III-nitride light-emitting diodes (LEDs) exhibit an injection-dependent emission blueshift and linewidth broadening that is severely detrimental to their color purity. By using first-principles multi-scale modeling that accurately captures the competition between polarization-charge screening, phase-space filling, and many-body plasma renormalization, we explain the current-dependent spectral characteristics of polar III-nitride LEDs fabricated with state-of-the-art quantum wells. Our analysis uncovers a fundamental connection between carrier dynamics and the injection-dependent spectral characteristics of light-emitting materials. For example, polar III-nitride LEDs offer poor control over their injection-dependent color purity due to their poor hole transport and slow carrier-recombination dynamics, which forces them to operate at or near degenerate carrier densities. Designs that accelerate carrier recombination and transport and reduce the carrier density required to operate LEDs at a given current density lessen their injection-dependent wavelength shift and linewidth broadening.

Although III-nitride light-emitting diodes (LEDs) have been highly successful in producing blue light efficiently, they face several challenges for longer green and red wavelengths.1 Their wall-plug efficiency decreases as the emission wavelength increases and becomes worse for high-power operation, a phenomenon known as the green gap.2–6 Another challenge is the blueshift of the emission wavelength and the broadening of the spectral linewidth with increasing carrier injection. These effects change the perceived hue, which severely deteriorates the color purity of LEDs at high operating powers.7 In many cases, the perceived hue is blueshifted, and this worsens the efficiency gap by requiring even longer wavelength devices to compensate for the perceived blueshift. Despite the overwhelming technological importance of this problem, a quantitative understanding of the injection-dependent spectral blueshift and linewidth broadening has been missing.

The band-edge emission of polar InGaN quantum wells is determined by the interplay of competing mechanisms that contribute to the emission by shifting the bandgap or by filling the bands (Fig. 1). To date, the most widely accepted explanation of the injection-dependent blueshift is the screening of polarization fields by free carriers, with a smaller role attributed to phase-space filling.8,9 Meanwhile, there is no widely accepted explanation for the origin of the linewidth broadening. III-nitride quantum wells exhibit strong piezoelectric and spontaneous polarization fields, which contribute to a quantum-confined Stark shift of the bandgap.10–12 As free carriers are injected into the quantum well, they screen the polarization charges, which results in a blueshift of the bandgap as the bands flatten [Fig. 1(a)]. A competing, and often overlooked, effect that redshifts the energy is the renormalization of the bandgap by many-body effects in the free-carrier plasma,13–16 an effect that has been directly measured in bulk samples.17,18 At carrier densities exceeding 1018 cm−3 relevant for LED operation, excited carriers exist predominantly in the correlated plasma state rather than as bound excitons,19 due to Pauli blocking and screening of the Coulomb interaction.20 An electron (hole) in a plasma repels other electrons (holes), creating a surrounding region of positive (negative) charge, called the exchange-correlation hole.21 The net result is an effective attractive potential for the carriers, which lowers the conduction band and raises the valence band as the carrier density increases [Fig. 1(b)]. In contrast to band-gap shift effects, phase-space filling contributes to a blueshift of the peak-emission energy by changing the occupancies of the bands.10–12 As the carrier density increases and the quasi-Fermi levels penetrate deeper into the bands [Fig. 1(c)], the emission occurs from the states that are further away from the band edge. This effect becomes pronounced only if both carriers are degenerate. Therefore, the emission of InGaN quantum wells is influenced by the complex interplay of band-gap shift and band-filling effects in the free-carrier plasma.

FIG. 1.

Schematic illustrations of the three primary effects that contribute to the band-edge emission of polar III-nitride quantum wells at carrier densities relevant to LED operation. Band-gap shift effects such as polarization-charge screening [panel (a)] and plasma renormalization [panel (b)] contribute to the emission spectrum by shifting the bandgap EG. Band-filling effects such as phase-space filling [panel (c)] contribute to the emission spectrum by changing the finite occupation of carriers (indicated by the electron and hole quasi-Fermi levels Ef,n and Ef,p, and their difference ΔEf), which, in turn, determines the region of phase-space from which the carriers recombine to produce light.

FIG. 1.

Schematic illustrations of the three primary effects that contribute to the band-edge emission of polar III-nitride quantum wells at carrier densities relevant to LED operation. Band-gap shift effects such as polarization-charge screening [panel (a)] and plasma renormalization [panel (b)] contribute to the emission spectrum by shifting the bandgap EG. Band-filling effects such as phase-space filling [panel (c)] contribute to the emission spectrum by changing the finite occupation of carriers (indicated by the electron and hole quasi-Fermi levels Ef,n and Ef,p, and their difference ΔEf), which, in turn, determines the region of phase-space from which the carriers recombine to produce light.

