Generic picture of the emission properties of III-nitride polariton laser diodes: Steady state and current modulation response

The main emission characteristics of electrically driven polariton lasers based on planar GaN microcavities with embedded InGaN quantum wells are studied theoretically. The polariton emission dependence on pump current density is first modeled using a set of semiclassical Boltzmann equations for the exciton polaritons that are coupled to the rate equation describing the electron-hole plasma population. Two experimentally relevant pumping geometries are considered, namely the direct injection of electrons and holes into the strongly coupled microcavity region and intracavity optical pumping via an embedded light-emitting diode. The theoretical framework allows the determination of the minimum threshold current density $J_{\text{thr},\text{min}}$ as a function of lattice temperature and exciton-cavity photon detuning for the two pumping schemes. A $J_{\text{thr},\text{min}}$ value of 5 and 6 A cm${}^{-2}$ is derived for the direct injection scheme and for the intracavity optical pumping one, respectively, at room temperature at the optimum detuning. Then an approximate quasianalytical model is introduced to derive solutions for both the steady-state and high-speed current modulation. This analysis makes it possible to show that the exciton population, which acts as a reservoir for the stimulated relaxation process, gets clamped once the condensation threshold is crossed, a behavior analogous to what happens in conventional laser diodes with the carrier density above threshold. Finally, the modulation transfer function is calculated for both pumping geometries and the corresponding cutoff frequency is determined.

Figure 1

Schematic 3D cross section of an InGaN/GaN MQW polariton LD based on an intracavity pumping geometry.

Figure 2

Results of a transfer matrix simulation of the field intensity profile of a polariton LD centered at $\lambda =415$ nm along with the corresponding refractive index profile.

Figure 3

(a)–(c) Occupancy of the polariton ground state vs pump current density calculated for the two pumping geometries at various temperatures and detunings (see text for details).

Figure 4

Plots of $J_{\text{thr}}$ vs detuning and temperature for (a) the direct electrical and (b) the intracavity optical pumping schemes. The red dashed line in each plot corresponds to the evolution of $J_{\text{thr},\text{min}}$ as a function of temperature. (c) Evolution of the optimum detuning (black solid and black dashed lines) and condensation threshold current density (red solid and red dashed lines) at the optimum detuning as a function of lattice temperature for the direct electrical (solid lines) and the intracavity (dashed lines) pumping schemes.

Figure 5

Averaged scattering rates $a$, $b$, and $c$ as a function of detuning obtained by fitting the full semiclassical Boltzmann system of equations at various temperatures for the two pumping schemes: electrical (solid lines) and intracavity (dashed lines) pumping.

Figure 6

Left-hand side vertical scale: evolution of $n_{x_{\infty }}$ as a function of temperature calculated at the optimum detuning using the exact expressions for the electrical (connected black dots) and the intracavity (connected black circles) pumping geometries (see text for details). Right-hand side vertical scale: relative deviation between the exact and the approximated expressions for the electrical (connected red dots) and the intracavity (connected red circles) pumping geometries.

Figure 7

(a) Polariton condensate occupation number vs current density for the electrical (red line) and intracavity (black line) pumping geometries determined at 320 K and at the optimum detuning. (b) Frequency dependence ($\nu =\omega /2\pi$) of the square modulus of the modulation transfer function, ${\left|H\left(\omega \right)\right|}^2$. Each curve corresponds to one of the steady-state solutions indicated in Fig. 7a.