Effects of the Spatial Extension of the Edge Channels on the Interference Pattern of a Helical Josephson Junction

TL;DR

This paper investigates how the spatial extension of edge states in topological Josephson junctions influences interference patterns, revealing oscillations with periodicity approaching twice the flux quantum when edge states overlap and crossed Andreev reflections dominate.

Contribution

It introduces a model considering extended edge states in topological JJs, showing how spatial extension affects interference patterns and periodicity, especially under crossed Andreev reflection dominance.

Findings

Interference pattern periodicity can approach 2$oldsymbol{ imes}$ the flux quantum.

Extended edge states cause a decay in the critical current with magnetic field.

Crossed Andreev reflections lead to oscillations with doubled periodicity.

Abstract

Josephson junctions (JJs) in the presence of a magnetic field exhibit qualitatively different interference patterns depending on the spatial distribution of the supercurrent through the junction. In JJs based on two-dimensional topological insulators (2DTIs), the electrons/holes forming a Cooper pair (CP) can either propagate along the same edge or be split into the two edges. The former leads to a SQUID-like interference pattern, with the superconducting flux quantum $\phi_0$ (where $\phi_0=h/2e$) as a fundamental period. If CPs' splitting is additionally included, the resultant periodicity doubles. Since the edge states are typically considered to be strongly localized, the critical current does not decay as a function of the magnetic field. The present paper goes beyond this approach and inspects a topological JJ in the tunneling regime featuring extended edge states. It is here considered the possibility that the two electrons of a CP propagate and explore the junction independently over length scales comparable to the superconducting coherence length. As a consequence of the spatial extension, a decaying pattern with different possible periods is obtained. In particular, it is shown that, if crossed Andreev reflections (CARs) are dominant and the edge states overlap, the resulting interference pattern features oscillations whose periodicity approaches $2\phi_0$.