thickness of 20 nm magnetized by the field He = 900 Oe. (© 2015 IEEE. Reprinted, with permission, from [47]) S. O. Demokritov and A. N. Slavin A1=1, A2=0.3, A3=0 0.4 0 -0.4 z, μm c) A1=1, A2=0, A3=0.15 0.4 0 z, μm -0.4 A1=1, A2=0, A3=0.3 0.4 0 -0.4 0 1 2 y, μm 3 4 5 normally not excited in axially symmetric guiding systems and can only be observed if this symmetry is broken. In contrast, a significant contribution of the mode with n = 3 is detected in most of the experiments. As seen from Fig. 15c, the co-propagation of this mode and the fundamental mode of the waveguide results in a periodic spatial beating pattern, where the spin-wave energy is periodically concentrated in the middle of the waveguide, while the transverse width of the spinwave beam shows a periodic modulation. By analogy with the light focusing in optics, this effect was given a name of “spin-wave focusing” [56]. Figure 16a shows a typical measured spin-wave intensity map for a 2.4 μm wide and 36 nm thick Py waveguide clearly demonstrating this effect (compare with Fig. 15c). In order to highlight the details of the interference pattern in Fig. 16a, the spatial decay of spin waves is numerically compensated by multiplying the experimental data by exp.(2y/ξ), where ξ is the spin-wave decay length – the distance over which the wave amplitude decreases by a factor of e. The latter is determined from the dependence of the BLS intensity integrated across the transverse waveguide section versus the propagation coordinate (solid symbols Fig. 16b). As seen from Fig. 16b, spin waves in the studied waveguide exhibit clear exponential decay characterized by ξ = 6.4 μm. Figure 16b also shows by open symbols the transverse width of the spin-wave beam versus the propagation coordinate, which can be used to quantitatively characterize the strength of the focusing effect. In particular, for the data of Fig. 16b, the modulation of the beam width caused by the focusing is equal to about 70%, while the smallest width observed at the focal point is equal to 0.65 μm. 305 Propagation z, μm a) 1 1 0 0.5 -1 0 0 1 2 3 4 5 6 7 BLS intensity, a.u. 6 Spin Waves y, μm 1.5 Integral intensity, a.u. 1 1.0 0.1 Beam width, μm b) 0.5 0 1 2 3 4 5 6 7 y, μm Fig. 16 (a) Measured map of the spin-wave intensity for a waveguide with the width of 2.4 μm and the thickness of 36 nm magnetized by the field of 900 Oe. Excitation frequency is 9.4 GHz. Spatial decay of spin waves is numerically compensated. (b) Solid symbols – BLS intensity integrated across the transverse waveguide section versus the propagation coordinate in the loglinear scale. Line is the exponential fit to the experimental data. Open symbols – transverse width of the spin-wave beam measured at one half of the maximum intensity versus the propagation coordinate. (© 2015 IEEE. Reprinted, with permission, from [47]) In the above discussion on the normal waveguide modes, we neglected the nonuniformity of the magnetic field inside the waveguide caused by the demagnetization effects. On one side, this nonuniformity does not qualitatively affect the structure of the modes evolving from plane spin waves due their geometrical confinement. As discussed above, the nonuniformity is known [49] to result in the appearance of the regions of strongly reduced internal field close to the edges of the stripe [50], which gives rise to additional spin-wave modes having no analogue in the case of extended magnetic films [23]. According to [49], the distribution of the internal field across the width of a magnetic stripe magnetized perpendicular to its axis can be approximated as (cf. Eq. 18): d d 4π MS atan − atan Hi (z) = He − π 2z + w 2z − w (22) Figure 17a shows this distribution calculated for the waveguide with the width of w = 2.1 μm and the thickness of d = 20 nm magnetized by the static field He = 1100 Oe. As seen from this data, close to the edges of the waveguide, the internal field is drastically reduced resulting in the appearance of field-induced channels, where spin waves can be localized, as discussed in previous subsection. 306 S. O. Demokritov and A. N. Slavin Internal field, Oe a) 1000 500 Field-induced channels 0 -1.0 -0.5 0.0 0.5 1.0 z, μm z, μm 1 Propagation 9.8 GHz 0 1 -1 z, μm 1 0.5 9.0 GHz 0 0 BLS intensity, a.u. b) -1 Distance between the beams, μm c) 4 y, μm 6 8 10 1.5 0.6 1.0 0.5 Beam width, μm 2 0 0.4 0.5 9.0 9.2 9.4 9.6 9.8 Frequency, GHz Fig. 17 (a) Calculated distribution of the internal static magnetic field across the width of a waveguide with the width of 2.1 μm and the thickness of 20 nm magnetized by the static field of 1100 Oe. Horizontal dashed line marks the value of the external magnetic field. (b) Measured maps of the spin-wave intensity for two excitation frequencies, as labeled. Spatial decay of spin waves is numerically compensated. (c) Distance between the centers of the spin-wave beams and their transverse width measured at one half of the maximum intensity versus the spin-wave frequency. (© 2015 IEEE. Reprinted, with permission, from [47]) Since these modes are mostly concentrated in the areas of the reduced field, their typical frequencies are lower than the frequencies of the “center” modes as discussed above. Therefore, in order to address them experimentally, one needs to excite the waveguide at frequencies below the frequency of the uniform ferromagnetic resonance f0 (see Fig. 14a). Figure 17b shows two decay-compensated spin-wave intensity maps measured for a waveguide with the parameters given above by applying the excitation signal at the frequencies of 9.0 and 9.8 GHz, which are smaller than f0 equal to about 10 GHz for the used experimental conditions. 6 Spin Waves 307 The data of Fig. 17b show that, in agreement with the simple qualitative model, at these frequencies, the spin waves do not occupy the entire cross-section of the waveguide. Instead, they form two narrow beams with the submicrometer width whose spatial positions depend on the spin-wave frequency. The quantitative analysis (see Fig. 17c) shows that, in the wide frequency interval, the widths of the beams vary moderately staying in the range 400–500 nm, while the distance between their centers monotonously increases with the decrease of the frequency from 0.8 μm to 1.4 μm, i.e., by more than 70%. Control and Conversion of the Propagating Waves One of the great advantages of spin waves for implementation of signal-processing devices is their controllability by the static magnetic field, which allows one to efficiently manipulate the spin-wave propagation. Although this control mechanism is straightforward, its implementation on the macroscopic scale requires the use of electromagnets making the resulting devices extremely space and power consuming. The downscaling of spin-wave devices provides a route for overcoming this drawback, since, in microscopic systems, the control magnetic field has to be created over small distances and sufficiently large local magnetic fields can be created by using relatively small electric currents [55]. This approach is schematically illustrated in Fig. 18a. Instead of using external electromagnets, the control magnetic field is created by the electric current flowing in a control line, which is directly integrated into the waveguide. The composite waveguide consists of two layers: the upper 20 nm thick Py layer guiding spin waves and the bottom 100 nm thick Cu layer used as a current-carrying line to generate controlling magnetic fields. Because of the large difference in conductivities of Cu and Py, the shunting of the control current through the Py layer is negligible, which makes electrical isolation between the two layers unnecessary. The Oersted field in the Py layer produced by the current in the Cu layer can be approximated as H = I/(2w + 2d), where I is the current strength and w = 0.8 μm and d = 0.1 μm are the width and the thickness of the Cu line, respectively. Due to the small cross-section of the control line, one achieves sufficiently high efficiency of the magnetic field generation of about 7 Oe/mA. The effect of the current on the dispersion characteristics of spin waves in the waveguide is illustrated by Fig. 18b. By applying I = ± 12 mA one creates controlling magnetic field H = ± 83 Oe, which adds or subtracts from the static magnetic field He = 520 Oe resulting in the shift of the dispersion curve by more than ±500 MHz. This shift leads to a significant variation of the longitudinal wavevector qy at the given spin-wave frequency. Figure 18c demonstrates the controllability of the wavevector and the wavelength of sin waves with the frequency of 7 GHz. These data show that by applying I = ±12 mA one can change these parameters by more than 50%. Note that the variation of qy with the current is nearly linear. Since the phase accumulated by the spin wave over a propagation distance L is proportional to qy : ϕ = Lqy , this also implies a linear controllability of the phase accumulation, which is attractive 308 z a) y He Py (20 nm) DH dc Frequency, GHz 8 I=12 mA 7 6 I=-12 mA 5 I=0 0 c) 1 2 3 4 -1 qy, μm 5 6 5 2.5 qy, μm 2.0 4 1.5 3 Wavelength, μm b) Cu (100 nm) -1 Fig. 18 (a) Schematic of a composite waveguide with the integrated control Cu line. (b) Dispersion curves of the fundamental waveguide mode for different currents in the control line, as labeled. Lines show the calculated dispersion curves, and symbols show the results of measurements by phase-resolved BLS technique. (c) Longitudinal wavevector qy and the wavelength of the spin wave at the frequency of 7 GHz versus the control current. Symbols – experimental data, lines – guides for the eye. (© 2015 IEEE. Reprinted, with permission, from [47]) S. O. Demokritov and A. N. Slavin 1.0 -15 -10 -5 0 5 10 15 Current, mA for technical applications. Based on the data of Fig. 18c, one can conclude that by applying the control current of ±12 mA, the phase accumulated by the spin wave can be changed by ±π radians at the propagation distance of about 3.2 μm which is smaller than the spin-wave propagation length. This makes the proposed mechanism well suited for implementation of magnonic logic devices [57], where the digital information is coded into the phase of propagating spin waves. In addition to the use of magnetic fields created by electric currents, the control of spin-wave propagation can also be realized by using demagnetizing fields. Since the demagnetizing field depends on the ratio between the width and the thickness of the waveguide (Eq. 22), simple variation of one of these parameters enables efficient manipulation of spin waves [58, 59]. A waveguide with the varying width is schematically shown in Fig. 19a: while the thickness of the waveguide d = 36 nm remains constant, its width w varies from 1.3 to 2.4 μm over a transition region with the length L. According to Eq. (22), such a variation results in a spatial variation of the internal field, which changes from 870 Oe in the narrow part of the waveguide to 950 Oe in the wide part (Fig. 19b). Due to this variation, the dispersion curves of the fundamental center waveguide mode are shifted in the two parts by about 500 MHz in the frequency domain (Fig. 19c). If the excitation frequency is chosen to be located between the 6 Spin Waves 309 b) y Internal field, Oe 1.3 μm He L 8.7 GHz e) L=3 μm Frequency, GHz 0 1 11 2 3 4 y, μm 5 6 Wide waveguide f2 10 Narrow waveguide f1 9 0 d) 900 850 2.4 μm c) 950 Wide waveguide z Narrow waveguide a) 1 2 -1 qy, μm Propagation 3 9.1 GHz 1 μm L=1 μm Fig. 19 (a) Schematic of a spin-wave waveguide with a varying width. (b) Calculated distribution of the internal static magnetic field in the section along the axis of the waveguide with the thickness of 36 nm and the geometrical parameters given in (a). The external static magnetic field He = 1000 Oe. (c) Calculated dispersion curves for the fundamental center waveguide mode in the wide and the narrow parts of the waveguide. (d) Maps of the spin-wave intensity measured at the excitation frequencies of 8.7 and 9.1 GHz, as labeled. Width of the transition region L = 2 μm. (e) Maps of the spin-wave intensity measured at the excitation frequency of 9.7 GHz in waveguides with L = 3 and 1 μm, as labeled. In (d) and (e) the spatial decay of spin waves is numerically compensated. (© 2015 IEEE. Reprinted, with permission, from [47]) cut-off frequencies of the two dispersion curves, i.e., between 8.7 and 9.2 GHz (f1 in Fig. 19c), the center mode propagating in the narrow part of the waveguide cannot pass into the wide part. Instead, it should be transformed into the edge mode, whose frequency range is located below that of the center modes. This case is illustrated in Fig. 19d showing two spin-wave intensity maps measured for the excitation frequencies of 8.7 and 9.1 GHz. These maps clearly demonstrate the conversion of the center mode into the edge mode characterized by two narrow spin-wave beams with frequency-dependent spatial separation (see Sect. “Implementation of Micro-Focus BLS for Laterally Patterned Magnetic Systems”). Since the two beams 310 S. O. Demokritov and A. N. Slavin propagate in the field-induced channels and are independent from each other, the observed transformation can be used for implementation of a spin-wave splitter. One also observes interesting behaviors in the case, when the frequency of spin waves is larger than cut-off frequencies in both parts of the waveguide (f2 in Fig. 19c). In the waveguides with the relatively long transition region (L = 3 μm in Fig. 19d) the propagation of spin waves from the narrow to the wide part is quasi-adiabatic. It is only accompanied by the increase in the wavelength, while the spatial structure of the spin-wave beam remains unchanged. However, in systems with shorter transitions (e.g., L = 1 μm in Fig. 19e), the propagation is accompanied by an appearance of a complex intensity pattern, which can be recognized as an interference pattern created by several waveguide modes with comparable amplitudes (see Fig. 15c). This is due to the strong coupling of the waveguide modes mediated by the spatial nonuniformity in the waveguide, which results in the efficient energy transfer from the fundamental mode to the higherorder modes. As seen from Fig. 19e, this effect causes a strong concentration of the spin-wave energy in the middle of the waveguide at a certain distance from the transition region, which can be treated as an enhanced spin-wave focusing. A particularly interesting case is a junction between a 1D waveguide and 2D film [60, 61] (Fig. 20a). Because of the abrupt transition in such a system, the wavevector of spin waves is not conserved during the conversion of the waveguide modes into the modes of the extended film. In other words, being radiated from the open end of the waveguide, the waveguide mode excites spin waves within a large range of wavevectors. Because of the temporal translation symmetry, the frequency of radiated spin waves is equal to that of the waveguide mode. Therefore, the characteristics of the radiated waves can be obtained by considering constantfrequency contours of the two-dimensional dispersion surface (Fig. 14a), as shown in Fig. 20b. Figure 20c shows such contours projected onto the qz − qy plane, calculated for the conditions used in the experiment: d = 36 nm, He = 690 Oe. In this representation the vector of the group velocity of spin waves Vg is directed along the normal to the constant-frequency contour: Vg = 2π ∇ f (qy , qz ). As seen from Fig. 20c, except for the region of small qz , the direction of the group velocity is practically constant and builds a well-defined angle with the direction of the static magnetic field He . This indicates that a large group of spin waves with different wavevectors transmits energy in the same direction. Therefore, one expects predominant radiation of the spin-wave energy from the waveguide along this direction. Figure 20d shows an experimental spin-wave intensity map corresponding to the frequency of 8.2 GHz. The shown experimental data confirm the conclusions of the above analysis: the spin waves are radiated in a form of two narrow beams and their directions coincide well with the direction of the group velocity obtained from the analysis of the 2D dispersion surface (arrows in Fig. 20d). Since the direction of the beams is determined by He , it can be electronically steered. In general, the phenomenon enables a directional, 1D transmission of spin waves in an extended 2D magnetic film without utilization of the geometrical confinement. Recently, this phenomenon was also observed for spin waves radiated by a spin-torque nanooscillator [62], which, due to its small size, also emits spin waves within large interval of wavevectors. 311 a) b) z y Waveguide Frequency 6 Spin Waves DE f2 f1 f0 He BV f0 qy Extended film qz 8.2 GHz qz, μm -1 c) 10.2 GHz d) 4 Waveguide Vg 2 0 He -2 -4 0 2 4 6 1 μm -1 qy, μm Fig. 20 (a) Schematic of a junction between 1D waveguide and 2D extended film. (b) Twodimensional dispersion surface of spin waves in an extended magnetic film with marked constantfrequency contours. (c) Constant-frequency contours projected onto the qz -qy plane calculated for t = 36 nm and He = 690 Oe. (d) Measured map of the intensity of spin waves with the frequency of 8.2 GHz radiated from a waveguide with the width of 2 μm into an extended magnetic film. The spatial decay of spin waves is numerically compensated. (© 2015 IEEE. Reprinted, with permission, from [47]) Inductive Excitation of Spin Waves in 1D Waveguides In spite of the recent progress in studies of spin-transfer torque excitation of propagating spin waves (see Sect. “Spin Waves in 0D”), the inductive excitation mechanism shortly introduced above still remains the most widely used in experimental investigations of spin-wave phenomena in both 2D and 1D magnetic systems, because its implementation does not require complex nanolithography techniques. This method is also characterized by the full control over the frequency of excited spin waves, which makes it attractive for research purposes. The inductive excitation by means of spin-wave antennae was widely used in the past for implementation of macroscopic-scale devices (see, e.g., [63, 64]) and was subsequently transferred onto the microscopic scale without significant modifications. Figure 21a shows schematics of a spin-wave waveguide with a spin-wave antenna on top. A microwave-frequency electric current transmitted through the antenna creates the dynamic magnetic field h, which couples to the dynamic magnetization in the waveguide and excites propagating spin waves. Experimentally, the excitation 312 S. O. Demokritov and A. N. Slavin process can be efficiently characterized by micro-focus BLS by placing the probing laser spot onto the waveguide in the vicinity of the antenna and recording the BLS intensity as a function of the frequency of the excitation current. Such an excitation curve is shown in Fig. 21b. The curve exhibits a fast drop of the spin-wave intensity at frequencies below the frequency of the ferromagnetic resonance f0 . However, the intensity of excited spin waves still remains noticeable in this region and shows a tail extending far into the low-frequency spectral interval. Within this frequency region, propagating edge modes are excited. The part of the excitation curve at frequencies above f0 corresponding to the center waveguide modes shows a nonmonotonous behavior with several oscillations. This oscillatory behavior can be understood based on the spin-wave excitation theory adapted for the case of microscopic waveguides [65]. According to this theory, the amplitudes of the waveguide modes An excited by the antenna are determined by the spatial overlap of the dynamic magnetic field h(x,y,z) created by the antenna with the dynamic magnetization in the waveguide m(x,y,z). As seen from Fig. 21a, the former has x- and y-components. Both these components are perpendicular to the direction of the static magnetization and, therefore, can linearly couple to the dynamic magnetization. However, because of the strong dynamic demagnetization in the thin-film waveguides, the effect of the out-of-plane component hx is relatively small and can be neglected in the first approximation. Then the excitation problem is reduced to the consideration of the spatial overlap of the hy component with the corresponding component of the magnetization. Neglecting the variations of the field and the dynamic magnetization across the waveguide thickness, the amplitudes of excited spin-wave modes can be expressed as: w/2 An ∝ ∞ hy (z)mny (z)dz −w/2 · hy (y)mny (y)dy (23) −∞ where w is the width of the waveguide. The corresponding profiles of components of the field and the dynamic magnetization are schematically shown in the inset in Fig. 21a: hy (z) = const, and hy (y) can be approximated by a rectangular pulse function with the width equal to that of the antenna d. As discussed in Sect. “Propagating Waves in 1D Magnetic Structures” (Eq. 21), the transverse profiles of the dynamic magnetization corresponding to the center modes can approximated as mny (z) ∝ sin (nπ (z + w/2)), while the longitudinal profile represents a propagating wave: mny (y) ∝ exp −iq ny y , where qyn are the longitudinal wavevectors of the modes at the given spin-wave frequency. Substituting these expressions into Eq. (22) one obtains: n 1 − (−1)n sin qy b An ∝ n qyn (24) 6 Spin Waves 313 x a) Antenna Microwave current y hy(y) z b h He Magnetic waveguide n=2 n=1 hy(z) my(y) y n=3 n z my (z) b) BLS Intensity, a.u. Edge Modes DE Modes f0 8 qy, μm -1 c) 6 n=3 2π/d 4 n=1 2 0 An, a.u. d) 1.0 n=1 0.5 n=3 0.0 7 8 9 10 11 Frequency, GHz 12 13 Fig. 21 (a) Schematic of a waveguide with the spin-wave antenna on top. Inset schematically shows the y- and z-profiles of the dynamic field of the antenna and those of the dynamic magnetization in the spin-wave modes. (b) Spin-wave excitation curve measured in a waveguide with the width of 2 μm and the thickness of 36 nm magnetized by the static field of 900 Oe. (c) Calculated dispersion curves for the first two symmetric waveguide modes. Horizontal dashed line shows the value of qy , at which the excitation efficiency vanishes. Arrows show the corresponding frequencies for the two modes. (d) Calculated amplitudes of the first two symmetric waveguide modes versus the excitation frequency. The vertical dashed line in (b)–(d) marks the frequency of the uniform ferromagnetic resonance f0 . (© 2015 IEEE. Reprinted, with permission, from [47]) 314 S. O. Demokritov and A. N. Slavin The first term in Eq. (24) shows that the amplitudes of the modes decrease with the increase of the mode number as 1/n and that only modes with symmetric transverse profiles can be excited. The second term shows that the excitation efficiency has a maximum at qy = 0, exhibits an oscillatory behavior in agreement with the experimental data of Fig. 21a, and vanishes for qy = 2π m/b or b = mλ, where m = 1,2,3 . . . and λ is the wavelength of the spin wave. Figure 21d presents the frequency dependences of the amplitudes of the first two symmetric waveguide modes calculated based on Eq. (23) and the dispersion curves obtained by using Eqs. (4) and (5) (shown in Fig. 21c). The calculations were performed for the conditions used to acquire the experimental excitation curve shown in Fig. 21b: w = 2 μm, d = 36 nm, He = 900 Oe, and b = 1.5 μm. As seen from Fig. 21d, the fundamental mode with n = 1 clearly dominates over the mode with n = 3 and the nodes of the corresponding curve match well with those seen in the experimental data. We note that, due to the shift of the dispersion curves corresponding to different modes (Fig. 21c), the frequencies, at which the excitation efficiency vanishes, are different for the modes with n = 1 and 3. Since the contribution of the mode with n = 3 is relatively small, vanishings of its amplitude cannot be seen in the experimental curve in Fig. 21b. Nevertheless, this vanishing can be proven by the spatial imaging of the spin-wave propagation. These measurements show a strong reduction in the spatial transverse modulation of the spin-wave beam at the frequency of about 10 GHz in agreement with the data of Fig. 21d. We emphasize that the dependence of the relative amplitudes of the waveguide modes on the excitation frequency allows one to control the strength of the spin-wave focusing effect. If the transverse modulation of the spin-wave beam is not desired, one can choose the spin-wave frequency in the vicinity of the point of the vanishing excitation efficiency of the mode with n = 3, while by choosing the frequency in the region, where the excitation efficiencies of both modes are approximately equal, one obtains the strongest mode interference. The above-described simple model can be extended by additionally taking into account the out-of-plane component of the dynamic magnetic field of the antenna [65]. This extension does not significantly modify the discussed frequency dependence of the excitation efficiency. However, it results in different excitation efficiencies of spin waves propagating in the positive and the negative direction of the y-axis. This excitation non-reciprocity is often confused with the intrinsic nonreciprocity of DE modes [19], which has no effect for spin waves with qy < <1/d addressed in most of the experiments utilizing the inductive spin-wave excitation. For typical parameters of the Py micro-waveguides, the excitation non-reciprocity results in the factor of 4–5 difference between the intensities of waves propagating in the opposite directions from the antenna. The above analysis shows that by using an inductive spin-wave antenna one can only efficiently excite spin waves within a certain interval of wavevectors corresponding to the first oscillation lobe of the excitation efficiency function (Fig. 21d), whose spectral width is determined by the width of the spin-wave antenna b. This limits the applicability of standard inductive excitation, since, in order efficiently to excite short-wavelength spin waves necessary for most of the magnonic applications, one has to reduce the width of the antenna, which inevitably 6 Spin Waves 315 causes the microwave impedance matching problems and increases the microwave losses in magnonic devices. This drawback can be partially overcome [59] by utilizing the spin-wave controllability by the demagnetizing fields discussed in Sect. “Propagating Waves in 1D Magnetic Structures”. Figure 22a schematically shows a system enabling inductive excitation of spin waves with the wavelength smaller than the width of the antenna: the width of the waveguide equal to 2 μm in the excitation area gradually reduces to 0.5 μm, while the thickness of the waveguide d = 40 nm stays unchanged. As discussed in detail in Sect. “Propagating Waves in 1D Magnetic Structures”, the variation of the waveguide width leads to the variation of the internal field along the waveguide axis, which causes the shift of the dispersion curves of the waveguide modes in the frequency domain. As a result, spin waves excited in the 2 μm-wide part of the waveguide continuously decrease their wavelength propagating in the tapered part. This process is illustrated by the two-dimensional phase map (Fig. 22b) and its section along the waveguide axis (Fig. 22c) obtained by using the phase-resolved micro-focus BLS technique. The experimental data clearly show a significant reduction of the wavelength, which, after passing the tapered part, becomes smaller than the width of the antenna equal to 2.2 μm. Figure 22d further characterizes this wavelength conversion process. It shows the dependence of the spin-wave wavelength at the output of the tapered part on the wavelength in the excitation area. These data show that the discussed conversion process enables the reduction of the wavelength by up to a factor of 15 and allows one to efficiently excite spin waves with the wavelength below 1 μm, which is more than by a factor of 2 smaller than the antenna width. We also note that, as shown in [59], the conversion process is not accompanied by noticeable additional losses associated with the spinwave reflection in the transition region, which makes it favorable for technical applications. Spin-Torque Transfer Effect and Spin Waves Since the first demonstration [25, 27–29, 66, 67] of the possibility to excite magnetization dynamics by spin-polarized electric currents due to the spin-transfer torque (STT) effect [6, 7], dynamic spin-torque phenomena have become the subject of intense research. The ability to control high-frequency magnetization dynamics by dc currents is promising for the generation of microwave signals [68–72] and propagating spin waves [16, 30, 73–75] in magnetic nanocircuits. Electronically controlled local generation of spin waves is particularly important for the emerging field of nanomagnonics, which utilizes propagating spin waves as the medium for the transmission and processing of signals, logic operations, and pattern recognition on nanoscale [3–5]. Initially, STT phenomena have been studied in 0D-nanodevices based on the tunneling or giant magnetoresistance spin-valve structures, where STT is induced by the electric current flowing through a multilayer that consists of a “fixed” magnetic spin-polarizer and the active magnetic layer, separated by a nonmagnetic 316 S. O. Demokritov and A. N. Slavin a) 2 μm z b) y He Pr op ag ati on L 0.5 μm c) cos (ϕ) 1 2.2 μm 0 -1 cos (ϕ) 1 0 -1 0 2 4 6 8 10 d) Converted wavelength, μm y, μm 1.5 1.0 0 5 10 15 20 25 Excited wavelength, μm Fig. 22 (a) Schematic of a tapered waveguide with the spin-wave antenna located in the wide part. (b) Measured phase map of spin waves propagating in a 40 nm thick tapered waveguide magnetized by the static field He = 900 Oe. The length of the tapered part L = 5 μm. Excitation frequency is 9.8 GHz. (c) Section of the phase map along the waveguide axis. (d) Dependence of the spin-wave wavelength at the output of the tapered part on the wavelength in the excitation area. (© 2015 IEEE. Reprinted, with permission, from [47]) metallic or insulating spacer. In these structures, the electric charges must cross the active magnetic layer to excite its magnetization dynamics. To enable current flow through the active magnetic layers, STT devices operating with spin-polarized electric current require that current-carrying electrodes are placed both on top and on the bottom of the spin valve. To keep the electric current below a reasonable limit, the devices should have sub-100-nm dimensions in both lateral directions. An alternative approach to the implementation of STT devices that avoids these shortcomings utilizes pure spin currents – flows of spin not accompanied by directional transfer of electrical charge. This approach does not require the flow of electrical current through the active magnetic layer, resulting in reduced Joule heating and electromigration effects. One can also eliminate the electrical leads attached to the magnetic layer to drain the electrical current, enabling novel geometries and functionalities of the STT devices. Moreover, it becomes possible to use insulating magnetic materials such as Yttrium Iron Garnet (YIG) [12, 76]. 