Kinetics and Boltzmann kinetic equation for fluctuation Cooper pairs

Abstract

The Boltzmann equation for excess Cooper pairs above T_c is derived in the framework of the time-dependent Ginzburg-Landau (TDGL) theory using Langevin's approach of the stochastic differential equation. The Newton dynamic equation for the momentum dependent drift velocity is obtained and the effective drag force is determined by the energy dependent life time of the metastable Cooper pairs. The Newton equation gives just the Drude mobility for the fixed momentum of Cooper pairs. It is shown that the comparison with the well-known result for Aslamazov-Larkin paraconductivity and BCS treatment of the excess Hall effect can give the final determination of all the coefficients of TDGL theory. As a result the intuitive arguments used for an interpretation of the experimental data for fluctuation kinetics are successively introduced. The presented simple picture of the degenerated Bose gas in tau-approximation near to Bose-Einstein condensation temperature can be used for analysis of fluctuation conductivity for the cases of high frequency and external magnetic field for layered and bulk superconductors. The work of the Boltzmann equation is illustrated by frequency dependent Aslamazov-Larkin conductivity in nanowires, two dimensional case and the case of strong electric field where the TDGL equation is solved directly. There are also derived explicit formulas for the current in the case of arbitrary time dependence of electric field up to THz range, the distribution of fluctuation Cooper pairs for non-parabolic dispersion, the influence of the energy cut-off and the self consistent equation for the reduced temperature. The general theory is illustrated by formulae for fluctuation conductivity in nanowires and nanostructured superconductors.