Suppression of the order parameter in homogeneous disordered superconductors

We calculate the first-order correction to the order parameter Δ in a disordered superconductor, using a formalism that treats electron-electron repulsion and order parameter fluctuations on an equal footing. We find that this correction is plagued by the same low-momentum singularities from the screened Coulomb potential as the correction to the transition temperature, and that these singularities are cancelled between terms in both cases. The cancellation of leading-order terms means that we must consider all terms of the same perturbation order in the calculation of Δ and ${\mathit{T}}_{\mathit{c}}$ if we are not to obtain qualitatively incorrect results. High momentum and frequency fluctuations then dominate, leading to a suppression of Δ proportional to ${\ln }^3$(Δτ). We therefore expect that the ratio Δ/${\mathit{T}}_{\mathit{c}}$ will be roughly constant, which is confirmed by detailed numerical evaluation. We show the general utility of our formalism in the evaluation of physical quantities in the disordered superconductor. Finally we comment on the appropriateness of the dirty-boson approach in the region where our calculations are applicable.