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Volume 969

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More information about this series at http://www.springer.com/series/5304 Sergio M. Rezende

Fundamentals of Magnonics Sergio M. Rezende Departamento de Fisica Universidade Federal de Pernambuco Recife, Brazil

ISSN 0075-8450 ISSN 1616-6361 (electronic) Lecture Notes in Physics ISBN 978-3-030-41316-3 ISBN 978-3-030-41317-0 (eBook) https://doi.org/10.1007/978-3-030-41317-0

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This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland To Adélia Preface

Magnetism and magnetic materials constitute one of the oldest fields of science and technology that keeps renewing itself with new discoveries and unique technological applications. Centuries before the time of Christ, ancient civilizations studied the wondrous properties of magnetite, the famed loadstone, and used it for orientation in the Earth’s magnetic field. The magnetic compass became an essential instrument of navigation for the Chinese in ancient times and for the Europeans in the Late Middle Ages. This application motivated one of the oldest books in experimental physics, De Magnete, written by William Gilbert and published in 1600. After the discoveries of the fundamental laws of electromagnetism in the nineteenth century by Ampére, Oersted, Faraday, and Henry, magnetic materials became essential for the fabrication of generators, motors, and transformers, building blocks of electricity that revolutionized the costumes of humanity. Then, in the twentieth century, they made possible the invention of loudspeakers, phones, relays, and other magnetic devices essential for telegraphy and telephony, as well as the development of magnetic recording for audio, video, and digital information. With the understanding of the atomic origin of magnetism of matter, made possible by the formulation of quantum mechanics in the beginning of the twentieth century, new magnetic phenomena were proposed theoretically or discovered experimentally, and new applications were developed. In 1930, Felix Bloch showed that the low-lying excitations of the spin system consisted of nonlocalized, collective spin deviations, which he named spin waves, whose quanta are called magnons. Few decades later, spin waves were observed experimentally and became the subject of intense research. Phenomena involving the k ¼ 0 magnon, or ferromagnetic resonance, laid the groundwork for novel applications in microwave ferrite devices used in telecommunications, in radar systems, and in a variety of industrial equipment. Despite this long history of scientific activity and technological applications, recently Professor Chia-Ling Chien of the University of Baltimore stated that in the last 30 years, activity in magnetism is facing its Golden Age. The recent vigorous impulse to magnetism was provided by spin-based phenomena that occur in nanoscale magnetic structures, such as the giant magnetoresistance (GMR) discovered in 1989 by Albert Fert and Peter Grünberg, Physics Nobel Prize winners in 2007. The discovery of GMR and other spin-dependent phenomena led to new sensing devices that boosted the capacity of storage media and made possible

vii viii Preface new storage devices such as the magnetic random-access memory. These and other developments led to the new field of spintronics, devoted to the investigation of basic phenomena and their application in devices for transport, storage, and processing of information, in which the main physical entity is the electron spin. The subfield of spintronics in which the phenomena are based on the properties of magnons is called magnonics, or magnon spintronics, which is the subject of this book. The unique properties of magnons in magnetic materials with very low damping, such as yttrium iron garnet (YIG), make them suitable for use as information carriers and logic processing without the need of electric current, overcoming an important fundamental limitation of conventional electronics: a power consumption which scales linearly with increasing number of individual processing elements.1,2 This book is intended to serve as a text for beginning engineering and physics graduate students in the areas of magnetism and spintronics. The level of presentation assumes only basic knowledge of the origin of magnetism, electromagnetism, and quantum mechanics. The book utilizes relatively simple mathematical derivations, aimed mainly at explaining the physical concepts involved in the phenomena studied and for the understanding of the experiments presented. We use in the book both SI and CGS units, because they are equally employed in research articles. Key topics include the basic phenomena of ferromagnetic resonance in bulk materials and in thin films, semi-classical theory of spin waves, quantum theory of spin waves and magnons, magnons in antiferromagnets, parametric excitation of magnons, magnon nonlinear dynamics and chaotic phenomena, Bose–Einstein condensation of magnons, and the very recent field of magnon spintronics. This breath of topics is not covered in any other single book. Also, no other textbook on magnetism has the material presented in the last three chapters. I would like to thank many collaborators with whom I worked in several magnonic phenomena presented in the book. In particular, I thank Frederick R. Morgenthaler who introduced me to spin wave phenomena over 50 years ago, during my Ph.D. program at MIT. I am also grateful to Nicim Zagury, Robert M. White, Vincent Jaccarino, Carlos A. dos Santos, Wido R. Schreiner, Stuart S. P. Parkin, and Doug L. Mills, with whom I had profitable collaborations for many years. I also thank my colleagues, students, and post-docs at UFPE, my coauthors in papers on magnonics and other fields, Cid B. de Araújo, Ivon P. Fittipaldi, Mauricio D. Coutinho-Filho, José Rios Leite, Jairo R. L. de Almeida, Sandra S. Vianna, Fernando L. A. Machado, Erivaldo Montarroyos, J. Marcílio Ferreira, Eduardo Fontana, Flávio M. de Aguiar, Osiel A. Bonfim, Antonio

