We theoretically investigate the emergence of topological superconductivity in dc-SQUIDs in the presence of Rashba spin-orbit coupling and an in-plane magnetic field. The transmission of the individual planar Josephson junctions (JJs) can be controlled by top gates, switching from a single JJ to a dc-SQUID behavior. The transition to the topological phase in the single JJ configuration is sensible to the direction of the in-plane magnetic field and we show that it is accompanied by minima in the critical current, serving as experimental signatures to identify the phase transition. Furthermore, the topological phase transitions in each of the JJs can be individually tuned by top gates. We show that there are distinctive signatures in the critical current and phase shift of the dc-SQUID for cases when none, either or both junctions are in the topological regime. We also investigate the effects of electrostatic disorder on the topological superconducting state of single JJs.
We theoretically investigate the crystalline anisotropy of topological superconductivity (TS) in phase controlled planar Josephson junctions (JJs) subjected to Rashba and Dresselhaus spin-orbit couplings and in-plane magnetic fields. We show how the interplay between the magnetic field direction and the orientation of the junction with respect to its crystallographic axes can affect the TS. Our results explain previous experiments demonstrating the high sensibility of TS to the in-plane magnetic field direction. The anisotropy can be used to electrically tune between BDI and D symmetry classes in a controlled fashion and thereby optimize the stability and localization of Majorana bound states in planar JJs. Our findings can be used as a guide for achieving the most favorable conditions when engineering TS in planar JJs and can be particularly relevant for setups containing non-collinear junctions, which have been proposed for fusion and braiding operations on multiple Majorana pairs.
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