#  Efficiency and Forward Voltage of Blue and Green Lateral LEDs with V-shape
defects and Random Alloy Fluctuation in Quantum Wells

Cheng-Han Ho  Graduate Institute of Photonics and Optoelectronics, National
Taiwan University, Taipei 10617, Taiwan  James S. Speck  Materials Department,
University of California, Santa Barbara, California 93106, USA  Claude
Weisbuch  Materials Department, University of California, Santa Barbara,
California 93106, USA  Laboratoire de Physique de la Matière Condensée, Ecole
Polytechnique, CNRS, IP Paris, 91128 Palaiseau Cedex, France  Yuh-Renn Wu
Author to whom correspondence should be addressed<EMAIL_ADDRESS>Graduate
Institute of Photonics and Optoelectronics, National Taiwan University, Taipei
10617, Taiwan  Electronic and Optoelectronic System Research Laboratories,
Industrial Technology Research Laboratories, Hsinchu, Taiwan.

###### Abstract

For nitride-based blue and green light-emitting diodes (LEDs), the forward
voltage $V_{\text{for}}$ is larger than expected, especially for green LEDs.
This is mainly due to the large barriers to vertical carrier transport caused
by the total polarization discontinuity at multiple quantum well and quantum
barrier interfaces. The natural random alloy fluctuation in QWs has proven to
be an important factor reducing $V_{\text{for}}$. However, this does not
suffice in the case of green LEDs because of their larger polarization-induced
barrier. V-defects have been proposed as another key factor in reducing
$V_{\text{for}}$ to allow laterally injection into multiple quantum wells
(MQWs), thus bypassing the multiple energy barriers incurred by vertical
transport. In this paper, to model carrier transport in the whole LED, we
consider both random-alloy and V-defect effects. A fully two-dimensional
drift-diffusion charge-control solver is used to model both effects. The
results indicate that the turn-on voltages for blue and green LEDs are both
affected by random alloy fluctuations and V-defect density. For green LEDs,
$V_{\text{for}}$ decreases more due to V-defects, where the smaller
polarization barrier at the V-defect sidewall is the major path for lateral
carrier injection. Finally, we discuss how V-defect density and size affects
the results.

††preprint: AIP/123-QED

##  I Introduction

Blue nitride-based light-emitting diodes (LEDs) with phosphor have become the
major white-light source. However, high-efficiency or micro-LED display
applications require green- and red-light sources and, for ultimate
efficiency, white-light lamps based on RGB color mixing. Currently, the peak
internal quantum efficiency ($IQE$) of blue LEDs is over 90$\%$ at a current
density of 1–10 A/cm2.[1] However, the $IQE$ of green LEDs remains low, which
is likely due to the strong quantum-confined Stark effect induced by the
polarization charge and the large defect density[1, 2, 3]. In addition, the
forward voltage $V_{\text{for}}$ of green LEDs also exceeds the expected 2.3
V, [4] which also limits the wall plug efficiency.

III-nitrides in the common (0001) orientation have polarization-induced
barriers at the InGaN/GaN quantum well/quantum barrier (QW/QB) interface due
to spontaneous and piezoelectric polarization differences between the well and
barrier materials.[5] The polarization discontinuity at the QW/QB interface
results in significant electric fields in the QWs and QBs—often referred to as
“piezofields.” For green LEDs, the polarization discontinuity at the QW/QB
interface increases due to the increased lattice mismatch between the InGaN
and GaN layers.

The influence of polarization-induced barriers has been discussed in many
studies[6, 7, 4] where carriers are easily blocked by the extra potential
barrier. Our previous studies of blue LEDs [8, 9] show that fluctuations in
indium composition provide extra paths for carrier injection. Including random
indium composition fluctuations in models thus helps to explain the lower
$V_{\text{for}}$ in blue LEDs compared with models without fluctuations. [10,
11]

To further reduce $V_{\text{for}}$ in situations with larger barriers, such as
in green LEDs, it is important to either reduce the polarization field or find
an alternative way to inject carriers. The most significant method is by
controlling the V-shaped defect (V-defect) opened at the InGaN/GaN MQW active
regions to provide an alternative way to inject carriers. Most V-defects are
the result of dislocation lines that form under certain growth conditions.
[12, 13, 14] Unlike the threading dislocation (TD) centers, which are typical
nonradiative centers, the inclined sidewalls of V-defects reportedly assist
carrier injection into QWs. [15, 16, 17] Since both random alloy fluctuations
and V-defects provide concurrent paths for carrier injection, it is important
to model both effects in carrier injection simultaneously. The importance of
considering both mechanisms is obvious as the paper leads to the conclusion
that random alloy fluctuations suffice to lead to efficient low-voltage
injection in blue LEDs while V-defects are needed to achieve this in longer
wavelength LEDs in the green spectral range.

