Two-band superconductivity in (AlMg)B2: Critical temperature and isotope exponent as a function of carrier density

We consider as a minimal model for superconductivity in (AlMg)B2 a two-band superconductor, having an anisotropic electronic structure described by two tight binding bands, considering a pairing scenario driven by an attractive interaction in which the interband pairing terms assume an important role. We solve the two-gap equations at the critical temperature T=Tc and calculate Tc and the chemical potential μ as a function of the number of carriers n for various values of pairing interaction, V, and cut-off energy, ωc. Using a self-consistent approach developed in a previous paper by two of the present authors, we calculate the isotope exponent α as a function of μ. We find that the isotope exponent shows a minimum in the energy range around a dimensional electronic topological transition (ETT) where the Fermi surface of one of the bands changes from 2D to 3D dimensionality. We have been able to fix the parameters of the theory, namely, the attractive interaction, V, and the cut-off frequency, ωc, by imposing experimental constraints on α, Tc and μ for undoped MgB2, specifically, 0.3, 40 K and 1.8 eV, respectively.