# 1D - InGaAs Multi-quantum well laser diode¶

*Section author: Takuma Sato*

In this tutorial, we simulate optical emission of a 1D InGaAs multi-quantum well laser diode grown on InP substrate. The blue region, seen in Figure 2.5.3.1, is the separate confinement heterostructure (SCH), which forms an optical waveguide in the transverse direction to confine the emitted light (red arrow). The multi-quantum wells and SCH are clad by InP on both sides. A voltage bias is applied to the gray edges.

**Input files**

LaserDiode_InGaAs_1D_cl_nnp.in

LaserDiode_InGaAs_1D_qm_nnp.in

**References**

**Table of contents**

## Current equation¶

The properties of optoelectronic devices are governed by Poisson equation, Schroedinger equation, drift-diffusion and continuity equations. We denote by \(n\) and \(p\) the carrier number density per unit volume. The continuity equations in the presence of creation (generation, \(G\) ) or annihilation (recombination, \(R\) ) of electron-hole pairs read

where the current is proportional to the gradient of quasi Fermi levels \(E_{F,n/p}(\boldsymbol{x})\)

Here the charge current has the unit of (area)\(^{-1}\)(time)\(^{-1}\). \(\mu_{n/p}\) are the mobilities of each carrier. In nextnano++, \(\mu_{n/p}\) are determined using the mobility model specified in the input file under currents{}. Hereafter we consider stationary solutions and set \(\dot{n}=\dot{p}=0\). The governing equations then reduce to

which we call
**current equation** (generation \(G=0\) in the present case).
nextnano++ solves this equation and Poisson equation self-consistently
when one specifies it in the input file as:

```
run{
current_poisson{}
}
```

## Recombination of carriers and emission spectrum¶

The generation/recombination rate, \(R(\boldsymbol{x})\), originates from several physical processes. In nextnano++, the following mechanisms are

implemented (cf. Recombination in currents{})

**Schockley-Read-Hall recombination**\(R_{\mathrm{SRH}}\) – carrier trapping by impurities.**Auger recombination**\(R_{\mathrm{Auger}}\) – a collision between two carriers results in the excitation of one and the recombination of the other with a carrier of opposite charge.**radiative recombination**\(R_{\mathrm{rad}}\) – emission/absorption of a**photon**.

Each mechanism can be turned on and off in the input file.

**Radiative recombination** describes the recombination of electron-hole
pairs at a position \(\boldsymbol{x}\) by emitting a photon and is given by

where \(C(x)\) [\(\mathrm{cm}^3\mathrm{s}^{-1}\)] is the (material-dependent) radiative recombination parameter which is propotional to the one specified in the database (Radiative recombination) and \(n(\boldsymbol{x}, E),p(\boldsymbol{x}, E)\) [\(\mathrm{cm}^{-3}\mathrm{eV}^{-1}\)] are the charge densities as a function of energy and position.

In nextnano++, this radiative recombination whose rate is calculated as above is regarded as **spontaneous emission**.
On the other hand, **the net amount of the stimulated emission rate** is given by:

This is consistent with eq.(9.2.39) in [ChuangOpto1995]. We note that here it is assumed that photon modes occupied by one photon each, i.e. takes into account neither energy-dependent photon density of states nor Bose-Einstein distribution.

Since the radiative recombination process involves no phonons, this transition is vertical and therefore this contribution is only relevant for semiconductors with a direct band gap such as GaAs.

**Absorption coefficient** is calculated from \(R_{rad,net}^{stim}(E)\) as

where \(n_r\) is the refractive index and \(V\) is the total volume of the device. The unit is [cm\(^{-1}\)]. In case of 1D simulation, calculated \(R_{rad,net}^{stim}(E)\) has the unit [\(\mathrm{cm}^{-2}\mathrm{s}^{-1}\mathrm{eV}^{-1}\)] and is divided by the total length instead of the volume. This formula is consistent with eq (9.2.25) in [ChuangOpto1995].

## Input file¶

In the beginning of the input file, we define several variables for the structure and parameters for simulation. The variable definitions are shown in Figure 2.5.3.2.

