Josephson current between topological and conventional superconductors

TL;DR

This paper investigates the Josephson current in junctions between topological and conventional superconductors, revealing how Majorana modes and broken time-reversal symmetry affect the current-phase relation and critical current scaling.

Contribution

It derives a general expression for the supercurrent in topological-superconductor junctions and analyzes the impact of Majorana modes on current behavior, contrasting with traditional relations.

Findings

Majorana modes cause asymmetry in current-phase relation.

Critical current is strongly suppressed in tunneling limit.

Scaling law for critical current differs from conventional Josephson junctions.

Abstract

We study the stationary Josephson current in a junction between a topological and an ordinary (topologically trivial) superconductor. Such an S-TS junction hosts a Majorana zero mode that significantly influences the current-phase relation. The presence of the Majorana state is intimately related with the breaking of the time-reversal symmetry in the system. We derive a general expression for the supercurrent for a class of short topological junctions in terms of the normal state scattering matrix. The result is strongly asymmetric with respect to the superconducting gaps in the ordinary ($\Delta_0$) and topological ($\Delta_{\mathrm{top}}$) leads. We apply the general result to a simple model of a nanowire setup with strong spin-orbit coupling in an external magnetic field and proximity-induced superconductivity. The system shows parametrically strong suppression of the critical current $I_c \propto \Delta_{\mathrm{top}}/R_N^2$ in the tunneling limit ($R_N$ is the normal state resistance). This is in strong contrast with the Ambegaokar-Baratoff relation applicable to junctions with preserved time-reversal symmetry. We also consider the case of a generic junction with a random scattering matrix and obtain a more conventional scaling law $I_c \propto \Delta_{\mathrm{top}}/R_N$.