  We study the differential conductance for charge tunneling into a
semiconductor wire--superconductor hybrid structure, which is actively
investigated as a possible scheme for realizing topological superconductivity
and Majorana zero modes. The calculations are done based on a tight-binding
model of the heterostructure using both a Blonder-Tinkham-Klapwijk approach and
a Keldysh non-equilibrium Green's function method. The dependence of various
tunneling conductance features on the coupling strength between the
semiconductor and the superconductor, the tunnel barrier height, and
temperature is systematically investigated. We find that treating the parent
superconductor as an active component of the system, rather than a passive
source of Cooper pairs, has qualitative consequences regarding the low-energy
behavior of the differential conductance. In particular, the presence of
sub-gap states in the parent superconductor, due to disorder and finite
magnetic fields, leads to characteristic particle-hole asymmetric features and
to the breakdown of the quantization of the zero-bias peak associated with the
presence of Majorana zero modes localized at the ends of the wire. The
implications of these findings for the effort toward the realization of
Majorana bound states with true non-Abelian properties are discussed.
