Impact of Carrier Injector Design on the Threshold of Interband Cascade Lasers

TL;DR

The paper addresses how interband cascade laser (ICL) injector design affects threshold by linking carrier balance to Auger recombination. It introduces a microscopic NEGF framework with an 8-band $\mathbf{k}\cdot\mathbf{p}$ Hamiltonian, Poisson electrostatics, and a GW-based calculation of Auger rates, avoiding phenomenological coefficients; threshold is determined from the gain–loss balance $\Gamma g_{th} = \frac{1}{L}\ln\left(\frac{1}{R}\right) + (1-\Gamma)\alpha_{clad}$. Key findings show that heavy electron injector doping reduces threshold mainly by suppressing the $ehh$ Auger channel and parasitic intra-valence absorption, but Auger alone cannot explain the rise in threshold currents at high doping. Raising hole injector levels via Ga$_{1-x}$In$_x$Sb does not outperform electron injector doping due to increased intra-valence absorption. The work provides design principles for optimizing ICL injectors and lays groundwork for future non-equilibrium transport studies beyond quasi-equilibrium assumptions.

Abstract

We investigate theoretically how the injector region design of interband cascade lasers (ICLs) impacts the threshold carrier and current densities. The model combines a polarization-sensitive 8-band $\mathbf{k}\cdot\mathbf{p}$ calculation, electrostatics, and a microscopic calculation of Auger recombination rates. The carrier-carrier scattering is included to lowest order within the non-equilibrium Green's function formalism. It captures the combined effects of charge carrier redistribution, parasitic absorption and bias voltage on the Auger recombination rate. We show that heavily doping the electron injector suppresses the dominant multi-hole Auger recombination by reducing the hole population of the recombination quantum wells. This agrees with the experimental observation that the heavy doping reduces threshold currents. Yet, our model suggests that the Auger recombination alone is not sufficient to explain the increase of threshold currents at high doping concentrations. Furthermore, by introducing indium to the conventional GaSb hole injector wells, we explain the rule of thumb from experiments that raising the hole injector levels does not outperform the doping strategy. Our model provides physical insights toward optimization of ICL carrier injectors.

Figures (5)

• Figure 1: One active region period of the reference ICL in Table \ref{['table:design_variations']} at the potential drop per period of $375\ \mathrm{meV}$. (a) In-plane dispersion in the entire reciprocal space included in the simulations. (b) Density of states integrated over the in-plane momentum. The horizontal lines indicate the quasi Fermi levels. The dashed, dotted, and solid curves represent the conduction, heavy-hole, and light-hole bandedges of the bulk materials, respectively. (c) Energy-resolved electron and hole densities in quasi-equilibrium (colormap). Red and blue curves show the relative magnitude of the energy-integrated electron and hole densities, respectively.
• Figure 2: Simulation flow for calculating the threshold carrier densities and Auger recombination current. $V$ and $l$ are the potential drop per period and period length, respectively. The other quantities are defined in the main text.
• Figure 3: Optical spectra of the reference design in Table \ref{['table:design_variations']} at the threshold voltage $V_\mathrm{th}=375\ \mathrm{mV}$. (a) Net gain spectrum and its decomposition into the interband, intra-conduction and intra-valence contributions. Black line indicates the losses outside of the active region. (b) Intraband absorption coefficients at maximum gain as a function of the polarization angle with respect to the $x$--$y$ plane. In both panels, squares mark the values extracted for the analysis of the parasitic absorption.
• Figure 4: Impact of the injector design variations on ICL performance figures. Red stars indicate the reference structure in Table \ref{['table:design_variations']}. (a) Threshold carrier densities in the W-QW and injectors in the doping and indium series. The W-QW carrier densities are defined by Eqs. \ref{['eq:definition_average_carrier_density']}, whereas the injector counterparts are simply averaged by the injector lengths. (b) Optical absorption coefficients at the photon energies of maximum gain and their breakdown to the intra-conduction and intra-valence contributions. (c) Potential drop per period at the lasing thresholds. (d) Interband currents due to the $eeh$ and $ehh$ Auger mechanisms. (e--g) Corresponding data from the indium series at a fixed bias of 358 $\mathrm{mV}$. (h) Peak values of the interband gain without parasitic absorptions.
• Figure 5: Threshold current densities from the measurements Vurgaftman2011WeihPhDVurgaftmanBandsPhotons (triangles) and Auger currents from the simulations (curves) against sheet doping density.