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Michael Coey
Stuart S. P. Parkin
Handbook of
and Magnetic
Handbook of Magnetism and Magnetic
J. M. D. Coey • Stuart S. P. Parkin
Handbook of Magnetism
and Magnetic Materials
With 618 Figures and 157 Tables
J. M. D. Coey
School of Physics
Trinity College
Stuart S. P. Parkin
Max Planck Institute of Microstructure Physics
Halle (Saale)
ISBN 978-3-030-63208-3
ISBN 978-3-030-63210-6 (eBook)
ISBN 978-3-030-63209-0 (print and electronic bundle)
© 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
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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
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
Volume 1
Part I Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
History of Magnetism and Basic Concepts . . . . . . . . . . . . . . . . . . . . .
J. M. D. Coey
Magnetic Exchange Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ralph Skomski
Anisotropy and Crystal Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ralph Skomski, Priyanka Manchanda, and Arti Kashyap
Electronic Structure: Metals and Insulators . . . . . . . . . . . . . . . . . . . .
Hubert Ebert, Sergiy Mankovsky, and Sebastian Wimmer
Quantum Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gabriel Aeppli and Philip Stamp
Spin Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sergej O. Demokritov and Andrei N. Slavin
Micromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Lukas Exl, Dieter Suess, and Thomas Schrefl
Magnetic Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Rudolf Schäfer
Magnetotransport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Michael Ziese
Magneto-optics and Laser-Induced Dynamics of Metallic Thin
Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mark L. M. Lalieu and Bert Koopmans
Magnetostriction and Magnetoelasticity . . . . . . . . . . . . . . . . . . . . . . .
Dirk Sander
Magnetoelectrics and Multiferroics . . . . . . . . . . . . . . . . . . . . . . . . . . .
Jia-Mian Hu and Long-Qing Chen
Magnetism and Superconductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Ilya M. Eremin, Johannes Knolle, and Roderich Moessner
Part II
Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Magnetism of the Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plamen Stamenov
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
Metallic Magnetic Thin Films . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
D. Wu and X.-F. Jin
Volume 2
Magnetic Oxides and Other Compounds . . . . . . . . . . . . . . . . . . . . . . .
J. M. D. Coey
Dilute Magnetic Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Alberta Bonanni, Tomasz Dietl, and Hideo Ohno
Single-Molecule Magnets and Molecular Quantum Spintronics . . .
Gheorghe Taran, Edgar Bonet, and Wolfgang Wernsdorfer
Magnetic Nanoparticles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1011
Sara A. Majetich
Artificially Engineered Magnetic Materials . . . . . . . . . . . . . . . . . . . . 1047
Christopher H. Marrows
Part III
Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1081
Magnetic Fields and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . 1083
Oliver Portugall, Steffen Krämer, and Yurii Skourski
Material Preparation and Thin Film Growth . . . . . . . . . . . . . . . . . . . 1153
Amilcar Bedoya-Pinto, Kai Chang, Mahesh G. Samant, and
Stuart S. P. Parkin
Magnetic Imaging and Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1203
Robert M. Reeve, Hans-Joachim Elmers, Felix Büttner, and
Mathias Kläui
Magnetic Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1255
Jeffrey W. Lynn and Bernhard Keimer
Electron Paramagnetic and Ferromagnetic Resonance . . . . . . . . . . . 1297
David Menard and Robert Barklie
Magnetization Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1333
Andrew D. Kent, Hendrik Ohldag, Hermann A. Dürr, and
Jonathan Z. Sun
Part IV
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367
Permanent Magnet Materials and Applications . . . . . . . . . . . . . . . . . 1369
Karl-Hartmut Müller, Simon Sawatzki, Roland Gauß and
Oliver Gutfleisch
Soft Magnetic Materials and Applications . . . . . . . . . . . . . . . . . . . . . . 1435
Frédéric Mazaleyrat
Magnetocaloric Materials and Applications . . . . . . . . . . . . . . . . . . . . 1489
Karl G. Sandeman and So Takei
Magnetic Sensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1527
Myriam Pannetier-Lecoeur and Claude Fermon
Magnetic Memory and Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1553
Wei Han
Magnetochemistry and Magnetic Separation . . . . . . . . . . . . . . . . . . . 1593
Peter Dunne
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.
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
Michael Coey married Wong May, a writer, in 1973;
they have two sons and a grand-daughter.