Close modal

An experimental understanding of the band-edge emission of InGaN LEDs has been impeded by the difficulty in distinguishing the competing effects. For example, Kuokstis et al. compared the luminescence of bulk films against quantum wells to isolate the effects of phase-space filling from polarization-charge screening.22 However, this approach assumes that polarization fields do not affect phase-space filling, which is not true as we will show later. On the other hand, several experimental works have attempted to explain the injection-dependent broadening of the high-energy tail of the luminescence spectrum in terms of carrier delocalization.23–27 Although these works reveal interesting correlations, it is difficult to establish causation from their data. On the theoretical front, previous studies have not explained the experimentally observed injection dependence of the blueshift and linewidth broadening. Della Sala et al. used self-consistent tight-binding simulations in the virtual-crystal approximation to conclude that polarization-charge screening is responsible for the injection-dependent blueshift; however, they neglected phase-space filling, carrier localization, and many-body renormalization.28 On the other hand, Peng et al.29 neglected alloy disorder, and it is unclear what simulation parameters they used to match the experimental data since the work dates from a time when various fundamental parameters, e.g., the bandgap of InN30 and polarization constants,31 were not accurately known. Therefore, a theory of the injection dependence of the emission spectrum of III-nitride LEDs is entirely missing.

In this work, we use first-principles multi-scale modeling to explain the carrier-injection dependence of the emission blueshift and linewidth broadening of III-nitride quantum wells. We benchmark our calculations against electroluminescence (EL) measurements of a polar InGaN quantum-well device and show that our calculation explains the experimentally observed injection dependence of the EL spectrum. In the context of these results, we identify the design strategies that minimize the wavelength shift and linewidth broadening of III-nitride emitters.

We self-consistently solved the Schrödinger and Poisson equations by using nextnano++32 and an in-house code, with parameters determined from first-principles density-functional theory (DFT) calculations.31,33–36 We provide details of our calculations, which account for alloy disorder and carrier localization, in the supplementary material. We calculated the spontaneous-emission spectrum at first considering only band-filling effects in the disordered landscape of the quantum well, later shifting the spectrum energies to account for band-gap shift effects. We verified the validity of such a shift by checking that polarization-charge screening and plasma renormalization lead predominantly to a rigid shift of the bands (Fig. S2 in the supplementary material). To calculate the band-gap shift, we solved the one-dimensional Schrödinger and Poisson equations by using an in-house code. We treated many-body exchange-correlation effects of the free carriers in the local-density approximation by using the Perdew–Wang parameterization37 of the Monte Carlo calculation by Ceperley and Alder.38 This treatment of exchange and correlation accurately describes the experimentally measured17 band-gap renormalization of bulk GaN (Fig. S3 in the supplementary material). In the supplementary material, we provide the details of the spectrum calculation and a discussion on the impact of alloy disorder on the numerical modeling of free-carrier screening and many-body renormalization, which explains our choice of calculating the band-gap shift effects with a one-dimensional rather than a three-dimensional solver.

To validate the accuracy of our calculations, we performed experimental measurements of the current-dependent EL spectrum of an InGaN LED packaged at Lumileds. These LEDs were designed so that practically all of the recombination occurs over a single quantum well, thus allowing us to determine the carrier density, which is needed to compare experiment with simulation. We discuss details of this “quasi-single-quantum-well” LED in the supplementary material. We performed electroluminescence measurements of the quasi-single-quantum-well LED under pulsed operation to minimize Joule heating while ensuring that the time-averaged current density is only 1% of the peak current density. Our measurements exhibit both a current-dependent blueshift of the peak emission energy and broadening of the spectral linewidth [Fig. 2(a)]. The injection-dependent broadening is stronger on the high-energy side of the luminescence spectrum, which other groups have observed as well.23–26,39 To compare our measurements with theory, we measured the recombination lifetime and carrier density using a previously developed small-signal RF technique,40,41 in which we acquired and simultaneously fit the input impedance and modulation response to an equivalent circuit model of the LED to obtain the differential carrier lifetime. We then integrated the differential carrier lifetime to obtain the full carrier lifetime.42,43Figure 2(b) shows the recombination lifetime as a function of the current density; we also show the equivalent carrier density calculated from the relation J = en2D/τ, where J is the current density, n2D is the two-dimensional carrier density, and τ is the recombination lifetime. By measuring the recombination lifetime at various current densities, we converted the current dependence of the EL spectra to a carrier-density dependence, which is directly accessible in our calculations.

