Blue LED Simulation
1. Introduction
It is very important to understand the operation and underlying physics of InGaN/GaN materials based electronic device in modern display industry such as flatpaneldisplay for backlight illumination and high efficiency light bulbs. For these reasons, the numerical device simulation is adapted to study the improvement of LED efficiency and to understand the basic operation of multiple quantum well LEDs.
But the main problem in device simulation is that there are so many unknown physical parameters which cannot be easily measured and is also still in debate regarding the actual polarization in the layer after screening by defects(dislocation or Vdefect) and uncertain polarity which depends on substrate condition and growth condition.
The basic simulation study for GaN LEDs utilizes the conventional InGaN/GaN MQW (multiple quantum well) LED structure and then extends the efficiency limit of this conventional LED structure to further incorporate n+SPS (shortperiodsuperlattice) for pure ohmic contact and current spreading layer for both nside and pside layers. Regarding active emission regions, we may consider various techniques to improve LED efficiency such as bandgap engineered cascade quantum well design or tunneling junction which will dramatically improve carrier injection into the quantum well regions and to increase efficiency by above 100%.
Despite the fact that all of these LED structure are basically based on the conventional LED structure, it is still not known about the correct simulation models and how to take into account experimental observation such as trapassisted tunneling through trap states in the quantum well region, the amount of actual polarization, and polarity.
These motivate this article to study more details about basic forward characteristic of current injection and various effects on the forward voltage shift.
We will briefly explain the new captureescape model for an accurate description of carrier distribution in the barrier, quantum well region and importance of polarity in AlN/GaN/InN material growth, and finally how to simulate the forward characteristic of LED operation.
2. Simulation Models
Practical approach to simulate LED device is to use the classical driftdiffusion solver with the selfconsistent SchrodingerPoisson solver for the quantum well bound state energy and then use this result to calculate spontaneous emission rate by developing effective mass approximation from k.p theory. We need to know that this approach is only valid in parabolic band approximation. With this assumption we simulated a conventional led with the following structure.
(1) Simulation LED Structure
0.4um pGaN – acceptor: 12x10^{18} / cm^{3}
0.045um thick EBL (Al_{x}sub>GaN_{(1x)} x=0.15) acceptor: 12x10^{18} / cm^{3}
0.015um thick GaN spacer
8 pairs of In_{x}GaN_{(1x)}/GaN well(x=0.17)
3um nGaN donor5x10^{18} / cm^{3}
* spacer and well/barrier have unintentionally doped donor level~2x10^{18} / cm^{3}
(2) Bandoffset Ratio
We studied bandoffset effect on the carrier profile with driftdiffusion solution.
The default hetero junction bandoffset ratio is 70% in ATLAS but in this case it shows very low electron carrier concentration in the first few quantum well/barrier regions (top in Figure 1.) even though the injected current density is 200mA/cm2. When we used 50% bandoffset ratio it showed reasonable electron concentration and radiative rate in the wells (bottom in Figure 1.) within the driftdiffusion solution.
Figure 1. Carrier density profile with different bandoffset ratio(left: 70% bandoffet, right:50% bandoffet) red color is electron concentration and blue color is hole concentration. 
(3) Limitation of Driftdiffusion Simulation
From previous simulation results, even though we adjusted the bandoffset to a lower value to get high electron concentration in the first few quantum well regions, the hole and electron profile in barrier regions is very low and abrupt at the interface. This is related to lower carrier penetration to barrier from well regions. We will discuss this in more details in section 24.
We studied the forward current behavior depending on the number of quantum well with driftdiffusion without captureescape model. It shows that the IV shifts to higher forward voltage when the number of quantum well increases at fixed positive polarization and is related to increased series resistance of quantum well/barrier regions caused by low carrier density in the barrier regions. Also when we increase the polarization charge from 0 to 50% then it will also increase the forward voltage (Figure 2). This is unrealistic behavior as we can see in a real LED operation. So we can conclude that the driftdiffusion solution cannot alone explains current behavior very well.