Given that V-defects play an important role in carrier injection, it is
important to understand the role of random alloy fluctuations and of V-defects
in carrier injection. The V-defect and random alloy effects may be simulated
separately in three-dimensional simulations. [8, 15] On the one hand,
V-defects range from a few hundred nanometers in size to the $\mu$m scale,
allowing for large mesh sizes. On the other hand, random alloy fluctuations
occur on a scale of a few nanometers, requiring mesh elements to be as small
as possible to account for such small-scale fluctuations. Including both
effects in a three-dimensional simulation requires unreasonable computer
memory (> a few TB) and computing times. However, to understand the influence
of random alloy fluctuation and V-defect on the efficiency of LEDs, especially
for green LEDs, it is essential to include both effects simultaneously. Recent
results on the lateral carrier diffusion[18] in QW show that the random alloy
fluctuations strongly limits carrier diffusion. Hence, if the carriers are
only injected through V-defects, the current spreading to the whole QW area
from the V-defect will be limited, which is observed without considering
random alloy fluctuation. Furthermore, the improvement in turn-on voltage
induced by V-defects, critical to the wallplug efficiency, can be jeopardized
by the adverse action of the threading dislocation within the V-defects as
nonradiative centers on the $IQE$. The balance between the reduction of
$V_{\text{for}}$ by increasing the V-defect density and the simultaneous
decrease in $IQE$ must be explored. A compromise to the present difficulties
of 3D simulations is to work on two-dimensional (2D) simulations that include
random alloy fluctuations and V-defects. Some issues might be missed in the 2D
model, such as the estimation of carrier quantum confinement effects occurring
in 3D. This may affect the estimation of carrier overlap in random alloy
fluctuating potentials. The V-defect filling factor of 2D and 3D models may
not be equal and needs some translation. However, the 2D model enables us to
calculate the whole LED structure, including the current crowding effect. This
helps us to understand the simultaneous influences of V-defect density and
random alloy fluctuation to $V_{\text{for}}$ or and $IQE$. In this paper, we
discuss the influence of V-defects and random alloy fluctuations with such an
alternative 2D model.

V-defects may be controlled by tuning the growth conditions. Often, V-defects
are formed in the widely used InGaN/GaN short periodic superlattice that is
grown before growing active InGaN/GaN QWs. [19] The general V-defect density
is about 1$\times$107 cm${}^{-2}\sim 5\times 10^{8}$ cm-2, and the typical
diameter is about 30–250 nm for five QWs in blue LEDs. [12, 13, 14, 19] Hence,
in this paper, we will cover the V-defect density from 0 to $\sim$ 6.25$\times
10^{8}$ cm-2. Furthermore, V-defects form an inverted hexagonal pyramid with
six inclined $\left\\{10\bar{1}1\right\\}$ semipolar sidewalls. [20] Studies
show that the indium composition in $c$-plane QWs is greater than in the
inclined QWs in the V-defect region. [21] Therefore, the bandgap of QWs in the
inclined sidewall of V-defects is larger than the bandgap of QWs in the $c$
plane. The polarization discontinuity between the semipolar
$\left\\{10\bar{1}1\right\\}$ InGaN sidewall QW and QB is much smaller than
the $c$-plane QW and is in the opposite direction. [22] Thus, holes can be
injected into the QWs in the $c$ plane through the sidewall V-defect QWs with
lower barrier potentials. [23] The additional paths provided decreased
$V_{\text{for}}$ and increase the carrier-injection efficiency. [24, 25, 26]
However, V-defects form at TDs, and a high density of nonradiative centers at
the center of V-defects could be a drawback. Thus, the density and diameter of
the V-defects in the LED should be optimized to minimize this counter-acting
effect. [27]

In this paper, we use simulations to investigate how random fluctuations and
V-defects affect InGaN-based LEDs. As mentioned above, computer-memory
limitations oblige us to use 2D simulations with fluctuations instead of
three-dimensional simulations. In 2D simulations, the size of the simulated
structure may be increased to the micron scale to approach the real device
structure of LEDs. In the following, we discuss the 2D structure with
different V-defect densities and diameters (lateral size). The forward voltage
$V_{\text{for}}$ and efficiencies (electrical and $IQE$) are discussed and
explained by the interplay between lateral carrier injection through the
V-defect sidewalls and subsequent carrier diffusion within the planar QW
regions.

##  II Methodology

To simulate random alloy fluctuations in devices, we use an in-house-developed
2D drift-diffusion charge control solver to determine the electrical carrier
distribution, recombination rates, and other properties. First, to account for
the random alloy fluctuations and V defects, we construct a mesh for a device
with a V-defect. The mesh size at the QW region is 0.5 nm in the $x$ direction
and 0.1 nm in the $z$ direction. The mesh size is larger in the $p$ and $n$
layers to save computer memory. We then use a random number generator to
create the random atom distribution in the alloy regions [28] and subsequently
use Gaussian averaging to assign the local indium composition in the QWs. [29,
30, 31]

Figure 1 shows the indium composition fluctuations in the QWs, where the
bandgap depends on the indium composition. Note that, in the 2D simulation, we
cannot calculate the real strain distribution with a strain solver because we
have neglected the third dimension. Thus, fully compressive strain is assumed
when calculating the lattice size of the GaN buffer layer based on the local
indium composition. We then calculate the local material parameters such as
bandgap, conduction- and valence-band energies, polarization according to the
local indium composition, and strain state. In addition, we use the
localization landscape (LL) model [29, 32, 33] shown in Eq. (2) instead of the
Schrödinger equation solver to obtain the effective quantum potential seen by
the carriers. [31]

###  II.1 Methodology of analyzing two-dimensional solver

To simulate the 2D Poisson, drift-diffusion, and LL equations, the following
equations were used:

$\displaystyle\bm{\nabla}\cdot\left(\varepsilon\bm{\nabla}\varphi\right)=q\left(n-p+N_{\text{A}}^{-}-N_{\text{D}}^{+}\pm\rho_{\text{pol}}\right),$
(1)
$\displaystyle\left[-\frac{\hbar^{2}}{2}\bm{\nabla}\left(\frac{1}{m_{\text{e,h}}^{\ast}}\bm{\nabla}\right)+E_{\text{c,v}}\right]u_{\text{e,h}}=1,$
(2) $\displaystyle n$ $\displaystyle=$
$\displaystyle\int_{1/u_{\text{e}}}^{+\infty}{\text{LDOS}\left(E\right)\frac{1}{1+\exp\left(\frac{E-E_{\text{F}n}}{k_{\text{B}}T}\right)}dE},$
(3) $\displaystyle p$ $\displaystyle=$
$\displaystyle\int_{1/u_{\text{h}}}^{-\infty}{\text{LDOS}\left(E\right)\frac{1}{1+\exp\left(\frac{E_{\text{F}p}-E}{k_{\text{B}}T}\right)}dE},$
(4) $\displaystyle J_{n,p}$ $\displaystyle=$
$\displaystyle\mu_{n,p}\left(n,p\right)\bm{\nabla}E_{\text{F}_{n,p}},$ (5)
$\displaystyle\frac{1}{q}\bm{\nabla}\left(J_{n,p}\right)$ $\displaystyle=$
$\displaystyle R_{n,p}-G_{n,p},$ (6) $\displaystyle R$ $\displaystyle=$
$\displaystyle R_{\text{SRH}}+B_{0}np+C_{0}\left(n^{2}p+np^{2}\right),$ (7)
$\displaystyle R_{\text{SRH}}$ $\displaystyle=$ $\displaystyle\frac{np-
n_{i}^{2}}{\tau_{n0}\left(p+n_{i}\right)+\tau_{p0}\left(n+n_{i}\right)}.$ (8)

