# 1D - Photoluminescence of Quantum Well¶

Attention

This tutorial is under construction.

- Input Files:
1D_PL_of_QW_absorption_nnp.in1D_PL_of_QW_nnp.in- Output Files:
bias_00000\Optics\spont_emission_power_region_longitudinal_nm.dat

- Scope:
In this tutorial, we show an approach how to model photoluminescence (PL) in 1D QW structures. The following is covered:

Short overview of the most essential groups which are needed in the input file for PL simulations

How to compute the absorption spectrum, when no experimental data is available

Results: photoluminescence spectra

Limitations of the simulation

- Important keywords:

`classical{ energy_resolved_density{} energy_distribution{} }`

`optics{ irradiation{} emission_spectrum{} quantum_region{} }`

`quantum`

,`current{}`

## Introduction¶

What are we modelling? In short, light impinges on the surface of the structure parallel to the growth direction.
A certain fraction of the total photon flux penetrates into the material and is absorbed,
which leads to generation of mobile charge carriers (*photogeneration*). These carriers are lifted into excited states.
If an excited electron recombines radiatively with a hole, light is emitted (*spontaneous emission*). As depicted in Figure 2.5.5.8,
the recombination process happens mainly in the quantum well.

The quantum well structure under consideration in this tutorial consists of the following material layers:

Layer |
Material |
Thickness (nm) |

1 |
Al |
500 |

2 |
GaAs |
7 |

3 |
Al |
500 |

Substrate |
GaAs |
1000 |

## Simulation scheme¶

In our model we treat the absorption and generation of charge carriers within a semiclassical approach. The current equation is calculated self-consistently within the Schrödinger and Poisson equations in order to get accurate charge carrier densities. Afterwards, the luminescence spectra are calculated quantum mechanically based on the occupied states.

**General approach**

One of the most important process in our simulation is the generation of charge carriers, which is governed by the generation rate \(G(x,E)\). The dependency on energy is described by the absorption coefficient \(\alpha(E)\). Since we assume not having experimental data for the absorption spectra available, we have to calculate \(\alpha (E)\).

Figure 2.5.5.9 visualizes the idea of our procedure. The **1. step** is running the input file
1D_PL_of_QW_absorption_nnp.in, which does not include any optical phenomena (photogeneration, emission, … ).
Then the **2. step** is to run the normal input file 1D_PL_of_QW_nnp.in, which includes generation of carriers,
using the imported absorption spectrum from the **1. step**.
Normally, one has to repeat the whole cycle, until the absorption spectra fully converge.
For simplicity, we assume that no additional repetition is needed.

**Input file**

The optical phenomena related to the irradiation, absorption and spontaneous emission
processes, which should be taken into account in the simulation, have to be specified in the `optics{}`

block.
The absorption process is modelled within a semiclassical approach calling `irradiation{}`

and `emission_spectrum{}`

.
The spontaneous emission is treated quantum mechanically inside the block `quantum_region{}`

:

```
optics{
irradiation{
illumination{ # specification of the light source, i.e. illumination spectrum
direction_x=1
gaussian_spectrum{
irradiance = $irradiance*1e4
wavelength = $peak_wavelength
gamma = 0.01
}
absorption{ # specification of absorption spectrum
import_spectrum{# choice of imported file with previously calculated abs spectra
import_from = "my_abs"
cutoff = yes
}
}
photo_generation{ # enabling photogeneration
output_energy_resolved = yes
}
output_spectra{ # output options
illumination = yes
absorption = yes
}
}
emission_spectrum{ # important group for absorption spectrum
refractive_index = 3.14768486
output_spectra{
absorption = yes
emission = yes
}
quantum_region{ # calculate emission spectrum quantum mechanically for the quantum region
name = "quantum_region"
intraband = no
interband = yes
polarization{ name = "longitudinal" re = [1,0,0] }
k_integration{
relative_size = 0.3
num_subpoints = 5
num_points = 10
#force_k0_subspace = yes
}
output_spectra{
spectra_over_energy = yes
spectra_over_wavelength = yes
emission = yes
power_spectra = yes
}
# settings for output spectra
energy_min = 0.001
energy_max = 5.0
energy_resolution = 0.001
spontaneous_emission = yes
energy_broadening_lorentzian= 1.0e-2
}
}
```

