Download Thermally driven magnon transport in the magnetic insulator Yttrium

Thermally driven magnon transport
in the magnetic insulator
Yttrium Iron Garnet
Dissertation
Milan Agrawal
Vom Fachbereich Physik der Technischen Universität Kaiserslautern zur
Verleihung des akademischen Grades Doktor der Naturwissenschaften“
”
genehmigte Dissertation
Betreuer: Prof. Dr. Burkard Hillebrands
Zweitgutachter: Prof. Dr. YoshiChika Otani
Datum der wissenschaftlichen Aussprache: 18. February 2014
D 386
“Basic research is what I’m doing
when I don’t know
what I am doing.”
—Wernher Von Braun
To my friends,
all we need in life is friends. . .
Kurzfassung
Die Forschungsergebnisse der vorliegenden Arbeit beinhalten die Untersuchung spinkalorischer
Phänomene an Ferromagnet-Metall-Heterostrukturen. Diese Phänomene erschließen die Wechselwirkung von Wärme mit einem Spinsystem und behandeln hauptsächlich die Erzeugung und Manipulation von Spinströmen durch Wärmeströme (Phononen). Spinströmen wird dabei nicht nur eine große Bedeutung bei der Entwicklung neuer fundamentaler Konzepte der Physik sondern auch
für die Herstellung magnetischer Speicherelemente zugesprochen. In Analogie zum klassischen
Seebeck-Effekt wird die Erzeugung eines Spinstroms durch das Anlegen eines Wärmestroms in
einem Spinsystem als Spin-Seebeck-Effekt (SSE) bezeichnet. Da der Spin-Seebeck-Effekt in einer
Vielzahl magnetischer Materalien (Spinsystemen) existiert, vom elektrischen Isolator bis hin zu
Metallen, hat diese Art der Generierung von Spinströmen in letzter Zeit große Aufmerksamkeit in
der Forschung geweckt. Das Potenzial zur Anwendung dieses Effekts zur Energieerzeugung durch
überschüssige Wärme macht ihn dabei umso attraktiver.
Im Allgemeinen unterteilt man Spinsysteme in Systeme, in denen ein Spintransport über freie
Elektronen stattfindet, sowie in Spinwellensysteme, d.h. in kollektive Anregungen des magnetischen Momente im Wellenvektorraum. Spinwellen in einem elektrisch isolierenden, ferromagnetischen System, welches sich durch die Abwesenheit freier elektronischer Ladungen auszeichnet,
überwinden dabei die begrenzten Reichweiten reiner Spinströme in Metallen. Ausschlaggebend
für den Aufbau spin-elektronischer (spintronischer) Schaltungen und einer Spinlogik mit hohen
Rechengeschwindigkeiten sind die großen Reichweiten von Spinströmen, welche durch Spinwellen übermittelt werden. Hierfür ist das ferrimagnetische und isolierende Material Yttrium-EisenGranat (YIG) ein vielversprechender Kandidat, da es wegen seiner geringen Dämpfung makroskopische Propagationslängen für Spinströme ermöglicht.
Im Rahmen dieser Arbeit wurden detaillierte Untersuchungen der Wechselwirkung zwischen Phononen mit Magnonen, den Quasiteilchen von Spinwellen, in monokristallinen YIG-Filmen durchgeführt. Zur erstmaligen direkten Messung der räumlichen Verteilungen von Magnonen- und Phononentemperatur in einem YIG-Film mit einem lateral angelegten thermischen Gradienten wurde ein neues Verfahren zur Messung der lokalen Magnetisierung in einem magnetischen Film
mittels Brillouin-Lichtstreuspektroskopie (BLS) entwickelt. Die Ergebnisse belegen eine große
i
Übereinstimmung zwischen den räumlichen Temperaturverteilungen für Magnonen und Phononen
und zeigen damit eine starke Wechselwirkung beider Subsysteme auf. Die gewonnenen Erkenntnisse werden anschließend verwendet, um den Ursprung des transversalen Spin-Seebeck-Effekts
zu verstehen, bei dem ein Spinstrom generiert wird, welcher senkrecht zum Wärmestrom bzw. dem
Temperaturgradienten fließt. Die Ergebnisse unterstreichen die Formulierung eines Konzepts der
spektralen Inhomogenität der Magnonentemperatur zur Erklärung des transversalen SSE anhand
der aktuell vorhandenen Theorien. Diese Ergebnisse bieten einen neuen Ansatz für ein tieferes
theoretisches Studium des Ursprungs des SSE.
Um den Ursprung des Spin-Seebeck-Effekts aufzudecken und industrielle Anwendungen zu entwickeln, bedarf es einer genauen Kenntnis der zugrunde liegenden Zeitskalen. In dieser Arbeit wird
erstmalig die zeitliche Entwicklung des longitudinalen Spin-Seebeck-Effekts, bei dem der Spinstrom parallel zum Wärmestrom fließt, an einer YIG-Platin(Pt)-Heterostruktur gemessen. Hierbei
wird die starke Spin-Bahn-Kopplung im Platin ausgenutzt, um den Spinstrom mithilfe des inversen
Spin-Hall-Effekts nachzuweisen. Für die zeitaufgelöste Messung des longitudinalen SSE werden
zur Erhitzung der Probe zum einen ein Laserstrahl und alternativ Mikrowellen verwendet und die
Vorteile dieser beiden Heizmethoden gegenüber konventionellen Methoden diskutiert.
Die zeitaufgelösten Messungen des longitudinalen SSE zeigen, dass dieser Effekt auf einer SubMikrosekunden-Zeitskala abläuft und dienen weiterhin zum Verständnis seiner Ursprünge. Im Speziellen lösen sie die bislang offene Frage, ob der longitudinale SSE ein reiner Oberflächeneffekt
ist oder ob ein Volumenbeitrag gegeben ist. Die Ergebnisse offenbaren, dass die Zeitentwicklung
des Temperaturgradienten in der Umgebung der YIG-Platin-Grenzfläche die Zeitskala des longitudinalen SSE bestimmt. Die Resultate zeigen, dass dieser Effekt ein Volumeneffekt ist, in dem
thermische Magnonen involviert sind, welche im magnetischen Material und innerhalb eines bestimmten Volumens nahe der Grenzfläche lokalisiert sind.
Weiterhin illustriert das Experiment des Mikrowellen-gestützten Heizens einer YIG-Pt-Heterostruktur das Konzept einer asymmetrischen Anregung von Spinwellen im YIG-Film. Die relative
Positionierung der Mikrowellenantenne bezüglich des YIG-Films begünstigt das asymmetrische
Spinpumpen im Metall über die nichtreziproken Spinwellenmoden des YIG, welche als magnetostatische Oberflächen-Spinwellen (MSSW) bekannt sind. Die Ergebnisse sind für die Entwicklung
lokaler Spinstrom-Quellen für spintronische Anwendungen sehr hilfreich.
Zusammenfassend trägt die vorliegende Dissertation zum fundamentalen Verständnis thermisch
angeregter Magnonen in Heterostrukturen bei, welche sich aus einem magnetischen Isolator und
einem nichtmagnetischen Metall zusammensetzen. Die Untersuchung einer neuen Methode zur
direkten Messung der Magnonentemperatur und die Ergründung des Ursprungs und der zeitlichen
Entwicklung des Spin-Seebeck-Effekts erweitern nicht nur den aktuellen Wissensstand auf diesem
Gebiet sondern eröffnen auch neue Gelegenheiten zur Entwicklung zukünftiger Anwendungen.
ii
Abstract
The research work presented in this thesis covers the investigation of spincaloric phenomena in
ferromagnetic|normal metal heterostructures. These phenomena explore the interaction of heat
with spin systems and mainly deal with the generation and the manipulation of spin currents by
means of heat currents (phonons). The significance of spin currents is widely seen in developing
new fundamental concepts of physics as well as in the industry of magnetic memories. Analogous
to the classical Seebeck effect, the generation of a spin current in a spin system by the application
of heat currents is known as the spin Seebeck effect (SSE). This mode of spin current generation
has recently attracted much scientific attention due to the existence of the spin Seebeck effect in
a wide variety of magnetic materials (spin systems), considering from insulators to metals. The
potential applications of this effect, in particular to generate electricity out of waste heat, make the
effect even more attractive.
Generally, spin systems can be classified into either a system constituting the traveling spins carried by free electrons or into a system of spin waves, collective excitations of magnetic moments
in the wavevector space. Having the advantage of being free from free-electronic charges, an
electrical-insulating-ferromagnetic system of spin waves overcomes the limitation of short propagation lengths of pure spin currents in metals. The long propagation length of spin currents carried
by propagating spin waves is crucial for building-up spin-electronic (spintronic) circuits and spin
logics for fast computation. For such purposes, the ferrimagnetic insulator Yttrium Iron Garnet
(YIG) is a promising material candidate due to its lowest known magnetic damping which offers
macroscopic propagation lengths of spin currents.
In the framework of this thesis, a detailed investigation of the interaction of phonons with magnons,
the quanta of spin waves, in single crystalline YIG films are carried out. The first direct measurement of the spatial distribution of the magnon and the phonon temperatures in the YIG film, subject
to a lateral thermal gradient, is realized by developing a novel technique to measure the local magnetization in the magnetic film using Brillouin light scattering (BLS) spectroscopy. The findings
reveal a close correspondence between the spatial dependencies of magnon and phonon temperatures which represents the strong interaction between the magnon and the phonon subsystems.
Subsequently, the findings are utilized to understand the origin of the transverse spin Seebeck
iii
effect (SSE), where a spin current flowing perpendicularly to the heat currents or the temperature gradient is generated. The results emphasize on the formulation of the concept of spectral
non-uniformity of magnon temperature to explain the transverse spin Seebeck effect with contemporary theories. The outcomes provide a new direction for a deeper theoretical investigation on the
origin of the spin Seebeck effect. In order to unfold the origin of the spin Seebeck effect as well
as to develop industrial applications, the knowledge about the timescales of the effect is essential.
In this thesis, very first measurements of the temporal evolution of the longitudinal spin Seebeck
effect, where the spin current flows parallel to the heat current, are carried out on a YIG|Platinum
(Pt) heterostructure. Here, the high spin-orbital coupling material Pt is employed to measure the
spin current via the inverse spin Hall effect. Two heating techniques, laser irradiation and microwave heating, are utilized to perform the time-resolved measurements of the longitudinal spin
Seebeck effect. The advantages of these heating techniques over the conventional heating methods
are explored as well.
The time-resolved measurements on the longitudinal spin Seebeck effect reveal that this effect
takes place on a sub-microsecond timescale. Further, these measurements assist in understanding
the origin of this effect. In particular, the findings answer the open question whether the effect is a
pure interface effect or if bulk activities are contributing to it. The results depict that the temporal
evolution of the temperature gradient in the vicinity of the YIG-film-interface with the normal
metal (Pt) controls the timescale of the longitudinal spin Seebeck effect. The findings show that
this effect is a bulk effect in which thermal magnons, laying up to a certain volume underneath the
interface inside the magnetic material, participate.
Moreover, the microwave heating experiment performed on the YIG|Pt heterostructure illustrates
the concept of an asymmetric excitation of spin-waves in the YIG film. The relative position of
the microwave antenna with respect to the YIG film stimulates asymmetric spin-current pumping
(spin pumping) in the normal metal via nonreciprocal spin-wave modes known as magnetostatic
surface spin-waves (MSSW). These outcomes assist in developing local spin-current sources for
spintronic applications.
In conclusion, this work contributes to the fundamental understanding of thermally driven magnons
in magnetic insulator|normal metal heterostructures. The investigation of a new technique for direct measurement of the magnon temperature and the study of the origin and the temporal evolution
of the spin Seebeck effect not only improve the contemporary knowledge of the field but also open
a new window for development of future applications.
iv
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
Theoretical concepts of magnetism and spintronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1 Ferromagnetism and magnetic interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.1.1 Exchange interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.1.2 Dipolar interaction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2 Magnetization dynamics and Landau-Lifshitz-Gilbert equation . . . . . . . . . . . . . . . . . . . . 10
2.3 Spin waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.1 Ferromagnetic resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.2 Walker’s equation and spin waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.3 Spin-wave theory of Kalinikos and Slavin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Spin transfer torque and spin pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.1 Spin transfer torque. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.2 Spin pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5 Spin Hall and inverse spin Hall effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.6 Spin caloritronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Spin-dependent Seebeck and Peltier effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8 Spin Seebeck effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8.1 Experimental Configuration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.8.2 Theory of the spin Seebeck effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.8.3 Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.9 Anomalous and Planar Nernst effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3
Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.1 Yttrium Iron Garnet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 YIG film deposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 YIG film surface treatments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.3 Pt film deposition on YIG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
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CONTENTS
3.2 Brillouin light scattering spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Principles of Brillouin light scattering spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.2 Working principle of the tandem Fabry-Pérot interferometer . . . . . . . . . . . . . . . 41
3.2.3 Conventional Brillouin light scattering spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.4 Brillouin light scattering microscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.3 Microwave techniques and spin-wave excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4 COMSOL Multiphysics simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4
Experimental results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1 Direct measurement of magnon temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Magnon-phonon interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.1.2 Magnons and their temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1.3 Back-scattering magnon mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.4 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.5 Measurement of magnon temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.1.6 Theoretical modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2 Temporal evolution of the spin Seebeck effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.2.2 Time-resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.2.3 Pt resistance measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.2.4 Thermal magnon diffusion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2.5 2D COMSOL simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2.6 Effective thermal-magnon diffusion length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.3 Microwave heat originated spin Seebeck effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.3.1 Sample structure design and experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.3.2 Asymmetry in spin pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3.3 Time resolved measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5
Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
A
Microwave spin currents in lateral spin valves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
A.1 Nonlocal spin injection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.2 Device structure and fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
A.3 Microwave excitation of pure spin current . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
A.4 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Own Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .119
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CHAPTER 1
Introduction
Spintronics is a rapidly evolving field of contemporary science and technology that has the potential to replace the world of electronics [1–4]. Unlike conventional electronics based on the
electron charge, in spintronics, the spin, another fundamental property of an electron, is explored.
The race of spintronics started with the discovery of the giant magnetoresistance (GMR) in the
late 1980s by Albert Fert [5] and Peter Grünberg [6] in epitaxial Fe|Co multilayers, as well as
with work by Parkin et al. [7] in polycrystalline multilayers. The initial work was carried out
at low temperature and high magnetic fields; however, the investigations of the GMR effect in
Co|Cu multilayers at room temperature and low magnetic fields [8], and the interface engineering
to enhance the GMR effect at very small fields [9, 10] made spintronics a technological reality by
developing a revolutionary nonvolatile magnetic random access memory (MRAM) for magnetic
hard disk drives (HDD) [11,12]. Although the GMR based magnetic media dramatically improved
the storage density, the technology was replaced by tunneling magnetoresistance (TMR) [13, 14]
just within ten years. In the TMR, the metallic spacer like Cu is substituted with a nonmetallic
material like MgO to enhance the magnetoresistance by multiple times. Other promising developments for further scaling of MRAM are the spin transfer torque based spin momentum transfer
(SMT) magnetization switching [15–17] and the magnetic domain wall racetrack memory [18].
In spite of advancements in the magnetic-recording industry [19], the designing of spin logics has
been a challenging task [2]. Shortly after the discovery of the GMR, Datta and Das [20] proposed
an electron spin based field effect transistor (spin-FET) where the electric field at the gate electrode
controls the orientation of spins. These spin-FETs have certain advantages over electronic FETs.
A spin-FETs can be used for fast reprogrammable logic devices with their own memory. They can
even eliminate the time delay between reading the data from magnetic memories and processing it
with the semiconductor based electronics. Recently, the spin-FET has been realized experimentally
[21, 22]; however, its functioning at ambient conditions has yet to be demonstrated.
Despite various advantages of spintronics, the basic architecture of information transfer and processing relies on the transport of electrons; though, there may not be a net charge transport.
In spintronic devices, a spin current—antiparallel spin polarized electrons moving in opposite
1
directions—carries the information in the form of spin angular momentum. The transport of electrons accompanies unavoidable Joule heating due to the intrinsic nature of the electronic transport.
The heat consumption and energy losses are considered to be major limitations of these technologies.
The emerging field of magnon spintronics provides another route for information transfer and
processing via spin waves [23–28]. Spin waves, entirely decoupled from the electron spin, are collective excitations of magnetic moments in magnetically ordered materials, and their quasiparticles
are known as magnons. Magnons are capable of carrying the spin current (spin angular momentum) and can be potential information carriers. Although spin waves have certain disadvantages
like slow group velocities and high attenuation compared to photons, the miniaturization of device
chips compensates these weaknesses and makes fast processing and modulation the most valuable features of a spin wave circuitry. Other advantages of spin waves are: (i) their wave nature,
thereby, they can propagate through magnetic waveguides, (ii) possibilities of external modulation
by a magnetic field, (iii) very short wavelengths (nanometers for the exchange spin waves), and
(iv) propagation over macroscopic distances even at ambient conditions.
The first spin-wave based logic device was experimentally demonstrated by Kostylev et al. in
Kaiserslautern [23]. The main challenges in the realization of magnon spintronic circuits are to
convert an electronic signal into a spin-wave signal and vice-versa, and to manipulate or control the
spin-wave signal using external means. Dynamic Oersted field based techniques like microstrips
and coplanar waveguides, are the most general techniques for spin wave excitation and detection.
In addition to this, during recent developments of the field, new phenomena are discovered to
deal with these requirements. For example, in heterostructures consisting of a ferromagnetic layer
and a paramagnetic metal layer, a direct current in the structure can excite spin dynamics in the
ferromagnetic layer. In this process, the direct current converts into the spin current (spin polarized
electrons) in the paramagnetic layer by the spin Hall effect (SHE) [29, 30], and subsequently, the
spin current excites the magnetization dynamics in the ferromagnetic layer by means of the spin
transfer torque (STT) [15, 16] and generates spin waves. The reverse effects like the inverse spin
Hall effect (ISHE) and the spin pumping effect have also been demonstrated experimentally. A
complete cycle of such conversion processes, i.e., an electric signal to a spin signal and back
to an electric signal, has been demonstrated experimentally in magnetic-insulator|normal-metal
(YIG|Pt) heterostructures [31]. These progresses bring the spin logics one step closer to real world
applications.
In recent years, another sub-field of spintronics known as spin caloritronics has emerged where the
interplay between heat and spin currents is in the main focus of research. Though thermoelectric
and thermomagnetic effects are well explored for the last two centuries, only recently thermomagnetic effects in meso- and nano-scopic devices get much attention due to discoveries of new
2
fascinating phenomena. Among these phenomena, the spin Seebeck effect is the most intriguing
phenomenon in the magnon spintronics. Analogous to the conventional Seebeck effect, Uchida
et al. [32] discovered that a thermal gradient applied to a ferromagnet generates a spin current in
the ferromagnet. The generated spin current is detected as an electric signal via the inverse spin
Hall effect. This effect is seen as of great interest due to its universal nature of existence for all
ferromagnets, and various industrial applications [33] such as the generation of electricity out of
waste heat [34].
Despite the fact that the spin Seebeck effect has been widely studied both theoretically [35–37] and
experimentally [32, 38–44], the underlying physics of this effect is not well understood. Uchida et
al. [32] in their very first investigation on Py|Pt structures predicted that the effect is caused by the
diffusion of spin polarized electrons in the magnetic metal. However, this concept was not well
taken as the typical spin diffusion lengths in metals are found to be around a few nanometers while
the effect was observed at millimeter length-scale. This concept failed again when the transverse
spin Seebeck effect was observed in magnetic insulators [41].
Within the framework of this thesis, a detailed investigation of the interaction of magnons with
phonons in single crystalline ferrimagnetic insulator yttrium iron garnet (Y3 Fe5 O12 , YIG) films
are carried out for a deeper understanding of the physics of spin caloric phenomena, in particular
the spin Seebeck effect. The material YIG has been selected for these investigations because of its
the lowest known spin-wave damping and electrically insulating nature.
The thesis is divided into four main chapters, an Introduction Chapter 1 and the Appendix.
In Chapter 2, the theoretical background relevant to understand the documented work is presented.
The chapter starts with a brief summary of the magnetism and various interactions involved. Subsequently, the fundamental of the magnetization dynamics and the Landau-Lifshitz-Gilbert equation
are described. A theoretical description of spin waves and their dispersion relations, is presented
next. The remaining chapter deals with spintronic phenomena. The fundamental principles of
the spin transfer torque and spin pumping in ferromagnetic|normal-metal heterostructures are discussed first. The concept of the spin Hall and inverse spin Hall effects is reported next. In the
last part of the chapter, various spin caloric phenomena are presented. Firstly, a comprehensive
overview of the spin-dependent Seebeck and Peltier effects is reported. A detailed description of
the spin Seebeck effect and its theory is provided next. At last, the anomalous and planar Nernst
effects are presented in order to draw a clear picture of various thermomagnetic effects involved in
conducting and nonconducting ferromagnetic materials.
Chapter 3 deals with the experimental methods and their setups used in the coursework of this thesis. The first part of the chapter presents the physical properties and thin film deposition techniques
of the ferrimagnetic material YIG. The procedure of the YIG|Pt sample preparation is described
3
next. The second part of the chapter is dedicated to Brillouin light scattering (BLS) spectroscopy.
Detailed descriptions of the working principles of a tandem Fabry-Pérot interferometer, conventional BLS, and BLS microscopy are discussed there. The third part of the chapter is dedicated
to the microwave techniques used for the spin-waves excitation in the YIG|Pt system. In the last
part, a brief summary of the COMSOL Multiphysics simulation tool is presented which has been
employed to study the thermodynamics of the YIG|Pt system.
In Chapter 4, the experimental results obtained within the framework of this thesis are presented.
In the first part of the chapter, the experiment about direct measurements of the magnon temperature is described in detail. Firstly, the concept of the magnon temperature and its measurement
using the back-scattering magnon mode of spin waves is described. Subsequently, the results of
magnon temperature measurements in a YIG film subject to a thermal gradient will be reported.
An attempt is made to understand the underlying physics of the spin Seebeck effect.
The second part of the chapter is dedicated to the study of the temporal evolution of the longitudinal
spin Seebeck effect. The experimental setup designed using a laser heating technique is explained.
The obtained results and their explanations with the thermal magnon diffusion model are presented
in this part. The last part addresses the microwave heat originated spin Seebeck effect in the
YIG|Pt system. In this part, additional to the longitudinal spin Seebeck effect measurements,
the experiments on the asymmetric excitation of spin waves and spin pumping measurements are
presented .
In Chapter 5, the experimental results obtained within the framework of this thesis are summarized, and a future perspective of the investigations is provided.
In Appendix A, the experiments on the microwave-spin-currents excitation and detection in lateral
spin valves are presented. The studies were carried out with the team of Prof. YoshiChika Otani
at the Center for Emergent Matter Science, RIKEN, Wako-shi, Japan. A brief description of the
theory of lateral spin valves is provided first. The procedure of spin valve device fabrication is
presented next. Subsequently, the technique of the excitation of a pure spin current by dynamics
magnetic fields, and the experimentally obtained results are discussed. At last, the conclusions are
drawn on the basis of these studies.
4
CHAPTER 2
Theoretical concepts of magnetism and spintronics
The first scientific discussion about magnetism is recorded in the Greek civilization around 600
B.C. [45]. Philosopher Thales of Miletus experimented with amber and lodestone to know about
electricity and magnetism [45]. Apparently, he associated these two phenomena together, although
more than twenty-four hundred years had elapsed before actual theoretical and experimental relationships were proven in the nineteenth century. In this chapter, the theoretical background of
magnetism is presented. Although it is impossible to cover entire theoretical literature here, the
most relevant theories will be presented in this chapter in order to understand the documented
research work.
In the first section, the concepts of ferromagnetism are introduced, and a general description about
the important magnetic interactions will be presented. The fundamental principles of magnetization dynamics with the Landau-Lifshitz-Gilbert equation are described in Section 2.2. Subsequently, a detailed descriptive theory of spin waves is discussed, and their dispersion relations for
ferromagnetic films are derived. Once the basic knowledge about the fundamentals of magnetism
has been developed, the field of spintronics will be the main focus of the chapter. The phenomena
of spin pumping, spin transfer torque, spin Hall effect, and inverse spin-Hall effect will be presented in the following sections. The last part of this chapter is dedicated to thermoelectric effects
in magnetic materials. The phenomenon of the spin Seebeck effect will be extensively discussed
for various experimental geometries and materials.
Over time, a common practice about physical units in magnetism has been altering. In the twentieth
century, the cgs system of units was used mostly. Throughout this thesis, the international standard
unit system SI (Systéme international d’unités, in French) has been adopted [46]. A unit conversion
table between the cgs system and the SI system can be found in Ref. [47].
2.1. Ferromagnetism and magnetic interactions
On the basis of the magnetic moment of an individual atom, materials can be categorized in three
classes. The materials which have zero magnetic moment for each of its constituent atom due to
5
2.1 Ferromagnetism and magnetic interactions
completely filled electron shells are named as diamagnetic materials. These materials induce an
opposite magnetization in the presence of a magnetic field. Hence, their magnetic susceptibility
is negative. Diamagnetism is an inherent characteristic of all materials except hydrogen atom
[48–50]. The second class of materials is paramagnetic materials. In paramagnetic materials,
the magnetic moment of an individual atom is non-zero, though, the average moment over many
atoms is zero in the absence of a bias magnetic field. The presence of a magnetic field aligns
atomic moments along the field direction and induces a positive magnetization. Therefore, these
materials have positive susceptibility which overcomes the negative diamagnetic susceptibility.
The third class of materials is known as ferromagnetic materials. Unlike paramagnetic materials,
the magnetic moments of individual atoms interact with each other very strongly. This strong
interaction creates a certain ordering of the atomic moments even in the absence of an external
magnetic field. This ordering causes a nonzero magnetic moment at zero magnetic field. Therefore, the susceptibility of these materials is very high [49, 51]. On the basis of the ordering of
the magnetic moments, this kind of materials can be further classified further as ferromagnetic,
ferrimagnetic, and antiferromagnetic. An antiferromagnetic material has its magnetic moments
opposing each other in the absence of a magnetic field. These opposing moments are equal in
magnitude; therefore, no magnetization is persisted by an antiferromagnetic material. In the case,
when opposing moments are unequal; a net magnetization exists, and this subclass of materials is
known as ferrimagnetic materials. A detailed description of these classified materials can be found
in Ref. [49, 51–53].