Figure 2. Polarization effect on IV shift by polarization(left) and by the number of quantum well(right). 
At this point, we need to take into account of more general carrier transport models to describe the accurate carrier dynamics in both the quantum well and barrier regions.
(4) CaptureEscape Model
It is well known fact that the conventional LED simulation can be easily done by classical drift diffusion with selfconsistent quantum well SchrodingerPoisson solver and spontaneous emission rate from multiple band kp theory. But the classical driftdiffusion approach have shown very high forward voltage even at low polarization, low series resistance(high doping), and show unrealistic carrier profile between barrier and quantum well. It is worse in case of multiple quantum well in that when the number of quantum well is increased then the IV curve tends to shift to higher forward voltage.
For these reasons, we recently developed a new captureescape model which calculates the exact 2D bound carrier density in the well regions with detailed carrier capture from the bulk barrier and escape from the well into the barrier by adding probability of carrier captureescape mechanism into the carrier rate equation.
The purpose of this article is to show the validation of new captureescape model through comparison with experimental data and to give some guidelines for optimizing new models for real LED simulation.
General 3D carrier density rate equations are now modified into the following form.
n^{3D} is bulk carrier density and Cn3D>2D is captureescape rate from this bulk carrier to 2D carrier in the well.
(1)
(2)
2D carrier density rate equations in the well regions become
(3)
(4)
In the 2D inplane quantum well regions, net recombination rate is the sum of spontaneous emission rate and total captureescape rate.
Here, x,z plane is in quantum well inplane and y is assumed to be normal growth direction.
The 2D bound carrier density becomes
(5)
The bulk carrier density which is used to calculate the current density in the driftdiffusion formalism is modified to consider this confined 2D carrier in the quantum well regions.
(6)
The captureescape rate is calculated by the following equation.
(7)
Here, tn is capture time in the well. ( material well.taun=1e12 s well.taup=1e12 s )
To consider this additional 2D bound carriers, we also have to consider SRH recombination as well as Auger recombination in the well region. Atlas takes into account these recombinations in the quantum well regions including surface and interface trap recombinations.
material capt.augern=1e31 capt.augerp=1e31 ;Auger in the well
material capt.srh.n=1e9 capt.srh.p=1e9 ;SRH in the well
inttrap qwell s.i donor e.level=0.2 density=1e12 ;interface trap
interface s.i s.well.n=1e3 s.well.p=1e3 ;surface recombination
To activate the above recombinations we need to turn on the following model flags.
model capt.srh capt.auger well.capt well.inplane well.selfcon
In the above model, well.inplane consider 2D inplane current density equation.
In the effective mass and parabolic approximation, the radiative recombination in quantum well is
(8)
where
• _{r}^{2D} = m_{r}/h^{2}L^{2} is the twodimensional density of states.
• m_{r} is a reduced electronhole mass.
• n_{r} is a material refractive index.
• I_{m,n} = (_{m}_{n}) in a overlap integral of electron and hole wavefunctions.
Now, when we consider this new captureescape model to the forward current dependence on number of quantum well as in section 23, the forward voltage shift is less pronounced as compared to Figure 2.
If polarization is further decreased by taking into account of more defect screening effect or is zero then, forward voltage change is less pronounced with increasing quantum well or no further forward voltage shift in 4QW case (Figure 3). The carrierprofile by captureescape model well explain why forward voltage is lower than by driftdiffusion solution. The carrier has some probability to penetrate into barrier regions which is much higher than in the case of noncaptureescape model (Figure 4).
Figure 3. Forward current behavior by captureescape model(left figure shows IV shift by the number of quantum well with +50%polarization of theoretical value and right figure shows less IV shift plot by the number of quantum well without polarization). 
Figure 4. carrier distribution at 200A/cm2 ( left is without captureescape model and right figure shows carrier distribution with captureescape model). 