6 Spin Waves 317 Among the physical mechanisms for creation of pure spin currents, the spinHall effect (SHE) [9–11, 77] plays the most important role so far. The effect is generally significant in nonmagnetic materials with strong spin-orbit interaction, such as Pt and Ta. An electrical current in these materials produces a spin current in the direction perpendicular to the charge flow, due to a combination of spinorbit splitting of the band structure (intrinsic SHE), and the spin dependence of the electron scattering on phonons and impurities (extrinsic SHE) [11, 77]. When a SHE layer is brought in contact with a ferromagnetic film, the spin current flows through the interface into the ferromagnet and exerts STT on its magnetization [78]. The ability to exert STT on ferromagnets over extended areas is a significant benefit of SHE. Indeed, when an in-plane current flows through an extended bilayer formed by a SHE material and a magnetic film, the spin current produced by SHE is injected over the entire area of the sample, which can be as large as several millimeters [78]. This feature makes SHE uniquely suited for the control of the spatial decay of propagating spin waves in 1D waveguides, when STT partially compensates the natural magnetic damping. The effect of pure spin current on the magnetization is similar to that of spin-polarized electric currents. Both can be described by the Slonczewski-LandauLifshitz-Gilbert equation [8]: dM β α dM = −γ M × H eff + + 2 M × M × ŝ M× dt Ms dt Ms (25) where α is the Gilbert damping parameter, β is the strength of the spin-transfer torque proportional to the spin current density, and ŝ is the unit vector in the direction of the spin-current polarization. All other notations are similar to those of Eqs. (1) and (2). In fact, Eq. (25) is an extension of Eq. (1), which takes into account magnetic damping and the STT effect. Within this model, the third term is mathematically very similar to the second one. Correspondingly, one expects that spin current with an appropriate polarization can reduce magnetic damping of propagating spin waves [78]. The above spin-wave damping compensation has been recently demonstrated experimentally [79]. Figure 23 shows the schematic of the test devices. They are based on a 20 nm thick YIG film grown by the pulsed laser deposition on Gadolinium Gallium Garnet (GGG) (111) substrate. The film is covered by an 8 nm thick layer of Pt deposited using dc magnetron sputtering and the YIG/Pt bilayer is patterned by e-beam lithography into a stripe waveguide with the width of 1 μm. The system is insulated by a 300 nm thick SiO2 layer, and a broadband 3 μm wide microwave antenna made of 250 nm thick Au is defined on top of the system by the optical lithography. The waveguide is magnetized by the static magnetic field He = 1000 Oe applied in its plane perpendicular to the long axis. A dc electrical current I flowing in the plane of the Pt film is converted by the SHE into the transverse spin accumulation (see inset in Fig. 23). The associated pure spin current IS is injected into the YIG film resulting in a spin-transfer torque on its magnetization. Depending on the relative orientation of the current and the 318 S. O. Demokritov and A. N. Slavin Pt Microwave current Spin-wave antenna YIG I IS He M Spin wave I z x y Probing laser light Pt (8 nm) / YIG (20 nm) waveguide GGG substrate Fig. 23 Schematic of the experiment on compensation of the spin-wave damping by pure spin current. Inset illustrates the generation of the pure spin current by the spin-Hall effect. (Reprinted from [79], with the permission of AIP Publishing) static magnetic field, the spin current either compensates or enhances the effective magnetic damping in the YIG film. The effects of spin current on the decay length of propagating spin waves were performed by applying a microwave signal at the frequency corresponding, for the given conditions, to a spin wave with the wavelength of about 5 μm, which can be efficiently excited by the used 3 μm wide inductive antenna and possess sufficiently large group velocity. The propagation of spin waves was mapped by rastering the probing laser spot over the surface of the YIG waveguide with step sizes of 200 and 250 nm in the transverse and the longitudinal directions, respectively. Figure 24a shows a representative map of the BLS intensity, proportional to the local intensity in the spin wave, obtained for I = 2.55 mA. As seen from these data, the spin wave propagates along the waveguide nearly uniformly without changing its transverse profile (inset in Fig. 24a), which is a clear signature of the singlemode propagation regime caused by the strong separation of the transverse modes in a narrow waveguide. The intensity of the wave decreases by only 60% over the propagation path of 10 μm. To characterize the decay length of spin waves and its dependence on the current, we plot in Fig. 24b the dependences of the spin-wave intensity on the propagation coordinate obtained for different dc currents in the Pt layer. These data show that spin waves in the waveguide experience well-defined exponential decay (note the logarithmic vertical scale) ∼ exp(−2y/ξ), where ξ is the decay length defined as a distance over which the wave amplitude decreases by a factor of e. As seen from the figure, the decay length strongly increases with the increase of the dc current, as expected for the effect of spin current on the effective magnetic damping. 6 Spin Waves 319 Fig. 24 (a) Normalized spatial intensity map of the propagating spin wave excited by the antenna. The map was recorded for I = 2.55 mA. The mapping was performed by rastering the probing spot over the area 1.6 by 10 μm, which is larger than the waveguide width of 1 μm. Dashed lines show the edges of the waveguide. Inset shows the transverse profile of the spin-wave intensity. (b) Dependences of the spin-wave intensity on the propagation coordinate for different currents, as labeled, in the log-linear scale. Lines show the exponential fit of the experimental data. (c) Current dependences of the decay length and the decay constant. Vertical dashed line marks IC . Solid line is the linear fit of the experimental data at I < IC . The data were obtained at He = 1000 Oe. (Reprinted from [79], with the permission of AIP Publishing) 320 S. O. Demokritov and A. N. Slavin Figure 24c summarizes the results of the spatially resolved measurements. The decay length (up-triangles) monotonously increases with the increase of I < IC and then shows an abrupt decrease at I > IC in contradiction to naive expectations that for large values of I the magnetic damping should be overcompensated by the spin current, and the propagating spin wave should be amplified. This experimental observation can be attributed to the strong nonlinear scattering of the propagating spin waves from large-amplitude current-induced magnetic fluctuations, which have been observed independently. To characterize the variation of the decay length with current in detail, we plot in Fig. 24c its inverse value – the decay constant (down-triangles), which is proportional to effective Gilbert damping constant αeff . In agreement with the simple theoretical model assuming the linear variation of αeff with current, the decay constant shows a linear dependence on I. By extrapolating this dependence to I = 0, we obtain the propagation length at zero current ξ0 = 2.4 μm. Additionally, one expects the linear dependence in Fig. 24c to cross zero at I = IC , which corresponds to an infinitely large decay length under conditions of the complete damping compensation. The data of Fig. 24c show, however, that the linear fit yields the intercept value larger than IC . This disagreement can be attributed to the Joule heating of the waveguide by the electric current in Pt resulting in the significant reduction of the effective magnetization. Since the decay length is proportional to the group velocity, which is known to decrease with the decrease in Meff , the effects of the heating on the propagation length counteract those of the spin current and do not allow one to achieve the decay-free propagation regime. We note, however, that the maximum achieved propagation length of 22.5 μm is nearly by a factor of two larger compared to the value of 12 μm estimated for a waveguide made of a bare YIG film without Pt on top (α = 5 × 10−4 ). Spin Waves in 0D The STT effect discussed above is of a particular importance for magnetic systems fully confined in all three directions. It is now well established that a spin-polarized electric current or, alternatively, a pure spin current, injected into a ferromagnetic layer through a nanocontact exerts a torque on the magnetization, leading to a strongly localized microwave-frequency precession of magnetization, which can be considered as a 0D spin-wave mode. This phenomenon can serve as a basis for the development of tunable nanometer-size microwave oscillators, the so-called spintorque nano-oscillators (STNO) [25, 27–29, 66, 67]. The density of magnetic energy in auto-oscillations excited by STT in a magnetic nanocontact could be very high. Therefore, the STT-induced precession modes are, usually, strongly nonlinear. Also, since the spin precession excited in a magnetic nanocontact is, usually, surrounded by a 2D film or is coupled to a 1D waveguide, it may radiate propagating spin waves. All these makes the phenomena connected with the STT-driven magnetization dynamics multifarious and intriguing. This section is devoted to the 0D spin-wave modes driven by spin-polarized electric or pure spin currents. 6 Spin Waves 321 Probing laser light Top electrode Current flow 200 nm Insulator 500 nm Py film Nanopillar Bottom electrode Fig. 25 Schematic of the studied STNO with an AFM image superimposed. The devices consist of an extended 6 nm thick Permalloy free layer and an elliptical nanopillar formed by a 9 nm thick Co70 Fe30 polarizing layer and a 3 nm thick Cu spacer. The nanopillar is located close to the edge of the top electrode enabling optical access to the free layer for BLS microscopy. Magnetic precession in the device is induced by dc current flowing from the polarizer to the free layer. The spatially resolved detection of spin waves is accomplished by focusing the probing laser light into a 250 nm spot, which is scanned over the surface of the Py film. (Reprinted from [31], with the permission of Springer Nature) Spin-Torque Nano-Oscillator (STNO) and Emitted Spin Waves Let us consider an STNO shown in Fig. 25 [30]. The device is formed by a nanocontact on an extended Permalloy (Py) film. The nanocontact is shaped as an elliptical nanopillar formed by the nanopattered polarizing Co70 Fe30 layer and a Cu spacer. A dc current I flowing from the polarizer to the Py film induces local magnetization oscillations in this film. The nanocontact is located within 200 nanometers from the edge of the top device electrode, enabling optical access to the Py film at larger distances. The spatially resolved detection of spin waves emitted by STNO was performed by micro-focus BLS spectroscopy, as described above. The probing laser light was focused onto the surface of the Py film and scanned in plane to record two-dimensional maps of the spin-wave intensity. The oscillation characteristics of STNOs were determined from the microwave signals generated due to the magnetoresistance effect as shown in Fig. 26. The plots of the power spectral density (PSD) illustrate the dependence of the oscillation frequency on the bias current I for three different angles ϕ between the in-plane bias magnetic field He = 900 Oe and the easy axis of the nanostructured polarizer. The microwave generation starts at an onset current I = 2.5–3.5 mA that depends on ϕ. The dependence of the generation frequency on current above the onset is caused by the nonlinear frequency shift, due to a combination of the demagnetizing 322 S. O. Demokritov and A. N. Slavin Frequency, GHz He He He 5° 8.2 8.2 8.2 8.0 8.0 8.0 7.8 7.8 7.8 7.6 7.6 7.6 7.4 7.4 7.4 7.2 7.2 2 3 4 I, mA 5 45° 25° 7.2 2 3 PSD, -5 pW/MHz 10 4 2 5 -4 10 10 -3 3 10 -2 4 10 5 -1 Fig. 26 Pseudo-color logarithmic maps of the power spectral density (PSD) of the signal generated by the device due to the magnetoresistance effect at different angles ϕ between the in-plane magnetic field and the easy axis of the nanopillar, as labeled. (Reprinted from [31], with the permission of Springer Nature) effects in Py and the dipolar field of the structured Co70 Fe30 polarizer. For small ϕ, the nonlinear shift is strongly negative. It becomes less pronounced with increasing ϕ and changes to positive at small I and ϕ > 20◦ . The region of positive nonlinear frequency shift is reduced at larger He and eventually disappears for He > 1200 Oe, suggesting its origin from the dipolar field of the polarizer. The possibility to control the nonlinear behaviors by varying the angle ϕ makes the studied STNOs uniquely suited for the analysis of the effects of the nonlinearity on the spin-wave emission. Figure 27 shows two-dimensional intensity maps of spin waves emitted by STNO at I = 5 mA, measured for different in-plane directions of the applied field He = 900 Oe. As seen in Fig. 27, the emission mainly occurs in the direction perpendicular to the in-plane field, regardless of its orientation, the generation frequency, or the magnitude of the nonlinear frequency shift. We note that although the sign of the nonlinear shift is expected to be important for the efficiency of spin-wave emission, the maps of Fig. 27 corresponding to significantly different nonlinear behaviors of the STNO (see Fig. 26) differ predominantly by the direction of emission, which rotates together with the field. Figure 28 illustrates the spin-wave characteristics determined at the location of the maximum spin wave intensity. Figure 28a–c show the BLS spectra of the emitted spin waves for I increasing from 3 to 5 mA, together with the spectrum of the thermal spin waves. The spectra exhibit a small nonlinear frequency shift at ϕ = 45◦ , which increases as ϕ is reduced, in agreement with the electrical measurements shown in Fig. 26. Figure 28d summarizes the dependences of the frequencyintegrated spin-wave intensity on the current I. As seen from these data, the intensity of the emitted spin waves increases linearly with current for the angle ϕ = 45◦ characterized by a small nonlinear frequency shift. In contrast, the data for ϕ = 25◦ 6 Spin Waves 323 a b He He He He 500 nm c d He He He He Normalized intensity 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 27 Normalized color-coded intensity maps of spin waves emitted by the STNO, recorded at different angles ϕ between the in-plane magnetic field He = 900 Oe and the easy axis of the elliptical nanopillar: (a) ϕ = 5◦ o, (b) ϕ = 25◦ o, (c) ϕ = 45◦ o, (d) ϕ = − 45◦ o. The bias current is I = 5 mA. The schematic of the top electrode is superimposed on each map, with a cross indicating the location of the nanocontact. The intensity maps acquired at I = 0 were subtracted to eliminate the contribution from the thermal spin waves. Arrows show the direction of the static magnetic field, and the dashed lines indicate the direction of the spin-wave emission. (Reprinted from [31], with the permission of Springer Nature) and ϕ = 5◦ exhibit a decrease of the spin-wave intensity starting from a certain value of current that decreases with decreasing ϕ. These findings are correlated with a larger nonlinear frequency shift, resulting in more significant reduction of the emission frequency far below FMR. In contrast, magnetoresistance measurements (Fig. 26) showed similar monotonic increases of generated power for all three configurations. Therefore, the decrease in the BLS intensity is associated with a decreased emission efficiency rather than a reduced amplitude of the oscillation in the nano-contact area. However, one should admit that decay length of the emitted waves was rather small, below 500 nm. Further studies [16, 73, 80] have shown that the spin waves emitted in these experiments have an evanescence nature, since their frequency were slightly below the spin-wave spectrum of the surrounding Py film. As demonstrated in Fig. 29, microwave parametric pumping can be used as a mechanism for the transfer of the generated microwave energy into the desirable spectral range above the FMR frequency [80]. This effect enables an increase of the propagation length of a c 7 45° 8 9 10 11 Frequency, GHz He 25° 7 8 9 10 11 Frequency, GHz He b 8 9 10 11 Frequency, GHz d Integral intensity BLS intensity, a.u. 7 He 5° BLS intensity, a.u. S. O. Demokritov and A. N. Slavin BLS intensity, a.u. 324 3 4 I, mA 5 Fig. 28 (a–c), Dependence of BLS spectra on the current for different in-plane directions of He = 900 Oe, as indicated. Shadowed regions show the spectrum of the thermally excited spin waves determined by measurement at I = 0. Color lines show the spectra acquired at the currents of 3.0 mA (black), 3.5 mA (blue), 4.0 mA (green), 4.5 mA (red), and 5.0 mA (pink). Dashed vertical lines mark the frequency of the ferromagnetic resonance (FMR). (d) Dependences of the integrated intensity of emitted spin waves on current for ϕ = 5◦ (triangles), 25◦ (squares), and 45◦ (dots). (Reprinted from [31], with the permission of Springer Nature) spin waves emitted by STNOs: the decay length of 540 nm for the auto-emission was increased to 940 nm for the pumping-induced emission. Moreover, the phenomenon of the pumping-induced emission does not disturb the unique directionality found for the emission in the auto-oscillation regime, as illustrated by Fig. 29. Spin-Hall Nano-Oscillator (SHNO) In the previous section, a STNO driven by spin-polarized electric current was considered. Another possibility to inject angular momentum into a magnetic system is utilization of pure spin current. As it has been already mentioned above, the application of pure spin current has numerous advantages compared to the spinpolarized electric current when the excitation of a large-amplitude 0D spin-wave modes is discussed. A complete compensation of damping by the spin current 6 Spin Waves 325 Fig. 29 Pseudocolor spatial intensity maps of the emitted spin waves, acquired at I = 5 mA. A schematic of the top electrode and a cross indicating the location of the nanocontact is superimposed on each map. (a) Spin-wave auto-emission, in the absence of the external pumping microwaves. (b) Spin-wave emission under influence of parametric pumping. Note an extended spin-wave propagation area for (b). (Reprinted with permission from [80], © 2011 by the American Physical Society) appears to be a straightforward extension of the damping reduction, described in Sect. “Spin-Torque Transfer Effect and Spin Waves”. However, as the compensation point is approached, additional nonlinear damping emerges due to the nonlinear interactions among different dynamical modes enhanced simultaneously by the spin current, preventing the onset of auto-oscillation. Since magnon-magnon scattering rates are proportional to the populations of the corresponding modes, detrimental effects of nonlinear damping can be avoided by selectively suppressing all the modes, except for the ones that can be expected to auto-oscillate. To achieve selective suppression, the frequency-dependent damping caused by the spin-wave radiation was used. To take advantage of this radiative damping, the spin current was locally injected into an extended magnetic film, in contrast with the geometry described in Sect. “Spin-Torque Transfer Effect and Spin Waves”. In fact, the local spin current enhances a large number of dynamical modes, but those having higher frequencies, and, consequently, higher group velocities, quickly escape from the active region, which results in their efficient suppression by the radiation losses. Meanwhile, the modes at frequencies close to the bottom of the spin-wave spectrum have a much smaller group velocity, and, therefore, minimum radiation losses. The scheme of our experiment with pure spin current is shown in Fig. 30a [31]. The studied device is formed by a bilayer of a 8 nm thick film of Pt and a 5 nm thick film of Py patterned into a disk with a diameter of 4 μm. Two 150 nm thick Au electrodes with sharp points separated by a 100 nm wide gap are placed on top of the bilayer, forming an in-plane point contact. The sheet resistance of the bilayer is nearly two orders of magnitude larger than that of the Au electrodes. Consequently, the electrical current induced by voltage between the electrodes should be strongly localized in the gap. Indeed, a calculation of the current distribution through a 326 a DC Current Probing laser light Py(5)Pt(8) Disk He 1μ b Normalized current density Fig. 30 (a) Scanning-electron microscopy image of the test spin-Hall nano-oscillator. The device consists of a 4 μm diameter disk formed by a 8 nm thick Pt on the bottom and a 5 nm thick Py layer on top, covered by two pointed Au(150 nm) electrodes separated by a 100 nm gap. (b) Normalized calculated distribution of current through the section of the device shown in the inset by a dashed line. (Reprinted from [30], with the permission of Springer Nature) S. O. Demokritov and A. N. Slavin Au(150) Top electrodes m 1 z 0.5 250 nm 0 -2 -1 0 z, μm 1 2 section across the middle of the gap (Fig. 30b) shows that most of the current flows through a 250 nm wide Pt strip. This electric current creates a pure spin current flowing into Py, due to the spin-Hall effects. The spin current injected into Py exerts spin-transfer torque on its magnetization. As a result, the damping is compensated, and the dynamic magnetic modes are enhanced. Figure 31 shows the BLS spectra obtained with the probing spot positioned in the center of the gap between the electrodes, at different values of the dc current I. At I = 0, the BLS spectrum exhibits a broad peak corresponding to incoherent thermal magnetization fluctuations in the Py film (Fig. 31a). As this thermal peak grows with increasing current, its rising front becomes increasingly sharper than the trailing front, consistent with the preferential enhancement of the low-frequency modes. Analysis of the dependence of the frequency-integrated BLS intensity on current (Fig. 31b) shows that the intensity of magnetic fluctuations diverges as the current approaches a critical value of Ic ≈ 16.1 mA. In contrast to confined systems driven by spatially uniform spin currents [81], the intensity of fluctuations does not saturate as the current approaches Ic , indicating that the nonlinear processes preventing the onset of auto-oscillations are avoided. At I ≥ Ic , a new peak appears in the BLS spectrum below the thermal peak, as indicated in Fig. 31a by an arrow. The calculated current density in the center of the gap at the onset is 3 × 108 A/cm2 , which is only slightly larger than the extrapolated value 1 × 108 A/cm2 obtained for a similar system without radiation losses [81]. Since this peak is not present in the thermal fluctuation spectrum, we can conclude that it corresponds to a new auto-oscillation mode that does not exist at I < Ic . The peak rapidly grows and then saturates above 16.3 mA (Fig. 31c–d). Comparing the Fig. 31 (a) BLS spectra of thermal fluctuation amplified by the spin current at currents below the onset of auto-oscillation. (b) Integral intensity of amplified thermal fluctuations and its inverse versus current. Both dependencies are normalized by their values at I = 0. (c) BLS pectra of the magnetization autooscillation driven by the spin current. Filled areas are the results of fitting by the Gaussian function. Note, that the spectral widths are determined by the resolution of the BLS setup. (d) The intensity and the center frequency of the auto-oscillation peak versus current. Curves are guides for the eye. Reprinted from [30], with the permission of Springer Nature 6 Spin Waves 327 328 S. O. Demokritov and A. N. Slavin spectra for I = 16.1 mA and 16.3 mA, we see that the onset of auto-oscillations is accompanied by a decrease in the intensity of thermal fluctuations, suggesting that the energy of the spin current is mainly channelled into the auto-oscillation mode. The spectral width of the auto-oscillation peak characterizing the coherence of auto-oscillations decreases just above Ic , and stabilizes above 16.3 mA. Note that the linewidth in the spectra shown in Fig. 31a and c is determined by the spectral resolution of our optical technique under usual conditions. Additional measurements at our instrument’s ultimate spectral resolution of 60 MHz show that the actual linewidth in the saturated regime is below this value, suggesting a high degree of coherence of the observed auto-oscillation mode. The frequency of the auto-oscillation peak monotonically decreases with increasing I (Fig. 31d). We note that the generated frequency is significantly below the frequencies of magnetic fluctuations even at the onset of auto-oscillations. We draw three important conclusions based on this observation. First, the auto-oscillation mode does not belong to the thermal spin-wave spectrum. Second, this mode is formed abruptly at the onset current, and not by gradual reduction of frequency from the spin-wave spectrum due to the red nonlinear frequency shift. Third, since the energy can be radiated only by propagating spin waves and there are no available spin-wave spectral states at the auto-oscillation frequency, the auto-oscillation mode is not influenced by the radiation losses. To determine the spatial profile of the auto-oscillation mode, we performed two-dimensional mapping of the dynamic magnetization