Impact of inhomogeneous broadening on optical polarization of high-inclination semipolar and nonpolar ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}\text{/}\mathrm{GaN}$ quantum wells

We investigate the influence of inhomogeneous broadening on the optical polarization properties of high-inclination semipolar and nonpolar ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN quantum wells. Different planar m-plane and $\left(20\overline{2}\overline{1}\right)$ samples were grown (including core-shell microrods) and have been characterized by excitation-dependent polarization-resolved confocal micro-photoluminescence. The measured degree of linear polarization (DLP) is compared to theoretical predictions obtained by Fermi-Dirac statistical filling of the electronic band structure calculated by the $\mathbf{k}·\mathbf{p}$ envelope function method. We show that our measured DLP at room temperature, as well as values reported by other groups, are systematically higher than the theoretical predictions. We propose to solve this discrepancy between theory and experiment by introducing inhomogeneous broadening in our calculations. Considering indium content fluctuations and the localization lengths of electrons and holes, different effective broadenings are applied to different subsets of subbands. We thereby show that inhomogeneous broadening leads to an increase of the DLP at room temperature. Furthermore, the dependence of the optical properties on the excitation density is better reproduced. Looking at the DLP as a function of the temperature gives us insight into the thermalization dynamics of charge carriers.

Figure 1

LT (10 K) and RT polarization-resolved PL spectra at low excitation density of the 3.3 nm thick $m$-plane ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ QW. A large FWHM of 194 meV and a high DLP of 0.9 are measured at LT. The RT DLP of 0.8 is slightly lower. The inset shows the projection $c^{\prime }$ of the $c$ axis on the growth plane $x^{\prime }{y}^{\prime }$ for an arbitrary growth direction $z^{\prime }$ $(\theta =\pi /2$ for the $m$-plane).

Figure 2

$\mathbf{k}·\mathbf{p}$ band structure computation results for an $m$-plane ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN SQW. Energy splitting $\mathrm{\Delta }E$ between A1 and B1 valence subbands at $\mathrm{\Gamma }$ and optical polarization $P_{A1}$ of the A1 valence subband versus the indium content $x_{In}$ of the QW for different QW thicknesses $L_{QW}$.

Figure 3

LT and RT DLPs of all investigated planar ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN SQW samples at low excitation density versus their FWHM.

Figure 4

RT DLP at low excitation density versus the average emission wavelength found in the literature and of all samples investigated in this work. Data obtained on the $m$-plane facets of ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN core-shell microrods are plotted as a false color distribution. Dashed lines show the theoretical predictions of the DLP obtained by Fermi-Dirac statistical filling of the electronic band structure calculated by the $\mathbf{k}·\mathbf{p}$ envelope function method. Continuous lines show the theoretical predictions of the DLP including inhomogeneous broadening in the calculations.

Figure 5

Valence band structure of a 3.3 nm thick $m$-plane ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ QW. The state composition expressed in the growth-plane momentum operator basis $\left\{|X^{\prime }\rangle ,|Y^{\prime }\rangle ,|Z^{\prime }\rangle \right\}$ is color coded. Anisotropic strain leads to two sets of subbands A and B which have almost pure $|Y^{\prime }\rangle$ and $|X^{\prime }\rangle$ character, respectively. Two different effective broadening parameters ${\gamma }_A$ and ${\gamma }_B$ apply to these sets of subbands.

Figure 6

State composition of the A1 valence subband at the $\mathrm{\Gamma }$ point expressed in the growth plane momentum operator basis $\left\{|X^{\prime }\rangle ,|Y^{\prime }\rangle ,|Z^{\prime }\rangle \right\}$ as a function of the crystal orientation $\theta$ of a 3.3 nm thick ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$/GaN QW for different indium content $x_{In}$.

Figure 7

Indium content fluctuations in the QW generate different potential landscapes (dashed lines) for the first conduction subband CB1 and the A1 and B1 valence subbands with ${\gamma }_{CB1}>{\gamma }_{A1}>{\gamma }_{B1}$. Charge carriers average the potential fluctuations over their localization length resulting in ${\gamma }_A>{\gamma }_B\gg {\gamma }_{CB}$ (continuous lines).

Figure 8

Dependence of the DLP on the two effective broadening parameters ${\gamma }_A$ and ${\gamma }_B$ for a 3.3 nm thick $m$-plane ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ QW at low carrier density $\left(n=1\times {10}^9{\text{cm}}^{-2}\right)$ and RT with $E_U=10$ meV. The contour corresponding to the measured DLP at RT (0.8) is highlighted.

Figure 9

Density of states of A1 and B1 valence subbands of a 3.3 nm thick $m$-plane ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ QW for different effective broadening parameters ${\gamma }_A$ and ${\gamma }_B$. Hole densities in each subband and occupancy (Fermi-Dirac distribution) at carrier density $n=1\times {10}^9{\text{cm}}^{-2}$ are shown for each set of broadening parameters.

Figure 10

Simulation results for a 3.3 nm thick $m$-plane ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ QW at low carrier density $\left(n=1\times {10}^9{\text{cm}}^{-2}\right)$. (a) Dependence of the DLP on the Urbach energy $E_U$ at LT and RT for different effective broadening parameters ${\gamma }_A$ at a fixed ratio ${\gamma }_B/{\gamma }_A=0.85$. (b) Dependence of the LT DLP on the effective broadening parameter ${\gamma }_A$ (with fixed ratio ${\gamma }_B/{\gamma }_A=0.85)$ and the Urbach energy $E_U$. The contour corresponding to the measured DLP at LT (0.9) is highlighted.

Figure 11

(a) Dependence of the DLP on the temperature and the effective broadening parameter ${\gamma }_A$ (with fixed ratio ${\gamma }_B/{\gamma }_A=0.85)$ for a 3.3 nm thick $m$-plane ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ QW at low carrier density $\left(n=1\times {10}^9{\text{cm}}^{-2}\right)$ and $E_U=10\text{meV}$. The highlighted white contour shows the measured DLP versus temperature from which ${\gamma }_A\left(T\right)$ can be extrapolated. (b) Measured peak energy and FWHM as a function of the temperature. The Varshni shift of ${\mathrm{In}}_{0.18}{\mathrm{Ga}}_{0.82}\mathrm{N}$ is reported as guide to the eye. Extrapolated ${\gamma }_A\left(T\right)$ and ${\gamma }_{\mathrm{non}\text{−}\mathrm{th}}\left(T\right)$ are plotted.

Figure 12

Measured DLP versus the excitation density (top $x$ axis) for the 3.3 nm thick $m$-plane ${\mathrm{In}}_x{\mathrm{Ga}}_{1-x}\mathrm{N}$ SQW at LT and RT. Dashed (solid) lines show the theoretical predictions of the DLP without (with) inhomogeneous broadening versus the carrier density (bottom $x$ axis).