  We investigate Bardeen-Cooper-Schrieffer (BCS)- Bose-Einstein condensation
(BEC) crossover in a two-band superfluid Fermi gas with an energy shift between
the bands. When the intraband coupling in the cold (first) band is fixed as
weak, we find that in the case of vanishing interband interaction and in the
strong-coupling limit of the hot (second) band the system undergoes a
transition to a single-component configuration with the full suppression of the
first energy gap and with the full redistribution of particles between bands.
For non-vanishing interband interaction we reveal the non-monotonic dependence
of the energy gap in the first band vs intraband coupling in the second band
with the presence of a hump. In the case of weak interband coupling the system
shows a significant amplification of the intrapair correlation length of the
condensate in the first band in the strong-coupling regime of the second band,
which clearly indicates the coexistence of giant Cooper pairs and a bosonic
condensate even for nonzero temperatures. This can lead to a non-monotonic
temperature dependence of the second energy gap with a peak. Here predicted
coexistence of the giant Cooper pairs and bosonic molecules can be verified by
means of the visualization of vortex cores in the two-component atomic
condensates as well as in some iron-based superconductors.