at the frequency of autooscillations, by rastering the probing laser spot in the two lateral directions and simultaneously recording the BLS intensity. An example of the obtained maps is presented in Fig. 32. These data show that the auto-oscillations are localized in a very small area in the gap between the electrodes. The spatial distribution Fig. 32 Normalized color-coded map of the measured BLS intensity over the auto-oscillation area, and two orthogonal sections through its center. Symbols are the experimental data, and filled areas under solid curves are the results of fitting by a Gaussian function. Dashed lines on the map show the contours of the top electrodes. The data were recorded at I = 16.2 mA. (Reprinted from [30], with the permission of Springer Nature) 6 Spin Waves 329 of the BLS intensity is well described by a Gaussian function with the width of 250 ± 10 nm, close to the diameter of the probing laser spot. The measured spatial distribution is a convolution of the actual spatial profile with the instrumental function determined by the shape of the laser spot. Therefore, we estimate that the size of the auto-oscillation region is less than 100 nm, significantly smaller than the characteristic size of the current localization (Fig. 30b). Therefore, we conclude that the auto-oscillation area is determined not by the spatial localization of the driving current, but by the nonlinear self-localization processes defining the geometry of a standing spin-wave “bullet” [82]. We emphasize that the observed quick saturation of the intensity of the auto-oscillation peak above the onset and its monotonic red frequency shift are the intrinsic characteristics of the “bullet” mode. Only one “bullet” mode exists at the frequency of the auto-oscillations and this frequency is well separated from the continuous spectrum of non-localized spin waves, Therefore, our findings provide strong evidence for that auto-oscillations involve only a single mode in the studied system. Nature of Spin Wave Modes Excited in 0D Magnetic Nanocontacts The nature of the auto-oscillation spin wave mode excited by either spin-polarized or pure spin current in magnetic nanocontacts (0D objects) is of a fundamental importance for the current-induced magnetization dynamics. The first theoretical analysis of the nature of the spin-wave eigen-mode excited by spin-polarized current in a nano-contact geometry was performed by J. Slonczewski [8]. He developed a spatially nonuniform linear theory of spin wave excitations in a nano-contact, where the “free” ferromagnetic layer is infinite in plane, while the spin-polarized current traversing this layer has a finite cross-section S = π Rc2 , where Rc is the contact radius. Considering a perpendicularly magnetized nano-contact Slonczewski showed that in the linear case the lowest threshold of excitation by spin-polarized current is achieved for an exchange-dominated propagating cylindrical spin wave mode having wave number q0 = 1.2/Rc and frequency [8]: ω (q0 ) = ω0 + Dex q02 . (26) Here ω0 is the ferromagnetic resonance (FMR) frequency in the magnetic film, 2 , ω ≡ 4π γ M , γ is the gyromagnetic ratio for electron spin, l Dex = ωM lex M S ex = 1/2 2 A/2π MS is the exchange length, A is the exchange constant, and MS is the value of the saturation magnetization. It was also shown that the threshold current Ith in such a geometry consists of two additive terms: the first one arises from the radiative loss of energy carried by the propagating spin wave out of the region of current localization, while the second one is caused by the usual energy dissipation in the current-carrying region: 330 S. O. Demokritov and A. N. Slavin lin Ith = 1.86 D (H ) . + σ σ Rc2 (27) Here σ = εgμB /2eMS dS whereε is the spin-polarization efficiency defined in [8], g is the spectroscopic Lande factor, μB is the Bohr magneton, e is the modulus of the electron charge, d is the thickness of the “free” magnetic layer, S is the crosssection area of the nano-contact), and (H) is the spin wave damping dependent on the bias magnetic field H. It turns out that for a typical nano-contact of the radius Rc = 20 − 30 nm the radiative losses are about one order of magnitude larger than the direct energy dissipation, and should give the main contribution into the threshold current. This result, however, contradicts experimental observations (see, e.g., [83]): the experimentally measured magnitude of the threshold current in an in-plane magnetized nano-contact is much smaller than the value predicted by Eq. (27), although the dependence of this current on the magnetic field H is satisfactory described by this equation. In this section we present a spatially nonuniform nonlinear theory of spin wave excitation by spin-polarized current in a nano-contact geometry for the case of the in-plane magnetization [82]. We show that in an in-plane magnetized magnetic film the competition between the nonlinearity and exchange-related dispersion leads to the formation of a stationary two-dimensional self-localized nonpropagating spin wave mode. Such nonlinear self-localized wave modes in two- or three-dimensional cases are conventionally called wave “bullets” [84]. The frequency of this spin wave “bullet” is shifted by the nonlinearity below the spectrum of linear spin waves and, therefore, this nonlinear mode has an evanescent character with vanishing radiative losses, which leads to a substantial decrease of its threshold current Ith in comparison to the linear propagating mode shown in Eq. (27). To describe the generation of a spin wave bullet by the spin-polarized current we consider a “free” ferromagnetic layer, infinite in y − z plane and having finite thickness d in the x direction (d is assumed to be sufficiently small for us to consider that the magnetization M is constant along the film thickness, and that the dipoledipole interaction can be described by a simple demagnetization field). We assume that the internal magnetic field H = Happ + Hex , consisting of the applied Happ and interlayer exchange Hex fields, is applied in the z direction in the film plane. Using the standard Hamiltonian spin-wave formalism [33], which has been successfully used to develop a spatially uniform nonlinear model of spin wave generation by spin-polarized current [85, 86], one can derive an approximate equation for the dimensionless complex spin wave amplitude b ≡ b(t, r): ∂b = −i ω0 b − DD b + N |b|2 b − b + f (r/Rc ) σ I b − f (r/Rc ) σ I |b|2 b. ∂t (28) √ Here ω0 ≡ ωH (ωH + ωM ) is the linear FMR frequency, (ωH ≡ γ H, DD ≡ (2A/MS ) ∂ω0 /∂H = (2γ A/MS )(ωH + ωM /2)/ω0 is the dispersion coefficient 6 Spin Waves 331 for spin waves, is the two-dimensional Laplace operator in the film plane, N = − ωH ωM (ωH + ωM /4)/ω0 (ωH + ωM /2) is the coefficient describing nonlinear frequency shift, and ≡ α G (ωH + ωM /2) is the spin wave damping rate (α G is the dimensionless Gilbert damping parameter). The dimensionless function f (x) describes the spatial distribution of the spin-polarized current. The dimensionless spin wave amplitude b is connected with the z-component of the magnetization by the equation |b|2 = (MS − Mz )/2MS . Equation (28) differs from the Eq. (9) in [8] (which resulted in the solution (27)) by the presence of two additional nonlinear terms: the term containing the coefficient N and describing a nonlinear frequency shift of the excited mode, and the last term describing the current-induced positive nonlinear damping that stops the increase of the amplitude of the excited mode at relatively large currents. Also, since the Eq. (28) was obtained as a Taylor expansion it is exactly correct only for sufficiently small spin wave amplitudes |b| < 1. Without damping and current terms ( = 0, I = 0) Eq. (28) coincides with the well-known (2 + 1)-dimensional nonlinear Schrödinger equation (NSE) [87]. In the considered case of an in-plane magnetized film the nonlinear coefficient N is negative, and the nonlinearity and dispersion satisfy the well-known Lighthill criterion ND < 0 (i.e., they act in opposite directions), and the NSE has a nonlinear self-localized radially symmetric standing solitonic solution (or the solution in the form of a standing spin wave bullet) b (t, r) = B0 ψ (r/) e−iωt , (29) where dimensionless function ψ(x), having maximum value of 2.2 at x = 0, describes the profile of the bullet. This function is the localized solution of the equation ψ + 1 ψ + ψ 3 − ψ = 0, x (30) which has to be found numerically (see e.g., [84]). In Eq. (29) B0 , , and ω are the characteristic amplitude, characteristic size, and frequency of the bullet, respectively. Among these three parameters only one is independent. Taking the amplitude B0 as an independent parameter, we can express the two other parameters as √ ω= ω0 + NB 20 , = |D/N| . B0 (31) We would like to stress that the frequency of the spin wave bullet lies below the linear frequency ω0 of the ferromagnetic resonance (see Eq. (31), and note that N < 0), i.e., outside the spectrum of linear spin waves. This is the main reason for the self-localization of the spin wave bullet, as the effective wave number of the spin 332 S. O. Demokritov and A. N. Slavin wave mode with frequency (6) is purely imaginary. It also follows from Eq. (29) and the expansion condition |b| < 1 that the maximum magnitude of B0 for which our perturbative approach is still correct is B0 = 0.46. It is well known [87] that the bullet-like solutions of (2 + 1)-dimensional NSE are unstable with respect to the small perturbations: the wave packets having the bullet shape Eq. (29), but amplitudes smaller than B0 , decay due to the dispersion spreading, while the wave packets having amplitudes higher than B0 collapse due to the nonlinearity. At the same time, Eq. (28) with both Gilbert dissipation and current I is a two-dimensional analog of a Ginzburg-Landau equation that is known to have stable localized solutions (see, e.g., review [88]). One can assume that for a small damping rate and current I the full nonconservative eq. (28) will have a bullet-like solution, only slightly different from the exact solution Eq. (29) of the conservative NSE equation. It is clear, however, that not all of such solutions can be supported in our case. For example, small-amplitude bullets, for which > > Rc , practically do not interact with the spatially localized current and will decay due to the linear dissipation. The large-amplitude (B0 ≥ 1) bullets, on the other hand, will also decay because the effective damping − σ I(1 − |b|2 ) for them changes sign and becomes positive. The excitation threshold of the spin wave bullet mode was calculated in [82] and the minimum value of this threshold turned out to be equal to the second term in Eq. (27), i.e., sobstantially lower than the threshold of excitation of the propagating spin-wave mode in the perpendicularly magnetized magnetic nanocontact Eq. (27). To find the spatial profile of the spin-wave bullet mode Eq. (28) was solved numerically. The results of comparison of the spin-wave excitation profiles at the threshold obtained for a typical set of experimental parameters [83] from the analytical solution Eq. (29) (solid black line) and numerical solution of Eq. (28) (black dots) are shown in Fig. 33. One clearly sees that the numerical profile of the nonlinear eigen-mode is practically indistinguishable from the approximate “bullet-like” profile, so the “bullet” model works exceptionally well in this case. For comparison we present in Fig. 33 the spatial profile of the Slonczhewski-like [8] linear mode, that is obtained from the solution of Eq. (28) where the nonlinear terms (terms containing |b|2 ) are omitted (red line). The amplitude of this linear mode at the threshold is vanishingly small, |b(r)|2 → 0. We also present the normalized spatial profile of a bullet mode above the thershold numerically calculated form Eq. (28), to show that with the increase of the bullet amplitude its width decreases in accordance with the experssion shown in Eq. (31) (blue line). As it was mentioned above, the bullet mode is excited in an in-plane magetized nanocontact, while the linear propagating spin wave mode (Slonczewski mode [8]) could be excited in a perpendicularly magnetized nanocotact. It is also well-known that in the case when the direction of the external bias magnetic field varies from inplane to the perpendicular the coefficient of the nonlinear frequency shift N changes its sign from negative to positive [89, 90]. Therefore, it is interesting to see the nature of the spin-wave mode excited by a spin-polarized (or pure spin) current under the oblique magnetization of a nanocontact. 333 Normalized power |b(r)|2/|b(0)|2 6 Spin Waves 1.0 Linear mode 0.8 Nonlinear "bullet" 0.6 Nonlinear "bullet" above the threshold 0.4 0.2 0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Normalized distance r/Rc Fig. 33 Normalized profiles of the spin wave mode generated by spin-polarized current at (the threshold: solid black line – bullet profile (29), circles – result of the numerical solution of Eq. (28), red line – profile of the linear eigen-mode calculated from the linearized Eq. (28). Blue line demonstrates the numerically calculated profile of the bullet mode above the thershold. Vertical dash-dotted line shows the region of the current localization. The parameters are: 4π M0 = 16.6 kG, Happ = Hex = 5 kOe, A = 2.85 · 10−6 erg/cm, α G = 0.015, d = 1.2 nm, Rc = 25 nm, ε = 0.3 The analytic [91], numerical [92], and experimental [93] studies of the spin wave excitation in current-driven magnetic nanocontacts were successfully performed, and confirmed the conclusions of the above-presented theory. In particular, the experimental study [93] was performed on a nanocontact 2Rc = 40 nm to the thin film tri-layer Co81 Fe19 (20 nm)/Cu(6 nm)/Ni80 Fe20 (4.5 nm), patterned into a 8 μ m × 26μ m mesa. On top of this mesa, a circular Al nanocontact was defined through SiO2 using e-beam lithography and an external magnetic field of a constant magnitude (μ0 He = 1.1 T) was applied to the sample at an angle θ e with respect to the film plane. Microwave excitations were only observed for a single current polarity, corresponding to electrons flowing from the “free” (thinner) to the “fixed” (thicker) magnetic layer. All measurements were performed at room temperature. Figure 34 shows the detailed angular dependence of the microwave frequencies generated at a constant current of I = 14 mA and a constant magnetic field amplitude of μ0 He = 1.1 T. The generated frequencies are approximately independent of the magnetic field angle up to about θ e = 35◦ , and then decrease from about 35 GHz to 10 GHz with the increasing angle. The most important feature of the results presented in Fig. 34 is the existence of two distinct and different modes for sufficiently small values of θ e – linear propagating spin wave Slonczewski mode, having a higher frequency, and a nonlinear self-localized spin-wave bullet mode having a lower frequency. The frequencies of these two modes differ by about 2.5 GHz at angles up to θ e = 40◦ , and then they start to approach each other up to θ e ≈ 55◦ where the mode having lower frequency completely disappears. 334 S. O. Demokritov and A. N. Slavin 40 Propagating mode Spin wave bullet 40 20 f [GHz] f [GHz] 30 30 20 10 10 0 0 0 0 15 30 45 60 75 Applied field agle θe [deg] 15 30 45 90 60 75 90 Applied field agle θe [deg] Fig. 34 Experimental frequencies of the current-induced spin wave modes as a function of the applied field angle θ e between the external bias magnetic field and the nanocontavt plane at I = 14 mA and μ0 He = 1.1 T (symols). Inset: theoretically calculated frequencies of the propagating (upper curve) and “bullet” (lower curve) modes at the threshold of their excitation for nominal parameters of the nanocontact sample. (Reprinted with permission from [82], © 2005 by the American Physical Society) The behavior of the frequencies of the current-induced spin wave modes shown in Fig. 34 remains the same for any magnitude of the bias current that is larger than I ≈ 10 mA. This behavior is also qualitatively similar to the behavior of the mode frequencies derived from analytic theory (see Fig. 2 in [91] and the inset in Fig. 34) and from the numerical simulation (see Fig. 4 in [92]). The experimental threshold currents for the two excited spin wave modes shown in Fig. 35 as functions of θ e were determined using the method proposed in [90]. The graph in Fig. 35 only shows the threshold currents determined from experiment for the magnetization angles 20◦ < θ e < 80◦ , since outside this range the signal was too weak to allow reliable determination of the excitation threshold. It is clear that the lower-frequency (bullet) mode has a lower threshold current at low magnetization angles. As the angle increases, the threshold currents for the two modes gradually approach each other and become essentially equal close to the critical angle θ e ≈ 50◦ , where the low-frequency mode disappears. These experimental data are also qualitatively similar to the threshold curves calculated analytically in [91] (see solid lines in Fig. 35) and the similar curves simulated numerically (see Fig. 3 in [92]). The inset in Fig. 35 shows the numerically calculated profile of the both modes: nonlinear self-localized bullet mode (left frame) and linear propagating mode (right frame). Thus, the results of the laboratory experiment [93] and the results of the numerical simulations [92] fully confirmed the theoretical ideas [8, 82] about the possibility of current-induced excitation of two qualitatively different spin wave modes in magnetic nanocontacs. 6 Spin Waves 335 Threshold current lth [mA] 30 25 100 100 y [nm] 20 0 −100 −100 0 x [nm] 200 y [nm] 0 −200 200 0 x [nm] −200 15 10 Propagating mode Spin wave bullet 5 0 0 15 30 45 60 75 90 Applied field agle qe [deg] Fig. 35 Measured threshold current for the propagating (empty triangles) and bullet (filled circles) modes as a function of the applied field angle θ e . Solid lines: thresholds for the same modes versus applied field angle theoretically calculated using the formalism of [91] (red line – bullet mode, blue line- propagating mode). Upper inset demonstrates the numerically calculated spatial profiles of the bullet mode (left) and propagating mode (right). (Reprinted with permission from [82], © 2005 by the American Physical Society) It is important to note that the excitation of self-localized evanescent spin wave bullet modes by pure spin current was subsequently observed in several independent experiments [31, 94, 95]. Another example of an interesting and unusual solitonic spin wave mode that can be excited by spin-polarized current is given by the so-called spin-wave droplet (or spin-wave droplet soliton) existing in perpendicularly magnetized nanocontacts having a large perpendicular magnetic anisotropy (PMA) [96–98]. As it was first demonstrated analytically in [99, 100], the Landau-Lifshitz equation for magnetic films can sustain a family of so-called magnon drop solitons, provided there is no spin wave damping. While any realistic magnetic system always exhibits some spin wave damping, hence making magnon drops unrealistic, it was demonstrated theoretically in [96] that in a current-driven magnetic nanocontact with PMA, where the spin wave damping is completely compensated by the STT effect [8], it would be possible to excite a magnon-drop-like excitations. In contrast to the conservative magnon drops, these so-called magnetic droplets are strongly dissipative, relying not only on the zero balance between the exchange and anisotropy, but also on the balance between the negative damping created by the STT effect and the positive nonlinear damping in the current-driven magnetic material. As a consequence, out of a large family of magnon drops, the additional net zero damping condition singles out a particular magnetic droplet with both a well-defined frequency and a well-defined direction of the dynamic magnetization at the center of the excited droplet. Note 336 S. O. Demokritov and A. N. Slavin that at large droplet amplitudes the direction of magnetization at the droplet center could be almost completely opposite to the similar direction at the droplet periphery (see Fig. 2 in [96]). These rather exotic nonlinear dynamical magnetic modes were observed experimentally in [97], and a more detailed description of magnetic droplets is presented in [98]. Coupling of a STNO and 1D Spin-Wave Waveguide to Each Other In the previous sections, we have demonstrated that STNO devices, employing 0D spin-wave modes can convert the energy of direct electrical current into propagating spin waves. We have also noticed that it is difficult to achieve frequency matching of STNO with the propagating spin waves, since the large-amplitude spin wave modes in STNOs are frequency shifted due to nonlinear properties of spin waves with respect to characteristic frequencies of 1D and 2D propagating spin waves. However, if one uses a spin-wave waveguide of a particular geometry as described below, efficient matching between such waveguides and STNOs can be achieved. This matching is realized by taking advantage of the dipolar magnetic field within the waveguide, which acts on 1D propagating spin-wave modes [16]. Figure 36a shows the layout of the studied device. A point-contact STNO is comprised of a multilayer Cu(4)/Co70 Fe30 (4)/Au(150) shaped into an elliptic nanopillar with dimensions of 120 nm × 40 nm fabricated on top of an extended 5 nm thick Permalloy (Py) film. Additionally, the device incorporates a 5 nm thick and 200 nm wide Co70 Fe30 nanostripe below the Py film. The device is magnetized by a static magnetic field He = 800 – 1200 Oe applied in the plane of the Py film perpendicular to the CoFe nanostripe. Figure 36b shows the characteristics of the oscillation of STNO determined by the standard electronic spectroscopy measurements. Above the onset current of about 3.5 mA, both the amplitude and the frequency of the auto-oscillations exhibit a smooth dependence on current, indicating a single-mode operation of the STNO. Correspondingly, Fig. 36c shows representative BLS spectra recorded with the probing laser spot positioned above the CoFe nanostripe. Note here that by comparing Fig. 36b and c, one can conclude that the frequency of the microwave signal is twice the frequency of the magnetization oscillation measured by BLS, since the former is due to the quadratic magnetoresistance effect. While the BLS spectra acquired above the CoFe nanostripe clearly show the signals resulting from the STNO oscillation, no such signals were detected away from the nanostripe. This observation indicates that the STNO can efficiently generate 1D spin waves, propagating along the CoFe nanostripe, but a radiation of 2D spin waves into the free Py film is inefficient. To understand this phenomenon, one has to consider the effects of the dipolar field of the CoFe nanostripe on the internal field in the magnetic layers. Both micromagnetic simulations and studies of spin-wave spectra of thermal fluctuations 6 Spin Waves 337 a Top electrode Current 200 nm Py (5) film z CoFe (5) nano-wire Nanopillar y He Bottom Cu(40) electrode c b 2.5 1.0 BLS intensity, a.u. PSD, pW/MHz 7 mA 2.0 1.5 6 mA 1.0 5 mA 0.5 4 mA 0 7 mA 0.8 0.6 5.5 mA 0.4 0.2 4 mA 0.0 13 14 Frequency, GHz 15 Frequency, GHz Fig. 36 (a) Layout of the studied STNO with an incorporated waveguide. Inset: SEM micrograph of the device. He is the static magnetic field. Numbers in parentheses indicate the thicknesses of the layers in nanometres. (b) Spectra of the current-induced oscillations of the STNO measured by a spectrum analyzer at different driving dc currents, as indicated. (c) BLS spectra recorded at different driving currents measured by positioning the probing laser spot on the nano-waveguide. Note that the spectral widths are determined by the resolution of the BLS setup. (Reprinted from [16], with the permission of Springer Nature) show that the internal field is significantly reduced in the magnetic film in the region of the CoFe nanostripe, as compared to the magnitude away from the nanostripe. The reduction of the internal field results in lowering of the local spin-wave spectrum, creating a one-dimensional channel with allowed spin-wave frequencies below the bottom of the spectrum in the free Py film. Low-frequency magnons excited by STNO are directionally guided along the CoFe nanostripe, since there are no states available at these frequencies in the free Py film. Thus, the staticfield channel induced by the CoFe nanostripe plays the role of a compound dipolar spin-wave waveguide formed by the strongly exchange-coupled bilayer of the CoFe nanostripe and the Py film on top of it. 338 S. O. Demokritov and A. N. Slavin Fig. 37 (a) Normalized decay-compensated spatial map of the spin-wave intensity. The positions of the top device electrode and the CoFe nanostripe are schematically shown. (b) Measured dependence of the integral spin-wave intensity on the propagation coordinate (symbols), on the log-linear scale. The line shows the result of the fitting of the experimental data by the exponential function. (c) Distribution of the spin-wave intensity in the section transverse to the nano-waveguide. Symbols are experimental data, curve is a fit by the Gaussian function. w is the full width at half maximum of the transverse intensity profile. (d) Dependence of w on the propagation coordinate. Symbols are experimental data, horizontal line is the mean value. (Reprinted from [16], with the permission of Springer Nature) 6 Spin Waves 339 The measured propagation characteristics of spin waves in the nano-waveguide are illustrated in Fig. 37. Figure 37a shows the normalized spatial map of the BLS intensity, which is proportional to the local spin wave intensity. The map was recorded at a constant dc current of 5 mA by rastering the probing laser spot over a 1.6 μm by 1.6 μm area with the step size of 100 nm. To highlight the transverse profile of the propagating wave, the spatial decay in the direction of propagation was compensated by normalizing the signal with the integral over the transverse section of the map (along the z-coordinate). The map of Fig. 37a clearly shows that the spin wave energy is concentrated entirely in the nano-waveguide, i.e., spin waves are guided by the field-induced channel without noticeable losses associated with the radiation of energy into the surrounding free Py film. The BLS intensity integrated over the transverse section of the map exhibits a simple exponential spatial decay in the direction of propagation (shown on the log scale in Fig. 37b). We define the decay length ξ as the distance over which the wave amplitude decreases by a factor of e. By fitting the data of Fig. 37b with the function exp(−2y/ξ), we obtain ξ = 1.3 μm. We note that this value is close to the best spinwave propagation characteristics obtained in low-loss Py films with comparable thickness, despite the higher dynamical losses expected due to the stronger damping in CoFe. By analyzing transverse cross-sections of the BLS intensity map (Fig. 37c), we determine the transverse full width at half maximum w of the spin wave intensity distribution for different positions along the waveguide. The obtained value w = 320 nm is independent of the propagation coordinate (Fig. 37d), which confirms that the spin wave is efficiently localized in the waveguide without spreading out. We note that the measured spatial profile (Fig. 37c) represents a convolution of the actual profile of the spin wave intensity with the distribution of intensity in the diffraction-limited probing light spot whose estimated diameter is 250 nm. The value w = 320 nm is therefore in a reasonable agreement with the measured waveguide width of 200 nm (inset in Fig. 36a). Conclusion and Outlook The post-CMOS information technology will require radically new solutions for digital and analog information processing. One promising approach is to employ the spin degree of freedom of electron for information storage and computing, which is the main focus of the rapidly growing field of spintronics [101–104]. In this new signal-processing paradigm signals will be codes in terms of a spin angular momentum that can be carried by either polarized electrons or spin waves. The use of spin waves (magnons) as carriers of spin angular momentum is preferable to the use of spin-polarized electrons, because Gilbert magnetic damping, associated with the transport of spin waves, is, usually, lower than the Ohmic losses associated with the transport of electrons. The typical medium for the spin wave propagation in the existing nano-scale magnonics is a soft magnetic metal such as permalloy (Py). This material choice is mainly dictated by the relative ease of 340 S. O. Demokritov and A. N. Slavin magnetic information readout via various types of magneto-resistance observed in metallic ferromagnetic heterostructures. A substantial progress has been made in this field during the last two decades. In particular, generation of self-sustained microwave magnetic oscillations by STT effect from spin-polarized currents [27, 29] as well as pure spin currents arising from spin-Hall effect [31, 94, 95, 105] have been demonstrated. Novel techniques for precise characterization of magnetization dynamics in nano-scale metallic magnetic systems, such as the technique of spin-torque ferromagnetic resonance (ST-FMR) [106, 107], have been developed. In spite of the rapid research progress in the field of metal-based magnonics, several significant limitations of the metal-based magnonic systems are very evident. One of them is the relatively large magnetic damping of ferromagnetic metals, which translates into large spin current densities needed to induce magnetization switching or self-generation of spin waves in ferromagnetic metals. The large magnetic damping also results in short propagation lengths (typically ∼1 μm) of magnons in metallic magnetic nanostructures, which critically hinders the transition from single magnonic elements to large-scale spintronic circuits based on propagating magnons. In addition, high electric conductivity of metallic magnets and the corresponding short charge screening length do not allow to employ magneto-electric effects such as the recently predicted flexoelectric effect [108–110] for manipulation of spin waves using electric field. These drawbacks are absent in magnetic dielectrics, the most common of which is yttrium iron garnet (YIG) – a ferrimagnetic insulator with very low magnetic damping (magnon lifetimes reaching 1 μs and magnon propagation length exceeding 1 cm) [35]. However, the technique of liquid phase epitaxy typically employed for the growth of high-quality YIG crystals does not allow for deposition of films sufficiently thin for observation of interfacial spin-dependent phenomena, which will determine the future of manipulation of spin waves at nanoscale. The pioneering experiments in YIG-based spintronics performed on the relatively thick (∼1–3 μm) epitaxial YIG films revealed some weak effects, but failed to demonstrate reproducible excitation or/and manipulation of propagating magnons by interfacial spintronic effects [12, 13, 111, 112]. The recently developed methods for growth of ultra-thin (∼ 10 nm) high-quality YIG films (ferromagnetic resonance (FMR) linewidth ∼3–5 Oe) by pulsed laser deposition (PLD) [113–115] and patterning of thin YIG films [116] remove major roadblocks for using magnetic dielectrics in nano-scale spintronic devices and open a new field of magnon-based spintronics of magnetic dielectrics. Pioneering experiments in this field performed in the last two years demonstrated excitation of magnonic signals in magnetic dielectrics by interfacial spin orbit torques, compensation of magnetic damping in magnetic dielectrics by pure spin currents, and resulting self-sustained generation of microwave magnetization oscillations in YIG film samples [117, 118]. 