Disorder effects in planar semiconductor-superconductor structures: Majorana wires versus Josephson junctions

Disorder effects in hybrid semiconductor-superconductor (SM-SC) nanowires, widely recognized as the main obstacle to realizing stable Majorana zero modes (MZMs) in these structures, have been systematically investigated theoretically in recent years. However, there are no corresponding detailed studies of disorder effects in planar Josephson junction (JJ) structures, which represent a promising alternative to the Majorana nanowire platform. In this paper, we perform a numerical analysis of the low-energy physics of JJ structures based on an effective microscopic model that includes two types of disorder, charge impurities inside the semiconductor and roughness on the surface of the superconducting film. We consider different parameter regimes, including low and high chemical potential values, weak and strong effective SM-SC coupling strengths, and weak and strong disorder strengths. 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 JJ structure is, generally, more robust against disorder than the topological superconductivity realized in a hybrid nanowire with similar parameters. On the other hand, we find that operating the JJ in a regime characterized by large values of chemical potential results in huge finite-size effects that can destroy the stability of MZMs.