Anisotropic topological superconductivity in Josephson junctions

We investigate the effects of magnetic and crystalline anisotropies on the\ntopological superconducting state of planar Josephson junctions (JJs). In\njunctions where only Rashba spin-orbit coupling (SOC) is present, the\ntopological phase diagram is insensitive to the supercurrent direction, but\nexhibits a strong dependence on the magnetic field orientation. However, when\nboth Rashba and Dresselhaus SOCs coexist, the topological phase diagram\nstrongly depends on both the magnetic field and junction crystallographic\norientations. We examine the impact of the magnetic and crystalline anisotropy\non the current-phase relation (CPR), energy spectrum, and topological gap of\nphase-biased JJs, where the junction is connected in a loop and the\nsuperconducting phase difference is fixed by a loop-threading magnetic flux.\nThe anisotropic CPR can be used to extract the ground-sate phase (i.e. the\nsuperconducting phase difference that minimizes the system free energy)\nbehavior in phase-unbiased JJs with no magnetic flux. Under appropriate\nconditions, phase-unbiased JJs can self-tune into or out of the topological\nsuperconducting state by rotating the in-plane magnetic field. The magnetic\nfield orientations at which topological transitions occur strongly depend on\nboth the junction crystallographic orientation and the relative strength\nbetween Rashba and Dresselhaus SOCs. We find that for an optimal practical\napplication, in which the junction exhibits topological superconductivity with\na sizable topological gap, a careful balancing of the magnetic field direction,\nthe junction crystallographic orientation, and the relative strengths of the\nRashba and Dresselhaus SOCs is required. We discuss the considerations that\nmust be undertaken to achieve this balancing for various junction types and\nparameters.\n