The absorption spectrum used in the group `irradiation{}`

should be imported from 1D_PL_of_QW_absorption_nnp.in.

```
import{
directory = "...1D_PL_of_QW_absorption_nnp\bias_00000\Optical\" # location of the file with absorption spectrum - it should be changed accordingly
file{
name = "my_abs" # rename filename
filename = "computed_absorption_spectrum_nm.dat" # reference desired filename
format = DAT
}
}
```

Inside the group `classical{}`

one has to specify **energy resolved densities** n(x,E) and p(x,E), which
are required for the semiclassical absorption and emission spectra. More information on the underlying
equations can be found here

```
classical{
Gamma{} HH{} LH{} # bands involved in 1 band calculation
energy_resolved_density{
# calculate position and energy resolved electron and hole densities: n(x,E), p(x,E)
# required for calculation of semiclassical emission and absorption spectra
min = -5.0
max = 5.0
energy_resolution = 0.001
only_quantum_regions = no
}
energy_distribution{
# settings for energy resolved density
min = -4.0
max = 4.0
energy_resolution = 0.001
only_quantum_regions = no
}
}
```

To calculate the quantum mechanical emission spectra, one has to include the group **quantum{}**. The group **quantum{}** as well as **current{}** and **poisson{}**
are also required for self-consistent quantum-current-poisson calculations. Inside these group proper convergence parameters have to be chosen.
In this part, one has to think about proper convergence parameters for the solvers.

```
poisson{ ... }
currents{ ... }
quantum{ ... }
```

Note that proper boundary conditions are needed for Poisson and current equation. These are imposed by contact regions.
In our simulation, we apply `ohmic{}`

contacts only to the bottom of the substrate, i.e. to the not illuminated side of the structure.

```
contacts{
ohmic{ name = "whatever" bias = 0.0 }
}
```

## Simulation¶

In the simulation a light source with Gaussian spectrum with central wavelength
\(\lambda\)_{peak} = 530 nm (2.34 ev) and linewidth of 10 meV is used.
The intensity \(\Phi\)_{intensity} is varied between the two values
0.5 \(\cdot\) 10^{4} W/cm^{2} and 0.05 \(\cdot\) 10^{4} W/cm^{2}.
The temperature in this simulation is swept between the three values 200 K, 250 K and 300 K.

**Results**

First, we have to calculate suitable absorption spectra with the input file 1D_PL_of_QW_absorption_nnp.in.
Figure 2.5.5.10 shows the calculated absorption spectrum at each temperature.
For all temperatures, the absorption coefficient at \(\lambda\) = 530 nm is of the order of 10^{6}.

Now we can run the main input file 1D_PL_of_QW_nnp.in, which imports and uses the computed absorption spectrum.

**Band edges**

Figure 2.5.5.11 shows the energy profiles with electron- and hole-Fermi levels. It is visible that boundary conditions (contacts) are only imposed on the right side of the structure. This set up was found to have better convergence behavior.

**Electron/ hole density**

Figure 2.5.5.12 illustrates the spatial and energy distribution of electrons and holes with respect to the band edges for
case: P_{illumination} 0.5 \(\cdot\) 10^{4} W/cm^{2} at 300 K.
Both, electrons and holes, are localized inside the quantum well, thus exhibit discrete energy levels.
The occupation of the energy levels gives us insight about possible transitions (*recombination*) between electron states in the
conduction band and hole states in the valence band. From Figure 2.5.5.12 we can deduce that
most transition energies are in the interval 1.4eV-1.6eV of magnitude. For the emission spectrum, we assume to find its peak energy
in this energy interval.

**Photogeneration**

Figure 2.5.5.13 depicts the spatial and energy resolved generation rate inside the structure for the case:
P_{illumination} 0.5 \(\cdot\) 10^{4} W/cm^{2} at 300 K.

**Spontaneous emission spectrum**

Figure 2.5.5.14 shows the normalized spontaneous emission spectra at
different temperatures. The peak of the emission spectra are primarily attributed to the E_{e1} - E_{h1} transition inside the quantum well.
Due to band gap shrinking the peak shifts to higher wavelengths with increasing temperatures. At higher temperatures the contribution
from other transitions, such as E_{e2} - E_{h1} to the spectra becomes visible. Thus, the spectrum exhibit a
broadening.

**Temperature dependence of emitted intensity**