About the Editors
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-
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
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,
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,
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,
Gerhard Fecher Max-Planck-Institute für Chemische Physik fester Stoffe, Dresden, Germany
Claudia Felser Max-Planck-Institute für Chemische Physik fester Stoffe, Dresden,
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,
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
Bernhard Keimer Max-Planck Institute for Solid State Research,
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,
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,
Priyanka Manchanda Howard University, Washington, DC, USA
Sergiy Mankovsky München,
Universität, München, Germany
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,
Karl-Hartmut Müller IFW Dresden, Institute for Metallic Materials, Dresden,
Hendrik Ohldag Advanced Light Source, Lawrence Berkeley National Laboratory,
Berkeley, CA, USA
Department of Physics, University of California Santa Cruz, Santa Cruz, CA,
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,
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),
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
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,
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
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
History of Magnetism and Basic Concepts
J. M. D. Coey
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
J. M. D. Coey
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.
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
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
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
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
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
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.
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
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
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
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
E = −m.B
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
σ m+
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
∇ × 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
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
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
∇ × 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)
J. M. D. Coey
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
H (r) =
∇ · M r − r
|r − r |3
3 M · en r − r d r +
|r − r |3
2 d r
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:
B (r) =
∇ × M × r − r
|r − r |3
3 (M × en ) × r − r d r +
|r − r |3
2 d r
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
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
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
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
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
virgin curve
initial susceptibility
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
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.
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
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
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
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
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
0 −i
1 0
i 0
0 −1
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 .
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
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
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
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
T C ,qp
T fN
T fN
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
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
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
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
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 + . . . ..
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 =
μ0 Ms 2 (1 − 3N )
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-
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
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
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
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
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
1 History of Magnetism and Basic Concepts
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
3+ (3d4 )
0.7 M0.3 )MnO3 where M = Ba, Ca, or Sr. The resulting mixture of Mn
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
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
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
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
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
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
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.
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
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
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 )
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.
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
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.
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
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
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.
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
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
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
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
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
J. M. D. Coey
MR = (R↑↓−R↑↑)/R↑↑
Spin valve sensor
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
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
1 History of Magnetism and Basic Concepts
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
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
J. M. D. Coey
and magnetic defects is providing new insight into magnetism at the atomic and
mesoscopic scales.
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
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
H-field (magnetic field
Magnetic moment
Specific magnetization
Magnetic energy density
susceptibility M/H
SI to cgs conversion
1 tesla = 10 kilogauss
1 kAm−1 = 12.57 oersted
1 Am2 = 1000 emu
1 Am−1 = 12.57 gauss†
1 Am2 kg−1 = 1 emu g−1
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
1 oersted§ = 79.58
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)
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.
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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.
Magnetic Exchange Interactions
Ralph Skomski
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
R. Skomski
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.
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
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
U (r 1 , r 2 ) =
4πεo | r 1 − r 2 |
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
R. Skomski
± (r 1 , r 2 ) = √ (φL (r 1 )φR (r2 ) ± φR (r 1 )φL (r 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
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
2 Magnetic Exchange Interactions
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)
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 >
R. Skomski
So =< Lo VR Ro >
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
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 ∗ >
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
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
2 Magnetic Exchange Interactions
The diagonalization of this Hamiltonian is trivial, reproducing E± = Eo ± T
and yielding
|σ > = √ (|L> + |R>) and |σ ∗ > = √ (|L> − |R>)
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 )
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
H= ⎜
T ⎟
T ⎠
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
n(r 1 ) n(r 2 )
dr 1 dr 2
| r1 − r2 |
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
|↑↑> = √ |LR> − √ |RL>
R. Skomski
cos χ
sin χ
|↑↓> = √ (|LR> + |RL>) + √ (|LL> + |RR>)
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
E↑↑ = –JD
= +D −
4T 2 +
Defining an effective exchange as J = E↑↑ –E↑↑ /2 yields
J = JD + −
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 =
+ JD − |T |
U ), Eq. (17) becomes
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
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)
|AFM > = σ (r1 ) σ (r2 ) (↑(1) ↑(2) −↑ (1) ↑(2))
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”).
