Paper Information

Journal:   JOURNAL OF THEORETICAL AND APPLIED PHYSICS (IRANIAN PHYSICAL JOURNAL)   MARCH 2016 , Volume 10 , Number 1; Page(s) 1 To 6.
 
Paper: 

PHYSICAL AND MATHEMATICAL JUSTIFICATION OF THE NUMERICAL BRILLOUIN ZONE INTEGRATION OF THE BOLTZMANN RATE EQUATION BY GAUSSIAN SMEARING

 
 
Author(s):  ILLG CHRISTIAN, HAAG MICHAEL, TEENY NICOLAS, WIRTH JENS, FAHNLE MANFRED*
 
* MAX PLANCK INSTITUTE FOR INTELLIGENT SYSTEMS, HEISENBERGSTR. 3, 70569 STUTTGART, GERMANY
 
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 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.

 
Keyword(s): ELECTRON SCATTERING, BOLTZMANN RATE EQUATIONS, BRILLOUIN ZONE INTEGRATION, TREATMENT OF DIRACS DELTA DISTRIBUTION
 
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