  We develop a theory for the proximity effect in
superconductor-semiconductor-normal metal tunneling structures, which have
recently been extensively studied experimentally, leading to the observation of
transport signatures consistent with the predicted zero-energy Majorana bound
states. We show that our model for the semiconductor nanowire having multiple
occupied subbands with different transmission probabilities through the barrier
reproduces the observed "soft-gap" behavior associated with substantial subgap
tunneling conductance. We study the manifestations of the soft gap phenomenon
both in the tunneling conductance and in local density of states measurements
and discuss the correlations between these two quantities. We emphasize that
the proximity effect associated with the hybridization between low-lying states
in the multiband semiconductor and the normal metal states in the lead is an
intrinsic effect leading to the soft gap problem. In addition to the intrinsic
contribution, there may be extrinsic effects, such as, for example, interface
disorder, exacerbating the soft gap problem. Our work establishes the generic
possibility of an ubiquitous presence of an intrinsic soft gap in the
superconductor-semiconductor-normal metal tunneling transport conductance
induced by the inverse proximity effect of the normal metal.