In these equations, $\varphi$ is the electrostatic potential in the structure,
$n$ and $p$ are the free electron and hole concentrations, and
$N_{\text{A}}^{-}$ and $N_{\text{D}}^{+}$ are the ionized acceptor and donor
concentrations determined by the ionization energy (and position in the
junction and junction electric field), respectively. $\rho_{\text{pol}}$ is
the polarization charge that has been computed from the divergence of the
total polarization, and $\frac{1}{u_{\text{e}}}$ and $\frac{1}{u_{\text{h}}}$
are the effective quantum potentials of electrons and holes calculated by the
LL equations, [33, 29] respectively.

First, we calculate $\varphi$ by using the Poisson equation (1) and obtain the
local band extrema $E_{\text{c}}$ and $E_{v}$ through the local $\varphi$ and
alloy composition. Then, the LL equations (2) are solved to yield
${u_{\text{e}}}$ and ${u_{\text{h}}}$ and the effective quantum potentials
$\frac{1}{u_{\text{e}}}$ and $\frac{1}{u_{\text{h}}}$. The carrier
distributions are calculated by using Eqs. (3) and (4) , the effective quantum
potentials $\frac{1}{u_{\text{e}}}$ and $\frac{1}{u_{\text{h}}}$, and the
quasi-Fermi levels $E_{\text{F}n}$ and $E_{\text{F}p}$. Finally, we determine
$E_{\text{F}n}$ and $E_{\text{F}p}$ from Eqs. (5) and (6). The recombination
rate is given by Eq. (7), and includes radiative recombination ($IQE$), Auger
recombination, and Shockley–Read–Hall (SRH) recombination. $B_{0}$ is the
radiative recombination rate. Equation (8) is the SRH recombination equation,
where $\tau_{n}$ and $\tau_{p}$ are the nonradiative carrier lifetimes of
electrons and holes, respectively. $C_{0}$ is the Auger recombination rate
coefficient, which is the recombination of three particles, either directly or
mediated by phonons. [34]

Figure 1: Flow chart of simulation model. The mesh with V-defect shapes is put
in the simulation program. The random alloy map is generated by using a random
number generator and then put in the solver to realize convergence after the
required number of feedback loops. The right panel shows the indium
composition map in the MQWs.

Next, we construct the mesh according to the position in the structure; each
mesh point has different parameters. To simulate V-defects in the structure,
the mesh is readjusted according to the V-defect diameter and density.
Finally, these equations are solved self-consistently. Figure 1 shows a
detailed flowchart of the simulation.

###  II.2 Device structures and parameters

To discuss the V-defect, we analyze the blue and green LEDs with the structure
shown in Fig. 2(a). The chip lateral size is 11 $\mu$m and the region of
multiple quantum wells (MQWs) is 10 $\mu$m wide, which is also defined as the
total region. Since our mesh element size is quite small compared to the size
of the device, we use about $1.1\times 10^{7}$ nodes to simulate the 11 $\mu$m
chip, which requires 55 GB of memory to run the 2D simulation. The $p$ contact
(in common $p$-side LEDs, this layer is made from indium tin oxide) is about 9
$\mu$m wide at the center of the active region, and the $n$ pad is about 1
$\mu$m wide.

Figure 2: (a) Simulated structure of lateral LED. (b) Definition of parameters
for V-defect structure.

For the blue and green LEDs, we consider a MQW LED with a 0.14 $\mu$m top
$p$-GaN layer doped at $3\times 10^{19}$ cm-3, a $p$-AlGaN electron blocking
layer (EBL) and $p$-GaN cladding layer both doped at $2\times 10^{19}$ cm-3,
five QWs and four QBs between the QWs are unintentionally doped $n$ type at
$1\times 10^{17}$cm-3, one 10 nm $n$-cladding layer and then a 2.5 $\mu$m
$n$-GaN layer doped at $5\times 10^{18}$ cm-3. The doping density the of
$n$-type cladding layer is $1.0\times 10^{19}$ cm-3. For the green LED, the
first two QBs are $n$ doped at $1.0\times 10^{18}$ cm-3 to partially screen
the piezoelectric field for better vertical carrier injection. We also ran a
simulated case without doping in the first two QBs (and with no V-defects) for
comparison. In addition, $\tau_{n}$ and $\tau_{p}$ for the blue and green LEDs
are 100 and 10 ns, respectively, because the green LED has a higher In content
in the InGaN than the blue LED and thus a lower growth temperature, which
leads to a higher density of SRH centers in the real device. The detailed
parameters of the structures are shown in Table 1. For the blue LED, the
average indium composition in the QWs is 13$\%$ (In0.13Ga0.87N). For the green
LED, the average indium composition in the QWs is 22$\%$ (In0.22Ga0.78N).
Additionally, alloy fluctuations in the InGaN layers are included in both the
blue and green LEDs. The $B_{0}$ is $2$ $\times$ $10^{-11}$ cm3/Vs and the
$C_{0}$ is $2$ $\times$ $10^{-31}$ cm6/Vs. These two parameters are not
changed in the simulation since we focus on V-defect and random alloy
fluctuation influences. The influence of overlap due to the different QCSE and
disorder is included in the term of $n(r)\times p(r)$.

