Theory of Nonequilibrium Superconductivity *

Transcription

1 Theory of Nonequilibrium Superconductivity * NIKOLAI B.KOPNIN Low Temperature Laboratory, Helsinki University of Technology, Finland and L.D. Landau Institute for Theoretical Physics, Moscow, Russia CLARENDON PRESS OXFORD 2001

2 CONTENTS I GREEN FUNCTIONS IN THE BCS THEORY Introduction Superconducting variables Ginzburg-Landau theory Example: Vortices in type II superconductors Bogoliubov-de Gennes equations Quasiclassical approximation Nonstationary phenomena Time-dependent Ginzburg-Landau theory Microscopic argumentation Boltzmann kinetic equation Outline of the contents 25 Green functions Second quantization Schrödinger and Heisenberg operators Imaginary-time Green function Definitions Example: Free particles The Wick theorem The real-time Green functions Definitions Analytical properties 39 The BCS model BCS theory and Gor'kov equations Magnetic field Frequency and momentum representation Order parameter of a d-wave superconductor Derivation of the Bogoliubov-de Gennes equations Thermodynamic potential Example: Homogeneous state Green functions Gap equation for an s-wave superconductor Perturbation theory Diagram technique Electric current 61

3 X CONTENTS Superconducting alloys Averaging over impurity positions Magnetic impurities Homogeneous state of an s-wave superconductor 71 II QUASICLASSICAL METHOD General principles of the quasiclassical approximation Quasiclassical Green functions Density, current, and order parameter Homogeneous state Real-frequency representation Example: Homogeneous state Eilenberger equations Self-energy Normalization Dirty limit. Usadel equations Boundary conditions Diffusive surface 96 Quasiclassical methods in stationary problems s-wave superconductors with impurities Small currents in a uniform state Ginzburg-Landau theory The upper critical field in a dirty alloy Gapless s-wave superconductivity Critical temperature Gap in the energy spectrum Aspects of d-wave superconductivity Impurities and d-wave superconductivity Impurity-induced gapless excitations The Ginzburg-Landau equations Bound states in vortex cores Superconductors with s-wave pairing d-wave superconductors 122 Quasiclassical method for layered superconductors Quasiclassical Green functions Eilenberger equations for layered systems Lawrence Doniach model Order parameter Free energy and the supercurrent Microscopic derivation of the supercurrent Applications of the Lawrence Doniach model Upper critical field Intrinsic pinning 137

4 CONTENTS XI III NONEQUILIBRIUM SUPERCONDUCTIVITY 8 Nonstationary theory The method of analytical continuation Clean superconductors Impurities Order parameter, current, and particle density The phonon model Self-energy Order parameter Particle particle collisions Transport-like equations and the conservation laws The Keldysh diagram technique Definitions of the Keldysh functions Dyson equation Keldysh functions in the BCS theory Quasiclassical method for nonstationary phenomena Eliashberg equations Self-energies Order parameter, current, and particle density Normalization of the quasiclassical functions Generalized distribution function s-wave superconductors with a short mean free path Stimulated superconductivity Kinetic equations Gauge-invariant Green functions Equations of motion for the invariant functions Quasiclassical kinetic equations Superconductors in electromagnetic fields Discussion Observables in the gauge-invariant representation The electron density and charge neutrality Collision integrals Impurities Electron-phonon collision integral Electron-electron collision integral Kinetic equations for dirty s-wave superconductors Small gradients without magnetic impurities Heat conduction The time-dependent Ginzburg Landau theory Gapless superconductors with magnetic impurities Generalized TDGL equations TDGL theory for d-wave superconductors d.c. electric field in superconductors. Charge imbalance 226

5 Xll CONTENTS IV VORTEX DYNAMICS 12 Time-dependent Ginzburg-Landau analysis Introduction Energy balance Moving vortex Force balance Flux flow Single vortex: Low fields Dense lattice: High fields Direction of the vortex motion Anisotropic superconductors Low fields High fields Flux flow in layered superconductors Motion of pancake vortices Intrinsic pinning Flux flow within a generalized TDGL theory Dirty superconductors d-wave superconductors Discussion: Flux flow conductivity Flux flow Hall effect Modified TDGL equations Hall effect: Low fields High fields Discussion: Hall effect Vortex dynamics in dirty superconductors Microscopic derivation of the force on moving vortices Variation of the thermodynamic potential Force on vortices Diffusion controlled flux flow Discussion Vortex dynamics in clean superconductors Introduction Boltzmann kinetic equation approach Forces in s-wave superconductors Spectral representation for the Green functions Useful identities Distribution function Localized excitations Delocalized excitations Flux flow conductivity Discussion Conductivity: Low temperatures 292

6 CONTENTS xiii Conductivity: Arbitrary temperatures Forces Boltzmann kinetic equation Canonical equations Uniform order parameter Boltzmann equation in presence of vortices Quasiparticles in the vortex core Transformation into the Boltzmann equation Vortex mass Equation of vortex dynamics Vortex momentum Vortex dynamics in d-wave superconductors Distribution function Conductivity 318 References 320 Index 325