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Demokritov received his Ph.D at Kapitsa Institute for Physical Problems, Moscow, Russia. In the 1990s, he moved to Germany to start to work with P. Grünberg at Research Center Jülich. Since 2004, he is a Professor at Münster University, Germany. His main directions of research are dynamics and quantum thermodynamics of magnetic structures, spin-wave research, and magnonics. Andrei N. Slavin is a Distinguished Professor and Chair of the Physics Department, Oakland University, Michigan, USA. He received his Ph.D from the St. Petersburg Technical University, Russia. Andrei is Fellow of the American Physical Society and Fellow of the IEEE. He is a specialist in magnetization dynamics and spin waves and published over 280 research papers in this field. 7 Micromagnetism Lukas Exl , Dieter Suess , and Thomas Schrefl Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Micromagnetics Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Gibbs Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spin, Magnetic Moment, and Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetostatics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zeeman Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetostatic Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Crystal Anisotropy Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetoelastic and Magnetostrictive Energy Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Characteristic Length Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exchange Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Critical Diameter for Uniform Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wall Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mesh Size in Micromagnetic Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 349 350 350 351 356 357 357 363 366 372 372 373 375 376 L. Exl () University of Vienna Research Platform MMM Mathematics – Magnetism – Materials, University of Vienna, and Wolfgang Pauli Institute, Wien, Austria e-mail: [email protected] D. Suess University of Vienna Research Platform MMM Mathematics – Magnetism – Materials, and Physics of Functional Materials, Faculty of Physics, University of Vienna, Wien, Austria e-mail: [email protected] T. Schrefl Christian Doppler Laboratory for Magnet Design Through Physics Informed Machine Learning, Department of Integrated Sensor Systems, Danube University Krems, Wiener Neustadt, Austria e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_7 347 348 L. Exl et al. Brown’s Micromagnetic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler Method: Finite Differences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ritz Method: Finite Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376 377 380 382 386 Abstract Computational micromagnetics is widely used for the design and development of magnetic devices. The theoretical background of these simulations is the continuum theory of micromagnetism. It treats magnetization processes on a significant length scale which is small enough to resolve magnetic domain walls and large enough to replace atomic spins by a continuous function of position. The continuous expression for the micromagnetic energy terms are either derived from their atomistic counterpart or result from symmetry arguments. The equilibrium conditions for the magnetization and the equation of motion are introduced. The focus of the discussion lies on the equations that form the basic building blocks of micromagnetic solvers. Numerical examples illustrate the micromagnetic concepts. An open-source simulation environment was used to address the ground state of thin film magnetic elements, initial magnetization curves, stress-driven switching of magnetic storage elements, the grain size dependence of the coercivity of permanent magnets, and damped oscillations in magnetization dynamics. Introduction Computer simulations are essential tools for product design in modern society. This is also true for magnetic materials and their applications. The design of magnetic data storage systems such as hard disk devices [1, 2, 3, 4, 5] and random access memories [6, 7] relies heavily on computer simulations. Similarly, the computer models assist the development of magnetic sensors [8, 9] used as biosensors or position and speed sensors in automotive applications [10]. Computer simulations give guidance for the advance of high performance permanent magnet materials [11, 12, 13] and devices. In storage and sensor applications, the selection of magnetic materials, the geometry of the magnetically active layers, and the layout of current lines are key design questions that can be answered by computations. In addition to the intrinsic magnetic properties, the microstructure including grain size, grain shape, and grain boundary phases is decisive for the magnet’s performance. Computer simulations can quantify the influence of microstructural features on the remanence and the coercive field of permanent magnets. The characteristic length scale of the abovementioned computer models is in the range of nanometers to micrometers. The length scale is too big for a description by spin polarized density functional theory. Efficient simulations by atomistic spin dynamics [14] are possible for nano-scale devices only. On the other hand, macroscopic simulations using Maxwell’s equations hide the magnetization 7 Micromagnetism 349 processes that are relevant for the specific functions of the material or device under consideration. Micromagnetism is a continuum theory that describes magnetization processes on significant length scales that are • large enough to replace discrete atomic spins by a continuous function of position (the magnetization), but • small enough to resolve the transition of the magnetization between magnetic domains For most ferromagnetic materials, this length scale is in the range of a few nanometers to micrometers. The first aspect leads to a mathematical formulation which makes it possible to simulate materials and devices in a reasonable time. Instead of billions of atomic spins, only millions of finite elements have to be taken into account. The second aspect keeps all relevant physics so that the influence of structure and geometry on the formation of reversed domains and the motion of domain walls can be computed. The theory of micromagnetism was developed well before the advance of modern computing technology. Key properties of magnetic materials can be understood by analytic or semi-analytic solutions of the underlying equations. However, the future use of powerful computers for the calculation of magnetic properties by solving the micromagnetic equations numerically was already proposed by Brown [15] in the late 1950s. The purpose of micromagnetics is the calculation of the magnetization distribution as function of the applied field or the applied current taking into account the structure of the material and the mutual interactions between the different magnetic parts of a device. Micromagnetics Basics The key assumption of micromagnetism is that the spin direction changes only by a small angle from one lattice point to the next [16]. The direction angles of the spins can be approximated by a continuous function of position. Then the state of a ferromagnet can be described by a continuous vector field, the magnetization M(x). The magnetization is the magnetic moment per unit volume. The direction of M(x) varies continuously with the coordinates x, y, and z. Here we introduced the position vector x = (x, y, z). Starting from the Heisenberg model [17, 18] which describes a ferromagnet by interacting spins associated with each atom, we derive the micromagnetic equations whereby several assumptions are made: 1. Micromagnetism is a quasi-classical theory. The spin operators of the Heisenberg model are replaced by classical vectors. 2. The The length of the magnetization vector is a constant that is uniform over each material of the ferromagnetic body and only depends on temperature. 3. The temperature is constant in time and in space. 4. The Gibbs free energy of the ferromagnetic body is expressed in terms of the direction cosines of the magnetization. 350 L. Exl et al. 5. The energy terms are derived either by the transition from an atomistic model to a continuum model or phenomenologically. In classical micromagnetism, the magnetization can only rotate. A change of the length of M is forbidden. Thus, a ferromagnet is in thermodynamic equilibrium, when the torque on the magnetic moment MdV in any volume element dV is zero. The torque on the magnetic moment MdV caused by a magnetic field H is T = μ0 MdV × H , (1) where μ0 is the permeability of vacuum (μ0 = 4π × 10−7 Tm/A). The equilibrium condition (1) follows from the direct variation of the Gibbs free energy. If only the Zeeman energy of the magnet in an external field is considered, H is the external field, H ext . In general additional energy terms will be relevant. Then H has to be replaced by the effective field, H eff . Each energy term contributes to the effective field. In section “Magnetic Gibbs Free Energy”, we will derive continuum expressions for the various contributions to the Gibbs free energy functional using the direction cosines of the magnetization as unknown functions. In section “Characteristic Length Scales”, we discuss the different characteristic length scales used to describe magnetic phenomena. In section “Brown’s Micromagnetic Equation”, we show how the equilibrium condition can be obtained by direct variation of the Gibbs free energy functional. Magnetic Gibbs Free Energy We describe the state of the magnet in terms of the magnetization M(x). In the following we will show how the continuous vector field M(x) is related to the magnetic moments located at the atom positions of the magnet. Spin, Magnetic Moment, and Magnetization The local magnetic moment of an atom or ion at position x i is associated with the spin angular momentum, h̄S, μ(x i ) = −g |e| h̄S(x i ) = −gμB S(x i ). 2m (2) Here e is the charge of the electron, m is the electron mass, and g is the Landé factor. The Landé factor is g ≈ 2 for metal systems with quenched orbital moment. The constant μB = 9.274 × 10−24 Am2 = 9.274 × 10−24 J/T is the Bohr magneton. The constant h̄ is the reduced Planck constant, h̄ = h/(2π ), where h is the Planck constant. The magnetization of a magnetic material with N atoms per unit volume is 7 Micromagnetism 351 M = Nμ. (3) The magnetic moment is often given in Bohr magnetons per atom or Bohr magnetons per formula unit. The magnetization is M = Nfu μfu , (4) where μfu is the magnetic moment per formula unit and Nfu is the number of formula units per unit volume. The length of the magnetization vector is assumed to be a function of temperature only and does not depend on the strength of the magnetic field: |M| = Ms (T ) = Ms , (5) where Ms is the saturation magnetization. In classical micromagnetism the temperature, T, is assumed to be constant over the ferromagnetic body and independent of time t. Therefore Ms is fixed and time evolution of the magnetization vector can be expressed in terms of the unit vector m = M/|M| M(x, t) = m(x, t)Ms . (6) The saturation magnetization of a material is frequently given as μ0 Ms in units of Tesla. Example. The saturation magnetization is an input parameter for micromagnetic simulations. In a multiscale simulation approach of the hysteresis properties of a magnetic material, it may be derived from the ab initio calculation of magnetic moment per formula unit. In NdFe11 TiN, the calculated magnetic moment per formula unit is 26.84 μB per formula unit [19]. The computed lattice constants were a = 8.537 × 10−10 m, b = 8.618 × 10−10 m, and c = 4.880 × 10−10 m [19] which give a volume of the unit cell of v = 359.0×10−30 m3 . There are two formula units per unit cell and Nfu = 2/v = 5.571 × 1027 . With (4) and (5), the saturation magnetization of NdFe11 TiN is Ms = 1.387 × 106 A/m (μ0 Ms = 1.743 T). Exchange Energy The exchange energy is of quantum mechanical nature. The energy of two ferromagnetic electrons depends on the relative orientation of their spins. When the two spins are parallel, the energy is lower than the energy of the antiparallel state. Qualitatively this behavior can be explained by the Pauli exclusion principle and the electrostatic Coulomb interaction. Owing to the Pauli exclusion principle, two electrons can only be at the same place if they have opposite spins. If the spins are parallel, the electrons tend to move apart which lowers the electrostatic energy. The corresponding gain in energy can be large enough so that the parallel state is preferred. 352 L. Exl et al. The exchange energy, Eij , between two localized spins is [18] Eij = −2Jij S i · S j , (7) where Jij is the exchange integral between atoms i and j and h̄S i is the angular momentum of the spin at atom i. For cubic metals and hexagonal closed packed metals with ideal c over a ratio there holds Jij = J . Treating the exchange energy for a large number of coupled spins, we regard Eij as a classical potential energy and replace S i by a classical vector. Let mi be the unit vector in direction −S i . Then mi is the unit vector of the magnetic moment at atom i. If ϕij is the angle between the vectors mi and mj , the exchange energy is Eij = −2J S 2 cos(ϕij ), (8) where S = |S i | = |S j | is the spin quantum number. Now, we introduce a continuous unit vector m(x) and assume that the angle ϕij between the vectors mi and mj is small. We set m(x i ) = mi and expand m around x i m(x i + a j ) =m(x i )+ ∂m ∂m ∂m aj + bj + cj + ∂x ∂y ∂z 1 ∂ 2m 2 ∂ 2m 2 ∂ 2m 2 + .... a + b + c 2 ∂x 2 j ∂y 2 j ∂z2 j (9) Here a j = (aj , bj , cj )T is the vector connecting points x i and x j = x i + a j . We can replace cos(ϕij ) by cos(ϕij ) = m(x i ) · m(x j ) in (8). Summing up over the six nearest neighbors of a spin in a simple cubic lattice gives (see Fig. 1) the exchange energy of the unit cell. The vectors a j take the values (±a, 0, 0)T , (0, ±a, 0)T , (0, 0, ±a)T . For every vector a, there is the corresponding vector −a. Thus the linear terms in the variable a in (9) vanish in the summation. The same holds for mixed second derivatives in the expansion (9). The constant term, m · m = 1, plays no role for the variation of the energy and will be neglected. The exchange energy of a unit cell in a simple cubic lattice is 6 j =1 Eij = − J S 2 6 ∂ 2 mi 2 ∂ 2 mi 2 ∂ 2 mi 2 mi · a + m · b + m · c i i ∂x 2 j ∂y 2 j ∂z2 j j =1 ∂ 2 mi ∂ 2 mi ∂ 2 mi + m · + m · = − 2J S 2 a 2 mi · i i ∂x 2 ∂y 2 ∂z2 (10) To get the exchange energy of the crystal, we sum over all atoms i and divide by 2 to avoid counting each pair of atoms twice. We also use the relations 7 Micromagnetism 353 Fig. 1 Nearest neighbors of spin i for the calculation of the exchange energy in a simple cubic lattice m· ∂ 2m ∂m 2 = − , ∂x ∂x 2 (11) which follows from differentiating m · m = 1 twice with respect to x. Thus we can write Eex ∂mi 2 J S2 3 ∂mi 2 ∂mi 2 = a + + . a ∂x ∂y ∂z (12) i The sum in (12) is over the unit cells of the crystal with volume V . In the continuum limit, we replace the sum with an integral. The exchange energy is Eex = A V ∂m ∂x 2 + ∂m ∂y 2 + ∂m ∂z 2 dV . (13) Expanding and rearranging the terms in the bracket and introducing the nabla operator, ∇, we obtain A (∇mx )2 + ∇my Eex = V 2 + (∇mz )2 dV . (14) 354 L. Exl et al. In equations (13) and (14), we introduced the exchange constant A= J S2 n. a (15) In cubic lattices, n is the number of atoms per unit cell (n = 1, 2, and 4 for simple cube, body-centered cubic, and face-centered cubic lattices, respectively). In a hexagonal closed packed structures, n is the ideal nearest neighbor distance √ (n = 2 2). The number N of atoms per unit volume is n/a 3 . At non-zero temperature, the exchange constant may be expressed in terms of the saturation magnetization, Ms (T ). Formally we replace S by its thermal average. Using equations (2) and (3), we rewrite A(T ) = J [Ms (T )]2 n. (NgμB )2 a (16) The calculation of the exchange constant by (15) requires a value for the exchange integral, J . Experimentally, one can measure a quantity that strongly depends on J such as the Curie temperature, TC ; the temperature dependence of the saturation magnetization, Ms (T ); or the spin wave stiffness parameter, in order to determine J. According to the molecular field theory [20], the exchange integral is related to the Curie temperature given by J = 3 k B TC S n 3 k B TC 1 or A = . 2 S(S + 1) z 2 a(S + 1) z (17) The second equation follows from the first one by replacing J with the relation (15). Here z is the number of nearest neighbors (z = 6, 8, 12, and 12 for simple cubic, body-centered cubic, face-centered cubic, and hexagonal closed packed lattices, respectively) and kB =1.3807×10−23 J/K is Boltzmann’s constant. The use of (17) together with (15) underestimates the exchange constant by more than a factor of 2 [21]. Alternatively one can use the temperature dependence of the magnetization as arising from the spin wave theory Ms (T ) = Ms (0)(1 − CT 3/2 ). (18) Equation (18) is valid for low temperatures. From the measured temperature dependence Ms (T ), the constant C can be determined. Then the exchange integral [21, 22] and the exchange constant can be calculated from C as follows: J = 0.0587 nSC 2/3 kB or A = 2S 0.0587 n2 S 2 C 2/3 kB . 2a (19) This method was used by Talagala and co-workers [23]. They measured the temperature dependence of the saturation magnetization in NiCo films to determine 7 Micromagnetism 355 the exchange constant as function of the Co content. The exchange constant can also be evaluated from the spin wave dispersion relation (see Chapter SPW) which can be measured by inelastic neutron scattering, ferromagnetic resonance, or Brillouin light scattering [24]. The exchange integral [22] and the exchange constant are related to the spin wave stiffness constant, D, via the following relations: J = D D 1 or A = NS. 2 2 Sa 2 (20) For the evaluation of the exchange constant, we can use S = Ms (0)/(NgμB ) [25] for the spin quantum number in equations (17), (19), and (20). This gives the relation between the exchange constant, A, and the spin wave stiffness constant, D, A= DMs (0) , 2gμB (21) when applied to (20). Using neutron Brillouin Scattering, Ono and co-workers [26] measured the spin wave dispersion in a polycrystalline Nd2 Fe14 B magnet, in order to determine its exchange constant. Ferromagnetic exchange interactions keep the magnetization uniform. Depending on the sample, geometry external fields may lead to a locally confined non-uniform magnetization. Probing the magnetization twist experimentally and comparing the result with the computed equilibrium magnetic state (see section “Brown’s Micromagnetic Equation”) is an alternative method to determine the exchange constant. The measured data is fitted to the theoretical model whereby the exchange constant is a free parameter. Smith and co-workers [27] measured the anisotropic magnetoresistance to probe the fanning of the magnetization in a thin permalloy film from which its exchange constant was calculated. Eyrich and co-workers [24] measured the field-dependent magnetization, M(H ), of a trilayer structure in which two ferromagnetic films are coupled antiferromagnetically. The M(H ) curve probes the magnetization twist within the two ferromagnets. Using this method the exchange constant of Co alloyed with various other elements was measured [24]. The interplay between the effects of ferromagnetic exchange coupling, magnetostatic interactions, and the magnetocrystalline anisotropy leads to the formation of domain patterns (for details on domain structures, see Chapter Domains). With magnetic imaging techniques, the domain width, the orientation of the magnetization, and the domain wall width can be measured. These values can be also calculated using a micromagnetic model of the domain structure. By comparing the predicted values for the domain width with measured data, Livingston [28, 29] estimated the exchange constant of the hard magnetic materials SmCo5 and Nd2 Fe14 B. This method can be improved by comparing more than one predicted quantity with measured data. Newnham and co-workers [30] measured the domain width, the orientation of the magnetization in the domain, and the domain wall width in foils 356 L. Exl et al. of Nd2 Fe14 B. By comparing the measured values with the theoretical predictions, they estimated the exchange constant of Nd2 Fe14 B. Input for micromagnetic simulations: The high temperature behavior of permanent magnets is of utmost importance for the applications of permanent magnets in the hybrid or electric vehicles. For computation of the coercive field by micromagnetic simulations, the exchange constant is needed as input parameter. Values for A(T ) may be obtained from the room temperature value of A(300 K) and Ms (T ). Applying (16) gives A(T ) = A(300 K) × [Ms (T )/Ms (300 K)]2 . Magnetostatics We now consider the energy of the magnet in an external field produced by stationary currents and the energy of the magnet in the field produced by the magnetization of the magnet itself. The latter field is called demagnetizing field. In micromagnetics, these fields are treated statically if eddy currents are neglected. In magnetostatics, we have no time-dependent quantities. In the presence of a stationary magnetic current, Maxwell’s equations reduce to [31] ∇ ×H = j (22) ∇ ·B = 0 (23) Here B is the magnetic induction or magnetic flux density, H is the magnetic field, and j is the current density. The charge density fulfills ∇ · j = 0 which expresses the conservation of electric charge. We now have the freedom to split the magnetic field into its solenoidal and nonrotational part H = H ext + H demag . (24) By definition, we have ∇ · H ext = 0, (25) ∇ × H demag = 0. (26) Using (22) and (24), we see that the external field, H ext , results from the current density (Ampere’s law) ∇ × H ext = j . (27) On a macroscopic length scale, the relation between the magnetic induction and the magnetic field is expressed by B = μH , (28) 7 Micromagnetism 357 where μ is the permeability of the material. Equation (28) is used in magnetostatic field solvers [32] for the design of magnetic circuits. In these simulations, the permeability describes the response of the material to the magnetic field. Micromagnetics describes the material on a much finer length scale. In micromagnetics, we compute the local distribution of the magnetization as function of the magnetic field. This is the response of the system to (an external) field. Indeed, the permeability can be derived from micromagnetic simulations [33]. For the calculation of the demagnetizing field, we can treat the magnetization as fixed function of space. Instead of (28), we use B = μ0 (H + M) . (29) The energy of the magnet in the external field, H ext , is the Zeeman energy. The energy of the magnet in the demagnetizing field, H demag , is called magnetostatic energy. Zeeman Energy The energy of a magnetic dipole moment, μ, in an external magnetic induction Bext is −μ · Bext . We use B ext = μ0 H ext and sum over all local magnetic moments at positions x i of the ferromagnet. The sum, Eext = −μ0 μi · H ext , (30) i is the interaction energy of the magnet with the external field. To obtain the Zeeman energy in a continuum model, we introduce the magnetization M = Nμ, define the volume per atom, Vi = 1/N, and replace the sum with an integral. We obtain Eext = −μ0 (M · H ext )Vi → −μ0 (M · H ext )dV . (31) V i Using (6), we express the Zeeman energy in terms of the unit vector of the magnetization Eext = − μ0 Ms (m · H ext )dV. (32) V Magnetostatic Energy The magnetostatic energy is also called dipolar interaction energy. In a crystal each moment creates a dipole field, and each moment is exposed to the magnetic field created by all other dipoles. Therefore magnetostatic interactions are long range. The magnetostatic energy cannot be represented as a volume integral over the magnet of an energy density dependent on only local quantities. 358 L. Exl et al. Demagnetizing Field as Sum of Dipolar Fields The total magnetic field at point x i , which is created by all the other magnetic dipoles, is the sum over the dipole fields from all moments μj = μ(x j ) μj 1 (μj · r ij )r ij − 3 . 