1Serga, A. A., Chumak, A. V., Hillebrands, B.: YIG Magnonics. J. Phys. D Appl. Phys. 43, 264002 (2010). 2Chumak, A. V., Vasyuchka, V. I., Serga, A. A., Hillebrands, B.: Magnon spintronics. Nat. Phys. 11, 453 (2015). Preface ix

Azevedo, José R. Fermin, Carlos Chesman, Roberto L. Rodríguez-Suárez, Eduardo Padrón-Hernández, Joaquim B. S. Mendes, Rafael O. R. Cunha, Luis H. Vilela- Leão, Javier D. C. López Ortiz, Gabriel A. Fonseca Guerra, José Holanda, Daniel S. Maior, Matheus Gamino, Obed Alves Santos, Pablo R. T. Ribeiro, and Mercedes Arana. Last but not least, I am also grateful to Alberto P. Guimarães and Anderson S. L. Gomes for helpful suggestions about the book, to Pavel Kabos for a careful review of the manuscript, and to Sérgio Mascarenhas for his constant stimulus and support.

Recife, Brazil Sergio M. Rezende October 3, 2019 About the Book

Fundamentals of Magnonics is intended to serve as a text for beginning engineering and physics graduate students in the areas of magnetism and spintronics. The level of presentation assumes only basic knowledge of the origin of magnetism, electromagnetism, and quantum mechanics. The book utilizes relatively simple mathematical derivations, aimed mainly at explaining the physical concepts involved in the phenomena studied and for the understanding of the experiments presented. Key topics include the basic phenomena of ferromagnetic resonance in bulk materials and in thin ﬁlms, semiclassical theory of spin waves, quantum theory of spin waves and magnons, magnons in antiferromagnets, parametric excitation of magnons, magnon nonlinear dynamics and chaotic phenomena, Bose–Einstein condensation of magnons, and the very recent ﬁeld of magnon spintronics. This breadth of topics is not covered in any other single book. Also, no other textbook on magnetism has the material presented in the last three chapters. Also, end-of-chapter problems are included.

xi Contents

1 The Zero Wave Number Magnon: Ferromagnetic Resonance ...... 1 1.1 Brief Introduction to Magnetic Materials ...... 1 1.2 Magnetization Dynamics ...... 4 1.2.1 Free Precession in a Magnetic Field ...... 4 1.2.2 rf-Driven Magnetic Resonance ...... 7 1.3 Ferromagnetic Resonance ...... 10 1.3.1 FMR Frequency with Demagnetizing Effects ...... 10 1.3.2 Ferromagnetic Resonance with Damping ...... 12 1.4 General Equation for the Ferromagnetic Resonance Frequency . . . 16 1.4.1 Effective Fields and FMR Frequency ...... 16 1.4.2 Effect of Crystalline Anisotropy on the FMR Frequency . . 18 1.5 Ferromagnetic Resonance in Thin Films ...... 22 1.5.1 Experimental Measurement of Ferromagnetic Resonance ...... 22 1.5.2 FMR in Crystalline Films with Cubic Anisotropy ...... 24 1.5.3 FMR in Ferromagnetic Films with Exchange Bias ...... 25 References ...... 29 2 Spin Waves in Ferromagnets: Semiclassical Approach ...... 31 2.1 Spin Waves in a Linear Ferromagnetic Chain ...... 31 2.2 Macroscopic Treatment of Spin Waves in 3D ...... 35 2.2.1 Spin Waves with Exchange and Zeeman Energies ...... 36 2.2.2 Effect of the Volume Magnetic Dipolar Fields on Spin Waves ...... 38 2.3 Magnetoelastic Waves ...... 42 2.4 Energy and Momentum Conservation Relations for Magnetoelastic Waves ...... 49 2.5 Coupled Spin–Electromagnetic Waves: Magnetic Polariton ...... 52 2.6 Experimental Techniques to Study Long-Wavelength Spin Waves ...... 54 2.6.1 Microwave Techniques ...... 54 2.6.2 Brillouin Light Scattering Techniques ...... 62 References ...... 68

