  We study the electronic structure of full-shell superconductor-semiconductor
nanowires, which have recently been proposed for creating Majorana zero modes,
using an eight-band $\vec{k} \cdot \vec{p}$ model within a fully
self-consistent Schr\"{o}dinger-Poisson scheme. We find that the spin-orbit
coupling induced by the intrinsic radial electric field is generically weak for
sub-bands with their minimum near the Fermi energy. Furthermore, we show that
the chemical potential windows consistent with the emergence of a topological
phase are small and sparse and can only be reached by fine tunning the diameter
of the wire. These findings suggest that the parameter space consistent with
the realization of a topological phase in full-shell InAs/Al nanowires is, at
best, very narrow.