3 H dip (x i ) = 4π rij5 rij j =i (33) The vectors r ij = x i − x j connect the source points with the field point. The distance between a source point and a field point is rij = |r ij |. In order to obtain a continuum expression for the field, we split the sum (33) into two parts. The contribution to the field from moments that are far from x i will not depend strongly on their exact position at the atomic level. Therefore we can describe them by a continuous magnetization and replace the sum with an integral. For moments μj which are located within a small sphere with radius R around x i , we keep the sum. Thus we split the dipole field into two parts [34]: H dip (x i ) = H near (x i ) + H demag (x i ). (34) Here 1 H near (x i ) = 4π rij <R 3 (μj · r ij )r ij rij5 − μj rij3 (35) is the contribution of the sum of the dipoles within the sphere (see Fig. 2). For the dipoles outside, the sphere we use a continuum approximation. Introducing the magnetic dipole element MdV , we can replace the sum in (33) with an integral for rij ≥ R Fig. 2 Computation of the total magnetostatic field at point atom i. The near field is evaluated by a direct sum over all dipoles in the small sphere. The atomic moments outside the sphere are replaced by a continuous magnetization which produces the far field acting on i 7 Micromagnetism 359 1 H demag (x i ) = 4π V* M(x ) · (x i − x ) (x i − x ) M(x ) − 3 dV . |x i − x |5 |x i − x |3 (36) The integral is over V *, the volume of the magnet without the small sphere around the field point x i . The sum in (35) is the contribution of the dipoles inside the sphere to the total magnetostatic field. The corresponding energy term is local. It can be expressed as an integral of an energy density that depends only on local quantities [34]. The term depends on the symmetry of the lattice and has the same form as the crystalline anisotropy. Therefore it is included in the anisotropy energy. When the anisotropy constants in (58) are determined experimentally, they already include the contribution owing to dipolar interactions. Magnetic Scalar Potential The demagnetizing field is nonrotational. Therefore we can write the demagnetizing field as gradient of a scalar potential H demag = −∇U. (37) Applying −∇ to U (x) = − 1 4π M(x ) · V* x − x dV |x − x |3 (38) gives (36). In computational micromagnetics, it is beneficial to work with effective magnetic volume charges, ρm = −∇ · M(x ), and effective magnetic surface charges, σm = M(x ) · n. Using 1 x − x 1 1 and ∇ = −∇ = −∇ |x − x | |x − x | |x − x | |x − x |3 we obtain 1 U (x) = 4π V* M(x ) · ∇ 1 dV . |x − x | (39) (40) Now we shift the ∇ operator from 1/|x − x | to M. We use ∇ · ∇ · M(x ) 1 M(x ) = + M(x ) · ∇ , |x − x | |x − x | |x − x | apply Gauss’ theorem, and obtain [31] 1 ρm (x ) 1 U (x) = dV + 4π V * |x − x | 4π ∂V * σm (x ) dS . |x − x | (41) (42) 360 L. Exl et al. Magnetostatic Energy For computing the magnetostatic energy, there is no need to take into account (35). The near field is already included in the crystal anisotropy energy. We now compute the energy of each magnetic moment μi in the field H demag (x i ) from the surrounding magnetization. The sum over all atoms is the magnetostatic energy of the magnet Edemag = − μ0 μi · H demag (x i ). 2 (43) i The factor 1/2 avoids counting each pair of atoms twice. Similar to the procedure for the exchange and Zeeman energy, we replace the sum with an integral Edemag = − μ0 2 M · H demag = − V 1 2 μ0 Ms m · H demag dV . (44) V Alternatively, the magnetostatic energy can be expressed in terms of a magnetic scalar potential and effective magnetic charges. We start from (44), replace H demag by −∇U , and apply Gauss’ theorem on ∇ · (MU ) to obtain Edemag = μ0 2 ρm U dV + V μ0 2 σm U dS. (45) ∂V Equation (45) is widely used in numerical micromagnetics. Its direct variation (see section “Brown’s Micromagnetic Equation”) with respect to M gives the cell averaged demagnetizing field. This method was introduced in numerical micromagnetics by LaBonte [35] and Schabes and Aharoni [36]. For discretization with piecewise constant magnetization only, the surface integrals remain. In a uniformly magnetized spheroid, the demagnetizing field is antiparallel to the magnetization. The demagnetizing field is H demag = −N M, (46) where N is the demagnetizing factor. For a sphere the demagnetizing factor is 1/3. Using (44), we find μ0 NMs2 V (47) Edemag = 2 for the magnetostatic energy of a uniformly magnetized sphereoid with volume V . In a cuboid or polyhedral particle, the demagnetizing field is nonuniform. However we still can apply (47) when we use a volume averaged demagnetizing factor which is obtained from a volume-averaged demagnetizing field. Interestingly the volume-averaged demagnetizing factor for a cube is 1/3 the same value as for the sphere. For a general rectangular prism, Aharoni [37] calculated the volume averaged demagnetizing factor. A convenient calculation tool for the demagnetizing 7 Micromagnetism 361 factor, which uses Aharoni’s equation, is given on the Magpar website [38]. A simple approximate equation for the demagnetizing factor of a square prism with dimensions l × l × pl is [39] N= 1 , 2p + 1 (48) where p is the aspect ratio and N is the demagnetizing factor along the edge with length pl. Magnetostatic Boundary Value Problem Equation (42) is the solution of the magnetostatic boundary value problem, which can be derived from Maxwell’s equations as follows. From (23) and (29), the following equation holds for the demagnetizing field ∇ · H demag = −∇ · M. (49) Plugging (37) into (49), we obtain a partial differential equation for the scalar potential ∇ 2 U = ∇ · M. (50) Equation (50) holds inside the magnet. Outside the magnet M = 0 and we have ∇ 2 U = 0. (51) At the magnet’s boundary, the following interface conditions [31] hold U (in) = U (out) , ∇U (in) − ∇U (out) · n = M · n, (52) (53) where n denotes the surface normal. The first condition follows from the continuity of the component of H parallel to the surface (or ∇ ×H = 0). The second condition follows from the continuity of the component of B normal to the surface (or ∇ ·B = 0). Assuming that the scalar potential is regular at infinity, |U (x)| ≤ C 1 for |x| large enough and constant C > 0 |x| (54) the solution of equations (50) to (53) is given by (42). Formally the integrals in (42) are over the volume, V *, and the surface, ∂V *, of the magnet without a small sphere surrounding the field point. The transition from V *→ V adds a term −M/3 to the field and thus shifts the energy by a constant which is proportional to Ms2 . This is usually done in micromagnetics [34]. 362 L. Exl et al. The above set of equations for the magnetic scalar potential can also be derived from a variational principle. Brown [16] introduced an approximate expression = μ0 Edemag V M · ∇U dV − μ0 2 (∇U )2 dV (55) for the magnetostatic energy, Edemag . For any magnetization distribution M(x), the following equation holds Edemag (M) ≥ Edemag (M, U ), (56) where U is an arbitrary function which is continuous in space and regular at infinity [16]. A proof of (56) is given by Asselin and Thiele [40]. The inequality (56) is sharp in the sense that if maximized with respect to the variable U , equality holds in (56) and U is the scalar potential owing to M. Then equality holds and Edemag reduces to the usual magnetostatic energy Edemag . Equation (55) is used in finite element micromagnetics for the computation of the magnetic scalar potential. The EulerLagrange equation of (55) with respect to U gives the magnetostatic boundary value problem (50) to (53) [40]. Examples Magnetostatic energy in micromagnetic software: For physicists and software engineers developing micromagnetic software, there are several options to implement magnetostatic field computation. The choice depends on the discretization scheme, the numerical methods used, and the hardware. Finite difference solvers including OOMMF [41], MuMax3 [42], and FIDIMAG [43] use (45) to compute the magnetostatic energy and the cell-averaged demagnetizing field. For piecewise constant magnetization only, the surface integrals over the surfaces of the computational cells remain. MicroMagnum [44] uses (42) to evaluate the magnetic scalar potential. The demagnetizing field is computed from the potential by a finite difference approximation. This method shows a higher speed up on Graphics Processor Units [45] though its accuracy is slightly less. Finite element solvers compute the magnetic scalar potential and build its gradient. Magpar [46], Nmag [47], and magnum.fe [48] solve the partial differential equations (50) to (53). FastMag [49], a finite element solver, directly integrates (42). Finite difference solvers apply the Fast Fourier Transforms for the efficient evaluation of the involved convolutions. Finite element solvers often use hierarchical clustering techniques for the evaluation of integrals [50]. Magnetic state of nano-elements: From (45), we see that the magnetostatic energy tends to zero if the effective magnetic charges vanish. This is known as pole avoidance principle [34]. In large samples where the magnetostatic energy dominates over the exchange energy, the lowest energy configurations are such that ∇ · M in the volume and M · n on the surface tend to zero. The magnetization is aligned parallel to the boundary and may show a vortex. These patterns are known as flux closure states. In small samples, the expense of exchange energy 7 Micromagnetism 363 Fig. 3 Computed magnetization patterns for a soft magnetic square element (K1 = 0, μ0 Ms = 1 T, A = 10 pJ/m, mesh size h = 0.56 A/(μ0 Ms2 ) = 2 nm) as function of element size L. The dimensions are L × L × 6 nm3 . The system was relaxed multiple times from an initial state with random magnetization. The lowest energy states are the leaf state, the C-state, and the vortex state for L = 80 nm, L = 150 nm, and L = 200 nm, respectively. For each state, the relative contributions of the exchange energy and the magnetostatic energy to the total energy are given for the formation of a closure state is too high. As a compromise the magnetization bends towards the surface near the edges of the sample. Depending on the size, the leaf state [51] or the C-state [52] or the vortex state has the lowest energy. Figure 3 shows the different magnetization patterns that can form in thin film square elements. The results show that with increasing element size the relative contribution of the magnetostatic energy, Fdemag /(Fex + Fdemag ) decreases. All micromagnetic examples in this chapter are simulated using FIDIMAG [43]. Code snippets are given in the appendix. Crystal Anisotropy Energy The magnetic properties of a ferromagnetic crystal are anisotropic. Depending on the orientation of the magnetic field with respect to the symmetry axes of the crystal, the M(H ) curve reaches the saturation magnetization, Ms , at low or high field values. Thus easy directions in which saturation is reached in a low field and hard directions in which high saturation requires a high field are defined. Figure 4 shows the magnetization curve, measured parallel to the easy and hard direction, of a uniaxial material with strong crystal anisotropy. The initial state is a two domain state with the magnetization of the domains parallel to the easy axis. The snapshots 364 L. Exl et al. Fig. 4 Initial magnetization curves with the field applied in the easy direction (dashed line) and the hard direction (solid line) computed for a uniaxial hard magnetic material (Nd2 Fe14√ B at room temperature: K1 = 4.9 MJ/m3 , μ0 Ms = 1.61 T, A = 8 pJ/m, the mesh size is h = 0.86 A/K1 = 1.1 nm). The magnetization component parallel to the field direction is plotted as a function of the external field. The field is given in units of HK . The sample shape is thin platelet with the easy axis in the plane of the film. The sample dimensions are 200 × 200 × 10 nm3 . The insets show snapshots of the magnetization configuration along the curves. The initial state is the two domain state shown at the lower left of the figure of the magnetic states show that domain wall motion occurs along the easy axis and rotation of the magnetization occurs along the hard axis. The crystal anisotropy energy is the work done by the external field to move the magnetization away from a direction parallel to the easy axis. The functional form of the energy term can be obtained phenomenologically. The energy density, eani (m), is expanded in a power series in terms of the direction cosines of the magnetization. Crystal symmetry is used to decrease the number of coefficients. The series is truncated after the first two non-constant terms. Cubic Anisotropy Let a, b, and c be the unit vectors along the axes of a cubic crystal. The crystal anisotropy energy density of a cubic crystal is eani (m) = K0 + K1 (a · m)2 (b · m)2 + (b · m)2 (c · m)2 + (c · m)2 (a · m)2 + K2 (a · m)2 (b · m)2 (c · m)2 + . . . . (57) 7 Micromagnetism 365 The anisotropy constants K0 , K1 , and K2 are functions of temperature. The first term is independent of m and thus can be dropped since only the change of the energy with respect to the direction of the magnetization is of interest. Uniaxial Anisotropy In hexagonal or tetragonal crystals, the crystal anisotropy energy density is usually expressed in terms of sin θ , where θ is the angle between the c-axis and the magnetization. The crystal anisotropy energy of a hexagonal or tetragonal crystal is eani (m) = K0 + K1 sin2 (θ ) + K2 sin4 (θ ) + . . . . (58) In numerical micromagnetics, it is often more convenient to use eani (m) = −K1 (c · m)2 + . . . . (59) as expression for a uniaxial crystal anisotropy energy density. Here we used the identity sin2 (θ ) = 1 − (c · m)2 , dropped two constant terms, namely, K0 and K1 , and truncated the series. When keeping only the terms which are quadratic in m, the crystal anisotropy energy can be discretized as quadratic form involving only a geometry-dependent matrix. The crystalline anisotropy energy is Eani = eani (m)dV , (60) V whereby the integral is over the volume, V , of the magnetic body. Anisotropy Field An important material parameter, which is commonly used, is the anisotropy field, HK . The anisotropy field is a fictitious field that mimics the effect of the crystalline anisotropy. If the magnetization vector rotates out of the easy axis, the crystalline anisotropy creates a torque that brings M back into the easy direction. The anisotropy field is parallel to the easy direction, and its magnitude is such that for deviations from the easy axis, the torque on M is the same as the torque by the crystalline anisotropy. If the energy depends on the angle, θ , of the magnetization with respect to an axis, the torque, T , on the magnetization is the derivative of the energy density, e, with respect to the angle [20] T = ∂e . ∂θ (61) Let θ be a small angular deviation of M from the easy direction. The energy density of the magnetization in the anisotropy field is 366 L. Exl et al. eK = −μ0 Ms HK cos(θ ) (62) TK = μ0 Ms HK sin(θ ) ≈ μ0 Ms HK θ. (63) the associated torque is For the crystalline anisotropy energy density eani = K1 sin2 (θ ) (64) the torque towards the easy axis is Tani = 2K1 sin(θ ) cos(θ ) = K1 sin(2θ ) ≈ 2K1 θ. (65) From the definition of the anisotropy field, namely, TK = Tani , we get HK = 2K1 μ0 Ms (66) Anisotropy field, easy and hard axis loops: K1 and – depending on the material to be studied – K2 are input parameters for micromagnetic simulation. The anisotropy constants can be measured by fitting a calculated magnetization curve to experimental data. Figure 4 shows the magnetization curves of a uniaxial material computed by micromagnetic simulations. For simplicity we neglected K2 and described the crystalline anisotropy with (59). The M(H ) along the hard direction is almost a straight line until saturation where M(H ) = Ms . Saturation is reached when H = HK . The above numerical result can be found theoretically. A field is applied perpendicular to the easy direction. The torque created by the field tends to increase the angle, θ , between the magnetization and the easy axis. The torque asserted by the crystalline anisotropy returns the magnetization towards the easy direction. We set the total torque to zero to get the equilibrium condition −μ0 Ms H cos(θ ) + 2K1 sin(θ ) cos(θ ) = 0. The value of H that makes M parallel to the field is reached when sin(θ ) = 1. This gives H = 2K1 /(μ0 Ms ). If higher anisotropy constants are taken into account the field that brings M into the hard axis is H = (2K1 + 4K2 )/(μ0 Ms ). Magnetoelastic and Magnetostrictive Energy Terms When the atom positions of a magnet are changed relative to each other the crystalline anisotropy varies. Owing to magnetoelastic coupling a deformation produced by an external stress makes certain directions to be energetically more 7 Micromagnetism 367 favorable for the magnetization. Reversely, the magnet will deform in order to minimize its total free energy when magnetized in certain directions. Spontaneous Magnetostrictive Deformation Most generally the spontaneous magnetostrictive deformation is expressed by the 0 as symmetric tensor strain εij 0 εij = (67) λij kl αk αl , kl where λij kl is the tensor of magnetostriction constants. Measurements of the relative change of length along certain directions owing to saturation of the crystal in direction α = (α1 , α2 , α3 ) give the magnetostriction constants. For a cubic material, the following relation holds εii0 3 1 2 , = λ100 αi − 2 3 0 εij = (68) 3 λ111 αi αj for i = j. 2 (69) The magnetostriction constants λ100 and λ111 are defined as follows: λ100 is the relative change in length measured along [100] owing to saturation of the crystal in [100]; similarly λ111 is the relative change in length measured along [111] owing to saturation of the crystal in [111]. The term with 1/3 in (68) results from the definition of the spontaneous deformation with respect to a demagnetized state with the averages αi2 = 1/3 and αi αj = 0. Magnetoelastic Coupling Energy All energy terms discussed in the previous sections can depend on deformations. The most important change of energy with strain arises from the crystal anisotropy energy. Thus the crystal anisotropy energy is a function of the magnetization and the deformation of the lattice. We express the magnetization direction in terms of the direction cosines of the magnetization α1 = a · m, α2 = b · m, and α3 = c · m (a, b, and c are the unit lattice vectors) and the deformation in terms of the symmetric strain tensor εij to obtain eani = eani (αi , εij ). (70) A Taylor expansion of (70) eani = eani (αi , 0) + ∂eani (αi , 0) ij ∂εij εij (71) 368 L. Exl et al. gives the change of the energy density owing to the strain εij . Owing to symmetry, the expansion coefficients ∂eani (αi , 0)/∂εij do not dependent on the sign of the magnetization vector and thus are proportional to αi αj . The second term on the right-hand side of (71) is the change of the crystal anisotropy energy density with deformation. This term is the magnetoelastic coupling energy density. Using ij Bij kl αi αj as expansion coefficients, we obtain eme = ij Bij kl αi αj εkl , (72) kl where Bij kl is the tensor of the magnetoelastic coupling constants. For cubic symmetry, the magnetoelastic coupling energy density is eme,cubic = B1 (ε11 α12 + ε22 α22 + ε33 α32 )+ 2B2 (ε23 α2 α3 + ε13 α1 α3 + ε12 α1 α2 ) + . . . (73) with the magnetoelastic coupling constants B1 = B1111 and B2 = B2323 . Equation (72) describes change of the energy density owing to the interaction of magnetization direction and deformation. The magnetoelastic coupling constants can be derived from the ab initio computation of the crystal anisotropy energy as function of strain [53]. Experimentally the magnetoelastic coupling constants can be obtained from the measured magnetostriction constants. When magnetized in a certain direction, the magnet tends to deform in a way that minimizes the sum of the magnetoelastic energy density, eme , and of the elastic energy density of the crystal, eel . The elastic energy density is a quadratic function of the strain eel = 1 cij kl εij εkl , 2 ij (74) kl where cij kl is the elastic stiffness tensor. For cubic crystals the elastic energy is 1 2 2 2 + ε22 + ε33 )+ eel,cubic = c1111 (ε11 2 c1122 (ε11 ε22 + ε22 ε33 + ε33 ε11 )+ (75) 2 2 2 2c2323 (ε12 + ε23 + ε31 ). Minimizing eme + eel with respect to εij under fixed αi gives the equilibrium strain or spontaneous magnetostrictive deformation 0 0 εij = εij (Bij kl , cij kl ). (76) 7 Micromagnetism 369 in terms of the magnetoelastic coupling constants and the elastic stiffness constants. Comparison of the coefficients in (76) and the experimental relation (67) allows to express the magnetoelastic coupling coefficients in terms of the elastic stiffness constants and the magnetostriction constants. For cubic symmetry the magnetoelastic coupling constants are 3 B1 = − λ100 (c1111 − c1122 ) 2 B2 = −3λ111 c1212. (77) (78) External Stress A mechanical stress of nonmagnetic origin will have an effect on the magnetization owing to a change of magnetoelastic coupling energy. The magnetoelastic coupling energy density owing to an external stress σ ext is [54] eme = − 0 σijext εij . (79) ij For cubic symmetry, this gives [20] 3 eme,cubic = − λ100 (σ11 α12 + σ22 α12 + σ33 α22 ) 2 (80) − 3λ111 (σ12 α1 α2 + σ23 α2 α3 + σ31 α3 α1 ) The above results can be derived from the strain induced by the external stress which is ext εij = sij kl σklext , (81) kl where sij kl is the compliance tensor. Inserting (81) into (72) gives the magnetoelastic energy density owing to external stress. For an isotropic material, for example, an amorphous alloy, we have only a single magnetostriction constant λs = λ100 = λ111 . For a stress σ along an axis of a unit vector a, the magnetoelastic coupling energy reduces to 3 eme,isotropic = − λs σ (a · α)2 . 2 (82) This equation has a similar form as that for the uniaxial anisotropy energy density (59) with an anisotropy constant Kme = 3λs σ/2. Magnetostrictive Self-Energy A nonuniform magnetization causes a nonuniform spontaneous deformation owing to (67). As a consequence, different parts of the magnet do not fit together. To 370 L. Exl et al. el , will occur. The compensate this misfit, an additional elastic deformation, εij associated magnetostrictive self-energy density is emagstr = 1 el el cij kl εij εkl . 2 ij (83) kl el we have to solve an elasticity problem. The total strain, To compute εij el 0 + εij , εij = εij (84) can be derived from a displacement field, u = (u1 , u2 , u3 ), according to [55] 1 εij = 2 ∂uj ∂ui + ∂xj ∂xi . (85) We start from a hypothetically undeformed, nonmagnetic body. If magnetism is 0 causes a stress which we treat as virtual body forces. Once these switched on, εij forces are known, the displacement field can be calculated as usual by linear elasticity theory. The situation is similar to magnetostatics where the demagnetizing field is calculated from effective magnetic charges. The procedure is as follows [56]. First we compute the spontaneous magnetostrictive strain for a given magnetization distribution with (67) or in case of cubic symmetry with (68) and (69). Then we apply Hooke’s law to compute the stress σij0 = 0 cij kl εkl (86) kl owing to the spontaneous magnetostrictive strain. The stress is interpreted as virtual body force fi = − ∂ σ0. ∂xj ij (87) j The forces enter the condition for mechanical equilibrium ∂ σij = fi with σij = cij kl εkl . ∂xj j (88) kl Equations (85) to (88) lead to a system of partial differential equations for the displacement field u(x). This is an auxiliary problem similar to the magnetostatic boundary value problem (see section “Magnetostatic Boundary Value Problem”) which is to be solved for a given magnetization distribution. 7 Micromagnetism 371 Based on the above discussion, we can identify two contributions to the total magnetic Gibbs free energy: The magnetoelastic coupling energy with an external stress 0 Eme = − σijext εij dV (89) V ij and the magnetostrictive self-energy Emagstr = 1 2 V ij 0 0 cij kl (εij − εij )(εkl − εkl )dV . (90) kl Artificial multiferroics: The magnetoelastic coupling becomes important in artificial multiferroic structures where ferromagnetic and piezoelectric elements are combined to achieve a voltage controlled manipulation of the magnetic state [57]. For example, piezoelectric elements can create a strain on a magnetic tunnel junction of about 10−3 causing the magnetization to rotate by 90 degrees [58]. Breaking the symmetry by a stress-induced uniaxial anisotropy, which can be created by a piezoelectric element, the deterministic switching between two metastable states in square nano-element is possible as shown in Fig. 5. 3 Fig. 5 Simulation of the stress-driven switching of a CoFeB nano-element (Ku = 1.32 kJ/m , μ0 Ms = 1.29 T, A = 15 pJ/m, λs = 3 × 10−5 , mesh size h = 0.59 A/(μ0 Ms2 ) = 2 nm, the magnetostrictive self-energy is neglected). The sample is a thin film element with dimensions 120 × 120 × 2 nm3 . The system switches from 0 to 1 by a compressive stress (−0.164 GPa) and from 1 to 0 by a tensile stress (0.164 GPa) 372 L. Exl et al. Characteristic Length Scales To obtain a qualitative understanding of equilibrium states, it is helpful to consider the relative weight of the different energy terms towards the total Gibbs free energy. As shown in Fig. 3, the relative importance of the different energy terms changes with the size of the magnetic sample. We can see this most easily when we write the total Gibbs free energy Etot = Eex + Eext + Edemag + Eani + Eme + Emagstr , (91) in dimensionless form. From the relative weight of the energy contributions in dimensionless form, we will derive characteristic length scales which will provide useful insight into possible magnetization processes depending on the magnet’s size. Let us assume that Ms is constant over the magnetic body (conditions 2 and 3 in section “Micromagnetics Basics”). We introduce the external and demagnetizing field in dimensionless form hext = H ext /Ms and hdemag = H demag /Ms and rescale the length x̃ = x/L, where L is the sample extension. Let us choose L so that tot = Etot /(μ0 Ms2 V ). The L3 = V . We also normalize the Gibbs free energy E 2 normalization factor, μ0 Ms V , is proportional to the magnetostatic self-energy of the fully magnetized sample. The energy contributions in dimensionless form are ext E 2 lex x 2 + ∇m y ∇m 2 L V , =− m · hext d V ex = E demag = − 1 E 2 ani = − E V z + ∇m 2 , dV (92) (93) , m · hdemag d V (94) K1 , (c · m)2 d V μ0 Ms2 (95) V V 2 is the domain after transformation of the length. Further, we assumed where V uniaxial magnetic anisotropy and neglected magnetoelastic coupling and magnetostriction. The constant lex in (92) is defined in the following section. Exchange Length In (92) we introduced the exchange length lex = A . μ0 Ms2 (96) It describes the relative importance of the exchange energy with respect to the magnetostatic energy. Inspecting the factor (lex /L)2 in front of the brackets in (92), 7 Micromagnetism 373 we see that the exchange energy contribution increases with decreasing sample size L. The smaller the sample, the higher is the expense of exchange energy for nonuniform magnetization. Therefore small samples show a uniform magnetization. If the magnetization remains parallel during switching, the Stoner-Wohlfarth [59] model can be applied. In the literature, the exchange length is either defined by (96) = 2A/(μ M 2 ) [61]. [60] or by lex 0 s Critical Diameter for Uniform Rotation In a sphere with uniaxial anisotropy, the magnetization reverses uniformly if its diameter is below D ≤ Dcrit = 10.2lex [60]. During uniform rotation of the magnetization, the exchange energy is zero, and the magnetostatic energy remains constant. It is possible to lower the magnetostatic energy during reversal by magnetization curling. Then the magnetization becomes nonuniform at the expense of exchange energy. The total energy will be smaller than for uniform rotation if the sphere diameter, D, is larger than Dcrit . Nonuniform reversal decreases the switching field as compared to uniform rotation. The switching fields of a sphere are [60] Hc = 2K1 for D ≤ Dcrit . μ0 Ms (97) Hc = 2K1 1 34.66A − Ms + for D > Dcrit . μ0 Ms 3 μ0 Ms D 2 (98) In cuboids and particles with polyhedral shape, the nonuniform demagnetizing field causes a twist of the magnetization near edges or corners [62]. As a consequence nonuniform reversal occurs for particle sizes smaller than Dcrit . The interplay between exchange energy and magnetostatic energy also causes a size dependence of the switching field [63, 64]. Grain size dependence of the coercive field. The coercive field of permanent magnets decreases with increasing grain size. This can be explained by the different scaling of the energy terms [64, 65]. The smaller the magnet, the more dominant is the exchange term. Thus it costs more energy to form a domain wall. To achieve magnetization reversal, the Zeeman energy of the reversed magnetization in the nucleus needs to be higher. This can be accomplished by a larger external field. Figure 6 shows the switching field a Nd2 Fe14 B cube as a function of its edge length. In addition we give the theoretical switching field for a sphere with the same volume according to (97) and (98). Magnetization reversal occurs by nucleation and expansion of reversed domains unless the hard magnetic cube is smaller than 6lex . 