Tuning Majorana zero modes with temperature in $\pi$-phase Josephson junctions

TL;DR

This paper investigates how temperature influences Majorana zero modes in $1$-phase Josephson junctions on quantum spin Hall insulators, revealing a temperature-driven phase transition affecting the presence of zero-energy modes.

Contribution

It demonstrates a temperature-induced transition from phase-winding to real junctions hosting Majorana zero modes, highlighting the role of proximity effects in these systems.

Findings

At zero temperature, phase-winding junctions are energetically favored.

Increasing temperature suppresses the proximity effect, leading to a transition to real junctions with Majorana zero modes.

The transition is driven by thermal effects on the superconducting order parameter.

Abstract

We study a superconductor-normal state-superconductor (SNS) Josephson junction along the edge of a quantum spin Hall insulator (QSHI) with a superconducting $\pi$-phase across the junction. We solve self-consistently for the superconducting order parameter and find both real junctions, where the order parameter is fully real throughout the system, and junctions where the order parameter has a complex phase winding. Real junctions host two Majorana zero modes (MZMs), while phase-winding junctions have no subgap states close to zero energy. At zero temperature we find that the phase-winding solution always has the lowest free energy, which we establish being due to a strong proximity-effect into the N region. With increasing temperature this proximity-effect is dramatically decreased and we find a phase transition into a real junction with two MZMs.