Eliashberg theory of the critical temperature and isotope effect. Dependence on bandwidth, band-filling, and direct Coulomb repulsion

The author investigates the dependence of the superconducting critical temperature and the isotope coefficient on bandwidth, band-filling, and the direct Coulomb repulsion, within Eliashberg theory. The Migdal approximation is assumed throughout, and the Coulomb repulsion is modeled by the Hubbard U and treated in the simplest approximation. A constant density of states with a finite bandwidth is assumed. While, in principle, small isotope coefficients are possible, it is unlikely that the isotope coefficient can ever be negative within this model. Furthermore, it is difficult to achieve small isotope coefficients for realistic parameters. Finally, a possible means by which large isotope coefficients can occur at low filling is discussed