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1D InGaAs Multi-quantum well laser diode
Author: Takuma Sato, nextnano GmbH
In this tutorial, we simulate optical emission of a 1D InGaAs multi-quantum well laser diode grown on InP substrate. The blue region 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.
The corresponding input files are:
- LaserDiode_InGaAs_1D_cl_nnp.in
- LaserDiode_InGaAs_1D_qm_nnp.in
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
$$
-e\frac{\partial n}{\partial t} + \mathbf{\nabla}\cdot (-e\mathbf{j}_n(\mathbf{x})) = -e(G(\mathbf{x})- R(\mathbf{x})),
e\frac{\partial p}{\partial t} + \mathbf{\nabla}\cdot e\mathbf{j}_p(\mathbf{x}) = e(G(\mathbf{x})- R(\mathbf{x})),
$$
where the current is proportional to the gradient of quasi Fermi levels $E_{F,n/p}(\mathbf{x})$
$$
\mathbf{j}_n(\mathbf{x}) = -\mu_n(\mathbf{x})n(\mathbf{x})\nabla E_{F,n}(\mathbf{x}),
\mathbf{j}_p(\mathbf{x}) = \mu_p(\mathbf{x})p(\mathbf{x})\nabla E_{F,p}(\mathbf{x}).
$$
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
$$
\nabla\cdot\mu_n(\mathbf{x})n(\mathbf{x})\nabla E_{F,n}(\mathbf{x})=-(G(\mathbf{x})-R(\mathbf{x})),
\nabla\cdot\mu_p(\mathbf{x})p(\mathbf{x})\nabla E_{F,p}(\mathbf{x})=G(\mathbf{x})-R(\mathbf{x}),
$$
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{
solve_current_poisson{}
}
Recombination of carriers and emission spectrum
The generation/recombination rate, $R(\mathbf{x})$, originates from several physical processes. In nextnano++, the following mechanisms are implemented (cf. documentation)
- 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 $\mathbf{x}$ by emitting a photon and is given by
$$
R_{\mathrm{rad}}(\mathbf{x}) = C\left[n(\mathbf{x}) p(\mathbf{x}) – n_i^2\right],
$$
where $n_i$ is the intrinsic density of the charge carriers. $C$ is a material dependent constant given in the database and has the unit of cm$^3$/s. $R_{\mathrm{rad}}(\mathbf{x})$ is written in emitted_photon_density.dat.
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.
Input file
In the beginning of the input file, we define several variables for the structure and parameters for simulation. The variables are shown below.
<figure structure>
<caption>The definition of variables. The gray regions are contacts of 1nm thickness. $NUMBER_OF_WELLS determines the repetition of quantum wells. The program automatically sweeps the bias voltage starting from $BIAS_START until $BIAS_END, at intervals of $BIAS_STEPS.</caption>
</figure>
Energy-dependent charge density and emission spectrum are calculated when the following is specified (see classical{} documentation for details):
classical{
energy_distribution{
min = -1.5 # Integrate from
max = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
only_quantum_regions = yes # (default: no)
}
emission_spectrum{
min = -1.5 # Integrate from
max = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
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{
solve_strain{} # solves the strain equation
solve_current_poisson{ # solves the coupled current and Poisson equations self-consistently
output_log = yes
}
solve_quantum{} # solves the Schroedinger equation
outer_iteration{ # solves the Schroedinger, Poisson and current equations self-consistently
iterations = 2000
current_repetitions = 5 #10
alpha_fermi = 0.01
residual = 1e6
residual_fermi = 1e-8
output_log = yes
}
}
# cl
run{
solve_strain{} # solves the strain equation
solve_current_poisson{ # solves the coupled current and Poisson equations self-consistently
output_log = yes
}
}
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), solve_quantum{} and outer_iteration{} are commented out to restrict the calculation to the current-Poisson equations only.
Results
The bandstructure and emission power spectrum of the system are stored in bandedges.dat. Figures search?q=bandedge&btnI=lucky shows the case for the bias $0.2$ V. Here the quasi Fermi level of electrons is lower than the quantum wells.
<figure bandedge>
<caption>Bandstructure of the laser diode system for a low bias of $0.2$ V. </caption>
</figure>
For the bias $0.8$ V (Figure search?q=bandedge2&btnI=lucky), 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 search?q=emission&btnI=lucky shows the emission spectrum in this case. When the bias is too small, e.g. Figure search?q=bandedge&btnI=lucky, the intensity is much smaller, as can be seen in Figure search?q=IV&btnI=lucky.
