  Hybrid superconductor-semiconductor devices are currently one of the most
promising platforms for realizing Majorana zero modes. Their topological
properties are controlled by the band alignment of the two materials, as well
as the electrostatic environment, which are currently not well understood.
Here, we pursue to fill in this gap and address the role of band bending and
superconductor-semiconductor hybridization in such devices by analyzing a gated
single Al-InAs interface using a self-consistent Schrodinger-Poisson approach.
Our numerical analysis shows that the band bending leads to an interface
quantum well, which localizes the charge in the system near the
superconductor-semiconductor interface. We investigate the hybrid band
structure and analyze its response to varying the gate voltage and thickness of
the Al layer. This is done by studying the hybridization degrees of the
individual subbands, which determine the induced pairing and effective
$g$-factors. The numerical results are backed by approximate analytical
expressions which further clarify key aspects of the band structure. We find
that one can obtain states with strong superconductor-semiconductor
hybridization at the Fermi energy, but this requires a fine balance of
parameters, with the most important constraint being on the width of the Al
layer. In fact, in the regime of interest, we find an almost periodic
dependence of the hybridization degree on the Al width, with a period roughly
equal to the thickness of an Al monolayer. This implies that disorder and shape
irregularities, present in realistic devices, may play an important role for
averaging out this sensitivity and, thus, may be necessary for stabilizing the
topological phase.