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3 Quantum Theory of Spin Waves: Magnons ...... 71 3.1 Ferromagnetic Magnons ...... 71 3.1.1 Bloch Spin Waves ...... 71 3.1.2 Magnon Operators ...... 74 3.2 Magnons with Anisotropy and Dipolar and Interactions ...... 80 3.3 Properties of Magnons: Coherent Magnon States ...... 87 3.3.1 Eigenstates of the Hamiltonian ...... 87 3.3.2 Thermal Magnons ...... 88 3.3.3 Coherent Magnon States ...... 90 3.4 Magnon Interactions and Relaxation Mechanisms ...... 93 3.4.1 Magnon Interactions ...... 93 3.4.2 Magnon Energy Renormalization ...... 96 3.4.3 Magnon Relaxation ...... 96 3.5 Magnon Hybrid Excitations with Phonons and Photons ...... 99 3.5.1 Quantization of Elastic Waves: Phonons ...... 99 3.5.2 Interacting Magnons and Phonons ...... 102 3.5.3 Eigenstates of the Magnon–phonon System ...... 103 3.5.4 Magnon–photon Hybrid Excitation ...... 106 3.6 Light Scattering by Magnons ...... 108 3.6.1 Magneto-Optical Interaction ...... 108 3.6.2 One-Magnon Light Scattering in Ferromagnets ...... 112 3.7 Magnons in Yttrium Iron Garnet ...... 115 3.7.1 Magnon Dispersion Relations in YIG ...... 116 3.7.2 Magnon Relaxation in YIG ...... 119 3.7.3 Thermal Properties of Magnons in YIG ...... 125 References ...... 132 4 Magnonics in Ferromagnetic Films ...... 135 4.1 Magnetostatic Spin Waves in Ferromagnetic Films ...... 135 4.1.1 Walker Equation ...... 136 4.1.2 Magnetostatic Waves in Unbounded Films Magnetized in the Plane ...... 137 4.1.3 Volume Magnetostatic Modes ...... 140 4.1.4 Surface Magnetostatic Modes ...... 142 4.1.5 Volume Magnetostatic Waves in Perpendicularly Magnetized Films ...... 145 4.1.6 Experiments with Magnetostatic Spin Waves ...... 146 4.2 Quantum Theory of Spin Waves in Thin Ferromagnetic Films . . . . 149 4.3 Spin Wave Relaxation in Ultrathin Films by Two-magnon Scattering ...... 153 4.3.1 Two-magnon Relaxation in Single Ferromagnetic Films . . . 154 4.3.2 Enhanced Two-magnon Relaxation in Exchange-biased Films ...... 160 4.4 Spin Waves in Coupled Magnetic Films ...... 169 4.4.1 Interlayer Coupling in Magnetic Trilayers ...... 169 4.4.2 Equilibrium Conﬁguration of the Magnetizations ...... 171 Contents xv

