InGaAs Multi-quantum well laser diode
Header
- Files for the tutorial located in nextnano++\examples
LaserDiode_InGaAs_1D_cl_nnp.in
LaserDiode_InGaAs_1D_qm_nnp.in
Introduction
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 98, 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.
Figure 98 Structure overview
Current equation
The properties of optoelectronic devices are governed by Poisson equation, Schrödinger 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 \(E_{F,n/p}(\mathbf{x})\)
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
which we call current equation (generation \(G=0\) in the present case). The nextnano++ tool 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, \(R(\mathbf{x})\), originates from several physical processes. In nextnano++, the following mechanisms are implemented (cf. recombination_model{ } )
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
where \(C(x)\) [\(\mathrm{cm}^3\mathrm{s}^{-1}\)] is the (material-dependent) radiative recombination parameter which is proportional to the one specified in the database (Radiative recombination) and \(n(\mathbf{x}, E),p(\mathbf{x}, E)\) [\(\mathrm{cm}^{-3}\mathrm{eV}^{-1}\)] 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 99.
Figure 99 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.
Charge density as a function of position \(n(\mathbf{x})\) 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 \(n(\mathbf{x}, E),p(\mathbf{x}, E)\) and emission spectrum are calculated when the followings are specified (see classical{ } for details):
grid{
...
energy_grid{
min_energy = -1.5 # Integrate from
max_energy = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
}
}
classical{
...
energy_distribution{ # Calculation of carrier densities as a function of energy
min_energy = -1.5 # Integrate from
max_energy = 0.5 # Integrate to
energy_resolution = 0.005 # Integration resolution
only_density_quantum_regions = yes
}
energy_resolved_density{
only_density_quantum_regions = yes
output_energy_resolved_densities{}
}
semiclassical_spectra{
output_spectra{
emission = yes
gain = yes
absorption = 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 Schrödinger, 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 band edges. Then the program solves the coupled
current-Poisson-Schrödinger 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 band structure and emission power spectrum of the system are stored in bandedges.dat.
Figure 100 shows the case for the bias \(0.2\) V.
Here the quasi Fermi level of electrons is lower than the quantum wells.
Figure 100 Band structure of the laser diode system for a low bias of \(0.2\) V.
For the bias \(0.8\) V (Figure 101), 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 109 shows the emission spectrum in this case. When the bias is too small,
e.g. Figure 100, the intensity is much smaller, as can be seen in Figure 113.
Figure 101 Band structure 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).
Energy eigenstates and eigenvalues
In the input file LaserDiode_InGaAs_1D_qm_nnp.in, the single-band Schrödinger 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 102 and Figure 103).
The light hole and split-off states are out of the quantum wells and not of our interest here.
Figure 102 Probability distribution \(|\psi(x)|^2\) of the lowest localized modes of electrons and holes for the band structure Figure 100. Horizontal lines are the corresponding eigenenergies.
Figure 103 Eigenvalues of the Gamma-band up to 5th and heavy-hole-band states up to 13th in relation to band edges. The Eigenvalues above these are higher than the barrier energy of the quantum wells. The Gamma band has single “miniband”, whereas the heavy-hole band has three. The 1st heavy-hole miniband consists of the 1st~5th eigenvalues, the 2nd heavy-hole miniband consists of the 6th~11th eigenvalues and the 3rd consists of the 12th and 13th eigenvalues.
Charge densities
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.
Figure 104 Energy-resolved electron and hole density, Gamma conduction band edge, HH valence band edge and quasi-Fermi levels at bias 0.2 V in quantum calculation.
Figure 105 Energy-resolved electron and hole density, Gamma conduction band edge, HH valence band edge and quasi-Fermi levels at bias 0.4 V in quantum calculation.
Figure 106 Energy-resolved electron and hole density, Gamma conduction band edge, HH valence band edge and quasi-Fermi levels at bias 0.6 V in quantum calculation.
Figure 107 Energy-resolved electron and hole density, Gamma conduction band edge, HH valence band edge and quasi-Fermi levels at bias 0.8 V in quantum calculation.
Figure 108 Energy-resolved electron and hole density, Gamma conduction band edge, HH valence band edge and quasi-Fermi levels at bias 1.0 V in quantum calculation.
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 109 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 110 with higher resolution.
Figure 109 Electron (red) and hole (blue) densities integrated over the device as a function of energy.
Figure 110 Hole density integrated over the device from classical (dashed) and quantum (solid) calculation.
The energy resolution in Figure 110 has been increased by a factor of 10 from Figure 109.
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 and absorption spectra
The spontaneous and stimulated emission spectra are written in
\Optical\semiclassical_spectra_photons.dat and
\Optical\stim_semiclassical_spectra_photons.dat, respectively (Figure 111).
The peak is at around 0.7-0.8eV, which is consistent with the charge distribution in Figure 109.
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.
Figure 111 Emission spectrum of the laser diode for the bias \(0.8\) V.
The absorption spectra are 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 spectra \(\alpha(E)\) and gain spectra \(g(E)\) 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.
Figure 112 Classically calculated absorption and gain spectra. The sign of the spectrum switches at the energy corresponding to the quasi Fermi-level separation in the active region.
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 112, we compare the hole current and photocurrent.
Figure 113 Charge current and photocurrent as a function of bias voltage (IV characteristics).
Figure 113 clearly shows the consequence of the difference in band structures Figure 100 and Figure 101. 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.
Figure 114 Conversion efficiency of the InGaAs laser diode.
Last update: 16/07/2024