Effect of spin-orbit coupling on the electron-phonon interaction of the superconductors Pb and Tl

The influence of spin-orbit coupling (SOC) on the electron-phonon interaction and on the pairing strength is investigated for Pb and Tl within density-functional theory. For Pb we find that SOC increases the coupling constant lambda by 44%, solving the longstanding puzzle of too small lambda values obtained consistently in previous first-principles calculations. The origin of the SOC-induced enhancement of lambda lies both in a softening of the phonon spectrum and in an increase in the electron-phonon coupling matrix elements. In contrast, for the neighboring element Tl only weak effects of SOC on the phonon spectrum and the electron-phonon interaction is obtained, leading to an overall reduction in lambda by 12%. In both cases, the inclusion of SOC improves the agreement with experiment.