Theory for impurity scattering in a superconducting film on a normal substrate

Abstract

The effects of impurity scattering on the transition temperature of a superconducting film of thicknessL are analyzed. The film is in contact with a normal substrate (with zero BCS interaction parameter). The analysis is carried out with the aid of a gap equation whose kernel is a solution of a linearized version of Eilenberger's transport equations. The kernel may be written as a sum of two terms. One of these terms dominates in the dirty limit. In this limit, this term is equal to the kernel that satisfies the diffusion equation of de Gennes. The significance of these two terms in determining the transition temperatureT _cis compared, for several values ofL, by varying the impurity scattering time. It is shown that the correction to the diffusion approximation may give quantitatively, and in certain limits qualitatively, different results forT _c. This correction in general leads to a different critical thicknessL _cfrom that determined in the diffusion approximation (T _c=0 forL<L _c).