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

الموضوعات : Journal of Theoretical and Applied Physics

Christian Illg ¹ (Max Planck Institute for Intelligent Systems)
Michael Haag ² (Max Planck Institute for Intelligent Systems)
Nicolas Teeny ³ (Max Planck Institute for Nuclear Physics)
Jens Wirth ⁴ (Institut für Analysis, Dynamik und Modellierung, Universität Stuttgart)
Manfred Fähnle ⁵ (Max Planck Institute for Intelligent Systems)

تاريخ الإصدار : 21 الثلاثاء , جمادى الأولى, 1437

الکلمات المفتاحية: Electron scattering, Boltzmann rate equations, Brillouin zone integration, Treatment of Diracs delta dist,

ملخص المقالة :

AbstractScatterings 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 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.