Mesoscopic effects in small superconducting grains: a semiclassical approach to BCS theory

We present a novel approach to mesoscopic superconductivity based on a semiclassical approximation for both the density of states and the interaction matrix elements entering the BCS gap equation. In this way, the discretness of the single-particle spectra and the inhomogeneity of the single particle eigenfunctions are included consistenly as mesoscopic fluctuations around the bulk values. We apply the theory to 3D cubic grains, finding good agreement with exact numerical calculations. For grains with irregular shape we predict a novel dependence of the gap with the excitation energy, present in an experimentally relevant region of parameters