  Topological insulator/superconductor two-dimensional heterostructures are
promising candidates for realizing topological superconductivity and Majorana
modes. In these systems, a vortex pinned by a pre-fabricated antidot in the
superconductor can host Majorana zero-energy modes (MZMs), which are exotic
quasiparticles that may enable quantum information processing. However, a major
challenge is to design devices that can manipulate the information encoded in
these MZMs. One of the key factors is to create small and clean antidots, so
that the MZMs, localized in the vortex core, have a large gap to other
excitations. If the antidot is too large or too disordered, the level spacing
for the subgap vortex states may become smaller than temperature. In this
paper, we numerically investigate the effects of disorder, chemical potential,
and antidot size on the subgap vortex spectrum, using a two-dimensional
effective model of the topological insulator surface. Our model allows us to
simulate large system sizes with vortices up to 1.8 $\mu$m in diameter. We also
compare our disorder model with the transport data from existing experiments.
We find that the spectral gap can exhibit a non-monotonic behavior as a
function of disorder strength, and that it can be tuned by applying a gate
voltage.
