Combined rate equation and Monte Carlo studies of electron transport in a GaAs/Al_0.45Ga_0.55As quantum-cascade laser

Abstract

Comparison of the Monte Carlo and rate equation methods as applied to the study of electron transport in a mid-infrared quantum cascade laser structure initially proposed by Page et al (2001 Appl. Phys. Lett. 78 3529) is presented for a range of realistic injector doping levels. An analysis of the difference between these two methods is given. It is suggested that justified approximations of the rate equation method, originated by imposing Fermi-Dirac statistics and the same electron effective temperature for each of the energy sub-bands, can be interpreted as partial inclusion of electron-electron interactions. Results of the rate equation method may be used as good initial conditions for a more precise Monte Carlo simulation. An algorithm combining rate equation and Monte Carlo simulations is examined. A reasonable agreement between the introduced method and a fully self-consistent resolution of Monte Carlo and Schrödinger coupled with Poisson equations is demonstrated. The computation time may be reduced when the combined algorithm is used.

Monte Carlo modeling applied to studies of quantum cascade lasers

“…We would like to notice that this particular structure design has been a basis for fabrication of QCL by Prof. M. Bugajski research group at Institute of Electron Technology, Warsaw, Poland (Kosiel et al 2009(Kosiel et al , 2011, and extensively investigated experimentally (Pierściński et al 2010(Pierściński et al , 2012Bugajski et al 2013Bugajski et al , 2014. In collaboration with this project microscopic description of electron behavior (Borowik et al 2010) and investigation of space-charge effect (Konupek et al 2011;Borowik et al 2012) have been performed by MC technique.…”

“…Approximations (Cassan 2000;Li et al 2008e) based on Fermi-Dirac statistics are used to determine electron sheet density on subbands and such estimate value is used to solve Schrödinger-Poisson equations self-consistently only once prior to MC simulation (Manenti et al 2003;Bonno et al 2005;Li et al 2008e). Modified algorithm has been proposed (Borowik et al 2012) and it requires preliminary calculation of subbands populations by rate equation method. More precise studies require self-consistent solution of Schrödinger-Poisson equations and MC simulation of electron population on energy subbands Jirauschek et al , 2010.…”

“…However, also in mid-infrared structures, the importance of e-e interactions cannot be neglected. Even if pure intersubband transport by e-e scattering is minimal, this mechanism leads to thermalization of electron distribution within subband, as well as pure energy transfer between various subbands (Iotti and Rossi 2001b;Borowik et al 2012), which significantly modifies the number of electron scattering events and, as a sconsequence, the resulting electron transfer between subbands. Hence, influences all device operating parameters.…”

Monte Carlo studies of the band-bending in GaAs/Al0.45Ga0.55As quantum-cascade laser

“…This effect can be included in modeling by solving selfconsistently coupled Schrödinger-Poisson equations, where in the Poisson equation the actual electrical charge distribution is used. Our calculations were performed using combined MC and RE algorithm [5] when the RE method is self-consistently coupled to the Schrödinger/ Poisson equation and after reaching the convergence, the MC simulation is executed.We studied the structure of mid-infrared QCL initially proposed by Page et al [12], and also recently fabricated at ITE, Warsaw [13]. The original device was prepared with an electron sheet density equal to 3.8×10 11 cm -2 , but the dependence of the device behavior on the injector doping density was also studied for higher values, as this parameter has a significant influence on understanding the operation and optimization on these devices [14].…”

“…The description of used simulation methods can be found in Ref. [2][3][4][5], and the detailed description of the electron microscopic behavior in this structure was presented elsewhere [3].Although, as will be shown later, the band-bending is also observed for lower densities, in Fig. 1 we present the shape of an electron wave-function in the QCL structures for an electron density of 8×10 11 cm -2 , as obviously for higher densities this effect is clearly visible in the scale of Fig.…”