Charge density as a function of position \(n(\boldsymbol{x})\) is always calculated by default. On the other hand, charge density as a function of energy \(n(E),p(E)\), charge density as a function of both position and energy \(n(\boldsymbol{x}, E),p(\boldsymbol{x}, E)\) and emission spectrum are calculated when the followings are specified (see classical{} for details):

```
classical{
...
energy_distribution{ # Calculation of carrier densities as a function of energy
min = -1.5 # Integrate from
max = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
only_quantum_regions = yes
}
energy_resolved_density{
min = -1.5 # Integrate from
max = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
only_quantum_regions = yes
}
emission_spectrum{
output_spectra{
emission = yes
gain = yes
absorption = yes
stimulated_emission = yes
spectra_over_energy = yes
spectra_over_wavelength = yes
spectra_over_frequency = yes
spectra_over_wavenumber = yes
photon_spectra = yes
power_spectra = yes
}
output_photon_density = yes
output_power_density = yes
}
}
```

The mobility model and recombination models for the current equation are specified in currents{} as

```
currents{
mobility_model = constant
# mobility_model = minimos
recombination_model{
SRH = yes # 'yes' or 'no'
Auger = yes # 'yes' or 'no'
radiative = yes # 'yes' or 'no'
}
}
```

The run{} flag specifies which equations to solve.
This is the main difference between `LaserDiode_*_qm_nnp.in`

and `LaserDiode_*_cl_nnp.in`

.

```
# qm
run{
strain{} # solves the strain equation
current_poisson{ # solves the coupled current and Poisson equations self-consistently
output_log = yes
iterations = 1000
alpha_fermi = 0.5
}
quantum_current_poisson{ # solves the Schroedinger, Poisson (and current) equations self-consistently
iterations = 1000
alpha_fermi = 0.9
residual = 1e6
residual_fermi = 1e-8
output_log = yes
}
}
# cl
run{
strain{} # solves the strain equation
current_poisson{ # solves the coupled current and Poisson equations self-consistently
output_log = yes
iterations = 1000
alpha_fermi = 0.7
residual_fermi = 1e-10
}
}
```

In this case nextnano++ first solves the strain equation from the
crystal orientation to decide the polarization charges (piezoelectric
effect) and shifted bandedges. Then the program solves the coupled
current-Poisson-Schroedinger equations in a self-consistent way (input
file: `LaserDiode_InGaAs_1D_qm_nnp.in`

). For the classical calculation
(`LaserDiode_InGaAs_1D_cl_nnp.in`

), `quantum_current_poisson{}`

is commented out to restrict the
calculation to the current-Poisson equations only.

## Results¶

### Band structure¶

The bandstructure and emission power spectrum of the system are stored in `bandedges.dat`

.
Figure 2.5.3.3 shows the case for the bias \(0.2\) V.
Here the quasi Fermi level of electrons is lower than the quantum wells.

For the bias \(0.8\) V (Figure 2.5.3.4), in contrast, it lies above the red line,
allowing electrons to flow into the quantum wells.
An electron trapped in the quantum wells is likely to recombine with a hole in the valence band, emitting a photon.
In the input file `\Optical\emission_photon_density.dat`

,
one can see that the photons are emitted from this active region (not shown).
Figure 2.5.3.12 shows the emission spectrum in this case. When the bias is too small,
e.g. Figure 2.5.3.3, the intensity is much smaller, as can be seen in Figure 2.5.3.16.

### Energy eigenstates and eigenvalues¶

In the input file `LaserDiode_InGaAs_1D_qm_nnp.in`

, the single-band Schroedinger equation is coupled
to the current-Poisson equation and solved self-consistently.
The wave functions of electrons and holes along with eigenvalues are written in
`\Quantum\probabilities_shift_quantum_region_Gamma_0000.dat`

and
`\Quantum\probabilities_shift_quantum_region_HH_0000.dat`

(Figure 2.5.3.5 and Figure 2.5.3.6).
The light hole and split-off states are out of the quantum wells and not of our interest here.