FIG. 2.

(a) Experimentally measured electroluminescence spectra of the InGaN quantum-well LED exhibiting a current-dependent blueshift and linewidth broadening. (b) Experimentally measured recombination lifetime (left axis) and the carrier density (right axis) calculated from the recombination lifetime, as a function of the injected current density.

FIG. 2.

(a) Experimentally measured electroluminescence spectra of the InGaN quantum-well LED exhibiting a current-dependent blueshift and linewidth broadening. (b) Experimentally measured recombination lifetime (left axis) and the carrier density (right axis) calculated from the recombination lifetime, as a function of the injected current density.

Close modal

Our modeling shows that we can accurately describe the carrier-density dependence of the peak-emission blueshift if we include the contributions of phase-space filling, polarization-charge screening, and many-body renormalization. In Fig. 3, we show that our calculated carrier-density dependence of the peak-emission energy is in excellent agreement with the experiment. We found that we needed to rigidly shift the band gap by −0.4 eV to quantitatively match the experimental gap, which suggests the presence of a systematic band-gap error in the modified k · p model.44 We refer the reader to the supplementary material for a discussion on how we solved for the relative contribution of each effect to the net peak shift. We also find that for quantum wells with thicknesses of ∼3 nm, the band-gap blueshift due to polarization-charge screening is compensated by a redshift due to plasma renormalization. We discuss the dependence of this cancellation on the quantum-well thickness in the supplementary material. Importantly, we show that polarization screening, phase-space filling, and plasma renormalization do not independently describe the shape of the carrier-density dependence curve. Therefore, our results show that it is crucial to accurately capture the contribution of all three effects to correctly model the emission spectra of InGaN emitters.

FIG. 3.

Theoretical carrier-density dependence of the peak emission energy of an InGaN quantum well (solid black curve) compared to the experiment (scatter points). We show the relative contributions from polarization-charge screening (blue curve), phase-space filling (green curve), and plasma renormalization (red curve). There is an excellent agreement between theory and experiment only if all three effects are included.

FIG. 3.

Theoretical carrier-density dependence of the peak emission energy of an InGaN quantum well (solid black curve) compared to the experiment (scatter points). We show the relative contributions from polarization-charge screening (blue curve), phase-space filling (green curve), and plasma renormalization (red curve). There is an excellent agreement between theory and experiment only if all three effects are included.

Close modal

Furthermore, we find that phase-space filling of carriers in the disordered potential landscape of the InGaN quantum well accurately describes the experimentally measured linewidth broadening. In Fig. 4(a), we show that our calculations of phase-space filling in the rigid-band approximation predict the relative increase in the full-width at half-maximum (FWHM) of the EL spectrum as a function of the carrier density. We report only the relative change to the FWHM rather than the exact value since only the former is physically meaningful due to the use of a constant energy-broadening parameter in calculating the joint density of states. As shown in Fig. 4(b), a signature of phase-space filling is the broadening of the high-energy luminescence tail, which is visible in the experimental EL spectrum of Fig. 2(a) as well. According to the van Roosbroeck–Shockley relation,45 the low-energy tail of the luminescence spectrum corresponds to the shoulder of the joint density of states, while the high-energy tail corresponds to the tail of the product of the electron and hole occupation functions. Since the electrons are lighter than holes in the III-nitrides, the onset of hole degeneracy determines the onset of the broadening of the high-energy tail since both carriers need to be degenerate for phase-space filling to contribute to the peak wavelength blueshift and linewidth broadening. Since strongly localized carriers have a smaller density of states than extended states, carrier localization exacerbates phase-space filling. However, localization is not a requirement for linewidth broadening, as previously conjectured,23,25,27 since broadening of the Fermi tail is a general feature of degenerate-carrier statistics. Our observation that polarization-charge screening and plasma renormalization lead predominantly to a rigid shift of the bands (see Fig. S2 of the supplementary material) further supports the argument that these two effects are less important than phase-space filling in explaining the linewidth broadening. We refer the reader to the supplementary material for further discussion on the impact of polarization fields and many-body effects on the linewidth broadening. Therefore, while the injection dependence of the peak-emission energy is due to the interplay of various physical effects, the injection-dependent linewidth broadening is predominantly due to phase-space filling.

