Effect of inhomogeneous coupling on BCS superconductors

We investigate the influence of inhomogeneity in the pairing coupling constant U(Pr) on dirty BCS superconductors, focusing on Tc, the order parameter Δ(Pr), and the energy gap Eg(Pr). Within mean-field theory, we find that when the length scale of the inhomogeneity is comparable to or larger than the coherence length, the ratio 2Eg∕Tc is significantly reduced from that of a homogeneous superconductor, while in the opposite limit, this ratio stays unmodified. In two dimensions, when strong phase fluctuations are included, the Kosterlitz-Thouless temperature TKT is also studied. We find that when the inhomogeneity length scale is much larger than the coherence length, 2Eg∕TKT can be larger than the usual BCS value. We use our results to qualitatively explain recent experimental observation of a surprisingly low value of 2Eg∕Tc in thin films.