Table 1: Basic parameters of blue and green LEDs. Area | Material | Thickness |  Doping blue , green | $E_{\text{a}}$ |  e, h mobility |  $\tau_{n}$, $\tau_{p}$ blue |  $\tau_{n}$, $\tau_{p}$ green
---|---|---|---|---|---|---|---
| unit | nm | 1018cm-3 | meV | cm2/V s | ns | ns
$p$ layer |  $p$-GaN | 140 | 30, 30 | 180 | 100, 5 | 100 | 10
$p$ EBL |  $p$-AlGaN | 20 | 20, 20 | 264 | 100, 5 | 100 | 10
$p$ cladding layer |  $p$-GaN | 10 | 20, 20 | 180 | 100, 5 | 100 | 10
QW(3,4,5) |  $n$-InGaN | 3 | 0.1, 0.1 | 25 | 150, 10 | 100 | 10
QB(3,4) |  $n$-GaN | 10 | 0.1, 0.1 | 25 | 350, 10 | 100 | 10
QW(1,2) |  $n$-InGaN | 3 | 0.1, 0.1 | 25 | 150, 10 | 100 | 10
QB(1,2) |  $n$-GaN | 10 | 0.1, 1 | 25 | 350, 10 | 100 | 10
$n$ cladding layer |  $n$-GaN | 10 | 10, 10 | 25 | 200, 10 | 100 | 10
$n$ layer |  $n$-GaN | 2500 | 5, 5 | 25 | 200, 10 | 100 | 10

Figure 2(b) shows the geometrical definition of V-defects, where H and D are
the depth and diameter of the V-defect, respectively. We use these parameters
in the following discussion. The region between the vertical black dotted
lines correspond to the TD along the direction $\left\\{0001\right\\}$ with a
30 nm width at the center of the V-defect, and the V-defect sidewall QWs are
inclined about 60∘ according to both transmission electron microscopy (TEM)
and scanning TEM,[35, 36] which is consistent with early work showing that the
sidewalls are $\left\\{10\bar{1}1\right\\}$ planes. In addition, the thickness
of the $c$-plane QWs is 3 nm. The inclined sidewall QW is also 3 nm thick in
the z direction, which corresponds to a thickness of 1.73 nm ($3\text{
nm}\times\cos 60^{\circ}$) along the normal to the inclined plane. [37] The
indium composition in the inclined QWs depends on the composition in $c$-plane
QWs. For example, for blue LEDs, the indium composition of inclined QWs is
8$\%$. [38, 39] For green LEDs, the indium composition in the inclined QWs is
about 16$\%$. [40] In addition, the TDs are defects induced by the lattice and
chemical mismatch between the sapphire substrate and GaN. [41, 42] Thus, we
include TD-associated trap states in the TDs regions, which trap either
electron or holes and are shown as the cross-hatched areas in Fig. 2(b). The
density of TD-associated trap states is about $1\times 10^{18}$ cm-3
corresponding to one trap state per $c$ translation along the dislocation line
within the 30 nm diameter TD ranges.[43, 44] The trap energy level
($E_{\text{t}}$) is 1.14 eV below $E_{\text{c}}$ for donor-like traps and 2.5
eV below $E_{\text{c}}$ for acceptor-like traps, as shown in Table 2. The trap
lifetimes are also given in Table 2. [44]

Table 2: Trap parameters in the threading dislocation (TD) center. Type |  |
---|---|---
Donor-like |  |
| Trap density (1/cm3) | $1\times 10^{18}$
| Trap $\tau_{n}$,$\tau_{p}$ (ns) | 1.01
|  $E_{\text{t}}$ below $E_{\text{c}}$ (eV) | 1.14
Acceptor-like |  |
| Trap density (1/cm3) |  $1\times$1018
| Trap $\tau_{n}$,$\tau_{p}$ (ns) | 0.38
|  $E_{\text{t}}$ below $E_{\text{c}}$ (eV) | 2.5

##  III Results and discussion

###  III.1 Influence of random alloy fluctuations, V-defects, and threading
dislocations for the current path

Figure 3 shows the calculated $E_{\text{c}}$ and effective quantum potential
$\frac{1}{u_{\text{e}}}$ for the blue and green LEDs, taking into account
random alloy fluctuations at a bias voltage $V=3$ V. A cross section with
minimum potential barrier was chosen for the full curves. They show the best
case for carrier transport across the QWs. The dashed curves for disorder
averaged levels show slightly larger barriers. Without considering the random
alloy fluctuation, the voltage voltage would be even higher which has been
discussed in ref. [6, 8, 31].Hence we will not discuss the cases with and
without random alloy fluctuation. Although Figs. 3(a) and 3(b) show that
random alloy fluctuations already change the depth of the QWs for blue LEDs
(compare the $E_{\text{c}}$ curves with and without fluctuations), the
potential barrier induced by piezoelectric field between QWs is still about
0.4 eV for the blue LED, whose carriers can overcome the barrier through
thermal excitation. However, for the green LED, the potential barrier is about
0.7 eV, which strongly impedes carrier transport across the MQWs at the bias
$V=3$ V. While the current density at that bias is a few A/cm2 for the blue
LED [see Fig. 6(a)], it is negligible in the green LED [Fig. 9(a)]. Thus, even
when considering random alloy fluctuations, carriers are hardly injected
directly into the $c$-plane QW for the green LED, despite the large 0.7 V
excess voltage. We also observe that the inclusion of the quantum disorder
correction through the use of the LL theory (thus yielding the effective
potential $\frac{1}{u_{\text{e}}}$) only weakly diminishes the potential
barriers, in particular for the green LED. Reducing $V_{\text{for}}$ in these
LEDs requires the use of another injection mechanism, namely, injection
through V-defects.

