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We provide a new analytical tool to calculate the energies of Andreev boundstates (ABS) in long imperfect SNS junctions, at present these can only bedescribed by numerical tools. We model an NS junction as a delta-function"Andreev" impurity, i.e., a localized potential which scatters an electron intoa hole with opposite spin. We show using the scattering matrix formalism that,quite surprisingly, an "Andreev" impurity is equivalent to an NS junctioncharacterized by both Andreev reflection and a finite amount of normalscattering. The ABS energies are then calculated using the T-matrix formalismapplied to a system with two Andreev impurities. Our results lie between thosefor a perfect long SNS junction limit described by the Andreev approximation(ABS energies depend linearly on the phase and are independent of the chemicalpotential) and the particle-in-the-box limit (bound state energies areindependent of the phase and have a linear dependence on the chemicalpotential). Moreover, we recover a closed-form expression for the ABS energiesby expanding around the particle-in-the-box limit.

Modeling long imperfect SNS junctions and Andreev bound states using two impurities and the T -matrix formalism