Theory of Superconducting Contacts

Abstract

The BCS theory of superconductivity is generalized to the case of a position-dependent energy gap (at the absolute zero of temperature and in the absence of magnetic fields). The BCS integral equation for the energy gap goes over into an integro-differential equation. The latter has nontrivial solutions (i.e., finite energy gap) even for the case of normal material ($V=0\mathrm{}$). Expressions are obtained for the energy gap, for the volume energy density, and for the surface energy density at an interface, for both normal and superconducting material. These results are applied to a number of problems involving superconducting contacts. When a thin slice of normal material is sandwiched between bulk superconductors, it is found that the slice acts superconducting for thicknesses less than about ${10}^{-5\mathrm{}}$ cm. When a thin slice of superconductor is sandwiched between bulk normal material, the slice acts like normal material for thicknesses less than about ${10}^{-5\mathrm{}}$ cm. The energy gap at the free surface of a bulk superconductor may differ by as much as thirty percent from its constant value deep inside the material, the former being either larger or smaller than the latter, depending on the value of $N\left(0\right)V$, where $N\left(0\right)\mathrm{}$ is the density of one-electron states of a given spin at the Fermi level in the normal metal.