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

TL;DR

This paper provides a rigorous physical and mathematical justification for using Gaussian smearing in Brillouin zone integrations of the Boltzmann rate equation, ensuring accurate numerical treatment of electron scattering processes.

Contribution

It offers a formal justification for the common numerical practice of Gaussian smearing in Brillouin zone integrations, clarifying its validity from both physical and mathematical perspectives.

Findings

Gaussian smearing is mathematically justified for energy conservation in Fermi's golden rule.

The procedure accurately approximates the Dirac delta distribution in numerical integrations.

Critical points and limitations of the smearing method are discussed.

Abstract

Scatterings of electrons at quasiparticles or photons are very important for many topics in solid state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings Fermi's golden rule is used in order to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.