  Majorana zero modes, with prospective applications in topological quantum
computing, are expected to arise in superconductor/semiconductor interfaces,
such as $\beta$-Sn and InSb. However, proximity to the superconductor may also
adversely affect the semiconductor's local properties. A tunnel barrier
inserted at the interface could resolve this issue. We assess the wide band gap
semiconductor, CdTe, as a candidate material to mediate the coupling at the
lattice-matched interface between $\alpha$-Sn and InSb. To this end, we use
density functional theory (DFT) with Hubbard U corrections, whose values are
machine-learned via Bayesian optimization (BO) [npj Computational Materials 6,
180 (2020)]. The results of DFT+U(BO) are validated against angle resolved
photoemission spectroscopy (ARPES) experiments for $\alpha$-Sn and CdTe. For
CdTe, the z-unfolding method [Advanced Quantum Technologies, 5, 2100033 (2022)]
is used to resolve the contributions of different $k_z$ values to the ARPES. We
then study the band offsets and the penetration depth of metal-induced gap
states (MIGS) in bilayer interfaces of InSb/$\alpha$-Sn, InSb/CdTe, and
CdTe/$\alpha$-Sn, as well as in tri-layer interfaces of InSb/CdTe/$\alpha$-Sn
with increasing thickness of CdTe. We find that 16 atomic layers (3.5 nm) of
CdTe can serve as a tunnel barrier, effectively shielding the InSb from MIGS
from the $\alpha$-Sn. This may guide the choice of dimensions of the CdTe
barrier to mediate the coupling in semiconductor-superconductor devices in
future Majorana zero modes experiments.
