Superconductivity of the Kronig-Penney model

We clarify the basic properties of superconductivity in the Kronig-Penney model, which mimics a layered superconductor or superconducting superlattice, taking into consideration the electron’s motion parallel to the layer. The exact integral kernel of the Gor’kov equation is obtained by the analytical Green’s function and based on it a Friedel-type microscopic oscillation of the pair potential at the boundary is correctly treated for the first time. The transition temperature $T_c$ is calculated with a rigorous treatment of the interband effect. It is found that this quantity can only be correctly determined by solving the Gor’kov equation with proper inclusion of the spatial dependence of the pair potential. It cannot be determined only by densities of states obtained from band calculations with the use of a simple BCS equation. The dependence of $T_c$ on the superlattice period and the spatial dependence of the pair potential are also discussed. The critical temperature shows nonmonotonic behavior as a function of the thickness period, which cannot be explained by the usual theory of the proximity effect by de Gennes.