Our simulations are performed at room temperature with a two-dimensional numerical transport model based on the continuity equations for charge carriers and the standard drift-diffusion currents, effectively coupling the partial differential equations for the electrostatic potential
\(\phi \) and the quasi-Fermi levels
\(E_{Fp}\) and
\(E_{Fn}\) of the valence band holes and conduction band electrons, respectively, given by
$$\begin{aligned}&\nabla \cdot \left( -\epsilon \nabla \phi \right) = q \left( p - n + N_d - N_a \right) , \end{aligned}$$
(1)
$$\begin{aligned}&\nabla \cdot J_p = \nabla \cdot \left( \mu _p p \nabla E_{Fp} \right) = -q R, \end{aligned}$$
(2)
$$\begin{aligned}&\nabla \cdot J_n = \nabla \cdot \left( \mu _n n \nabla E_{Fn} \right) = q R. \end{aligned}$$
(3)
Here
\(\epsilon \) is the permittivity,
q is the elementary charge,
p is the hole density in the valence band,
n is the electron density in the conduction band,
\(N_d\) is the ionized donor density, and
\(N_a\) is the ionized acceptor density. Also,
\(J_p\) (
\(J_n\)) is the hole (electron) current density,
\(\mu _p\) (
\(\mu _n\)) is the hole (electron) mobility, and
R is the net recombination rate per unit volume. More detailed description of the model can be found in Kivisaari et al. (
2015), Kivisaari et al. (
2017b), Sadi et al. (
2014). The recombination rate
R, consisting of Shockley–Read–Hall (SRH)
\(R_{SRH}\), radiative
\(R_{rad}\), and Auger recombination
\(R_{Aug}\), is modeled using the well-known parameterized formula given by Kivisaari et al. (
2015)
$$\begin{aligned} \begin{aligned} R&= R_{SRH} + R_{rad} + R_{Aug} \\&= \left[ \frac{A}{n+p+2n_i} + B + C (n+p) \right] (n p - n_i^2), \end{aligned} \end{aligned}$$
(4)
where
\(n_i\) is the intrinsic carrier density, and
A,
B and
C are recombination constants for SRH, radiative and Auger processes, respectively. Also, the non-radiative surface recombination rate (Chen et al.
2016) was calculated with
$$\begin{aligned} R_{surf} = \left[ \frac{V_{surf}}{n+p+2n_i} \right] (n p - n_i^2) , \end{aligned}$$
(5)
where
\(V_{surf}\) is the recombination velocity at the given surface. Similar equation was used for the interface recombination at the top and bottom interfaces of the AR. The recombination parameters used in our simulations are shown in Table
1.
Table 1
The recombination parameters used for calculating the recombination rates
A
| \( 3 \times 10^5\) (s \( ^{-1} \)) Bender et al. ( 2013) |
B
| \( 2 \times 10^{-10} \) ( \( {\text{cm}}^{{{3}}} \;{\text{s}}^{{ - {{1}}}} \)) Schubert ( 2006) |
C
| \( 1 \times 10^{-30} \) ( \( {\text{cm}}^{{{6}}} \;{\text{s}}^{{ - {{1}}}} \)) Strauss et al. ( 1993) |
\( V_{interface} \) (GaInP/GaAs) | 1.5 ( \( {\text{cm}}\;{\text{s}}^{{ - {{1}}}} \)) Levinshtein et al. ( 1999) |
\( V_{surface} \) (AlGaAs/air) | \( 4 \times 10^5 \) ( \( {\text{cm}}\;{\text{s}}^{{ - {{1}}}} \)) Levinshtein et al. ( 1999) |
For analyzing our results we use uniformity of the total recombination within the active region of the LED that was determined as
$$ Uniformity = \frac{{{\text{min}}(R)}}{{{\text{max}}(R)}}, $$
(6)
where min(
R) and max(
R) are the minimum and maximum rates for the total recombination, respectively. With this definition, a value of 1 corresponds to a perfect uniformity, and a value of 0 corresponds to a completely non-uniform situation. Additionally, we use the internal quantum efficiency (IQE) of the devices defined as the ratio between the radiative and the total recombination rates integrated through the device volume. Furthermore, the current density
J is defined as the integrated total current across the whole structure divided by the total horizontal cross-section area of the device by assuming a 100
\(\mu \)m device length along the unsimulated axis.