Nonequilibrium Electrons and Phonons in Superconductors (Selected Topics in Superconductivity)

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This book introduces the main concepts of nonequilibrium phenomena in superconductors. The authors cover both experimentally well-understood topics and problems which physicists could challenge more in view of current theoretical understanding. Some of these topics include thermoelectric phenomena, influence of laser radiation as well as fluctuations in superconductors.

Author(s): Armen M. Gulian, Gely F. Zharkov
Series: Selected Topics in Superconductivity
Edition: 1
Publisher: Springer
Year: 1999

Language: English
Pages: 413
Tags: Физика;Электродинамика / Электричество и магнетизм;

NONEQUILIBRIUM ELECTRONS AND PHONONS IN SUPERCONDUCTORS......Page 1
Springerlink......Page 0
Half-title......Page 2
Selected Topics in Superconductivity......Page 3
Title Page......Page 4
Copyright Page......Page 5
Foreword......Page 6
Preface......Page 8
Contents......Page 12
1.1.1. Infinite Conductivity......Page 21
1.1.3. Energy Gap......Page 22
1.1.4. Analogy with Relativistic Quantum Theory......Page 23
1.1.5. Andreev Reflection......Page 25
1.1.6. Electron Density of States......Page 27
1.2. PHENOMENOLOGICAL GINZBURG–LANDAU THEORY......Page 28
1.2.1. Free Energy Functional......Page 29
1.2.2. London Penetration Depth......Page 32
1.2.3. Coherence Length......Page 33
1.2.4. Sign of Surface Energy......Page 34
1.2.5. Superheating in a Magnetic Field......Page 36
1.2.6. Flux Quantization......Page 39
1.3. BCS–GOR'KOV THEORY......Page 40
1.3.1. Equations for Ψ Operators......Page 41
1.3.2. Off-Diagonal Long-Range Order......Page 42
1.3.3. Spin–Singlet Pairing......Page 43
1.3.4. Solutions in Momentum Representation......Page 44
1.3.5. Self-consistency Equation......Page 45
1.3.7. Gauge Invariance......Page 46
1.3.8. Description at Finite Temperatures......Page 47
1.3.9. Weak-Coupling Ratio 2Δ(T=0)/T c......Page 48
1.4. SELF-CONSISTENT PAIR FIELD: MICROSCOPIC DERIVATION OF GINZBURG–LANDAU EQUATIONS......Page 49
1.4.1. Iterated Equations......Page 50
1.4.2. Magnetic Field Inclusion......Page 51
1.4.3. Slow Variation Hypothesis......Page 52
1.4.4. Evaluation of Phenomenological Parameters......Page 53
1.4.5. Failure of the "Quantum-Mechanical Generalization" for Time-Dependent Problems......Page 55
REFERENCES......Page 56
2.1.1. Magnetic and Nonmagnetic Impurities......Page 59
2.1.2. Diagram Expansion and Spatial Averaging for Normal Metals......Page 60
2.1.3. Born's Approximation......Page 62
2.1.4. Equations for a Superconducting State......Page 65
2.2. MAGNETIC IMPURITIES......Page 67
2.2.1. Averaging over Spin Directions......Page 68
2.2.3. Depression of Transition Temperature......Page 69
2.2.4. Energy Gap Suppression......Page 72
2.2.5. Gapless Superconductivity......Page 73
2.3.2. Equations on an Imaginary Axis......Page 75
2.3.3. Analytical Continuation Procedure......Page 77
2.3.4. Anomalous Propagators and Dyson Equations......Page 78
2.3.5. Regular Terms......Page 81
2.3.6. Nonlocal Kernels......Page 83
REFERENCES......Page 84
3.1.1. Fröhlich's Hamiltonian......Page 87
3.1.2. Migdal Diagram Expansion......Page 89
3.1.3. Eliashberg Equations in Weak-Coupling Limit......Page 90
3.1.4. Comparison with the BCS–Gor'kov Model......Page 91
