Rigorous results for the electrical conductivity due to electron–phonon scattering

Journal of Mathematical Physics, Volume 62, Issue 2, February 2021. We present a rigorous solution of the Boltzmann equation for the electron–phonon scattering problem in three spatial dimensions in the limit of low temperatures. The different temperature scaling of the various scattering rates turns the temperature into a control parameter that is not available in classical kinetic theory and allows for a rigorous proof of Bloch’s T5 law. The relation between the Boltzmann equation and the Kubo formula is also discussed, as well as implications for the scattering of electrons by excitations other than phonons.