Onset of surface superconductivity beyond the Saint-James-de Gennes limit

We revisit the problem of the surface superconductivity nucleation focusing on the detailed study of the critical field \(H_{c3}\) as a function of temperature and disorder. Using the semiclassical Eilenberger formalism we find that away from the Ginzburg-Landau region the ratio between the nucleation critical field \(H_{c3}\) and the upper critical field \(H_{c2}\) deviates strongly from the Saint-James-de Gennes limit. In particular, the \(H_{c3}/H_{c2}\) is found to be a nonmonotonic function of temperature, which reaches the maximum for a set of parameters corresponding to a crossover region from ballistic to diffusive scattering, when the mean free path in a bulk of a superconductor is of the same order as zero-temperature superconducting coherence length. We also analyze the robustness of the nucleated phases with respect to diffusive scattering off the sample boundary by solving exactly corresponding eigenvalue problem of an integral equation for the critical field. The implications of these results for the transport in superconductors of various geometries near \(H_{c3}\) are briefly discussed. In particular, we present results for the mechanism of magnetoconductivity oscillations due to surface superconductivity effects.