Memory Function Representation for the Electrical Conductivity of Solids

We derive a formula for the electrical conductivity of solids that includes relaxation and dissipation due to scattering such as by disorder, phonons, or others which do not conserve momentum, while also including quantum coherence. The derivation is based on the Kubo formula, with a Mori memory function approach to include dissipation effects at all orders in the relaxation interaction. It offers a practical method to evaluate the conductivity with electronic-structure codes, avoiding the complications and limitations of both the Kubo formula in the thermodynamic limit and the semiclassical nature of Boltzmann transport. The derivation of our formula provides a method applicable to other transport coefficients and correlation functions. In the present case, it captures interband contributions to the conductivity as well as the metal-insulator phase transition.