Spin Waves (eBook)

Spin Waves 2
Preface 6
Acknowledgments 8
Contents 9
Introduction to Magnetism 14
Magnetic Properties of Materials 14
Diamagnetism 16
Paramagnetism 16
Ferromagnetism 16
Ferrimagnetism and Antiferromagnetism 17
Spinning Top 18
Magnetism 21
Equation of Motion 21
Gyromagnetic Ratio 23
Angular Momentum in Quantum Mechanics 25
Basic Postulates of Quantum Mechanics 26
Eigenvalue Equations 27
Angular Momentum 27
Addition of Angular Momenta 33
Magnetic Moments of Atoms and Ions 36
Construction of Ground States of Atoms and Ions 36
Elements Important to Magnetism 41
Problems 41
References 44
Quantum Theory of Spin Waves 45
Charged Particle in an Electromagnetic Field 45
Zeeman Energy 48
Larmor Precession 50
Origins of Exchange: The Heisenberg Hamiltonian 51
Spin Wave on a Linear Ferromagnetic Chain 58
Harmonic Oscillator 62
Harmonic Oscillator Eigenfunctions 62
Raising and Lowering Operators 64
Magnons in a 3D Ferromagnet: Method of Holstein and Primakoff 67
Magnon Dispersion Relation 67
Magnon Interactions 72
Problems 76
References 77
Magnetic Susceptibilities 79
Diamagnetism 79
Paramagnetism 82
Weiss Theory of Ferromagnetism 85
Néel Theory of Ferrimagnetism 88
Exchange Field 93
Uniform Magnetization 94
Non-uniform Magnetization 95
Magnetocrystalline Anisotropy 96
Uniaxial Anisotropy 96
Cubic Anisotropy 98
Coordinate Transformations 99
Polder Susceptibility Tensor 103
Equation of Motion for the Magnetization 103
Susceptibility Without Exchange or Anisotropy 103
Susceptibility with Exchange and Anisotropy 105
Magnetic Damping 106
Magnetic Switching 114
Stoner--Wohlfarth Particle 114
Damped Precession 116
Problems 118
References 120
Electromagnetic Waves in Anisotropic-Dispersive Media 123
Maxwell's Equations 123
Constitutive Relations 124
Instantaneous Poynting Theorem 126
Complex Poynting Theorem 128
Energy Densities in Lossless Dispersive Media 129
Wave Equations 131
Polarization of the Electromagnetic Fields 134
Group and Energy Velocities 136
Plane Waves in a Magnetized Ferrite 139
Propagation Parallel to the Applied Field 140
Propagation Perpendicular to the Applied Field 142
The Magnetostatic Approximation 144
Problems 146
References 149
Magnetostatic Modes 150
Walker's Equation 150
Spin Waves 152
Uniform Precession Modes 155
Normally Magnetized Ferrite Film 155
Tangentially Magnetized Ferrite Film 156
Ferrite Sphere 157
Normally Magnetized Film: Forward Volume Waves 162
Tangentially Magnetized Film: Backward Volume Waves 169
Tangentially Magnetized Film: Surface Waves 173
Problems 177
References 178
Propagation Characteristics and Excitation of Dipolar Spin Waves 180
Energy Velocities for Dipolar Spin Waves 180
Propagation Loss 182
Relaxation Time for Propagating Modes 182
Surface Waves 184
Volume Waves 185
Summary of the Phenomenological Loss Theory 187
Mode Orthogonality and Normalization 189
Forward Volume Waves 189
Backward Volume Waves 191
Surface Waves 193
Excitation of Dipolar Spin Waves 194
Common Excitation Structures 194
Forward Volume Waves 199
Backward Volume Waves 205
Surface Waves 206
Discussion of Excitation Calculations 208
Problems 210
References 212
Variational Formulation for Magnetostatic Modes 214
General Problem Statement 214
Calculus of Variations 215
Formulation for One Independent Variable 215
Extensions to Three Independent Variables 217
Small-Signal Functional for Ferrites 219
Interpretation of the Functional 221
Stationary Formulas 223
Stationary Formula Examples with Forward Volume Waves 225
Large k-limit 226
Improved Approximation 227
Effect of Medium Inhomogeneity 229
Finite Element Analysis 229
Problems 230
References 232
Optical-Spin Wave Interactions 234
Symmetric Dielectric Waveguides 235
TE Modes 235
TM Modes 238
Optical Mode Orthogonality and Normalization 239
Magneto-Optical Interactions 242
Can You Tell the Difference Between and ? 242
Definition of Magnetization at High Frequencies 245
Symmetry Requirements on the Permittivity 246
Coupled-Mode Theory 247
Coupled-Mode Equations 248
Energy Conservation 249
Solutions to the Coupled-Mode Equations 250
Scattering of Optical-Guided Modes by Forward VolumeSpin Waves 252
Coupled-Mode Equations 252
Coupling Coefficients 256
Tightly Bound Optical Mode Approximation 261
Cotton--Mouton Effect 263
Anisotropic Bragg Diffraction 264
Problems 267
References 271
Nonlinear Interactions 273
Large-Amplitude Spin Waves 273
Foldover and Bistability 277
Hamiltonian Equations of Motion 281
Spin Wave Interactions 283
Decay Instability 290
H(2) Coefficients 292
Nonlinear Schrödinger Equation 294
Modulational Instability and Solitons 295
Split-Step Fourier Method 297
Anomalous Dispersion 299
Other Aspects 301
Routes to Chaos 303
Center Manifold Theory 303
Quantizing Low-Dimensional Chaos 306
Problems 312
References 315
Novel Applications 319
Nano-Contact Spin-Wave Excitations 319
Current-Induced Spin Torque 320
Magnetic Precession 325
Magnetic Precession in Patterned Structures 332
Inverse Doppler Effect in Backward Volume Waves 335
Problems 339
References 340
Appendix A Properties of Yttrium–Iron–Garnet (YIG) 343
References 344
Appendix B Currents in Quantum Mechanics 345
Density of States 345
Electric and Spin Current Densities 347
Reflection and Transmission at a Boundary 348
Tunneling Through a Barrier 349
References 351
Appendix C Characteristics of Spin Wave Modes 352
Constitutive Tensors 352
Polder Susceptibility Tensor 352
Permeability Tensor 353
Uniform Precession Mode Frequencies 353
Spin Wave Resonance Frequencies 353
General Magnetostatic Field Relations 353
Forward Volume Spin Waves 354
Backward Volume Spin Waves 355
Surface Spin Waves 356
Appendix D Mathematical Relations 358
Trigonometric Identities 358
Vector Identities and Definitions 358
Fourier Transform Definitions 359
Index 360