Fate of Topological Superconductivity in Epitaxial Thin-Film Majorana Setups with Disorder

We study the proximity effect in a quantum wire strongly coupled to a superconductor with a thickness which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al layers which form a very strong contact with either InAs or InSb. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects essentially render the topological phase inaccessible when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor, the “hard gap” regime which is seen ubiquitously in the experiments is accompanied by a large shift in the effective chemical potential of the nanowire that pushes the topological phase transition to magnetic field strengths which greatly exceed the critical field of Al. We also show that this result is robust with respect to disorder scattering within the thin superconducting film. Our results demonstrate the significant importance of treating the parent superconductor on a microscopic level.