Spherical Impurity in an Infinite Superconductor

Abstract

The modifications in a superconducting medium due to a single, spherically symmetric, nonmagnetic impurity are examined using the BCS theory of superconductivity. The energy spectrum of quasiparticles has both a discrete portion (bound states) and a continuous portion (scattering states). In the scattering region ($E>\mathrm{}\Delta$), a given energy corresponds to two distinct momentum states, one above and one below the Fermi level. The calculation of the $S$ matrix for the impurity scattering is thus a problem of two coupled channels. The electron density $n\left(r\right)\mathrm{}$ and the order parameter $\Delta \mathrm{}\left(r\right)\mathrm{}$ far from the impurity are evaluated asymptotically in terms of the eigenphase shifts and mixing parameter of the two channels. Two soluble models for the impurity are considered. With a hard-sphere, long-range spatial oscillations are found in $\Delta \mathrm{}\left(r\right)\mathrm{}$ as well as in $n\left(r\right)\mathrm{}$. With a delta-shell potential, a resonant enhancement occurs in the scattering of quasiparticles with momentum near the Fermi momentum. Both the spatial oscillations and the resonant enhancement are expected to appear for more general impurity potentials.