Electron relaxation in metals: Theory and exact analytical solutions

TL;DR

This paper develops an exact analytical approach to model electron relaxation in metals by transforming the Boltzmann equation into a Schrödinger-like equation, enabling precise solutions for electron-electron and electron-phonon interactions.

Contribution

It introduces a novel analytical method transforming the Boltzmann equation into a Schrödinger-like form for exact solutions in electron relaxation studies.

Findings

Exact relaxation rates derived and compared with quasi-equilibrium rates.

Analytical solutions applicable to both electron-electron and electron-phonon relaxation.

High and low temperature regimes analyzed for relaxation dynamics.

Abstract

The non-equilibrium 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 "quasi-equilibrium" (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 Schroedinger-like 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 relaxation. The exact relaxation rates are compared with the QE relaxation rates at high and low temperatures.