Figure 3: Cross section plot for the three top QWs of $E_{\text{c}}$ without
fluctuations ($E_{\text{c}}$ w/o fluc), $E_{\text{c}}$ with fluctuations
($E_{\text{c}}$ w/ fluc),effective potential $\frac{1}{u_{\text{e}}}$ ,
$E_{\text{F}n}$ along z-direction with random alloy fluctuations but without
V-defects. (a) is for the blue LED, and (b) is for the green LED. The applied
bias is 3V. The full curves w/ fluc are for the cross section with minimum
barrier and the dashed lines are for the fluctuation-averaged potential
$E_{\text{c}}$, $\frac{1}{u_{\text{e}}}$ and $E_{\text{F}n}$.

We first focus on hole injection through a V-defect. Figures 4(a) and 4(b)
show the valence-band potential of blue and green LEDs in a V-defect,
respectively. First, in the middle of the V-defect, within $\pm$15 nm of the
TD in the $p$-GaN layer region, holes are trapped in the hole traps associated
with the TD, which repels holes from the TD center [the repelling potential is
shown in a lighter color in the TD region in Figs. 4(a) and 4(b)]. Next, holes
transport toward the $c$-plane QWs through the inclined plane, as shown in
Figs. 5(c) and 5(d). Given the position of the electron-injecting region
relative to the $c$-plane QWs, electrons are first injected into the side-wall
QWs, from which they diffuse toward the lower-energy $c$-plane QWs.

Figures 4(c) and 4(d) show the radiative recombination map for the blue and
green LEDs at 20 A/cm2, respectively. For blue LEDs, recombination occurs
throughout the structure as electrons and holes (at least four of the
$c$-plane QWs for holes) are injected vertically [Figs. 5(a) and 5(c)] in the
$c$-plane QWs. For the green LED, while electrons are injected both vertically
and through the V-defect [Fig. 5(b)], holes are not injected vertically
because of the much larger polarization-induced barrier in the $c$ plane,
Holes are therefore injected through the V-defect sidewall near which
recombination occurs and decays laterally within a characteristic diffusion
length. In addition, only two to three $c$-plane QWs receive holes through the
V-defect. Both effects limit the effectiveness of V-defects as a solution to
excite the whole volume of the $c$-plane QWs. For the blue LED, the crowding
effect of carriers near V-defects is weaker because holes can still be
injected vertically into the whole $c$-plane area due to random alloy
fluctuations. The relative importance of vertical injection versus V-defect
injection is discussed in more detail below along with how the V-defect size
and density affects $V_{\text{for}}$ and the $IQE$ for blue and green LEDs.

A remarkable phenomenon is seen in Fig. 5: both electrons and holes are
preferentially injected into QWs near the middle of the MQW stack, QWS #2, 3,4
for the blue LED, and QW #4 for the green LED. This is dominantly due to the
peculiarity of hole injection through V-defects. Due to the diminished
piezoelectric induced barrier at the sidewall (which is a semipolar plane at
60 degree), the electrons and holes are more easily injected sideways than
vertically, especially for holes, which have an heavier effective mass and
lower mobility. Due to the p-doping of the last barrier, holes are easily
injected in QW5 through the sidewall. Overcoming one more barrier is still
relatively easy for holes, compared to the c-plane vertical injection. Holes
then first meet electrons in QW4 and recombine there. Why aren’t electrons
injected in QW5? Close inspection of electron current in Fig. 5 (b) for green
LEDs show that they experience a significant energy barrier between the
sidewall QWs 4 and 5 (blue color of the barrier). This is due to the nearby p
doping inside the V-defect which raises the barrier. The behavior is more
significant for the green case as can be seen in Figs. 5(b) and (d), as well
as on the recombination current in Fig. 4(d). This was not observed without
V-defect since it is much harder for holes to overcome the large piezoelectric
barrier in the c-plane direction especially in the green case. Then, holes are
preferentially injected in QW # 5 where they meet electrons. It is well known
that holes in the planar LED case are mainly injected in the QW nearest to the
p-side in planar MQW LEDs, and this is of course well reproduced in our other
simulation papers which include disorder effects without V-defects.

Figure 4: Valence-band energy and radiative recombination for blue LED and
green LED at V-defect density $2.5\times$107 cm-2 and $D=100$ nm at a current
density of 20 A/cm2. (a) Valence band for the blue LED, (b) valence band for
the green LED, (c) radiative recombination for the blue LED at 20 A/cm2, and
(d) radiative recombination for the green LED at 20 A/cm2. The plot is at log
scale. White dashed lines show the location of V-defects on the $x$ axis.
Insets in panels (a) and (b) at different color scale emphasize the higher
potential energy for holes at the center due to trap charging. The scale color
bar ranges from -3.0eV (red) to -3.1 eV(blue)

Since carriers are mainly going through V-defects for the green LED, the
density of V-defects play an important role in the LED current-voltage
characteristics. Using a high V-defect density to obtain more injection sites,
the active $c$-plane QW area becomes smaller and the carrier density in the
$c$ plane increases for the same current. This adversely affects the $IQE$ and
droop behavior. [45] Therefore, it is important to discuss in detail how the
V-defect density and size affect the LED structure.

Figure 5: Current distribution for blue LED and green LED at V-defect density
of $2.5\times$107 cm-2 and $D=100$ nm at a current density of 20 A/cm2. (a)
Electron current density |$J_{n}$| for blue LED, (b) electron current density
|$J_{n}$| for green LED, (c) hole current density |$J_{p}$| for blue LED, and
(d) hole current density |$J_{p}$| for green LEDs. The plot is at log scale.
The vector sense for the electrons [panels (a) and (b)] and hole [panels (c)
and (d)] are shown by the red arrows.