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
R. Skomski
negative for JD = 0. Ferromagnetic coupling requires
|T | <
+ JD
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 > =
(|LL> + |LR> + |RL> + |RR>)
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
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
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
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> =
(|LR> + |RL>) (|↑↓> − |↓↑>)
and a FM triplet
|FM ↑↑> = √ (|LR> − |RL>) |↑↑>
|FM0> =
(|LR> − |RL>) (|↑↓> + |↓↑>)
R. Skomski
|FM ↓↓> = √ (|LR> − |RL>) |↓↓>
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
σx ⊗ σx + σy ⊗ σy + σz ⊗ σz = ⎜
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
Jij S i · S j − g μo μB
S1 · H i
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 .
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
because S · S
= 3/16 −
nonzero for S ≥ 1.
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↓
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 )
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
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+ .
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
|ψ> = √ |↑ ◦ ↓> + √ |↓ ◦ ↑> − √ |◦ ↑↓ ◦>
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)
The result of the calculation is
3 1 S (S + 1) − L (L + 1)
2 2
J (J + 1)
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
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 .
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
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
=⎝ 0 1 0 ⎠
√1 0 √1
partially diagonalizes the Hamiltonian and yields
Q+ HQ = ⎝ 2Tpd
2Tpd 0
0 ⎠
0 EM
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
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
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
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
R. Skomski
J (R) = Jo
2kF R cos (2kF R) − sin (2kF R)
(2kF R)4
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
+ 2
(0.507 ln (kF ao ) − 0.162)
kF ao
kF ao 2
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
The Stoner theory replaces Eq. (37) by the semiphenomenological expression
= 1 − I D(EF )
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
2 2
∇ +
Vo (r − r j )
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
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
|I D (EF ) − 1 + f (k)|
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
2 Magnetic Exchange Interactions
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]
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
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
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
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
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
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
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
Vi (r) = VLSDA (r) + U
− ni
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
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
2 Magnetic Exchange Interactions
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
R. Skomski
Table 2 Kondo
temperatures (in kelvin) for
some transition-metal
impurities in nonmagnetic
hosts (gray column) [59, 65]
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 0
0 T ⎟
0 T ⎠
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
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
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
H = −2 z J S · <S> − 2μo μB S · H
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 ),
R. Skomski
and self-consistently evaluating <S> yields the Curie temperature
Tc =
2 S(S + 1)
3 kB
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 >
2 Magnetic Exchange Interactions
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 >
3 kB T <sB > =
j JBA <sB > + JBB <sB >
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 =
6 kB
(JAA + JBB ) ±
(JAA − JBB )2 + 4 JAB JBA
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 .
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
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
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
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
cos (θn+1 –θn ) –J
n cos (θn+2 –θn )
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
Aside from including FM (δ = 0) and AFM (δ = π) states,
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.
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
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
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
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
R j 1– cos k · R j
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
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) =
SJ R 2 k 2
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
2 Magnetic Exchange Interactions
Table 3 Spin-wave stiffness
D and exchange stiffness A
for some materials [78]
Ni80 Fe20
Co2 MnSn
Fe3 O4
A = 2 c S2
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
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
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
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].
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
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)>=| ↑↓↑↓↑↓↑↓ · · · >
|AFM (2)>=| ↑↓↑↓↑↓↑↓ · · · >
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− |↑↓↑↓↑↓↑↓ · · · >=|↑↓↑↑↓↓↑↓ · · · >
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
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
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
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
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
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 )
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
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
H = J ⎝0 1 0⎠ − J ⎝1 1 1⎠
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 >= √ (2| ↑↑↓ >−| ↑↓↑ >−| ↓↑↑ >)
2 Magnetic Exchange Interactions
|2 > = √ (| ↑↓↑ >−| ↓↑↑ >)
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
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,
|>= | ↑↓↑↓ >+ | ↓↑↓↑ >− √ (| ↑↑↓↓ >+ . . . )
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
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
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
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
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. Sellmyer, and A. Ullah. The underlying research
in Nebraska has been supported by DOE BES (DE-FG02-04ER46152) NSF EQUATE (OIA2044049), the NU Collaborative Initiative, HCC and NCMN.
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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
Anisotropy and Crystal Field
Ralph Skomski, Priyanka Manchanda, and Arti Kashyap
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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.
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
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
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
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.
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
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
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
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.