FIG. 4.

(a) Carrier-density dependence of the luminescence full-width at half-maximum due to phase-space filling of carriers in the disordered potential landscape of the InGaN quantum well. (b) Theoretical luminescence curve of a representative InGaN quantum well, with the peak-emission energy centered at zero. The signature of phase-space filling is the broadening of the high-energy tail of the luminescence spectrum.

FIG. 4.

(a) Carrier-density dependence of the luminescence full-width at half-maximum due to phase-space filling of carriers in the disordered potential landscape of the InGaN quantum well. (b) Theoretical luminescence curve of a representative InGaN quantum well, with the peak-emission energy centered at zero. The signature of phase-space filling is the broadening of the high-energy tail of the luminescence spectrum.

Close modal

One important question that remains to be answered is why III-nitride LEDs grown on polar planes suffer from more severe injection-dependent linewidth broadening than III-phosphide and semipolar/non-polar III-nitride LEDs even though phase-space filling is a universal phenomenon that is present in all materials. The answer is simply that polar III-nitride LEDs operate at higher carrier densities due to their weaker oscillator strengths and correspondingly smaller radiative recombination (B) coefficients,46 and are, thus, more susceptible to phase-space filling. In Fig. 5, we show the carrier density required to operate 3 nm single-quantum-well LEDs at radiative current densities of 1, 50, and 1000 A/cm2 as a function of the B coefficient. We also show experimentally measured B coefficients for various (0001) polar47 and (202̄1̄) semipolar48 LEDs. Polar LEDs have low B coefficients due to their strong polarization field, which separates electrons and holes to opposite sides of the quantum well and lowers the probability of recombination. The B coefficient of polar LEDs decreases with increasing emission wavelength (or indium content); therefore, longer wavelength emitters undergo more severe injection-dependent spectral broadening. In contrast, semipolar LEDs have higher B coefficients due to their smaller polarization fields; consequently, they can operate at much lower carrier densities for a given current density. For this reason, semipolar LEDs exhibit less injection-dependent linewidth broadening than polar LEDs, a conclusion that is directly supported by optical measurements of semipolar LEDs in the literature.49–52 The B coefficient of III-phosphide LEDs tends to be even higher than semipolar III-nitride LEDs, with typical B coefficients of the order ∼10−10 cm3 s−1.53 In fact, such high radiative recombination coefficients mean that III-phosphide LEDs are more likely to experience stimulated emission before undergoing significant linewidth broadening, which may explain why luminescence broadening is typically not observed in the III-phosphide system. Our results also explain why some non-polar LEDs exhibit an (often small) injection-dependent blueshift and linewidth broadening despite the absence of a polarization field.26,54,55 Because there is no quantum-confined Stark effect in non-polar LEDs, higher indium compositions are required to obtain a given wavelength. Carrier localization due to stronger alloy disorder reduces the density of states and lowers the B coefficient (if electrons and holes are not co-localized),4,56 which makes phase-space filling important in non-polar LEDs as well. Although our analysis has been for InGaN LEDs, it applies equally well to AlGaN quantum-well LEDs, which also have strong polarization fields57 and carriers localized by alloy disorder.58 Hence, we have shown that recombination coefficients, and in particular the B coefficient, are important parameters that determine the likelihood of a device undergoing phase-space filling and injection-dependent linewidth broadening.

FIG. 5.

Effect of the B coefficient on the carrier density required to obtain a given radiative current density. The circles correspond to experimental B coefficients for polar (0001) LEDs measured by David et al. for blue (450 nm), green (535 nm), orange (600 nm), and red (645 nm) emitters.47 The star is the experimental B coefficient measured by Monavarian et al. for a semi-polar (202̄1̄) blue LED (430 nm).48 LEDs with lower B coefficients are more susceptible to phase-space filling, and consequently to stronger spectral broadening, because they operate at higher carrier densities for a given current density.

FIG. 5.

Effect of the B coefficient on the carrier density required to obtain a given radiative current density. The circles correspond to experimental B coefficients for polar (0001) LEDs measured by David et al. for blue (450 nm), green (535 nm), orange (600 nm), and red (645 nm) emitters.47 The star is the experimental B coefficient measured by Monavarian et al. for a semi-polar (202̄1̄) blue LED (430 nm).48 LEDs with lower B coefficients are more susceptible to phase-space filling, and consequently to stronger spectral broadening, because they operate at higher carrier densities for a given current density.

