Generalization of Anderson's Theorem for Disordered Superconductors

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, $\xi_0$, 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 $[\xi_0/a]^d$); however, when this ratio is $\sim 1$, these considerations are important. The linearized gap equation is solved numerically for various disorder ensembles to illustrate this general principle.