The degree of magnetic ordering depends on the temperature of the material. Below a certain critical temperature known as the Curie temperature, the ordered magnetic moments in ferromagnets
generate a long-range ordering to build up a macroscopic magnetization. This interaction has a
quantum mechanical origin and is named as the exchange interaction. The exchange interaction is
the basis for the formation of ferromagnetism and is discussed in the next section. Another kind of
coupling commonly known as dipolar interaction has a classical origin and describes the mutual
interaction among magnetic dipoles of a material. The dipolar interaction is a long-range interaction which also leads to the ordering of magnetic moments, although its strength is too weak to
describe the ferromagnetic ordering at room temperature.
2.1.1 Exchange interaction
In metals and insulators, a direct exchange interaction between spins at the lattice sites has no
classical analogue and is the origin of the ferromagnetic interaction. Introduced by Heisenberg in
1928 [54], this interaction is caused by the overlap of the electronic wave functions. Electrons are
spin-half particles (fermions) and, so, obey the Pauli exclusion principle [55], i.e., no two identical
6
2.1 Ferromagnetism and magnetic interactions
fermions may occupy the same quantum state simultaneously. Consequently, electrons must have
an antisymmetric wave function with respect to exchange of particles.
In a two-electron system like a hydrogen molecule, the total electronic wave function is a product
of a spin-dependent and a space-dependent wave function. An antisymmetric total wave function
leads to two possible combinations: a symmetric spin-dependent wave function (triplet state S =
1) with an antisymmetric spatial wave function or antisymmetric spin-dependent wave function
(singlet state, S = 0) with symmetric spatial wave function. Symmetric and antisymmetric spatial
wave functions attribute to different Coulomb energies. The difference in the Coulomb energies is
known as the exchange energy. Of course, the lower energy state is more favorable.
Quantum mechanically, in the Heisenberg model, the Hamiltonian of the exchange interaction for
an n-electron system can be written as
H=−
n
2 X ex
J Si · Sj ,
~2 i6=j i, j
(2.1)
where the exchange energy for a two-spin system Si and Sj , with the single-electron wavefunctions
ψi , ψj , can be expressed as an exchange integral:
Z
e2
ex
Ji, j = 2 ψi∗ (r1 )ψj∗ (r2 )
ψi (r1 )ψj (r2 ) dr1 dr2 .
|r1 − r2 |
(2.2)
A positive J ex represents a ferromagnetic coupling which tends to align spins parallel to each other
while a negative J ex means an antiferromagnetic coupling. The exchange integral is non-zero if
the electronic wave functions of spins overlap each other. As the spatial overlapping between the
wave functions decreases rapidly with the distance (Eq. (2.2)), the exchange interaction is a shortrange interaction. Thereby, in the first approximation, the exchange energy can be calculated over
the nearest neighbors (N N ). The exchange energy of a spin Si can be expressed in terms of its
neighboring spins Sj :
NN
NN
Eiex
while for the entire material
X
X
2J ex
2J ex
µi ·
Sj .
Sj =
= − 2 Si ·
~
gµB ~
j
j
1 X ex
Ei .
2 i
E ex =
(2.3)
(2.4)
Here, the relation between spins and magnetic moments, µ = −gµB S/~, is used, where g is the
Landé g-factor, and µB is the Bohr magneton. Further, the exchange energy can be interpreted in
terms of the Zeeman energy as Ez = −µi · Bex . So, Eq. (2.3) can be described as the interaction
of the i-th spin with an effective magnetic field
NN
Bex = −
2J ex X
Sj .
gµB ~ j
(2.5)
7
2.1 Ferromagnetism and magnetic interactions
It is important to mention here that the alignment of the neighboring spins is not always exactly
parallel or anti-parallel to each other. In the case of spin-waves and domain walls, spins are slightly
tilted with respect to each other leading to an increase in the exchange energy. The derivation of a
general expression of the exchange field considering a macroscopic magnetization M can be found
in textbooks [56].
In a semiclassical approach for macroscopic calculations, the exchange energy of the system can
be written in terms of the averaged value as [56]
E ex = −2
X
Ji, j S 2 cos θi, j .
(2.6)
i>j
Here θi, j is the angle between the direction of the nearest spins Si and Sj . For long wavelengths,
where θi, j is small, the above equation can be expanded as a Taylor series, and the exchange energy
density can be quantified as
ǫex =
A
(∇ · M)2 ,
2
Ms
(2.7)
where A, Ms denote the exchange constant and the saturation magnetization, respectively. Using
the relationship of magnetic field and energy, i.e., Bex = −∇M ǫex , the exchange field can be
calculated as
Bex =
2A 2
D 2
∇M=
∇ M.
2
Ms
Ms
(2.8)
Here D = 2A/Ms is the exchange stiffness.
2.1.2 Dipolar interaction
The dipolar interaction is a long-range interaction which has a classical origin. The interaction is
basically the influence of one magnetic moment on another and is the origin of the shape anisotropy
and the dipolar mode dispersion. The dipolar energy between two magnetic moments µi and µj
depends on their separation rij and relative orientation, given as [57]
µ0 µi · µj 3(µi · rij )(µj · rij )
dp
dp
.
−
E = −µi · µ0 Hj (rij ) =
4π
rij3
rij5
(2.9)
The field Hdp
j describes the magnetic field generated by the magnetic moment µj . Eq. (2.9) can
be summed up over all magnetic moments considering their mutual interaction with each other to
obtain the dipolar energy of the system. The strength of the dipolar interaction can be estimated by
considering a simple example. The total dipolar energy of a system of two moments of one Bohr
magneton (1 µB ) each, separated by a distance of 0.1 nm apart, is around 9.6 × 10−26 J, which
8
2.1 Ferromagnetism and magnetic interactions
is equivalent to a temperature of 7 mK. This estimation shows that the dipolar interaction has a
negligible contribution to the ferromagnetism at room temperature.
In comparison to the exchange interaction (Eq. (2.8)), which exists at the atomic scale, the dipolar
interaction is a long-range interaction proportional to ∝ 1/rij3 . Therefore, the dipolar magnetic
field should be evaluated over all magnetic moments in a solid. In a homogeneously magnetized
infinite long sheet, the dipolar fields of constituting magnetic moments compensate each other and
generate no net magnetic field around it. However, for a spatially confined magnetic structure, an
uncompensated magnetic field exists in its vicinity, known as the stray field or demagnetizing field
Hd .
The demagnetizing field Hd of a system can be calculated using Maxwell’s equations. Mathematics becomes simpler in the magnetostatic limits, where the system is free from current and charges.
For these systems, Maxwell’s equations are reduced to
∇ × Hd = 0 ,
(2.10)
∇ · B = µ0 ∇ · (Hd + M) = 0 .
(2.11)
The assumptions are valid for a system where variations can be considered as quasi-static, i.e.,
slow compared to the speed of light. These approximations are justified for spin waves as their
frequencies for a given wave vector are much smaller than the frequency of the light for the same
wave vector [58, 59]. According to Eq. (2.10), the demagnetizing field Hd is an irrotational vector
field and can be written as the gradient of a scalar field, magnetic potential ΦM , as Hd = −∇ΦM .
Substitution of the magnetic potential into Eq. (2.11) leads to a magnetostatic analog of the Poisson
equation in electrostatics:
∇2 ΦM = −ρM ,
(2.12)
ρM = −∇ · M
(2.13)
where
is the magnetic charge density analog to the electric charge density and regarded as a source of the
magnetic stray field. Any nonzero ρM give rise to an inhomogeneous magnetic field. The solution
of Eq. (2.12) can be found in textbooks [57] as
Z
Z
′
1
1
ρM
∇ · M(r′ ) ′
′
ΦM (r) =
dr = −
dr .
4π
|r − r′ |
4π
|r − r′ |
(2.14)
For finite magnetization distributions, the integral can be divided into two terms—one describes the
stray field generated by magnetic charges in the volume and the other term describes the influence
of surface charges:
1
ΦM (r) = −
4π
Z
′
V
1
∇ · M(r′ ) ′
dr +
′
|r − r |
4π
I
S
n(r′ ) · M(r′ ) ′
dA .
|r − r′ |
(2.15)
9
2.2 Magnetization dynamics and Landau-Lifshitz-Gilbert equation
Here, n(r′ ) represents the normal to the surface. It is important to notice here that the surface
magnetic charges n(r′ ) · M(r′ ) will be nonzero when the magnetization is out of the surface plane.
′
In the case of a homogeneous magnetization distribution, ∇ · M(r′ ) will be zero and the demagnetization field will exist due to the surface magnetic charges as
I
1
n(r′ )
′
Hd (r) = − ∇ M ·
dA .
′
4π
S |r − r |
(2.16)
Since, the demagnetizing field Hd (r) depends linearly on the magnetization M of the system, a
demagnetization tensor N can be introduced as
Hd (r) = N M(r) .
(2.17)
Although the calculation of the demagnetization field is difficult, it can be calculated analytically
for regular geometries like ellipsoid, sphere, cylinder, and infinite plane. In the case of infinitely
extended films magnetized in-plane, the net demagnetizing field of the system vanishes. Generally, this case is applicable for thin magnetic films where the thickness of the film is very small
compared to their lateral dimensions. However, when such a magnetic film is kept in an oscillating
magnetic field, its magnetization starts to precess and generates a dynamic demagnetization magnetic field. This dynamic demagnetization field plays a very important role in the magnetization
dynamics of thin films.
2.2. Magnetization dynamics and Landau-Lifshitz-Gilbert equation
In the last section, different magnetic interactions have been introduced. It is seen that magnetic
moments in magnetic materials are coupled via these interactions. When a magnetic material
is exposed to an external magnetic field, its magnetic moments tend to orient themselves along
the bias field to minimize their energy. The kinetic motion of the magnetic moments disturbs this
alignment. Therefore, the moments do not orient themselves directly to the magnetic field; instead,
they execute a precessional motion around the direction of the field. The precession of magnetic
moments is referred to as magnetization dynamics. This behavior of magnetization under the
influence of an external magnetic field is described phenomenologically by the Landau-Lifshitz
and Gilbert equation of motion [60].
The Landau-Lifshitz-Gilbert (LLG) equation is a torque equation which was first introduced by
Lev Landau and Evgeny Lifshitz in 1935 as Landau-Lifshitz equation [61]. The Landau-Lifshitz
(LL) equation was a damping-free equation. Later, Gilbert modified it by inserting a magnetic
damping term [60]. In this section, first the LL equation will be derived by a semiclassical approach. Later, this equation will be modified by introducing the Gilbert damping term.
10
2.2 Magnetization dynamics and Landau-Lifshitz-Gilbert equation
When a magnetic moment µm is placed in an effective magnetic field Beff , it experiences a torque:
τ = µm × Beff .
(2.18)
In a semiclassical approach, the magnetic moment µm of an atom can be written in terms of its
angular momentum J as
gµB
J = −γJ ,
(2.19)
~
where γ = gµB /~ is the gyromagnetic ratio. As τ = dJ/dt, Eq. (2.18) can be re-written as
µm = −
dJ
1 dµm
=−
= µm × Beff .
dt
γ dt
(2.20)
In the continuum limit, the atom magnetic moment can be replaced by the macroscopic magnetization M resulting in the equation of motion, i.e, Landau-Lifshitz (LL) equation:
dM
= −γM × Beff .
dt
(2.21)
Here, the effective magnetic field Beff is a sum of all external and internal magnetic fields:
Beff = B0 + BM (t) + Bex + Bd + Bani + · · ·
(2.22)
Here, B0 is the static component of the applied magnetic field while BM (t) is the dynamic component. Bex is a magnetic field originating from the exchange interaction. Bd represents the
demagnetization field created by the dipolar interaction of magnetic surface and volume charges.
The field Bani includes all kinds of anisotropic fields like shape anisotropy, crystalline anisotropy,
etc.
The physical meaning of Eq. (2.21) explores two important features of the equation of motion. A
dot-product of Eq. (2.21) with M results that the magnitude of M is conserved. Secondly, a dotproduct with Beff concludes that the orientation of M with respect to Beff will not vary over time.
Further, the system is non-dissipative which means that the magnetization will continue to precess
for infinitely long time. Definitely, in the real world no such system exists. To rectify this issue,
Gilbert [60] phenomenologically introduced a proper damping term to the LL equation leading to
a Landau-Lifshitz-Gilbert equation as
dM
αG
= −γM × Beff +
dt
Ms
dMs
Ms ×
dt
.
(2.23)
Here, αG denotes the dimensionless Gilbert damping parameter. An important feature of the
Gilbert damping parameter is its viscous nature, i.e, an increase in the rotation of magnetization
dMs /dt increases the damping of the system.
A schematic of the interplay between different torques on the magnetization is depicted in Fig. 2.1.
The damping torque always acts perpendicular to the magnetization as well as the precessional
11
2.3 Spin waves
Figure 2.1: The precessional motion of the magnetization around the effective magnetic field direction,
governed by the Landau-Lifshitz-Gilbert equation (2.23). The torque −γM × Beff provides the rotational
motion around Beff while the Gilbert damping term forces the magnetization to be aligned along the effective
field Beff . Figure from Ref. [62].
term and tries to align the magnetization along the effective magnetic field. Therefore, the damping torque provides a dissipative mechanism which eventually, transfers the energy and the angular
momentum of the spin system (magnon system) to the phonon system via spin-orbit (spin-lattice)
interaction [63]. Other than spin-orbital interaction, in metallic systems, magnon-electron scattering [64–66] and eddy currents [67] provide extra channels for magnetization relaxation. Additionally, damping channels like spin pumping [68, 69], multi-magnon scattering [58, 70, 71], and
Cherenkov scattering processes (Section 4.1.7) also exist in magnetic materials.
2.3. Spin waves
In the last section, it has been observed that, within the macro-spin approximation, the LandauLifshitz-Gilbert equation describes the magnetization dynamics in homogeneously magnetized
structures. A ferromagnet is perfectly ordered at T = 0 K. An increase in the temperature causes
thermal fluctuations and reduces the magnetization. The magnetization vanishes at the critical temperature Tc . In order to explain the temperature dependence of the magnetization in ferromagnetic
materials, Bloch introduced the concept of spin waves as collective excitations of magnetic moments in magnetically ordered materials [72]. At temperatures significantly below Tc , low energy
transverse fluctuations of the magnetization in ferromagnets are known as spin waves or magnons.
Analog to lattice vibrations, spin waves are quantized in nature, and their quanta are known as
magnons, which are the Goldstone bosons of a ferromagnetic system [73]. Goldstone bosons are
quasi-particles which appear as a consequence of continuous global symmetry breaking.
Unlike phonons, spin waves (magnons) are oscillations of magnetic moments (transverse deviation
12
2.3 Spin waves
of moments) in a wave-vector space (k-space) or in real space. For a given spin wave, all magnetic
moments precess with a fixed frequency depending on the effective magnetic field in the system,
and their wavelengths are determined by the variation of phase among the neighboring magnetic
moments. Since the wavelength of a spin wave is determined by the phase shift of the precessing magnetic moments, the mutual interactions between the moments, discussed in Section 2.1,
play a crucial role for spin waves. On the basis of these interactions—exchange interaction and
dipolar interaction—spin waves are classified into two branches: dipolar-dominated spin waves,
and exchange-dominated spin waves. For long-wavelength spin waves, the difference between
neighboring moments is rather small; therefore, the impact of exchange interaction is negligible
for these spin waves. The energy of long-wavelength spin waves is determined by the dipolar energy of the system; thus, these spin waves are known as dipolar-dominated spin waves. In contrast,
the exchange interaction is very important for short-wavelength spin waves. The energy of these
exchange-dominated spin waves is determined by the exchange energy of the system and given as
~ω ∝ k 2 [58, 74].
The propagation properties of spin waves are elucidated by their dispersion relation, i.e., a relationship between the angular frequency ω and wave vector k. In the next sections, the dispersion
relation will be described for a ferromagnetic ellipsoid and an in-plane magnetized thin films without anisotropy. The derivations of the dispersion relation in thin films are based on the theory
purposed by Damon and Eshbach [75] as well as by Hurben and Patton [59, 76, 77]. A modified
theory by Kalinikos and Slavin [78] will be presented later.
2.3.1 Ferromagnetic resonance
In 1946, Griffiths observed that microwaves are strongly absorbed by ferromagnetic films at a
frequency different from the Larmor frequency of the electron spin [79]. Later, Kittel [80] theoretically explained this phenomenon as the ferromagnetic resonance in a magnetic material. The magnetization dynamics of a uniform precession motion is described by the Landau-Lifshitz-Gilbert
equation. The magnetic moments are assumed to precess coherently with the same frequency and
phase. The frequency of a uniform precession mode is known as ferromagnetic resonance (FMR)
frequency ωFMR . The FMR mode describes spin waves with infinite wavelength, i.e., their wave
vector |ksw | = 0. Therefore, all magnetic moments are parallel to each other and precess in phase.
Hence, the exchange energy to the total spin-wave system equals to zero.
In order to calculate the FMR frequency, the Landau Lifshitz equation (2.21) is solved for small
dynamic magnetic fields. In this approach, the magnetization is assumed to have a static as well as
a dynamic component written as M(t) = M0 +m(t). The dynamic magnetization m is assumed to
have a harmonic time-dependence as m = m exp(iωt). The dynamic magnetization is considered
13
2.3 Spin waves
to be a response of the effective magnetic field using Eqs. (2.17), (2.22), given as
B(t) = B0 − µ0 Ñm(t) .
(2.24)
Assuming the dynamic magnetization to be very small in comparison with the static magnetization
(m(t) ≪ M0 ), the LLG equation (2.23) can be linearized. For an ellipsoid with the demagnetizing
tensor N obtained from analytical solutions, the ferromagnetic resonance frequency [81] will be
q
ωFMR = γ [Beff + µ0 Ms (Nyy − Nzz )][Beff + µ0 Ms (Nxx − Nzz )] ,
(2.25)
and is known as the Kittel formula. Here, the effective magnetic field is Beff = µ0 (Hext + Hani ),
and the external magnetic filed Bext = µ0 Hext act along the z-axis. For an infinitely long thin film,
magnetized in the xy-plane (Nxx = 1, Nyy = Nzz = 0), the Kittel formula is simplified to
p
ωFMR = γ Beff (Beff + µ0 Ms ) .
2.3.2
(2.26)
Walker’s equation and spin waves
Unlike FMR, for spin waves with a finite wave vector, neighboring spins posses phase differences
in their gyration motion. Therefore, the magnetic interactions (Eqs. (2.8), (2.16)) can no longer
be neglected for these spin waves. As a consequence, the dynamic magnetization is not uniform
anymore and becomes space-dependent, i.e., m(r, t). The dispersion relation of dipolar spin waves
is derived here using the field dynamics and magnetostatic Maxwell’s equations.
The magnetostatic Maxwell’s equations for the dynamic fields are as follows [59]:
∇×h=0,
(2.27)
∇·b=0,
(2.28)
b = µ0 (1 + χ̄)h ,
(2.29)
with
where χ̄ is the Polder susceptibility tensor [59]. By Eq. (2.27), h can be written in terms of a scalar
potential as h = −∇Ψ. Using, the above equations one obtains:
2
∂ 2Ψ
∂ Ψ ∂ 2Ψ
+
+
= 0,
(1 + χ)
∂x2
∂y 2
∂z 2
(2.30)
where
ω0 ωM
,
ω02 − ω 2
ωM = −γµ0 Ms ,
χ=
ω0 = −γµ0 H0 .
14
(2.31)
(2.32)
(2.33)
2.3 Spin waves
Equation (2.30) is known as the Walker equation. The case χ = −1 describes ferromagnetic
resonance (FMR) while χ 6= −1 represents spin waves with nonzero wave vectors. The dispersion
relation for dipolar spin waves in an infinitely long in-plane (yz-plane) magnetized thin film (see
Fig. 2.2) of thickness d is calculated here. Assuming the potential as a function of plane waves
with their wave vector inclined at an angle φ with respect to the magnetization M, it has a general
form:
Figure 2.2: Thin film geometry for spin-wave dispersion relation calculations. A film of thickness d is
placed in the yz-plane, magnetized along the z-axis. The in-plane wave vector k is inclined at an angle φ to
the magnetization M.
Ψi = A(x) exp[−ik(z cos φ + y sin φ)], for − d/2 ≤ x ≤ d/2
(2.34)
Ψo = B(x) exp[−ik(z cos φ + y sin φ)], for other x
(2.35)
The potential Ψi will be chosen such that it satisfies the Walker equation inside the magnetic film
while potential Ψo will be obtained from the Laplace equation (∇2 Ψ = 0) since the magnetization
outside the film is zero. Further, at boundaries x ± d/2, the tangential component of h, and the
normal component of b must be continuous. Using the above mentioned conditions, a general
dispersion relation for dipolar spin waves in an in-plane magnetized film can be formulated [59,
77, 82] as
k 2 − (1 + χ)2 q 2 − κk 2 sin2 (φ) + 2(1 + χ)qk cot(qd) = 0 ,
where χ is given by Eq. (2.31). The out-of-plane wave vector q is defined as follows:
s
1 + χ sin2 (φ)
.
q=k −
1+χ
And, κ is given by:
κ=
ωωM
.
− ω2
ω02
(2.36)
(2.37)
(2.38)
Equation 2.36 is an implicit function; so, it needs to be solved numerically. The results of the
numerical calculations by Damon and Eshbach [75] are shown in Fig 2.3.
On the basis of the characteristics of Ψ along the thickness of thin films—oscillatory or exponentially decaying—spin waves are categorized in two classes:
15
2.3 Spin waves
Figure 2.3: (a) Spin-wave dispersion for an in-plane magnetized thin film (from Ref. [75]) calculated by
using Eq. (2.36). The volume-wave dispersion (k|| ) consists of various thickness modes according to
Eq. (2.37). The surface mode exists only above a critical angle φs . (b) The amplitudes of the surface
waves, having their maximum at the film surface, for two oppositely propagating spin waves. Figures from
Ref. [82].
1. When the out-of-plane wave vector q is real, the solutions of Eq. (2.36) are periodic in
nature due to the cotangent (cot(qd)) function. These spin waves are known as backward
volume magnetostatic spin waves (BVMSW). The word volume appears as the magnetization
varies over the entire thickness of the thin film which means spin waves exist in the entire
volume [84]. The dispersion of these spin waves decreases monotonically, and thus their
group velocity vg = ∂ω/∂k is negative (see Fig. 2.4). This is the reason why these waves
are called backward volume modes. It is important to note that the solutions of Eq. (2.36)
for a real q are periodic; so, there can exist an infinite number of wave vectors along the
thickness of the film. Furthermore, by solving Eq. (2.37), it is possible to determine the
frequency band in which BVMSW exist.
2. On the other hand, when the wave vector q is imaginary, the solution of Eq. (2.36) consists of an exponential decaying amplitude function along the film thickness. So, these spin
waves are localized at the surface of the film and known as magnetostatic surface spin waves
(MSSW). These spin waves exhibit a non-reciprocal propagation with a positive group velocity (see Fig. 2.4). Their frequency band exists above the BVMSW band; however, their
propagation is permitted only for propagation angles above a certain critical angle [77],
shown in Fig. 2.3. The MSSW are studied in the framework of this thesis in YIG films, and
will be presented in Section 4.3.2.
The spin-wave modes discussed above have their static magnetization in the plane of the film.
16
2.3 Spin waves
Figure 2.4: Schematic illustration of dipolar spin waves. (a) Backward volume magnetostatic spin waves
(BVMSW) with k||H0 . (b) Magnetostatic surface spin wave (MSSW) with k⊥H0 . The red arrows represent
the direction of the dynamic magnetization. The magnetic field H0 is pointing outwards here. (c) The
dispersion relations for the MSSW-mode with k⊥H0 , and the BVMSW-mode with k||H0 . Figure (c) from
Ref. [83].
When the magnetization is out of the film-plane, the spin-wave mode is known as forward volume
magnetostatic spin waves (FVMSW). The FVMSW are not relevant to the work presented in this
thesis. The reader can refer to literature [58, 59, 85] for a detailed description.
2.3.3
Spin-wave theory of Kalinikos and Slavin
In the last section, the dispersion relation for dipolar spin waves has been derived. Since the relation is an implicit function, numerical analysis is required to get a solution. In 1986, an analytic
dispersion relation was formulated for thin magnetic films by Kalinikos and Slavin [78], accounting both for dipolar and exchange interactions on the basis of classical perturbation theory. The
analytical solutions are not generalized solutions; these solutions are rather restricted for thin films
with thicknesses much smaller than the wavelength of spin waves, i.e., d ≪ λsw . The reliability of
this theory is very high as many experimental results support the theoretical calculations [86–89].
According to this theory, the spin-wave dispersion relation for a thin film lying in the yz-plane (see
Fig. 2.2) is the following:
q
ω(k) = γ (B0 + Dk 2 )(B0 + Dk 2 + µ0 Ms Fpp (k|| , φ, d) ) ,
(2.39)
where B0 is the effective magnetic field, D is the exchange stiffness, φ is the angle between the
magnetization and the in-plane wave vector. The in-plane wave vector k|| is defined by:
pπ ,
k 2 = k||2 +
d
(2.40)
17
2.4 Spin transfer torque and spin pumping
which indicates that the total wave vector is quantized in the x-direction. The dimensionless function Fpp is known as dipole-dipole matrix element and accounts for the dipolar fields created by
the magnetic surface charges of the p-th perpendicular standing spin-wave mode. For the first
spin-wave mode, i.e., p = 0, the dipole-dipole matrix element is given as:
µ 0 Ms
F00 (k, φ, d) = 1 + P00 (k, d)[1 − P00 (k, d)]
sin2 (φ) − P00 (k, d) cos2 (φ) , (2.41)
B0 + Dk 2
where
1 − e−kd
P00 (k, d) = 1 −
.
(2.42)
kd
On neglecting the exchange interaction, the dispersion relation for the magnetostatic surface spin
waves (MSSW) at φ = π/2, and the backward volume mode (BVMSW) at φ = 0, can be obtained
from Eq. (2.39) as
and
v
u
u
ωMSSW (k) = γ tB0
ωBVMSW (k) = γ
B 0 + µ 0 Ms +
µ 0 Ms
2
2
(1 − e−2kd )
!
s
1 − e−kd
B 0 B 0 + µ 0 Ms
.
kd
(2.43)
(2.44)
2.4. Spin transfer torque and spin pumping
In the previous sections, a basic understanding about magnetism, magnetization dynamics, and
spin waves has been developed. An emphasis was given to the theoretical concepts of spin waves
and their modes in thin magnetic films. In the remaining part of this chapter, the focus will be on
the development of a comprehensive conceptual knowledge about various phenomena related to
electronic charge, electronic spins, spin waves, and heat currents.