Close modal

Our results demonstrate that device designs that reduce the carrier density required to operate the device at a given current density reduce the injection-dependent blueshift and linewidth broadening. Improving the inter-well hole transport and spreading the number of carriers over more quantum wells enables the same light-power output for a lower carrier density. 3D engineering of the active region by using V-pits has recently been shown to be a practical way of improving hole transport,59 as evidenced by state-of-the-art multi-quantum-well LEDs fabricated with 3D V-pit engineering, which show improved efficiency droop as well as smaller wavelength shift and linewidth broadening compared to LEDs with poor inter-well hole transport.60 Designs that minimize the polarization field, e.g., semi-polar, non-polar, and thinner polar LEDs, minimize the injection-dependent wavelength blueshift because they allow the device to be operated at a lower carrier density for a given current density. Such designs simultaneously reduce the injection-dependent linewidth broadening and reduce efficiency droop, although at the expense of also requiring higher indium concentrations, which may inadvertently lead to a broader linewidth at low carrier density. In contrast, inefficient designs with more defects also operate at lower carrier densities for a given current density due to their higher non-radiative recombination rates and, thus, exhibit less linewidth broadening. In general, it is important to identify the origin of small injection-dependent linewidth broadening, particularly in devices that are more susceptible to defects, e.g., micro-LEDs, as it can be a reflection of their high non-radiative recombination rate, which is highly undesirable. We highlight that designs that minimize efficiency droop by reducing the operating carrier density of LEDs also lead to better color purity.

In summary, we have calculated the carrier-density dependence of the emission spectrum of InGaN LEDs. In contrast to the widely accepted hypothesis that the injection-dependent emission blueshift in III-nitride LEDs is primarily due to polarization-charge screening, we have shown that the emission shift depends on a complex interplay between polarization-charge screening, exchange-correlation effects, and phase-space filling of carriers in the disordered potential landscape of the quantum well. We have also shown that the injection-dependent linewidth broadening is caused primarily by phase-space filling, which is exceptionally prominent in polar III-nitride quantum wells due to their weaker oscillator strengths and lower radiative recombination coefficients. This emphasizes the innate connection between carrier dynamics and the current-dependent spectral characteristics of LEDs. In particular, emitters with poor transport and recombination dynamics offer poorer control over the injection-dependent color purity. Hence, designs that reduce the carrier density required to operate the LED at a given current density simultaneously reduce the efficiency droop and improve the high-power color purity of III-nitride LEDs.

See the supplementary material for (1) details of our modified k · p calculations, (2) details of how we calculated the spontaneous emission spectrum, (3) details of the quasi-single-quantum-well InGaN LED on which we performed EL measurements, (4) a discussion on the impact of localization and alloy disorder on the modeling of free-carrier screening, (5) details of how we obtained the relative contributions of phase-space filling, polarization-charge screening, and plasma renormalization to the peak emission shift in Fig. 3, (6) a discussion of the competition between polarization-charge screening and plasma renormalization in shifting the bandgap, (7) a discussion of the effects of polarization-charge screening and plasma renormalization on the linewidth broadening, (8) a discussion of the merits and drawbacks of various designs that reduce the carrier density of LEDs at a given current density, (9) evidence that polarization-charge screening and plasma renormalization lead predominantly to a rigid shift of the bands (Fig. S2), and (10) a comparison of the plasma renormalization of GaN calculated in the local-density approximation to experimental measurements of bulk GaN (Fig. S3).

We thank Siddharth Rajan for the useful discussions. This project was funded by the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy, under Award No. DE-EE0009163. Computational resources were provided by the National Energy Research Scientific Computing Center, a Department of Energy Office of Science User Facility, supported under Contract No. DEAC0205CH11231. N.P. acknowledges the support of the Natural Sciences and Engineering Research Council of Canada Postgraduate Doctoral Scholarship.

The authors have no conflicts to disclose.

