Effects of the superconducting film edge and surface roughness on the topological superconductivity of planar semiconductor-superconductor Josephson junctions

Planar Josephson junctions have been intensely investigated in recent years as a promising platform for topological superconductivity and Majorana zero modes. The vast majority of the theoretical studies involve pristine systems characterized, in particular, by a proximity-induced pairing potential (typically introduced “by hand”) that is constant throughout the proximitized regions separated by a junction of constant width. On the other hand, it is known that disorder (in the superconductor) plays a crucial role in inducing robust superconductivity and suppressing the dramatic dependence on the film thickness associated with Fermi surface mismatch. In turn, disorder (in the superconductor) may affect the low-energy properties of the hybrid structure. Furthermore, the junction width is expected to exhibit variations generated by imperfect patterning (i.e., etching of the superconducting film), which also impacts the low-energy physics. In this work, we investigate the effect of the superconducting film edge and surface roughness on the topological superconductivity of planar Josephson junctions based on a numerical analysis of a microscopic effective model of the hybrid structure. We provide semi-quantitative estimates of the edge roughness (i.e., variations of the effective junction width) and surface roughness (i.e., variations of the effective superconducting film thickness) consistent with the presence of a topological superconducting phase that supports Majorana zero modes.