A Consistent Treatment of Strong Coupling in Disordered Superconducting Films

We develop a consistent treatment of disorder effects in strong coupling superconductors. We use two different approaches, starting either from above or below the transition temperature, and show their equivalence. The normal state approach is a natural extension of a recently developed non-perturbative resummation technique. For the superconducting state we extend the standard Eliashberg theory to include disorder effects. We obtain a compact set of equations which are physically transparent and easy to use. They correctly reproduce the usual strong coupling results and disordered perturbation theory results in the appropriate limits. We numerically solve these equations to obtain transition temperature as a function of disorder for different phonon couplings.