Josephson diode effect with Andreev and Majorana bound states

Abstract

Superconductor-semiconductor hybrids are useful for realizing the Josephson diode effect, where nonreciprocity in the supercurrents occurs due to the interplay of the Josephson effect and applied magnetic fields. These junctions can host Andreev and Majorana states with the same ingredients, though their interplay with the Josephson diode effect is unclear. In this work, we consider short Josephson junctions based on superconductor-semiconductor systems under homogeneous Zeeman fields and investigate the Josephson diode effect in the presence of Andreev and Majorana states. Under generic conditions, the Zeeman field component parallel to the spin-orbit axis promotes an asymmetric low-energy spectrum as a function of the superconducting phase, which persists in the trivial and topological phases hosting Andreev and Majorana bound states, respectively. Interestingly, this asymmetry creates supercurrents that are not odd functions of the phase difference, leading to a nonreciprocal behaviour and the Josephson diode effect. We show that the Josephson diode effect is particularly promoted under the presence of both zero-energy Andreev and Majorana bound states, revealing that Josephson diodes can be realized in the trivial and topological phases of superconductor-semiconductor hybrids. We then demonstrate that the Zeeman field evolution of the diode's efficiencies can map the topological phase transition and the formation of Majorana bound states via an oscillatory behavior that becomes more visible in long superconductors. While Josephson diodes generally exist in the trivial and topological phases of Josephson junctions, we discover that in the tunneling regime only a Josephson diode effect in the topological phase remains due to the finite contribution of Majorana bound states. Our findings clarify the Josephson diode effect and aid in realizing Majorana-only Josephson diodes.