The dc Josephson current in a long multi-channel quantum wire

TL;DR

This paper derives a comprehensive formula for the dc Josephson current in long multi-channel SNS junctions, linking it to scattering amplitudes at the Fermi level, applicable to various channel configurations and useful for interaction studies.

Contribution

It provides a general analytical framework for calculating the Josephson current in multi-channel SNS junctions without simplifying assumptions about scattering processes.

Findings

Current expressed in terms of scattering amplitudes at Fermi level

Formulas derived for channels with same dispersion relations

Integral representation for channels with different dispersions

Abstract

The dc Josephson current across a multi-channel SNS junction is computed by summing contributions from sub-gap Andreev bound states, as well as from continuum states propagating within the superconducting leads. We show that, in a long multi-channel SNS-junction, at low temperatures, all these contributions add up, so that the current can be entirely expressed in terms of single-particle normal- and Andreev reflection amplitudes at the Fermi level at both SN interfaces. Our derivation applies to a generic number of channels in the normal region and/or in the superconducting leads, without assumptions about scattering processes at the SN interfaces: if the channels within the central region have the same dispersion relation, it leads to simple analytical formulas for the current at low temperatures; if the channels within the central region have different dispersion relations, it allows for expressing the current in terms of a simple integral involving only scattering amplitudes at the Fermi level. Our result motivates using a low energy effective boundary Hamiltonian formalism for computing the current, which is crucial for treating Luttinger liquid interaction effects.