1D - Optics: Optical gain and spontaneous emission rate of strained GaN quantum well

Section author: Naoki Mitsui

Warning

This tutorial is under construction

In this tutorial, we calculate the optical gain and spontaneous emission rate of strained GaN quantum well using 8-band k.p model implemented in our optics{} (optional) section. This tutorial aims to reproduce the results obtained in [ChuangIEEE1996]:

• “Optical gain of strained wurtzite GaN quantum-well lasers” S. L. Chuang, IEEE Journal of Quantum Electronics (1996)

Related files

• Chuang_1996_IEEE_GaN_QW_nnp.in

• Chuang_1996_IEEE_GaN_QW_postprocess.py (python script using nextnanopy)

Table of contents

nextnano++ can calculate the spontaneous emission rate and optical gain in 2 different models.

1. “Semiclassical” calculation corresponds to classical{}

2. “Quantum” calculation corresponds to optics{} (optional)

For the 1st model, please refer to 1D - InGaAs Multi-quantum well laser diode. Roughly speaking, this model calculates the carrier densities either quantum mechanically or classically and the emission rate is calculated from them in a phenomenological way (2.5.4.4).

The calculation described here is the 2nd model. This starts from the Fermi’s golden rule (time-dependent perturbation theory) and electrons in a condensed matter are treated fully quantum mechanically. This model has the following characteristics:

• able to take into account the band-bending and band-mixing effect by strain

• distinguishes the different polarization

• requires less phenomenological parameter

• require the k.p parameters instead

(For most of the important materials, the parameters are already included in our database file.)

Structure

Figure 2.5.11.87 The band edges and Fermi energies for Al_0.3Ga_0.7N-GaN quantum well with the carrier concentration \(n=3\times 10^{19}\) cm\(^{-3}\) inside the well region.

The above figures show the Gamma band edge of the Al_0.3Ga_0.7N-GaN quantum well.

Please see the input file for the details.

The parameters used in this simulation are as follows.

Property	Symbol	Value [unit]
quantum well width	\(L_w\)	2.6, 5.0 [nm]
doping concentation	\(N_D\)	0 [cm-3]
carrier concentration in the well	\(n\)	1, 2, 3 \(\times\) 1019 [cm-3]
linewidth (FWHM)	\(\Gamma\)	0.0132 [eV]
temperature	\(T\)	300 [K]

Note

The piezo- and pyroelectricity are not yet taken into consideration here for the simplicity.

Results

Spontaneous emission rate

The formula used for the spontaneous emission calculation in optics section is as follows:

(2.5.11.11)\[r^{spon}(\vec{\epsilon}, \omega)=\frac{n_re^2E}{\pi\hbar^2c^3\varepsilon_0m_0^2}\frac{2}{V}\sum_{n>m}\sum_{\mathbf{k}_\parallel}|\vec{\epsilon}\cdot\vec{\pi}_{nm}(\mathbf{k}_\parallel)|^2 \mathcal{L}(E_n(\mathbf{k}_\parallel)-E_m(\mathbf{k}_\parallel)-E)f_n(\mathbf{k}_\parallel)(1-f_m(\mathbf{k}_\parallel)),\]

For the detail of the definition of each quantity and calculation scheme, please see our 1D - Optical absorption for interband and intersubband transitions.

Here we show this \(r^{spon}(\vec{\epsilon}, \omega)\) calculated for \(L_w=2.6\) [nm], \(L_w=5.0\) [nm] and each polarization. These results well agree with Fig.7 of [ChuangIEEE1996].

Figure 2.5.11.88 \(r^{spon}\) for an Al_0.3Ga_0.7N-GaN quantum well with the carrier concentration \(n=3\times 10^{19}\) cm\(^{-3}\) on each polarization TE (x or y) and TM (z). \(L_w=2.6\) [nm]

Figure 2.5.11.89 \(r^{spon}\) for an Al_0.3Ga_0.7N-GaN quantum well with the carrier concentration \(n=3\times 10^{19}\) cm\(^{-3}\) on each polarization TE (x or y) and TM (z). \(L_w=5.0\) [nm]

When we don’t apply the linewidth broadening, the result shows the exact energy where the emission by each pair of state starts.

Figure 2.5.11.90 TE emission rate in Figure 2.5.11.89 with (red dashed line) and without (blue line) line width broadening.

Optical Gain

The optics section can calculate the absorption spectra \(\alpha(\vec{\epsilon},\omega)\). This can be understood as a negative gain, i.e.

(2.5.11.12)\[\alpha(\vec{\epsilon},\omega)=-g(\vec{\epsilon},\omega)\]

For the details of the calculation scheme of \(\alpha(\vec{\epsilon},\omega)\), please see our 1D - Optical absorption for interband and intersubband transitions.

Here we show this \(g(\vec{\epsilon}, \omega)\) calculated for \(L_w=2.6\) [nm], \(L_w=5.0\) [nm] and polarization.

Figure 2.5.11.91 \(g(\vec{\epsilon},\omega)\) for a Al_0.3Ga_0.7N-GaN quantum well with the carrier concentration \(n=1,2,3\times 10^{19}\) cm\(^{-3}\) on each polarization TE (x or y) and TM (z). \(L_w=2.6\) [nm]

Figure 2.5.11.92 \(g(\vec{\epsilon},\omega)\) for a Al_0.3Ga_0.7N-GaN quantum well with the carrier concentration \(n=1,2,3\times 10^{19}\) cm\(^{-3}\) on each polarization TE (x or y) and TM (z). \(L_w=5.0\) [nm]

These results almost agrees with Fig.8 of [ChuangIEEE1996] except for the case when the gain peak is relatively low. This is because the models used here and [ChuangIEEE1996] apply the linewidth broadening in different steps.

Please help us to improve our tutorial. Should you have any questions or comments, please send to support [at] nextnano.com.