In the field of spintronics [1–3, 90–93], most phenomena can roughly be classified into two categories: (i) independent-electron spin based phenomena, (ii) collective excitation (spin-wave) based
phenomena. In metallic systems, both of these categories can exist simultaneously while, in magnetic insulators, spin currents are carried only by spin waves; so, only spin-wave based phenomena
exist. Phenomena like the spin Hall effect and its Onsager reciprocal the inverse spin Hall effect utilize the spin property of independent electrons [29, 94–98], whereas in phenomena like the
spin pumping process, spin waves generate a spin current [99–101]. In a particular phenomenon
known as the spin transfer torque, the spin-polarized electrons excite the magnetization precess
and, thereby, convert a spin polarized current to a spin-wave current [102–106].
The field of spintronics is emerging very rapidly. New effects and phenomena are being invented
frequently. In this thesis, only the relevant effects are discussed. A brief introduction to the spin
18
2.4 Spin transfer torque and spin pumping
transfer torque and spin pumping will be given in this section. The next section will be dedicated
to spin Hall effects. In the subsequent sections, the field of spin caloritronics, which focuses
on the interaction of a spin with heat currents, will be described. An overview about various
thermoelectric effects will be given first. Later, the phenomenon of the spin Seebeck effect will be
discussed in detail.
2.4.1 Spin transfer torque
When electrons flow through ferromagnetic materials, they become partially spin polarized due to
spin-based interactions between the ferromagnets and electrons. By means of the same interactions, spin-polarized electrons can also influence the orientation of the magnetization in ferromagnets. This effect is known as spin transfer torque (STT). In 1996, Slonczewski [15] and
Berger [16] independently predicted that a current flowing perpendicular to the plane (CPP) in
a metallic ferromagnet|normal-metal|ferromagnet multilayer1 can generate a spin transfer torque
via spin-polarized electrons, strong enough to reorient the magnetization in one of the ferromagnet
layer. When current flows through one magnetic layer, it becomes spin polarized and carries
angular momentum with it. The current remains spin polarized inside the normal metal, and, subsequently, interacts with the magnetization of the next adjacent magnetic layer. The exchange
interaction between the injected spins and the magnetization induces a torque. This torque is
known as spin transfer torque which can result in magnetization precession when the spin current
is large enough [107, 108].
The existence of the spin transfer torque has been demonstrated in many challenging experiments [17, 106, 109, 110]. Moreover, magnetization precession and, even, magnetization reversal has been achieved in nano-oscillators [111–113] and magnetic tunnel junctions (MTJ) [114].
Recent progress of the STT phenomenon overcomes the only way to switch or excite magnetic moments using magnetic fields in the last 50 years, by developing magnetic random access memory
(MRAM) which enables to scale memory storage devices [115, 116].
In the macroscopic approximation, using the conservation of angular momentum, the so-called
Slonczewski torque, which equals to the rate of change of the total magnetic moment of a magnet,
reads [102, 107, 117]
γ~
m × (m × Is ) ,
(2.45)
2eMs V
where Ms is the saturation magnetization of the ferromagnetic layer of volume V , and m is the
τs = −
unit vector along the magnetization direction. The spin current in the ferromagnet is represented by
Is . Since the torque influences the magnetization dynamics of the ferromagnetic layer, it appears
as an additional contribution to the LLG equation and has a similar form as the Gilbert damping
1
Throughout this thesis, normal metal stands for a diamagnetic or a paramagnetic metal.
19
2.4 Spin transfer torque and spin pumping
torque. Depending on the direction of the spin current Is , the Slonczewski torque either enhances
the Gilbert damping or reduces it. The direction of the spin current depends on the geometry and
materials combination of the device [102, 108].
Besides the magnetization precession and reversal in uniformly magnetized layered systems, the
spin transfer torque has an important application for domain wall motion in magnetic structures.
On the basis of STT, Berger first proposed the theory of domain wall motion by spin polarized
electrons [16, 118], and later many laboratories demonstrated it experimentally [119–122]. These
results stimulated the development of new storage devices on the basis of current-induced domain
wall motion, for example, racetrack memory [18].
2.4.2 Spin pumping
In the late 1990s, the physics community was motivated to perform spin-current induced magnetization switching in ferromagnet|normal metal layered structures. During the investigations,
it was found that in Cu-Co and Pt-Co layered structures, the Gilbert damping parameter of the
ferromagnetic metal was considerably larger than the bulk values [17, 109, 123]. Tserkovnyak
et al. [100, 124] described the enhancement of the Gilbert damping parameter as a leakage of the
spin current from the ferromagnet to the adjacent normal metal. The mechanism is an inverse
effect of the spin-current induced magnetization switching, i.e., a precessing magnetization loses
torque by emitting a spin current. Therefore, magnetization precession acts as a spin pump which
injects angular momentum into a normal metal.
Figure 2.5: Schematic illustration of the spin-pumping process. The precessing magnetization m in a
ferromagnet (F) injects a spin current Is into the adjacent normal metal (N).
The spin-pumping theory is developed in the adiabatic approximation, in which the electrons in
the normal metal are always in equilibrium with the magnetization precession at the interface of
the ferromagnet. Tserkovnyak et al. explained the injection of a spin current by the scattering of
the time-dependent spin potential at the interface. According to the spin pumping theory, the spin
current Is flows from the ferromagnet to the normal metal perpendicularly to the interface. If the
ferromagnet is thicker than its ferromagnetic coherence length, the spin current injected into the
20
2.5 Spin Hall and inverse spin Hall effects
normal metal is determined by the spin mixing conductance g↑↓ [100, 101, 124, 125] as
~
∂ m̂
,
Is =
Re(g↑↓ ) m̂ ×
4π
∂t
(2.46)
where m̂ is a unit vector along the magnetization in the ferromagnet. Due to the emission of a
spin current, the magnetization precession loses its spin angular momentum which leads to an additional damping. The additional damping term is analogous to the Gilbert damping parameter
in the Landau-Lifshitz-Gilbert equation (Eq. (2.23)). Therefore, experimentally the spin mixing
conductance can be determined by measuring the additional line-width broadening ∆H in ferromagnetic resonance experiments [126, 127]. In the limit of a normal metal being thicker than its
spin diffusion length, the spin mixing conductance is given as [127, 128]
g↑↓ =
4π γ Ms t ∆H
,
g µB ω
(2.47)
where Ms is the saturation magnetization of the ferromagnetic thin film, and t is its thickness.
The origin of the spin pumping lies in the time retarded response of the interlayer exchange coupling [129]. The static interlayer exchange coupling—which leads to RKKY oscillations in the
normal metals—is explained by the instantaneous response of electrons in the normal metal to the
interface exchange coupling between the ferromagnet and the normal metal. On the other hand,
the time retarded response of the interlayer exchange coupling leads to the spin pumping [130].
This theory is in agreement with the standard spin pumping theory [100].
2.5. Spin Hall and inverse spin Hall effects
In 1879, Edwin Hall discovered that conductors develop a transverse voltage in the presence of a
perpendicular magnetic field due to the Lorentz force acting on the moving electrons [131]. The
generated voltage (Hall voltage) is proportional to the applied magnetic field. This effect is known
as Hall effect. Shortly thereafter, he found that, in magnetic metals, the transverse voltage is not
proportional to the applied magnetic field, rather there exists an additional contribution. Nowadays,
this effect is named as anomalous Hall effect (AHE) [132]. Experimentally, it was observed that
the Hall resistivity ρx,y in ferromagnets on application of a magnetic field Hz , initially increases
steeply for weak Hz and attains a saturation level at large values of Hz . So, an empirical relation
can be established as [132–134]
ρx,y = R0 Hz + Rs Mz ,
(2.48)
where R0 , Rs are the characterizing strength constants of the ordinary- and the anomalous Hall
effects, respectively. Essentially, in ferromagnets moving electrons attain transverse velocities in
opposite directions for different spin polarizations. Since, in ferromagnets, most spins are polarized for one direction, depending on the direction of magnetization, the spin-dependent transverse
21
2.5 Spin Hall and inverse spin Hall effects
Figure 2.6: Schematic representation of the (a) Spin Hall effect and the (b) inverse spin Hall effect. In the
spin Hall effect a spin polarized current ( js ) is generated by a charge current ( jq ) in normal metals. A
reverse phenomenon happens for the inverse spin Hall effect.
velocity leads to an additional Hall voltage. A similar spin-dependent transverse velocity exists in
the normal metals [29, 135]; however, due to the absence of spin polarized electrons, it does not
lead to an observable voltage. Instead, this mechanism separates the opposite spin polarities (spinup and spin-down) in normal metals resulting in an accumulation of opposite spins on the opposite
transverse edges of the normal metal. The effect is known as the spin Hall effect (SHE) [29, 30].
The effect was first predicted by D’yakonov and Perel in 1971 [135] and the resurgence was triggered by the theoretical work of Hirsch [29] and Zhang [30] about 30 years later. The reverse
effect, where a transverse unpolarized electric charge current is generated in a normal metal by
the injection of a spin polarized current is referred to as inverse spin Hall effect (ISHE) [96, 136].
Schematic illustrations of these effects are depicted in Fig. 2.6.
In the past, many theoretical attempts have been made to explain the anomalous Hall effect [137–
139]. Recently, microscopic theory unifies all these approaches [132]. The SHE is a consequence of spin-orbital coupling. The effect originates from two mechanisms: the Berry phase
mechanism which leads to the intrinsic SHE, and the skew-scattering [137, 138] and side-jump
scattering [139] mechanisms which jointly result in the extrinsic SHE. The intrinsic SHE can be
understood in terms of the Berry-phase curvatures; therefore, it can be imagined as an intrinsic
quantum-mechanical property of perfect crystals [132, 140]. In this effect, transverse spin dependent velocities originate from interband coherence induced by an external electric field. The
extrinsic SHE can be understood in terms of asymmetric scattering over defects and impurities.
The existence of the SHE and the ISHE has been demonstrated in both semiconductors [141, 142]
and metals [94,96,136,143]. In semiconductors, the intrinsic mechanism dominates while, in metals, the intrinsic mechanism can be neglected; thereby, the dominating mechanism is the extrinsic
SHE [132].
In the research work presented here, the inverse spin Hall effect in Pt is utilized to detect spin
currents in YIG|Pt bilayers. Therefore, the extrinsic spin Hall effect will be discussed in next
22
2.6 Spin caloritronics
paragraphs. For theoretical formulations of the intrinsic SHE, literature [132, 140, 144] is referred.
The reader can get an overview of the subject in review references [95, 145].
The extrinsic spin Hall effect mechanism holds two scattering processes: skew scattering and
side-jump scattering. Spin skew scattering, also known as Mott scattering, is based on the idea
that the spin-orbital interaction creates an effective magnetic field gradient within the scattering
plane. The field gradient produces a net force acting towards or away from the scattering center
depending upon the direction of the spin polarization [132]. The side-jump scattering occurs due
to different acceleration and retardation during the scattering process, which ultimately results in
a net displacement of spins in the transverse direction after multiple scatterings. The contribution
of these two scattering mechanisms to the total spin Hall resistivity ρH depends on the electric
resistivity ρ of the material as [97]
ρH = ass ρ + bsj ρ2 ,
(2.49)
where ass , bsj are material parameters for the skew scattering and the side-jump scattering, respectively.
In the spin Hall effect, the charge current density jq , and the spin current density js are coupled in
the following relationship [97]:
js = j0s + θSHE
2e
(jq × σ) ,
~
(2.50)
jq = j0q + θSHE
2e
(js × σ) ,
~
(2.51)
where j0q , j0s are initial current densities. σ is the spin polarization tensor, and θSHE is known as the
spin Hall angle. Clearly, the above equations imply that the flow of charge current generates a spin
current in the transverse direction and vice-versa. Equation (2.50) is a mathematical representation
of the spin Hall effect while Equation (2.51) represents the inverse spin Hall effect. These effects
are illustrated in Fig. 2.6.
2.6. Spin caloritronics
Spintronics is a field which emerged in the last two decades. Though the fundamental theoretical
works are rather old, the technology of fabrication of meso- and nano-scopic devices reinvigorated
the field. Spintronics mainly engages with the coupling of electron spin with the charge transport
in condensed matter. Recently, a new sub-field of spintronics—spin caloritronics—has spawned
whereby the interaction of spins with heat currents is in the main focus of research. Though
thermoelectric and thermomagnetic effects are well explored for two centuries [48, 146], only
recently thermomagnetic effects in meso- and nano-scopic devices get much attention. On the
23
2.7
Spin-dependent Seebeck and Peltier effect
other side, a significant amount of research work has been done to investigate thermoelectric effects
in nanostructures [147, 148].
The phenomena in spin caloritronics can be categorized into three classes, namely: (i) spindependent effects, (ii) spin-wave-dependent effects, and (iii) relativistic effects [91]. The first
class about spin-dependent effects deals with the thermoelectric generalization of spintronic phenomena. For example, the magneto-Seebeck effect is a generalization of the magnetoresistance
effect in layered structures. The spin-dependent Seebeck effect is the equivalent to a spin current generation in nonlocal spin-valves with the heat current as an origin of driving spin polarized
electrons. The second class addresses the effects related to spin waves—also known as magnon
spintronics. In these effects, thermally excited spin waves couple with the electronic spin through
the spin pumping and the spin transfer torque phenomena. Examples of this class are the spin
Seebeck effect, the spin Peltier effect, the thermal spin transfer torque, etc. The last class handles
the generalization of relativistic corrections such as thermal Hall effects. An equivalent of thermal
Hall effects (see Sec. 2.9) in the spin-world could be the spincaloric Hall effect [91].
In next sections, the phenomena of spin-dependent Seebeck and Peltier effects from the first class
of effects, and spin Seebeck and Peltier effect from the second class of effects will be discussed.
The main focus will be on the spin Seebeck effect which has been investigated in the framework of
this thesis. The reader is suggested to refer to the review reference [91] for a further overview of
various spincaloric effects. A special journal issued on spin caloritronics can be cited for detailed
knowledge of the subject [149].
2.7. Spin-dependent Seebeck and Peltier effect
The spin-dependent effects exist in metallic and semiconductor materials where the transport channels are spin carriers. The two-current model [150]—currents carried by the spin-up electrons
and the spin-down electrons—describes most of the spintronics phenomena in these materials.
Bauer et al. extended this model for spin-dependent thermoelectric effects [91] which is briefly
presented here. Theoretical [151–153] as well as experimental works [154–157] show that thermoelectric properties of magnetic metals depend on the spin-dependent properties of electrons. In
metallic systems, the conventional charge conductivity σ and the Seebeck coefficient S are determined by the corresponding properties of the majority (↑) and the minority (↓) spin carriers and
read as
σ = σ↑ + σ↓ ,
and
S=
24
σ↑S ↑ + σ↓S ↓
.
σ↑ + σ↓
(2.52)
(2.53)
2.8 Spin Seebeck effect
Furthermore, in a linear response, the total current densities, i.e., total spin current density js =
j↑s − j↓s , total charge current density jq = j↑q + j↓q , and total heat current density Q = Q↑ + Q↓ , are
coupled together as [91]
 



jq
1
P
ST
∇µq /e
 



 



 js  = σ(ǫF )  P
1
P ′ ST   ∇µs /2e 
 



′
Q
ST P ST κT /σ
−∇T /T
(2.54)
where P (= (σ ↑ − σ ↓ )/σ) and P ′ denote the spin polarization of the conductivity (σ) and its energy
derivative, respectively. Here µq = (µ↑ + µ↓ )/2 stands for the charge electrochemical potential,
µs = µ↑ − µ↓ is the spin accumulation, and T is the temperature of the system. The above
equation physically can be understood as follows. A spin current in a system can be generated
by polarized spins, a spin accumulation, and the most importantly by a thermal gradient. The
generation of a spin current by a thermal gradient in metallic systems or semiconductors is known
as spin-dependent Seebeck effect. Further, the charge current in a system can also be driven by
a spin accumulation ∇µs —this effect is termed as spin-dependent Peltier effect—and a thermal
gradient (conventional Seebeck effect).
Experimentally, Slachter et al. [158] have demonstrated the spin-dependent Seebeck effect in
permalloy-copper (Py-Cu) nonlocal spin valves whereby a thermal gradient in Py injects a spin
current in the adjacent normal metal Cu. In the normal metal, the injected spin current is measured
using an additional ferromagnet. Recently, the spin-dependent Peltier effect has also been realized
experimentally [159].
2.8. Spin Seebeck effect
In ferromagnet|normal-metal bilayer systems, on the application of a thermal gradient in the ferromagnet, the injection of a spin current from the ferromagnet into the normal metal is known as
spin Seebeck effect (SSE) [32,38–44]. The injected spin current in the normal metal is transformed
into electric voltage by means of the inverse spin Hall effect (discussed in Section 2.5). This effect
has been observed first in a metallic system (Py), thereafter in magnetic insulators (YIG [160],
ferrites [44], garnets [161]), magnetic semiconductors (GaMnAs [39]), and Heusler alloys [162];
hence it is a universal effect rather than material specific. However, it is important to know that the
physics of the SSE is entirely different from the spin-dependent Seebeck effect observed in metallic
systems. The contribution of the conduction electron spins to the SSE is negligibly small [163].
Currently, this effect has attracted much scientific attention due to its potential applications. For
example, recent progresses show that based on this effect thin-film structures can be fabricated to
25
2.8 Spin Seebeck effect
Figure 2.7: Schematic illustration of (a) the transverse and (b) the longitudinal spin Seebeck effect. In the
transverse configuration, the spin current (Is ) and the temperature gradient (∇T ) act perpendicular to each
other while in the longitudinal configuration, the spin current flows along with the applied thermal gradient
in the ferromagnetic film.
generate electricity from waste heat sources [34]. Further advancements in industrial applications
like temperature sensors, temperature gradient sensors, and thermal spin-current generators require
an in-depth understanding of this effect.
2.8.1 Experimental Configuration
In ferromagnet| normal-metal bilayer structures, the spin Seebeck effect has been studied in two
configurations namely: (i) the transverse configuration and (ii) the longitudinal configuration. Corresponding to these configurations, the SSE is classified as the transverse spin Seebeck effect
(TSSE) [32, 39, 40] and the longitudinal spin Seebeck effect (LSSE) [42, 44, 111]. Schematically,
these effects are illustrated in Fig. 2.7. In the transverse SSE, the applied thermal gradient to the
magnetic field acts along the film plane. The generated spin current is observed in the normal
metal, usually Pt, via the inverse spin Hall effect. Since, in this configuration, the spin current
flows transverse to the applied thermal gradient, the effect is named as transverse SSE. This configuration can be used for both metallic as well as non-metallic magnetic materials.
In the longitudinal SSE, the applied thermal gradient and the thermal spin injection act along the
same direction. In order to measure the thermal spin current via ISHE, the spin current has to
be perpendicular to the magnetic field; so in the longitudinal configuration, the magnetic field is
applied along the film plane. The configuration is illustrated in Fig. 2.7(b). It is important to
note that the longitudinal configuration can only be used for insulating ferromagnets, because, in
metallic systems, a parasitic contribution due to the anomalous Nernst effect [162,166,167] appears
in this configuration (discussed in Section 2.9). An important difference between the longitudinal
SSE and the transverse SSE is that the former has a direction of spin current injection (N→F or
26
2.8 Spin Seebeck effect
F→N) is opposite to the transverse configuration [33].
2.8.2 Theory of the spin Seebeck effect
Although there have been numerous experimental and theoretical studies about this effect, the
underlying physics is yet not fully understood. The most accepted theory predicts that the SSE is
driven by the difference in the local temperatures of magnon-, phonon-, and electron baths of the
system [35–37]. The theories suggest that the injection of the spin current is due to the thermal
nonequilibrium at the interface of the normal metal and the ferromagnet. An imbalance between
the thermal spin currents at the interface results in this effect which can be explained by a simple
model discussed here [91].
Consider a normal-metal|ferromagnet|normal-metal (N|F|N) sandwich system. At thermal equilibrium, via the spin pumping process (Eq. (2.46)), a thermal spin current (Ips ) will be injected
from the ferromagnet to the normal metal due to the thermal fluctuation of the magnetization.
However, at a finite temperature, according to the fluctuation-dissipation theorem [164], normal
metals generate thermal noise known as Johnson-Nyquist noise in the form of current fluctuations.
The Johnson-Nyquist noise scales linearly with the electron temperature, and the currents generated by this noise are partially spin polarized. Therefore, a back injection of a spin current (Its ) via
the spin transfer torque takes place. According to the second law of thermodynamics, at thermal
equilibrium, these two currents acting oppositely cancel each other, and no net spin current flows
across the F|N interfaces.
Taking one step further, it is assumed that the thickness of the ferromagnet is smaller than the
magnetic domain wall width. In this case, the behavior of the ferromagnet can be described by a
macrospin model. In the macrospin model, all magnetic moments are considered to be precessing
in unison, so, their motion can be characterized by a single magnon temperature2 [58, 165]. Now,
in order to study the impact of the temperature on the system, it is important to understand the
behavior of the phonons and the electrons in the system. Generally, the electrons are strongly
coupled with the phonons compared to the spins in the system. Therefore, the assumption that
the temperature of the electron system is the same as that of the phonons is feasible. Hence, the
thermodynamics of the system can be described by two temperatures: phonon temperature (Tp )
and magnon temperature (Tm ).
Consider that the N|F|N system is subjected to a thermal gradient which increases the temperature
of the left normal-metal and decreases the temperature of the right normal-metal3 . The temperature
2
The magon temperature depends on the population of magnons in the system and is defined by Bose-Einstein
statistics.
3
The geometrical configuration is very similar to the longitudinal SSE (Fig. 2.7); however, the theoretical description is valid for both the transverse and the longitudinal SSE.
27
2.8 Spin Seebeck effect
Figure 2.8: A normal-metal|ferromagnet|normal-metal layered system (N|F|N) subject to a thermal bias
TpL − TpR . The ferromagnet has a magnon temperature Tm . The thermal fluctuations of the magnetization
m pump a spin-current (Ips ) into the normal metal. A spin current (Its ) via the spin transfer torque generated
by the Johnson-Nyquist noise flows back into the ferromagnet from the normal metal. A bias temperature
imbalances the spin currents, and a net spin current flows from the left side to the right side. On basis of
Ref. [91].
profiles are shown in Fig. 2.8. Since the phonon temperature (TpL ) at the left F|N interface is higher
than the magnon temperature (Tm ), the injected spin current (Its ) by the Johnson-Nyquist noise
will be larger than the spin pumping generated spin current (Ips ) by the ferromagnet. So, a net
spin current injects into the ferromagnet from the normal metal at the hot interface. Contrarily,
at the right interface, the conditions are opposite. At this end, a net spin current flows from the
ferromagnet to the normal metal. When the interfaces are identical and in equilibrium, the spin
current injected at the left interface must be the same as the spin current ejected from the right
interface. So, the magnon temperature can be expressed as the average of phonon temperatures,
i.e., Tm = (TpL + TpL )/2. This mechanism explains the role of magnetization fluctuation in the spin
Seebeck effect.
In the above mentioned mechanism, the ferromagnet is considered to be a macrospin. However,
this assumption is not valid for large magnetic samples, usually millimeters long [32]. So, an
extension of the theory is necessary to explain the ongoing experiments. Xiao et al. generalized
the theory by releasing the assumption of the macrospin model [35]. They employed the magnonphonon interactions in their theory and defined a space-dependent magnon temperature Tm (x)
considering the theory of Sanders and Walton, Ref. [35], as a foundation.
As early as in 1977, Sanders and Walton studied the magnon heat transport and magnon-phonon
thermal relaxation mechanisms in ferromagnetic insulators [165]. They found that, in thermally
isolated systems, the magnon temperature obtains equilibrium with the phonon temperature in a
finite relaxation time interval τmp . The relaxation time τmp is inversely proportional to the strength
of the magnon-phonon interaction. Moreover, they conclude that the steady-state magnon temperature gradient differs from the phonon temperature gradient. According to their theory, the magnon
28
2.8 Spin Seebeck effect
and the phonon temperatures are related to each other via a diffusion equation [165]:
1
Cp Cm
d2 Tm (x)
+
[Tp (x) − Tm (x)] = 0 ,
2
dx
Cp + Cm Km τmp
(2.55)
where Cp , Cm are the specific heats of the phonon and magnon systems, and Km is the thermal
conductivity of magnons. The prefactor of the temperature difference term is a material dependent
parameter known as characteristic length scale parameter λ for magnons and defined as
λ2 =
Cp + Cm
Km
Km τmp ≈
τmp .
Cp Cm
Cm
(2.56)
The specific heat of magnons is much smaller than that of phonons, i.e., Cm ≪ Cp [289]. Using
scattering theory, Xiao et al. predicted the spin current in transverse configuration of the SSE
(Fig. 2.7(a)) [35]:
Is (x) =
~γ gr
kB (Tm (x) − Tp (x))
2π Ms Vcoh
(2.57)
where Ms is the saturation magnetization, and Vcoh is the magnetic coherence volume which is a
material dependent parameter standing for the range over which a given perturbation can be felt.
A similar expression is obtained by linear response theory [36, 37]. For further details about linear
response theory, the reader is referred to the review article [33].
In the framework of this thesis, the magnon and phonon temperatures are measured for a YIG film
subject to a thermal gradient. The value of the characteristic length scale parameter λ is estimated
on the basis of the experimental results. The experimental setup, measurement technique, and the
results will be presented in Section 4.1.
2.8.3 Open questions
Since the discovery of the spin Seebeck effect by Uchida et al. in a NiFe|Pt metallic bilayer
system [32], a tremendous amount of experimental and theoretical work has been going on. The
universal nature of the effect is confirmed by observing it in various metallic-, semiconductor-, and
insulating magnetic materials. However, no universally accepted understanding about this effect
has been built up. For example, in ferromagnetic metals, the mean free path of magnons is too small
to explain the observed length scale of the transverse SSE in the experiments by Uchida et al. on
the NiFe|Pt system [32]. Very recent experimental results by Meier et al. claim that the transverse
SSE signal observed for the NiFe|Pt system might be a consequence of experimental errors [168].
