The influence of the transverse width on the Andreev bound states and self-consistent gap function in clean SNS system

Abstract

Andreev bound states in clean, ballistic superconductor-normal metal-superconductor (SNS) junctions are calculated exactly and by using the Andreev approximation (AA). The AA appears to break down for junctions with transverse dimensions chosen such that the motion in the longitudinal direction is very slow. The doubly degenerate states typical for the traveling waves found in the AA are replaced by two standing waves in the exact treatment and the degeneracy is lifted.
A multiple scattering Green's function formalism is used, from which the bound-state energies are found through the local density of states. The scattering by the interfaces in any layered system of ballistic normal metals and clean superconducting materials is taken into account exactly. The formalism allows, in addition, for a self-consistent determination of the gap function in which a proximity effect is exhibited. In the numerical calculations, the pairing coupling constant for aluminum is used.