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
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)
= K1 sin2 θ + K1 sin θ cos 2φ
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
K1 c 2
= 4 x 2 y 2 + y 2 z2 + z2 x 2 + 6 x 2 y 2 z2
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-
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
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
Point group
C2h (2/m)
C2h (2/m)
D2h (mmm)
Space group
D4h (4/mmm)
D4h (4/mmm)
D4h (4/mmm)
P42 /mnm
D3 (32)
D3d (3m)
P32 12
D3d (3m)
C6v (6 mm)
D6h (6/mmm)
P63 mc
P63 /mmc
D6h (6/mmm)
T (23)
Th (m3)
Td (43m)
Td (43m)
Oh (m3m)
P21 3
Oh (m3m)
Oh (m3m)
Oh (m3m)
Oh (m3m)
Oh (m3m)
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
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
or cos6φ terms. Up to the sixth order, the anisotropy of magnets with a tetragonal
crystal structure is described by
= K1 sin2 θ + K2 sin4 θ + K2 sin4 θ cos 4φ + K3 sin6 θ + K3 sin6 θ cos 4φ (5)
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
V = K1 sin θ + K2 sin θ + K2 sin θ cos θ cos (3 φ)
+ K3 sin φ + K3 sin θ cos (6φ) + K3 sin3 θ cos3 θ cos (3
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.
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:
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
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
Ha =
μo Ms
3 Anisotropy and Crystal Field
Table 2 First- and
second-order anisotropy
constants at room temperature
Fe3 O4
Nd2 Fe14 B
Sm2 Fe17 N3
Sm2 Fe17 C3
Y2 Co17
Tm2 Co17
Sm2 Fe14 B
K1 (MJ/m3 )
K2 (MJ/m3 )
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
Ms 2
+ K2
Ms 4
− μo MH
where M = Ms sinθ . Minimizing the energy, ∂η/∂M = 0, and dividing the result by
μo M yields
R. Skomski et al.
2 K1
4 K2
μo Ms 2
μo Ms 4
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
M(H ) = Ms
Ha 2
15H 2
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
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
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:
V (r) n(r) dV
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
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
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
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
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) = −
| Ri − r |
This sum is easily converted into a sum of spherical harmonics by exploiting the
4 π rl
l (2 l + 1) R l+1
|R−r |
Y1 m∗ (Θ, Φ) Y1 m (θ, φ)
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
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
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
3 Anisotropy and Crystal Field
Table 4 Crystal-field
parameters for some noncubic
rare-earth transition-metal
intermetallics [10]
R2 Fe14 B
R2 Fe17
R2 Fe17 N3
A2 0
K/ao 2
A4 0
K/ao 4
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
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:
ρ (r) ρ r dV dV | r − r |
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
Cubic CF
Tetragonal CF
A + 2B + E
A + 2B + 2E
3A + 2B + 2E
3A + 2B + 3E
Trigonal CF
A + 2E
3A + 2E
3A + 3E
3A + 4E
3 Anisotropy and Crystal Field
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
Ti3+ , V4+
Ti2+ , V3+
V2+ , Cr3+
Cr2+ , Mn3+
Mn2+ , Fe3+
Fe2+ , Co3+
Co2+ , Ni3+
Ni2+ , Pd2+
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
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
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
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
= me c 2 + me v 2 − me v 4
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 =
s · ∇V × k
2me c
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 =
2me c 4πεo r 3
Using Appendix C, we can evaluate the average <1/r3 > and obtain Hso = ξ l · s.
Here ξ is the spin-orbit coupling constant:
Z 4 e2
2me c ao 4πεo n3 l l + 1/ (l + 1)
3 Anisotropy and Crystal Field
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 +
= me c 2 + me α 2 c 2 − me α 4 c 4
Similarly, Eq. (21) becomes
me 4 4 2
Z α c +
n l l + /2 (l + 1)
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
ξ (meV)
ξ (meV)
ξ (meV)
Table 8 Spin-orbit coupling
constants ξ for tripositive 4f
and 5f transition-metal ions
[32, 33]
R. Skomski et al.
ξ (meV)
ξ (meV)
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
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.
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]
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
aK c
for TbFe2
K1 (RT)a
μo Ms (RT)
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
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-
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
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).