They find that the observed signals for the NiFe|Pt system subject to in-plane thermal gradient
and magnetic field can be fully explained by the Planar Nernst effect (discussed in Section 2.9).
However, they find that the creation of an in-plane thermal gradient is a challenging task; any small
non-uniformity in the thermal gradient can lead to a signal that mimics the transverse SSE signal.
29
2.9 Anomalous and Planar Nernst effect
Additionally, the two temperature model (Tm and Tp ) [35, 165] is experimentally tested within
the coursework of this thesis; however, no clear evidence of any difference in the magnon and
the phonon temperatures has been observed when an in-plane thermal gradient is applied to the
insulating magnetic material YIG. So, the origin of the SSE is still unclear and under discussion.
Some experimental studies show that the interface proximity effect in the YIG|Pt system could
exhibit similar behavior as observed for the SSE [169]. But, very recent measurements claim no
such proximity effects [170, 171]. Even a more important question whether the SSE is an interface
or bulk effect is still open [172, 173].
To shed light on this controversial physics, an entirely new experimental approach has been developed here, in this thesis, where the temporal evolution of the longitudinal SSE in YIG|Pt bilayer
structures is investigated. The findings reveal that the LSSE signal evolves at sub-microsecond
timescale, and a certain thickness of the YIG film effectively contributes to the LSSE. The experimental setup and results will be presented in Sections 4.2 and 4.3.
2.9. Anomalous and Planar Nernst effect
Analogous to the Hall effect, when a thermal gradient (∇Tx ) drives the electrons in a conductor
or semiconductor in the presence of a perpendicular magnetic field (Hz ), a transverse electric
field (Ey ) is generated, and the effect is known as Nernst effect. Likewise the anomalous Hall
effect (AHE), in magnetic materials, this effect is also influenced by the magnetization (M) of
the material; thereby, in magnetic material, it is named as anomalous Nernst effect (ANE) and
expressed as
EANE = NANE ∇T × M ,
(2.58)
where NANE is a material-dependent proportionality constant. The origin of these two effects, ANE
and AHE, is the same and has been discussed in Section 2.5. It is important to notice that EANE
resembles with the inverse spin Hall signal, given by Eq. (2.51) as EISHE ∝ js × σ whenever ∇T
lies along the direction of the spin current density js . The direction of both the spin polarization σ
and the magnetization M is given by the external bias magnetic field. In the case of the longitudinal
SSE in metallic systems, ∇T lies along js (see Fig. 2.7(b)); thereby the longitudinal SSE signal is
distorted by the anomalous Nernst effect.
Table 2.1: A classification of Hall and Nernst effects
Field Hz 6= 0, Mz = 0
Hz 6= 0, Mz 6= 0
Hz = 0, Mz = 0
Hxy 6= 0, Mxy 6= 0
Ex
Hall effect
Anomalous Hall effect
Spin Hall effect
Planar Hall effect
∇Tx
Nernst effect
Anomalous Nernst effect
Spin Nernst effect
Planar Nernst effect
30
2.9 Anomalous and Planar Nernst effect
In ferromagnetic metals, it is not necessary to have a perpendicular magnetic field (Hz ) acting on
the moving electrons (along x-axis), a transverse electric field (Ey ) exists even in the case when
the magnetization of the material lies in the plane (xy-plane). If the electrons are driven by an
electric field (Ex ), the effect is known as planar Hall effect [174, 175], else if they are driven by a
thermal gradient (∇Tx ), the effect is named as planar Nernst effect [168, 175–177]. These effects
are caused by the spin-dependent scattering like the anisotropic magnetoresistance (AMR). The
transverse electric field generated by the planar Nernst effect (PNE) reads [175]
Ey = αN sin φ cos φ ∇Tx
(2.59)
where φ is the angle between ∇Tx and the magnetization M of the material, and αN is a material
dependent parameter. In Table 2.1, a classification of different Hall and Nernst effects is pro-
vided. The spin Nernst effect [178, 179] is analogous to the spin Hall effect and yet to be observed
experimentally.
In both the planar Nernst effect and the transverse spin Seebeck effect (Fig. 2.7(a)) in ferromagnetic
metallic systems, a thermal gradient (∇Tx ) generates a transverse electric field (Ey ) in the presence
of an in-plane magnetization. In the case of the transverse SSE, the electric field is generated via
the inverse spin Hall effect (EISHE ∝ js × σ), thereby, has a cos φ dependence with respect to
the in-plane magnetization rotation while the planar Nernst effect has a sin φ cos φ dependence.
Therefore, the magnetization rotation measurements can distinguish the planar Nernst effect and
the transverse spin Seebeck effect in metallic systems [168].
31
CHAPTER 3
Experimental methods
Magnetism always remains a fascinating topic since its discovery about 2500 years ago [180]. An
experimental investigation of magnetic materials requires (i) a means to magnetize the material
and/or to excite its magnetization dynamics, and (ii) a means of measuring the resulting impact
on the magnetic material. For the investigation of static magnetization, usual means to magnetize
a specimen are electromagnets and permanent magnets. Their magnetic properties can be measured using Hall effect [180, 181], vibrating-sample magnetometer (VSM) [182], SQUID [183],
and magnetoresistance [184, 185] techniques. Techniques like x-ray magnetic circular dichroism (XMCD) [186], magnetic neutron scattering [187, 188], and polarized neutron reflectometry
(PNR) [189], are used to study the magnetic structures of materials. Magneto-optical Kerr effect
(MOKE) [190, 191] and Faraday effect [192] techniques empower to study the magnetization by
optical means [193].
The investigation of magnetization dynamics in magnetic materials can be realized with various
techniques like ferromagnetic resonance (FMR) spectroscopy, time-resolved magneto-optical Kerr
effect (TR-MOKE) [194–196], and Brillouin light scattering (BLS) spectroscopy [197,198]. In the
last decade, a few new techniques like inverse Faraday effect [199, 200], spin pumping and inverse
spin Hall effect [127, 201–203], spin-transfer-torque nano-oscillators [204–206] have been developed to study different aspects of the magnetization dynamics in thin films and nanostructures.
Each of these technique has its own advantage and disadvantage. FMR spectroscopy is employed
to access low-energy spin waves (wave vectors are close to zero) while BLS spectroscopy, additionally, enables to access high-energy spin waves with information about their wave vector.
TR-MOKE provides information about the time-domain of the magnetization dynamics, whereas
BLS spectroscopy measures frequencies of magnetization dynamics in materials [207].
In the scope of this thesis, FMR spectroscopy, BLS spectroscopy, spin pumping, and inverse spin
Hall techniques were employed to study the magnetization dynamics of a ferrimagnetic material—
Yttrium Iron Garnet (YIG). In this chapter, the preparation methodology of the investigated sample
structures, and the investigation techniques mentioned above have been explained with all necessary technical details. The first section will address the physical and the magnetic properties of
33
3.1 Yttrium Iron Garnet
the material YIG. Later, YIG film deposition techniques and the process of preparing the YIG|Pt
bilayer structure, used for the longitudinal spin Seebeck effect measurements, will be presented.
In Section 3.2, Brillouin light scattering spectroscopy (BLS) and its applications for the detection
of spin-wave intensity, phase, frequency, and wave vector, will be introduced. Special emphasis will be given to the conventional BLS technique, which has been utilized to investigate the
magnon temperature in YIG. In Section 3.3, microwave techniques will be explained in order to
understand the spin-wave excitation processes in magnetic materials. Section 3.4 is dedicated to
the introduction of the COMSOL Multiphysics simulation package which was utilized to solve
heat thermodynamics in the YIG|Pt system in this thesis.
3.1. Yttrium Iron Garnet
Yttrium Iron Garnet (Y3 Fe5 O12 , YIG) is a marvelous man-made material which belongs to the
family of microwave ferrites. First synthesized by Bertaut and Forrat [208] in 1956, the role of
this material in the magnetic world is similar to that of germanium in the semiconductor world.
YIG is a high-grade dielectric insulator with excellent high-frequency magnetic properties having
the narrowest known ferromagnetic resonance line—generally smaller than 39.79 A/m (0.5 Oe)—
and the lowest spin-wave magnetic damping—the Gilbert damping parameter is of the order of
10−5 [27, 209, 210]. YIG has tremendous applications in the microwave industry and has been
extensively studied to discover new effects and phenomena in magnetism [209].
Figure 3.1: The crystal structure of yttrium iron garnet (Y3 Fe5 O12 , YIG). A YIG crystal is a body-centered
cubic lattice with 80 atoms per unit cell. Fe3+ ions are situated at octahedral (16a, brown) and tetrahedral
(24d, teal) sites. The indirect superexchange interaction mediated by O2− ions aligns spins of two Fe3+ sublattices antiparallel. The Y3+ ions are positioned at dodecahedral (24c, white) sites. Figure from Ref. [211].
The crystalline structure of YIG coincides with natural garnets. YIG lattice has a body-centered cubic structure with 80 atoms in its unit cell [210]. Its primitive elementary cell, with lattice constant
34
3.1 Yttrium Iron Garnet
a = 1.2376 nm, is a half cube consisting of 4 octants and one molecule of Y3 Fe5 O12 [209, 212].
3+
The space group of garnet is O10
h = Ia3d, and its point group is m3m. Yttrium ions (Y ) oc-
cupy dodecahedral sites (c) and have the electronic configuration of noble gas Krypton; therefore, they are diamagnetic in nature and do not contribute to the magnetic properties of YIG.
The unbalanced position of Fe3+ ions in the YIG crystal structure is responsible for its magnetic properties. Fe3+ ions occupy antiferromagnetically-coupled eight-octahedral sites (8a) and
twelve-tetrahedral sites (12d). As the spin of Fe3+ ion is 5/2, the total magnetic moment (gΣS)
of a half YIG unit cell is 2 × (12 − 8) × 5/2, i.e, 20 µB , where µB is the Bohr magneton, at
zero temperature [210]. By definition, the saturation magnetization of YIG can be calculated as
Ms = 20 µB /(a3 /2) = 195 kA/m, thus, 4πMs = 2445 kA/m which closely matches with the
experimentally measured value—2470 kA/m [209, 213] at 0 K.
Table 3.1: Material parameters of bulk YIG
Saturation magnetization Ms at 0 K
196 kA/m [59]
Saturation magnetization Ms at 298 K
140 kA/m [59]
Gyromagnetic ratio γ
Gilbert damping parameter α
2π · 2.8 × 1010 T−1 s−1 [58]
5 × 10−5 [58]
A primitive cell of YIG has 80 atoms out of which 20 atoms are magnetic. Therefore, YIG consists
of 20 ferrimagnetic-lattice sites, and accordingly, 20 magnetic branches, in the energy range from
0 − 1000 K [214,215]. Only one branch out of these 20 branches is acoustical—the rest are optical
(0.7−26 THz [216])—and lies in the microwave frequency range (below 100 GHz). Therefore, for
microwave frequencies, YIG can be treated as a ferromagnetic material, instead of ferrimagnetic,
for small magnetic fields below the Curie temperature (560 K). A large band gap of 2.85 eV makes
YIG an insulator [217].
3.1.1 YIG film deposition
The YIG films investigated in this research work were grown by Liquid Phase Epitaxy (LPE) [218,
219] on single crystalline Gadolinium Gallium Garnet (Gd3 Ga5 O12 , GGG) substrates. GGG has
a cubic crystalline structure having eight molecules per unit cell of lattice constant a = 1.2383 nm.
For epitaxial growth of a defect-free thick YIG film (up to 15 µm), GGG is the most suitable,
commercially available, wafer material with a lattice mismatch of 6 × 10−5 nm [219].
LPE is a well established YIG-film growth technique. In a typical growth process, oxides of
yttrium and iron, Y2 O3 and Fe2 O3 , are dissolved into a mixture of molten PbO and B2 O3 [218,
220, 221]. A typical composition of these materials is chosen such that the saturation temperature
35
3.1 Yttrium Iron Garnet
of the mixture is around 1000◦ C so as to avoid excess evaporation of lead oxide PbO. A rotating
GGG substrate, which acts as a seed to deposit the garnet, is submerged in the molten mixture.
During the growth process, the temperature of the mixture is kept constant. The rotation of the
substrate provides a uniform fluid flow which assists in maintaining an even concentration of the
solution and uniform mass transfer to the substrate. The film growth terminates when the substrate
is taken out of the mixture. The film thickness depends on the growth rate and the deposition time.
In this LPE process, the YIG film grows on both sides of the substrate.
The LPE grown YIG films employed for investigations in the framework of this thesis are obtained
commercially from the Scientific Research Company Carat, R&D Institute of Materials, Division
of Crystal Growth and Technology, Lviv, Ukraine. As discussed above, in the LPE-YIG fabrication process, the film grows on both sides of the GGG substrate; double-sided YIG films were
obtained from the production house. Generally, due to a large thickness of the GGG substrate
(≈ 500 µm), the magnetic properties of the two YIG-layers do not influence each other. However,
for the magnon temperature experiment, presented in Section 4.1, where the spin waves are probed
optically by BLS spectroscopy, one YIG-film was removed from the GGG substrate by mechanical
polishing using diamond paste to avoid all undesired signal accessible by optical means1 .
Although the LPE deposition technique produces high quality YIG films, this technique can only
grow thick (order of microns) films. Deposition techniques like pulsed laser deposition are useful
to grow thin YIG film. A detailed description of this technique can be found in the literature
[209, 222–224].
3.1.2 YIG film surface treatments
In a different experiment, where the temporal evolution of the longitudinal SSE is studied (Section 4.2), a Pt film is grown over the YIG film. Very recent studies about the spin pumping in
YIG|Pt heterostructures show that the spin current injection efficiency in the normal metal Pt from
YIG depends on a parameter known as spin-mixing conductance which essentially quantifies the
spin transport characteristics of the YIG|Pt interface [126, 225] (also see Section 2.4.2). The study
shows that the spin pumping efficiency can be enhanced by two orders of magnitude by improving
the YIG|Pt interfacial qualities [126]. In order to improve the interface quality, a detailed YIGsurface cleaning process discussed in Ref. [126, 226] was followed and is briefly described here.
The surface cleaning process was carried out in assistance of Mr. Viktor Lauer, now a Ph.D. scholar
in the group of Prof. B. Hillebrands at TU Kaiserslautern, Kaiserslautern, Germany.
A 6.7 µm thick double-side YIG film, grown over a 500 µm thick GGG substrate by the LPE
1
The GGG substrate is transparent for green laser light (532 nm), used for BLS spectroscopy. As the green light
can penetrate through 6.7 µm thick YIG, it was mandatory to remove the YIG film from one side.
36
3.1 Yttrium Iron Garnet
technique, discussed above, was pre-cleaned with acetone and isopropanol in an ultrasonic bath
to remove particles like dust, etc. An ex situ surface treatment was carried out by cleaning the
YIG surface with piranha etch2 , a 3:2-by-volume mixture of sulfuric acid (H2 SO4 ) and hydrogen
peroxide (H2 O2 ). The mixture is a very strong oxidizing agent which removes most of organic
matter from the YIG surface. After treating the surface with piranha, a plasma ion etching using
O+ /Ar+ ions in a ratio of 1:1, and energies around 50 − 100 eV, was performed for 10 minutes to
oxidize and remove any leftover organic matter.
3.1.3 Pt film deposition on YIG
In order to study the longitudinal SSE effect, as many as 7 Pt micro-strips were grown on the
piranha-cleaned YIG film. A photolithography technique was used to engrave a micro-strip pattern,
depicted in Fig. 3.2, on the YIG film. A brief description of the patterning process is presented
here. Details about the involved lithographic techniques can be found in the literature [227–230].
The piranha-and-plasma-cleaned YIG film was first coated with a 3.5 µm-thick layer of photoresist
AZ2030 using a spray coater—EVG 101 Advanced Resist Processing System. The coated film was
baked softly at 100◦ C. After the soft-baking, the photoresist was exposed to a pattern—depicted in
Fig. 3.2(a)—of intense light of wavelength 365 nm using the EVG 620 Double Side Mask Aligner.
A hard baking was performed afterwards at 110◦ C for 1 minute. The exposure to intense light
causes a chemical change in the photoresist so that it can be removed by special solutions known
as developers. The light-exposed YIG film was developed for 45 seconds in a solution of MIF-726
developer. The process engraves the mask on the photoresist layer. The film was etched again with
O+ ions plasma for 20 seconds to clean the YIG surface inside the patterned area.
Figure 3.2: (a) The photolithography masks (a) to grow as many as 7 Pt strips of length l = 3 mm, centralwidth wc = 100 µm, and center-to-center separation d = 1 mm, and (b) to produce 100 nm thick gold
contact-pads of length l = 300 µm and width w = 200 µm, at the edges of the Pt strips.
2
Special precautions must be taken while handling a piranha solution.
37
3.1 Yttrium Iron Garnet
The deposition of Pt strips on the YIG film was carried out by growing a polycrystalline Pt film on
the patterned photoresist-covered YIG film by Molecular Beam Epitaxy (MBE). The Pt deposition
process was performed by Dr. E. Papaioannou at TU Kaiserslautern, Kaiserslautern, Germany.
MBE is a monolayer-controlled high-quality, epitaxial thin-film growth technique based on the
interaction of atoms/molecules adsorbed from molecular beams on a heated crystalline substrate
under ultra-high vacuum (UHV) conditions [231, 232]. The UHV is required to minimize the
contamination at the growth surface and in the epitaxial layer. Until the UHV conditions are
achieved, the growth surface remains covered with a monolayer of residual gas inside the growth
chamber. To drive off this monolayer, the background pressure should be as low as 10−11 mbar.
Another reason for UHV conditions arises from the purity of thin films. Since the growth rate
in the MBE is usually around 1 MLs−1 (one monolayer per second), to achieve an impurity level
smaller than the growth rate, UHV conditions are necessary [232]. Further details about the MBE
technique can be found in the literature [231, 232].
In the growth process, molecular beams are produced by evaporating or sublimating high-purity
materials. During the growth process, a Pt molecular beam was produced by evaporating a Pt target
using an electron-beam. By keeping the YIG sample at a temperature of 300 K, a sufficient amount
of energy was provided for the mobility of the deposited atoms. A 10 nm thick polycrystalline Pt
thin film was grown by this process at a pressure of 5 × 10−11 mbar with a growth rate of 0.05 Å/s.
Figure 3.3: A microscopic image of the YIG|Pt structure depicting 10 nm thick Pt strips patterned by
photolithography. Gold contact-pads were fabricated on the edges of the Pt strips.
As a next step, a lift-off process was performed using acetone to remove the photoresist from the
YIG surface. This process detaches the Pt film deposited over the resist, leaving the pattern of Pt
on the YIG surface. The whole process of photoresist-patterning was repeated to grow 100 nm
thick gold contact-pads on the edges of the Pt strips. The pattern of the gold layer is depicted
in Fig. 3.2(b). The 100 nm thick gold thin film was deposited by an electron beam physical vapor deposition technique [233] using an electron beam evaporator Pfeiffer Classic 500 L. Again,
the lift-off process was performed to remove unnecessary gold from the YIG surface. The final
38
3.2 Brillouin light scattering spectroscopy
YIG|Pt sample structure is shown in Fig. 3.3. This sample was used to perform the time-resolved
measurements in Sections 4.2 and 4.3.
3.2. Brillouin light scattering spectroscopy
Brillouin light scattering (BLS) spectroscopy is a technique to detect phonons and magnons by
optical means. In the last two decades, this technique has emerged as a most powerful tool to
investigate magnetization dynamics in magnetic thin films and multilayers [234, 235]. BLS spectroscopy not only empowers to measure the frequency of spin waves, but also provides information
about their wave vector [203, 236, 237], and their phase [238–240]. The investigation about the
spatial [241, 242] and temporal [242, 243] distribution of spin waves has also been realized by incorporating new extensions to BLS spectroscopy. These developments enhance the scope of BLS
spectroscopy and lead to discoveries of new physical phenomena. For example, the time-resolved
BLS technique enables to investigate the evolution of parametrically excited magnons in the YIG
film and the formation of a Bose–Einstein condensate of magnons at room temperature [244, 245].
The space-resolved BLS technique leads to discoveries in the field of nonlinear wave phenomena
like spin-wave solitons [246], bullets [247] and caustics [248]. Recent developments have demonstrated a direct mapping of spin waves generated by a single nano-oscillator [205, 249]. Very
recently, this technique is employed to optical detection of spin transport even in nonmagnetic
metals like silver and copper [250].
Undoubtedly, Brillouin light scattering is a very versatile technique to investigate various aspects of
magnetization dynamics in magnetic materials. In the framework of this thesis, this technique has
been assigned to measure the magnon temperature in a YIG film. The measurements are presented
in the next chapter under Section 4.1. The principles and functionality of BLS spectroscopy are
discussed in the following sections.
3.2.1 Principles of Brillouin light scattering spectroscopy
Quantum mechanically, Brillouin light scattering is a process of inelastic scattering of photons by
phonons3 , or photons by magnons. Photons are the quasiparticles of electromagnetic waves while
phonons are collective excitations of atoms and molecules in condensed matter. The term magnons
denotes the quasiparticles of collective excitations of spin lattices in magnetically ordered materials. In the inelastic scattering process, where the kinetic energy of the photons is not conserved,
either a magnon is created (Stokes process) resulting in a decrease in the energy of the scattered
photon or a magnon is annihilated (anti-Stokes process) which results in an increase in the energy
3
The study of photon-phonon scattering is beyond the scope of this thesis. The reader can refer to Ref. [251]
39
3.2 Brillouin light scattering spectroscopy
Figure 3.4: Schematic diagram of the Stokes and anti-Stokes process. In the Stokes process, spin-waves
(magnons) are created while, in the anti-Stokes process, spin-waves (magnons) are annihilated.
of the scattered photon. Assuming that the scattering process is invariant against translation and
time reversal, the energy and momentum of the system will be conserved. Therefore, the energy
~ ωsc and the linear momentum ~ ksc of scattered photons (light) can be expressed as
~ ωsc = ~ ωin ± ~ ωsw ,
(3.1)
~ ksc = ~ kin ± ~ ksw ,
(3.2)
where ~ ωin , ~ ωsw , are the energies of the incident photon and spin wave, and ~ kin , ~ ksw are their
linear momentum, respectively. The process of spin wave creation (−) and annihilation (+) are
pictorially presented in Fig. 3.4. For finite temperatures (T ≫ 5 K), both of these processes have
almost equal probability [252].
In the classical picture, the process of inelastic scattering can be explained by the Bragg reflection
of the incident light by a moving phase grating created by spin waves as a result of magneto-optical
effects [235, 253]. The spin-wave density is coupled with the fluctuating term of the dielectric
permittivity δǫ of the medium via spin-orbital interaction as
X
δǫij (r, t) =
Kijk Sk (r, t) ,
(3.3)
k
where Sk (r, t) is the k-component of spin density S(r, t) at the position r and time t, and Kijk
are the elements of the linear spin-orbit coupling tensor. Thereby, the interaction between photons
and the medium is modulated by the presence of spin waves. As a result the frequency of the
reflected light from the medium gets shifted by the frequency of spin waves, given by the Doppler
shift [235, 253] as
ωsc = ωin ± ksw · vsw .
40
(3.4)
3.2 Brillouin light scattering spectroscopy
Here v indicates the phase velocity of the spin wave in the medium. The mathematical signs
depend on the direction of the propagating Bragg grating with respect to the wave vector of the
incident light. The phase velocity of a spin wave is given by
v=
ω
ksw .
2
ksw
(3.5)
Considering the Bragg condition ksw = kin − ksc , the energy and momentum conservation con-
ditions in Eq. (3.1) and Eq. (3.2) can be obtained. In order to determine the wave vector of spin
waves, it is crucial to notice that the translation invariance is broken along the normal of thin films,
so the conservation of linear momentum is not valid along the film normal. Only the in-plane components of wave vectors are conserved. Therefore, the wave vectors of light should be replaced by
their projection along the film plane such that
||
||
ksc
= kin ± ksw .
(3.6)
So, by changing the projection of the light beam, i.e., the angle of incidence of light with respect
to the film plane, wavevector-resolved BLS measurements can be performed. The wavevectorresolved BLS technique is discussed in Section 3.2.3.
The conservation of energy in Eq. (3.1) allows to determine the frequency of spin waves. Typically,
the frequency of light is in the THz regime while acoustic spin waves exist in the GHz frequency
range. Therefore, to observe a GHz variation over the THz frequency, a high-spectral resolution is
needed. Another crucial factor is that the cross section of the inelastic photon-magnon scattering
is very small compared to the cross section of the elastic scattering process [235]. As a result,
only a fraction of the incident photons can be inelastically scattered as a BLS signal. Therefore, to
detect spin waves, a very high spectral resolution of the interferometer with a frequency stabilized
laser, and a high sensitive photodetector are required to design a BLS spectrometer. The working
principle of such an interferometer called tandem Fabry-Pérot interferometer is discussed in the
next section.
3.2.2 Working principle of the tandem Fabry-Pérot interferometer
As discussed in the last section, the investigation of spin waves requires a device with high dynamical range, good contrast, and good frequency resolution. In order to fulfill these requirements,
a tandem Fabry-Pérot interferometer (TFPI) [254–256] is incorporated in the BLS spectrometer.
A TFPI consists of two single Fabry-Pérot interferometers (FPI) assembled in series such that the
light can pass through the interferometer mirrors several times. The configurational arrangement
of such interferometers is shown in Fig. 3.6. An FPI consists of two flat mirrors (etalon) separated
by a distance d with reflecting dielectric coatings facing each other.
41
3.2 Brillouin light scattering spectroscopy
Figure 3.5: Ray diagram of a Fabry-Pérot
interferometer. The incident light encounters multiple reflections inside the etalon
comprising two parallel mirrors having a
high-reflection coating on their inner sides
and mounted at a distance d apart.
The highly reflecting dielectric coatings lead to multiple reflections between the mirroring planes.
The efficiency is improved by coating the backsides of the mirrors with an anti-reflection coating.
Since, the dielectric coating is not fully reflective, a small part of the light always transmits through
the mirror at each reflection. The intensity of the transmitted light through the etalon is described
by the Airy function [257] as
T = T0
1
.
1 + F sin2 (∆φ/2)
(3.7)
Here, T is the transmittance coefficient and T0 represents the maximum possible transmission. The
parameter F is known as coefficient of finesse and defined as
F =
4R
,
(1 − R)2
(3.8)
where R is the reflection coefficient. The phase difference ∆φ between two successive transmitted
rays will be given by
∆φ =
2π
λ
2nd cos(θ) .
