Electron relaxation in metals: theory and exact analytical solutions

The nonequilibrium dynamics of electrons is of a great experimental and theoretical value, providing important microscopic parameters of the Coulomb and electron-phonon interactions in metals and other cold plasmas. Because of the mathematical complexity of collision integrals, theories of electron relaxation often rely on the assumption that electrons are in a “quasiequilibrium” QE with a time-dependent temperature, or on the numerical integration of the time-dependent Boltzmann equation. We transform the integral Boltzmann equation to a partial differential Schrödinger-type equation with imaginary time in a one-dimensional “coordinate” space reciprocal to energy which allows for exact analytical solutions in both cases of electron-electron and electron-phonon relaxations. The exact relaxation rates are compared with the QE relaxation rates at high and low temperatures.