The carrier profile without captureescape model shows that electron concentration in the barrier region is far below 1x10^{10}/cm^{3} in 1st barrier and hole concentration is decreased to below 1x10/cm^{3} at the 3rd barrier region. But the carrier concentrations in the barrier by captureescape model are smoothly decayed from quantum well region and have 1x10^{14}/cm^{3} in the first two quantum well/barrier pairs. We can conclude from this result that the captureescape model accurately describes the exact carrier profile and so explains why driftdiffusion solution always produces high forward voltage when the number of quantum well is increased.
3. Material Growth and TrapAssisted Tunneling (TAT)
So far we have used the positive polarization scheme, in which polarization is directed toward to bottom interface as in Gaface growth condition. But polarity is still uncertain because it is very dependent on which substrate buffer layer is used and also on growth condition.
Figure 5. Polarity inversion by substrate and buffer layer. 
The default polarity in Atlas uses the positive convention which is positive charge at the bottom interface and negative charge at the top interface. So direction of polarization is downward and builtin electric field is opposite direction. Figure 6. shows that polarity inversion significantly affects the forward voltage. The actual amount of polarization by screening effect from various defects(dislocation or Vdefect) is very hard to estimate and optimize to fit the experimental forward current. Most simulation tasks so far normally use polarization between 20% and 80% from the theoretical value but it is a very broad range to adjust to fit forward current.
Figure 6. Forward current behavior by polarity (red: minus polarity, blue: positive polarity). 
When we use polarization we should keep in mind that the polarization is globally scaled from theoretical calculation which is not true in actual polarization of each layer. To simulate more accurate polarGaN LEDs it is very important to understand underlying polarization in each layer.
Also, if nonnegligible traps in the quantum well exist, there are some possibilities that electron can tunnel from nside to pside region via trapassisted tunneling mechanisms.
For this purpose, we simulated trapassisted tunneling to see the effect on forward tunneling current in a single quantum well case.
Figure 7 shows forward current behavior caused by existence of traps in quantum well regions and trapassisted tunneling effects in low bias regions. In a positive polarity case, we can see the conventional trapassisted tunneling current at low bias range.
Figure 7. forward current behavior by donor trap and trapassisted tunneling (red: without traps and TAT, blue: with trap only, light blue: traps + TAT). Top plot is negative polarization case and bottom plot is positive polarization case. 
4. Bandgap Reduction by Ambient Temperature
We studied the ambient temperature effect on forward current because LED performance is significantly affected by the operation temperature. Normally, high efficiency and high power LED operates in fairly high temperature above 300K and bandgap reduction by temperature is very important to analyze forward current behavior. The default bandgap reduction model by temperature is Varshini model and model parameters are listed in Piprek’s book[5].
(9)
In negative polarity, low bias region is much more affected by temperature but in positive polarity whole region is increased by evaluated ambient device temperature.
Figure 8. Ambient temperature effect on forward current (top: negative polarity, bottom: positive polarity). 
5. Validation
(1) Captureescape Model
The new captureescape model in Atlas was validated through experimental data from reference[1].
Figure 9. Simulation of forward current using captureescape model (top figure is linear plot and bottom is log plot). 
The forward voltage is 3.24V at 20mA and it is very close to the reported value in the reference.
The ~V/Rs region(ohmic) is well reproduced and at medium bias range it shows very close to the experimental data. Thereafter, red color is the experimental data and blue color is the fitted simulation data.
The power curve is fitted by assuming 80% extraction efficiency and effective mass is adjusted to fit luminious power from spontaneous emission spectrum (Figure 10).

Figure 10. IL curve fitting result to experimental data. 
Figure 11 shows apparent Auger droop efficiency curve and Auger coefficient value of 2.4e30 is adjusted to fit high current regions.
Figure 11. EQE plot (left is linear scale and right is log scale). 
To fit the low and medium bias range SRH life time is most dominant fitting parameters which are well explained in ABC model and Auger coefficient is the most important to fit the efficiency droop curve in the high injection range. The external quantum efficiency (EQE) is defined as the ratio of the emitted number of photons to the number of injected carriers. From the simulation results, the total EQE ratio is obtained through dividing the total radiative rate in the quantum wells by injection current.
