Multiband Superconductors Close to a 3D–2D Electronic Topological Transition

Within the two-band model of superconductivity, we study the dependence of the critical temperature T c and of the isotope exponent α in the proximity to an electronic topological transition (ETT). The ETT is associated with a 3D–2D crossover of the Fermi surface of one of the two bands: the σ subband of the diborides. Our results agree with the observed dependence of T c on Mg content in A $$_1-x Mg_x B_2$$ (A = Al or Sc), where an enhancement of T c can be interpreted as due to the proximity to a ‘shape resonance.’ Moreover we have calculated a possible variation of the isotope effect on the superconducting critical temperature by tuning the chemical potential.