Onset of surface superconductivity

We examine the onset of superconductivity in the surface region of a metal. Surface effects are particularly important in systems with a short bulk coherence length ${\xi }_0$. We show that, to the accuracy of the calculation, the surface transition temperature ${\mathit{T}}_{\mathit{c}\mathit{S}}$ equals the bulk transition temperature ${\mathit{T}}_{\mathit{c}\mathit{B}}$ if the electron-electron interaction is of the standard BCS form, i.e., a single attractive square well, extending up to some critical energy ${\mathrm{\omega }}_0$ much smaller than the Fermi energy ${\mathrm{\varepsilon }}_{\mathit{F}}$. If one takes into account, in addition, the repulsive part of the interaction extending beyond ${\mathrm{\omega }}_0$ up to energies of order ${\mathrm{\varepsilon }}_{\mathit{F}}$, then one may have ${\mathit{T}}_{\mathit{c}\mathit{S}}$>${\mathit{T}}_{\mathit{c}\mathit{B}}$ in certain cases, although, due to restrictions imposed on the parameter values by various physical conditions, the relative increase of ${\mathit{T}}_{\mathit{c}}$ is very small, typically ${10}^{\mathrm{-}3}$, at least in the weak coupling limit. However, we also find a considerable gap enhancement, of order 20%, near the surface which could be of interest for critical-current measurements. Therefore we suggest an experimental reexamination of systems with short ${\xi }_0$, i.e., superconducting degenerate semiconductors and the new high-${\mathit{T}}_{\mathit{c}}$ oxides in confined geometries where the surface-to-volume ratio is non-negligible.