Mott's formula for the thermopower and the Wiedemann-Franz law

Abstract

The Mott formula for the thermopower, $S=\frac{\left(\frac{{\pi }^2}{3}\right)\left(\frac{k_B^{2}T}{e}\right){\sigma }^{\prime }}{\sigma }$, and the Wiedemann-Franz law, $\frac{K}{\sigma T}={\left(\frac{k_B}{e}\right)}^2\left(\frac{{\pi }^2}{3}\right)$, are shown to be exact for independent electrons interacting with static impurities and phonons treated in the adiabatic approximation. This is true irrespective of the interaction strength. These results are derived using a Green's-function technique that emphasizes the importance of corrections to the free-electron heat current operator. These corrections have frequently been neglected in the past. The Green's-function technique is well suited for going beyond the adiabatic phonon approximation, and the implications of doing so are briefly discussed.