3 Fig. 6 Computed grain size √ dependence of the coercive field of a perfect Nd2 Fe14 B cube at room temperature (K1 = 4.9 MJ/m , μ0 Ms = 1.61 T, A = 8 pJ/m, the mesh size is h = 0.86 A/K1 = 1.1 nm, the external field is applied at an angle of 10−4 rad with respect to the easy axis). The sample dimensions are L × L × L nm3 . Left: Switching field as function of L in units of HK . The squares give the switching field of the cube. The dashed line is the theoretical switching field of a sphere with the same volume. A switching field smaller than HK indicates nonuniform reversal. Right: Snapshots of the magnetic states during switching for L = 10 nm and L = 80 nm 374 L. Exl et al. 7 Micromagnetism 375 Wall Parameter The square root of the ratio of the exchange length and the prefactor of the crystal anisotropy energy gives another critical length. The Bloch wall parameter δ0 = A K (99) denotes the relative importance of the exchange energy versus crystalline anisotropy energy. It determines the width of the transition of the magnetization between two magnetic domains. In a Bloch wall, the magnetization rotates in a way so that no magnetic volume charges are created. The mutual competition between exchange and anisotropy determines the domain wall width: Minimizing the exchange energy favors wide transition regions, whereas minimizing the crystal anisotropy energy favors narrow transition regions. In a bulk uniaxial material the wall width is δB = π δ0 . Single Domain Size With increasing particle, the prefactor (lex /L)2 for the exchange energy in (92) becomes smaller. A large particle can break up into magnetic domains because the expense of exchange energy is smaller than the gain in magnetostatic energy. In addition to the exchange energy, the transition of the magnetization in the domain wall √ also increases the crystal anisotropy energy. The wall energy per unit area is 4 AK1 . The energy of uniformly magnetized cube is its magnetostatic energy, Edemag1 = μ0 Ms2 L3 /6. In the two domain states, the magnetostatic energy is roughly one half value, Edemag2 = μ0 Ms2 L3 /12. The energy of the wall √ of this 2 is Ewall2 = 4 AK1 L . Equating the energy of the single domain state, Edemag1 , with the energy of the two domain state, Edemag2 + Ewall2 , and solving for L give the single domain size of a cube LSD ≈ √ 48 AK1 . μ0 Ms2 (100) The above equation simply means that the energy of a ferromagnetic cube with a size L > LSD is lower in a the two domain state than in the uniformly magnetized state. A thermally demagnetized sample with L > LSD most likely will be in a multidomain state. We have to keep in mind that the magnetic state of a magnet depends on its history and whether local or global minima can be accessed over the energy barriers that separate the different minima. The following situations may arise: (1) A particle in its thermally demagnetized state is multidomain although L < LSD [66]. When cooling from the Curie temperature, a particle with L < LSD may end up in a multidomain state. Although the single domain state has a lower energy, it cannot be accessed because it is separated from the multidomain 376 L. Exl et al. state by a high energy barrier. This behavior is observed in small Nd2 Fe14 B particles [66]. (2) An initially saturated cube with L > LSD will not break up into domains spontaneously if its anisotropy field is larger than the demagnetizing field. The sample will remain in an almost uniform state until a reversed domain is nucleated. (3) Magnetization reversal of a cube with L < LSD will be nonuniform. Switching occurs by the nucleation and expansion of a reversed domain for a particle size down to about 5lex . For example in Nd2 Fe14 B, the single domain limit is LSD ≈ 146 nm, and the exchange length is lex = 1.97 nm. The simulation presented in Fig. 6 shows the transition from uniform to nonuniform reversal which occurs at L ≈ 6lex . Mesh Size in Micromagnetic Simulations The required minimum mesh size in micromagnetic simulations depends on the process that should be described by the simulations. Here are a few examples: (1) For computing the switching field of a magnetic particle, we need to describe the formation of a reversed nucleus. A reversed nucleus is formed near edges or corners where the demagnetizing field is high. We have to resolve the rotations of the magnetization that eventually form the reversed nucleus. For the computation of the nucleation field the required minimum mesh size has to be smaller than the exchange length [61] at the place where the initial nucleus is formed. (2) For the simulation of domain wall motion, the transition of the magnetization between the domains needs to be resolved. A failure to do so will lead to an artificial pinning of the domain wall on the computational grid [67]. For the study of domain wall motion in hard magnetic materials, the required minimum mesh size has to be smaller than the Bloch wall parameter. (3) In soft magnetic elements with vanishing crystal or stress-induced ansisotropy, the magnetization varies continuously [68]. The smooth transitions of the magnetization transitions can be resolved with a grid size larger than the exchange length. Care has to be taken if vortices play a role in the magnetization process to be studied. Then artificial pinning of vortex cores on the computational grid [67] has to be avoided. Brown’s Micromagnetic Equation In the following, we will derive the equilibrium equations for the magnetization. The total Gibbs free energy of a magnet is a functional of m(x). To compute an equilibrium state, we have to find the function m(x) that minimizes Etot taking into account |m(x)| = 1. In addition the boundary conditions 7 Micromagnetism 377 ∇mx · n = 0, ∇my · n = 0, and ∇mz · n = 0 (101) hold, where n is the surface normal. The boundary conditions follow from (11) and the respective equations for y and z and applying Green’s first identity to each term of (14). The boundary conditions (101) can also be understood intuitively [15]. To be in equilibrium, a magnetic moment at the surface has to be parallel with its neighbor inside when there is no surface anisotropy. Otherwise there is an exchange torque on the surface spin. Most problems in micromagnetics can only be solved numerically. Instead of solving the Euler-Lagrange equation that results from the variation of (91) numerically, we directly solve the variational problem. Direct methods [69, 70] represent the unknown function by a set of discrete variables. The minimization of the energy with respect to these variables gives an approximate solution to the variational problem. Two well-known techniques are the Euler method and the Ritz method. Both are used in numerical micromagnetics. Euler Method: Finite Differences In finite difference micromagnetics, the solution m(x) is sampled on points (xi , yj , zk ) so that mij k = m(xi , yj , zk ). On a regular grid with spacing h, the positions of the grid points are xi = x0 + ih, yj = y0 + j h, and zk = z0 + kh. The points (xi , yj , zk ) are the cell centers of the computational grid. The magnetization is assumed to be constant within each cell. To obtain an approximation of the energy functional, we apply the trapezoidal rule; more precisely, we replace m(x) by the values at the cell centers mij k and the spatial derivatives of m(x) with the finite difference quotients. The approximated solution values mij k are the unknowns of an algebraic minimization problem. The indices i, j , and k run from 1 to the number of grid points Nx , Ny , Nz in x, y, and z direction, respectively. In the following, we will derive the equilibrium equations whereby for simplicity we will not take into account the magnetoelastic coupling energy and the magnetostrictive self-energy. We can approximate the exchange energy (14) on the finite difference grid as [71] Eex 2Ai+1j k Aij k mx,i+1j k − mx,ij k 2 ≈ h3 + ··· , Ai+1j k + Aij k h (102) ij k where we introduced the notation Aij k = A(xi , yj , zk ). The prefactor in (102) is the harmonic mean of the values for the exhange constants in cells i + 1j k and ij k. This follows from the interface condition Ai+1j k (mx,i+1j k − mx,interface )/(h/2) = Aij k (mx,interface − mx,ij k )/(h/2), where the minterface is the magnetization at the interface between the two cells. Eext ≈ −μ0 h3 ij k Ms,ij k (mij k · H ext,ij k ). (103) 378 L. Exl et al. To approximate the magnetostatic energy, we use (42) and (45). Replacing the integrals with sums over the computational cell, we obtain Edemag ≈ μ0 Ms,ij k Ms,i j k 8π ∂Vij k ij k i j k ∂Vi j k (mij k · n)(mi j k · n ) dSdS . |x − x | (104) The volume integrals in (42) and (45) vanish when we assume that m(x) is constant within each computational cell ij k. The magnetostatic energy is often expressed in terms of the demagnetizing tensor Nij k,i j k Edemag ≈ μ0 3 h Ms,ij k mTij k Nij k,i j k mi j k Ms,i j k 2 (105) ij k i j k We approximate the anisotropy energy (60) by Eani ≈ h3 (106) eani (mij k ). ij k The total energy is now a function of the unknowns mij k . The constraint (5) is approximated by |mij k | = 1 (107) where ij k runs over all computational cells. We obtain the equilibrium equations from differentiation ⎤ ⎡ Lij k ∂ ⎣ (mij k · mij k − 1)⎦ = 0, Etot (. . . , mij k , . . . ) + ∂mij k 2 ⎡ ∂ ⎣ Etot (. . . , mij k , . . . ) + ∂Lij k ij k Lij k ij k 2 (108) ⎤ (mij k · mij k − 1)⎦ = 0. (109) In the brackets we added a Lagrange function to take care of the constraints (107). Lij k are Lagrange multipliers. From (108) we obtain the following set of equations for the unknowns mij k −2Aij k h3 2Ai−1j k mi+1j k − mij k mi−1j k − mij k 2Ai+1j k + + · · · Ai+1j k + Aij k Ai−1j k + Aij k h2 h2 −μ0 Ms,ij k h3 H ext,ij k (110) 7 Micromagnetism +μ0 Ms,ij k h3 379 N ij k,i j k mi j k Ms,i j k i j k +h3 ∂eani = −Lij k mij k . ∂mij k The term in brackets is the Laplacian discretized on a regular grid. First-order equilibrium conditions require also zero derivative with respect to the Lagrange multipliers. This gives back the constraints (107). It is convenient to collect all terms with the dimensions of A/m to the effective field H eff,ij k = H ex,ij k + H ext,ij k + H demag,ij k + H ani,ij k. (111) The exchange field, the magnetostatic field, and the anisotropy field at the computational cell ij k are H ex,ij k = 2Aij k μ0 Ms,ij k H demag,ij k = − 2Ai−1j k mi+1j k − mij k 2Ai+1j k + Ai+1j k + Aij k Ai−1j k + Aij k h2 mi−1j k − mij k + · · · (112) h2 Nij k,i j k mi j k Ms,i j k (113) i j k H ani,ij k = − ∂eani 1 , μ0 Ms,ij k ∂mij k (114) respectively. The evaluation of the exchange field (112) requires values of mij k outside the index range [1, Nx ] × [1, Ny ] × [1, Nz ]. These values are obtained by mirroring the values of the surface cell at the boundary. This method of evaluating the exchange field takes into account the boundary conditions (101). Using the effective field, we can rewrite the equilibrium equations μ0 Ms,ij k h3 H eff,ij k = Lij k mij k . (115) Equation (115) states that the effective field is parallel to the magnetization at each computational cell. Instead of (115) we can also write μ0 Ms,ij k h3 mij k × H eff,ij k = 0. (116) The expression Ms,ij k h3 mij k is the magnetic moment of computational cell ij k. Comparison with (1) shows that in equilibrium the torque for each small volume element h3 (or computational cell) has to be zero. The constraints (107) also have to be fulfilled in equilibrium. 380 L. Exl et al. Ritz Method: Finite Elements Within, the framework of the Ritz method the solution is assumed to depend on a few adjustable parameters. The minimization of the total Gibbs free energy with respect to these parameters gives an approximate solution [15, 16]. In the following we describe a famous and computationally efficient Ritz method, namely, the finite element ansatz. Most finite element solvers for micromagnetics use a magnetic scalar potential for the computation of the magnetostatic energy. This goes back to Brown [16] who introduced an expression for the magnetostatic energy, Edemag (m, U ), in terms of the scalar potential for the computation of equilibrium magnetic states using the Ritz method. We replace Edemag (m) with Edemag (m, U ), as introduced in (55), in the expression for the total energy. The vector m(x) is expanded by means of basis functions ϕi with local support around node x i mfe (x) = (117) ϕi (x)mi . i Similarly, we expand the magnetic scalar potential U fe (x) = (118) ϕi (x)Ui . i The index i runs over all nodes of the finite element mesh. The expansion coefficients mi and Ui are the nodal values of the unit magnetization vector and the magnetic scalar potential, respectively. We assume that the constraint |m| = 1 is fulfilled only at the nodes of the finite element mesh. We introduce a Lagrange function; Li are the Lagrange multipliers at the nodes of the finite element mesh. By differentiation with respect to mi , Ui , and Li , we obtain the equilibrium conditions ∂ ∂mi ∂ ∂Ui ∂ ∂Li Etot (. . . , mi , Ui . . . ) + Li i Etot (. . . , mi , Ui . . . ) + Li i Etot (. . . , mi , Ui . . . ) + 2 2 Li i 2 (mi · mi − 1) = 0, (119) (mi · mi − 1) = 0, (120) (mi · mi − 1) = 0. From (119) we obtain the following set of equations for the unknowns mi (121) 7 Micromagnetism 381 2 A∇ϕi · ∇ϕj dV mj V j − μ0 Ms H ext ϕi dV V + (122) μ0 Ms ∇U ϕi dV V ∂eani ( + V j ϕj mj ) ∂mi dV = −Li mi . Equation (120) is the discretized form of the partial differential equation (50) for the magnetic scalar potential. Equation (121) gives back the constraint |m| = 1. In the following, we introduce the effective field at the nodes of the finite element mesh H eff,i = − 2 μ0 M j A∇ϕi · ∇ϕj dV mj V + H ext,i + H demag,i 1 − μ0 M V (123) ∂eani dV , ∂mi where M = V Ms ϕi dV . H demag,i is the demagnetizing field averaged over the finite elements surrounding node i. This average can be computed by plugging (118) into the third line of (122) and dividing the resulting expression by −μ0 M. The equilibrium equations are μ0 MH eff,i = Li mi . (124) We can write the equilibrium conditions in terms of a cross product of the magnetic moment, Mmi , and the effective field at node i μ0 Mmi × H eff,i = 0. (125) The system is in equilibrium if the torque equals zero and the constraint |mi | = 1 is fulfilled on all nodes of the finite element mesh. Instead of a Lagrange function for keeping the constraint |m| = 1, projection methods [72] are commonly used in fast micromagnetic solvers [73]. In the iterative scheme for solving (125), the search direction d k+1 is projected onto a plane i perpendicular to mki , corresponding to first-order approximation of the constraint is normalized. at node i. After each iteration k, the vector mk+1 i 382 L. Exl et al. Magnetization Dynamics Brown’s equations describes the conditions for equilibrium. In many applications, the response of the system to a time varying external field is important. The equations by Landau-Lifshitz [74] or Gilbert [75] describes the time evolution of the magnetization. The Gilbert equation in Landau-Lifshitz form |γ |μ0 ∂m |γ |μ0 α =− m × H eff − m × (m × H eff ) ∂t 1 + α2 1 + α2 (126) is widely used in numerical micromagnetics. Here |γ | = 1.76086 × 1011 s−1 T−1 is the gyromagnetic ratio and α is the Gilbert damping constant. In (126) the unit vector of the magnetization and the effective field at the grid point of a finite difference grid or finite element mesh may be used for m and H eff . The first term of (126) describes the precession of the magnetization around the effective field. The last term of (126) describes the damping. The double cross product gives the motion of the magnetization towards the effective field. The interplay between the precession and the damping terms leads to damped oscillations of the magnetization around its equilibrium state. In the limiting case of small deviations from equilibrium and uniform magnetization, the amplitude of the oscillations decay as [76] a(t) = Ce−t/t0 . (127) For small damping, the oscillations decay time is [76] t0 = 2 . αγ μ0 Ms (128) Switching of magnetic nano-elements. Small thin film nano-elements are key building blocks of magnetic sensor and storage applications. By application of a short field pulse, a thin film nano-element can be switched. After reversal, the system relaxes to its equilibrium state by damped oscillations. Figure 7 shows the switching dynamics of a NiFe film with a length of 100 nm, a width of 20 nm, and a thickness of 2 nm. In equilibrium the magnetization is parallel to the long axis of the particle (x axis). A Gaussian field pulse (dotted line in Fig. 7) is applied in the (-1,-1,-1) direction. After the field is switched off the magnetization oscillates towards the long axis of the film. From an exponential fit to the envelope of the magnetization component, My (t), parallel to the short axes, we derived the characteristic decay times of the oscillation which are t0 ≈ 0.613 ns and t0 ≈ 0.204 ns for a damping constant of α = 0.02 and α = 0.06, respectively. According to (128), the difference between the two relaxations times is a factor of 3, given by the ratio of the damping constants. Fig. 7 Switching of a thin film nano-element by a short field pulse in the (-1,-1,-1) direction for α = 0.06 (top row) and α = 0.02 (bottom row). (K1 = 0, μ0 Ms = 1 T, A = 10 pJ/m, mesh size h = 0.56 A/(μ0 Ms2 ) = 2 nm). The sample dimensions are 100 × 20 × 2 nm3 . The sample is originally magnetized in the +x direction. Left: Magnetization as function of time. The thin dotted line gives the field pulse, Hext (t). Once the field is switched off damped oscillations occur which are clearly seen in My (t). The bold grey line is a fit to the envelope of the magnetization component parallel to the short axis. Right: Transient magnetic states. The numbers correspond to the black dots in the plot of My (t) on the left 7 Micromagnetism 383 384 L. Exl et al. Acknowledgments The authors thank the Austrian Science Fund (FWF) under grant No. F4112 SFB ViCoM and grant No. P31140-N32 for financial support. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged. Appendix The intrinsic material properties listed in Table 1 are taken from [77]. The exchange lengths and the wall parameter are calculated as follows: l = A/(μ0 Ms2 ), δ0 = ex √ A/|K1 |. Table 1 Intrinsic magnetic properties and characteristic lengths of selected magnetic materials Material Fe Co Ni Ni0.8 Fe0.2 CoPt Nd2 Fe14 B SmCo5 Sm2 Co17 Fe3 O4 TC (K) 1044 1360 628 843 840 588 1020 1190 860 μ0 Ms (T) 2.15 1.82 0.61 1.04 1.01 1.61 1.08 1.25 0.6 A(pJ/m) 22 31 8 10 10 8 12 16 7 K1 (kJ/m3 ) 48 410 -5 -1 4900 4900 17200 4200 -13 lex (nm) 2.4 3.4 5.2 3.4 3.5 2.0 3.6 3.6 4.9 δ0 (nm) 21 8.7 40 100 1.4 1.3 0.8 2.0 23 The examples given in Figs. 3 to 7 were computed using the micromagnetic simulation environment FIDIMAG [43]. FIDIMAG solves finite difference micromagnetic problems using a Python interface. The reader is encouraged to run computer experiments for further exploration of micromagnetism. In the following we illustrate the use of the Python interface for simulating the switching dynamics of a magnetic nano-element (see Fig. 7). The function relax_system computes the initial magnetic state. The function apply_field computes the response of the magnetization under the influence of a time varying external field. import numpy a s np from f i d i m a g . m i c r o import Sim from f i d i m a g . common import CuboidMesh from f i d i m a g . m i c r o import UniformExchange , Demag from f i d i m a g . m i c r o import TimeZeeman mu0 = 4 ∗ np . p i ∗ 1 e−7 7 Micromagnetism A Ms 385 = 1 . 0 e −11 = 1 . / mu0 d e f r e l a x _ s y s t e m ( mesh ) : sim = Sim ( mesh , name= ’ r e l a x ’ ) sim . d r i v e r . s e t _ t o l s ( r t o l =1e −10 , a t o l =1e −10) sim . d r i v e r . a l p h a = 0 . 5 sim . d r i v e r . gamma = 2 . 2 1 1 e5 sim . Ms = Ms sim . d o _ p r e c e s s i o n = F a l s e sim . s e t _ m ( ( 0 . 5 7 7 3 5 0 2 6 9 , 0 . 5 7 7 3 5 0 2 6 9 , 0 . 5 7 7 3 5 0 2 6 9 ) ) sim . add ( U n i f o r m E x c h a n g e (A=A) ) sim . add ( Demag ( ) ) sim . r e l a x ( ) np . s a v e ( ’m0 . npy ’ , sim . s p i n ) d e f a p p l y _ f i e l d ( mesh ) : sim = Sim ( mesh , name= ’ dyn ’ ) sim . d r i v e r . s e t _ t o l s ( r t o l =1e −10 , a t o l =1e −10) sim . d r i v e r . a l p h a = 0 . 0 2 sim . d r i v e r . gamma = 2 . 2 1 1 e5 sim . Ms = Ms sim . s e t _ m ( np . l o a d ( ’m0 . npy ’ ) ) sim . add ( U n i f o r m E x c h a n g e (A=A) ) sim . add ( Demag ( ) ) s i g m a = 0 . 1 e−9 def gaussian_fun ( t ) : r e t u r n np . exp ( −0.5 ∗ ( ( t −3∗ s i g m a ) / s i g m a ) ∗ ∗ 2 ) mT = 0 . 0 0 1 / mu0 zeeman = TimeZeeman ([ −100 ∗ mT, −100 ∗ mT, −100 ∗ mT] , t i m e _ f u n = g a u s s i a n _ f u n , name= ’H ’ ) sim . add ( zeeman , s a v e _ f i e l d = T r u e ) sim . r e l a x ( d t = 1 . e −12 , m a x _ s t e p s = 10000) i f __name__ == ’ __main__ ’ : mesh = CuboidMesh ( nx =50 , ny =10 , nz =1 , dx =2 , dy =2 , dz =2 , u n i t _ l e n g t h =1e −9) r e l a x _ s y s t e m ( mesh ) a p p l y _ f i e l d ( mesh ) 386 L. Exl et al. References 1. Fukuda, H., Nakatani, Y.: Recording density limitation explored by head/media cooptimization using genetic algorithm and GPU-accelerated LLG. IEEE Trans. Magn. 48(11), 3895–3898 (2012) 2. 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Z. Sowjetunion 8(153), 101–114 (1935) 75. Gilbert, T.: A lagrangian formulation of the gyromagnetic equation of the magnetization field. Phys. Rev. 100, 1243 (1955) 76. Miltat, J., Albuquerque, G., Thiaville, A.: An introduction to micromagnetics in the dynamic regime. In: Hillebrands, B., Ounadjela, K. (eds.) Spin Dynamics in Confined Magnetic Structures, pp. 1–34. Springer, Berlin (2002) 77. Coey, M.D.J.: Magnetism and Magnetic Materials:. Cambridge University Press, Cambridge 001 (2001) Lukas Exl studied mathematics and computational physics and received his PhD from TU-Wien in 2014. He is currently running the project “Reduced Order Approaches in Micromagnetism” at WPI. He works on computational methods in magnetism and quantum mechanics with emphasis on (data-driven) PDEs and model reduction. He is Senior Scientist at the University of Vienna and lectures numerical methods. Dieter Suess received his PhD from the TU-Wien in 2002 where he completed his Habilitation in 2007 in “Computational Material Science.” In 2006 he proposed “Exchange Spring Media” for recording. Since 2018 he is assoc. Prof. and Group Speaker of the “Physics of Functional Materials” group at the University of Vienna. 390 L. Exl et al. Thomas Schrefl received his PhD from the TU-Wien in 2002 where he completed his Habilitation in 1999 in “Computational Physics.” He worked on the development of numerical micromagnetic solvers for application in magnetic recording and permanent magnet. He his head of the Center for Modelling and Simulation at Danube University Krems, Austria 8 Magnetic Domains Rudolf Schäfer Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relevance of Domains and Domain Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Magnetic Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Driving Forces for Domain Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interplay of Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain Walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain Wall Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Domain Wall Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Current-Driven Domain Wall Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 393 398 399 404 406 409 414 415 420 426 432 Abstract Magnetic domains are the basic elements of the magnetic microstructure of magnetically ordered materials. They are formed to minimize the total energy, with the stray field energy being the most significant contribution. The reordering of domains in magnetic fields determines the magnetization curve, domains can be engineered on purpose, and they can be applied in devices. In this chapter a review of the basics of magnetic domains is presented. It will be shown how the magnetic energies act together to determine the domain character and how R. Schäfer () Institute for Metallic Materials, Leibniz Institute for Solid State and Materials Research (IFW) Dresden, Dresden, Germany Institute for Materials Science, Dresden University of Technology, Dresden, Germany e-mail: [email protected] © Springer Nature Switzerland AG 2021 J. M. D. Coey, S. S. P. Parkin (eds.), Handbook of Magnetism and Magnetic Materials, https://doi.org/10.1007/978-3-030-63210-6_8 391 392 R. Schäfer domains can be classified. Domain walls and their dynamics, both field- and current-driven, will be addressed. Introduction According to Fig. 1, magnetically ordered materials may be described by five scaledependent hierarchic levels [1]: atomic level theory, level (1), explains the origin and magnitude of magnetic moments, crystal anisotropy, or magnetoelastic interactions, and it deals with the arrangement of spins on the crystal lattice sites. The theory works on a microscopic, sub-nanometer scale level. The other extreme level, the magnetization Curve in level (5), describes the average magnetization of a specimen as a function of applied magnetic field and may be seen as a macroscopic descriptive level. These two extreme levels are interlinked by magnetic microstructure analysis, levels (2) to (4) in Fig. 1. Here the individual atomic magnetic moments, defined in 5. Magnetization Curve (independent of scale) M/Ms 1 H/Ha 1 sis c eti gn Ma eA tur 2. Micromagnetic Analysis (1 - 1000 nm) uc str ly na cro Mi H 3. Domain Analysis (1 - 1000 µm) 1. Atomic Level Theory (< 1 nm) 4. Phase Analysis (> 100 µm) Fig. 1 Five descriptive levels of magnetically ordered materials, illustrating the link-function of magnetic microstructure between atomic foundations and technical applications of magnetic materials. The anisotropy field Ha , used in the magnetization curve, is defined as Ha = 2K1 /μ0 Ms with K1 and Ms being the first-order cubic anisotropy constant and saturation magnetization, respectively. Indicated are the sample dimensions for which the five concepts are applicable. The domain image in level (3) was obtained by magneto-optical Kerr microscopy on the (100) surface of an Fe3wt%Si-sheet of 0.5 mm thickness, and the M(H )-loop in level (5) was calculated by magnetic phase theory for the configuration in level (4). (The domain image at level (3) is adapted by permission from Ref. [1] (c) Springer 1998) 8 Magnetic Domains 393 level (1), are no longer considered. One rather sums them in a certain neighborhood and represents the local average over a small volume of magnetic moments by a classical magnetization vector M – the discrete, quantum mechanical, and statistical properties of the spins and elementary moments are thus ignored. For a constant temperature this vector has a constant length, the technical saturation magnetization Ms . Then one can divide the M-vector by the saturation magnetization, leading to the unit vector of magnetization m(r) = M(r)/Ms with m2 = 1. For this mesoscopic approach in the description of magnetic materials, it does not matter whether a material is ferromagnetic or ferrimagnetic, as the latter is also characterized by a net magnetization vector. The purpose of magnetic microstructure analysis is the determination of the vector field m(r) and its response to a magnetic field. The magnetization vectors are typically arranged as magnetic domains, so Domain Analysis – level (3) – is in the center of magnetic microstructure analysis. Level (2), the continuum theory of micromagnetics, deals with the connecting elements between the domains, the domain walls, and their substructures. Level (4), phase theory, ignores the specific arrangement of the domains and rather focuses on their