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Home , Josephson effect

Laurent-Patrick Levy

Magnetism and

Translated by Stephen Lyle With 129 Figures and 13 Tables

Springer Contents

Part I. Magnetism

1. and Induction. Thermodynamics 3 1.1 Magnetostatics 3 1.2 Demagnetising Factors 7 1.3 The Reciprocity Theorem 9 1.4 Magnetic Energy and Work 11 1.5 Thermodynamic Potentials 12 1.6 Magnetocaloric Effects 19 1.7 Einstein-de Haas Effect 20

2. Orbital and Spin Magnetism Without Interactions 23 2.1 Electrons in a Magnetic Field. The Hamiltonian 23 2.2 Bohm-Aharonov Effect and Permanent Currents 26 2.3 Landau Levels 30 2.4 The Quantum 34 2.5 Orbital Magnetism and Landau 39 2.6 The Haas-van Alphen Effect 41 2.7 Pauli of Conduction Electrons 43 2.8 The Zeeman Effect 44 2.9 The Spin-Orbit Interaction 47 2.10 Thermodynamics of Localised Spins 48 2.11 Van Vleck Paramagnetism 50 2.12 The Ionic Anisotropy Field 52 2.13 Crystalline Field 53 2.14 Jahn-Teller Distortion 56 2.15 Disappearance of Orbital Angular Momentum in Transition Ions 58

3. Exchange Interactions 61 3.1 Direct Exchange Between Spins 61 3.2 Atoms with Two Valence Electrons 62 3.3 The Hund Rules 67 VIII Contents

3.4 The Hydrogen Molecule 68 3.5 Effective Exchange Hamiltonian for a 72 3.6 Hartree-Fock Methods 74 3.7 Superexchange 77 3.8 The Energy of Magnetic States 79

4. Phase Transitions 83 4.1 Second Order Transitions 83 4.2 Correlation Length, Fluctuations and the Ornstein-Zernike Theory 90 4.3 Critical Exponents and Renormalisation of Fluctuations 93 4.4 Scaling Laws 98

5. Mean Field 101 5.1 Molecular Field 101 5.2 Susceptibility and Spontaneous Magnetisation 106 5.3 Reaction Field 108 5.4 110 5.5 Exotic Magnets 113

6. The Ising Model 117 6.1 One-Dimensional Partition Function 118 6.2 Zero Field Solution 118 6.3 Transfer Matrices 119 6.4 Correlation Functions 121 6.5 Glauber Dynamics 127 6.6 Critical Slowing Down 129 6.7 Lattice 130

7. The XY Model 133 7.1 The Spin 1/2 XY Chain 135 7.2 The Classical One-Dimensional XY Model 139 7.3 Kosterlitz-Thouless Transition in the Two-Dimensional XY Model 141

8. Linear Response 147 8.1 Isothermal Response 150 8.2 Adiabatic Response and Fluctuation-Dissipation Theorem . . 152 8.3 Comparing Isothermal and Adiabatic Susceptibilities 159 8.4 Properties of Response Functions 161 8.4.1 Kubo Formulas 161 8.4.2 Kramers-Kronig Relations 162 8.4.3 Time Reversal Symmetries 164 8.4.4 Non-Linear Response 165 Contents IX 8.5 Applications to Resonance Phenomena 166 8.5.1 Bloch Equations 166 8.5.2 Relaxation Between Electron and Nuclear Spins 170 8.5.3 Dipolar Relaxation 171 8.5.4 Exchange Narrowing 172 8.5.5 Ferromagnetic Resonance 173 8.5.6 Ferrimagnetic and Antiferromagnetic Resonance 176 9. Spin Waves 179 9.1 Spin Hydrodynamics 179 9.2 Elementary Excitations and 184 9.3 Dipolar Interactions and Magnetostatic Modes 192 9.4 Thermodynamics 195 9.5 Non-Linear Terms 197 9.6 Spin-Wave Spectroscopy 198 9.7 Parallel Pumping 199 9.8 Other Representations 200 9.9 Hydrodynamic Methods 201 10. Quantum Spin Chains 205 10.1 One-Dimensional Planar XY Systems 208 10.1.1 Field Theory. The Sine-Gordon Equation 208 10.1.2 XY Phase Correlation Functions. Instantons 214 10.2 Some Theorems 216 10.2.1 The Lieb-Schultz-Mattis Theorem 216 10.2.2 Marshall's Theorems 217 10.3 Valence Bond States 218 10.3.1 Solid Valence Bond States 221 10.4 The Non-Linear a Model for Antiferromagnetic Chains 226 11. Itinerant Magnetism 235 11.1 The Kohn Singularity 239 11.2 The Stoner Model. Magnons in Metals . 242 11.3 The Excitonic . Spin Density Waves 248 11.4 The Hubbard Model 252 11.4.1 Electron Correlations 256 11.4.2 Destruction of Antiferromagnetism by Holes 258 11.4.3 High Temperature Superconductivity 259 Contents

