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We analyze the interplay between superconductivity and the formation of boundpairs of fermions (BCS-BEC crossover) in 2D models of interacting fermions withsmall Fermi energy EF and weak attractive interaction, which extends toenergies well above EF. The 2D case is special because one has to distinguishbetween bound state formation and superconductivity already at weak coupling.We briefly review the situation in the one-band model and then consider twodifferent two-band models: the one with a hole and an electron band, and theone with two electron bands. In each case we obtain the bound state energy 2 E0for two fermions in a vacuum and solve the set of coupled equations for thepairing gaps and the chemical potentials to obtain the onset temperature of thepairing, Tins and the quasiparticle dispersion. We then compute the superfluidstiffness and obtain the actual Tc. We show that, at EF >> E0 the behavior ofboth two-band models is BCS-like in the sense that Tc almost coincides withTins and both are much smaller than EF. At EF < E0, the two models behavedifferently: 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 pairbehavior. In the model with a hole and an electron band, Tins ~ Tc and bothremain finite at EF=0 The preformed pair behavior still exists in this modelbecause Tc is numerically smaller than Tins. For both models we re-express Tinsin terms of the fully renormalized two-particle scattering amplitude byextending to two-band case the method pioneered by Gorkov and Melik-Barkhudarovback in 1961. We apply our results to to Nb-doped SrTiO3 and to Fe-pnictidesand Fe-chalcogenides, particularly to FeSe, in which superconducting gap iscomparable to the Fermi energy.

physical review. b./physical review. b

Superconductivity vs bound state formation in a two-band superconductor  with small Fermi energy -- applications to Fe-pnictides/chalcogenides and  doped SrTiO3