Gulian A.M., Zharkov G.F. Nonequilibrium Electrons and Phonons in Superconductors
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Contents
Chapter 1. Basic Equilibrium Properties
1.1. Introductory Concepts .......................
1.1.1. Infinite Conductivity .....................
1.1.2. Ideal Diamagnetism ....................
1.1.3. Energy Gap ........................
1.1.4. Analogy with Relativistic Quantum Theory . .......
1.1.5. Andreev Reflection ....................
1.1.6. Electron Density of States .................
1.1.7. Coherence Factors .....................
1.2. Phenomenological Ginzburg–Landau Theory ...........
1.2.1. Free Energy Functional ..................
1.2.2. London Penetration Depth .................
1.2.3. Coherence Length .....................
1.2.4. Sign of Surface Energy ..................
1.2.5. Superheating in a Magnetic Field .............
1.2.6. Flux Quantization .....................
1.3. BCS–Gor’kov Theory .......................
1.3.1. Equationsfor Operators. ...............
1.3.2. Off-Diagonal Long-Range Order .............
1.3.3. Spin-Singlet Pairing ....................
1.3.4. Solutions in Momentum Representation ........
1.3.5. Self-Consistency Equation ................
1.3.6.
Isotope Effect
.......................
1.3.7. Gauge Invariance .....................
1.3.8. Description at Finite Temperatures ............
1.3.9. Weak-CouplingRatio (T = 0)/T
C
...........
1.4. Self-Consistent Pair Field: Microscopic Derivation of
Ginzburg–Landau Equations ..............
1.4.1. Iterated Equations .....................
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1.4.2.
1.4.3.
1.4.4.
1.4.5.
Magnetic Field Inclusion .................
Slow Variation Hypothesis ................
Evaluation of Phenomenological Parameters ......
Failure of the “Quantum-Mechanical Generalization” for
Time-Dependent Problems ................
References ............................
Chapter 2. Dynamics of Gapless Superconductors
2.1. Scattering on Impurities .....................
2.1.1.
2.1.2.
2.1.3.
2.1.4.
2.1.5.
2.1.6.
Magnetic and Nonmagnetic Impurities .........
Diagram Expansion and Spatial Averaging for Normal
Metals ...........................
Born’s Approximation ..................
Equations for a Superconducting State ..........
Anderson’s Theorem ...................
“Londonization” by Elastic Scattering ..........
2.2. Magnetic Impurities ........................
2.2.1.
2.2.2.
2.2.3.
2.2.4.
2.2.5.
Averaging over Spin Directions .............
Spin-Flip Time .....................
Depression of Transition Temperature ..........
Energy Gap Suppression .................
Gapless Superconductivity ................
2.3. Nonstationary Ginzburg–Landau Equations ...........
2.3.1.
2.3.2.
Causality Principle and Nonlinear Problems .......
Equations on an Imaginary Axis ...........
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2.3.3.
2.3.4.
2.3.5.
2.3.6.
Analytical Continuation Procedure ...........
Anomalous Propagators and Dyson Equations .....
Regular Terms ......................
Nonlocal Kernels .....................
References .............................
Chapter 3. Nonequilibrium General Equations
3.1. Migdal–Eliashberg Phonon Model ................
3.1.1.
3.1.2.
3.1.3.
3.1.4.
Fröhlich’s Hamiltonian ..................
Migdal Diagram Expansion ...............
Eliashberg Equations in Weak-Coupling Limit .....
Comparison with the BCS–Gor’kov Model .......
3.2. Equations for Nonequilibrium Propagators ...........
3.1.1.
3.2.2.
3.2.3.
3.2.4.
Phonon Heat-Bath: Applicability .............
Expansion Over External Field Powers .........
Phonon Heat-Bath: Consequences ............
Analytical Continuation: Anomalous Functions .....
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CONTENTS xiii
3.2.5.
3.2.6.
Complete Set of Equations ................
Keldysh Technique Approach ...............
3.3. Quasi-Classical Approximation ..................
3.3.1.
3.3.2.
3.3.3.
3.3.4.
3.3.5.
3.3.6.
3.3.7.
Eilenberger Propagators ..................
Eliashberg Kinetic Equations ...............
Normalization Condition .................
Gauge Transformation ...................
Electron and Hole Distribution Functions .........
Kinetic Equations: Keldysh Option ............
Expressions for Charge and Current ............
References .............................
Chapter 4. Electron and Phonon Collision Integrals
4.1. Collision Integral Derivation ....................
4.1.1.
4.1.2.
4.1.3.
4.1.4.
4.1.5.
Spatially Homogeneous States ..............
Separation of Real and Virtual Processes .........
Nondiagonal Channel ...................