volume distribution by collecting all domains that are magnetized along a specific direction in a domain phase. The rearrangement of the phases in a magnetic field finally leads to the (idealized) magnetization curve. So the domains are at the end responsible for magnetization curves. In a conventional definition, magnetic domains are uniformly magnetized regions that appear spontaneously in otherwise unstructured ferro- or ferrimagnetic samples [1]. In the example presented in the center of Fig. 1, the domains are well-ordered owing to the facts that the surface of the crystal is “well”-oriented, meaning that it contains two easy anisotropy axes for the magnetization, that the crystal is largely free from internal mechanical stress, and that no magnetic field is applied to the specimen. In general, however, magnetic domains do not have to be uniformly magnetized, and they can be more or less complex depending on many circumstances. To give an impression of the variability of domain patterns, a collage of selected domain images of various magnetic materials is shown in Fig. 2. In this chapter a review of some basic aspects of magnetic domains is presented that is based on an earlier textbook on magnetic domains [1]. The magneto-optical Kerr images were partly taken and adapted from the book. Relevance of Domains and Domain Analysis The magnetization curve in Fig. 1 is calculated under the assumption of an infinitely extended specimen, the arrangement of domain phases rather than individual domains is considered, and the properties of domain walls are completely ignored. In this phase-theoretical approach, it is further assumed that the domain phase volumes can freely reach their optimum equilibrium values, thus ignoring coercivity and irreversibility effects. The result is a reversible, vectorial magnetization 394 R. Schäfer NiFe film NiFe film dCo /Si/G Co Co crysta l Fe lass cg talli film me et he s Si Fe NiFe FeS i sh Ga rne t fi eet lm FeSi shee t Fig. 2 Collection of domain images, obtained by Kerr microscopy on various magnetic materials. (Most domain images are adapted by permission from Ref. [1] (c) Springer 1998) curve that may be used as a theoretical reference. In practice, however, those conditions are rarely met. Applied materials have a polycrystalline-, amorphous-, or nanocrystalline microstructure; they are of finite size in the shape of particles, films, ribbons, sheets, or bulk magnets; domain walls may be pinned at defects and grain boundaries; or they may interact with each other in case of thin films; mechanical stress can influence the local preference of the magnetization direction; and the domains and effective magnetic field are strongly influenced by surfaces and the sample geometry. All these features may have strong effects on the measured magnetization curve leading to coercivity and irreversibility. Therefore the correct interpretation of measured hysteresis curves often requires the experimental analysis of the domains that are responsible for the loop and of their (in general) irreversible response to magnetic fields, known as magnetization process. Figure 3 demonstrates such processes for four different magnetic films with strong perpendicular anisotropy, i.e., they all have an easy axis for the magnetization that is aligned perpendicular to the film plane. In the course of the experiments, a strong positive magnetic field was applied along the anisotropy axis, and then the field was inverted and successively increased in the opposite direction, thus following the upper branch of the shown hysteresis loops. The magnetization process is initiated by domain nucleation, followed by domain wall motion (upper three rows 8 Magnetic Domains 1 395 -15.5 mT M/Ms 0 -16.2 mT -17.5 mT -17.9 mT -7.8 mT -7.9 mT -8.1 mT -300 mT -370 mT -400 mT -20 mT -67 mT -106 mT µ0H in mT 20 µm -1 -40 -20 0 20 40 1 -7.0 mT 0 100 µm -1 -40 -20 0 20 40 1 -200 mT 0 -1 10 µm -1000 -500 0 500 1000 1 -1 mT 0 10 µm -1 -1000 -500 0 500 1000 Fig. 3 Domain nucleation and growth in magnetic films with strong perpendicular anisotropy, together with the hysteresis loops along which the domains were imaged. First row: [Co (0.3 nm)/Pt (0.7 nm)]3 multilayer (sample courtesy D. Makarov, Dresden). Second row: Pt (3 nm)/Co (1 nm)/Pt (1.5 nm) trilayer (sample courtesy P.M. Shepley and T.A. Moore, Leeds). Third row: FePt film, 16 nm thick (sample courtesy P. He and S.M. Zhou, Tongji [4]). Fourth row: FePd(11 nm)/FePt (24 nm) double layer (Sample courtesy L. Ma and S.M. Zhou, Tongji [3]) in the figure) or proceeding nucleation (lower row). For such polycrystalline thin films, the domain character is more determined by domain wall coercivity effects due to defects and roughness, randomness of the sample nanostructure, etc. rather than by an equilibrium of magnetic energies [2, 3]. Typical for such films is a slow creeping of the nucleated reversed domains into the still saturated area, indicating thermally activated processes. The domains shown can therefore hardly be predicted by domain theory – the only way to interpret the hysteresis curves is by observation of the domains that are responsible for the loops. This is also true for the M(H )-loop of the amorphous ribbon in Fig. 4. Rather than revealing a rectangular loop as expected for such material, a predominantly flat curve is measured inductively with two distinct steps at small field. Domain observation immediately discloses the reason for such behavior: perpendicularly magnetized domains across most of the ribbon’s cross section are seen from the side, whereas longitudinal 180◦ domains are found on the surface. The perpendicular domains are magnetized by reversible rotational processes in an applied magnetic 396 R. Schäfer Volume M(H)curve (inductively measured) –8 20 –1 µm M/Ms Surface M(H)curve (by MOKE magnetometry) M ag fie net ld ic 1 –6 –4 –2 0 2 4 6 H in kA/m 8 Fig. 4 Hysteresis curves and domains of a surface-crystallized, 20-μm-thick amorphous ribbon of composition Fe84.3 Cu0.7 Si4 B8 P3 . Together with E. Lopatina, IFW Dresden (Adapted and reprinted from Ref. [5] with permission from Elsevier) field along the ribbon axis, leading to the flat portions of the loop, while the surfaces are magnetized by the coercive motion of 180◦ domain walls, which is responsible for the low-field behavior. In fact, a rectangular surface loop is measured by magneto-optical Kerr (MOKE) magnetometry with the same coercivities as found inductively. From the relative induction amplitude at the switching fields of the surface, a depth of about one micrometer for the longitudinally magnetized surface regions can be derived. The reason for this behavior is a crystallized surface that sets the volume under planar compression stress, whereas the surface itself is under tension. By magnetoelastic interaction, this inhomogeneous stress state leads to a perpendicular anisotropy in most of the volume and longitudinal anisotropy at the surfaces. There are also cases where the interpretation of a hysteresis loop is apparently simple: the rectangular loop measured on a permalloy film that is in direct exchange contact with an antiferromagnetic NiO film (Fig. 5) suggests 180◦ wall motion as dominating magnetization process. In fact this can be proven by domain imaging. However, the domain walls surprisingly change character between symmetric Néel and cross-tie wall (see section “Domain Walls”) on the descending and ascending branches of the loop, respectively. This subtle difference can by no means be derived from the magnetization loop. These reversible wall alterations indicate the existence of bi-modal coupling strengths due to pinned and unpinned spins at the interface between ferro- and antiferromagnet [6]. Surprising domain phenomena may also be found when magnetic samples are excited by high-frequency magnetic fields. The ground domain state of a nanocrystalline FeNbSiCuB tape wound core with a week circumferential anisotropy, for example, consists of 180◦ domains that are running along the circumferential direction (Fig. 6a). Excited quasistatically along the easy axis, i.e., in a slowly changing magnetic field well below the Hertz regime, 180◦ walls are nucleated, 8 Magnetic Domains 397 M/Ms 1 10 µm 0 Heb –1 –40 –20 0 20 µ0H in mT 40 Fig. 5 Field-shifted hysteresis loop in a Ni81 Fe7 (30 nm) / NiO (30 nm) exchange-bias double film together with high-resolution domain wall images along the forward and backward branch of the loop. A symmetric Néel wall (upper image) and an asymmetrically distorted cross-tie wall (lower image) are observed. Obtained at IFW Dresden together with J. McCord. (Adapted and reproduced from Ref. [6] by permission from IOP Publishing) At 1 kHz At 0.04 Hz Ground state H M/Ms 1 Ku M/Ms 1 –10 H in A/m 5 –5 –5 5 H in A/m 10 –1 0.2 mm a) b) Fig. 6 Difference between quasistatic and dynamic magnetization process, demonstrated for a nanocrystalline Fe73 Cu1 Nb3 Si16 B7 tape wound core with circumferential anisotropy. The quasistatic hysteresis loop in (a) is governed by the motion of 180◦ walls along the easy axis, whereas dynamic excitation with a sinusoidal ac field of 1 kHz frequency results in a domain nucleation-dominated reversal (b). Together with S. Flohrer, IFW Dresden. (Adapted and reprinted from Ref. [7] with permission from Elsevier) shifted and annihilated, leading to a square-type hysteresis loop. If excited at high frequency, however, the domain character surprisingly changes to a patch-type pattern (b) and the area of the hysteresis loop, and thus the energy loss increases due to eddy current effects. 398 R. Schäfer After laser scribing Without domain refinement 2 mm 2 µm a) b) Fig. 7 Engineering and application of domains: (a) Domain refinement in a grain-oriented transformer steel sheet by laser scribing for the purpose of energy loss reduction. (b) Currentinduced motion of domain walls in a CoNi multilayer microwire with perpendicular anisotropy, an experimental test structure for the race-track memory [8]. (Courtesy S.S.P. Parkin, IBM and Halle) From these examples we see that the correct interpretation of hysteresis curves may indeed be strongly supported by the analysis of the domains that are responsible for the loop. But magnetic domains can also be engineered on purpose to obtain favorable properties in devices, and they can be actively used in devices as functional units. The best-known example for domain engineering is the intended domain refinement in transformer steel sheets by scratching or laser scribing (Fig. 7a). The stress introduced locally in this way interrupts the basic domains, acting like an artificial grain boundary. This mechanism works for well-textured material, in which the basic domains would otherwise be wide thus causing large anomalous eddy current losses when operated at power frequency. A classical example for the application of magnetic domains is the bubble memory which was developed back in the 1970s. In this device, cylindrical domains as carriers of information were shifted in a magnetic garnet film along deposited ferromagnetic structures on top of the film by applying a rotating magnetic field. A modern concept of such a domain shift register device is the race track memory (Fig. 7b), in which perpendicularly magnetized domains are shifted in a magnetic nanowire by electrical current. Domain Formation Magnetic domains are formed to minimize the total free energy [9]. Under ideal conditions (i.e., ignoring coercivity), a vector field of magnetization directions m(r) arises in a ferro- or ferrimagnetic specimen so that the total energy reaches an absolute or relative minimum under the constraint of a constant magnetization, i.e., m2 = 1. In this section the energies are reviewed, and it is demonstrated how they act together to define the domain character. 8 Magnetic Domains 399 Magnetic Energies The relevant magnetic energies are summarized in an illustrative way in Fig. 8. They can be classified in local terms, which depend on the local magnetization direction (anisotropy, applied field, and magnetoelastic coupling energy) and nonlocal terms that give rise to torques on the magnetization vector, which depend at any point on the magnetization direction at every other point. The stray field and magnetostrictive self-energies belong to this class. The exchange energy may be seen as local, but it depends on the derivatives of the magnetization direction. In the following the most important aspects of the energy terms are listed. Given are energy densities Ex , which by integration lead to the energies εx = Ex dV with V being the sample volume. Exchange energy The alignment of magnetic moments occurs via exchange coupling. Deviations from a constant magnetization direction therefore invoke the penalty of exchange energy Exchange energy Anisotropy energy E ex = 0 M M M Easy axis H ext M Ea > 0 S N S N N M Hd S Magnetostrictive self energy m divH d M H ext Magneto-elastic coupling energy [100] [100] E ms = 0 N 100 >0 0] [10 N H ext = 0 Ea = 0 E ex > 0 Stray field energy External field energy 0] [01 0] [10 Tensile stress E ms > 0 Fig. 8 Summary of magnetic energies that are relevant for the formation and character of magnetic domains 400 R. Schäfer Eex = A(grad m)2 , (1) where A is the exchange stiffness constant, a temperature-dependent material constant in units J/m. Anisotropy energy Most magnetic materials are anisotropic, i.e., there are easy axes along which the magnetization vector is preferably aligned and along which the saturated state can be “more easily” obtained than along other directions. These can be preferred crystal axes (magnetocrystalline anisotropy) or an axis that arises from some short-range ordering of atoms like Ni-Ni and Fe-Fe atomic pairs in NiFe alloys. The driving force for this induced anisotropy is the magnetization of the material that is present at an annealing temperature below the Curie temperature and which can intentionally be aligned by an applied external magnetic field. Also annealing under mechanical stress may result in a uniaxial anisotropy, called creep-induced anisotropy. In amorphous and nanocrystalline ribbons, this type of anisotropy may be dominating, and often an easy plane of magnetization transverse to the stress axis is created by stress annealing. Shape effects are part of the stray field energy, and they do not belong to the anisotropy terms. Deflecting the magnetization out of an easy axis requires additional energy, called anisotropy energy. In the most simple case of a uniaxial anisotropy (as it occurs in crystal lattices with hexagonal or tetragonal symmetry or in case of an induced anisotropy), the energy density is written as Ea = Ku1 sin2 ϑ + Ku2 sin4 ϑ , (2) where ϑ is the angle between anisotropy axis and magnetization direction and Ku1 and Ku2 are the anisotropy constants of first and second order – higher orders can usually be neglected. An easy axis is described by a large positive Ku1 , whereas ‘planar’ and ‘conical’ anisotropies are found for large negative Ku1 and intermediate Ku2 values. The anisotropy constant Ku1 corresponds to the energy needed to saturate the sample in the so-called “hard” direction (ϑ = 90◦ ). In multiaxial materials such as iron, all 100 directions are easy, whereas in nickel the 111 axes are the preferred crystal axes. External field energy, also called Zeeman energy, is added to a magnet if an external magnetic field H ext is applied. It is given by EZ = −μ0 H ext · M = −μ0 Hext · M · cos(ϕ) , (3) where ϕ is the angle between magnetization and field. This interaction energy of external field and magnetization vector field m(r) causes domain wall motion and rotational processes and finally leads to saturation along the field direction if the field is strong enough. The minimum of the Zeeman energy is achieved when the magnetization is aligned to the magnetic field (ϕ = 0). 8 Magnetic Domains 401 Stray field energy Sinks and sources of the magnetization vector field (div m) lead to magnetic poles, which act as sources and sinks for a magnetic stray field. Magnetic poles can be present as volume or as surface poles if the magnetization vector M is not parallel to the surface. The stray field H d , arising from the poles, is illustrated in Fig. 8 for a finite ellipsoidal magnet that is homogeneously magnetized to the right. This leads to north (N) and south (S) poles at the edges and a stray field from (N) to (S). Within the magnet the stray field is called demagnetizing field as it opposes the magnetization. The presence of poles and stray fields causes the stray field energy 1 Ed = − μ0 H d · M with H d = −N · M . 2 (4) The demagnetizing factor N (a tensor in general) is zero for infinitely extended bodies and becomes the larger the closer the specimen edges along M. The stray field energy thus scales with N and with the average magnetization. A particularly unfavorable case is an infinitely extended plate that is magnetically saturated perpendicular to its surface. The demagnetizing factor along M is 1 then, and the demagnetizing or stray field energy is written Ed = 1 μ0 Ms2 = Kd . 2 (5) The stray field energy coefficient Kd is a measure for the maximum energy densities which may be connected with stray fields. Independent of the complexity of real stray fields, their energy always scales with the material parameter Kd . As the demagnetizing field of a body along a short axis is stronger than along a long axis, the applied magnetic field along the short axis has to be stronger to produce the same field inside the specimen. The shape of the magnet is thus the source of magnetic anisotropy (shape anisotropy). For an infinitely extended body that is magnetized along an infinite direction, the demagnetizing factor N is zero, and there will be no stray field energy at all. Magnetostrictive self-energy In magnetostrictive material, the crystal lattice is spontaneously elongated or contracted along the magnetization direction if the magnetostriction constant λ is positive or negative, respectively. As magnetostriction is quadratic in the magnetization vector, this lattice distortion is not important for 180◦ domains as all domains will lead to the same deformation. However, for the 90◦ domain configuration shown in Fig. 8, it will cause elastic energy (called magnetostrictive self-energy) as the spontaneous deformations of various parts of the domain pattern, indicated by ellipses in the figure, do not fit together elastically. The energy density of this incompatible domain configuration is 9 Ems = − Cλ2100 , 8 (6) 402 R. Schäfer λs > 0 λs = +35·10–6 λs = +24·10–6 λs = 0 100 µm λs = +8·10 a) λs < 0.2·10–6 –6 b) Fig. 9 Illustration of magnetostrictive and magnetoelastic energies: (a) Closure domains, observed on the surfaces of two amorphous ribbons with positive and zero magnetostriction as indicated. A backward pointing magnetic field along the ribbon axis was applied. (b) Typical domains in the as-quenched state of amorphous ribbons with different magnetostriction constants. Frozen-in internal mechanical stress, which is present in all four materials after quenching, leads to different complexity in domains depending on the magnetostriction constant. The ribbon thickness is about 20 μm in each case where λ100 is the magnetostriction constant for the 100 directions and C is the relevant shear modulus for the given configuration. If not enforced for topological reasons or by magnetic fields, nature tries to avoid such incompatible domain arrangements. Figure 9a demonstrates this effect for amorphous ribbons with some induced anisotropy perpendicular to the ribbon surface. A moderate magnetic field along the ribbon axes was applied that enforces a certain longitudinal magnetization. If magnetostriction is positive, domains running transverse to the field direction are more favorable because here the sample elongation in the neighboring domains fits together elastically. For a material without magnetostriction, a longitudinal domain arrangement will cause no problem as the elastic distortion, indicated in Fig. 8, will not occur. In fact this arrangement is even more favorable because here the specific wall energy of the basic domain walls is lower than for the transverse case. Magnetoelastic interaction energy There is also an inverse effect: applied mechanical stress of nonmagnetic origin can act on the magnetization direction in materials with non-zero magnetostriction by adding magnetoelastic interaction energy. The stress can be an external stress or some nonmagnetic internal stress resulting from dislocations or inhomogeneities in composition, structure, and temperature. In the example shown in Fig. 8, a horizontal tensile stress favors the horizontal anisotropy axes in a material with otherwise dominating positive cubic crystal anisotropy. If magnetocrystalline anisotropy is low or absent like in 8 Magnetic Domains 403 amorphous materials, the magnetoelastic coupling may lead to dominating stressinduced anisotropies (Fig. 9b). This can be seen by writing the magnetoelastic coupling energy as Eme = 3 λs σ sin2 ϑ 2 (7) where σ is the uniaxial mechanical stress, λs the isotropic magnetostriction constant, and ϑ the angle between magnetization vector and stress axis. This energy term describes a uniaxial anisotropy along the stress axis with an anisotropy constant of Ku = 32 λs σ , compare Eq. (2). Although magnetostriction is a relatively weak effect with induced strains of typically 10−5 only, the examples in Fig. 9 demonstrate that its effect on domain patterns can be significant. Domain wall energy The specific energy of domain walls is not an independent term, but rather consists of exchange and anisotropy energy, owing to the deviation of magnetization from the anisotropy axes and non-parallel magnetic moments across the wall. In the most simple case of a Bloch√wall in an infinitely extended uniaxial material the specific wall energy is γw0 = 4 A/Ku . See section “Domain Walls” for more information on domain walls. For summary, the characteristic coefficients of the magnetic energy terms and their order of magnitude in typical magnetic materials are collected in Fig. 10. Magnetic Energy Energy Coefficient Range Exchange energy Exchange stiffness constant A Material constant 10 Anisotropy energy Anisotropy constant K Material constant K1: Crystal anisotropy Ku: Induced (uniaxial) anisotropy ±(102 External field energy µ0HextMs Hext = external field Ms = saturation magnetization Depends on field magnitude Hext, unit J/m3 Stray field energy Kd = 1/2 µ0Ms Magnetoelastic interaction energy J/m 7 2 λ = mechanical stress λ = magnetostriction constant 6 ) J/m3 J/m3 Depends on stress magnitude ext, unit J/m3 2 Magnetostrictive self energy Cλ C = shear modulus λ = magnetostriction constant Fig. 10 Summary of energy coefficients and their range of magnitude 3 J/m3 404 R. Schäfer Driving Forces for Domain Formation Primarily it is the stray field energy that is responsible for the development of magnetic domains: domains are formed to reduce or avoid stray field energy. This fact is illustrated in Fig. 11 by comparing an infinitely extended with a finite sample. The NiFe (permalloy) film was sputter-deposited in the presence of a magnetic field, so it has a weak uniaxial anisotropy along the vertical direction in the images. For Fig. 11a, a piece of the film was broken from the wafer which extends 30 mm along the easy axis. In view of the thickness of just 240 nm, the specimen may be considered as infinitely extended so that stray field energy does not play a role. In fact, domains are not visible in that case: when a magnetic field along the easy axis is applied, the film switches from magnetization up to down and vice versa, leading to a magnetization curve with two steep, discontinuous steps at the coercivity field. The switching occurs by the fast and abrupt motion of a 180◦ domain wall as indicated schematically in the figure. The multidomain states at the discontinuity fields cannot be captured in the experiment because the expense of domain wall energy makes them statically unfavorable. The situation changes if an open sample, prone to a demagnetizing field, is considered. For Fig. 11b the infinite permalloy film was replaced by a 100×100 μm2 film element. For this finite specimen, the in-plane demagnetizing factor N has raised to 0.0015, and a demagnetizing energy of N Kd m2 is added. A saturated state at remanence, like in Fig. 11a, would thus be highly unfavorable as it would cost 30 x 5 mm2 M/Ms 1 1 Finite sample ( = 0) Hext = Hint + µ0Hext in mT –5 a) 5 –1 M µ0Hext in mT –5 5 –1 b) Fig. 11 Comparison between infinitely extended and finite samples. Shown are the magnetooptically measured magnetization curves along the (vertical) induced anisotropy axis in a Ni80 Fe20 Permalloy film of 240 nm thickness, which is infinitely extend in (a) and of finite size in (b). The domain images in (b) show the full patterned element, whereas in (a, upper inset) only a part of the extended film is shown. The lower inset in (a) is a schematics 8 Magnetic Domains 405 maximum stray field energy (| m |= 1). At zero field, the film element is rather in a demagnetized, multidomain state that reveals a flux-closed, pole-free nature. In the magnetization curve, this is expressed by the shearing transformation: for a given magnetization value m, the external field Hext must be enlarged by the demagnetizing field −Hd = N Ms m to reach the same magnetization state. Thus the discontinuous magnetization curve is transformed into a finite-slope curve with a well-defined magnetization value for every field value. For the finite element, the domains can thus be followed along the sheared M(H )-loop. So for the existence of magnetic domains, a finite sample size along the magnetization direction (easy axis) is required as the stray field energy is the driving force for domain observation. Infinitely extended films will consequently also have domains if the easy axis is perpendicular to the film plane, compare Fig. 3. There are two further cases in which magnetic domains or related objects can exist even in the absence of demagnetizing effects: • Consider a sample that is embedded in an ideal soft magnetic yoke to obtain fluxclosure. With a coil, wrapped around the yoke, a certain average magnetization can be enforced in the yoke by some feedback mechanism. If a magnetization value is then enforced in the sample that lies within the range of a discontinuous jump in the magnetization curve, a multidomain state will be enforced in which the two states at the endpoints of the jump will be mixed in a certain volume ratio so that the enforced magnetization is achieved. In case of the previously discussed uniaxial material, this would be a 180◦ domain state in which the 180◦ domain walls move as the enforced magnetization is varied. Such circuits are realized in machines with an inductive load on a rigid voltage, such as an idling transformer. • The second case may be found in magnetic materials with broken inversion symmetry in the atomic lattice in which the crystallographic handedness induces a quantum-mechanical Dzyaloshinskii-Moriya interaction (DMI) [10] by spinorbit scattering. Unlike direct Heisenberg or superexchange, which favor parallel or antiparallel alignment of neighboring magnetic moments according to a Hamiltonian that is proportional to S i · S j , the DMI is proportional to S i × S j thus favoring perpendicularly aligned neighboring spins S. In competition with collinear coupling, the DMI can lead to nanoscale, homochiral magnetization modulations like long-period helical spin-spiral phases. Most prominent is the topologically stable skyrmion spin structure that was predicted theoretically by A. Bogdanov [11, 12] and which has been directly observed in nanolayers of cubic helimagnets with intrinsic DMI [13] and in Fe/Ir bilayers [14] with surface/interface-induced chiral interactions [15]. Magnetic skyrmions are axisymmetric vortex patterns with a homochiral rotation of spins that can exist as isolated entities in the saturated states of chiral magnets [14,16] or in form of skyrmionic condensates (two-dimensional lattices and other mesophases) [12, 17]. In modern literature, intrinsically caused magnetic modulations (e.g., chiral helicoids and skyrmions) are often classified as magnetic micro- or spintextures, 406 R. Schäfer whereas the modulated elements of multidomain states (domain