Effect of inhomogeneous coupling on BCS superconductors

We investigate the influence of inhomogeneity in the pairing coupling constant U(r) on dirty BCS superconductors, focusing on T-c, the order parameter Delta(r), and the energy gap E-g(r). 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 2E(g)/T-c 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 T-KT is also studied. We find that when the inhomogeneity length scale is much larger than the coherence length, 2E(g)/T-KT can be larger than the usual BCS value. We use our results to qualitatively explain recent experimental observation of a surprisingly low value of 2E(g)/T-c in thin films.