Theory of thin proximity-effect sandwiches

Abstract

A theory for $N-S$ sandwiches is developed for clean metals in perfect contact, where the thickness of $N$ is much less than that of $S$. Assuming spatially constant pair potentials in $N$ and $S$, the exact double-layer Green's function is obtained. From this, we calculate the local density of states at the interface between $N$ and an oxide tunneling barrier. This tunneling density of states is examined in detail and compared with experimental results. We find that at energies far above ${\Delta }_N$ and ${\Delta }_S$, the local density of states contains a BCS-like term depending on ${\Delta }_N$, as well as types of oscillatory terms. The pair potential ${\Delta }_N$ also leads to an energy gap, but produces no BCS behavior at ${\Delta }_N$. A large peak in the local density of states is found at the energy corresponding to a one-dimensional bound state in $N$. Between this bound state and the pair potential in $S$, no states exist. Qualitative agreement with experiment is demonstrated over wide energy regions, but quantitative agreement is unsatisfactory below ${\Delta }_S$; sharp structure appearing near ${\Delta }_S$ in the experimental second harmonic singa ($\frac{d^{2}V}{d{I}^2}$) is also inadequately explained by the present theory. We make further use of the double-layer Green's function in obtaining the self-consistency conditions for the two pair potentials. Considering the possibility of extracting detailed information on the electron-phonon interaction in $N$ from existing experiments, we conclude that the present theory must be modified in several respects, most notably by including normal scattering at the $N-S$ interface.