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

We model 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, disorder in the superconductor does not affect the induced superconductivity nor, therefore, the effective topological superconductivity that emerges above a critical magnetic field. Here 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 Bogoliubov-de Gennes Hamiltonian. We find that the combination of strong proximity coupling and superconductor disorder suppresses the (nontopological) induced gap at zero magnetic field by disordering the induced pair potential. In the topological superconducting phase at large magnetic field, strong proximity coupling also reduces the localization length of Majorana bound states, such that the induced disorder eliminates the topological gap while bulk zero modes proliferate, even for short wires.