  We theoretically consider superconducting proximity effect, using the
Bogoliubov-de Gennes theory, in heterostructure sandwich-type geometries
involving a normal s-wave superconductor and a non-superconducting material
with the proximity effect being driven by Cooper pairs tunneling from the
superconducting slab to the non-superconducting slab. Applications of the
superconducting proximity effect may rely on an induced spectral gap or induced
pairing correlations without any spectral gap. We clarify that in a
non-superconducting material the induced spectral gap and pairing correlations
are independent physical quantities arising from the proximity effect. This is
a crucial issue in proposals to create topological superconductivity through
the proximity effect. Heterostructures of 3D topological insulator (TI) slabs
on conventional s-wave superconductor (SC) substrates provide a platform, with
proximity-induced topological superconductivity expected to be observed on the
"naked" top surface of a thin TI slab. We theoretically study the induced
superconducting gap on this naked surface. In addition, we compare against the
induced spectral gap in heterostructures of SC with a normal metal or a
semiconductor with strong spin orbit coupling and a Zeeman splitting potential
(another platform for topological superconductivity). We find that for any
model for the non-SC metal (including metallic TI) the induced spectral gap on
the naked surface decays as $L^{-3}$ as the thickness ($L$) of the non-SC slab
is increased in contrast to the slower $1/L$ decay of the pairing correlations.
Our distinction between proximity-induced spectral gap (with its faster spatial
decay) and pairing correlation (with its slower spatial decay) has important
implications for the currently active search for topological superconductivity
and Majorana fermions in various superconducting heterostructures.
