# Full proximity treatment of topological superconductors in Josephson-junction architectures

## TL;DR

This paper investigates how proximity-induced topological superconductivity can be achieved in Josephson junction architectures with spatially separated superconductors, using self-consistent solutions of Bogoliubov-de Gennes equations to optimize conditions for Majorana modes.

## Contribution

It provides a microscopic, self-consistent analysis of proximity effects in Josephson junctions, revealing how to maximize the superconducting gap and conditions for topological phases.

## Key findings

- Proximity coupling is weaker but can be optimized in these architectures.
- Self-consistent solutions reveal conditions for topological phases.
- Fulde-Ferrell-Larkin-Ovchinnikov phase can appear in the 2DEG channel.

## Abstract

Experiments on planar Josephson junction architectures have recently been shown to provide an alternative way of creating topological superconductors hosting accessible Majorana modes. These zero-energy modes can be found at the ends of a one-dimensional channel in the junction of a two-dimensional electron gas (2DEG) proximitized by two spatially separated superconductors. The channel, which is below the break between the superconductors, is not in direct contact with the superconducting leads, so that proximity coupling is expected to be weaker and less well-controlled than in the simple nanowire configuration widely discussed in the literature. This provides a strong incentive for this paper which investigates the nature of proximitization in these Josephson architectures. At a microscopic level we demonstrate how and when it can lead to topological phases. We do so by going beyond simple tunneling models through solving self-consistently the Bogoliubov-de Gennes equations of a heterostructure multicomponent system involving two spatially separated $s$-wave superconductors in contact with a normal Rashba spin-orbit-coupled 2DEG. Importantly, within our self-consistent theory we present ways of maximizing the proximity-induced superconducting gap by studying the effect of the Rashba spin-orbit coupling, chemical potential mismatch between the superconductor and 2DEG, and sample geometry on the gap. Finally, we note (as in experiment) a Fulde-Ferrell-Larkin-Ovchinnikov phase is also found to appear in the 2DEG channel, albeit under circumstances which are not ideal for topological superconducting phase.