# Topological superconductivity in planar Josephson junctions: Narrowing down to the nanowire limit

## TL;DR

This paper theoretically investigates topological superconductivity in narrow planar Josephson junctions, revealing how lead width, magnetic field, and phase difference influence topological phases and critical current behavior, guiding experimental searches.

## Contribution

It extends the understanding of topological phases in Josephson junctions by analyzing the effects of narrow superconducting leads and symmetry classes, providing new phase diagrams and experimental insights.

## Key findings

- BDI phase diagram depends on magnetic field and phase difference
- Critical current minima indicate phase transitions with small spin-orbit coupling
- Narrow leads alter the relationship between phase transitions and topological index

## Abstract

We theoretically study topological planar Josephson junctions (JJs) formed from spin-orbit-coupled two-dimensional electron gases (2DEGs) proximitized by two superconductors and subjected to an in-plane magnetic field $B_\parallel$. Compared to previous studies of topological superconductivity in these junctions, here we consider the case where the superconducting leads are narrower than the superconducting coherence length. In this limit the system may be viewed as a proximitized multiband wire, with an additional knob being the phase difference $\phi$ between the superconducting leads. A combination of mirror and time-reversal symmetry may put the system into the class BDI. Breaking this symmetry changes the symmetry class to class D. The class D phase diagram depends strongly on $B_{\parallel}$ and chemical potential, with a weaker dependence on $\phi$ for JJs with narrower superconducting leads. In contrast, the BDI phase diagram depends strongly on both $B_\parallel$ and $\phi$. Interestingly, the BDI phase diagram has a "fan"-shaped region with phase boundaries which move away from $\phi = \pi$ linearly with $B_\parallel$. The number of distinct phases in the fan increases with increasing chemical potential. We study the dependence of the JJ's critical current on $B_\parallel$, and find that minima in the critical current indicate first-order phase transitions in the junction only when the spin-orbit coupling strength is small. In contrast to the case of a JJ with wide leads, in the narrow case these transitions are not accompanied by a change in the JJ's topological index. Our results, calculated using realistic experimental parameters, provide guidelines for present and future searches for topological superconductivity in JJs with narrow leads, and are particularly relevant to recent experiments [A. Fornieri et al., Nature (London) 569, 89 (2019)].