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

## TL;DR

This paper introduces an analytical method using the T-matrix formalism to calculate Andreev bound state energies in long, imperfect SNS junctions modeled as two impurities, bridging the gap between numerical and analytical approaches.

## Contribution

The authors develop a new analytical framework for ABS energies in long SNS junctions using two impurities and the T-matrix formalism, providing insights beyond numerical methods.

## Key findings

- Derived a closed-form expression for ABS energies.
- Showed the equivalence of an Andreev impurity to an NS junction with both Andreev and normal scattering.
- Bridged the gap between perfect junction and particle-in-the-box models.

## Abstract

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