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81850612.Pdf View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Springer - Publisher Connector J Theor Appl Phys (2016) 10:1–6 DOI 10.1007/s40094-015-0193-5 RESEARCH Physical and mathematical justification of the numerical Brillouin zone integration of the Boltzmann rate equation by Gaussian smearing 1 1 2 3 1 Christian Illg • Michael Haag • Nicolas Teeny • Jens Wirth • Manfred Fa¨hnle Received: 12 August 2015 / Accepted: 27 September 2015 / Published online: 7 November 2015 Ó The Author(s) 2015. This article is published with open access at Springerlink.com Abstract Scatterings of electrons at quasiparticles or that this procedure is justified. We show with physical and photons are very important for many topics in solid-state mathematical arguments that this numerical procedure is in physics, e.g., spintronics, magnonics or photonics, and general correct, and we comment on critical points. therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of Keywords Electron scattering Á Boltzmann rate these scatterings, Fermi’s golden rule is used to calculate equations Á Brillouin zone integration Á Treatment of Diracs the transition rate from an initial state to a final state in a delta distribution 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 Bril- Introduction louin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement In solid-state physics, scatterings of electrons at periodic of the Dirac delta distribution by a Gaussian. The Dirac perturbations (quasiparticles or photons) are very important delta distribution appears in Fermi’s golden rule where it for many research fields and we give three examples in the describes the energy conservation among the interacting following: particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view 1. In all-optical switching experiments [1] a thin ferri- magnetic film, e.g., GdFeCo, is irradiated by a femtosecond laser pulse which can be linearly or & Manfred Fa¨hnle circularly polarized and thereafter a demagnetization [email protected] with subsequent switching of the magnetization can be Christian Illg observed under certain preconditions. The fundamental [email protected] mechanisms are strongly debated at the moment, Michael Haag however, electron–photon scatterings, electron–pho- [email protected] non scatterings and electron–magnon scatterings cer- Nicolas Teeny tainly play a big role for the demagnetization of the [email protected] ferrimagnetic film. Jens Wirth 2. In ultrafast demagnetization experiments [2] a thin [email protected] ferromagnetic film, e.g., Ni or Fe, is irradiated by a 1 Max Planck Institute for Intelligent Systems, Heisenbergstr. femtosecond laser pulse which is normally linearly 3, 70569 Stuttgart, Germany polarized and thereafter an ultrafast demagnetization 2 Max Planck Institute for Nuclear Physics, Saupfercheckweg (on the timescale of about 100 fs) without switching of 1, 69117 Heidelberg, Germany the magnetization can be observed. The magnetization 3 Institut fu¨r Analysis, Dynamik und Modellierung, Universita¨t recovers on a timescale of several picoseconds. Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany Despite many years of research the fundamental 123 2 J Theor Appl Phys (2016) 10:1–6 mechanisms are still unclear but scatterings of elec- where Z Z trons at phonons [3, 4] or at magnons [5] or at electrons X ÂÃ 1 3 3 0 k [6] have been discussed intensively. Win ¼ d k d k nj0k0 1 À njk Wj0k0;jk 4 X2 ð Þ 3. Spin-polarized currents are important for devices in BZ j;j0;k BZ BZ Z Z spintronics [7], e.g., spin-transistors or spin-diodes. X ÂÃ 1 3 3 0 k W k k n n 0 0 W 0 : The lifetime of the spin-polarized electrons is crucial out ¼ 2 d d jk 1 À j k jk;j0k X 0 BZ BZ for the spintronics devices. The lifetimes are deter- BZ j;j ;k mined by scatterings of electrons at quasiparticles and ð5Þ at interfaces or defects. XBZ is the Brillouin zone (BZ) volume and n is the dis- A correct numerical calculation of the various scattering tribution function for the electrons. processes is important for the understanding of these Often one is also interested in the rate of change of the effects in solid-state physics. In quantum mechanics, Fer- distribution function njk due to scattering which is also k mi’s golden rule gives the transition rate W 0 from an calculated