  We study semiconductor nanowires coupled to a bilayer of a disordered
superconductor and a magnetic insulator, motivated by recent experiments
reporting possible Majorana-zero-mode signatures in related architectures.
Specifically, we pursue a quasiclassical Usadel equation approach that treats
superconductivity in the bilayer self-consistently in the presence of
spin-orbit scattering, magnetic-impurity scattering, and Zeeman splitting
induced by both the magnetic insulator and a supplemental applied field. Within
this framework we explore prospects for engineering topological
superconductivity in a nanowire proximate to the bilayer. We find that a
magnetic-insulator-induced Zeeman splitting, mediated through the
superconductor alone, cannot induce a topological phase since the destruction
of superconductivity (i.e., Clogston limit) preempts the required regime in
which the nanowire's Zeeman energy exceeds the induced pairing strength.
However, this Zeeman splitting does reduce the critical applied field needed to
access the topological phase transition, with fields antiparallel to the
magnetization of the magnetic insulator having an optimal effect. Finally, we
show that magnetic-impurity scattering degrades the topological phase, and
spin-orbit scattering, if present in the superconductor, pushes the Clogston
limit to higher fields yet simultaneously increases the critical applied field
strength.