The internal quantum efficiency (IQE) is defined as Rrad /( Rrad + Rnon_rad).
The radiative rate (Rrad) is the total radiative rate which is mostly coming from quantum well regions and the nonradiative rate (Rnon_rad) is the sum of each SRH and Auger recombination rate.
Figure 12. carrier distribution (left) and each recombination rate in the quantum well by the captureescape model. 
Figure 13. WallPlugEfficiency. 
As we can see from Figure 13, the carrier profile is very different with driftdiffusion solution (Figure 1.)
The carrier concentration is much higher in barrier regions than the profile by the driftdiffusion solution.
Because the standard LED has AlGaN EBL layer to block electron current from nside, there is small leakage current and most current flows into the quantum well and then recombines to give spontaneous or nonradiative rate. Some references define the external quantum efficiency as the following equation.
(10)
The injection efficiency() is the ratio of the total injected current into the quantum well to the total current. In this conventional LED example, the injection efficiency is assumed to be 1.0 and EQE equals to internal quantum efficiency.()
The wallplugefficiency(WPE) in Figure 13 is defined as ratio of the total output power(W) to the input power(I*V). There are many unknown physical parameters like SRH life time, Auger coefficients, and captureescape time. The SRH life time and Auger coefficients are adjusted to fit the experimental data. The overall simulation results are quite well reproduced with the captureescape model and physical parameters from the reference[1].
(2) Trapassist tunneling(TAT)
Because trapassisted tunneling is very important mechanism in low and medium bias ranges, we fitted to the experimental data[6] through Atlas trapassisted tunneling model with the captureescape approach. For this purpose, we took into account of midgap traps states in quantum well region where is the most probable path for tunneling via trap site. Because field enhancement term of TAT is strongly dependent on local field in SRH recombination rate, the most important fitting parameter is actual polarization charge which is also affected by well Indium composition, defects, and well thickness.
We used tunneling effective mass to fit the experimental data and Figure 14 shows reproducing the experimental forward current behavior except below 2.0V. According to the paper, difference below 2.0V could be due to heavyhole tunneling. We need to study more about this another trapassisted tunneling mechanism.
Figure 14. Simulation of Trapassisted tunneling. 
6. Summary
We can conclude that the new captureescape model is a very accurate description model for exact carrier distribution on both barrier and quantum well . We developed a selfconsistent solution from SchrodingerPoisson solver for 2D bound carrier density in quantum well and bulk carrier density equation is linked by proper captureescape rate equation to simulate the GaN based LED device. To simulate correctly EQE(or IQE) curve and efficiency droop, we have to consider the exact carrier density which will affect the radiative and nonradiative recombination rate in quantum well region through the captureescape model. Theoretical approaches using k.p and captureescape model are well incorporated into the recent ATLAS device simulation and it is a very accurate model for carrier dynamics in multiple quantum well design.
References.
 Blue light emitting diode exceeding 100% quantum efficiency, Joachim Piprek, Phys. Status Solidi RRL 8, No. 5, 424426(2014)
 Simulation of lightemitting diodes for new physics understanding and device design, K. A. Bulashevich et al., Proc. Of SPIE, vol. 8278(2012)
 Consistent set of band parameters for the groupIII nitrides AlN, GaN, and InN, Patrick Rinke et al. Physical Review B 77. (2008)
 Atlas User’s Manual, Silvaco , 2015
 Semiconductor Optoelectronic Devices, Introduction to Physics and Simulation, Joachim Piprek, University of California at Santa Barbara, Academic Press
 Trap tunneling in InGaN/GaN LEDs: experiments and physicsbased simulation, NUSOD 2014
 Trapassisted tunneling in InGaN/GaN single quantum well light emitting diodes, M. Auf der Maur, et al., Applied Physics Letters. 105 (2014)