Transition Temperature of Narrow-Band Superconductors

Abstract

The equation for the vertex part of a Cooper pair is developed in the Wannier representation to highlight the atomic nature of the electrons responsible for superconductivity in narrow energy bands. For a nondegenerate band and a short-range interaction between two electrons at sites ${\overrightarrow{\mathrm{n}}}_1$ and ${\overrightarrow{\mathrm{n}}}_2$, the transition temperature $T_c$ is determined by a small set of coupled integral equations in ${\overrightarrow{\mathrm{n}}}_1-{\overrightarrow{\mathrm{n}}}_2$ and in the energy variable $\omega$. With contact interaction, ${\overrightarrow{\mathrm{n}}}_1={\overrightarrow{\mathrm{n}}}_2$, only a single equation in $\omega$ remains as the defining equation for $T_c$. The solution has the Bardeen-Cooper-Schrieffer (BCS) form with an attractive interaction depending on a phonon Green's function in site space and with a repulsive interaction determined by the intra-atomic Coulomb integral $U$. The isotope effect is calculated as a function of $U$; the result can account for small negative or even positive effects, as observed in transition metals. For a degenerate band, the vertex part depends on the site variables $\overrightarrow{\mathrm{n}}$ and on the orbital indices $i$, the latter denoting a set of localized orbitals which transform according to a degenerate representation of the crystal group. $T_c$ is calculated in the contact model for a cubic ${\Gamma }_{25}^{\prime }$ band. The result contains the total density of states at the Fermi surface and the intraorbital and interorbital interactions weighted with factors $\frac{1}{3}$ and $\frac{2}{3}$, respectively. The lowering of $T_c$ by long-range Coulomb interactions due to exchange effects is also briefly discussed.