###  III.2 Influence of V-defects and random alloy fluctuations on $I$-$V$ and
$IQE$

The influence of V-defects for the specific lateral LED of Fig. 2 is now
discussed for green and blue LEDs. The effects on $IQE$ and $V_{\text{for}}$
of both random alloy fluctuations and V-defects are simulated for different
V-defect densities and diameters. We use V-defect densities of $1\times
10^{6}$, $2.5\times 10^{7}$, $1\times 10^{8}$, $2.25\times 10^{8}$, and
$6.25\times 10^{8}$ cm-2 and V-defect diameters of 50, 100, 220, 280, and 340
nm.

####  III.2.1 Performance of blue V-defect LED

We first discuss the dependence of $V_{\text{for}}$ on the V-defect density.
Here we define the forward voltage $V_{\text{for}}$ as the voltage needed for
a current density of 1 A/cm2. Figure 6(a) shows the results for the blue LED
structures without and with the V-defects. The V-defect diameter is 100 nm.
$V_{\text{for}}$ is about 2.88 V for the case without V-defects. The maximum
density of V-defects is $6.25\times 10^{8}$ cm-2, and $V_{\text{for}}$ for
this case is about 2.74 V at 1 A/cm2, which equates to about zero excess
voltage.

For a V-defect density of $1\times 10^{8}$ cm-2, Fig. 6(b) shows the
dependence on V-defect size. The results show that the voltage decreases as
the V-defect diameter increases from 50 to 100 nm, but $V_{\text{for}}$
remains constant for the larger V-defect diameters, with all the decrease in
$V_{\text{for}}$ already realized at the lateral V-defect size of 50 nm. One
factor contributing to this result is that the voltage is already close to
$\frac{h\nu}{e}$, leaving little possibility for further reduction of
$V_{\text{for}}$.

Figure 6(c) shows the $IQE$ of blue LEDs with different V-defect densities;
the $IQE$ peak slightly decreases as the V-defect density increases. For high
V-defect densities, the $IQE$ peak appears at higher current density. In the
low-current-density region (10-3 to 1.0 A/cm2), holes are injected mainly
through V-defects and electrons are injected vertically through the random
alloy fluctuating potential, whereas for higher current density, the higher
$V_{\text{for}}$ means that both carriers start to be injected directly from
the $c$-plane QW through the fluctuating potential barriers. Since the
V-defect center contains strong nonradiative recombination (NR) centers,
carriers are injected through V-defects experience more NR.

Figures 7(a) and 7(b) show the radiative recombination for current densities
of 10-2 and 20 A/cm2, respectively, for blue LEDs at a V-defect density of
$2.5\times 10^{7}$ cm-2. At low current density, the top QW shows weak
radiative recombination, which means that holes are injected from V-defects
due to the diminished piezoelectric field-induced barrier at the sidewall,
especially for holes with lower mobility. At high current density, the top QW
also emits light because holes are also injected vertically. Thus, at higher
current density, the $IQE$ and $V_{\text{for}}$ are less influenced by
V-defects and the dislocation region. More QWs emit light, and the impact of
NR diminishes, so all curves regroup.

Figures 7(b)–7(d) show the recombination distribution for different V-defect
densities at $J=20$ A/cm2. The crowding effect is limited because of vertical
injection through random alloy fluctuations but is significant at low V-defect
density. Note that if random alloy fluctuations are not considered in the
c-plane QW, carrier diffusion extends much further. Then, we do not observe
current crowding near V-defect for the lower V-defect density, making the
predicted $IQE$ higher. Increasing the V-defect density diminished the
crowding effect and thus moves the peak $IQE$ to a higher current density,
although the peak $IQE$ is diminished somewhat because of the increased
density of NR centers [Fig. 6(c)]. Figure 6(d) shows the $IQE$ for different
V-defect diameters at a V-defect density of $1.0\times 10^{8}$ cm-2. When the
diameter increases, the V-defect area increases and the active $c$-plane area
decreases. Thus, the $IQE$ decreases with increasing V-defect diameter because
of the increased Auger effect, which is consistent with earlier work. [15]

Figure 6: (a) Voltage versus current for blue LED at different V-defect
densities for V-defect diameter $D=100$ nm. (b) Voltage versus current for
blue LED at different V-defect diameters and a V-defect density of $1\times
10^{8}$ cm-2. (c) $IQE$ versus current for blue LED at different V-defect
densities and a V- defect diameter of $D=100$ nm. (d) $IQE$ versus current for
blue LED at different V-defect diameters and a V-defect density of $1\times
10^{8}$ cm-2.

Figure 7: Distribution of radiative recombination rate for blue LED with
$D=100$ nm. The plot is at log scale. (a) V-defect density $2.5\times 10^{7}$
cm-2 at current density 0.01 A/cm2 (b) V-defect density $2.5\times 10^{7}$
cm-2 at current density 20 A/cm2 (c) V-defect density $1\times 10^{8}$ cm-2 at
current density 20 A/cm2 (d) V-defect density $6.25\times 10^{8}$ cm-2 at
current density 20 A/cm2. White dashed lines indicate the location of
V-defects on $x$ axis. The red arrow in panel (b) indicates that the top QW is
not injected vertically at low current (bias) and injection occurs through the
V-defects. The red dashed lines indicate the QW positions.

