  We study the energy spectrum of a vortex core in a two-dimensional
semiconductor with Rashba spin-orbit interaction and proximity-coupled to a
conventional superconductor and a ferromagnetic insulator. We perform
self-consistent calculations using the microscopic tight-binding Bogoliubov-de
Gennes method on a lattice and confirm the existence of Majorana fermions in
the non-trivial topological phase. We also find two different topologically
trivial bulk superconducting phases, only differing in the type of vortex core
structure they support and separated by a zero-energy excitation. Furthermore,
we find an asymmetry in the energy spectrum with respect to both Zeeman
splitting and vortex rotation direction and explain its physical origin.