Nick Pant: Conceptualization (equal); Investigation (lead); Methodology (lead); Writing – original draft (lead); Writing – review & editing (lead). Xuefeng Li: Data curation (supporting); Methodology (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Elizabeth DeJong: Data curation (supporting); Methodology (supporting); Writing – review & editing (supporting). Daniel Feezell: Conceptualization (supporting); Funding acquisition (supporting); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Rob Armitage: Conceptualization (equal); Funding acquisition (lead); Project administration (lead); Supervision (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). Emmanouil Kioupakis: Conceptualization (equal); Funding acquisition (lead); Project administration (equal); Supervision (lead); Writing – original draft (supporting); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

1.
S.
Pimputkar
,
J. S.
Speck
,
S. P.
DenBaars
, and
S.
Nakamura
,
Nat. Photonics
3
,
180
(
2009
).
2.
Y. C.
Shen
,
G. O.
Mueller
,
S.
Watanabe
,
N. F.
Gardner
,
A.
Munkholm
, and
M. R.
Krames
,
Appl. Phys. Lett.
91
,
141101
(
2007
).
3.
E.
Kioupakis
,
P.
Rinke
,
K. T.
Delaney
, and
C. G.
Van de Walle
,
Appl. Phys. Lett.
98
,
161107
(
2011
).
4.
S. Y.
Karpov
,
Appl. Sci.
8
,
818
(
2018
).
5.
A.
David
,
N. G.
Young
,
C.
Lund
, and
M. D.
Craven
,
ECS J. Solid State Sci. Technol.
9
,
016021
(
2019
).
6.
D. S. P.
Tanner
,
P.
Dawson
,
M. J.
Kappers
,
R. A.
Oliver
, and
S.
Schulz
,
Phys. Rev. Appl.
13
,
044068
(
2020
).
7.
Y.
Robin
,
M.
Pristovsek
,
H.
Amano
,
F.
Oehler
,
R. A.
Oliver
, and
C. J.
Humphreys
,
J. Appl. Phys.
124
,
183102
(
2018
).
8.
J.-H.
Ryou
,
P. D.
Yoder
,
J.
Liu
,
Z.
Lochner
,
H.
Kim
,
S.
Choi
,
H. J.
Kim
, and
R. D.
Dupuis
,
IEEE J. Sel. Top. Quantum Electron.
15
,
1080
(
2009
).
9.
C.
Weisbuch
,
S.
Nakamura
,
Y.-R.
Wu
, and
J. S.
Speck
,
Nanophotonics
10
,
3
(
2021
).
10.
A.
Hangleiter
,
J. s.
Im
,
J.
Off
, and
F.
Scholz
,
Phys. Status Solidi B
216
,
427
(
1999
).
11.
S. F.
Chichibu
,
A. C.
Abare
,
M. S.
Minsky
,
S.
Keller
,
S. B.
Fleischer
,
J. E.
Bowers
,
E.
Hu
,
U. K.
Mishra
,
L. A.
Coldren
,
S. P.
DenBaars
, and
T.
Sota
,
Appl. Phys. Lett.
73
,
2006
(
1998
).
12.
R. A.
Oliver
,
S. E.
Bennett
,
T.
Zhu
,
D. J.
Beesley
,
M. J.
Kappers
,
D. W.
Saxey
,
A.
Cerezo
, and
C. J.
Humphreys
,
J. Phys. D: Appl. Phys.
43
,
354003
(
2010
).
13.
P.
Vashishta
and
R. K.
Kalia
,
Phys. Rev. B
25
,
6492
(
1982
).
14.
G.
Tränkle
,
H.
Leier
,
A.
Forchel
,
H.
Haug
,
C.
Ell
, and
G.
Weimann
,
Phys. Rev. Lett.
58
,
419
(
1987
).
15.
S.
Das Sarma
,
R.
Jalabert
, and
S.-R. E.
Yang
,
Phys. Rev. B
41
,
8288
(
1990
).
16.
F.
Caruso
and
F.
Giustino
,
Phys. Rev. B
94
,
115208
(
2016
).
17.
