Self-consistent calculations of the pair potential and the tunneling density of states in proximity contacts

By the use of the quasiclassical theory, the pair potential and the tunneling density of states of a so-called proximity contact, i.e., a superconducting layer N of thickness a on top of a very thick superconducting substrate S, are self-consistently calculated. Special attention is given to N layers with a thickness of the order of the coherence length, in which case the pair potential deviates appreciably from a step. Free parameters in the numerical calculation are the film thickness a, the ratio of the Fermi velocities of the two metals, the ratio of their transition temperatures, and a potential strength that describes possible imperfections of the contact plane. The calculated density of states is compared with the approximate results obtained from the often-used step model for the order parameter. This model reproduces the rigorous numerical results reasonably well, provided the parameters of the step are properly adjusted