  Recent experiments on certain Fe-based superconductors have hinted at a role
for paired electrons in "incipient" bands that are close to, but do not cross
the Fermi level. Related theoretical works disagree on whether or not
strong-coupling superconductivity is required to explain such effects, and
whether a critical interaction strength exists. In this work, we consider
various versions of the model problem of pairing of electrons in the presence
of an incipient band, within a simple multiband weak-coupling BCS
approximation. We categorize the problem into two cases: case(I) where
superconductivity arises from the "incipient band pairing" alone, and case(II)
where it is induced on an incipient band by pairing due to Fermi-surface based
interactions. Negative conclusions regarding the importance of incipient bands
have been drawn so far largely based on case(I), but we show explicitly that
models under case(II) are qualitatively different, and can explain the
non-exponential suppression of Tc, as well as robust large gaps on an incipient
band. In the latter situation, large gaps on the incipient band do not require
a critical interaction strength. We also model the interplay between phonon and
spin fluctuation driven superconductivity and describe the bootstrap of
electron-phonon superconductivity by spin fluctuations coupling the incipient
and the regular bands. Finally, we discuss the effect of the dimensionality of
the incipient band on our results. We argue that pairing on incipient bands may
be significant and important in several Fe-based materials, including LiFeAs,
FeSe intercalates and FeSe monolayers on strontium titanate, and indeed may
contribute to high critical temperatures in some cases.