To further to discuss the joint influence of V-defect density and diameter on
the $IQE$, we consider the ratio of V-defect area to the total chip area.
Figure 8(a) shows $V_{\text{for}}$ as a function of the V-defect area ratio,
where points on a given V-defect density curve correspond to the different
V-defect diameters (50, 100, 220, 280, and 340 nm), which change the V-defect
area ratio. At low V-defect area ratio, $V_{\text{for}}$ is affected by both
the V-defect area ratio [46] and the V-defect density. This result is
attributed to the density being too low, even for large V-defects, so the
carriers injected from V-defects need to travel through a long lateral region
to inject into the entire QW. In addition, the natural alloy potential
fluctuations in the QW limit the lateral transport distance of electrons and
holes. As a result, carriers become crowded near the V-defect at low V-defect
density, so the device performance is strongly limited by the lateral
transport distance from the V-defect within the QW. [18]

This result is again verified by Figs. 7(b)–7(d). The recombination becomes
less crowded when the V-defect density increases. Once the V-defect area ratio
grows larger, $V_{\text{for}}$ decreases and then saturates at the expected
value near 2.7 V. Another contribution comes from the modified vertical
injection due to the presence of V-defects: the fluctuations in the vertical
energy barriers induced by the random alloy fluctuations causes some carriers
to be directly injected from the $c$ plane in regions with lower barriers.
With V-defects, electrons and holes can flow laterally into all quantum wells.
[47] Once carriers flow into QWs, they screen the polarization-related
electric fields so that vertical current injection directly into $c$-plane QWs
increases at lower bias voltage compared with the case without V-defects.

The diameter and density of V-defects also affect the efficiency of the blue
LED. Since the optimized performance is decided by wall plug
efficiency($WPE$), we estimate the $WPE$ by assuming a 90% light extraction
efficiency (LEE), typical of LEDs grown on patterned sapphire substrates(PSS).
We simulated how the $WPE$ at a fixed 20 A/cm2 current density depends on the
V-defect area ratio, which corresponds to different diameters and densities of
V-defects. The results are shown in Fig. 8(b). Interestingly, the low V-defect
density of $1.0\times 10^{6}$ and $2.5\times 10^{7}$ cm-2 have a much lower
$WPE$ and $IQE$ than the others because the current paths from V-defects to
the whole chip area are too long so the constant injected current of 20 A/cm2
becomes crowded near the V-defect region. They also have a higher
$V_{\text{for}}$. When the V-defect density exceeds $1.00\times 10^{8}$ cm-2,
which is within the lateral diffusion length, the carriers are easily injected
into the whole active region, diminishing crowding and further canceling the
quantum-confined Stark effect. This is also seen in Figs. 7(b)–7(d).

This latter effect helps to increase electron-hole overlap and increase the
efficiency. However, the $IQE$ also gradually decreases as the V-defect area
ratio further increases because the active volume decreases, and the droop
effect starts to dominate at lower current density for larger V-defect area
ratio. The conclusion is obvious: it is advantageous (i) to have a high
V-defect density so that the V-defect separation is about the lateral carrier
transport distance (often referred to as a diffusion length), to inject
carriers homogeneously, and (ii) to have small V- defects to minimize
nonemitting, unproductive V-defects. The larger V-defects densities also have
a smaller $V_{\text{for}}$, but the reduction of $V_{\text{for}}$ saturates as
V-defects density becomes larger than $10^{8}$ cm-2. Hence, the $WPE$ reaches
a peak at the V-defect ratio around 5%.

Figure 8: (a) V-defect region ratio (V-defect area/chip area) versus turn-on
voltage for different V-defect densities at a current density of 1 A/cm2 for
blue LEDs. (b) V-defect region ratio versus $WPE$ for different V-defect
densities at a current density of 20 A/cm2 for blue LEDs. The 90% LEE is
assumed.

####  III.2.2  Performance of green V-defect LED

For the green LED without V-defects, $V_{\text{for}}=3.83$ V at 1 A/cm2
current density is greater than for the blue LED [Fig. 9(a)]. [6, 4] If we
further remove the barrier doping, as indicated by the black dotted line,
$V_{\text{for}}$ increases. Despite doping the QB barriers between the MQWs,
the polarization barrier induced by the high indium composition is not
completely offset. Therefore, V-defects are much more helpful to decrease the
turn-on voltage for green LEDs than for blue LEDs. The lowest V-defect density
of $1\times 10^{6}$ cm-2 decreases the turn-on voltage to 2.81 V, as shown in
Fig. 9(a).

We also simulated different V-defect densities and find that $V_{\text{for}}$
decreases as the V-defect density increases. The different diameters of
V-defects are also simulated at a density $1\times 10^{8}$ cm-2, as shown in
Fig. 9(b). The results are similar to the blue LED shown in Fig. 6(b). The
voltage decreases as the diameter increases and then saturates.

Figure 9: (a) Voltage versus current for green LED at different V-defect
densities and for a V-defect diameter of $D=100$ nm. (b) Voltage versus
current for green LED at different V-defect diameters and a V-defect density
of $1\times 10^{8}$ cm-2. (c) $IQE$ versus current density for green LED at
different V-defect densities and for a V-defect diameter of $D=100$ nm. (d)
$IQE$ versus current for green LED at different V-defect diameters and a
V-defect density of $1\times 10^{8}$ cm-2.

We also calculate $IQE$ versus V-defect density for $D=100$ nm [see Fig.
9(c)]. The results show that different V-defect densities produce different
performances. Unlike the blue LED where some carriers are injected from the
$c$ plane, in the green LED, most carriers are injected into the planar QWs
through the V-defect sidewall, as shown in Figs. 4 and 5. For the lower
V-defect density, the $IQE$ peaks slightly higher due to a lower TD density.
However, due to greater current crowding, where carriers are crowded in the
$c$-plane QW near V-defects, the droop effect appears at lower current
densities, as shown in Fig. 9(c). When the current density increases,
additional carriers flow into the whole active QW region for all V-defect
densities. Therefore, the $IQE$s reach a similar value for all V-defect
densities, as shown in Fig. 9(c).