T.
Nagai
,
T. J.
Inagaki
, and
Y.
Kanemitsu
,
Appl. Phys. Lett.
84
,
1284
(
2004
).
18.
D.
Hirano
,
T.
Tayagaki
, and
Y.
Kanemitsu
,
Phys. Rev. B
77
,
073201
(
2008
).
19.
A.
David
,
N. G.
Young
, and
M. D.
Craven
,
Phys. Rev. Appl.
12
,
044059
(
2019
).
20.
M.
Kira
and
S. W.
Koch
,
Semiconductor Quantum Optics
(
Cambridge University Press
,
2011
).
21.
R. M.
Martin
,
Electronic Structure: Basic Theory and Practical Methods
(
Cambridge University Press
,
Cambridge
,
2004
).
22.
E.
Kuokstis
,
J. W.
Yang
,
G.
Simin
,
M. A.
Khan
,
R.
Gaska
, and
M. S.
Shur
,
Appl. Phys. Lett.
80
,
977
(
2002
).
23.
N. I.
Bochkareva
,
V. V.
Voronenkov
,
R. I.
Gorbunov
,
A. S.
Zubrilov
,
P. E.
Latyshev
,
Y. S.
Lelikov
,
Y. T.
Rebane
,
A. I.
Tsyuk
, and
Y. G.
Shreter
,
Semiconductors
46
,
1032
(
2012
).
24.
M. J.
Davies
,
T. J.
Badcock
,
P.
Dawson
,
M. J.
Kappers
,
R. A.
Oliver
, and
C. J.
Humphreys
,
Appl. Phys. Lett.
102
,
022106
(
2013
).
25.
M. J.
Davies
,
T. J.
Badcock
,
P.
Dawson
,
R. A.
Oliver
,
M. J.
Kappers
, and
C. J.
Humphreys
,
Phys. Status Solidi C
11
,
694
(
2014
).
26.
M. J.
Davies
,
P.
Dawson
,
S.
Hammersley
,
T.
Zhu
,
M. J.
Kappers
,
C. J.
Humphreys
, and
R. A.
Oliver
,
Appl. Phys. Lett.
108
,
252101
(
2016
).
27.
G. M.
Christian
,
S.
Schulz
,
M. J.
Kappers
,
C. J.
Humphreys
,
R. A.
Oliver
, and
P.
Dawson
,
Phys. Rev. B
98
,
155301
(
2018
).
28.
F.
Della Sala
,
A.
Di Carlo
,
P.
Lugli
,
F.
Bernardini
,
V.
Fiorentini
,
R.
Scholz
, and
J.-M.
Jancu
,
Appl. Phys. Lett.
74
,
2002
(
1999
).
29.
L.-H.
Peng
,
C.-W.
Chuang
, and
L.-H.
Lou
,
Appl. Phys. Lett.
74
,
795
(
1999
).
30.
J.
Wu
,
W.
Walukiewicz
,
K. M.
Yu
,
J. W.
Ager
,
E. E.
Haller
,
H.
Lu
,
W. J.
Schaff
,
Y.
Saito
, and
Y.
Nanishi
,
Appl. Phys. Lett.
80
,
3967
(
2002
).
31.
C. E.
Dreyer
,
A.
Janotti
,
C. G.
Van de Walle
, and
D.
Vanderbilt
,
Phys. Rev. X
6
,
021038
(
2016
).
32.
S.
Birner
,
T.
Zibold
,
T.
Andlauer
,
T.
Kubis
,
M.
Sabathil
,
A.
Trellakis
, and
P.
Vogl
,
IEEE Trans. Electron Devices
54
,
2137
(
2007
).
33.
A. F.
Wright
,
J. Appl. Phys.
82
,
2833
(
1997
).
34.
Q.
Yan
,
P.
Rinke
,
A.
Janotti
,
M.
Scheffler
, and
C. G.
Van de Walle
,
Phys. Rev. B
90
,
125118
(
2014
).
35.
P. G.
Moses
,
M.
Miao
,
Q.
Yan
, and
C. G.
Van de Walle
,
J. Chem. Phys.
134
,
084703
(
2011
).
36.
P.
Rinke
,
M.
Winkelnkemper
,
A.
Qteish
,
D.
Bimberg
,
J.
Neugebauer
, and
M.
Scheffler
,
Phys. Rev. B
77
,
075202
(
2008
).
37.
J. P.
Perdew
and
Y.
Wang
,
Phys. Rev. B
45
,
13244
(
1992
).
38.
D. M.
Ceperley
and
B. J.
Alder
,
Phys. Rev. Lett.
45
,
566
(
1980
).
39.
C.
Frankerl
,
F.
Nippert
,
A.
Gomez-Iglesias
,
M. P.
Hoffmann
,
C.
Brandl
,
H.-J.
Lugauer
,
R.
Zeisel
,
A.
Hoffmann
, and
M. J.
Davies
,
Appl. Phys. Lett.
117
,
