Modeling of electron–electron scattering in Monte Carlo simulation of quantum cascade lasers

Abstract

A theoretical model of electron-electron scattering in multisubband systems is proposed and used to set up a Monte Carlo simulator of quantum cascade lasers. Special features of the electron-electron scattering model are the following: ͑i͒ A fast and accurate computation of bare potential matrix elements by means of Fourier analysis is developed. ͑ii͒ A screening model is proposed that allows us to describe intersubband matrix elements. ͑iii͒ Nonequilibrium screening factors, defined through an effective subband temperature for each subband, are periodically reevaluated. ͑iv͒ The developed algorithm makes use of rejection procedures in order to determine the correct number of scattering events as well as the distribution of the final states. Other characteristics of the model are the following: the energy levels and the wave functions are determined in a self-consistent way, the Pauli exclusion principle is included, and the periodicity of the structure is accounted for. This model is applied to the study of a terahertz resonant phonon quantum cascade laser. A large influence of the screening model on the subband population is demonstrated. For the considered design, emission at a frequency as low as 1 THz is confirmed. We have found that the magnitude of population inversion phenomena may be strongly sensitive to electron-electron scattering, reducing the possible performance near 1 THz.

Density-matrix theory of the optical dynamics and transport in quantum cascade structures: The role of coherence

“…The QCL has been the subject of extensive theoretical research. The stationary properties of QCLs were studied by a rate equation 10,11 and a Boltzmann-type approach 12,13,14,15,16,17 as well as a quantum theory employing both nonequilibrium Green's functions 18,19,20,21,22 and density-matrix theory. 23,24,25 Here, the gain, the current-voltage characteristics, and the stationary charge distributions have been established.…”

Monte Carlo modeling applied to studies of quantum cascade lasers

“…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.…”

“…This discrepancy has been explained as resulting from lack in the semiclassical MC model of dephasing effects. Bonno et al (2005Bonno et al ( , 2006 used this structure as a testbed for the MC research of e-e scattering and multi-subband screening models. In other research (Williams et al 2003b) reasonable agreement of maximal operating temperature has been reported: given by MC simulation $ 85 K for surfaceplasmon waveguide and $ 160 K for metal-metal waveguide versus experimentally observed values of 92 and 137 K, respectively.…”

Coherence and spatial resolution of transport in quantum cascade lasers

“…This defines the hopping model [14] which has been frequently applied to QCLs [9][10][11][12][13]. It is usually solved by the Monte Carlo technique and provides the stationary occupations f α (k).…”

“…Conventionally, QCLs are modeled by rate equation schemes either for the average electron densities in the subbands [6][7][8], or the occupation of the individual states [9][10][11][12][13]. The latter ones are often simulated with the Monte-Carlo technique, which suggests this denomination, albeit the term hopping transport [14,15] seems to be more appropriate for this type of models.…”