3.2.1. Phonon Heat-Bath: Applicability......Page 92
3.2.2.1. Analytical Continuation: Causal Propagators......Page 93
3.2.3. Phonon Heat Bath: Consequences......Page 94
3.2.4. Analytical Continuation: Anomalous Functions......Page 96
3.2.5. Complete Set of Equations......Page 98
3.2.6. Keldysh Technique Approach......Page 99
3.3.1. Eilenberger Propagators......Page 100
3.3.2. Eliashberg Kinetic Equations......Page 101
3.3.4. Gauge Transformation......Page 103
3.3.5. Electron and Hole Distribution Functions......Page 105
3.3.6. Kinetic Equations: Keldysh Option......Page 106
3.3.7. Expressions for Charge and Current......Page 107
REFERENCES......Page 108
4.1.1. Spatially Homogeneous States......Page 111
4.1.3. Nondiagonal Channel......Page 112
4.1.4. Impurities......Page 113
4.1.5. Effective Collision Integral......Page 114
4.2.1. Diagram Evaluation of Electron–Electron Self-Energy......Page 115
4.2.2. Analytical Continuation......Page 116
4.2.3. Transition to Energy-Integrated Propagators......Page 117
4.2.4. Derivation of the Canonical Form......Page 118
4.3.1. Application of Keldysh Technique......Page 122
4.3.2. Quasi-classical Approximation......Page 125
4.3.3. Phonon Distribution Function......Page 126
4.3.4. Polarization Operators in Keldysh's Technique......Page 127
4.3.5. Polarization Operators: Analytical Continuation Technique......Page 128
4.3.6. Equivalence of Keldysh and Eliashberg Approaches......Page 131
4.3.7. Transition to Energy-Integrated Propagators......Page 132
4.4.1. Electron–Phonon Self-Energy Parts......Page 133
4.4.2. Canonical Form for Electron–Phonon Collisions......Page 135
REFERENCES......Page 137
5.1.1. Estimate for Magnetic Field Depairing Effect......Page 139
5.1.2. Single-Quantum Transitions......Page 140
5.1.3. Excitation Source in Normal Metals......Page 141
5.1.4. "Dirty" Superconductors......Page 142
5.2.1. Nonequilibrium Self-Consistency Equation......Page 143
5.2.3. Solution for Distribution Function at T ≈ T c......Page 144
5.2.4. Enhancement of the Gap......Page 145
5.3. PHOTON–ELECTRON INTERACTIONS......Page 146
5.3.1. Quantum Description......Page 147
5.3.2. Collision Integral as a Nonequilibrium Single-Electron Source......Page 149
5.3.3. Classical Field Action in a "Dirty" Limit......Page 150
5.4.2. "Switching on" of Eliashberg Mechanism......Page 151
5.4.3. Multiparticle Channels of Photon Absorption......Page 152
REFERENCES......Page 155
6.1.1. Polarization Operators......Page 157
6.1.2. Consequences of Equilibrium Phonon Distribution......Page 158
6.2.1. Electron Distribution Function......Page 159
6.2.3. Phonon Heat-Bath Realization......Page 160
6.2.4. Induced and Spontaneous Processes......Page 161
6.2.5. Properties of the Recombination Channel......Page 162
6.3. VIOLATION OF DETAILED BALANCE......Page 166
6.3.1. Phonon Deficit and Order Parameter Enhancement......Page 167
6.3.2. Comparison with Alternative Approaches......Page 168
REFERENCES......Page 169
7.1. ORDER PARAMETER, ELECTRON EXCITATIONS, AND PHONONS......Page 171
7.1.2. Normalization Condition......Page 172
7.1.3. Definition of Order Parameter......Page 173
7.1.4. Nondiagonal Collision Channel......Page 174
7.1.6. Charge Density......Page 175
7.1.7. Gap-Control Term......Page 176
7.1.9. Determination of the f1-Function......Page 177
7.1.10. Determination of the f2-Function......Page 179