<figure bandedge2>
<caption>Bandstructure for a high bias $0.8$ V. Electrons flowing from the left and holes from the right recombine in the active zone (multi-quantum well structure).</caption>
</figure>
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\wf_probabilities_shift_quantum_region_Gamma_0000.dat and \Quantum\wf_probabilities_shift_quantum_region_HH_0000.dat (Figure search?q=wf&btnI=lucky and search?q=ev&btnI=lucky). The light hole and split-off states are out of the quantum wells and not of our interest here.
<figure wf>
<caption>Probability distribution $|\psi(x)|^2$ of the lowest localized modes of electrons and holes for the bandstructure search?q=bandedge2&btnI=lucky. Horizontal lines are the corresponding eigenenergies. </caption>
</figure>
<figure ev>
<caption>Eigenvalues of the Gamma-band and heavy-hole-band states in relation to bandedges. The Gamma band has single “miniband”, whereas the heavy-hole band has three. </caption>
</figure>
The charge density with- and without quantum calculation shows different features due to the discretization of energy levels in quantum wells. In the output integrated_densities_vs_energy.dat we find the electron and hole densities as a function of energy (Figure search?q=density&btnI=lucky and search?q=density_Eresolution&btnI=lucky). This quantity is integrated over the device and has the unit [cm$^{-2}$eV$^{-1}$]. We observe a good correspondence between the spectra and relevant energies of the present structure.
<figure density>
<caption>Electron (red) and hole (blue) densities integrated over the device as a function of energy. This figure 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 search?q=wf&btnI=lucky at bias of $0.8$ V. The hole density is shown in Figure search?q=density_Eresolution&btnI=lucky with higher resolution.</caption>
</figure>
The hole density has been calculated with higher energy resolution in Figure search?q=density_Eresolution&btnI=lucky. The ratchet-like shape of the quantum calculation within the wells results from the discretized density of states multiplied by (quasi) Fermi-Dirac distribution (cf. Figure 9.8 in [Chuang]).
<figure density_Eresolution>
<caption>Hole density integrated over the device from classical (dashed) and quantum (solid) calculation. The energy resolution has been increased by a factor of 10.</caption>
</figure>
The spontaneous and stimulated emission spectra are written in \Optical\emission_spectrum_photons.dat and stimulated_emission_spectrum.dat, respectively. Please note that the stimulated emission calculation in nextnano++ assumes photon modes occupied by one photon each, i.e. takes into account neither energy-dependent density of states nor Bose-Einstein distribution.
<figure emission>
<caption>Emission spectrum of the laser diode for the bias $0.8$ V. The peak is at around 0.7-0.8eV, which is consistent with the charge distribution in Figure search?q=density&btnI=lucky. The stimulated emission does not occur above the quasi Fermi level separation, $E_{Fn}-E_{Fp}$.</caption>
</figure>
The absorption coefficient and gain spectra are by definition independent of photon population. 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. With quantum calculation, we get qualitatively consistent results.
<figure cl_gain>
<caption>Classically calculated absorption coefficient and gain spectrum. The sign of the spectrum switches at the energy corresponding to the quasi Fermi-level separation in the active region. </caption>
</figure>
The output file IV_characteristics.dat contains right- and left-contact current in unit of [A/cm$^2$]. In the present case, the right-contact current is hole current, whereas the left-contact current is electron current. In Figure search?q=IV&btnI=lucky, we compare the hole current and photocurrent.
<figure IV>
<caption>Charge current and photocurrent as a function of bias voltage (IV characteristics). This figure clearly shows the consequence of the difference in Figure search?q=bandedge&btnI=lucky and search?q=bandedge2&btnI=lucky.</caption>
</figure>
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
$$
\eta = \frac{I_{\mathrm{photon}}}{I_{\mathrm{charge}}}.
$$
This quantity is written in internal_quantum_efficiency.dat and shown in Figure search?q=efficiency&btnI=lucky.
<figure efficiency>
<caption>Conversion efficiency of the InGaAs laser diode. </caption>
</figure>
- Please help us to improve our tutorial. Should you have any questions or comments, please send to support [at] nextnano.com.