Impurities .........................
Effective Collision Integral ................
4.2. Inelastic Electron-Electron Collisions ...............
4.2.1.
4.2.2.
4.2.3.
4.2.4.
4.2.5.
Diagram Evaluation of Electron-Electron
Self-Energy ........................
Analytical Continuation ..................
Transition to Energy-Integrated Propagators .......
Derivation of the Canonical Form .............
Essence of Elementary Acts ................
4.3. Kinetic Equation for Phonons ...................
4.3.1.
4.3.2.
4.3.3.
4.3.4.
4.3.5.
4.3.6.
4.3.7.
Application of Keldysh Technique ............
Quasi-Classical Approximation ..............
Phonon Distribution Function ...............
Polarization Operators in Keldysh’s Technique ......
Polarization Operators: Analytical Continuation
Technique .........................
Equivalence of Keldysh and Eliashberg
Approaches ........................
Transition to Energy–Integrated Propagators .......
4.4. Inelastic Electron–Phonon Collisions ...............
4.4.1.
4.4.2.
4.4.3.
Electron–Phonon Self-Energy Parts ............
Canonical Form for Electron–Phonon Collisions .....
Canonical Form for Phonon–Electron Collisions .....
References .............................
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xiv CONTENTS
Chapter 5. Microwave-Enhanced Order Parameter
5.1. Source of Excitations .......................
5.1.1.
5.1.2.
5.1.3.
5.1.4.
Estimate for Magnetic Field Depairing Effect ......
Single-Quantum Transitions ...............
Excitation Source in Normal Metals ...........
“Dirty” Superconductors .................
5.2.
Stimulation Effect
.........................
5.2.1.
5.2.2.
5.2.3.
5.2.4.
Nonequilibrium Self-Consistency Equation .......
Relaxation-Time Approximation .............
Solution for Distribution Function at .......
Enhancement of the Gap .................
5.3. Photon–Electron Interactions ...................
5.3.1.
5.3.2.
5.3.3.
Quantum Description ...................
Collision Integral as a Nonequilibrium
Single-Electron Source ..................
Classical Field Action in a “Dirty” Limit ........
5.4. Enhancement in Perfect Crystals .................
5.4.1.
5.4.2.
5.4.3.
Quasi-particle Scattering and Kinematic Conservation
Laws ...........................
“Switching on” of Eliashberg Mechanism ........
Multiparticle Channels of Photon Absorption ......
References .............................
Chapter 6. Phonon-Deficit Effect
6.1. Collision Integral as a Phonon Source ..............
6.1.1.
6.1.2.
Polarization Operators ..................
Consequences of Equilibrium Phonon Distribution . . .
6.2. Negative Phonon Fluxes .....................
6.2.1.
6.2.2.
6.2.3.
6.2.4.
6.2.5.
6.2.6.
Electron Distribution Function ..............
Phonon Source in Linear Approximation ........
Phonon Heat-Bath Realization ..............
Induced and Spontaneous Processes ...........
Properties of the Recombination Channel ........
Comparison with Relaxation ...............
6.3. Violation of Detailed Balance ..................
6.3.1.
6.3.2.
6.3.3.
6.3.4.
Phonon Deficit and Order Parameter Enhancement . . .
Comparison with Alternative Approaches ........
Recombination Peak ...................
Exclusion of Divergence .................
References .............................
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CONTENTS xv
Chapter 7. Time-Dependent Ginzburg–Landau Equations
7.1. Order Parameter, Electron Excitations, and Phonons .......
7.1.1.
7.1.2.
7.1.3.
7.1.4.
7.1.5.
7.1.6.
7.1.7.
7.1.8.
7.1.9.
7.1.10.
7.1.11.
7.1.12.
7.1.13.
7.1.14.
7.1.15.
Basic Kinetic Equations .................
Normalization Condition .................
Definition of Order Parameter ..............
Nondiagonal Collision Channel
.............
Spectral Functions and . . . .......
Charge Density ......................
Gap-Control Term ....................
Local-Equilibrium Approximation ............
Determination
ofthe
-Function
............
Determination of the -Function ............
Order Parameter Equation ................
Contribution of Nonequilibrium Phonons ........
Contribution of the Gauge-Invariant Potential ......
Charge Density and Invariant Potential .........
Phonons and Order Parameter Dynamics ........
7.2. Interference Current
. . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1.
7.2.2.
7.2.3.
7.2.4.
7.2.5.
7.2.6.
Usadel Approximation ...................
Current Components in the Vicinity of .........
More Complete Expressions ................
Interference Current in Complete Form ..........
Full Set of Equations ...................
Boundary Conditions ...................
7.3. Viscous Flow of Vortices ......................
7.3.1.