7.1.12. Contribution of Nonequilibrium Phonons......Page 180
7.1.14. Charge Density and Invariant Potential......Page 181
7.1.15. Phonons and Order Parameter Dynamics......Page 182
7.2.1. Usadel Approximation......Page 185
7.2.2. Current Components in the Vicinity of T c......Page 190
7.2.4. Interference Current in Complete Form......Page 192
7.2.5. Full Set of Equations......Page 193
7.2.6. Boundary Conditions......Page 194
7.3.1. Abrikosov Vortices......Page 195
7.3.3. Low-Velocity Approximation......Page 196
7.3.4. Linearized Equations......Page 198
7.3.5. Effective Conductivity: Results......Page 199
7.4.1. Ginzburg's Number......Page 200
7.4.2. Paraconductivity......Page 203
7.4.3. Aslamasov–Larkin Mechanism......Page 205
7.4.4. Maki–Thompson Mechanism......Page 207
REFERENCES......Page 209
8.1.1. Tinkham Expression for the Gauge-Invariant Potential......Page 215
8.1.2. Normal Metal—Superconductor Interface......Page 216
8.1.3. New Characteristic Length in Superconductors......Page 218
8.2.1. Damping of Collective Oscillations......Page 220
8.2.2. Dispersion of the Charge-Imbalance Mode......Page 221
8.3.1. Collisionless Dynamics for Spatially Homogeneous Modes......Page 222
8.3.2. Dispersion Equation......Page 224
8.3.3. Stability Analysis for Particle-Hole Symmetry......Page 226
8.3.4. Instability at Branch Imbalance......Page 228
REFERENCES......Page 230
9.1.1. Phase Slippage......Page 233
9.1.2. Initial Dimensionless Equations......Page 235
9.1.3. Boundary Conditions......Page 238
9.2.1. Nonsingular Representation......Page 239
9.2.2. Matrix Representation for "Sweeping" Method......Page 240
9.2.3. Recurrence Relations......Page 241
9.2.4. Solution Procedure......Page 242
9.3. ANALYSIS OF RESULTS......Page 243
9.3.1. Single Active Center......Page 244
9.3.2. Oscillation Frequency......Page 247
9.3.3. Temporal Behavior of the Phase Difference......Page 249
9.3.6. Two Active Centers......Page 251
9.3.7. Current-Voltage Relations: Galayko Model......Page 253
9.3.8. Shortcomings of the TDGL in the Absence of Relaxation......Page 260
9.3.9. More Features of Numeric Solutions......Page 263
9.3.10. Role of Interference Current Component......Page 264
9.4.1. Generation of Electromagnetic Radiation......Page 268
REFERENCES......Page 272
10.1.1. Josephson Effect......Page 275
10.1.3. Derivation of Excitation Source......Page 276
10.1.4. Expression for Tunnel Current......Page 278
10.1.6. Complete Set of Equations......Page 279
10.2.1. Clark's Branch Imbalance......Page 280
10.2.2. Oscillations of the Gauge-Invariant Potential......Page 281
10.2.3. Satellites in Scattered Radiation......Page 283
10.3.1. Analytic Solution with a Branch Imbalance......Page 294
10.3.2. Inclusion of Self-Consistency Equation......Page 295
10.3.3. Analysis of Numerical Solutions: Subthreshold Voltages......Page 296
10.3.4. Analysis of Numerical Solutions: Superthreshold Voltages......Page 297
10.4.1. Phonon Deficit in a Subthreshold Regime......Page 298
10.4.2. Superthreshold Regime: Preconditions of Deficit......Page 299
10.4.3. Microrefrigeration......Page 302
10.5.1. Modified Aslamasov–Larkin Model......Page 305
10.5.2. Cos φ-Term Paradox......Page 306
10.5.3. Excess Current......Page 309
REFERENCES......Page 310
11.1.1. Spectral Function of Electron–Phonon Interaction......Page 313
11.1.3. Separation of "Coherent" Contributions......Page 314
