# 2.4.7. 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{} section. This tutorial aims to reproduce the results obtained in [ChuangIEEE1996]:

Related files

• Chuang_1996_IEEE_GaN_QW_nnp.in

• Chuang_1996_IEEE_GaN_QW_postprocess.py (python script using nextnanopy)

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{}

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 (2.5.3.4).

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¶

The above figures show the Gamma bandedge of the Al0.3Ga0.7N-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:

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

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].

$$r^{spon}$$ for an Al0.3Ga0.7N-GaN quantum well with the carrier concentration $$n=3\times 10^{19}$$ cm$$^{-3}$$ on each polarization TE (x or y) and TM (z).

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

### Optical Gain¶

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

(2.4.7.2)$\alpha(\vec{\epsilon},\omega)=-g(\vec{\epsilon},\omega)$

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.

$$g(\vec{\epsilon},\omega)$$ for a Al0.3Ga0.7N-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).

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.