Andreev scattering and density of states in weakly coupled superconductors

Abstract

The Bogoliubov-de Gennes Equations (BdGE) for inhomogeneous superconductors are transformed by a WKBJ ansatz into a set of two coupled nonlinear differential equations for slowly varying complex functions η(z) and ξ(z) which determine the quasiparticle wave functions. These equations are solved both numerically and in successive iterations starting from the known values of η and ξ in the homogeneous superconductor. We find that for energies above the gap the probability of Andreev scattering from a self-consistently varying pair potential Δ(z) is considerably smaller than that for a step-like pair potential. In microbridges the density of states of the bound states agrees very well with that of a rectangular pair potential well with an effective normal layer thickness ∫ [ 1 − Δ ( z ) Δ ] d z ; Δ is the constant value of Δ(z) deep in the superconducting banks. In junctions with finite S-layer thickness the density of the continuum states exhibits oscillations which are due to electron-hole interferences.