7.3.2.
7.3.3.
7.3.4.
7.3.5.
Abrikosov Vortices .....................
Effective Conductivity: Definition ............
Low-Velocity Approximation ...............
Linearized Equations ...................
Effective Conductivity: Results ..............
7.4. Fluctuations ............................
7.4.1.
7.4.2.
7.4.3.
7.4.4.
Ginzburg’s Number ....................
Paraconductivity ......................
Aslamasov–Larkin Mechanism ..............
Maki–Thompson Mechanism ...............
References .............................
Chapter 8. A Longitudinal Electric Field and Collective Modes
8.1. Longitudinal Electric Field . ....................
8.1.1.
8.1.2.
Tinkham Expression for the Gauge-Invariant Potential . .
Normal Metal—Superconductor Interface ........
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8.1.3. New Characteristic Length in Superconductors .....
8.2. Carlson–Goldman Modes ....................
8.2.1.
8.2.2.
Damping of Collective Oscillations ...........
Dispersion of the Charge-Imbalance Mode .......
8.3. Stability and Breaking of Cooper Pairs ..............
8.3.1.
8.3.2.
8.3.3.
8.3.4.
Collisionless Dynamics for Spatially Homogeneous
Modes ...........................
Dispersion Equation ...................
Stability Analysis for Particle-Hole Symmetry .....
Instability at Branch Imbalance .............
References .............................
Chapter 9. Phase-Slip Centers
9.1. One-Dimensional Approach ...................
9.1.1.
9.1.2.
9.1.3.
Phase Slippage ......................
Initial
Dimensionless
Equations
.............
Boundary Conditions ...................
9.2. Calculations Procedure ......................
9.2.1.
9.2.2.
9.2.3.
9.2.4.
Nonsingular Representation ...............
Matrix Representation for “Sweeping” Method .....
Recurrence Relations ...................
Solution Procedure ....................
9.3. Analysis of Results ........................
9.3.1.
9.3.2.
9.3.3.
9.3.4.
9.3.5.
9.3.6.
9.3.7.
9.3.8.
9.3.9.
9.3.10.
Single Active Center ..................
Oscillation Frequency ..................
Temporal Behavior of the Phase Difference ......
Other Boundary Conditions ...............
Finite-Gap Results ...................
Two Active Centers ...................
Current-Voltage Relations: Galayko Model ......
Shortcomings of the TDGL in the Absence of Relaxation
More Features of Numeric Solutions ..........
Role of Interference Current Component ........
9.4. Emission from Phase-Slip Centers ................
9.4.1.
9.4.2.
Generation of Electromagnetic Radiation ........
Phonon Radiation .....................
References .............................
Chapter 10. Josephson Junctions
10.1. Tunnel Source of Excitations ..................
10.1.1.
Josephson Effect ....................
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CONTENTS xvii
10.1.2.
10.1.3.
10.1.4.
10.1.5.
10.1.6.
Tunneling Hamiltonian and Self-energies .......
Derivation of Excitation Source ............
Expression for Tunnel Current .............
“Tunnel Frequency” Parameter .............
Complete Set of Equations ...............
10.2. Oscillatory Properties of a Tunnel Source ............
10.2.1.
10.2.2.
10.2.3.
Clark’s Branch Imbalance ...............
Oscillations of the Gauge-Invariant Potential .....
Satellites in Scattered Radiation ............
10.3. Self-Consistent Solution of Kinetic Equations ..........
10.3.1.
10.3.2.
10.3.3.
10.3.4.
Analytic Soluction with a Branch Imbalance ......
Inclusion of Self-Consistency Equation ........
Analysis of Numerical Solutions: Subthreshold
Voltages .........................
Analysis of Numerical Solutions: Superthreshold
Voltages .........................
10.4. Phonon Emission from a Tunnel Junction ............
10.4.1.
10.4.2.
10.4.3.
Phonon Deficit in a Subthreshold Regime .......
Superthreshold Regime: Preconditions of Deficit . . .
Microrefrigeration ...................
10.5. Weakly Coupled Bridges .....................
10.5.1.
10.5.2.
10.5.3.
Modified Aslamasov-Larkin Model ..........
Cos -Term Paradox ..................
Excess Current .....................
References ............................
Chapter 11. Influence of Laser Radiation
11.1. Elesin Approach to Quasi-Particle Distributions .........
11.1.1.
11.1.2.
11.1.3.
11.1.4.
Spectral Function of Electron–Phonon Interaction . . .
Excess Quasi-particles: Normalization Condition . . .
Separation of “Coherent” Contributions ........
Analytic Solution for ...............
11.2. Order-Parameter Ambiguity ....................
11.2.1.
11.2.2.
