Phonon limited thermoelectric transport in Pb

We present a fully ab initio based scheme to compute transport properties, i.e. the electrical conductivity {\sigma} and thermopower S, in the presence of electron-phonon interaction. Therefore, we explicitly investigate the k-dependent structure of the Eliashberg spectral function , the coupling strength, the linewidth and the relaxation time {\tau}. We obtain a state-dependent {\tau} and show its necessity to reproduce the increased thermopower at temperatures below the Debye temperature, without accounting for the phonon-drag effect. Despite the detailed investigations of various k and q dependencies, the presented scheme can be easily applied to more complicated systems.