  We study the robustness of Majorana zero energy modes and minigaps of
quasiparticle excitations in a vortex by numerically solving
Bogoliubov-deGennes equations in a heterostructure composed of an \textit{s}
-wave superconductor, a spin-orbit-coupled semiconductor thin film, and a
magnetic insulator. This heterostructure was proposed recently as a platform
for observing non-Abelian statistics and performing topological quantum
computation. The dependence of the Majorana zero energy states and the minigaps
on various physics parameters (Zeeman field, chemical potential, spin-orbit
coupling strength) is characterized. We find the minigaps depend strongly on
the spin-orbit coupling strength. In certain parameter region, the minigaps are
linearly proportional to the \textit{s}-wave superconducting pairing gap
$\Delta_{s}$, which is very different from the $\Delta_{s}^{2}$ dependence in a
regular \textit{s-} or \textit{\p}-wave superconductor. We characterize the
zero energy chiral edge state at the boundary and calculate the STM signal in
the vortex core that shows a pronounced zero energy peak. We show that the
Majorana zero energy states are robust in the presence of various types of
impurities. We find the existence of impurity potential may increase the
minigaps and thus benefit topological quantum computation.
