Simple electron-electron scattering in non-equilibrium Green's function simulations

Abstract

Abstract. In this work we include electron-electron interaction beyond Hartree-Fock level in our non-equilibrium Green's function approach by a crude form of GW through the Single Plasmon Pole Approximation. This is achieved by treating all conduction band electrons as a single effective band screening the Coulomb potential. We describe the corresponding self-energies in this scheme for a multi-subband system. In order to apply the formalism to heterostructures we discuss the screening and plasmon dispersion in both 2D and 3D systems. Results are shown for a four well quantum cascade laser with different doping concentration where comparisons to experimental findings can be made. IntroductionSince the quantum cascade laser (QCL) was introduced more than 20 years ago [1] it has continuously been improved and redesigned to operate from the mid-infrared all the way down to the Terahertz (THz) range. Compact laser sources at these wavelengths are valuable for spectroscopic applications [2] but a major problem is that THz QCLs do not operate at room temperature. This can be overcome via difference frequency generation which recently has been demonstrated at powers in the milli-watt range [3]. Direct THz-QCLs still generate a lot of interest in the research community due to the promise of higher wall-plug efficiency and the prospects of miniaturization of cryo-coolers.The main temperature degrading mechanism of the THz QCLs is currently not fully understood and is still debated by the community [4]. This is a challenge to theory and there is a great need of realistic modeling tools that are able to treat all important quantum effects on the same footing. A summary of different methods for simulating these structures is found in [5]. Monte Carlo simulations have shown that electron-electron interaction, beyond the meanfield or Hartree approximation can influence the dynamics of THz QCLs [6,7,8].In this work we include and study the effects of a simple electron-electron scattering via the Single Plasmon Pole Approximation (SPPA) [9,10]. In this approximation we capture both the static limit as well as dynamic effects. This gives an energy dependent (non-local in time) interaction beyond the Hartree-Fock approximation. This has been studied in a similar model with promising results [11], and with this work we want to adapt the idea into our model described in Ref. [12]. In other methods based on Non-Equilibrium Green's Functions (NEGF)

Theory and measurements of harmonic generation in semiconductor superlattices with applications in the 100 GHz to 1 THz range

“…A temperature T = 300 K and an electron density of 1.5 × 10 18 cm −3 have been used as input to the NEGF calculations [22,24]. The contact area used to compare the measured current and NEGF calculated current density is 1.26 × 10 −12 m 2 .…”

“…In this paper we are not interested in circuit-equivalent approaches nor in phenomenological parameter fitting. Instead we focus on microscopic predictions of HHG in semiconductor superlattices, starting from nonequilibrium Green's functions (NEGF) methods [22][23][24]. Among the results presented here, we highlight: (i) Nonsymmetric current-voltage curves are explained with a microscopic NEGF approach, which can be adjusted by a nonsymmetric generalization of the Tsu and Esaki [25] formula.…”

Simulating terahertz quantum cascade lasers: Trends from samples from different labs

“…While our standard model evaluates all distribution functions self-consistently and thus does not require this concept at all, we need the electron temperature for the plasmon occupations in the single plasmon-pole approximation used to approximate the GW result. 32 In the following, we will model the conduction band of the quantum cascade laser as one effective band, with an electron temperature T e as one of its properties. A bias over this structure will heat the electrons, and they will subsequently relax emitting optical phonons.…”

Two-well quantum cascade laser optimization by non-equilibrium Green's function modelling

“…This is expected to increases the current density and decreases the gain. Since we cannot fully model the e-e interactions 36 , we restrict the doping concentration of the grown devices to an areal doping density of 4.5·10 10 cm −1 (corresponding to a volume doping density of 1.5 · 10 17 cm −3 of the doped 3 nm region, and an average period volume density of ∼ 1.4 · 10 16 cm −3 ), where we expect the effect to be moderate.…”

“…However, the simulations presented above do not include e-e scattering. In order to check the relevance of this scattering mechanism, we include it within a simplified GW approximation 36 . This results in a better thermalization of the electron distribution within the subbands, as seen in the right part of Fig.…”