11.1.4. Analytic Solution for Δ = 0......Page 315
11.2.1. First-Order "Coherent" Correction......Page 316
11.2.3. Multiple-Order-Parameter Solutions......Page 317
11.2.4. Stability of Solutions......Page 318
11.3.1. Inclusion of Thermal Phonons......Page 320
11.3.2. τ-Approximation......Page 321
11.3.4. Solution for Δ = 0......Page 322
11.3.6. Two Branches of a Nonzero-Order Parameter......Page 323
11.4.1. Stationary Solutions for Time-Dependent Problems......Page 324
11.4.2. Local Stability Against Space-Time Fluctuations......Page 325
11.4.3. Coexistence of Normal and Superconducting States......Page 327
11.5.1. Equilibrium Diamagnetic Response......Page 328
11.5.3. Role of Boundary Conditions......Page 329
11.5.4. Superheated States at External Pumping......Page 330
11.6.1. Finite Curvature of the Fermi Surface......Page 331
11.6.2. Photoinduced Potential and Owen–Scalapino μ* Model......Page 332
REFERENCES......Page 335
12.1. "WIDE" PUMPING SOURCE......Page 337
12.1.1. Elesin Theorem......Page 338
12.1.2. "Two-Peak" Approximation for α²(ω)F(ω)......Page 340
12.1.3. Numerical Analysis for Realistic Spectral Function......Page 341
12.2.1. Analytic Solution for Resonant Pumping Case......Page 345
12.2.2. Tunnel Injection of Electrons......Page 348
12.2.3. Simplifications for "Narrow" Distributions......Page 349
12.2.4. Analytic Solution for Symmetric Junctions......Page 350
12.2.5. Injection from Bulk Sample to a Thin Film......Page 351
12.3.1. Decoupling of Electron–Phonon Kinetics......Page 355
12.3.2. Phonon Absorption and Inverse Population......Page 356
12.3.3. Phonon Field Amplification in "Narrow" Electron Distributions......Page 357
12.3.4. Stability Against Order-Parameter Fluctuations......Page 359
12.3.5. Fluctuations of Superfluid Velocity......Page 360
12.3.6. Estimated Gain......Page 361
12.4.1. Two Channels of Electron–Photon Interaction......Page 362
12.4.2. Photons Versus Phonons......Page 363
12.4.4. Experimental Feasibility......Page 364
REFERENCES......Page 366
13.1.1. Thermopower of Normal Fermi Liquids......Page 369
13.1.2. Response of a Superconductor's Normal Component......Page 370
13.2. THERMOELECTRIC FLUX IN A SUPERCONDUCTING RING......Page 372
13.2.1. Meissner Effect and Incomplete Cancellation of Thermoelectricity......Page 373
13.2.2. "Gigantic" Flux Puzzle......Page 374
13.3.2. Spatially Inhomogeneous Kinetic Equation......Page 375
13.3.3. Influence of Heat Flux......Page 377
13.3.5. Calculation of Branch Imbalance Potential......Page 378
13.3.6. Branch Imbalance and Thermopower......Page 379
13.3.7. Thermopower in Optical Pumping......Page 380
REFERENCES......Page 381
14.1.1. Bean–Livingston Barrier......Page 385
14.1.2. Hollow Superconducting Cylinder......Page 386
14.1.3. General Consideration of Gibbs Free Energy......Page 388
14.1.4. In-Plate Penetration......Page 390
14.1.5. Penetration into a Hollow Cylinder......Page 393
14.2. VORTEX ORIGINATION BY THERMOELECTRIC CURRENT......Page 396
14.2.1. Free Energy Barrier......Page 397
14.3. VORTEX–ANTIVORTEX PAIR GENERATION......Page 399
14.3.1. Two-Vortice Free Energy......Page 400
14.3.2. Vortex–Antivortex Separation......Page 402
14.3.3. Threshold Temperature of Separation......Page 404
14.3.4. Comparison with Experiment and Discussion......Page 405
REFERENCES......Page 407
Index......Page 411