  Majorana fermions are predicted to play a crucial role in condensed matter
realizations of topological quantum computation. These heretofore undiscovered
quasiparticles have been predicted to exist at the cores of vortex excitations
in topological superconductors and in heterostructures of superconductors and
materials with strong spin-orbit coupling. In this work we examine topological
insulators with bulk s-wave superconductivity in the presence of a
vortex-lattice generated by a perpendicular magnetic field. Using
self-consistent Bogoliubov-de Gennes, calculations we confirm that beyond the
semi-classical, weak-pairing limit that the Majorana vortex states appear as
the chemical potential is tuned from either side of the band edge so long as
the density of states is sufficient for superconductivity to form. Further, we
demonstrate that the previously predicted vortex phase transition survives
beyond the semi-classical limit. At chemical potential values smaller than the
critical chemical potential, the vortex lattice modes hybridize within the top
and bottom surfaces giving rise to a dispersive low-energy mid-gap band. As the
chemical potential is increased, the Majorana states become more localized
within a single surface but spread into the bulk toward the opposite surface.
Eventually, when the chemical potential is sufficiently high in the bulk bands,
the Majorana modes can tunnel between surfaces and eventually a critical point
is reached at which modes on opposite surfaces can freely tunnel and annihilate
leading to the topological phase transition previously studied in the work of
Hosur et al.