(3.9)
Here, n is the refractive index of the medium in between the mirrors, and θ is the angle of incidence
which is zero for perpendicular incidence. The transmitted intensity in Eq. (3.7) varies periodically
with the phase difference ∆φ of two interfering rays and will be maximum when ∆φ is a multiple
integer of 2π. So, for a perpendicular incidence of light the condition of constructive interference,
using Eq. (3.9), will be
2nd = mλ, m ∈ N .
(3.10)
Therefore, by changing the mirror spacing d, a wavelength (or frequency) selective scanning of
the light intensity is possible. Since, the intensity of the transmitted light through the etalon varies
periodically (Eq. (3.9)), Stokes and anti-Stokes peaks of higher transmission orders (m) might
overlap each other, so an unambiguous determination of the spectrum is almost impossible. The
frequency separation between two neighboring transmission peaks is called free spectral range
42
3.2 Brillouin light scattering spectroscopy
Figure 3.6: (a) Schematic illustration of the tandem Fabry-Pérot interferometer consisting of two FabryPérot interferometers (FPI1 and FPI2) assembled in series. Their optical axes are inclined at an angle α. (b)
The transmission spectra of FPI1 and FPI2 in the upper panels. The inclination of optical axes shifts the
transmission spectrum. Consequently, the higher transmission orders of the TFPI spectrum are effectively
suppressed, and an unambiguous determination of the spectrum is possible. Note that the spin-wave signal
is much weaker than the reference signal. Figure from Ref. [226].
(FSR), and can be expressed by Eq. (3.10), for n = 1, as
∆fFSR =
c
.
2d
(3.11)
The FSR is related to the full-width at half-maximum (FWHM) δλ of a transmission band by a
quantity known as finesse:
√
∆λFSR
R
≈π
,
(3.12)
F =
δλ
1−R
which is a measure of the quality of an interferometer. In order to resolve the issue of the bandoverlapping for higher transmission orders (high m values), J. R. Sandercock developed a tandem
arrangement of two FPIs [254–256] connected in series. In this arrangement, the second interferometer FPI2 is set at an angle of α with respect to the optical axis of the first interferometer FPI1.
One mirror of each interferometer is mounted on a common scan stage which moves along the
direction of the optical axis of FPI1 during a frequency scan. The moving stage changes the mirror
spacing in FPI1 and FPI2 simultaneously; therefore, a variation of spacing between the mirrors of
FPI1 by δd results in a variation of δd cos(α) in FPI2. The different mirror spacings in FP1 and
FP2 have a significant influence on the transmission of the TFP interferometers. In order to transmit light through both FPIs, the condition in Eq. (3.10) must be fulfilled simultaneously for both
FPIs. This means that higher transmission orders of the second FPI2 will appear later than those
of FPI1. As a result, higher order transmission maxima are strongly suppressed. A schematic picture is presented in Fig. 3.6(b). The intensity of the central transmission order is always maximal.
Hence, a unique frequency assignment is feasible using two FPIs.
The sophisticated tandem interferometer configuration requires high contrast and precise frequency
43
3.2 Brillouin light scattering spectroscopy
stabilization. Since the inelastically scattered signal has weak intensity, high contrast is achieved
by passing the light beam as many as 3 times through each FPI before the beam is guided to the
photodetector. Moreover, as thermal drifts, vibration, and air circulation have a significant impact
on the parallelism of the interferometer mirrors, to ensure the highest possible stabilization, Piezo
actuators are attached at the interferometer mirrors which are actively controlled by a computer
program [258]. For further stabilization, the interferometer is kept on an active vibration damper.
3.2.3 Conventional Brillouin light scattering spectroscopy
In conventional or normal BLS spectroscopy [256, 259], a lens with a long focal length is used
to focus light on the sample under investigation. In such spectroscopes, spin waves can be observed in two geometrical configurations namely forward scattering and backward scattering. The
forward-scattering geometry is useful to investigate optically transparent magnetic materials while
the backward-scattering can be used for transparent as well as optically opaque materials. In the
forward-scattering geometry, a focused probing light passes through the ferrimagnetic sample and
is collected by the objective lens placed on the other side of the sample. Finally, the collected
light (signal beam) is directed to the TFP interferometer by analyzing its frequency. A wave vector
selectivity can be realized in this case using a movable diaphragm placed in between the objective
with a high numerical aperture and the interferometer. More details about the forward-scattering
geometry can be found in Ref. [83, 203, 245].
On the other hand, in the backward-scattering geometry, the probing beam is focused at the sample
and the back-scattered light from the sample is collected by the same objective (lens) and guided
to the interferometer [237,260,261]. The physics of back-scattering geometry is discussed here. A
classical picture of spin waves as a phase grating, presented in Section 3.2.1, explains the inelastically scattered light as a consequence of Bragg scattering. The Bragg condition [203, 257] for
maximal scattering will be satisfied when
ksw = 2nkin sin(θB ) ,
(3.13)
where θB is the angle between the phase grating and the wave vector of the scattered light. Pictorially, the backward scattering process in a film is depicted in Fig. 3.7. For a backward inelastic
scattering in a transparent film, a reflecting surface on the backside of the film is necessary to reflect back light at the moving phase grating (see Fig. 3.7(a)). Usually, the reflecting surface can be
a metallic layer or a microwave antenna—a metallic strip to excite spin waves in the sample. The
||
parallel component of the reflected-light wave vector kr reverses its direction on scattering by a
||
||
spin wave having ksw = 2nkr = 2nkin sin(θB ). On combining with kr⊥ , the kr component generates a scattered signal propagating antiparallel to the incident wave vector in transparent magnetic
films, shown in Fig. 3.7(b). Note that the light wave vector inside the film is nkin where n is
44
3.2 Brillouin light scattering spectroscopy
the refractive index of the material. Similarly, in metallic magnetic films, the inelastic scattering
process occurs at the surface of the film depicted in Fig. 3.7(c).
Figure 3.7: A classical explanation of the magnon-photon inelastic scattering process in thin films. The
incident light (kin ) scatters on a moving wave front (ksw ), created by spin waves. (a) In optically transparent
magnetic films, for example, YIG films for a light of wavelength 532 nm, the incident light (kin ) is first
reflected (kr ) by a reflecting layer, usually a microwave antenna, and thereafter, interacts with a moving
wave front. The vector diagrams for transparent (b) and reflecting (b) films show that only the parallel
||
component of the light wave vector kr interacts with spin waves.
Clearly, the concept of backward scattering geometry can be utilized to study the wave vector of
spin waves by varying the angle θB . The maximum wave vector of a spin wave can be accessed
at θB = π/2. This principle is implemented in designing the wavevector-resolved BLS technique
[83, 203].
In the above section, the conventional BLS spectroscopy is demonstrated as a tool to observe spin
waves in magnetic thin films where translation invariance is broken along the normal of the thinfilms plane. However, in thick magnetic films, the conventional BLS spectroscopy can be used to
access spin waves with a fixed wave vector at any angle of probing light incidence. These spinwave modes have their wave vector antiparallel to the light wave vector and therefore, inelastically
scatter back the incident light at the same angle as the angle of incidence. These spin waves
(magnons) are known as Back-scattering magnon (BSM) modes. They have a fixed wave vector,
defined by the wave vector of light, irrespective to the light incidence angle and given as
ksw = 2nkin .
(3.14)
Figure 3.8: Ray diagram of the backscattering magnon (BSM) mode. The incident light with wave vector kin scatters back
at 180◦ on interacting with a spin wave of
wave vector ksw (= 2nkin ). Here n is the
refractive index of the film.
45
3.2 Brillouin light scattering spectroscopy
Figure 3.9: Ray diagram of the BLS microscope. (a) The probing beam light path is illustrated. The beam
is guided to the sample through the microscope objective by mirrors and beam-splitters. The backscattered
light is analyzed by the TFP interferometer. A small part of the backscattered light is observed by an
additional photodiode so as to confirm the focal position of the sample by intensity measurements. (b) Ray
diagram for sample illumination. Figure from Ref. [258].
The extraordinary feature of the BSM mode is that its fixed wave vector for a given material and
the probing laser enables it to be a solely magnetization dependent spin-wave mode. This feature
is used to investigate the magnon temperature in YIG, in Section 4.1.
3.2.4 Brillouin light scattering microscopy
A Brillouin light scattering microscopy setup is nothing but a normal BLS spectroscopy setup
with a high-numerical-aperture microscope objective which enables to investigate microstructures
and nanostructures. BLS microscopy is used in this thesis to investigate nonlocal spin-valve nanodevices fabricated at RIKEN, Wako-shi, Japan in collaboration with Prof. Y. Otani. The sample
structure and experimental results are presented as Appendix A in the thesis.
The instrumental development of the BLS microscope was a subject of investigation for previous
research works [241, 258], thereby, the optical setup of the BLS microscope is similar to the previous ones up to a large extent. A detailed description about the instrument with technical details
can be found in the literature [241, 258, 262]. A brief description of BLS microscopy is presented
here, in the next paragraphs.
The BLS microscope employed for investigations in this thesis has a spatial resolution of 300 nm
achieved with a laser wavelength of 532 nm, and a microscope objective of numerical aperture
NA = 0.75. In contrast to normal BLS, the wave vector resolution is sacrificed to enhance the
46
3.3 Microwave techniques and spin-wave excitation
spatial resolution due to Heisenberg’s uncertainty principle. A microscope objective with high
numerical aperture allows to focus the laser beam on microstructures; however, the probing beam
does not reach the sample with a well-defined angle anymore; instead, the incident beam falls with
a wide angle of the light cone defined by the numerical aperture as θ = 2 arcsin(NA) ≈ 97◦ . So,
for a laser of wavelength 532 nm, backscattered photons accumulated by the objective can have a
shift of the wave vector within a range ∆ksc = 0 − 17 rad/µm.
A schematic Brillouin light scattering microscope setup is depicted in Fig. 3.9. A BLS spectrometer comprises a tandem Fabry-Pérot interferometer, a continuous-wave frequency-stabilized laser
(Spectra Physics, Excelsior, 200 mW), a microscope objective, an electromagnet, and a controlling
PC. The sample under investigation is mounted on a 3D micro-controlled positioning piezo-stage
having an accuracy of about 10 nm. The probing position on the sample is ensured by monitoring
the laser spot during a measurement process using a sample illustration unit comprising a CCD
camera and a halogen lamp. The illuminating light passes through the sample microscope objective; therefore, the optical resolution of the illustration unit is very high.
An important issue for microscopic measurements is that the long-term stability of the sample
position is disturbed by thermal drifts and mechanical vibrations. In order to achieve a perceivable
stability, an automated picture-recognition software is implemented to monitor and auto-correct the
position of the sample during a measurement process. The algorithm detects any spatial deviation
of the sample from the reference position in the CCD camera image and compensates this deviation
by moving the stage in the opposite direction. Data acquisition and its visualization, and the
stabilization of the interferometer are executed by a computer-controlled TFPDAS4 program [258].
3.3. Microwave techniques and spin-wave excitation
In magnetic materials, thermal fluctuations of the magnetization excite spin waves of energy
~ ωsw < kB T . However, these thermal spin waves have small amplitudes and incoherent nature;
thus, they cannot be used for application purposes in spintronics. An efficient excitation of coherently propagating spin waves is, therefore, a challenge. Microwave techniques are very powerful
tools to excite magnetization dynamics in magnetic materials [85, 263, 264]. The microwave magnetic field couples with the magnetization and exerts a torque on it which drives the magnetization
to precess and generates coherent spin waves. The intensity and the frequency of the excited spin
waves depend on the power and frequency of the driving microwaves, and on the microwave circuit
design.
The excitation of spin waves in magnetic films can be realized using different microwave components like microwave cavities, dielectric resonators, striplines, microstrip antennas, and coplanar
waveguides, etc. Microstrip antennas are generally preferred over other spin-wave excitation tech47
3.4 COMSOL Multiphysics simulations
niques due to their simple design, easy-handling, small size, and easy incorporation with other
techniques like BLS and other optical measurement techniques.
Figure 3.10: A schematic of the spin-wave
excitation by a microstrip antenna. An antenna of width 600 µm is patterned on a copper laminated duroid substrate. Microwave
current generated dynamic Oersted field h,
circulating around the antenna, excites spin
waves propagating along the x-axis in the
YIG film.
A microstrip antenna generates a dynamic Oersted field h circulating around the microstrip and
excites spin waves with a wave vector along the x-axis, perpendicular to its length, shown in
Fig. 3.10. When the external bias magnetic field in the YIG film is along the z-axis, both the
hx and hy components of the microwave magnetic-field act normal to the magnetization in YIG
and, thus, exert a torque to precess the magnetization. Since the wave vector of spin waves is
normal to the bias field, this configuration excites magnetostatic surface spin waves (MSSW). On
the other hand, if the bias field acts along the x-axis, only the hy component exerts a torque on
the magnetization in YIG. So, the excitation efficiency for backward volume magnetostatic spin
waves (BVMSW) is less compared to the MSSW.
The microwave spin wave excitation technique is applied to study YIG|Pt heterostructures in Section 4.3. For more details about microwave techniques references [265, 266] are suggested.
3.4. COMSOL Multiphysics simulations
COMSOL Multiphysics4 is a commercially-available powerful simulation tool for modeling and
solving scientific and engineering problems. COMSOL uses the finite element method (FEM) to
discretize the model geometry and runs finite element analysis to solve partial differential equations
(PDEs) [267]. It includes a complete environment for modeling any physical phenomenon that can
be described by ordinary or partial differential equations. The simulation package solves PDEs
defined by the physical problem using different model constraints and boundary conditions. The
simulator includes various material parameters relevant to a specific problem.
COMSOL Multiphysics offers an extensive and well-managed user interface with a large variety
of programming, preprocessing, and postprocessing possibilities. A simulation creates sequences
of steps that include the geometry and mesh creation, solver settings, visualization, and results presentation. It is easy to parameterize any part of the model by changing a node in the model tree and
4
48
http:www.comsol.com
3.4 COMSOL Multiphysics simulations
re-run the sequences. Defined on the basis of fundamental laws of science, COMSOL can solve
PDEs for a wide range of scientific and engineering phenomena including electrochemistry, electromagnetism, heat transfer, optics, microwave engineering, and fluid dynamics etc [267]. Within
the course of this thesis, the heat transfer module is used to explore the thermodynamics of the
YIG|Pt system, in Section 4.2.5.
The heat transfer module supports fundamental mechanisms of heat transfer, including conductive, convective, and radiative heat transfer, governed by the first law of thermodynamics, i.e., the
principle of conservation of energy. Instead of internal energy, it deals with a rather measurable
quantity—temperature—to solve the heating equation [268] for fluids as
ρCp
∂T
+ ρCp u · ∇T = ∇ · (k∇T ) + Q ,
∂t
(3.15)
where ρ is the density, Cp is the specific heat capacity at constant pressure, T is the absolute
temperature, u is the velocity of the fluid, k is the thermal conductivity, and Q is a heat source.
The conservation of mass is presumed for this equation. Since the YIG|Pt thermodynamic system
has fluid velocity u = 0, the above equation simplifies to
ρCp
∂T
− ∇ · (k∇T ) = Q .
∂t
(3.16)
The above mentioned module is utilized to study the YIG|Pt system thermodynamics in Section 4.2.5.
49
CHAPTER 4
Experimental results and discussion
In the last chapter, various configurations of the spin Seebeck effect (SSE) are analyzed. The spin
Seebeck effect has been studied in several magnetic materials like metals, semiconductors, and
insulators. Further, this effect has been investigated at room temperature as well as at low temperature. With the very first investigation on Py|Pt structures, it was predicted that this effect is caused
by the diffusion of spin polarized electrons in the magnetic metal [32]. However, this concept was
not well taken as the typical spin diffusion lengths in metals are found to be around a few nanometers while the effect was observed at millimeter length-scale [97, 269]. This concept failed again
when the spin Seebeck effect was observed in the magnetic insulator YIG [41]. Therefore, big
efforts are dedicated to the theoretical understanding of the phenomenon. A number of theories
have been proposed considering different possible/potential causes responsible for this effect. One
of the most accepted theory was developed by Xiao et al. [35], known as the theory of magnon
driven transverse spin Seebeck effect. This theory is based upon the difference in the magnon and
phonon temperatures at the interface of the normal metal and magnetic material, developed over
an old theory of the magnon-phonon heat relaxation effect on heat transport [165].
This chapter is dedicated to the investigation of the underlying physics of the spin Seebeck effect.
In the first section, a novel method to measure the magnon temperature in the transverse geometry
of SSE, in a magnetic insulator YIG, is discussed. In Section 4.2, the study on the longitudinal SSE
in the YIG|Pt bilayer is presented. The results about the temporal evolution of the longitudinal SSE
obtained by the laser heating experiment are depicted. In the last sections, microwave measurement
studies on YIG|Pt structures are presented.
4.1. Direct measurement of magnon temperature
4.1.1 Magnon-phonon interactions
A magnetic system can be considered as a combination of three subsystems: phonons, electrons,
and spins (magnons). These subsystems can be considered independent of each other if the mutual
51
4.1 Direct measurement of magnon temperature
interactions among them are weak. Generally, the spin-lattice relaxation time, i.e., the relaxation
time of magnons with phonons, is larger than the intra-relaxation times of magnons, electrons or
phonons [65]. Therefore, these subsystems establish fast internal equilibria within themselves and
can be characterized by their own temperatures, like phonon temperature Tp , electron temperature
Te , and spin temperature Tm . Usually, the relaxation time of the electron subsystem with the
phonon subsystem is also much less than the spin-lattice relaxation time. Therefore, these two
temperatures Tp and Te can be assumed equal within the framework of this thesis.
The interaction of magnons with phonons can be explained by scattering processes. This interaction will be strongest at the point of intersection of phonon and magnon dispersion curves because
the energy and the momentum for the one-phonon–one-magnon scattering process will be conserved at the intersection point. [47]. At this point, a coupled-mode known as magnetoelastic mode
exists. Higher-order interactions, like one-phonon–two-magnon processes, are other channels for
the interaction.
The interaction of magnons with phonons influences their transportation in the system. Above a
certain temperature, a few Kelvins, the magnon thermal conductivity is negligibly small compared
to the phonon thermal conductivity [270–273]. Therefore, the motion of phonons influences the
motion of magnons but not vice-versa. When a thermal gradient ∇T is applied along a ferro-
magnetic material, it creates a higher density of magnons in the hot regions because the magnon
density is proportional to the local phonon temperature of the system. These magnons at the hot
end diffuse towards the cold regions and increase the magnon population there, which essentially
means that the temperature of magnons at the cold end rises [35]. Therefore, at equilibrium, the
magnon temperature Tm goes higher than the phonon temperature Tp at the cold end and lower at
the hot end. The transport of the thermal magnons depends on the magnon-phonon coupling and
the magnon-magnon scattering parameters. In the case of a strong magnon-phonon coupling, the
magnons will rather quickly be in equilibrium with the phonon bath and follow the temperature
profile of the phonons. On the other hand, for weak coupling, the local Tp has a weaker influence
on Tm ; hence the difference between the magnon temperature and the phonon temperature at the
ends of the magnetic system will be larger [165].
4.1.2 Magnons and their temperature
As discussed in the last Chapter, the magnons are bosonic quasi-particles defined as the quanta
of spin waves in k-space. In thermal equilibrium, the population of magnons of frequency ω at
temperature Tm is given by the Bose-Einstein statistic:
n(Tm ) =
52
1
exp(~ω/kB Tm ) − 1
(4.1)
4.1 Direct measurement of magnon temperature
where ~ and kB are the reduced Planck constant and Boltzmann factor, respectively [58]. As each
thermal exchange magnon reduces the total magnetization of the system by one Bohr magneton,
the magnetization of the material is a measure of the magnon population. As the Eq. (4.1) implies
that the magnon population depends on the magnon temperature; one can state that the magnetization of a system is a measure of the magnon temperature. Hence, the spatial distribution of the
magnon temperature can be determined by measuring the local magnetization of the system.
The local magnetization can be measured using optical techniques like the magneto-optical Kerr
effect; however, the sensitivity of this technique is not so high to observe very small variations. The
Brillouin light scattering spectroscopy technique (BLS) enables us to measure the local magnetization of the system by measuring the frequency of the spin waves. Although BLS is best known as a
tool for the spatially resolved investigation of dipolar magnons which typically have characteristic
wavelengths between several microns to millimeters, it can also be used to detect highly spatially
localized thermal exchange magnons (wavelengths of the order of hundreds of nanometers).
In the previous paragraphs, it is shown that the spatial distribution of the magnon temperature
can be determined by measuring the local magnetization. Although by measuring the frequency
of exchange-dominated spin-waves (spatially localized due to their small wavenumbers) the local
magnetization of the system can be calculated, this concept cannot be used to measure the variation
in the local magnetization of the system caused by any external factor like thermal gradients. The
reason is that the spin-wave frequency is not only a function of the magnetization but of spin-wave
wavevector too [75, 78]. Therefore, any variation in the spin-wave frequency can be originated by
either the variation of the magnetization or a change in the wavenumber. A simultaneous variation
of both quantities, wavenumber as well as frequency, is also possible. Therefore, in order to utilize
the spin-wave frequency as a tool to measure the local magnetization, it is mandatory to keep the
wavenumber fixed for all spin waves measured by BLS spectroscopy. The spatial variation of
the magnetization caused by the thermal gradient is studied by measuring the frequency shift of
a particular spin-wave mode known as back-scattering magnon mode with the thermal gradient
at different local phonon temperatures. The wavenumber of the back-scattering magnon mode is
fixed as discussed in the next section.
4.1.3 Back-scattering magnon mode
Typically, in micrometer thick films, a broad spectrum of spin waves is excited by the thermal
energy at room temperature [27, 58]. As shown in Section 3.2.3, one can investigate the different spin-wave modes through BLS spectroscopy by changing the direction of the incident-photon
wave vector relative to the magnetization of the film [203, 245]. Nevertheless, a particular thermal
magnon mode km always exists which travels along the probing light inside the film and satisfies,
53
4.1 Direct measurement of magnon temperature
|km | = 2 n |kl |, where n (= 2.36 for YIG [274]) is the refractive index of the film, and kl is the
photon wave vector (|kl | = 1.18 × 105 rad cm−1 for the green laser light of wavelength 532 nm).
Conservation of momentum implies that the incident photon will be scattered back by a thermal
magnon satisfying k = km at an angle of 180◦ . This mode is assigned as a back-scattering magnon
(BSM) mode. The wavenumber of this mode is fixed due to the BLS scattering conditions with
a precision of 1 rad/cm. As a result, no wavenumber transformation, which can potentially occur
in a temperature gradient [89], can influence the BSM mode, and thus no wavenumber dependent
frequency change can influence the measurements [275].
Figure 4.1: (a) The orientation of the incident (kl ), reflected (kr ), and back-scattered (kl ) photon wave
vectors on the YIG film surface. (b) A typical BLS intensity spectrum to measure the back-scattering
magnon (BSM) mode. The spectrum was captured in the second FSR range; therefore, two elastic scattered
peaks were observed.
For the experiment, discussed in the next section, the BSM mode lies in the region of the exchangedominated spin waves with |km | = 5.67×105 rad cm−1 , on following the values mentioned above,
with a wavelength of 110 nm. The position of this mode on the spin-wave dispersion curve of an
in-plane magnetized YIG film at B = 250 mT is depicted in Fig. 4.3(a) at two temperatures: 300 K
and 400 K . Calculations of these dispersion curves were executed using the theory from Ref. [78]
and the magnetization versus temperature data from Ref. [276]. Other quantities like the exchange
constant are assumed to be temperature independent for these calculations. Due to their small
wavelength, these spin waves cannot recognize any variation in the magnetization away from their
local position. So, these spin waves are strongly dependent on the local magnetization. Therefore,
the back-scattering thermal magnon mode is well suited for measuring the local magnetization of
the system, and it is studied here in the backward scattering configuration using an incident light
at an angle of 45◦ to the film surface, shown in Fig. 4.1(a). This configuration allows us to measure
the spatial distribution of the phonon temperature, Tp , by the IR camera, normal to the YIG film
surface to avoid parallax error.
54
4.1 Direct measurement of magnon temperature
4.1.4 Experimental setup
The investigation of the magnon temperature was done on the low-magnetic-damping ferrimagnetic material Yttrium Iron Garnet (YIG, Y3 Fe5 O12 ). A commercially obtained single crystal
(111) YIG film of dimensions 3 mm × 10 mm with a thickness of 6.7 µm was grown by liquid
phase epitaxy techniques on a 0.5 mm thick Gallium Gadolinium Garnet (GGG, Gd3 Ga5 O12 ) substrate. The film was magnetized in-plane along its long axis by a magnetic field of B = 250 mT.
A schematic diagram of the experimental setup is shown in Fig. 4.2(a). Two Peltier elements for
heating and cooling purposes are kept apart by an edge-to-edge separation of 3.2 mm to create a
thermal gradient along the length of the film. The thermal gradient was measured by an infrared
(IR) camera with a temperature resolution of 0.1 ◦ C and a spatial resolution of 40 µm.
Figure 4.2: (a) In the experimental setup, two opposite ends of a YIG film (on a GGG substrate) were
placed on Peltier elements to create a lateral thermal gradient ∇T . In the schematic, the coldest regions are
shown in blue, the hottest in red. The film was magnetized in-plane with an externally applied magnetic field
B = 250 mT parallel to ∇T . An infrared (IR) camera was used to obtain thermal images of the system. (b)
Infrared image of the YIG film shown in (a). The white-dashed line indicates the path along which the laser
was scanned in order to perform the magnon temperature measurements.
The laser spot position, i.e., the magnetization measurement point was identified visually in the
infrared image. For this purpose, the sample was heated by a high power (40 mW) laser. However,
a small laser power of 7 mW was used for all the regular measurements to minimize the local
sample heating, and to avoid the formation of spurious thermal gradients around the laser spot.
During the measurement process, the laser spot temperature was kept stable with an accuracy of
± 0.3 ◦ C. All BLS measurements were performed in the second free spectral range (FSR) region
to improve the frequency resolution [256, 277].
