  We investigate effects of ordinary nonmagnetic disorder in the bulk of a
superconductor on magnetic adatom-induced Shiba states and on the
proximity-induced superconductivity in a nanowire that is tunnel coupled to the
bulk superconductor. Within the formalism of self-consistent Born
approximation, we show that, contrary to widespread belief, the
proximity-induced topological superconductivity can be adversely affected by
the bulk superconducting disorder even in the absence of any disorder in the
nanowire (or the superconductor-nanowire interface) when the proximity tunnel
coupling is strong. In particular, bulk disorder can effectively randomize the
Shiba-state energies. In the case of a proximate semiconductor nanowire, we
numerically compute the dependence of the effective disorder and pairing gap
induced on the wire as a function of the semiconductor-superconductor tunnel
coupling. We find that the scaling exponent of the induced disorder with
respect to coupling is always larger than that of the induced gap, implying
that at weak coupling, the proximity-induced pairing gap dominates, whereas at
strong coupling, the induced disorder dominates. These findings bring out the
importance of improving the quality of the bulk superconductor itself (in
addition to the quality of the nanowire and the interface) in the experimental
search for solid-state Majorana fermions in proximity-coupled hybrid structures
and, in particular, points out the pitfall of pursuing strong coupling between
the semiconductor and the superconductor in a goal toward having a large
proximity gap. In particular, our work establishes that the bulk superconductor
in strongly coupled hybrid systems for Majorana studies must be in the
ultraclean limit, since otherwise the bulk disorder is likely to completely
suppress all induced topological superconductivity effects.