11.2.3.
11.2.4.
11.2.5.
First-Order “Coherent” Correction ...........
Critical Pumping Intensity ...............
Multiple-Order-Parameter Solutions ..........
Stability of Solutions ..................
Physics of Coherent Instability .............
11.3. Finite Temperatures ........................
11.3.1.
11.3.2.
Inclusion of Thermal Phonons .............
-Approximation ....................
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xviii CONTENTS
11.3.3.
11.3.4.
11.3.5.
11.3.6.
Iterative Solution Procedure ..............
Solution for ...................
Coherent Contribution .................
Two Branches of a Nonzero-Order Parameter .....
11.4.
Dissipative Phase Transition ...................
11.4.1.
11.4.2.
11.4.3.
11.4.4.
Stationary Solutions for Time-Dependent
Problems ........................
Local Stability Against Space-Time Fluctuations . . .
Coexistence of Normal and Superconducting
States ..........................
Velocity of Phase-Boundary Motion .........
11.5.
Magnetic Properties .......................
11.5.1.
11.5.2.
11.5.3.
11.5.4.
Equilibrium Diamagnetic Response ..........
Paramagnetic Instability ................
Role of Boundary Conditions .............
Superheated States at External Pumping .......
11.6. Branch Imbalance Initiated by Absorption of High-Energy
Quanta ..............................
11.6.1.
11.6.2.
Finite Curvature of the Fermi Surface .........
Photoinduced Potential and Owen–Scalapino
-Model ........................
References ............................
Chapter 12. Inverse Population Instabilities
12.1. “Wide” Pumping Source ....................
12.1.1.
12.1.2.
12.1.3.
Elesin Theorem ...................
“Two-Peak” Approximation for ......
Numerical Analysis for Realistic Spectral Function .
12.2. “Narrow” Pumping .......................
12.2.1.
12.2.2.
12.2.3.
12.2.4.
12.2.5.
Analytic Solution for Resonant Pumping Case ....
Tunnel Injection of Electrons .............
Simplifications for “Narrow” Distributions ......
Analytic Solution for Symmetric Junctions ......
Injection from Bulk Sample to a Thin Film ......
12.3. Phonon Instability ........................
12.3.1.
12.3.2.
12.3.3.
12.3.4.
12.3.5.
12.3.6.
Decoupling of Electron–Phonon Kinetics .......
Phonon Absorption and Inverse Population ......
Phonon Field Amplification in “Narrow” Electron
Distributions ......................
Stability Against Order-Parameter Fluctuations ....
Fluctuations of Superfluid Velocity ..........
Estimated Gain .....................
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CONTENTS xix
12.4. Photon Instability .........................
12.4.1. Two Channels of Electron–Photon Interaction .....
12.4.2. Photons Versus Phonons ................
12.4.3. “Clean” Limit Kinematic Restrictions .........
12.4.4. Experimental Feasibility ................
References ............................
Chapter 13. Thermoelectric Phenomena
13.1. Linear Response to Thermal Gradient ..............
13.1.1. Thermopower of Normal Fermi Liquids ........
13.1.2. Response of a Superconductor’s Normal Component .
13.2. Thermoelectric Flux in a Superconducting Ring .........
13.2.1. Meissner Effect and Incomplete Cancellation of
Thermoelectricity
....................
13.2.2. “Gigantic” Flux Puzzle .................
13.3. Thermoelectricity and Charge Imbalance ............
13.3.1. Pethick–Smith Effect: Qualitative Description . ....
13.3.2. Spatially Inhomogeneous Kinetic Equation ......
13.3.3. Influence of Heat Flux .................
13.3.4. “Mutilated” Collision Operator .............
13.3.5.
Calculation
of
Branch Imbalance Potential
.
......
13.3.6. Branch Imbalance and Thermopower . .........
13.3.7. Thermopower in Optical Pumping ...........
References ............................
Chapter 14. Vortices and Thermoelectric Flux
14.1. Vortex Origination by a Magnetic Field .............
14.1.1. Bean–Livingston Barrier ................
14.1.2. Hollow Superconducting Cylinder ...........
14.1.3. General Consideration of Gibbs Free Energy .....
14.1.4. In-Plate Penetration ...................
14.1.5. Penetration into a Hollow Cylinder . ..........
14.2. Vortex Origination by Thermoelectric Current ..........
14.2.1. Free Energy Barrier ...................
14.3. Vortex–Antivortex Pair Generation ................
14.3.1. Two-Vortice Free Energy ................
14.3.2. Vortex–Antivortex Separation .............
14.3.3. Threshold Temperature of Separation .........
14.3.4. Comparison with Experiment and Discussion .....
References ............................
Index ...................................
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