55
4.1 Direct measurement of magnon temperature
4.1.5 Measurement of magnon temperature
In the first reference experiment, BLS measurements were carried out to obtain the temperature
dependence of the BSM mode on a uniformly heated sample without a thermal gradient. These
measurements were performed in the middle of the YIG film keeping both heat reservoirs at the
same temperature. In this configuration, a uniform phonon temperature was observed along the
length of the YIG strip in the gap between the Peltier elements. The measurements exhibit that
the frequency of the BSM mode declines monotonically with increasing temperature of the YIG
film which happens due to a decrease in the magnetization of the film. An increase of temperature leads to a rise in the thermal magnon population in YIG and, subsequently, decreases the
total magnetization of the system. The monotonic behavior of BSM frequency with temperature
is shown in Fig. 4.3(b). The monotonic decrease of the BSM frequency cannot be explained theoretically just by considering the magnetization as the only temperature-dependent parameter. The
exchange constant and anisotropic fields do contribute, but their impacts are small. Particularly,
the exchange constant is invariant over the temperature range of the measurements [278, 279].
Figure 4.3: (a) The dispersion relations of spin waves propagating perpendicular to the magnetization at
temperatures of 300 K and 400 K in a 6.7 µm thick YIG film magnetized by a field of B = 250 mT. Bullets
show the positions of the BSM mode at km = 5.67 × 105 rad cm−1 . (b) Measured and polynomial fitted
BSM frequency as a function of the temperature of the YIG film. The frequency decreases monotonically
with temperature as a result of the increasing magnon population.
Generally, in magnetic systems where the phonon temperature is uniform throughout the system,
magnons establish an equilibrium with phonons via magnon-phonon interaction and attain a temperature equal to the phonon temperature [165]. So, the phonon temperature in previous measurements performed over a uniformly heated YIG film is considered to be equal to the magnon temperature in the film. By fitting the BSM-frequency-versus-phonon-temperature curve, i.e., fBSM
versus Tp curve with a third order polynomial, shown in Fig. 4.3(b), the BSM frequency fBSM (in
56
4.1 Direct measurement of magnon temperature
Figure 4.4: Measured phonon temperature Tp and magnon temperature
Tm at different positions on the YIG
film along the laser scan line are
shown in Fig. 4.2(b).
GHz) can be expressed as a function of the magnon temperature Tm (in K) as
fBSM = 23.431 − 0.085 Tm + 2.663 × 10−4 Tm2 − 2.917 × 10−7 Tm3 .
(4.2)
Once the reference experiment was complete, a thermal gradient ∇T was created along the YIG
film by passing electric currents through the Peltier elements in opposite directions to each other
(Fig. 4.2(a)). The phonon temperature profile was observed to be practically linear between the hot
and the cold edges of the heat reservoirs (∆Tp = 85 K). The BLS frequency of the BSM mode
was captured simultaneously with the phonon temperature at different positions on the YIG film
along the thermal gradient in between and close to the inner edges of the heat reservoirs. To prove
the repeatability of the measurement, and to estimate the measurement errors, the experiment was
repeated four times and statistics were accumulated.
In Fig. 4.4, the phonon temperature Tp and the magnon temperature Tm , calculated from the frequency of the BSM mode, are plotted as a function of the probing position on the YIG film. One
can easily see that Tm follows the trend of Tp within the limit of experimental errors. The difference between these two temperatures as a function of position on the film is depicted in Fig. 4.5
with a 95 % confidence level. It is important to notice here that the maximum difference between
Tp and Tm is only about 2.8 % of ∆Tp , and in contrast to the theoretical expectations [35,165], this
difference does not change monotonically between the hot and the cold edges.
57
4.1 Direct measurement of magnon temperature
Figure 4.5: The difference between measured
Tp and Tm within a 95 % confidence level obtained using data from Fig. 4.4.
4.1.6 Theoretical modeling
Apparently, Tp and Tm are coupled with each other by the magnon-phonon interaction and, within
the framework of a one-dimensional model as described in Ref. [165], have the following relationship (Eqs. (2.55), (2.56)):
1
d2 Tm (x)
+ 2 [Tp (x) − Tm (x)] = 0 ,
2
dx
λ
(4.3)
where λ is a characteristic length parameter proportional to the square root of the magnon-phonon
relaxation time. The equation implies that in a magnetic system where the relaxation time is
large, therefore, a large λ, the difference in Tp (x) and Tm (x) will be pronounced at the boundaries.
However, no experiment has been reported so far to determine the value of λ in magnetic insulators.
Moreover, the theoretical estimation for λ in YIG is quite uncertain and varies from 0.85 mm to
8.5 mm [35].
In this work, the above equation is solved to determine the value of λ for the experimental conditions. To obtain a continuous dependence of the phonon temperature Tp (x), the experimentally
measured data are fitted with a Boltzmann sigmoid function1 as
1−p
p
+
Tp (x) = T0 + A
1 + exp((x − x1 )/k1 ) 1 + exp((x − x2 )/k2 )
(4.4)
using fitting parameters T0 = 286.487 K, A = 106.980 K, p = 0.617, x1 = −0.769 mm, x2 =
1.272 mm, k1 = −0.823 mm, and k2 = −0.473 mm.
The fitted function is substituted to the equation to get a second order differential equation only in
Tm (x) which has been solved numerically for different values of λ. In the calculation, dTm (x)/dx
is assumed to be zero at the sample boundaries because the heat can dissipate only through the
phonons at the boundaries, as the flow of magnons outside of the sample boundaries is not possible.
1
The phonon temperature can be calculated by solving the heat equations for such systems [62]; however, it is not
crucial for the theoretical analysis presented here.
58
4.1 Direct measurement of magnon temperature
Figure 4.6: Plotted are the measured phonon Tp (open triangles) and
magnon Tm (open circles) temperatures along the BLS laser scan line
indicated in Fig. 4.1(b). The Tp data
is fitted with a Boltzmann sigmoid
function.
From the experimental findings, the maximum possible difference between Tm (x) and Tp (x) at the
position x = ± 2 mm is determined by considering the maximal of the magnon temperature error
bar at x = − 2 mm and minimal of the magnon temperature error bar at x = + 2 mm. By using
these data, the value of λmax is numerically calculated ≈ 0.47 mm which is about one order of
magnitude smaller than the value estimated by Xiao et al. [35] for YIG using the experimental
data of the SSE experiments reported in Ref. [32].
4.1.7 Conclusions
From the findings, one could suppose that the magnon contribution to the transverse spin Seebeck effect, even in magnetic insulators where the magnon-phonon interaction is weaker compared to magnetic metals such as Permalloy, is negligible or rather small. However, the impact
of long-wavelength magnons, mainly the dipolar magnons, is not accounted in this experiment.
In the temperature range of our experiment, the population of exchange magnons in YIG is much
higher than the dipole magnons, therefore, the magnetization is primarily dependent on the exchange magnons. For the exchange (small-wavelength) magnons, having km & 105 rad cm−1 ,
three-particle processes, like the Cherenkov processes analogously with the Cherenkov radiation
phenomenon, provide extra channels for the spin-lattice (magnon-phonon) relaxation [58, 280].
These processes, when a magnon emits or absorbs a phonon, do not change the total number of
magnons, but contribute to establish the thermal equilibrium between the phonon and magnon
thermal baths.
59
4.2 Temporal evolution of the spin Seebeck effect
Figure 4.7: Schematic of the Cherenkov splitting and confluence processes.
The contribution of exchange determined Cherenkov processes to the total magnon damping in2
creases with km
and approaches to be equal to relativistic spin-lattice relaxation mechanisms in
YIG for km & 106 rad cm−1 [58]. Since the magnon-phonon coupling mechanisms for the exchange magnons are quite different from the dipolar magnons, the dipolar magnons must be described by their own temperature. As a result, we can assume that the temperature of dipolar
magnons might not be equal to the phonon temperature and this magnon group can contribute to
the SSE, but it is not possible to determine the temperature of dipolar magnons by the purposed
approach.
In conclusion, this experiment represents a channel to measure the spatial distribution of the
magnon temperature in a magnetic insulator YIG by studying the frequency shift of the BSM mode
with an in-plane temperature gradient. The experimentally determined length scale parameter λ is
found to be, rather less than previously expected [35]. Furthermore, the study reveals that in the
magnetic insulator YIG, having the lowest known magnetic damping, the contribution of exchange
magnons to the transverse spin Seebeck effect is negligible, and this phenomenon should be associated with the low-energy long-wavelength dipolar magnon group. However, the measurement
of magnon temperature for dipolar magnons is beyond the contemporary state-of-the-art. The results are useful to understand the functionality of thermal magnons in the spin Seebeck effect and
valuable for the development of spin-caloritronics.
4.2. Temporal evolution of the spin Seebeck effect
From the magnon temperature experiment, one can state that (1) either the transverse spin Seebeck
effect does not exist (2) or this effect is dominantly dependent on the dipolar magnons (3) or this
effect exists; however, the BLS measurements are not sensitive enough to detect the temperature
difference of magnons and phonons [173]. Therefore, it is very important to investigate the SSE
in another configuration, known as the longitudinal spin Seebeck effect (LSSE). As discussed in
the last chapter, the SSE is a phenomenon where a spin current is generated in spin-polarized
60
4.2 Temporal evolution of the spin Seebeck effect
materials like metals [32], semiconductors [39, 40], and insulators [41, 42, 44] on the application
of a thermal gradient. Generally, the generated spin current is measured by the inverse spin Hall
effect (ISHE) [136] in a normal metal like Pt, placed in contact with the spin-polarized material.
Usually, in LSSE experiments, conventional heating methods such as Peltier-element heaters or
resistive-heating elements are used. These methods have certain drawbacks. First of all, the designs of such experimental setups are rather complicated. There is always a high risk of sample
breaking due to the overload of supporting heat sinks in sandwich configurations to generate a vertical thermal gradient. Secondly, a precise control over heating time cannot be achieved in these
methods. Thirdly, the thermal gradient is rather limited by the maximal temperature of the Peltier
elements. Therefore, it was necessary to develop other heating methods, which can overcome all
these limitations. In the framework of this thesis, two such methods have been developed, namely
pulsed-laser heating and pulsed-microwave heating techniques. Both of these methods provide
a very precise control over heating time. Further, the pulsed-laser heating technique enables to
achieve a spatial control over the heating position.
The most important feature of these techniques is the precise control over the heating time. This
feature facilitates to investigate the dynamics of the SSE which are studied for the first time in the
scope of this thesis and reported here. In the following sections, the investigations of the pulsedlaser heating will be discussed first. Afterwards, results from the microwave heating are presented,
and a comparison of the findings of these two methods will be reported.
4.2.1 Experimental setup
The LSSE measurements were performed on a bilayer of a magnetic insulator, Yttrium Iron Garnet
(YIG), and a normal metal, Pt. A 6.7 µm thick YIG sample of dimensions 14 mm × 3 mm was
grown by liquid phase epitaxy on a 500 µm thick Gallium Gadolinium Garnet (GGG) substrate. To
achieve a good YIG|Pt interface quality, a detailed cleaning process as discussed in Section 3.1.2,
and in Ref. [70], was followed before the deposition of Pt, discussed in the last chapter. A strip
(3 mm × 100 µm) of 10 nm thick Pt was deposited on the cleaned YIG surface by MBE at a pres-
sure of 5 × 10−11 mbar with a growth rate of 0.05 Å/s. A schematic diagram of the sample is
shown in Fig. 4.8.
A laser heating technique [156,167,281] was implemented to heat the Pt strip from the top surface
to create a vertical thermal gradient along the y-direction perpendicular to the bilayer interface
(See Fig. 4.9). For this purpose, a continuous laser beam (wavelength 655 nm) was modulated by
an acousto-optical modulator (AOM, Isomet 1205C), and focused on the middle of the Pt strip
using a microscope objective (Leitz PL 16 x/0.30) down to a laser spot size of 30 µm − 70 µm.
To study the temporal profile of the laser beam in parallel, the transmitted laser beam through the
61
4.2 Temporal evolution of the spin Seebeck effect
Figure 4.8: Sketch of the sample structure. A 10 nm thick Pt strip was deposited on a 6.7 µm thick YIG
film. A vertical thermal gradient was created by heating Pt with a pulsed laser. An in-plane magnetic field
B = 20 mT was applied to the YIG film. The generated voltage across the Pt strip due to ISHE was
measured.
YIG sample was monitored by an ultrafast photo-diode. A (10% − 90%) rise time of 200 ns was
observed for the laser pulses (solid line in Fig. 4.10). The sample structure was mounted on a
copper block to provide a thermal heat sink. The electric connections to the Pt strip were made
using gold wires with silver-paste as an adhesive material. In order to avoid external noise in the
signal, coaxial cables were connected between the sample holder and the oscilloscope.
Figure 4.9: Sketch of the experimental setup. A continuous laser beam (wavelength 655 nm), modulated by
an acousto-optical modulator (AOM), was focused down on a 10 nm thick Pt strip, deposited on a 6.7 µm
thick YIG film, by a microscope objective (MO). The laser intensity profile was monitored by an ultrafast
photo-diode. An in-plane magnetic field B = 20 mT was applied to the YIG film. The heated Pt strip
created a thermal gradient perpendicular to the YIG|Pt interface (see the inset). The generated voltage
across the Pt strip due to the ISHE was amplified and measured by an oscilloscope.
62
4.2 Temporal evolution of the spin Seebeck effect
4.2.2 Time-resolved measurements
The time-resolved measurements of the LSSE were carried out using a 10 µs long laser pulse with
a repetition rate of 10 kHz. An in-plane magnetic field B = 20 mT was applied to saturate the YIG
film magnetization along the x-direction. As a result of the LSSE, a spin current flows along the
y-direction. By the inverse spin Hall effect, this spin current converts into an electric field along
the z-direction in Pt. The electric field was detected as a potential difference VLSSE between the
two short edges of the Pt strip (shown in Fig. 4.9). The LSSE voltage VLSSE was amplified by
40 dB (power) using a high input impedance preamplifier and monitored on an oscilloscope. The
measurements were performed for both ±x-directions of the magnetic field. The LSSE voltage
changes its polarity by reversing the direction of magnetic field [42]; an absolute average value of
VLSSE was evaluated to eliminate the thermal emf offset.
Figure 4.10: The time profiles of the laser intensity (solid line), and the LSSE voltage (VLSSE ) at various
laser powers of 75 mW (open square), 105 mW (open circle), and 131 mW (open triangle).
In Fig. 4.10, the temporal profile of the laser light intensity and VLSSE for different laser heating
powers are plotted. The LSSE signal rises sharply for the first 1µs and then gradually attains a
max
max
saturation level VLSSE
. With increasing laser power, VLSSE
increases linearly (Fig. 4.11(a)). This
linear behavior indicates that the laser heating is in the linear regime, and no nonlinear phenomena
are involved in this process. A comparison of the rising edges of the laser intensity and the LSSE
signal provides a clear signature that the LSSE has no direct correlation with the temporal profile
of the laser intensity.
For the understanding of the ongoing mechanism, the rise time of VLSSE was analyzed for different
laser powers. In Fig. 4.11(b), the rise time (10% − 90%), t, show that, within the limits of exper63
4.2 Temporal evolution of the spin Seebeck effect
max , with linear fitting, and (b) the rise time (10% − 90%) t as a function
Figure 4.11: (a) Measured VLSSE
of the applied laser power. The rise time is practically unchanged with the laser power.
imental error, t is independent of the laser power which depict that nonlinear spin Seebeck effect
contributions [282] are not involved in these experiments2 . Average value of t = 572 ± 27 ns was
obtained by using data shown in Fig. 4.11(b). The rise time is much different from the rise time of
the laser intensity (10% − 90% rise time ≈ 200 ns).
A similar temporal behavior like the one observed for VLSSE can be expected to originate from
a RC circuit. Therefore, the resistance and capacitor measurement on the sample structure were
performed to analyze the time constant of the sample structure. The electrical measurements were
executed by inserting additional resistances and capacitors in the original circuit and monitoring
the rise time of the circuit. By performing basic RC calculations on sample structure, a time
constant of 48 ns (R ≈ 508 Ω, C ≈ 95 pF) was obtained.
4.2.3 Pt resistance measurement
The first hunch to interpret the rise time was to study the temperature evolution in the YIG|Pt system. Fortunately, the Pt strip on top of the YIG film can be utilized as a resistance-temperature (RT)
2
Recently, it has been predicted that the spin current generated in the magnetic material via the SSE doesn’t scale
linearly with the thermal gradient, rather, it shows the signature of rectification and negative differential [282] at high
negative-thermal-gradients.
64
4.2 Temporal evolution of the spin Seebeck effect
detector to measure the temperature at the surface of the YIG film. The resistance measurements
of the Pt strip were performed to calculate the variation of the temperature in the YIG|Pt system
by the laser heating. To do so, a constant current3 Ic = 0.5 mA was passed through the Pt strip
(room temperature resistance RPt ≈ 508 Ω), and the potential drop (∆VPt = ∆RPt Ic ) due to the
heating of Pt was measured with the same experimental setup used for the VLSSE measurements.
A much longer laser pulse of 4 ms was used to heat the Pt strip. Note that the thermal emf (few
microvolts) has negligible influence on these measurements as the potential drop (∆VPt ) is very
large (≈ 0.25 V).
To evaluate the average temperature of the Pt strip, auxiliary measurements of the static resistance
versus temperature were performed on the same Pt strip. The YIG film was placed on a Peltier
element to heat uniformly. The temperature of YIG was measured using a thermocouple as well
as with an infrared camera. The resistance of the Pt strip was measured using a Keythley sourcemeter. The resistance of the uniformly heated Pt strip varies linearly with temperature. By fitting
this data, ∆VPt was expressed in terms of temperature (T ). It is important to note that the resistance
of the entire Pt strip was measured; therefore, the temperature of the Pt calculated using auxiliary
measurements evaluates only the average temperature of Pt. The temperature at the laser spot will
be much different from the average temperature.
Figure 4.12: (Color online) The time profile of the variation of temperature in Pt on heating with a 4 ms
long laser pulse with various powers.
In Fig. 4.12, the variation in the average temperature of the Pt strip (∆T ) is plotted for different
laser powers. A rise time of 2 ms, obtained by fitting the data with a single saturating exponential
function, is three orders of magnitude longer than the rise time of the LSSE signal. Since both,
VLSSE and ∆T (or ∆R) were measured over the entire Pt strip and are attributed to the heat dynam3
Although current itself can produce heating in the Pt strip, this will not influence the overall outcomes.
65
4.2 Temporal evolution of the spin Seebeck effect
ics, the comparison of dynamics of these two quantities is valid here. The saturating exponential
behavior of the temperature illustrates that the heat losses in the system dominantly control the
heat dynamics of Pt. Further, the measurements evident that, likewise the LSSE signal, the rise
time of the temperature is also independent of the laser power.
4.2.4 Thermal magnon diffusion model
From the last subsection on the measurements of the Pt resistance, it can be concluded that the temperature of the system has no direct correlation with the fast timescale of the LSSE. There should
be a separate thermodynamic quantity which can explain this rise time of the LSSE. In order to dig
out the cause contributing to the fast rising of the LSSE, a model is proposed where we consider
the thermally induced motion of magnons in a system of normal metal|magnetic material (e.g.,
Pt|YIG) subject to a thermal gradient perpendicular to the interface. In such a system, the spin
current flowing into/out of the normal metal depends on the temperature difference of the magnon
bath in ferromagnets and the phonon baths (or electron bath) in the normal metal [35, 37], and on
the magnon accumulation close to the interface in the magnetic material [172]. On the application
of a temperature gradient, thermal magnons having higher population at hotter regions—in local
equilibrium their population is proportional to the phonon temperature—propagate towards colder
regions having less magnon population. The propagation of magnons creates a magnon density
gradient in the system along with the phonon thermal gradient. This implies that the spatial distribution of the magnon density depends on the magnon population (phonon temperature) and their
propagation lengths.
The interface of the normal metal|magnetic material (e.g., Pt|YIG) also plays a crucial role in the
spin current injection into the normal metal. The interface quality can change the spin pumping
efficiency [70] which means that essentially the quality of the interface alters the spin mixing conductance in a bilayer system. Further, there could be other parasitic effects like the proximity effect
which states that some metals like Pt can show paramagnetic nature when it is placed in the vicinity of a ferromagnetic material [169]. Therefore, the spin Seebeck voltage can be considered as a
combination of interface effects and a bulk contribution from the magnon motions and, eventually,
can be expressed as
VLSSE (t) ∝ α
Z0
y=−δd
∇Ty (y, t) dy + β
Zl
∇Ty (y, t) exp(−y/L) dy ,
(4.5)
y=0
where ∇Ty is the phonon thermal gradient perpendicular to the interface, l the magnetic film
thickness, and L the effective magnon diffusion length. Here, y = 0 represents the position of
the interface. A thickness of δd = 5 nm in YIG is considered as a thickness of the interface. The
66
4.2 Temporal evolution of the spin Seebeck effect
parameter α defines the coupling between the electron bath in the normal metal and the phonon
bath in the magnetic material. The coupling parameter β specifies the magnon-magnon interaction
within the magnetic material. The first term of Eq. (4.5) includes interface parasitic effects like
proximity effect [169], interfacial thermal resistance [173] etc. It is important to note that Eq. (4.5)
does not include thermoelectric effects like normal or anomalous Nernst effect [177, 283, 284].
Since YIG is insulator, it is considered that no such effect exists for YIG|Pt system.
4.2.5 2D COMSOL simulations
In order to determine the phonon thermal gradient ∇Ty , a 2D phonon heat conduction equation is
solved for the YIG|Pt bilayer using the COMSOL Multiphysics simulation package [Section 3.4].
In the simulation model, a 10 nm thick and 300 µm wide Pt rectangular layer (2D) was placed on
a 6.7 µm thick and 300 µm wide YIG layer. The entire structure was mounted on a GGG substrate
(100 µm × 300 µm). The simulation parameters are indicated in Table 4.1. The YIG|Pt interfacial
thermal resistance [173] was implemented in the simulations over a thickness δd = 5 nm. As a
boundary condition, the temperature along the bottom edge of the GGG layer (see Fig. 4.13(a))
was kept fixed at room temperature, i.e., 293.15 K. This boundary condition resembles the heat
sink in the experimental setup. A 25 µm wide area at the middle of the Pt block was considered as
a heat source which replicated the laser heating in the experimental setup. The lateral dimension
of the heat source is comparable to the laser spot size and larger than the thickness of the YIG film.
Table 4.1: Material parameters used for COMSOL simulations.
Material
Density
Thermal conductivity
Heat capacity
(kg/m3 )
(W/mK)
(J/kgK)
21450 [285]
20 [286]
130 [285]
YIG
5170 [287]
6.0 [288]
570 [289]
GGG
7080 [288]
7.94 [288]
400 [288]
Pt
From the simulations, the distribution of the temperature and the thermal gradient was extracted.
In Fig. 4.13(b), the distribution of temperature in the Pt|YIG|GGG layers after 10 µs of heating is
depicted. The heat front takes nearly 10 µs to reach the bottom of the YIG layer. From simulations,
the temperature distribution along the Pt layer length was evaluated. In Fig. 4.14(a), the simulated
temporal evolutions of the average temperature in the whole Pt and at the laser heating spot are
shown. The temperature dynamics of the laser spot matches closely to the dynamics of the average
temperature. The temperature evolution in Pt (≈ 2 ms) was obtained as slow as it was observed
in the Pt-resistance-measurement experiment. From simulations, we find that a gradual increase
67
4.2 Temporal evolution of the spin Seebeck effect
Figure 4.13: (a) Sketch of the simulated geometry. A 10 nm thick Pt layer was placed on a 6.7 µm thick
YIG layer. A 100 µm thick GGG substrate was kept underneath the YIG layer. The width of each layer
was 300 µm. A 25 µm long area (red) was considered as a heat source. The average thermal gradient is
calculated along horizontal lines at various distances d from the interface. (b) Simulated distribution of
temperature in the Pt|YIG|GGG system after 10 µs. The Pt layer is not visible due to the scale of the figure.
in the average temperature is due to the large heat capacity and volume of the system. The heat
radially propagates away from the heated area on the Pt strip.
Figure 4.14: Numerically calculated time profiles of (a) the average temperature in Pt and at the laser heating
spot, (b) thermal gradients, ∇Tavg , in YIG at the interface and at d = 50 nm, 200 nm, 500 nm, 1 µm, and
2 µm distances away form the YIG|Pt interface.
On the other hand, the thermal gradient close to the interface shows fast dynamics. The average
thermal gradient ∇Tavg along lines parallel to the interface was evaluated for various distances d
from the interface in the YIG film by averaging ∇T along these lines. The average thermal gradient
∇Tavg is depicted in Fig. 4.14(b). These parallel lines essentially represent the parallel planes (xz)
in the experimental geometry. Contrary to the temperature in Pt, the average thermal gradient rises
very rapidly and saturates within microseconds. This timescale agrees with the timescale of the
LSSE. As d, i.e., the depth of the reference line (a plane in the 3D model) from the interface,
increases, the rise time of the temperature gradient increases due to a decrease in the heat flux
68
4.2 Temporal evolution of the spin Seebeck effect
caused by a finite thermal conductivity and the increasing thermal capacity (∝ volume).
4.2.6 Effective thermal-magnon diffusion length
The fast rise of the thermal gradient (≈ 20 ns) at the interface of the YIG|Pt bilayer, shown in
Fig. 4.14(b), leads to the conclusion that the timescale of the LSSE cannot be explained by examining only the time evolution of the thermal gradient at the interface. The timescale of the LSSE
must be influenced by a second, rather slower process. On the basis of this argument, the first
term of Eq. (4.5) can be considered as static over the timescale of our interest (> 20 ns). Using
the phonon thermal gradient data, obtained from the COMSOL simulations, the integral term of
Eq. (4.5) was calculated for various magnon propagation lengths L. The integral was computed
from the interface up to the thickness of the YIG film.
In Fig. 4.15, the normalized value of the experimentally and numerically calculated VLSSE for
L = 300 nm, 500 nm, and 700 nm are plotted as a function of time. Clearly, our model replicates
the experimentally observed timescale of the LSSE. On comparing the calculated values of VLSSE
with the experimental results, the effective magnon diffusion length is evaluated to be ≈ 500 nm.
Note that the very first slow increase in the normalized VLSSE (for time ≤ 0 µs) originates from
the switching time of the laser (≈ 0.25 µs), shown in the inset to Fig 4.15. The effective magnon
diffusion length exhibits the depth inside YIG over which the thermal gradient is crucial for the
LSSE.
4.2.7 Conclusions
The model indicates that the temporal evolution of the LSSE depends on the thermal gradient
in YIG. Further, thermal magnons up to a depth of a few hundreds of nanometer in YIG are
effectively contributing to the LSSE. The typical effective magnon diffusion length of 500 nm
agrees with recent theoretical calculations [290, 291]. These results support the findings in Ref.
