  We analyze the interplay between superconductivity and the formation of bound
pairs of fermions (BCS-BEC crossover) in 2D models of interacting fermions with
small Fermi energy EF and weak attractive interaction, which extends to
energies well above EF. The 2D case is special because one has to distinguish
between bound state formation and superconductivity already at weak coupling.
We briefly review the situation in the one-band model and then consider two
different two-band models: the one with a hole and an electron band, and the
one with two electron bands. In each case we obtain the bound state energy 2 E0
for two fermions in a vacuum and solve the set of coupled equations for the
pairing gaps and the chemical potentials to obtain the onset temperature of the
pairing, Tins and the quasiparticle dispersion. We then compute the superfluid
stiffness and obtain the actual Tc. We show that, at EF >> E0 the behavior of
both two-band models is BCS-like in the sense that Tc almost coincides with
Tins and both are much smaller than EF. At EF < E0, the two models behave
differently: in the model with two electron bands, Tins ~E0/log(E0/EF) and Tc
~EF << Tins. In between Tins and Tc the system displays preformed pair
behavior. In the model with a hole and an electron band, Tins ~ Tc and both
remain finite at EF=0 The preformed pair behavior still exists in this model
because Tc is numerically smaller than Tins. For both models we re-express Tins
in terms of the fully renormalized two-particle scattering amplitude by
extending to two-band case the method pioneered by Gorkov and Melik-Barkhudarov
back in 1961. We apply our results to to Nb-doped SrTiO3 and to Fe-pnictides
and Fe-chalcogenides, particularly to FeSe, in which superconducting gap is
comparable to the Fermi energy.