102107
(
2020
).
40.
A.
David
,
N. G.
Young
,
C. A.
Hurni
, and
M. D.
Craven
,
Appl. Phys. Lett.
110
,
253504
(
2017
).
41.
A.
Rashidi
,
M.
Monavarian
,
A.
Aragon
, and
D.
Feezell
,
Appl. Phys. Lett.
113
,
031101
(
2018
).
42.
L. A.
Coldren
,
S. W.
Corzine
, and
M. L.
Mashanovitch
,
Diode Lasers and Photonic Integrated Circuits
(
John Wiley & Sons
,
2012
).
43.
A.
Rashidi
,
M.
Monavarian
,
A.
Aragon
, and
D.
Feezell
,
Sci. Rep.
9
,
19921
(
2019
).
44.
D.
Chaudhuri
,
M.
O’Donovan
,
T.
Streckenbach
,
O.
Marquardt
,
P.
Farrell
,
S. K.
Patra
,
T.
Koprucki
, and
S.
Schulz
,
J. Appl. Phys.
129
,
073104
(
2021
).
45.
R.
Bhattacharya
,
B.
Pal
, and
B.
Bansal
,
Appl. Phys. Lett.
100
,
222103
(
2012
).
46.
E.
Kioupakis
,
Q.
Yan
, and
C. G.
Van de Walle
,
Appl. Phys. Lett.
101
,
231107
(
2012
).
47.
A.
David
,
N. G.
Young
,
C. A.
Hurni
, and
M. D.
Craven
,
Phys. Rev. Appl.
11
,
031001
(
2019
).
48.
M.
Monavarian
,
A.
Rashidi
,
A.
Aragon
,
S. H.
Oh
,
M.
Nami
,
S. P.
DenBaars
, and
D.
Feezell
,
Opt. Express
25
,
19343
(
2017
).
49.
Y.
Zhao
,
S.
Tanaka
,
C.-C.
Pan
,
K.
Fujito
,
D.
Feezell
,
J. S.
Speck
,
S. P.
DenBaars
, and
S.
Nakamura
,
Appl. Phys. Express
4
,
082104
(
2011
).
50.
C.-C.
Pan
,
S.
Tanaka
,
F.
Wu
,
Y.
Zhao
,
J. S.
Speck
,
S.
Nakamura
,
S. P.
DenBaars
, and
D.
Feezell
,
Appl. Phys. Express
5
,
062103
(
2012
).
51.
Y.
Zhao
,
S. H.
Oh
,
F.
Wu
,
Y.
Kawaguchi
,
S.
Tanaka
,
K.
Fujito
,
J. S.
Speck
,
S. P.
DenBaars
, and
S.
Nakamura
,
Appl. Phys. Express
6
,
062102
(
2013
).
52.
D. F.
Feezell
,
J. S.
Speck
,
S. P.
DenBaars
, and
S.
Nakamura
,
J. Disp. Technol.
9
,
190
(
2013
).
53.
O. A.
Fedorova
,
K. A.
Bulashevich
, and
S. Y.
Karpov
,
Opt. Express
29
,
35792
(
2021
).
54.
A.
Chakraborty
,
B. A.
Haskell
,
S.
Keller
,
J. S.
Speck
,
S. P.
DenBaars
,
S.
Nakamura
, and
U. K.
Mishra
,
Appl. Phys. Lett.
85
,
5143
(
2004
).
55.
A.
Chitnis
,
C.
Chen
,
V.
Adivarahan
,
M.
Shatalov
,
E.
Kuokstis
,
V.
Mandavilli
,
J.
Yang
, and
M. A.
Khan
,
Appl. Phys. Lett.
84
,
3663
(
2004
).
56.
S.
Schulz
,
M. A.
Caro
,
C.
Coughlan
, and
E. P.
O’Reilly
,
Phys. Rev. B
91
,
035439
(
2015
).
57.
Q.
Guo
,
R.
Kirste
,
S.
Mita
,
J.
Tweedie
,
P.
Reddy
,
S.
Washiyama
,
M. H.
Breckenridge
,
R.
Collazo
, and
Z.
Sitar
,
Jpn. J. Appl. Phys.
58
,
SCCC10
(
2019
).
58.
R.
Finn
and
S.
Schulz
, “
Impact of random alloy fluctuations on the electronic and optical properties of (Al,Ga)N quantum wells: Insights from tight-binding calculations
,” arXiv:2208.05337 (
2022
).
59.
C.-K.
Li
,
C.-K.
Wu
,
C.-C.
Hsu
,
L.-S.
Lu
,
H.
Li
,
T.-C.
Lu
, and
Y.-R.
Wu
,
AIP Adv.
6
,
055208
(
2016
).
60.
R.
Armitage
,
T.
Ishikawa
,
H. J.
Kim
, and
I.
Wildeson
,
Proc. SPIE
XXVI
,
PC120220M
(
2022
).

Supplementary Material