A Study of Carbon Substitutions in MgB_2 within the two-band Eliashberg   theory
  We study the effects of C substitutions in MgB_2 within the two-band model in the Eliashberg formulation. We use as input the B-B stretching-mode frequency and the partial densities of states N_{sigma}(EF) and N_{pi}(EF), recently calculated for Mg(B_{1-x}C_{x})_2 at various x values from first-principles density functional methods. We then take the prefactor in the Coulomb pseudopotential matrix, mu, and the interband scattering parameter, Gamma^{sigma pi}, as the only adjustable parameters. The dependence on the C content of Tc and of the gaps (Delta_{sigma} and Delta_{pi}) recently measured in Mg(B_{1-x}C_{x})_2 single crystals indicate an almost linear decrease of mu on increasing x, with an increase in interband scattering that makes the gaps merge at x=0.132. In polycrystals, instead, where the gap merging is not observed, no interband scattering is required to fit the experimental data. 
G. A. Ummarino, D. Daghero, R. S. Gonnelli and A. H. Moudden