nextnano - Software for
semiconductor nanodevices

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

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, $R<0$) or annihilation (recombination, $R>0$) of electron-hole pairs read $$ \frac{\partial p}{\partial t}=-\mathbf{\nabla}\cdot \mathbf{j}_p(\mathbf{x}) - R(\mathbf{x}),
\frac{\partial n}{\partial t}=-\mathbf{\nabla}\cdot \mathbf{j}_n(\mathbf{x}) - R(\mathbf{x}), $$ where the current is proportional to the gradient of quasi Fermi levels $E_{F,p/n}(\mathbf{x})$ $$ \mathbf{j}_p(\mathbf{x}) = \mu_p(\mathbf{x})p(\mathbf{x})\nabla E_{F,p}(\mathbf{x}),
\mathbf{j}_n(\mathbf{x}) = -\mu_n(\mathbf{x})n(\mathbf{x})\nabla E_{F,n}(\mathbf{x}). $$ Note that the charge current here is divided by $|e|$, i.e. has the unit of (area)$^{-1}$(time)$^{-1}$. $\mu_{p/n}$ are the mobilities of each carrier. In nextnano++, $\mu_{p/n}$ are determined using the mobility model specified in the input file under currents{}. Hereafter we consider stationary solutions and set $\dot{p}=\dot{n}=0$. The governing equations then reduce to $$ -\nabla\cdot\mathbf{j}_p(\mathbf{x})=R(\mathbf{x}),
-\nabla\cdot\mathbf{j}_n(\mathbf{x})=R(\mathbf{x}), $$ which we call current equation. nextnano++ solves this equation and Poisson equation self-consistently when one specifies it in the input file as:

run{
   solve_current_poisson{}
}

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.
• stimulated emission $R_{\mathrm{st}}$ – incoming photon stimulates the emission of another 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.

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>

Emission spectrum is calculated when the following data is specified:

classical{
   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 following flag specifies which equations to solve.

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
   }
}

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.

The bandstructure and emission power spectrum of the system are stored in bandedges.dat. Figures search?q=bandedge&amp;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&amp;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 emitted_photon_density.dat, one can see that the photons are emitted from this active region (not shown). Figure search?q=emission&amp;btnI=lucky shows the emission spectrum in this case. When the bias is too small, e.g. Figure search?q=bandedge&amp;btnI=lucky, the intensity is much smaller, as can be seen in Figure search?q=IV&amp;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>

With the input file LaserDiode_InGaAs_1D_qm_nnp.in, we also obtain the electron- and hole-wavefunctions. They are stored in wf_probabilities_shift_quantum_region_Gamma_0000.dat. <figure wf> <caption>Probability distribution $|\psi(x)|^2$ of the localized modes of electrons and holes for the bandstructure search?q=bandedge2&amp;btnI=lucky. Horizontal lines are the corresponding eigenenergies. </caption> </figure>

The spontaneous and stimulated emission spectra are written in emitted_photon_spectrum.dat and stimulated_emission_spectrum.dat, respectively.

<figure emission> <caption>Emission spectrum of the laser diode for the bias $0.8$ V.</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&amp;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&amp;btnI=lucky and search?q=bandedge2&amp;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&amp;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.