  We study the low-energy physics of topological insulator (TI) nanoribbons
proximity-coupled to s-wave superconductors (SCs) by explicitly incorporating
the proximity effects that emerge at the TI-SC interface. We construct a
low-energy effective theory that incorporates the proximity effect through an
interface contribution containing both normal and anomalous terms and an
energy-renormalization matrix. We show that the strength of the
proximity-induced gap is determined by the transparency of the interface and
the amplitude of the low-energy TI states at the interface. Consequently, the
induced gap is strongly band-dependent and collapses for bands containing
states with low amplitude at the interface. We find that states with energies
within the bulk TI gap have surface-type character and, in the presence of
proximity-induced or applied bias potentials, have most of their weight near
either the top or the bottom surface of the nanoribbon. As a result, single
interface TI-SC structures are susceptible to experiencing a collapse of the
induced gap whenever the chemical potential is far enough from the value
corresponding to the bulk TI Dirac point and crosses weakly coupled bands. We
also find that changing the chemical potential in single-interface structures
using gate potentials may be ineffective, as it does not result in a
significant increase of the induced gap. On the other hand, we find that
symmetric structures, such as a TI nanowire sandwiched between two
superconductors, are capable of realizing the full potential of TI-based
structures to harbor robust topological superconducting phases.
