  We theoretically consider the proximity effect in
semiconductor-superconductor hybrid nanostructures, which are being extensively
studied in the context of the ongoing search for non-Abelian Majorana fermions
in solid state systems. Specifically, we consider the dependence on the
thickness of the semiconductor in the direction normal to the interface, a
physical effect that has been uncritically neglected in all prior work on the
subject. Quite surprisingly, we find the completely unanticipated result that
increasing the semiconductor thickness leads to a drastic suppression of the
induced superconducting gap due to proximity-induced interband coupling. As a
result, in the limit of strong semiconductor-superconductor coupling, the
proximity-induced gap becomes much smaller than the bulk superconductor gap and
depends weakly on the interface transparency.
