Thermal conductivity and Lorenz ratio of metals at intermediate temperatures with mode-level first-principles analysis

The electrical and thermal transport properties of metals become complicated in the intermediate and low temperature range ($0.1{\mathrm{\ensuremath{\Theta}}}_{D}\ensuremath{-}{\mathrm{\ensuremath{\Theta}}}_{D}$, with ${\mathrm{\ensuremath{\Theta}}}_{D}$ being the Debye temperature) due to electron-phonon inelastic scattering. For the Wiedemann-Franz law, a notable feature is that the Lorenz ratio significantly deviates from the Sommerfeld value. Although qualitatively theoretical understanding has been developed for decades, a mode-level first-principles analysis is still lacking in this temperature range and a better understanding of inelastic scattering and thermal transport mechanisms is desirable. In this work, we take aluminum and copper as examples. We find that two factors are essential to correctly predict the thermal conductivity and Lorenz ratio in the intermediate temperature range. First, the momentum relaxation time should be used for electrical conductivity calculations, while the energy relaxation time should be used for electronic thermal conductivity calculations. Second, proper choice of broadening parameter and fine sampling in the Brillouin zone is vital. Using the mode-level description of inelastic electron-phonon scattering at intermediate temperatures, the correct Lorenz ratio can be obtained within the present scheme, while using only the energy or momentum relaxation time cannot capture the correct trend of Lorenz ratio. The calculation scheme can be expanded to other metallic systems and is valuable for a better understanding of the transport properties of metals.