  Interfacing a topological insulator (TI) with an $s$-wave superconductor (SC)
is a promising material platform that offers the possibility to realize a
topological superconductor through which Majorana-based topologically protected
qubits can be engineered. In our computational study of the prototypical SC/TI
interface between Nb and Bi$_2$Te$_3$, we identify the benefits and possible
bottlenecks of this potential Majorana material platform. Bringing Nb in
contact with the TI film induces charge doping from the SC to the TI, which
shifts the Fermi level into the TI conduction band. For thick TI films, this
results in band bending leading to the population of trivial TI quantum-well
states at the interface. In the superconducting state, we uncover that the
topological surface state experiences a sizable superconducting gap-opening at
the SC/TI interface, which is furthermore robust against fluctuations of the
Fermi energy. We also show that the trivial interface state is only marginally
proximitized, potentially obstructing the realization of Majorana-based qubits
in this material platform.