Part II. Superconductivity

12. Macroscopic Aspects of Superconductivity 263 12.1 Four Phenomena 263 12.2 Electron- Interactions 267 12.3 The Two-Fluid Model 270 12.4 The London Equations 271 12.5 London and Pippard Superconductors 272 12.6 Thermodynamics of the Transition 274 12.7 The Intermediate State 276 12.8 Critical Current in a Superconducting Wire 280 12.9 Two Types of Superconductor 283 13. Ginzburg—Landau Theory 285 13.1 Ginzburg-Landau Free Energy 287 13.2 The Ginzburg-Landau Equations 291 13.3 Flux Quantisation 295 13.4 The Little-Parks Effect 299 13.5 Critical Current in a 300 13.6 Energy of an Interface Between Normal and Superconducting States 301 13.7 Linearised Ginzburg-Landau Equations 303 13.8 Nucleation of Superconductivity at HC2 303 13.9 Surface Nucleation Hc3 305 14. The BCS Theory of Superconductivity 309 14.1 The Electron-Phonon Interaction 309 14.2 The BCS Hamiltonian 311 14.3 Mean Field Approximation and Diagonalisation of the BCS Hamiltonian 314 14.4 BCS and Coherent States 319 14.5 Finite Temperatures 322 14.6 Thermodynamic Properties 324 14.7 The Tunnel Effect 324 14.8 Nuclear Relaxation and Ultrasound Absorption 327 14.9 Electromagnetic Screening 331 14.10 Conclusion 336

15. Vortices in Type II Superconductors 337 15.1 Isolated Vortices 337 15.2 Interactions Between Vortices 340 15.3 The Lattice 342 15.4 Magnetisation Curves 344 15.5 Potential of a Vortex Near a Surface 344 Contents XI 15.6 Dissipation by Vortex Flow 346 15.7 Observing Vortex Motion 349 15.8 Vortex Pinning 350 15.9 Strong Pinning Limit. The Bean Model 351 15.10 Thermal Transport of Flux Lines 352 16. The Josephson Effect and Quantum Interferometers 355 16.1 Quasi-Particle Tunnelling Current 355 16.2 The Josephson Effect 358 16.3 Microscopic Origins of the Josephson Effect 361 16.4 The Josephson Effect in a Magnetic Field 363 16.5 The AC Josephson Effect 366 16.6 Quantum Interferometers. The AC SQUID 369 16.7 Flux Transformers 373 16.8 Mechanical Analogy for the AC SQUID 373 16.9 The DC SQUID 376 16.10 Electromagnetic Waves 378 17. Inhomogeneous Superconductivity 381 17.1 The Bogoliubov-de Gennes Equations 381 17.2 Semi-Classical Approximation 384 17.3 Vortex Core States 386 17.4 Interface Between a Superconductor and a Normal Metal. 390 17.5 S-N-S Junctions and Andreev States 397 17.6 Electromagnetic Properties of Proximity Structures 399 17.7 Gapless Superconductivity 400 17.8 Collective Modes 402 17.9 Conclusions and Perspectives 404

A. Representations of Continuous and Point Groups 407 A.I General Notions 407 A.2 Representations of the Rotation Group 410 A.3 Point Groups 412 A.4 Representations of Point Groups 416 A.5 Application to the Crystalline Field 416 B. Second Quantisation 421 B.I State Space for N Free Fermions 421 B.2 Other Representations 422 B.3 Representation of Operators in Second Quantisation 424 B.4 Perturbation Theory 426 XII Contents C. Exercises 429 C.I Magnetism 429 C.I.I Magnetic Textures 429 C.I.2 Long Range Exchange Interaction Between Spin 1/2 Particles and the Mean Field Approximation 430 C.1.3 Spin-Flop Transition for an Antiferromagnet 431 C.I.4 Spin Waves and Quantum Fluctuations in Antiferromagnets 433 C.1.5 Order by Disorder: Ground State Selection Processes in Antiferromagnets Frustrated by Thermal Fluctuations 435 C.2 Superconductivity 438 C.2.1 Superconductors Under Pressure 438 C.2.2 Superconductor Arrays 439 C.2.3 Superconductivity in Inhomogeneous Magnetic Field . 440 C.2.4 Intermediate State of a Type I Superconductor 441

Bibliography 443 Index 461