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
3 Anisotropy and Crystal Field
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
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)
corresponding to 3z2 – r2 and
O2 2 (J ) =
1 2
J+ + J− 2
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
20 x 4 + y 4 + z4 − 3r 4 /5 = 35z4 − 30 z2 r 2 + 3r 4 + 5 (x + iy)4 + (x–iy)4
and corresponds to O4 0 + 5 O4 4 . Here O4 0 = 35 Jz 4 −
30J (J + 1) Jz +
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
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
Ql = θl < r l >4f Ol 0
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
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
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)
O4 0 = 8 J · (J − 1/2) · (J –1) · (J –3/2)
O6 0 = 16 J (J − 1/2) · (J –1) · (J –3/2) · (J − 2) · (J − 5/2)
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
Q2 /ao 2
Q4 /ao 4
Q6 /ao 4
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
3cos2 − 1
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
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 =
Q2 A2 0 3 cos2 θ − 1
Comparison with Eq. (1) yields
K1 = −
A2 0 Q2
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
K1 = − A2 0 Q2 − 5 A4 0 Q4 −
A6 0 Q6
3 Anisotropy and Crystal Field
K2 =
35 0
A4 Q4 +
A6 0 Q6
K3 = −
231 0
A6 Q6
Tetragonal magnets also have
K2 =
A4 4 Q4 + A6 4 Q6
K3 = −
A6 4 Q6
whereas hexagonal magnets exhibit only one in-plane term
K3 = −
A6 6 Q6
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
231 0
A6 Q6
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.
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
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
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
expression simplify to Legendre polynomials, for example,
O2 0 ∼ 12 3cos2 θ − 1 = P2 (cos θ ). The thermal averages
< cos θ >= N
kB T
cos θ cosm θ sin θ dθ
are readily evaluated by a high-temperature expansion of the exponential function
and yield the rare-earth anisotropy [43].
K1 (T ) = K1 (0)
15 kB T 2
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)
R. Skomski et al.
Table 12 First and second-order anisotropy constants at low temperatures (LT) and at room
Nd2 Fe14 B
Pr2 Fe14 B
Sm2 Fe17 N3
K1 (MJ/m3 )
Table 13 Transition-metal
and rare-earth contributions
to the room-temperature
magnetocrystalline anisotropy
[10]. All values are in MJ/m3
(MJ/m3 )
Structure Refs.
K1 (MJ/m3 )
Nd2 Fe14 B
Sm(Fe11 Ti)
Sm2 Fe17 N3
Sm2 Co17
K2 (MJ/m3 )
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
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]
α-Fe2 O3
γ-Fe2 O3
Fe3 O4
MnFe2 O4
CoFe2 O4
NiFe2 O4
CuFe2 O4
MgFe2 O4
BaFe12 O19
SrFe12 O19
PbFe12 O19
BaZnFe17 O27
Y3 Fe5 O12
Sm3 Fe5 O12
Dy3 Fe5 O12
(La0.7 Sr0.3 )MnO3 −0.002
Sr2 FeMoO6
μo Ms
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)
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
Co (α)
K1 (RT)
μo Ms (RT)
Co (β)
Fe0.96 C0.04
Fe4 N
Fe16 N2
Fe3 B
Fe23 B6
Fe0.65 Co0.35
Fe0.20 Ni0.80
Co3 Pta
Mn2 Ga
Mn3 Ga
Mn3 Ge
Fe7 S8
a Extrapolation
Cubic (bcc)
Cubic (fcc)
Cubic (fcc)
(modified fcc)
(C6 Cr23 )
Cubic (bcc)
(L10 )
Cubic (fcc)
(L10 )
(L10 )
(L10 )
(L10 )
(D022 )
(D022 )
(D022 )
to fully ordered Co3 Pt has been suggested to yield 3.1
E1 (k) 0
0 E2 (k)
0 i
−i 0
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
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
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
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 | l · s | u> <u | l · s | o>
Eu −Eo
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 ,
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
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
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 ∼
Vo (Eo − Eu )n−1
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
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
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
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
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
2 2
∇ + j Vo r − Rj + j Hso Rj
where the matrix elements of Hso are those of Fig. 15. The lattice periodicity is
accounted for by the ansatz
ψkμ (r) = N
exp ik · Rj φμ r − Rj
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
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
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
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,
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
[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
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].
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
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
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
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
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
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
μo 3m · R m · R − m · m R 2
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
= − μo
M (r) · H (r) dV
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
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
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
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
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
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
3 Anisotropy and Crystal Field
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)
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
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
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 )
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
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
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:
λs E E
3 cos2 θ − 1 ε + ε2 − ε σ
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:
λs σ 3 cos2 θ − 1
3 Anisotropy and Crystal Field
and the magnetoelastic contribution to K1 KME = 3λs σ /2, which may be fairly
For cubic crystallites, there are two independent magnetostriction coefficients in
the lowest order, and the polycrystalline average over all possible orientations is
λs =
λ100 + λ111
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
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
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
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
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
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
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
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
− K1 √
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)
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)
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
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
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
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
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.
