In the following example we are going to show you, how can be the spectra of a Light emitting diode calculated with the nextano++ software.

In a Led the photons are emitted in the radiate recombination process:

$$R_{rad} = c_r (n p - n_i n_i)$$

Where the $n$, and $p$ correspond the density of the holes and the electrons in the volume element, and $n_i$ is the intrinsic density of the charge carriers. This recombination rate is coupled in the drift-diffusion solver of the nextnano++, and it calculates the stationary solution of the problem:

$$\frac{d}{dt}n = 0 $$ $$\frac{d}{dt}p = 0 $$

According to this simple model, where $c_r$ is just a material constant, the radiation recombination rate, which generates photons on energy $E_r$ could be written in the form:

$$R_{rad}(E_r) = c_r \int n(E_e) \cdot p(E_h) \delta(E_e-E_h-E_r) dE_e dE_h $$

Where the density $n(E_e)$, and $p(E_h)$ the electron and hole densities on the energy niveu $E_e$, and $E_h$ in the volume element.

In order to get the spectra of the emission the integral should be calculated for each volume element, and integrated over volume. This model calculates with constant $c_r$, which is in QW structures not correct, and also the re-absorption of photons is not included.

currents{
    mobility_model    = constant 
    recombination_model{
      SRH            = no        # Shockley-Read-Hall recombination
      Auger          = no        # Auger recombination
      radiative      = yes       # radiative recombination (direct recombination)
    }     
    output_fermi_levels{}
    output_currents{} 
}

In order to get physically valid results we have to calculate with radiative recombinations in the drift diffusion.

           radiative      = yes       # radiative recombination (direct recombination)

classical{
   Gamma{}
   LH{}
   HH{}
   SO{}

   output_bandedges{ averaged = no }
   output_carrier_densities{}
   output_intrinsic_density{}
   energy_distribution{              # Calculation of carrier densities in function of energy
	min = -5                     # Integrate from
	max = 5                      # Integrate to
	energy_resolution = 0.05     # Integration Resolution
   }
}<>

The density has to be calculated in the energy domain, which means we have to define the integration range

   energy_distribution{              # Calculation of carrier densities in function of energy
	min = -5                     # Integrate from
	max = 5                      # Integrate to
	energy_resolution = 0.05     # Integration Resolution
   }

The simulated test system was a p-i-n diode structure from pGaAs-pAlGaAs-InGaAs-nAlGaAs-nGaAs hetero-structure composition. In the center insulator part (InGaAs) the Fermi level reaches the band-edges of the Quantum well, which makes population in the well for both electrons, and holes. The applied bias in the drift diffusion equation results the splitting of the hole and electron Fermi levels. It can be seen on the band structure profile on figure 1

The carrier distribution in the energy can be seen on figure 2. The density is summed for the full device, it results the loss of the position information for the carrier densities.

The emission spectrum of the Led is plotted on figure 3. The dependence of the spectrum on the bias voltage can be calculated with voltage sweeps.