Three-Dimensional Modeling of Minority-Carrier Lateral Diffusion Length Including Random Alloy Fluctuations in ($\mathrm{In}$,$\mathrm{Ga}$)$\mathrm{N}$ and ($\mathrm{Al}$,$\mathrm{Ga}$)$\mathrm{N}$ Single Quantum Wells

For nitride-based $(\mathrm{In},\mathrm{Ga})\mathrm{N}$ and $(\mathrm{Al},\mathrm{Ga})\mathrm{N}$ quantum-well (QW) light-emitting diodes (LEDs), the potential fluctuations caused by natural alloy disorders limit the lateral intra-QW carrier diffusion length and current spreading. The diffusion length mainly impacts the overall LED efficiency through sidewall nonradiative recombination, especially for $\mu \mathrm{LEDs}$. In this paper, we study the carrier lateral diffusion length for nitride-based green, blue, and ultraviolet C (UVC) QWs in three dimensions. We solve the Poisson and drift-diffusion equations in the framework of localization landscape theory. The full three-dimensional model includes the effects of random alloy composition fluctuations and electric fields in the QWs. The dependence of the minority carrier diffusion length on the majority carrier density is studied with a full three-dimensional model. The results show that the diffusion length is limited by the potential fluctuations and the recombination rate, the latter being controlled by the spontaneous and piezoelectric fields in the QWs and by the screening of the internal electric fields by carriers.

Figure 1

(a) Simulation structure, with dashed line representing the virtual structure. The electrons and holes are generated at the red area with a width of 10 nm. (b),(c) Indium composition maps of single QW structures for blue and green LEDs with random alloy fluctuations. (d) Single QW structure for UVC. The red arrow marks the respective average composition in the QW to the color bar. (e)–(g) Side views of the calculated potential $E_c$ for blue, green, and UVC structures at $x=0$, respectively. The barrier doping of this case is $2.0\times {10}^{19}{\mathrm{cm}}^{-3}$. White dashed lines separate the QW from the QB.

Figure 2

(a) Simulation flowchart. The first step constructs the LED structure, then the alloy fluctuation model is used in the calculation. Finally, all parameters are input to the 3D-DDCC solver to obtain the results. The average indium composition is 15%. (b),(c) Indium maps with and without alloy fluctuations, respectively.

Figure 3

Blue LED for $n$-$i$-$n$ structure. (See Fig. 2) The QW in-plane view is at $z=11\mathrm{nm}$, as recombination at the 11-nm plane is stronger than in the middle of the QW at 11.5 nm. The QW is in the 10- to 13-nm region. (a) Radiative recombination and (b) minority carrier density (holes). (c) Band diagram affected by alloy fluctuation [along the red line in (a)], (d),(e) averaged radiative recombination and hole density along the $x$ direction, respectively.

Figure 4

(a),(b) Averaged radiative recombination rate distribution along the lateral direction for the cases with and without random alloy fluctuations, respectively. (c) Extracted diffusion length with hole (minority) diffusion in the $n$-$i$-$n$ structure. (d) Extracted diffusion length with electron (minority) diffusion in the $p$-$i$-$p$ structure. The black line indicates the case with random alloy fluctuations, the blue line indicates the case without fluctuations, the red line indicates the case without polarization, and the green line indicates the case without fluctuations and without the polarization effect. (e)–(g) Electron and hole overlap for the three different situations using 1D-DDCC [34] simulations in the $n$-$i$-$n$ structure.

Figure 5

Comparison of minority carrier diffusion lengths for different doping concentrations in the QB for different structures. (a),(b) Diffusion lengths for holes and electrons versus the majority carrier density in QWs, respectively. The indium compositions for green and blue QWs are 22% and 15%, respectively. The average $\mathrm{Al}$ composition of the UVC LED is 40% and 60% for the QW and QB, respectively. The generation rate representing light excitation is ${10}^{26}{\mathrm{cm}}^{-3}{\mathrm{s}}^{-1}$.

Figure 6

Comparison of different factors affecting the carrier diffusion length at a doping concentration of $2\times {10}^{19}{\mathrm{cm}}^{-3}$ in the QB for the blue and green structures. (a) Comparison of diffusion length versus local excitation density (generation rate) in blue LEDs. (b) Influence of nonradiative lifetime on diffusion length. In green and blue $n$-$i$-$n$ structures, the diffusion length is almost unchanged. (c),(d) Influence of mobility on diffusion length.

Figure 7

Comparison of minority diffusion lengths with and without fluctuations, without polarization effect, and with different doping concentrations in the QB. (a),(b) Green structure, (c),(d) UVC structure. The black line indicates the alloy fluctuation case, the red line indicates the case without polarization, the blue line indicates the case without fluctuations, and the green line indicates the case without fluctuations and without polarization.