Figure 9(d) shows the $IQE$ for different V-defect diameters and for a
V-defect density of $1.0\times 10^{8}$ cm-2. For the larger diameters, the
V-defect area increases and the active $c$-plane area decreases. Thus, the
$IQE$ decreases as the V-defect diameter increases. To further discuss this,
we consider how the performance depends on the V-defect area ratio.

Figure 10: Distribution of radiative recombination rate for green LED at
$D=100$ nm. The plot is at log scale.(a) V-defect density $2.5\times 10^{7}$
cm-2 at current density 0.01 A/cm2. (b) V-defect density $2.5\times 10^{7}$
cm-2 at current density 20 A/cm2. (c) V-defect density $1\times 10^{8}$ cm-2
at current density 20 A/cm2. (d) V-defect density $6.25\times 10^{8}$ cm-2 at
current density 20 A/cm2. Vertical white dashed lines indicate the location of
V-defects along the $x$ axis.

Figure 10 shows the recombination distribution for different current densities
and different V-defect densities. Unlike for blue LEDs, a strong crowding
effect occurs until the V-defect density exceeds $1\times 10^{8}$ cm-2,
independently of the current density. This again shows that the V-defect
density is the dominant factor for carrier injection. As mentioned in the blue
case, if the random alloy fluctuation is not considered, the current crowding
effect is not observed, which leads to a much higher $IQE$ and the delay of
droop behavior.

Figure 11(a) shows $V_{\text{for}}$ versus the V-defect area ratio and for
different V-defect densities. Figure 11(a) shows that the turn-on voltage for
green LEDs decreases as the V-defect area ratio increases. In addition, green
LEDs have a higher indium composition and a higher polarization-related
electric field, so only holes flow into the $c$ plane from V-defects and
become crowded at the last QWs, as do electrons [see Fig. 5(b)]. Therefore,
the turn-on voltage decreases as the V-defect area ratio increases. This
effect influences $WPE$ of the green LED.

Figure 11(b) shows how the V-defect area ratio affects the $WPE$ at 20 A/cm2.
Again a 90% LEE is assumed in estimating the $WPE$. As mentioned,
$V_{\text{for}}$ decreases as the V-defect area ratio increases, which improve
the $WPE$. At the same V-defect area ratio, a greater V-defect density
corresponds to a greater $IQE$. Because carriers are injected into the QW from
V-defects and the diffusion length in the fluctuating QWs is short, the $IQE$
depends much more on the V-defect density than on the V-defect diameter.
Combining both effects, the $WPE$ reaches a maximum near 5% V-defect area
ratio. For the same V-defect area ratio, a high defect density is preferable
in green LEDs since they provide a higher $IQE$ for a similar
$V_{\text{for}}$.

Figure 11: (a) V-defect region ratio (V-defect area/chip area) versus turn-on
voltage for different V-defect densities at a current density 1 A/cm2 for
green LED. (b) V-defect region ratio versus $WPE$ for different V-defect
densities at a current density of 20 A/cm2 for green LED. The 90% LEE is
assumed.

##  IV Conclusion

In contrast with previous analyses of V-defects [15] in LEDs, we computed the
influence of both random alloy fluctuations and of V-defect density and size.
Random alloy fluctuations help to inject carriers vertically, whereas weak
polarization-related electric fields in the inclined sidewalls of V-defects
provide an alternative path for lateral carrier injection into the $c$-plane
QWs. In blue LEDs, V-defects have only a small impact, mainly reducing
$V_{\text{for}}$ by 0.14 V because carriers are rather efficiently injected
vertically. This feature cannot be observed without including both random
alloy fluctuaion and V-defect. With considering only the V-defect model[15],
carriers can only diffuse into QW through the V-defect where the influence of
V-defect density might be overestimated. In green LEDs, the large
polarization-related barriers hamper vertical transport, so carriers are
preferentially injected from V-defects. $V_{\text{for}}$ is strongly reduced
by about 1.0 V, even at small V-defect density, but the limited lateral
diffusion length induces current crowding near V-defects. Therefore,
increasing V-defect density increases the peak $IQE$ at large current
densities. With the use of an alternative 2D model to 3D models, we could
simulate a much larger, realistic area of LEDs and observe the limits of
lateral carrier diffusion from V-defects due to random alloy fluctuations,
while these features cannot be seen in 3D simulations as the modeled LED size
is well below the diffusion length, severely limited by computing resources.

When carriers are injected into the $c$-plane QWs at high bias and current
density, they can screen the polarization field and thereby allow more
carriers to be injected either through V-defects or vertically through the
$c$-plane stack of barriers and wells, thus providing a welcome synergy
between V-defects and vertical carrier injection. However, optimization will
require further work. Although the increase of V-defect density decreases
$V_{\text{for}}$, it also reduces the $IQE$ due to a higher possibility to be
trapped and recombine nonradiatively in TDs around the center of V-defects.
Detailed measurements of these nonradiative parameters are required for
precise selection of V-defect density and size.

## Acknowledgments

This work was supported by the Ministry of Science and Technology, Taiwan
(Grant Nos. MOST 108-2628-E-002-010-MY3, MOST 110-2923-E-002-002, and MOST
111-2923-E-002-009). The work at UCSB was supported by the National Science
Foundation (Grant No. DMS-1839077), and by grants from the Simons Foundation
(Grant Nos. 601952 to J.S. and 601954 to C.W.). This work was also supported
by the TECCLON project, grant ANR-20-CE05-0037-01 of the French Agence
Nationale de la Re-cherche (ANR). Additional support for JSS was provided by
the U.S. Department of Energy (Award No. DE-EE0009691). The data used in this
paper can be requested by contacting the corresponding author. The modeling
tool was developed by the NTU laboratory, and more detailed information can be
found on our website (http://yrwu-wk.ee.ntu.edu.tw/).

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