Physical and mathematical justification of the numerical Brillouin zone integration of the Boltzmann rate equation by Gaussiansmearing

Scatterings of electrons at quasiparticles orphotons are very important for many topics in solid-statephysics, e.g., spintronics, magnonics or photonics, andtherefore a correct numerical treatment of these scatteringsis very important. For a quantum-mechanical description ofthese scatterings, Fermi’s golden rule is used to calculatethe transition rate from an initial state to a final state in afirst-order time-dependent perturbation theory. One cancalculate the total transition rate from all initial states to allfinal states with Boltzmann rate equations involving Brillouinzone integrations. The numerical treatment of theseintegrations on a finite grid is often done via a replacementof the Dirac delta distribution by a Gaussian. The Diracdelta distribution appears in Fermi’s golden rule where itdescribes the energy conservation among the interactingparticles. Since the Dirac delta distribution is a not afunction it is not clear from a mathematical point of viewthat this procedure is justified. We show with physical andmathematical arguments that this numerical procedure is ingeneral correct, and we comment on critical points.