with Boltzmann rate equations [10] jk;j0k Z X n ÂÃ initial electronic state Wjk in a solid with energy ejk to a dnjk 1 3 0 k 0 ¼ d k nj0k0 1 À njk Wj0k0;jk final electronic state W 0 0 with energy e 0 0 (j,j : band indi- dt X j k j k BZ j0;k BZ ð6Þ ces; k, k0: wave vectors) due to a periodic perturbation ÂÃo k arising from a (quasi)particle [8] À njk 1 À nj0k0 Wjk;j0k0 : 2 ÀÁ k 2p k So we have to calculate Brillouin zone integrals of the form W 0 0 ¼ M 0 0 Áde0 0 Àðe Æ hx Þ : ð1Þ jk;j k h jk;j k j k jk qk Z 3kgk d e k : Æhxqk is the energy of the involved (quasi)particle (q: d ð Þ ð ð ÞÞ ð7Þ BZ wave vector, k: polarization) which may be, e.g., photons, k k phonons, magnons, plasmons etc., with frequency xqk for Because the quantities ejk, ej0k0 , Wj0k0;jk, Wjk;j0k0 can be k absorption (plus sign) or emission (minus sign), and Mjk;j0k0 calculated numerically only for a finite number of k- is the scattering matrix element points, finite k-point grids have to be used for the numerical calculation of the total transition rate Wtotal or k 0 M 0 0 F W 0 0 W FW ; 2 jk;j k ¼h j k qk jki ð Þ of the rate of change of the distribution function dnjk=dt. Thereby, energy conservation e 0 0 e hx and where jFi and jF0i are the initial and final (quasi)particle j k ¼ jk Æ qk momentum conservation k Æ q ¼ k0 þ G have to be ful- states and Wqk is the scattering operator. Thereby, momentum conservation k Æ q ¼ k0 þ G is demanded (G: filled; however, energy conservation in combination with reciprocal lattice vector). Fermi’s golden rule is the first- momentum conservation is in general never fulfilled for a order approximation of the time-dependent quantum-me- finite k-point grid. Therefore, the Dirac delta distribution chanical perturbation theory. It implies that the scattering has to be replaced by a ‘‘smeared’’ delta function to processes are Markovian which means that a scattering obtain a result which approximates the integral (which is process does not depend on preceding scattering processes. done, e.g., in Refs. [3, 4, 11–13] and in very many Fermi’s golden rule is only valid in a time window where other papers). To do this, often the following equation is the perturbation time on the one hand must be short enough used Z Z because of the first-order approximation and on the other 1 e2ðkÞ d3kgðkÞ dðeðkÞÞ d3kgðkÞ pffiffiffi exp À hand must be long enough to replace the sinðxÞ=x-function 2 BZ BZ pr r appearing in the derivation of Fermi’s golden rule by the ð8Þ Dirac delta distribution. The validity of Fermi’s golden rule for a magnetization dynamics on the 100 fs timescale is and the smearing parameter r has to be chosen appropri- critically discussed in Ref. [4]. ately, see Sect. 3. This means that the contribution of a Normally, one is not interested in a specific transition certain grid point to the total transition rate Wtotal or to the k 0 rate of change of the distribution function dnjk=dt is small rate Wjk;j0k0 from an initial state Wjk to a final state Wj0k but if the energy conservation is fulfilled very badly, and vice in the total transition rate Wtotal from all initial states to all final states. Thereby k and k0 are related via k Æ q ¼ k0 þ G versa the contribution is large if the energy conservation is if the scattering is at a quasiparticle with wave vector q. fulfilled very well. However, from a mathematical point of This is calculated with Boltzmann rate equations [4, 9] view it is not obvious that Eq. (8) holds since the Dirac delta distribution is not a function and the smearing is with Wtotal ¼ Win À Wout ð3Þ 123 J Theor Appl Phys (2016) 10:1–6 3 respect to the energy e but the integration is with respect to Numerical integration of the Dirac delta the wave vector k. The problem is explained in more detail distribution in Sect. 2. Mathematical proofs of Eq. (8) under certain precondi- It is very well known that in integrals involving the Dirac tions can be found in Ref. [14], Theorem 7.2.1, and in delta distribution, the distribution can be replaced by a Ref. [15], Theorem 6.1.5; however, the proofs are for Gaussian for the limes r ! 0. It reads general distributions and are very abstract. We want to Z Z þ1 þ1 1 e2 show in this article that Eq. (8) is correct using also de gðeÞ dðeÞ¼lim de gðeÞ pffiffiffi exp À r!0 pr r2 physical arguments.
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