Quantum transport in Josephson junctions

Abstract

Josephson junctions consist of two superconductors separated by some barrier, be it superconducting, ferromagnetic, or semiconducting. The striking feature of Josephson junctions is that for a phase difference between the two superconducting reservoirs, electrical current will flow between them. To study this transport we utilise the techniques of non-equilibrium Green’s functions; capable of probing the interplay between coherent quantum effects over macroscopic distances. We begin by broadly discussing the quantum mechanical techniques required in this work, before deriving the relevant equations to be solved. We then introduce numerical discretisation and explore how in lattice systems spin-orbit coupling can produce intricate fractals in an electron’s bandstructure. We then introduce a numerical algorithm which, although creates numerical instabilities, improves the speed of conventional Green’s function calculations. Finally, we study the electrical current, and the bound states which carry this current, in various Josephson junction architectures. We first compare our calculations with previous results in the literature to verify our results and demonstrate the generality of the non-equilibrium Green’s functions technique. We then explore numerically the anomalous Josephson effect, where current flows even in the absence of a phase bias between the two junctions. We provide a universal condition to be satisfied for Josephson junctions to exhibit anomalous current before modelling some experimental data concerning this effect.