Generalization of Anderson's theorem for disordered superconductors

Abstract

We show that at the level of BCS mean-field theory, the superconducting ${T}_{c}$ is always increased in the presence of disorder, regardless of order parameter symmetry, disorder strength, and spatial dimension. This result reflects the physics of rare events---formally analogous to the problem of Lifshitz tails in disordered semiconductors---and arises from considerations of spatially inhomogeneous solutions of the gap equation. So long as the clean-limit superconducting coherence length, ${\ensuremath{\xi}}_{0}$, is large compared to disorder correlation length, $a$, when fluctuations about mean-field theory are considered, the effects of such rare events are small (typically exponentially in ${[{\ensuremath{\xi}}_{0}/a]}^{d}$); however, when this ratio is $\ensuremath{\sim}1$, these considerations are important. The linearized gap equation is solved numerically for various disorder ensembles to illustrate this general principle.