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

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 to. 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 to 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 to 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 to. 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 to, 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 on InAs 2DEGs proximitized by narrow Al superconducting leads [A. Fornieri et al., Nature (London) 569, 89 (2019)].