Effects of finite superconducting coherence lengths and of phase gradients in topological SN and SNS junctions and rings

TL;DR

This paper investigates how finite superconducting coherence lengths and phase gradients influence Majorana fermions in topological SN and SNS junctions and rings, revealing their impact on localization and system behavior.

Contribution

It introduces a detailed analysis of finite coherence lengths and phase gradients in topological junctions, highlighting their effects on Majorana fermion localization and system properties.

Findings

Finite coherence length determines the effective topological region size.

Non-perfect interface transmission affects Majorana fermion extension.

Phase gradients significantly influence Majorana localization.

Abstract

We study the effect of a finite proximity superconducting (SC) coherence length in SN and SNS junctions consisting of a semiconducting topological insulating wire whose ends are connected to either one or two s-wave superconductors. We find that such systems behave exactly as SN and SNS junctions made from a single wire for which some regions are sitting on top of superconductors, the size of the topological SC region being determined by the SC coherence length. We also analyze the effect of a non-perfect transmission at the NS interface on the spatial extension of the Majorana fermions. Moreover, we study the effects of continuous phase gradients in both an open and closed (ring) SNS junction. We find that such phase gradients play an important role in the spatial localization of the Majorana fermions.