[171] that the parasitic interface effect like the proximity Nernst effect has negligible influence on
the longitudinal spin Seebeck effect. In this model, magnon diffusion times are neglected because
their group velocities are much higher than the group velocities of phonons. The spectral nonuniformity of phonons [292] and/or magnons [A1] is also not necessary to consider in this model.
In conclusion, the time-resolved measurements of the longitudinal spin Seebeck effect in YIG|Pt
bilayers performed by the laser heating experiments reveal that the rise time of the LSSE is submicrosecond fast, and the LSSE signal attains its maximum within a few microseconds though the
temperature in the system establishes in milliseconds. The timescale of the LSSE is independent of
the strength of the heating source. The model of the thermal-magnon diffusion in thermal gradients
show that the LSSE is governed by the diffusion of thermal magnons from the interface toward the
69
4.3
Microwave heat originated spin Seebeck effect
Figure 4.15: Comparison of normalized spin Seebeck voltage VLSSE measured experimentally with the
numerical calculations for different effective magnon diffusion lengths L = 300 nm, 500 nm, and 700 nm.
The inset shows the laser intensity switching time (≈ 0.25 µs).
bulk. Moreover, the establishment of the thermal gradient in the vicinity of the YIG film interface
determines the timescale of the LSSE. The model estimates a typical diffusion length for thermal
magnons to be around 500 nm in the YIG|Pt system. Our results provide a very important piece
of information about the timescale of the spin Seebeck effect that shed light on the underlying
physics.
4.3. Microwave heat originated spin Seebeck effect
In the last section, laser heat induced longitudinal spin Seebeck effect was discussed. Laser heating
allows us to have a precise control over the heating time. However, as discussed in the last section,
lasers usually have a typical switching time. In the previous experiment, a switching time of
≈ 0.25 µs was found 4 . This switching time lies close to the timescale of the spin Seebeck effect.
Therefore, it is beneficiary to have a technique where the switching time of heating is much shorter.
The microwave heating technique is one of the suitable candidates for it. Unlike the laser heating or
conventional heating, the microwave heating is a volumetric heating technique which heats each
atom/molecule simultaneously at surface as well as in bulk of the specimen. Therefore, unlike
conventional heating, no heat diffusion mechanism is necessary here. In volumetric heating, the
4
Lasers with fast switching time are commercially available; however, usually, their powers are not large enough
to be used for heating purposes, and/or using these lasers are not economically viable.
70
4.3
Microwave heat originated spin Seebeck effect
Figure 4.16: (a) Sample structure. The YIG|Pt sample was placed along the microstrip-line with Pt facing
to it. The Pt strip was connected to the middle SMA-connector using a gold wire. (b) A typical thermal
image of the sample structure observed for a long MW pulse (microseconds).
average temperature of the system rises linearly with time as each atom/molecule absorbs heat
independently while in conventional heating, via diffusion mechanism, the average temperature of
the system asymptotically reaches the oven temperature. Principally, with volumetric heating the
average temperature of the system continues to rise as long as power is applied [293]. In microwave
heating, the heating time can be controlled by varying the duration of microwave pulses. Usually
the switching time of microwave pulses is around couples of nanoseconds. To verify the findings of
the temporal evolution of the longitudinal spin Seebeck effect, this technique is used and described
here.
4.3.1 Sample structure design and experimental setup
The microwave heating technique is employed on the same YIG|Pt sample utilized for the laser
heating experiment. A copper (Cu) microwave strip antenna was used to radiate the microwaves.
To design such an antenna, a 0.6 mm wide Cu microstrip was fabricated on a duroid substrate. The
bottom side of duroid was metalized to create a ground channel for the microwave strip-line. The
duroid substrate was glued on a metal block with a conducting silver paste. SMA (SubMiniature
version A) coaxial RF connectors were assembled to connect the microwave input/output cables.
To obtain the maximum microwave heating efficiency, the YIG sample was placed along the stripline with the Pt side facing towards the microstrip-line. The Pt strip was connected with gold wires
to recover the signal from Pt. In Fig. 4.16, the actual sample structure is shown which was utilized
to perform microwave measurements.
71
4.3
Microwave heat originated spin Seebeck effect
Figure 4.17: A schematic sketch of the experimental setup. Microwave pulses generated by a MW generator
were amplified and guided to the sample structure to heat the Pt strip placed on top of the microstrip-line.
The transmitted MW pulses were monitored on an oscilloscope. The ISHE voltage generated in the Pt strip
was amplified with a high-impedance-low-frequency amplifier and measured via a separate channel of the
oscilloscope.
A schematic diagram of the experimental setup is shown in Fig. 4.17. Microwave (MW) pulses
generated by a microwave generator (Anritsu MG3692 C) were amplified by a microwave amplifier
(MITEQ) and guided to the sample structure, discussed above, using thick coaxial cables. These
MW pulses were absorbed by the Pt metal and started to heat it up. As a result, a thermal gradient
along the z-axis was formed. A Y-circulator was connected to the MW circuit line to block the
reflected microwaves from the sample section. A DC-blocker was used to avoid any back-flow of
DC to the MW generator. The transmitted MW pulses through the sample section were rectified
using a high-frequency (HF) diode and monitored on an oscilloscope (Agilent DSO7034 A). This
signal provides the information about the MW pulse shape and its duration; therefore, this signal
is considered as the reference signal. The sample structure was mounted between the poles of a
water-cooled electromagnet creating a magnetic field along the x-axis.
The inverse spin Hall signal (ISHE) generated in the Pt strip along y-axis was passed through a
low-pass filter to block MW currents. The filtered signal was amplified by a high-input-impedancelow-frequency voltage amplifier. The high-impedance amplifier was selected to extract a maximum
voltage out of the Pt strip. A total power amplification of 40 dB was used which amplified the ISHE
voltage (VISHE ) by 100 times. At last, the amplified signal was observed on the oscilloscope via a
separate channel.
72
4.3
Microwave heat originated spin Seebeck effect
4.3.2 Asymmetry in spin pumping
A primary test on the spin-current injection efficiency was carried out using a continuous microwave current before performing the time-resolved measurements. Another purpose for performing these measurements was to study the spatial distribution of spin-wave pumped current in
the YIG film [294]. The frequency of the MW current was kept fixed to 6.8 GHz, and magneticfield-scanning was executed for both positive as well as negative field values. During the magneticfield-scanning process, the ferromagnetic resonance conditions were fulfilled two times—once for
the positive magnetic field and the second time with the negative. Corresponding to these magnetic
fields, a spin current produced by the spin pumping process is injected in the normal metal Pt. In
Pt, this spin current gets converted into an electric field EISHE , as discussed in previous sections,
and finally measured as a voltage VISHE on the oscilloscope.
In Fig. 4.18, the inverse spin Hall voltage versus the applied magnetic field µ0 H is plotted. Clearly,
three features can be noticed at very first glance. Firstly, there are two signals, which belong to
two spin-pumping signals at FMR, with opposite polarities. Secondly, these two signals at FMR
are not equal in magnitude. Thirdly, there is an offset for all non-resonance magnetic fields which
has an opposite polarity to the FMR signal.
The first feature, i.e., two signals at FMR with opposite polarities, originates from the inverse spin
Hall relationships
EISHE = θSHE ρPt (Js × σ) ,
Zl
VISHE = EISHE dy,
(4.6)
(4.7)
0
where θSHE denotes the spin Hall angle, ρPt and l are the resistivity and the length of the Pt, respectively. Js represents the spin current, and σ is the spin polarization. The direction of σ depends
on the direction of the magnetic field. Therefore, with inverting the magnetic field direction, the
direction of EISHE reverses; hence, VISHE .
In order to understand the second feature of the spectrum that the signals have unequal amplitude,
it is important to discuss the spin-wave modes excited in the experimental geometry. It is clear
from the sample orientation, shown in Fig. 4.17, that the spin waves excited by the Oersted field of
the microstrip-line propagate along the y-axis perpendicular to the applied magnetic field direction.
These kinds of spin waves where k⊥H is known as magnetostatic surface spin waves (MSSW) or
Damon-Eshbach (DE) spin waves. A detailed description about these spin waves is given in the
last chapter. The DE spin-waves are nonreciprocal spin waves and travel along a direction given
by
k = H × n,
(4.8)
73
4.3
Microwave heat originated spin Seebeck effect
Figure 4.18: The inverse spin Hall voltage (VISHE ) generated in the Pt strip as a function of the applied
magnetic field µ0 H. An asymmetry in the amplitude of VISHE at FMR-magnetic field appears due to an
asymmetric excitation of spin waves.
where n is the normal to the film surface. Essentially, this means that the position of the YIG|Pt
sample on the microstrip-line plays an important role in the spin wave excitation process [295].
In Fig. 4.19, the asymmetric excitation of DE spin waves for two-oppositely-directed magnetic
fields has been demonstrated. It becomes clear from Eq. (4.8) that when a magnetic field is applied
along the +x-direction, spin waves can be excited only along the −y-direction with respect to the
microstrip-line. On the other hand, when the magnetic field is applied along the −x-direction,
the spin waves can propagate along the y-axis. Since the YIG film is, intensely, not positioned
symmetrically around the microstrip-line, the excitation efficiencies for two opposite fields are
different, as seen in Fig. 4.18.
For a better understanding of the impact of the position of the YIG film on top of the microwavestrip antenna, spin-pumping measurements were carried out for various positions. Two cases have
been discussed here. For the first case, the lower edge of the YIG film was nearest to the microstrip
antenna while, for the other case, the upper edge was placed close to the antenna. Magnetic-field
scans were carried out for both situations and are plotted in Fig. 4.20.
In Fig. 4.20, one can see clearly that when the microstrip-antenna is at the upper edge of the YIG
film, the magnitude of the inverse spin Hall voltage (VISHE ) is higher for the positive magnetic
field than the negative field. The situation is exactly opposite when the antenna is at the lower
edge. This tendency can be understood as following. Since the EISHE is directly proportional
to the spin current Js (Eq. (4.6)), which depends on the intensity of spin waves (population of
magnons), one can say that EISHE is proportional to the intensity of spin waves excited in the
74
4.3
Microwave heat originated spin Seebeck effect
Figure 4.19: Schematic sketches of DE spin-wave excitation for (a) negative and (b) positive magnetic
fields. Since, for the negative magnetic field, the area of the spin-wave excitation is smaller than the positivefield excitation area, a small amplitude of the inverse spin Hall voltage (VISHE ) was obtained for the negative
field, shown in Fig. 4.18. In the figure, the normal to the YIG film plane (n) points towards the reader. The
spin current Js flows from YIG to Pt along the z-axis.
Figure 4.20: Comparison plots of the inverse spin Hall voltage (VISHE ) for microstrip-antenna placed close
to the upper edge and the lower edge of the YIG film for (a) negative and (b) positive magnetic fields.
system. When the antenna is placed close to the upper edge of the film, spin waves can be excited
in a larger area of the YIG film for a positive magnetic field in comparison to a negative magnetic
field (see Fig. 4.19). On the other hand, when the antenna is close to the lower edge of the YIG
film, the situation reverses and hence, the amplitude of VISHE . Therefore, it can be concluded that
the unidirectional nature of DE spin waves regulates the asymmetry of the ISHV signal.
The third feature seen in Fig. 4.20 indicates that there is an offset for both positive and negative
fields; moreover, the polarity of this offset changes with the direction of the magnetic field. Additionally, the offset has an opposite polarity than that of the signal at FMR for a same direction
of the magnetic field. In order to investigate more about this behavior, magnetic field scans for
various microwave input powers were carried out. In Fig. 4.21, the VISHE versus the magnetic
field data is plotted for different applied microwave powers. The peak-to-peak amplitude of VISHE
scales linearly with the applied microwave power.
75
Microwave heat originated spin Seebeck effect
8
6
VISHE (µV)
2
0
-4
-6
-20
372 mW
468 mW
741 mW
933 mW
1175 mW
-10
(b)
12
18.6 mW
74.1 mW
117.5 mW
186.2 mW
295.1 mW
4
-2
14
(a)
VISHE (µV)
4.3
10
8
6
4
2
0
10
20
μ0H (mT)
0
0
0.2
0.4
0.6
0.8
1.0
1.2
Power (W)
Figure 4.21: (a) Plotted is (VISHE ) as a function of the applied magnetic field for various microwave heating
powers. (b) The peak-to-peak amplitude of VISHE versus applied microwave power is plotted. The peak-topeak amplitude of VISHE scales linearly with the MW power.
The signature that this off-resonance VISHE signal scales linearly with MW power gives a hint that
the signal originates from the heating produced in Pt as shown in Fig. 4.16. This can’t be an offresonance spin pumping since the polarity of the signal is opposite to the signal at FMR. The entire
scenario indicates that the longitudinal spin Seebeck effect [70, 296] generates the off-resonance
VISHE signal. When the Pt strip is heated with microwave absorption, a thermal gradient from
YIG to Pt develops normal to the interface. The thermal gradient generates a spin current flowing
along with the gradient. Since the Pt strip is hot, the spin current generated via the longitudinal
SSE flows from Pt to YIG, in contrast to spin pumping where the spin current flows from YIG to
Pt. Hence, this argument explains the opposite polarities of the resonance (spin pumping) and the
off-resonance (longitudinal SSE) inverse-spin-Hall voltages (VISHE ) observed in Fig. 4.18. With
increasing the MW power, the temperature increases in the Pt strip which enlarges the thermal
gradient close to the YIG|Pt interface and hence injects a large spin current into the YIG film.
Impact of the large spin current appears as a higher VISHE signal highlighted in Fig. 4.21(b).
4.3.3 Time resolved measurements
In the last section, it is found that microwaves heat the Pt strip in the YIG|Pt bilayer structure and
result in the generation of a thermal spin current, via the longitudinal spin Seebeck effect. With
increasing the microwave power, the injected spin current, in YIG from Pt, increases, as a result,
the inverse spin Hall voltage generated in Pt rises. So, it is clearly evident that a microwave heating
can be used to generate the spin Seebeck effect. The most fascinating feature of MW heating is that
a precise control over the heating time of the sample is possible. This feature has been utilized here
to study the temporal evolution of the longitudinal SSE. The benefit of MW heating over a laser
76
4.3
Microwave heat originated spin Seebeck effect
irradiation is its fast switching times (< 10 ns). In this section, the time-resolved measurements
performed by employing MW heating are presented.
The time-resolved measurements were performed by using 10 µs long MW pulses of power ≈
0.32 mW (25 dBm), instead of continuous MW as used for the asymmetry spin pumping experiment. The frequency of the MW pulses (6.8 GHz) is chosen such that the magnetic system stays at
off-resonance condition for the range of the magnetic field of interest (± 25 mT). The experiment
was executed at various MW powers to get a better LSSE signal using the same setup as shown
in Fig. 4.17. The measurements were recorded for both positive and negative magnetic fields, and
an average value of off-resonance-VISHE was considered. As discussed in the last section, the offresonance VISHE is attributed to the longitudinal spin Seebeck effect, henceforth, the off-resonance
VISHE values will be denoted by VLSSE .
Figure 4.22: (a) Plotted is the temporal evolution of (VLSSE ) on the application of a 10 µs long microwave
pulse which creates a vertical temperature gradient in the YIG|Pt structure by heating the Pt strip. (b)
A comparison of experimentally measured VLSSE data with calculated values using Eq. (4.5) for various
effective magnon diffusion lengths L = 300, 500, 700 nm.
In Fig. 4.22, VLSSE is plotted versus time. The longitudinal SSE signal (VLSSE ) shows similar
features as seen in the laser heating experiment. The main difference observed here is that the
VLSSE signal appears as soon as the MW current runs . Contrarily, in the laser heating experiment a
time lag exists due to the laser switching time (see Fig. 4.15). Using the model of thermal magnon
diffusion, discussed in Section 4.2.4, the normalized experimental-VLSSE -signal was plotted against
the theoretical values in Fig. 4.22(b). Clearly, the microwave heating experiment reproduces the
results of the laser heating experiment, discussed in Section 4.2. In the case of the microwave
heating experiment, a typical thermal magnon diffusion length is found to be L ≈ 500 nm which
is the same as in the laser heating experiment. Therefore, both experiments support the thermal
magnon diffusion theory and give similar results.
77
4.3
Microwave heat originated spin Seebeck effect
In a new series of measurements, the study of the fall time of the longitudinal SSE signal, i.e.,
the signal when the microwave heating is terminated, was performed using rather longer, 50 µs
long pulses. The thermodynamics of the cooling process should be similar to the heating process.
This conclusion is drawn on the basis of the argument that the thermal gradient within the depth
d = 1 µm underneath the YIG|Pt interface achieves a steady state after 10 µs of the start-time
of heating (see Fig. 4.14). However, the temperature of the system still continues to increase for
hundreds of microseconds. When the heating of the Pt strip is turned off, the flow of heat in the
system still remains in the same direction as in the case of the heating. i.e., from Pt to YIG, which
means that the direction of the thermal gradient in the vicinity of the interface will not reverse, only
the magnitude of the thermal gradient will alter over time. Therefore, assuming that the dynamics
of the thermal gradient is likely to be the same, the experimental temporal reduction of the LSSE
signal can be fitted with the same theoretical data obtained in the Section 4.2.5 taking into account
the fact that the thermal gradient is diminishing over time.
Figure 4.23: A comparison of experimentally measured VLSSE curve, after switching-off the microwave current (heating) at time = 0 µs, with calculated values using Eq. (4.5) for various effective magnon diffusion
lengths L = 300, 500, 700 nm by taking into account the fact that the temperature gradient is diminishing
over time.
In Fig 4.23, the temporal reduction of VLSSE is plotted against the theoretical data. The time (t =
0 µs) is defined as the time at which the microwave current switches off. Clearly, the experimental
results can be explained by the thermal magnon diffusion model by considering a typical effective
magnon diffusion length (L) equal to 500 nm. This particular scenario can be understood as the
78
4.3
Microwave heat originated spin Seebeck effect
diffusion of thermal magnons from the interface of the YIG|Pt layers towards the interior (bulk) of
the YIG film, i.e., a spin current flows from Pt to the YIG film. As soon as the heating of the Pt strip
stops, the thermal gradient in the vicinity of the interface diminishes resulting in a retardation of
thermal magnons which eventually appears as a fall of the VLSSE signal. After a few microseconds,
the thermal gradient vanishes and the signal drops down to zero.
4.3.4 Conclusions
In the last section, the microwave heating technique is demonstrated to study the longitudinal spin
Seebeck effect. The microwave heating is rather a less-complicated volumetric-heating technique.
Due to microwave absorption in the Pt strip, a perpendicular thermal gradient to the YIG-film
plane develops in the YIG|Pt bilayer. When the applied microwave frequency hits the resonance
condition of the YIG film, magneto-static surface spin-waves (MSSW) are excited which inject a
spin current into Pt through the spin pumping process and are observed via the inverse spin Hall
effect in Pt. Due to the non-reciprocal nature of the MSSWs, an asymmetric inverse spin Hall
signal is recorded which can be explained by the unequal excitation efficiency of the microwave
antenna depending on its relative position with respect to the YIG film.
In the case of off-resonance conditions, a pure spin Seebeck effect signal is observed which scales
linearly with the microwave power. Subsequently, time-resolved longitudinal SSE measurements
are carried out at off-resonance conditions using microwave pulses. The findings confirm the
observations drawn in the laser heating experiment. The thermal magnon diffusion model explains
the results and leads to conclusions that (i) the LSSE is governed by the diffusion of the thermal
magnons from the interface toward the bulk of the magnetic material, that (ii) the establishment of
the thermal gradient in the vicinity of the YIG film interface determines the timescale of the LSSE,
and that (iii) thermal magnons have a typical diffusion length of 500 nm.
79
CHAPTER 5
Conclusions and outlook
The research work presented in this thesis addresses controversial physics of the spincaloric phenomenon, the spin Seebeck effect, in ferromagnetic|normal-metal heterostructures and improves
the fundamental understanding of the role of thermally driven magnons in these heterostructures.
The investigations are carried out on a ferrimagnetic insulator Yttrium Iron Garnet (YIG) to explore the interactions of the phonons and magnons in this kind of systems. To put light on the
controversial physics of the spin Seebeck effect, two questions have been answered in this thesis:
1. The first question is how the quasiparticles, magnons and phonons interact with each other
at thermal nonequilibrium, and how the individual temperatures of these two quasiparticle
systems are correlated with the spin Seebeck effect.
2. The second question deals with the time scale of the longitudinal spin Seebeck effect and
reveals the impact of the interface and bulk contributions.
In order to answer the first question about the interactions of the magnons and phonons, direct
measurements of the spatial distribution of the magnon and phonon temperatures in a YIG film,
subject to a lateral thermal gradient, are realized by developing a novel technique to measure
the local magnetization in the magnetic film using Brillouin light scattering (BLS) spectroscopy.
A specific exchange-dominated spin-wave mode, the back-scattering magnon mode, was studied
which solely depends on the local magnetization at the probing position in the material. By measuring the spatial distribution of the frequency of the BSM mode, the local magnon temperature
distribution in the system was calculated. The findings reveal a close correspondence between the
spatial dependencies of magnon and phonon temperatures which represents the strong interaction
between the magnon and the phonon subsystems.
YIG has the lowest magnetic damping; so it is anticipated that this material should have a weak
magnon-phonon interaction. The outcomes of the magnon temperature experiments raise a question about the strength of the interaction and emphasize on the formulation of the concept of
spectral non-uniformity of the magnon temperature. Unlike the dipolar magnons, the exchange
81
magnons have additional channels to interact with the phonons like the Cherenkov processes,
which enhance their interaction strength.
The findings of the magnon temperature experiment are utilized to understand the origin of the
transverse spin Seebeck effect (SSE). Theoretically, it is predicted that the two temperature model,
magnon temperature and phonon temperature, in magnetic materials can explain the transverse
SSE. Theories anticipate that a difference in the spatial distribution of these two temperatures leads
to the injection of a spin current into the normal metal placed in contact with the ferromagnet. A
close similarity in spatial dependencies of the temperatures reopens the question about the origin
of the transverse SSE. The concept of the spectral non-uniformity of the magnon temperature
provides a new direction for deeper theoretical investigations of the origin of the SSE.
In order to uncover the origin and for a better understanding of the underlying physics, the knowledge about the timescales of the spin Seebeck effect is essential. The second question deals with
this task. The very first measurement on the temporal evolution of the longitudinal spin Seebeck
effect were carried out on a YIG|Platinum (Pt) heterostructure. The high spin-orbital coupling
material Pt was employed to measure the spin current generated by the longitudinal SSE via the
inverse spin Hall effect. The spin injection efficiency in Pt was enhanced by treating the YIG film
surface with a special cleaning recipe using a piranha solution; subsequently, the Pt strips were
grown by MBE technique. The time resolved measurements of the longitudinal SSE have been
realized through two heating techniques, laser irradiation and microwave heating. The advantages
of these heating techniques over the conventional heating methods are that a precise control over
the heating time is possible in order to perform controlled measurements.
The time-resolved measurements reveal that the longitudinal SSE occurs at a sub-microsecond
timescale. The longitudinal SSE signal attains its maximum within the first few microseconds
though the temperature in the system establishes in milliseconds. The temporal evolution of the
longitudinal SSE depends on the thermal gradient in the vicinity of the YIG|Pt interface and the
diffusion of thermal magnons from the interface. The thermal magnons up to a depth of a few
hundreds of nanometers underneath the YIG interface are effectively contributing to the longitudinal SSE. The model of thermal magnon diffusion in temperature gradients estimates a typical
effective magnon diffusion length of 500 nm for the YIG|Pt system. These measurements assist
in understanding of the origin of the effect. Particularly, the outcomes answer the open question
whether the effect is a pure interface effect or bulk effect. The results reveal that the effect is a bulk
effect in which thermal magnons, laying up to a certain volume underneath the interface inside the
magnetic material, participate.
Additionally, the microwave heating experiments on the YIG|Pt bilayer structures were carried out
to avoid any artifact attributed to the heating time delay occurred due to a finite laser switching
time. The outcomes of these experiments confirm the observations drawn about the timescale
82
of the longitudinal SSE in the laser heating experiment. Moreover, in these experiments, a clear
picture about the direction of the spin current flow, from YIG→Pt or Pt→YIG, appears. The results
show that in the spin pumping experiment, a spin current flows from YIG to Pt while, in the spin
Seebeck effect experiment when Pt is hot, the spin flows reversely, i.e, from Pt to YIG, for the same
experimental configuration. The outcomes help in understanding of the spin current directions in
different experiments related to the spin pumping and the spin Seebeck effect.
In conclusion, this work contributes to the fundamental understanding of the role of thermally
driven magnons in magnetic-insulator|normal-metal heterostructures. The investigation of a new
technique for direct measurement of the magnon temperature, and the studies of the origin and
temporal evolution of the spin Seebeck effect not only improve the contemporary knowledge of
the field but also open a new window for development of future applications.
Outlook and future perspective
In the past ten years, the field of spintronics has gained an amazing pace of development. During
these years, the main objectives have been to discover new phenomena and explore their physics.
In the contemporary era, the motivation is diverting towards potential applications based on these
phenomena. The questions addressed in this thesis contribute to this development.
The magnon temperature measurement experiment demonstrates a very first technique to measure
the magnon temperatures in magnetic materials and assists in understanding of the magnon-phonon
interaction. The technique can be utilized to explore the potential future materials and guides the
direction for the synthesis of new materials. The proposed concept of spectral non-uniformity of
magnon temperature stimulates new theories of the spin Seebeck effect [292].
The knowledge of the temporal evolution of the longitudinal spin Seebeck effect is essential for
the development of future industrial applications like temperature sensors, temperature gradient
sensors, thermal spin-current generators, etc. The experiment reveals the factors responsible for the
timescale of the effect and, thereby, gives direction for improvement. Since, the temporal evolution
of the longitudinal SSE depends on the development of the thermal gradient in the magnetic film,
it is anticipated that the timescale of the effect can be controlled by varying the thickness of the
film. Therefore, one can perform these time-resolved measurements on high quality ultrathin YIG
films.
Another aspect of the spin Seebeck effect which remains untouched is its nonlinear nature predicted recently [282]; however, the experimental demonstration is yet to be realized. The microwave heating technique employed in this thesis empowers to generate high temperature gradients for such measurements.
