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Proximity effect and Majorana bound states in clean semiconductor nanowires coupled to disordered superconductors

We study a semiconductor wire with strong spin-orbit coupling, which is proximity-coupled to a superconductor with chemical potential disorder. When tunneling at the semiconductor- superconductor interface is very weak, it is known that disorder in the superconductor does not affect the induced superconductivity nor, therefore, the effective topological superconductivity that emerges above a critical magnetic field. We demonstrate nonperturbatively how this result breaks down with stronger proximity coupling by obtaining the low-energy (i.e., subgap) excitation spectrum through direct numerical diagonalization of an appropriate BdG hamiltonian. With strong proximity coupling, we find that disorder in the parent superconductor suppresses the (non- topological) induced gap at zero magnetic field by disordering the induced pair potential. In the topological superconducting phase at large magnetic field, strong proximity coupling reduces the localization length of Majorana bound states, such that the induced disorder eliminates the topological gap while bulk Majorana zero modes emerge, even for short wires.