Impact of disorder on the topological superconductivity realized in planar semiconductor-superconductor structures: Comparison between Majorana nanowires and Josephson junctions

Pristine semiconductor-superconductor (SM-SC) hybrid nanowires and Josephson junctions have been intensely studied in recent years as promising platforms for realizing topological superconductivity and Majorana zero modes (MZMs). Disorder effects have been widely identified as the major roadblock for realizing stable MZMs in hybrid nanowires. However, there are no systematic studies of disorder effects in planar Josephson junctions at the same modeling level. We perform a detailed numerical analysis of the low-energy physics of Josephson junction structures based on an effective microscopic model that incorporates two types of disorder, charge impurities inside the semiconductor and surface roughness on the superconducting film. We consider different parameter regimes, including weak and strong effective SM-SC coupling, low and high chemical potential values, and weak and strong disorder strength. The results are benchmarked using disordered hybrid nanowires realized in planar SM-SC structures similar to those involved in the fabrication of Josephson junctions and having similar model parameters and disorder strengths. We find that the topological superconducting phase hosted by a Josephson junction structure is, generally, more robust against disorder than the topological superconductivity realized in a hybrid nanowire with similar parameters. On the other hand, operating the Josephson junction in a regime characterized by large chemical potential values leads to huge finite-size effects that can destroy the stability of MZMs.