Time-dependent Ginzburg-Landau equations for a dirty gapless superconductor

Abstract

A complete set of charge-conserving, gauge-invariant, time-dependent Ginzburg-Landau equations is derived for a gapless, dirty, type-II superconductor using the method of Gor kov and Eliashberg. Pair breaking by magnetic impurities and by a static magnetic field is permitted in any ratio, in contrast to the equations of Hu and Thompson, where the magnetic impurities dominate. Our extension of the range of validity requires a sizable increase in complexity. The space and time derivatives of the order parameter and the dynamic electromagnetic fields must be included to one higher order than in previous work. The equations are solved for a plane geometry in perpendicular magnetic field to first order in a parallel time-independent electric field, including all screening or backflow effects. The nonlinear terms due to normal-state Joule heating found by Larkin and Ovchinnikov are also obtained, and a precise connection is made between the vertex functions appearing in our equations and the electron distribution function appearing in the Boltzmann equation in the normal state.

Keldysh technique and non-linear σ-model: basic principles and applications

“…Later Gor'kov and Eliashberg [143] extended application of the Ginzburg-Landau theory to include time dependent dynamical phenomena. It was revisited in a number of subsequent publications [144,145,146,147,148,149,150] and books [124,125,151]. Within the σ-model terminology the static GL functional may be obtained by means of supersymmetric [152] or replica [153] approaches.…”

Theoretical Study of Anomalous Forces Externally Induced by Superconductors

“…However, in order to increase the magnitude of the effect so that it was strong enough to allow a practical application, we suppose that the anomalous effect could be enhanced if we consider some technological practices adopted in other physical situations, as suggested in [51], as to use time-dependent electric fields [52], introduce one (or many) impurity in the system [53], with pulsed magnetic fields [54][55][56][57][58], by making the superconductor rotate more quickly [59][60][61] or by using a superconductor device made of layers of different materials [62].…”

Critical fluctuation conductivity in layered superconductors in a strong electric field

“…A short comment would be useful regarding the application of the cut-off procedure to the k-integrals in Eqs. (27) and (22). It could seem more appealing to apply the cut-off on the translated wave-vector magnitude, such as k 2 y + [k x + (e 0 Eτ /2 )] 2 < cξ −2 0 , as it was approximately done, although in a different gauge, by Kajimura and Mikoshiba, 38 while studying the paraconductivity in arbitrary electric field in the two-dimensional case.…”

“…23 and 26. In some more recent derivations of the TDGL, 22,27 however, the original appearance of these parameters is preserved, so that the order parameter relaxation rate is written as depending on the actual temperature. In this mathematically somewhat stricter sense, one could therefore write the life-time of the metastable Cooper pairs as τ (BCS) = π /16k B T ln (T /T 0 ).…”