walls, Bloch lines, Bloch points, magnetic swirls etc. – compare Fig. 18) are commonly addressed as being part of magnetic microstructure [1]. This different classification, however, is questionable: spatially inhomogeneous spin structures arising in magnetic nanolayers are formed under the mutual influence of intrinsic and dipolar forces [18]. This levels out the difference between the terms magnetic microstructure and magnetic micro- or spintextures [19]. Interplay of Energies Once the precondition for domain formation is fulfilled, the domain character is finally determined by an interplay of the magnetic energies. For demonstration of this principle, let us have a look at the prominent example of domain formation in grain-oriented Fe3wt%Si steel that is used as core material in transformers. Like for pure iron, the easy directions of magnetization are the 100 directions for this material. Transformer sheets are typically 0.3 mm thick, consist of wide grains in the centimeter regime, and are Goss-textured. In this [001](110) texture the [001] easy direction is oriented, within a few degrees deviation, along the rolling axis during the manufacturing process of the sheets. The grain surfaces are (110) oriented within the same accuracy, and the other two easy axes are oriented at angles of ±45◦ relative to the surface. As shown in Fig. 12, the domain structure of such sheets consists of simple slab domains that are separated by 180◦ walls in case of ideally oriented grains. Their existence may, e.g., be enforced by some demagnetization effects at the grain boundaries. For increasing out-of-plane misorientation of the [001] easy direction, fine lancet-shaped domains of increasing density are superimposed on the basic domains. The formation of those so-called supplementary domains is a consequence of energy optimization. Let us firstly assume an infinitely extended grain with ideal (110) orientation. It will be homogeneously magnetized along the surface-parallel easy axis (Fig. 13a), thus completely avoiding magnetic poles. As the grain is infinitely extended, domains are not to be expected. A different situation arises if the [001]-axis is misoriented by some degrees relative to the surface (b). Assuming that the magnetization strictly follows the [001] axis, magnetic surface poles will arise. The associated stray field energy can be reduced by forming ±180◦ basis domains (c) which leads to the presence of opposite poles on the same surface, thus allowing the field lines of the stray field to run along the surface. A further reduction of stray field energy could be achieved by reducing the basic domain spacing as this would bring the opposite poles in closer distance. However, the narrower the domain width, the higher the expense of domain wall energy associated with the rising wall area of the basic domain walls that extend all through the thickness. Nature finds a more economic way to keep the overall energy low by adding supplementary domains to the basic domains (d). The shallow lancet domains at the surface collect the net flux that is transported toward the surface in the basic domain. The lancets are oppositely magnetized to the basic domains, thus leading to 8 Magnetic Domains 407 (110) surface Goss texture 100 easy axes Out-of-plane misorientation 0.1 mm 2° miso riented 4° miso riented idealy oriente d 8° mis 30 mm oriente d Fig. 12 Domains on a Goss-textured transformer sheet. Shown are the domains of four grains with increasing out-of-plane misorientation as indicated. The ceramic insulation coating, by which such sheets are usually covered to avoid eddy currents between the sheets, was removed for domain imaging by Kerr microscopy. (The domain images are adapted by permission from Ref. [1] (c) Springer 1998) a narrow spacing of opposite surface poles as required for stray field reduction. The flux is then transported to a surface of opposite polarity and distributed again. This is achieved by internal domains that are magnetized along the internal, transverse easy axes. Those transverse domains can extend all through the volume, or they can be connected to a basic domain wall so that the neighboring basic domain is used to lead flux downwards. Because this system of compensating domains is superimposed on the basic domains that would be present without misorientation, these domains are called supplementary domains. If a (moderate) magnetic field is applied along the surface-parallel easy axis, Zeeman energy is added and those basic domains with magnetization along the field direction will grow on expense of the opposite basic domains by 180◦ wall motion. The rise in stray field energy, caused by the absence of oppositely magnetized basic domains, is then compensated by an increasing number of supplementary domains. At the same time those internal transverse domains, which are connected to the basic domain walls, have to extend across the whole sheet thickness. So the transverse domain volume is larger compared to the demagnetized state. This change in relative domain volumes has consequences for the stress state of the sheet: as the magnetostriction constant is positive for FeSi, the cubic crystal lattice is tetragonally distorted along the magnetization direction. The basic domains thus cause an 408 R. Schäfer N Easy axes S a) N N N N S S N S N S S N S N N d) N N N S S S S N S S N N S S S N c) N N N S S N N N N b) S S N N e) N Tensile stress µ*-corrected N N N N Without tensile stress g) With tensile stress f) Fig. 13 Interplay of magnetic energies, illustrated on the example of domain formation in FeSi transformer sheets with (110)-related surfaces. Shown is the introduction of basic and supplementary domains (b–d) in case of a slightly misoriented surface, starting from an ideally oriented surface in (a). Tensile stress leads to domain refinement (e, f). In (g) the μ*-effect is illustrated. (Image (d) is adapted by permission from Ref. [1] (c) Springer 1998) elongation of the sheet along the rolling direction, while in the transverse domains the sheet is transversely expanded. A change in the transverse domain volume will thus result in a magnetostrictive change in length during remagnetization along the [001] easy axis. Driven in a magnetic field at power frequency, the sheet will be set in mechanical vibration leading to acoustic transformer noise. Furthermore, the repeated destroying and rebuilding of supplementary domains forms an important part of hysteresis loss as the energy bound in the supplementary domains is lost in every cycle. Magnetostrictive interaction can, however, also be favorably used in transformer sheets. The supplementary domains are suppressed under tensile stress applied along the preferred axis, because tensile stress magnetostrictively disfavors the transverse domains that are attached to the supplementary domains. A domain state as in Fig. 13c would thus result. Rather than superimposing supplementary domains 8 Magnetic Domains 409 to lower the stray field energy, which is forbidden now, a similar effect is achieved by lowering the basic domain width (e). The domain images in (f) demonstrate this effect. Obviously even ideally oriented grains assume a small domain width if they are coupled to less well-oriented grains to achieve flux continuity. A narrow domain with is favorable if the domains are excited by AC magnetic fields. The larger the density of the walls, the smaller the velocity of every wall for a given induction level which lowers domain wall-related eddy current effects (so-called anomalous eddy current losses). In practice the tensile stress is created by the insulation coating that is at the same time stress-effective. The planar stress exerted by the coating is for the Goss texture equivalent to a uniaxial stress and will thus suppress the supplementary domains. Two further, energy-related aspects are worth to be noted: (i) so far it was assumed that the domains are strictly magnetized along the easy crystal axes in the demagnetized state and up to moderate applied magnetic fields. This is in fact true for most of the volume domains. By approaching the (110) surface, however, the magnetization bends toward the surface (Fig. 13g). So the surface poles are spread over a certain volume and not just at the surface which helps to reduce the stray field energy at the expense of some anisotropy energy, though. The phenomenon is known as μ*-effect. (ii) The basic ±180◦ walls are zigzag folded across the thickness as indicated in Fig. 13. Although the total wall area is larger than in case of straight, perpendicular (110) walls that would have the smallest area, the total wall energy is reduced by the folding. The reason is the specific wall energy, which is lower for {100} wall orientations. The (110) wall therefore tends to rotate toward these orientations, forming tilted or zigzag walls with a lower overall energy. Domain Classification The magnetic energy coefficients, listed in Fig. 10, can be combined in several ways to obtain dimensionless parameters that reflect the interplay of energies and thus the domain character. The ratio between anisotropy and stray field energy is the most important. This ratio is called the quality factor, defined by Q= Keff . Kd (8) Here Keff is the effective anisotropy constant and Kd the stray field energy coefficient defined in Eq. (5). If the anisotropy energy dominates over the stray field energy (Q > 1), domains are formed that avoid an expense of anisotropy energy while keeping the stray field energy as low as possible. If the stray field energy is dominant (Q 1), stray fields are avoided by flux-closed domain patterns that adapt to keep the anisotropy energy as low as possible. In the following discussion we use the quality factor as primary criterion as it leads to the most fundamental way of classifying domains and magnetic materials. In Fig. 14 a number of typical materials are listed in the order of decreasing quality factor. Further criteria are the 410 R. Schäfer Material µ0 Ms in Tesla SmCo5 1.05 CoPt (L10) 0 w 0 w K1 , Ku in J/m3 Q 12 1.7 107 hexagonal 39 0.84 57 1.0 10 4.9 106 tetragonal 12 1.5 28 Sm2Co17 1.29 14 4.2 106 rhombohed. 6.3 1.83 31 Nd2Fe14B 1.61 7 4.5 106 tetragonal 4.4 1.25 23 BaFe12O19 0.48 7 3.2 105 hexagonal 3.5 4.68 6 Cobalt (Co) 1.79 31 4.5 105 hexagonal 0.35 = –45 = –260 8.3 15 = +22 –21 20.9 4 –55 –23 42 0.8 1 300 A J/m in 10 10 11 44 in 10 m in mJ/m2 Iron (Fe) 2.15 21 4.8 104 bcc 0.03 Nickel (Ni) 0.60 8 –4.5 103 fcc 0.03 Permalloy film (Ni81Fe19wt%) 1.00 13 50 - 200 Ku induced 3 10 Fe74Cu1Nb3Si15B7 nanocryst. ribbon 1.24 6 ~20 Ku induced 4 10 s 0.2 550 0.04 Amorph. ribbon, Co-based 0.6 2.5 ~3 Ku induced 2 10 s 0.1 900 0.01 100 111 = 100 = 111 = s Fig. 14 Material parameters that are important for domain analysis. The listed materials are ordered in terms of decreasing quality factor Q. Listed are furthermore saturation polarization μ0 Ms , exchange stiffness constant A, first order anisotropy constant K1,u , magnetostriction constant λ, wall width parameter Δ0w [see Eq. (14)], and specific wall energy of a 180◦ Bloch wall γw0 [see Eq. (13)]. (Data are taken from Refs. [20, 21]) manifold of easy directions and the surface orientation of the investigated specimen, which we treat as secondary criteria to classify the wide variability of domain phenomena. In Fig. 15 the interplay of stray field and anisotropy energy is illustrated by comparing three material classes with uniaxial anisotropy but highly different Q-factors. In all cases the easy axis is perpendicular to the plate surface, on which domain observation was performed by Kerr microscopy, i.e., the specimens are extremely misoriented with respect to the imaged surface. Compared are the domains of a NdFeB single crystal (left column) with those in amorphous films and ribbons (right column). The strong magnetocrystalline anisotropy of 4.5 · 106 J/m3 8 Magnetic Domains 411 D Ku Dominating anisotropy energy (Q >1) D = 5 µm Film: W 1 7 µm N S N D = 1 µm N S N S Q= 0.01 e) Towards bulk: N 7 N S S N N Surface 5 µm S a) 14 µm Dominating stray-field energy (Q<<1) D = 25 µm Q= 0.0003 b) f) D = 25 µm 40 µm Q= 0.001 c) g) D = 25 µm 120 µm Q= 0.002 20 µm h) d) (Q polar magnetization 1) in-plane magnetization D = 1 mm 5 µm Fig. 15 Classification of magnetic domains for three extrem cases of the quality factor. (a–d) NdFeB single crystal, Q = 4.4. (e–h) FeBSi-based amorphous film and ribbons with stressinduced perpendicular anisotropy and Q-values as indicated. (i) Cobalt single crystal, Q = 0.35. In each case the domain images are taken on top surface, while the sketches show side views. The Kerr images in (a–h) are adapted by permission from Ref. [1] (c) Springer 1998, while (i) was obtained together with I. Soldatov, Dresden, and reproduced from Ref. [22] with the permission of AIP Publishing 412 R. Schäfer makes NdFeB a material with Q = 4.4, while Q 1 for amorphous material owed to some weak (stress-)induced anisotropy around the order of 10–100 J/m3 (compare Fig. 14). An intermediate anisotropy of 4.5 · 105 J/m3 results in a quality factor of 0.35 for hexagonal cobalt, shown on the bottom of the figure. In case of the high-anisotropy material, the domains are strictly magnetized along the easy axis to avoid anisotropy energy, even though this causes magnetic poles at the surface thus costing stray field energy. This rule is strictly followed for films and bulk specimens, though the domain character changes as function of thickness. For films (Fig. 15a) a simple plate domain state with up-and-down domains is observed. For rising √ sample thickness D, the domain width W increases according to W ∼ D. Beyond a critical thickness (about 5 μm in the example), the domain walls get corrugated close to the surface (b), which leads to a better intermixing of poles. This lowers the stray field energy compared to the hypothetical case of straight domain walls. For further rising thickness (c), domain branching sets in: close to the surface a fine domain pattern is enforced to minimize the stray field energy by bringing opposite poles close together, whereas in the bulk, wide domains are favored to save wall energy. With increasing thickness (d) further iterated generations of domains are added, leading to a progressive domain refinement toward the surface in a fractal way. Theory [1] yields a characteristic D 2/3 dependence of the basic domain width and a constant surface domain width within the branching regime. As only up-and-down domains are involved in high Q uniaxial material, the branching scheme is called two-phase branching. Different arguments apply to the low-anisotropy material presented in the right column of Fig. 15. In the limit of small thickness (not shown), a thin film with a weak perpendicular anisotropy would be in-plane magnetized, because the anisotropy energy density of this state would be smaller than the stray field energy density of a uniformly perpendicularly magnetized state. Beyond a critical thickness, however, the magnetization starts to oscillate out of the plane in a periodic manner to save part of the anisotropy energy (Fig. 15e). The oscillation modulation assumes the character of a two-dimensional flux-closed pattern, called dense stripe domains, the half period of which typically equals the film thickness. Due to flux closure, stray field energy is completely avoided as required for a Q 1 material, whereas anisotropy energy and exchange energy are spent due to the deviations from the easy axis and the non-parallel alignment of magnetization, respectively. For larger thickness, an oscillating magnetization would consume increasing anisotropy energy so that nature prefers a Landau pattern (f). Here the bulk is strictly magnetized along the easy axis, and the expenses of exchange and anisotropy energy are concentrated in regular domain walls and closure domains (being magnetized perpendicular to the easy axis), respectively. When the perpendicular anisotropy increases slightly, still within the Q 1 regime, the anisotropy energy in the closure domains would rise so that beyond a critical anisotropy level (or beyond a critical thickness – note that the closure domain volume increases with thickness √ due to a D-increase of the basic domain width) a three-dimensional branching scheme takes over (g): the closure domains become themselves modulated in a similar continuous manner as seen for films in (e), thus lowering their anisotropy energy density. These stripe oscillations are connected with the basic domains by 8 Magnetic Domains 413 assuming the corrugated shape visible in the photograph and model. With a further increase in anisotropy or sample thickness, the stripe domains do grow into regular domains, the closure domains of which now decay into a further generation of stripe pattern as shown in (h). This type of branching may be considered as multiphase branching which occurs despite the fact of having a uniaxial material rather than cubic or other multiaxial materials. In any case, the overall domain patterns (e - h) are completely free from stray field energy as required for such low-Q material. If anisotropy and stray field energy are competing with about equal magnitude (Q ≈ 1), a hybrid structure is formed as shown in Fig. 15i for a thick cobalt crystal. The character of the domain pattern is similar to that of NdFeB in the macroscopic aspects, i.e., the branching mode in the bulk agrees with that of the high-anisotropy material. The surface domain width, however, is smaller because of the smaller wall energy of cobalt and theory predicts closure domains with a tilted magnetization, owing to a lower anisotropy compared to NdFeB. By comparing the two domain photographs in Fig. 15i, which show the out-of-plane and in-plane magnetization components separately, it seems that the fine surface pattern of the in-plane component resembles the branched domains of the amorphous ribbon in image (h). Obviously the closure domains are modulated in a dense stripe domain pattern. According to these findings, cobalt with its intermediate anisotropy forms a kind of hybrid, following the high-anisotropy two-phase branching scheme in the bulk and a low-anisotropy multiaxial branching scheme at the surface. The classification principle discussed so far is generally valid – different crystal symmetries or sample orientations just add modifications. Consider the case of Fe3wt%Si material in which the three 100 axes are easy. The quality factor of 0.03 implies flux-closure domain configurations at zero field. Their character depends on the surface orientation, and by having two further easy axis compared to the uniaxial materials in Fig. 15, more degrees of freedom for the domain formation are available. This becomes immediately apparent by looking at the domains of the non-oriented sheet presented in Fig. 16a. The variety of flux-closed domain patterns can be sub-classified according to the surface orientation: on an ideally oriented surface, the principle of flux closure is immediately seen by wide domains that are separated by well-oriented domain walls. Two grains with (110) and (100) orientations are marked in the figure. A pole-free wall orientation requires that the component of the magnetization perpendicular to the wall is the same on both sides of the wall as indicated in Fig. 16b. For slight misorientation of a few degrees flux collection is achieved by supplementary domains as seen for the (100)-related surface in Fig. 16c (see Fig. 13 for a thorough explanation of this phenomenon for a (110)-related surface). For stronger misorientation a domain branching scheme is energetically preferred. In Fig. 16d this is illustrated for a (111) surface, i.e., the case of extreme misorientation. Here in most of the volume a domain structure is formed that occupies easy directions only, and these domains are joined so that no magnetic stray fields are generated as required by Q 1. Near the surface zones, however, the two requirements of using only easy directions and avoiding stray fields are incompatible as the surface does not contain an easy direction. By branching nature finds a compromise: the domains get finer toward the surface by adding several 414 R. Schäfer ~(111) ~(110) (100) 10 µm 1)(11 (110) 100 µm a) mn m2 m1 ace f sur ~(100) mn e.a. (100)-cut d) Pole-free wall orientation requires: (m1 m2) n = 0 n b) a) c) Fig. 16 The domains on a non-oriented Fe3wt%Si sheet (0.5 mm thick) reveal the influence of the surface orientation on the domain character. (a) Low-resolution Kerr image giving an overview. (b) Schematics of a 90◦ wall illustrating the condition for a pole-free wall orientation. (c) Sketch of the fir tree structure, a supplementary pattern that appears on slightly misoriented (100) surfaces of iron-like material. (d) High-resolution Kerr image on a (111) surface, together with a schematics showing the phenomenon of multiaxial branching in iron-like material with cubic crystal anisotropy. (The images in (a) are adapted by permission from Ref. [23] (c) Springer 2009) generations of echelon domains in a fractal way (the number of generations depends on the thickness). By the wide volume domains, wall energy is saved, and by the fine surface domains, the volume of the outermost closure domains, which cannot be magnetized along an easy axis, is minimized. Those closure domains are actually magnetized in a continuously varying way similar to the dense stripe domains in films (see Fig. 15e). So the unavoidable anisotropy energy of the surface zone is reduced and right at the surface the magnetization lies parallel to the surface as required by the low-quality factor. Domain Walls Domain walls form a continuous transition between neighboring domains. The domain wall structure and character primarily depends on the Q-factor and thickness of the specimen. Furthermore the wall angle (i.e., the relative angle of the 8 Magnetic Domains 415 Left domain (x) Bloch path 1 x S Néel path Right domain a) x sin (x) 0 –4 –2 0 2 4 x / A/Ku N b) W180 Fig. 17 (a) Bloch and Néel wall paths in an infinite uniaxial material. The magnetic poles for the Néel wall are indicated in the vector plot. (b) Wall profiles of a 180◦ Bloch wall. The indicated wall width, W180 , is defined on basis of the slope of the magnetization angle ϕ(x). (Adapted by permission from Ref. [1] (c) Springer 1998) neighboring domain magnetizations) and magnetostriction may have an influence. Here some basic aspects of domain walls are reviewed; for details and a thorough review, we refer to Ref. [1]. Domain Wall Types Two principle modes of magnetization rotation across a domain wall can be distinguished: the Bloch and Néel wall. In Fig. 17a the two paths are illustrated for the simplest of all domain walls, a planar 180◦ wall in a (hypothetic) infinitely extended medium with uniaxial anisotropy that separates two domains of opposite magnetization. If the domain magnetizations are parallel to the wall, there will be no global magnetic poles, meaning that the component of magnetization perpendicular to the wall is the same on both sides of the wall (compare Fig. 16b). In the Bloch wall the magnetization rotates parallel to the wall plane, so there are no poles inside the wall either (divm = dmx /dx = 0), and the stray field energy is zero. For the Néel path the stray field energy would be maximum, making this wall path inferior to the Bloch path in bulk material. Néel-type walls can nevertheless be favorable in magnetic thin films as shown below. Classical Bloch wall If magnetostriction and higher-order anisotropy constants are neglected, the specific wall energy γw0 of a Bloch wall is written as an integral over exchange energy (considering the non-parallel moments in the wall) and anisotropy energy (due to deviations from the easy axis): γw0 = ∞ −∞ [A(dϕ/dx)2 + Ku cos2 ϕ] dx , ϕ(−∞) = π π , ϕ(∞) = − . 2 2 (9) Here x is the coordinate perpendicular to the wall, the angle ϕ rotates in the wall from 90◦ to −90◦ , and ϕ(±∞) are the boundary conditions given by the 416 R. Schäfer neighboring domains (see Fig. 17a). The solution of this ansatz is obtained by variational calculus, which leads to a function ϕ(x) that minimizes γw0 under the boundary conditions. Starting with Euler’s equation 2A(dϕ/dx) = −2Ku sin ϕ cos ϕ , (10) multiplying it with (dϕ/dx) and integrating with respect to x leads to the first integral: A(dϕ/dx) 2 = Ku cos2 ϕ . (11) According to this equation, the exchange and anisotropy energy densities are equal at every point in the wall: at positions where the anisotropy energy is high, the magnetization rotates rapidly leading to a large exchange energy. From Eq. (11) we obtain dx = A/Ku dϕ/ cos ϕ , (12) which, inserted together with (11) into the total wall energy (9), yields γw0 = 2 ∞ −∞ Ku cos2 ϕ dx = 2 AKu π/2 −π/2 cos ϕ dϕ = 4 AKu . (13) √ Integration of (12) leads to the functional dependence sin ϕ = tanh(x/ A/Ku ), which is plotted in Fig. 17b. From the indicated definition, the classical Bloch wall width is derived as W180 = π Δ0w , with Δ0w = A/K (14) being the wall width parameter (see Fig. 14 for examples). Although calculated for infinite samples, the classical Bloch wall also occurs in “real” specimens if the quality factor of the material is larger than one. Then the magnetic surface poles, which are caused by the out-of-plane magnetic moments of the wall in case of in-plane magnetized domains, can be tolerated. For low anisotropy material (Q 1), however, the requirement of pole avoidance enforces different wall types. In Fig. 18 they are collected for materials with a (weak) uniaxial, in-plane anisotropy Ku . There is a difference between thin films, thick films, and bulk specimens. Walls in thin films with Q < 1 From the micromagnetic point of view, magnetic films are defined as “thin” if their thickness is below the classical Bloch wall width. Then wall modes using a predominantly in-plane rotation of magnetization, known as symmetric Néel wall and cross-tie wall, have a lower energy than the Bloch mode, although magnetic poles cannot be avoided in those walls. The characteristics of 8 Magnetic Domains 417 D = 60 nm D = 460 nm Cross-tie wall D = 10 nm Asymmetric Bloch wall Symmetric wall 20 µm Permalloy thickness in nm 100 200 375 Wall angle 50 Symmetric S S S S S Tail Asymmetric N N N N N Tail Core Cross-tie Asymmetric wall Bloch wall Symmetric Néel wall 1 2 3 4 5 6 7 10 12 15 20 25 30 40 50 60 // Vortex wall 5 D / A/K d Side view D / A/K u Cross Bloch line Circular Bloch line Swirl Fig. 18 Phase diagram for various types of domain walls that exist in low-Q thin and thick films at zero applied field and in a hard-axis field that causes magnetization rotation in the domains, thus reducing the wall angle. The corresponding thicknesses for permalloy are indicated. Shown are high-resolution Kerr images of permalloy together with sketches (symmetric Néel and cross-tie wall, Bloch lines and swirl) and micromagnetically simulated vector plots (asymmetric Bloch and Néel wall, calculated for permalloy). The contour lines in the calculated wall profiles indicate the center of the walls, i.e., the surfaces on which the magnetization is strictly aligned in the drawing plane. The pictures are taken and adapted from Ref. [1]. Since the anisotropy has only a moderate influence on the wall energy in films, the diagram is valid for a wider range of low-Q materials. (Adapted by permission from Ref. [1] (c) Springer 1998) the symmetric Néel wall is its decomposition in a sharply localized core and two extremely wide tails that take over a large part of the total rotation. A dipolar pole pattern appears at the core, which carries about half of the pole density, and the other half of the poles are displaced in the tails, both of them adding stray field energy. The two characteristic lengths of a 180◦ Néel wall in a film of thickness D are then given by Wcore = 2 A/(Ku + Kd ) and Wtail ≈ 0.56DKd /Ku . (15) Because Kd is much√larger than Ku for materials with Q 1, the core width roughly scales with A/Kd , i.e., in the core the exchange energy (A) is primarily balanced by the stray field energy (Kd ) that is connected with the magnetic poles. The tail is rather determined by a balance between stray field (Kd ) and anisotropy 418 R. Schäfer energy (Ku ). The stray field energy thus has an opposite effect on both parts of the wall. The poles in the extended tail

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