Theory of Dirty Superconductors in Weakly Localized Regime

Abstract

The superconducting transition temperature T_c and upper critical field H_c2 in dirty superconductors are studied by taking into account the dynamically screened Coulomb interaction and disorder, perturbatively. We find that the results depend on the relative magnitude of τ₀⁻¹ and ω_D, τ₀ and ω_D being the relaxation time of an electron and the phonon-Debye frequency, respectively, even in the weakly localized regime. We examine the case of τ₀⁻¹<ω_D exclusively in this paper. The Anderson localization enhances the Coulomb repulsion between electrons and modifies the density of states. In addition, disorder induces the depairing effect due to the retardation of the Coulomb interaction. We obtain that T_c decreases in proportion to the sheet resistance in thin films and to λ² in bulk systems where λ=h⁄2πε_Fτ₀, ε_F being the Fermi energy. The correction in T_c is a function of |ln T_cτ₀| in both bulk systems and thin films.