Generalization of Anderson's Theorem for Disordered Superconductors

We show that at the level of BCS mean-field theory, the superconducting Tc 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. We discuss the relevance of these effects for short coherence length superconductors and compare them with an exact treatment of disorder within dirty d-wave BCS theory to describe superfluid stiffness results on the overdoped cuprates.