83
APPENDIX A
Microwave spin currents in lateral spin valves
Spintronics is a research field where the additional degree of freedom of electrons, i.e., the spin
is utilized to play the same role as of the charge in electronics. In order to realize the spin world,
development of spin accumulation, manipulation, and detection techniques is a prime concern.
Over time, there have been many discoveries about spin-current generation methods. The spin
injection [297, 298] method is rather old compared to recent methods such as spin pumping and
the spin Hall effect. Most of the work on spin injection, on giant magnetoresistance and tunnel
junctions, has been performed on multilayered structures. These multilayered films or nanopillars
are vertical heterostructure where currents flow perpendicular to the film interfaces. In these twoterminal devices, the functionality of spin current can be explained by a quasi-one dimensional
spin transport model. However, the main constraint in these devices is that the spin current is
always accompanied by a charge current. Therefore, investigations of spin transport properties are
rather complex due to the involvement of charges in the transport.
Contrarily, in two-dimensional lateral spin valve structures, shown in Fig. A.1, is it possible to
separate the spin current from the accompanying charge current, which allows to generate a pure
spin current. The pioneer work on these lateral spin valves, also known as nonlocal spin valves,
was done by Johnson and Silsbee in the 1980s [297–300]. They reported that nonequilibrium
spins injected from a ferromagnet into a normal metal diffuses inside the normal metal over the
spin diffusion length. However, almost two decades later, Jedema et al. resurrected the concept
by performing the nonlocal spin valves measurement on meso-structures prepared by lithography
techniques [301,302]. These studies attracted the attention of scientists and geared up the research.
Although there have been different studies on nonlocal spin valves such as pure spin current induced magnetization reversal [303], direct and inverse spin Hall effects [94, 96, 304], Hanle effect [305], spin-dependent Seebeck and Peltier effects [158, 159], and so on, most of the investigations deal with the static component of the injected spin current in the nonmagnetic (normal) metal.
The motivation behind the studies performed in the course work of this thesis is to explore the dynamic component the spin current in the normal metals. To serve this purpose, the microwave field
excitation technique has been employed and discussed here with the experimental outcomes. In
85
A.1 Nonlocal spin injection
the next section, first, the theory of the nonlocal spin-current generation will be discussed briefly.
Thereafter, the device structure and its fabrication techniques will be presented. Subsequently, the
experiment results will be discussed in the next section. At last, the conclusions are drawn on the
basis of the experimental outcomes.
A.1. Nonlocal spin injection
A spin current is a flow of electron spins which compensates the imbalance between the population of spin-up and spin-down electrons in diffusive conductors. In transition metals, electrical
conduction is governed by only those electrons which are in the vicinity of the Fermi level. In
ferromagnetic transition metals (Co, Fe, Ni), a difference in density of states for spin-up and spindown electrons appears due to the exchange interaction. Therefore, the conduction properties of
electrons depend on their spin orientations. When a ferromagnet gets into a physical contact with
other materials, due the breaking of translational symmetry, spin accumulation takes place at the
physical interface and is quantified as the difference between the chemical potentials of spin-up
and spin-down electrons at that position in the material.
Figure A.1: A nonlocal spin valve device structure of ferromagnet/normal-metal/ferromagnet (F1 /N/F2 ).
The ferromagnets are separated by a distance L. An electric current Iq injects spin polarized electrons in N
which generates a spin current Is by spin diffusion process. A leakage of the injected spins occurs at the
ferromagnet F2 position, and a potential difference develops between N and F2 .
In a typical nonlocal spin valve device ferromagnet/normal-metal/ferromagnet (F1 /N/F2 ) structure,
as shown in Fig. A.1, F1 is a spin injector, N is a spin transport channel, and F2 acts as a spin
detector. In ferromagnets, the decoherence length of spin is of the order of their lattice constants
[269]; therefore, in typically spin-valve devices only the components of spin which are aligned
along the magnetization can survive. Accordingly, the electric current density j↑,↓ for spin-up (↑)
and spin-down (↓) electrons driven by the applied electric field E = −∇φ and carrier density n↑,↓
difference (diffusion model), can be expressed as [97]
j↑,↓ = σ↑,↓ E − eD↑,↓ ∇n↑,↓ ,
(A.1)
where σ↑,↓ is the electric conductivity and D↑,↓ the diffusion constant for spin-up and spin-down.
Using the relationship between Fermi energy (ǫF ) and density of states (N ), i.e, ∇n = N ∇ǫF ,
86
A.1 Nonlocal spin injection
and the Einstein relationship σ = e2 N D for both spin channels (↑, ↓), the current density can be
rewritten as
j↑,↓ = −
σ↑,↓
∇µ↑,↓ ,
e
(A.2)
where µ↑,↓ = ǫF↑,↓ + eφ is the electrochemical potential. Using the continuity equations for
charge and spin, and the detailed balance condition which ensures that no spin scattering occurs
at equilibrium, the spin diffusion equation can be derived in terms of electrochemical potentials
as [97, 110, 306]
∇2 (σ↑ µ↑ + σ↓ µ↓ ) = 0 ,
1
∇2 (µ↑ − µ↓ ) = 2 (µ↑ − µ↓ ) ,
λ
(A.3)
(A.4)
where the spin-diffusion length λ is given by
λ=
p
Dτsf .
(A.5)
Here τsf and D denote the the spin relaxation time and diffusion constant, respectively [97]. In
normal metals, the electric conductivity of spins is equal, i.e., σ↑ = σ↓ , while this no longer holds
for ferromagnets. In ferromagnets, the spin-diffusion lengths are very small compared to normal
metals. The spin diffusion length is a crucial parameter for spin valve devices. Materials with a
long spin diffusion length are preferred for spin transport studies. In the review article [269], the
spin diffusion length for different magnetic and nonmagnetic materials is presented.
When an electric current Iq = I↑ + I↓ passes through the interface F1 |N, a spin current Is = I↑ − I↓
is injected into the normal metal. The currents I↑,↓ depend on the interface resistance for spin-up
and spin-down electrons. The injected spins in the normal metal diffuses along its length. Here,
it is assumed that the other dimensions of the normal metal bar are much smaller than its spindiffusion length. The injected spin current is absorbed by the ferromagnet F2 at the interface N|F2 .
On solving Eqs. (A.3), (A.4), one can obtain the electrochemical potential in normal metal as
N
µN
↑,↓ = µ0 ± σ(a exp(−|x|/λN ) − b exp(−|x − L|/λN )) .
(A.6)
Here, the first term describes the charge transport, which will be zero for all x > 0 (see Fig. A.1).
The second term represent the exponential decay of the chemical potential in the normal metal.
Further, the term with the coefficient a stands for the spin accumulation due to the spin injection
from F1 , and the term with b is due to the spin depletion caused by F2 . It is important to note that,
for all x > 0, no charge transport is taking place. In this region, only the spin current exist; this
shows that nonlocal spin valves generate a pure spin current.
The mechanism can also be explained by a simple model of spin-dependent density of states in
(F1 /N/F2 ) lateral structures, as shown in Fig. A.2. In the absence of charge currents, the Fermi
87
A.2 Device structure and fabrication
Figure A.2: Illustration of spin transport in a F1 /N/F2 structure by the spin-dependent density of states
model. Illustration is simplified by considering a half-metallic density of states. Only the important subbands of metals are shown here. At equilibrium, all Fermi energy levels are equal. When a current is
applied between F1 and N, spins accumulate and diffuse in N. The spin transport channel between F2 and
N depends on the relative orientation of the magnetizations in the ferromagnets. Accordingly, a potential
difference appears for parallel (VP ) and antiparallel (VAP ) configurations. Figure from Ref. [308].
energies for all three metals are at an equal level. When a current passes from F1 to N, only one
spin sub-band in F1 is available for participating in the charge transport. Therefore, an imbalance
of the spin population occurs in N close to the interface, and due to the carrier density difference,
spins start to diffuse in N. When the magnetization of F2 is parallel to F1 , the majority spin electrons
in the s-band of N are allowed to scatter to the d-band of F2 . On the other hand, when they are
antiparallel, the majority spins of the d-band of F2 scatter to the minority spin band of N. The
process is depicted in Fig. A.2.
In both cases, an electric potential difference (VP = −VAP ) between N and F2 can be measured,
which reverses its sign according to the orientation of the magnetizations in F1 and F2 . A spin ac-
cumulation voltage Vacc can be defined as a voltage difference between the parallel and antiparallel
magnetization configurations:
Vacc = VP − VAP = ∆Rs Iq ,
(A.7)
where ∆Rs is the spin accumulation signal [306]. In the nonlocal spin valve experiments, usually
the spin accumulation signal is measured as a function of the bias magnetic field. An overview of
the recent progress of lateral spin valve devices can be found in literature [307].
A.2. Device structure and fabrication
In a nonlocal spin valve device, the spin accumulation signal is dependent on the F|N interface
resistance. Recent studies show that the spin accumulation can be enhanced by 100 times by
88
A.2 Device structure and fabrication
Figure A.3: (a) The scanning electron microscope (SEM) image of the e-beam patterned suspended resist mask. The mask consists 200 nm thick polymethyl methacrylate (PMMA) and 500 nm thick methyl
methacrylate (MMA). (b), (c) A schematic illustration of the shadow evaporation technique with suspended
resist mask. The inside view of the mask along the cut-plane indicated by the dotted-yellow line in the SEM
image. (b) First, Py wires were obliquely deposited at a disposition angle of 45◦ . (c) Subsequently, MgO
and Ag layer were deposited at normal incidence. Note that, for oblique deposition, the substrate was tilted
rather than Py beam.
using a MgO tunnel contact at the interface, instead of usual ohmic contacts [305]. To study the
microwave spin currents in lateral spin valves, the spin valve devices with MgO tunnel contact
were fabricated in the group of Prof. YoshiChika Otani at the Center for Emergent Matter Science,
RIKEN, Wako-shi, Japan. All fabrications, as well as measurements presented here were carried
out at RIKEN in assistance with Mr. Hiroshi Idzuchi, a D. Sc. candidate in the group of Prof. Otani.
In order to excite spins propagating in normal metal Ag, the designed nano-devices consist of two
parts: a typical nonlocal spin valve (NLSV) device, and a coplanar waveguide placed in the vicinity
of the NLSV. A coplanar waveguide is synthesized such that the dynamic Oersted field acts out-ofplane, perpendicular to the injected spins in silver. Thereby, the dynamic field can exert a torque
on the spin and can stimulate a precessional motion. A schematic view of the device structure in
shown in Fig. A.4.
The device fabrication was carried out mainly in two steps. In the first step, the coplanar waveguide of gold was produced by patterning a resist mask with the photolithography technique on a
Si/SiO2 substrate and, subsequently, by depositing a gold film by physical vapor deposition. Once
the gold waveguides were produced, the spin valve fabrication procedure discussed in Ref. [305]
was followed to produce the spin valve devices of NiFe(Py)/Ag/NiFe. The devices were fabricated
without breaking vacuum in the deposition chamber by means of a shadow evaporation technique
using a suspended resist mask. A resist mask pattern, having 500 nm thick methyl methacrylate and
200 nm thick polymethyl methacrylate, shown in Fig. A.3, was created by the e-beam lithography
89
A.3 Microwave excitation of pure spin current
Figure A.4: (a) A schematic view of the nonlocal spin valve device with an antenna. The Oersted dynamic
field (hd ) by the antenna acts perpendicular to the injected spins, by the left permalloy (Py1) wire, thereby,
exerts a torque on the spins. Consequently, the spins start precessing around the bias static magnetic field
(H0 ). Finally, these spins interact with Py2 and generate a potential difference V between Ag and Py2. The
precessional behavior of spins is studied by measuring V as a function of the bias magnetic field H0 and
the applied microwave frequency. (b) A typical SEM image of a NLSV device. Note that the groundline of
the coplanar waveguide (CPW) is not shown in the image. The CPW consists of ground-signal lines (G-S)
only rather than the G-S-G configuration.
technique. An e-beam deposition technique was used to deposit all magnetic and non-magnetic
thin layers at ultrahigh vacuum conditions. However, to prevent the contamination of non-magnetic
layers with magnetic particles, the depositions of magnetic and non-magnetic materials were performed separately in two interconnected evaporation chambers.
In the NLSV fabrication procedure, first the two ferromagnetic wires of Py were obliquely deposited by tilting the substrate at an angle of 45◦ . In the second step, a MgO layer of 3 nm thickness was deposited at the same angle of tilt. In the third step, the Ag wires were deposited normal
to the substrate. At last, a capping layer of MgO was deposited to prevent the oxidation and contamination of the devices. A schematic illustration is shown in Fig. A.3. After the lift-off process,
the devices were annealed in an N2 +H2 environment. Since various dimensions of the gold, silver and permalloy wires were fabricated, the actual dimensions are given while discussing the
experimental results in the next section.
A.3. Microwave excitation of pure spin current
In the last section, the NLSV device fabrication technique was discussed. Mainly, two kinds
of measurements were performed on these devices using Nagase Prober—a vacuum and lowtemperature probers station—at room temperature as well as at low temperature (10 K). In the
first measurement procedure, the spin accumulation signal (Eq. (A.7)) was measured as a function
90
A.3 Microwave excitation of pure spin current
Figure A.5: (a) The SEM image of the measured NLSV device with a 400 nm wide, 50 nm thick Ag antenna
placed along the Ag wire keeping a gap of 200 nm in between them. The Py wires are separated apart by
a distance of 1.5 µm. (b) The observed spin signal (V /I) as a function of the applied bias magnetic field.
The amplitude of the spin signal jumps between two levels when the magnetization of any of the Py wire
switches its orientation. A spin accumulation signal ∆Rs = 1.5 mΩ was observed for this deceive at 10 K.
of the applied static magnetic field along the length of the Py wires. No microwave current was
applied during these measurements. In the second measurement procedure, to observe the precessional motion of spins, the spin accumulation signal (∆Rs ) was measured as a function of the
microwave frequency. The measurements were carried out for various static magnetic fields with
varying microwave power. An abrupt change in the ∆Rs signal is expected when the resonance
condition of spins, most likely the Larmor frequency of electron spin, is satisfied by the microwave
frequency and the magnetic field. The Larmor frequency of an electron spin is given as
gµB
B.
ωL =
~
For electrons, at B = 1 T, νL = ωL /2π = 28.025 GHz.
(A.8)
The challenge to carry out these studies was to optimize the device structure. As spin current
decays exponentially in normal metals, it is good practice to arrange the injector (Py1) and detector (Py2) wires close to each other in order to obtain a large signal. However, if the distance
traveled by the spins from the injector to the detector is short, the spins will not have sufficient
time to interact with the excitation magnetic field (hd ). These two factors compete with each other.
Many devices were manufactured for the optimization propose. Devices with various thicknesses
of microwave antenna and its distance from the Ag wire were fabricated to achieve an efficient
excitation magnetic field. But no success is recorded in observing a resonance behavior of the spin
accumulation signal. A few measurements with device specifications are discussed here.
In Fig. A.5(a), a nonlocal spin valve device with an Ag coplanar1 antenna is shown. The Py wires
1
Ag antennas using the e-beam patterning were fabricated, instead of Au antennas (with photolithography), to
91
A.3 Microwave excitation of pure spin current
Figure A.6: The spin signal V /I versus the frequency of the applied microwave current to the antenna is
plotted for the device shown in Fig. A.5(a). The spin signal (a) for the microwave power (Pmw ) of 10 dBm
and 15 dBm at a bias magnetic field B = 100 mT, (b) for various magnetic fields B = 20, 50, 100, −20 mT
with a fixed microwave power Pmw = 15 dBm. At B = −20 mT, the Py wires have an antiparallel (AP)
magnetization configuration. All measurements were performed at 10 K.
were kept apart at a rather long distance of 1.5 µm to increase the dwell time of the spins. A 400 nm
wide, 50 nm thick Ag antenna was placed along the Ag wire of the NLSV device keeping a gap
of 200 nm in between them. In the measurement procedure, first, the spin accumulation signal
(∆Rs ) was measured as a function of the bias static magnetic field without any microwave current
though the antenna. A constant current I = 800 mA was passed through the Py1|Ag interface,
and the potential difference V was measured across the Py2|Ag as shown in Fig. A.4(a). The
measurement was performed at a low temperature of 10 K. In Fig. A.5(b), the V /I signal is plotted
versus the magnetic field. The value of V jumps between two levels for the parallel and antiparallel
configurations of the magnetizations of Py1 and Py2. A spin signal ∆Rs = 1.5 mΩ obtained for
this devices indicates the good quality of the device; thereby, the device was good enough to carry
out further measurements.
In order to investigate the dynamics of the injected spin current in Ag, the microwave power and
the magnetic field dependence of the V /I signal were studied at a low temperature of 10 K and are
plotted in Fig. A.6. It can be seen from the Fig. A.6(a) that on increasing the applied microwave
power through the antenna, the magnitude of the spin signal varies. These measurements were
performed at a magnetic field B = 100 mT, where the magnetization of both Py wires are parallel
to each other. However, no signature of any kind of resonance behavior is observed. A Larmor
resonance is anticipated at a microwave frequency of around 2.8 GHz. A series of measurements
were carried out for various bias magnetic fields as well as for the parallel and the antiparallel (AP)
configurations, but no inconsistency or peculiarity in the spin signal is noticed. For all of these
reduce the gap between the antenna and the Ag wire; hence, to improve the excitation efficiency of the antenna.
92
A.3 Microwave excitation of pure spin current
Figure A.7: Plotted is the spin signal V /I, at a magnetic field of 0.1 mT and microwave power of 15 dBm,
and the S21 -parameter (transmission losses) of the antenna as a function the applied microwave frequency.
The oscillatory behaviors of two signals match very well.
measurement, it is seen that on varying the microwave frequency, the spin signal first drops rapidly
and, subsequently, slowly bounces back to the origin level at higher frequencies. Furthermore, the
spin signal is embedded in oscillations. In order to understand the origin of these features, the
transmission characteristics of the antenna was analyzed.
In Fig. A.7, the transmission characteristic of the antenna, measured as S21 -parameter, and the spin
signal at a bias magnetic field of 0.1 mT and microwave power (Pmw = 15 dBm, are plotted as a
function of the applied microwave frequency. The transmission of microwaves is very weak for
sub-gigahertz frequencies and reaches a maximum at around 0.5 GHz. With further increasing the
frequency, S21 decreases gradually. The opposite trend can be seen in the response of the spin
signal with the microwave frequency. The positions of the oscillations in the transmission curve
reflect in the spin signal, which leads to draw a conclusion that most likely a capacitive coupling—
two metal wires placed parallel to each other—results in an almost identical oscillatory behavior
for both signals. Moreover, the microwave current through the antenna creates joule heating which
alters the temperature of the substrate, thereby, the temperature of NLSV device and the V /I
signal [309]. It is clear from Fig. A.6(a) that with increasing microwave power through the antenna,
the value of V /I signal decreases, which means that an increase in the temperature of the device
reduces the value of V /I signal. Similarly, when the transmission through the antenna is high,
more heat is produced; hence the V /I signal drops rapidly. As the transmission decreases with
increasing microwave frequency, the joule heating also decreases; hence, the V /I signal gradually
attains its initial level. The origin of the mismatch in the frequencies of peak positions of these two
signals (see Fig. A.7) is not clear.
93
A.4 Conclusions
A.4. Conclusions
In the last section, some of the results obtained from the microwave measurements of NLSV have
been discussed. Though the impact of the applied microwave signal is seen on the spin accumulation signal, no signature of a resonance behavior is observed. With current understanding, this can
be explained as follows. The spin diffusion length for Ag is around 1 − 1.5 µm [305, 310]. The
presented device in the last section has a separation of 1.5 µm in between the injector and detector
Py wires. A spin takes roughly around 15 ps to travel this distance. However, the experiment was
performed in a microwave frequency range below a few GHz (≤ 5 GHz). So, the time period of
dynamic magnetic field is of the order of nanosecond (≥ 0.2 ns). It indicates that the spins do
not have enough time to interact with the dynamic field. Essentially, during their flight, the diffusing spins experience a rather static magnetic field created by the antenna. Hence, no resonance
behavior is observed for these measurements.
In the experimental geometry, it is possible to excite the magnetization dynamics in the Py wires
by the antenna. However, no such excitation of Py wires is noticed. This raises a question about
the efficiency of the antenna in the real devices. It is highly probable that the excitation efficiency
of the antenna is poor due to the device geometry. A better device geometry has recently been
demonstrated by Kuhlmann et al. [311], where the NLSV devices are fabricated on the top of the
coplanar waveguide (G-S-G type) having an electrical insulating spacer in between them.
The experiment can be realized for materials with longer spin diffusion length and/or at high microwave frequencies.
94
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Own Publications
[A1] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. D. Karenowska, G. A. Melkov, and
B. Hillebrands, Direct measurement of magnon temperature: New insight into magnonphonon coupling in magnetic insulators, Physical Review Letters 111, 107204(2013).
[A2] M. Agrawal, V. I. Vasyuchka, A. A. Serga, A. Kirihara, P. Pirro, T. Langner, M. B. Jungfleisch,
A. V. Chumak, E. Th. Papaioannou, and B. Hillebrands, Sub-microsecond fast temporal evolution of the longitudinal spin Seebeck effect, submitted to Physical Review B.
119
Acknowledgment
I am sincerely thankful to everyone who directly or indirectly contributed to realize the dream of
my mother. First of all, I am grateful to Prof. Dr. Burkard Hillebrands for showing trust in me and
giving me the opportunity to be a part of his team. His guidance and supervision have been like
burning torches in my hands to enlighten the path of my journey.
I am grateful to my second supervisor Prof. Dr. YoshiChika Otani for guiding me and giving me
a fantastic opportunity to work in his labs at RIKEN, Japan. I thank Dr. Yasuhiro Fukuma for a
productive collaboration. I am thankful to Mr. Hiroshi Idzuchi for supporting me during my stay
in Japan in lab work, as well as in integrating me into social life.
I very much appreciate Dr. Oleksandr Serga and Dr. Vitaliy Vasyuchka for their professional advise
and support throughout my PhD duration. Without their guidance, it was unimaginable to achieve
this goal. Their support has been the backbone of this thesis.
I thank Prof. Dr. G. A. Melkov for his theoretical support and fruitful discussions.
I thank my thesis committee members, Prof. Dr. Martin Aeschlimann, Prof. Dr. Michael Fleischhauer, and Dr. Oleksandr Serga for evaluating the progress of my research work at different
occasions.
I thank Dr. Johannes Paul from Sensitec GmbH, Mainz for mentoring my thesis work and discussing on my career plans.
I am deeply indebted to Philipp Pirro and Dr. Björn Obry for teaching me the concepts of magnetism, helping in learning various experiments, and proof reading this thesis. Their supportive
advice have always been compelling and broadened my experimental horizon.
I especially thank Dr. Isabel Sattler for taking care of all administrative and financial issues. She
has always been kind and supportive. She has all sorts of arrows in her quiver.
I am thankful to all coauthors of my publications, Dr. Alexy Karenowska, Akihiro Kirihara,
Thomas Langner, Dr. Benjamin Jungfleisch, Dr. Andrii Chumak, Dr. Evangelos Papaioannou, for
sharing their experiences with me and contributing in the research work.
I appreciate Viktor Lauer and Frank Heussner for assisting me in device fabrication and computation, respectively.
121
I thank Thomas Meyer, Dr. Katrin Schultheiß, and Thomas Brächer for assisting me in BLS labs.
I thank Dr. Frederik Fohr for his support during the initial stage of my PhD life. He helped a lot in
making a fruitful collaboration with RIKEN.
I am grateful to Dieter Weller for his technical support. He assisted me in many ways. He has
always been very kind with me.
I applaud Roland Neb and Dr. Evangelos Papaioannou for sharing office with me. We had some
great discussions about philosophy and science.
I appreciate the unrestricted support of Dr. Britta Leven, Dr. Andrés Conca, Sibylle Müller, and
Peter Clausen in various matters.
I admire Dr. Katrin Schultheiß, Thomas Brächer, and Thomas Langner for sharing their cars with
me in many trips.
I appreciate Dr. Sandra Wolff, Dr. Bert Lägel, and Christian Dautermann for their help in the Nano
Structuring Center.
I am thankful to Dr. Benjamin Jungfleisch and Dr. Thomas Sebastian for their helping hands on
uncountable matters.
I thank all my group members for integrating me in German way of life.
I acknowledge the support of the MAINZ coordination office.
I acknowledge ISGS for supporting international students.
I acknowledge the financial support of DFG within Priority Program 1538 and Graduate School
Materials Science in Mainz (MAINZ).
I am obliged to Prof. Dr. Anjan Barman for guiding, advising, supporting, encouraging me. He
contributed a lot in shaping my career.
My heartfelt gratitude to my parents and brothers for their encouragements and moral support in
pursuing my academic career. Thank you mom for being mast of my life.
Last but not least, I am thankful to all my friends especially Shubh, Manoj, and Tyagi. I don’t
think I can count what they have done for me. I believe all we need in life is friends. . .
122
Curriculum vitae
Personal Information
Name:
Milan Agrawal
Date of Birth:
June 06, 1986
Place of Birth:
Changroad, India
Nationality:
Indian
Marital Status:
Single
Schooling
1991 – 2001
Secondary Education
Government High School, Changroad, Haryana, India
2001 – 2003
Senior Secondary Education (10+2)
Government Senior Secondary School, Charkhi Dadri, India
Academics
Jul 2003–Jun 2006
Bachelor of Science
Maharshi Dayanand University, Rohtak, India
Jul 2007–Jun 2009
Master of Science in Physics
Indian Institute of Technology Delhi, New Delhi, India
Thesis:“Synthesis and characterization of magnetic nanoparticles”
PhD Studies
Nov 2010–Feb 2014
PhD in Physics
under supervision of Prof. Dr. B. Hillebrands
Department of Physics, TU Kaiserslautern, Germany
Research
May 2008–Jul 2008
Visiting Student with Dr. Reji Philip
Light & Matter Physics, Raman Research Institute, Bangalore
Project:“Synthesis and optical characterization of
magnetic nanoparticles”
Sep 2009–Nov 2010
Project Assistant with Prof. A. Barman
S N Bose National Centre for Basic Sciences, Kolkata
India-EU Collaborative Program
Jan 2012–Mar 2012
Visiting Student with Prof. Y. Otani
Advanced Science Institute, RIKEN, Wako-shi, Japan
Project: “RF modulation of spin currents in non-local spin valve”
Scholarship
Nov 2010–Nov 2013
124
Graduate School of Excellence Materials Science in Mainz (MAINZ)