### Charge denstities¶

We can find the energy-resolved charge density \(n(x,E)\) and \(p(x,E)\)
in the output `electron_density_vs_energy.fld`

and `hole_density_vs_energy.fld`

.
The following figures represent \(n(x,E)\) and \(p(x,E)\) [\(\mathrm{cm}^{-3}\mathrm{eV}^{-1}\)]
with respect to the band edges and quasi-fermi levels at bias 0.2, 0.4, 0.6, 0.8 and 1.0 V.
We can see that the carrier densities around the quantum wells increase as the bias increases.

Note

These graphs are generated by nextnanopy.

We also have the charge densities integrated over the device \(n(E),p(E)\) [\(\mathrm{cm}^{-2}\mathrm{eV}^{-1}\)] and energy \(n(x),p(x)\) [\(\mathrm{cm}^{-3}\)].

\(n(E)\) and \(p(E)\) with and without quantum calculation shows different features
due to the discretization of energy levels in quantum wells. This is shown in `integrated_densities_vs_energy.dat`

.

Figure 2.5.3.12 illustrates the population inversion in stationary (quasi-equilibrium) state of the device under bias. Solid and dashed lines are for quantum and classical calculations, respectively. The black arrows mark the relevant energies of the structure 4 at bias of 0.8 V. The hole density is shown in Figure 2.5.3.13 with higher resolution.

The energy resolution in Figure 2.5.3.13 has been increased by a factor of 10 from Figure 2.5.3.12.

Note

Although these charge densities either with variable \(E\) or \(x\) are both obtained by integrating \(n(x,E)\) and \(p(x,E)\) over the corresponding variable, these are independently calculated in nn++ simulation. Hence it is possible to turn off the calculation only for \(n(x,E)\) and \(p(x,E)\) calculating the integrated charge densities. In this case it runs much faster and needs much less memory.

### Emission spectra and absorption coefficient¶

The spontaneous and stimulated emission spectra are written in
`\Optical\emission_spectrum_photons.dat`

and
`\Optical\stim_emission_spectrum_photons.dat`

, respectively (Figure 2.5.3.14).
The peak is at around 0.7-0.8eV, which is consistent with the charge distribution in Figure 2.5.3.12.
The stimulated emission does not occur above the quasi Fermi level separation, \(E_{Fn}-E_{Fp}\).

The formulas used for the calculation in the source code are specified above: Recombination of carriers and emission spectrum.

The absorption coefficient is calculated as

where \(n_r\) is the refractive index and \(V\) is the total volume of the device. The unit is [cm\(^{-1}\)]. In case of 1D simulation, calculated \(R_{rad,net}^{stim}(E)\) has the unit [\(\mathrm{cm}^{-2}\mathrm{s}^{-1}\mathrm{eV}^{-1}\)] and is divided by the total length instead of the volume. This formula is consistent with eq (9.2.25) in [ChuangOpto1995].

The absorption coefficient \(\alpha(E)\) and gain (coefficient) \(g(E)\) are essentially the same quantity with opposite signs,

These are by definition independent of the initial photon population.
**Please note that the gain spectrum in nextnano++ is cut off where it is negative**.
For details, see classical{}.

The spectrum changes its sign at the energy \(E_{Fn}-E_{Fp}\), that is,
the separation of the quasi Fermi levels. According to the output
`bandedges.dat`

, this value is -0.0001-(-0.7702)=0.7701eV. The
following result has been calculated classically. We also get
qualitatively consistent results from quantum mechanical simulation.

### Current and internal quantum efficiency¶

The output file `IV_characteristics.dat`

contains right- and left-contact current in unit of [Acm\(^{-2}\)].
In the present case, the right-contact current is hole current, whereas the left-contact current is electron current.
In Figure 2.5.3.15, we compare the hole current and photocurrent.

Figure 2.5.3.16 clearly shows the consequence of the difference
in bandstructures Figure 2.5.3.3 and Figure 2.5.3.4.
The holes and electrons recombine in the multi-quantum well layers,
emitting one photon per electron-hole pair. The efficiency of conversion
from charge current into photocurrent is called the *internal quantum efficiency*

This quantity is written in `internal_quantum_efficiency.dat`

and shown in Figure 16.

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