R. Skomski et al.
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
Yl m (θ, φ) = Nl exp (imφ) Pl m (cos θ )
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 =
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 =
5 3 cos2 θ –1
5 3z2 − r 2
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.
3 Anisotropy and Crystal Field
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)
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 · 3/2
1/2 · 3
1/2 · 3/2
x2 – y2
1/4 · 15/2
sinT cosT
– 1/2 · 15/2
3 z2 – r2
3 cos2T – 1
1/4 · 5
sinT cosT
1/2 · 15/2
2 xy
1/4 · 15/2
x3 – 3 xy2
– 1/8 · 35
(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
y (5 z2 – r2)
1/8 · 21
2 xyz
sin T cosT
1/4 · 105/2
3 x2y – y3
1/8 · 35
x4 – 6 x2y2 + y4
3/16 · 35/2
xz (x2 – 3 y3)
sin T cosT
– 3/8 · 35
– y2) (7 z2 – r2)
3/8 · 5/2
xz (7 z2 – 3 r2)
sinT cos T – 3 cosT
– 3/8 · 5
z4 – 30 z2r2 + 3 r4
35 cos T – 30 cos T + 3
yz (7 z2 – 3r2)
sinT cos3T – 3 cosT
3/8 · 5
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)
3/16 · 35/2
x (x4 – 10 x2y2 + 5 y4)
– 3/32 · 77
z (x4 – 6 x2y2 + y4)
sin4T cosT
3/16 · 385/2
– 3 y2)·(9 z2 – r2)
– 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 xyz (3z2 – r2)
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)
3/32 · 77
x6 – 15 x4y2 + 15 x2y4 – y6
1/64 · 3003
x (x4 – 10 x2y2 + 5 y4)
– 3/32 · 1001
(x – 6 x2y2 + y4)(11 z2 – r2)
sin T cos T – 1
3/32 · 91/2
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
xz (33 z4 – 30 z2r2 + 5 r4)
– 1/16 · 273/2 sinT (33 cos T – 70 cos T + 5 cosT )
z6 – 315 z4r2 + 105 z2r2 – 5 r6
1/32 · 13
yz (33z4 – 30z2r2 + 5r4)
sinT (33 cos T – 70 cos T + 5 cosT )
1/16 · 273/2
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
xy (6 x4 – 20 x2y2 – 6 y4)
sin T
1/64 · 3003
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
Point group
C1 (1)
Ci (1)
C2 (2)
Cs (m)
C2h (2/m)
D2 (222)
C2v (mm2)
D2h (mmm)
C4 (4)
S4 (4)
C4h (4/m)
D4 (422)
C4v (4 mm)
D2d (42m)
D4h (4/mmm)
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
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
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
3 Anisotropy and Crystal Field
Table A.17 (Continued)
Crystal system
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
|x 2 − y 2 > = N R3d (r) sin2 θ cos 2φ
|xz> = 2N R3d (r) sin θ cos θ cos φ
|z2 > = R3d (r) 3 sin2 θ − 1
|yz > = 2N R3d (r) sin θ cos θ sin φ
15/16π, ao = 0.529 Å, and
R3d (r) =
4Z 5/2 r 2
exp −
81ao2 30ao3
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:
R1s = exp −
ro 3
R. Skomski et al.
= 2 2ro 3
= exp −
2 6ro 3 ro
= 81 3ro 3
exp −
27 − 18 + 2 2 exp −
6 − 2 exp −
= 81 6ro 3
R3d =
81 30ro 3 ro 2
R4f =
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)
<r> =
ro 2
3 n − l (l + 1)
<1/r> =
<1/r 2 > =
<1/r 3 > =
n2 r
n3 ro 2 (2l + 1)
n3 ro 3 l (l + 1) (l + 2)
3 Anisotropy and Crystal Field
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.
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Electronic Structure: Metals and Insulators
Hubert Ebert, Sergiy Mankovsky , and Sebastian Wimmer
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Electronic Structure Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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
H. Ebert et al.
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.
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