  We investigate the effects of disorder characterising a superconducting thin
film on the proximity-induced superconductivity generated by the film (in,
e.g., a semiconductor) based on the exact numerical analysis of a
three-dimensional microscopic model. To make the problem numerically tractable,
we use a recursive Green's function method in combination with a patching
approach that exploits the short-range nature of the interface Green's function
in the presence of disorder. As a result of the Fermi surface mismatch between
the superconductor (SC) and the semiconductor (SM) in combination with the
confinement-induced quantization of the transverse SC modes, the proximity
effect induced by a clean SC film is typically one to three orders of magnitude
smaller that the corresponding quantity for a bulk SC and exhibits huge
thickness-dependent variations. The presence of disorder has competing effects:
on the one hand it enhances the proximity-induced superconductivity and
suppresses its strong thickness dependence, on the other hand it generates
proximity-induced effective disorder in the SM. The effect of proximity-induced
disorder on the topological superconducting phase and the associated Majorana
modes is studied nonperturbatively.
