Quantitative spin-dependent electron-electron interaction to calculate the superconducting parameters μ and λ

Abstract

The spin-dependent Kukkonen-Overhauser (KO) effective electron-electron interaction in electron gas with a deformable background is used to calculate the BCS superconducting parameters $\ensuremath{\mu}$, ${\ensuremath{\mu}}^{*}$, and $\ensuremath{\lambda}$. The density and spin local field factors are utilized to incorporate the quantitative effects of exchange and correlation. The repulsive parameter $\ensuremath{\mu}$ is compared to results using the historical Thomas-Fermi or nearly equivalent random phase approximation (RPA) interactions. The resulting $\ensuremath{\mu}$ using the KO interaction is $45%$ larger at ${r}_{s}=1.65$; rising to $153%$ larger at ${r}_{s}=5.62$. Retardation reduces the effective repulsion, but ${\ensuremath{\mu}}^{*}$ is still $20\ensuremath{-}25%$ larger. The predicted superconducting transition temperature would be reduced by $5\ensuremath{-}20%$ using the McMillan formula. The attractive superconducting parameter $\ensuremath{\lambda}$, which depends on the electron-test charge interaction and the phonon spectrum, is also calculated using simple Debye-based models for phonon dispersion. This leads to a larger value of $\ensuremath{\lambda}$ than the same calculation using Thomas-Fermi or RPA interactions. Modern calculations of the phonon dispersion relations are often done using density functional theory. With the proper exchange and correlation kernel, the self-consistency of the method should yield the correct phonon dispersion relation and electron-phonon matrix elements. If the Thomas-Fermi or RPA interactions are used at any stage, the results for $\ensuremath{\lambda}$ are probably quantitatively inaccurate.