4.4.3 Derivation of the Spin Wave Dispersion Relation ...... 173 4.4.4 FMR and BLS Experiments in Fe/Cr/Fe Trilayers ...... 180 References ...... 184 5 Magnons in Antiferromagnets ...... 187 5.1 Antiferromagnetic Materials ...... 187 5.2 Antiferromagnetic Resonance: The k ¼ 0 Magnons ...... 191 5.2.1 Easy-Axis Antiferromagnets ...... 192 5.2.2 Easy-Plane Antiferromagnets ...... 195 5.3 Magnons in Easy-Axis Antiferromagnets ...... 198 5.3.1 Magnons in Easy-Axis Antiferromagnets: AF Phase ...... 198 5.3.2 Magnon Energy Renormalization ...... 207 5.3.3 Magnons in Easy-Axis Antiferromagnets: SF Phase ...... 210 5.3.4 Magnons in Easy-Axis Antiferromagnets: Canted Phase . . . 216 5.4 Magnons in Easy-Plane Antiferromagnets ...... 217 References ...... 221 6 Magnon Excitation and Nonlinear Dynamics ...... 223 6.1 Linear Excitation of Magnons ...... 223 6.1.1 Theory ...... 223 6.1.2 Experiments for Linear Excitation of Magnons ...... 228 6.2 Parametric Excitation of Magnons ...... 232 6.2.1 Parallel Pumping Process ...... 232 6.2.2 Perpendicular Pumping: First-Order Suhl Process— Subsidiary Absorption ...... 236 6.2.3 Second-Order Suhl Process: Premature Saturation of the FMR...... 239 6.2.4 Three-Magnon Coincidence Process ...... 241 6.3 Magnon Interactions and Their Effects on Parametric Magnons . . . 243 6.4 Nonlinear Dynamics of Magnons and Chaotic Behavior ...... 248 6.4.1 Nonlinear Equations of Motion ...... 249 6.4.2 Self-Oscillations in the Parametric Magnon Populations . . . 251 6.4.3 Period Doubling, Strange Attractors, and Chaos ...... 253 References ...... 257 7 Bose–Einstein Condensation of Magnons ...... 259 7.1 Experimental Observation of Magnon Condensation ...... 259 7.1.1 Bose–Einstein Condensation ...... 259 7.1.2 Experiments with BEC of Parametrically Driven Magnons . . 261 7.2 Theory for the Dynamics of the Microwave-Driven Magnon Gas in k-Space ...... 265 7.3 Quantum Coherence of the Bose–Einstein Magnon Condensate . . . 274 7.4 Wave Function of the Magnon Condensate ...... 278 References ...... 283 xvi Contents

8 Magnon Spintronics ...... 287 8.1 Spin Current in Nonmagnetic Metals ...... 287 8.2 Conversions Between Charge and Spin Currents ...... 294 8.2.1 Spin Hall Effects ...... 294 8.2.2 Rashba–Edelstein Effects ...... 296 8.3 Magnon Excitation by Spin Transfer Torque ...... 300 8.3.1 Spin Transfer Torque ...... 300 8.3.2 Magnon Excitation by Spin-Polarized DC Currents in Magnetic Nanostrutures ...... 302 8.4 Spin Pumping by k ¼ 0 Magnons in Magnetic Bilayers ...... 309 8.4.1 Spin Pumping Damping ...... 309 8.4.2 Electric Spin Pumping Effect ...... 316 8.4.3 Electric Spin Pumping Experiments with 3D Metallic Layers...... 319 8.4.4 Electric Spin Pumping Experiments with 2D Electron Systems ...... 322 8.5 Magnonic Spin Current, Magnon Accumulation, and Diffusion of Magnons ...... 323 8.6 Magnonic Spin Seebeck Effect ...... 331 8.6.1 Model for the Magnonic Spin Seebeck Effect in Ferromagnetic Insulators ...... 331 8.6.2 Magnonic Spin Seebeck Effect in Antiferromagnetic Insulators ...... 337 8.6.3 Comparison of Theory with Experimental Results for theSSEinFMI...... 339 8.7 Concluding Remarks and Perspectives for Magnon Spintronics and Related Fields ...... 342 References ...... 345

Index ...... 353 About the Author

Sergio Machado Rezende graduated in Electrical Engineering in Rio de Janeiro (1963) and received MSc (1965) and PhD (1967) degrees from the Massachusetts Institute of Technology, both in Electrical Engineering and Materials Science. He is one of the founders and first chairman of the Physics Department of the Federal University of Pernambuco (UFPE) (1972–1976), in Recife, where he is a Full Professor. He was twice a Visiting Professor at the University of California, Santa Barbara (1975–1976 and 1982–1984). He has published over 270 scientific papers in international journals on dynamic excitation phenomena in magnetic materials, magnetic nanostructures, magnonics and spintronics and has supervised over 40 MSc and PhD students. His scientific activities have never been interrupted by science managing positions he held, Dean of the Center for Exact Sciences of UFPE (1984–1988), Scientific Director of the Pernambuco Science Foundation (1990–1993), State Secretary for Science and Technology of Pernambuco (1995– 1998), President of FINEP, the main federal agency for funding of S&T in Brazil (2003–2005), and Minister for Science and Technology of Brazil (2005–2010), under President Luiz Inácio Lula da Silva. He is a member of several scientific societies, Honorary President of the Brazilian Society for the Advancement of Science (SBPC), and has received several prizes and awards in Brazil and other countries.

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