Study of a superconductor—normal-metal interface in a finite geometry

A finite geometry of a superconductor—normal-metal ($S-N$) system is studied, assuming a constant BCS interaction in the superconductor. The Green's functions of the finite system are constructed from the wave-function solutions, satisfying boundary conditions at the free surfaces. The tunneling electronic density of states in $S$ and $N$ is calculated and it is shown that there are geometrical resonance effects, associated with the variation of the pair potential at the interface, due to the finiteness of both slabs. The effect of a finite mean free path is demonstrated. The result reproduces, as special cases, the Tomasch effect, the Rowell-McMillan effect, and the de Gennes-Saint James bound states below the energy gap. The transition temperature, in the case where both metals are much thinner than the coherence length (the Cooper limit) is calculated, and the effect of a finite mean free path on the Cooper-limit result is found.