First principles study of the spin-orbit coupling effect on the Tl-Pb superconducting alloy

We have studied the influence of spin-orbit coupling (SOC) on the phonon dispersion, the electron-phonon (e-ph) coupling and on the superconducting properties for the Pb-Tl alloy in the stable fcc-phase doping regime. This system have been studied within the framework of density functional perturbation theory, using a mixed-basis pseudopotential method and the virtual crystal approximation (VCA) for modeling the alloy. The Eliashberg spectral function ($\alpha^{2}F(\omega)$) and the electron-phonon coupling parameter ($\lambda$) have been calculated with and without SOC. The observed effects of SOC in the full phonon dispersion and $\alpha^{2}F(\omega)$ consist in a softening of the phonon frequencies and an increase of the e-ph coupling matrix elements, which become weaker on the Tl-rich side. SOC enhances $\lambda$ by as much as 48\% in some cases and improves its overall behavior as a function of the concentration for the alloy, leading to a very nice agreement with experimental data from tunneling measurements.