Thermal conductivity and charge imbalance in superconductors

We use the quasiclassical Green-function formalism to derive kinetic equations describing thermal transport and charge relaxation in superconductors. The effects of pair-breaking processes due to magnetic impurities or phonons are fully included, allowing the case of strong electron-phonon coupling to be considered. We calculate the temperature-dependent thermal conductivity of a superconductor by solving numerically the kinetic equation for the distribution function. In the strong-coupling case we solve in addition the nonlinear Eliashberg equations for the renormalization and gap functions. The resulting thermal conductivity shows a pronounced maximum at low temperatures in the case of pure Pb, and a gradual disappearance of the maximum with increasing amounts of impurity scattering. The charge-relaxation rate in the strong-coupling case shows near $T_c$ the same dependence on the gap function as in the case of weak coupling, but at lower temperatures there are pronounced differences between the temperature dependence of the relaxation rates in the two cases.