overshoot effects of electron on efficiency droop in ingan gan mqw light emitting diodes

China Received 30 December 2015; accepted 20 April 2016; published online 27 April 2016 To evaluate electron leakage in InGaN/GaN multiple quantum well MQW light emitting diodes LEDs, an

Overshoot effects of electron on efficiency droop in InGaN/GaN MQW light-emitting diodes

Yang Huang, Zhiqiang Liu, Xiaoyan Yi, Yao Guo, Shaoteng Wu, Guodong Yuan, JunXi Wang, Guohong Wang, and Jinmin Li

Citation: AIP Advances 6, 045219 (2016); doi: 10.1063/1.4948511

View online: http://dx.doi.org/10.1063/1.4948511

View Table of Contents: http://aip.scitation.org/toc/adv/6/4

Published by the American Institute of Physics

Overshoot effects of electron on efficiency droop

in InGaN/GaN MQW light-emitting diodes

Yang Huang,1,2,3Zhiqiang Liu,1,2,3,aXiaoyan Yi,1,2,3,aYao Guo,1,2,3

Shaoteng Wu,1,2,3Guodong Yuan,1,2,3JunXi Wang,1,2,3Guohong Wang,1,2,3

and Jinmin Li1,2,3

1R&D Center for Semiconductor Lighting, Chinese Academy of Sciences,

Beijing 100083, P R China

2State Key Laboratory of Solid State Lighting, Beijing 100083, P R China

3Beijing Engineering Research Center for the 3rd Generation Semiconductor Materials

and Application, Beijing 100083, P R China

(Received 30 December 2015; accepted 20 April 2016; published online 27 April 2016)

To evaluate electron leakage in InGaN/GaN multiple quantum well (MQW) light emitting diodes (LEDs), analytic models of ballistic and quasi-ballistic transport are developed With this model, the impact of critical variables effecting electron leakage, including the electron blocking layer (EBL), structure of multiple quantum wells (MQWs), polarization field, and temperature are explored The simulated results based on this model shed light on previously reported experimental observations and provide basic criteria for suppressing electron leakage, advancing the design

of InGaN/GaN LEDs.C 2016 Author(s) All article content, except where other-wise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).[http://dx.doi.org/10.1063/1.4948511]

INTRODUCTION

The origin of efficiency droop in InGaN/GaN light emitting diodes (LEDs) remains a topic of active debate.1,2Although many theories have attempted to explain these experimental observations, including device heating,3delocalization of carriers,4Auger recombination,5and electron leakage,6 there is still no consensus on their underlying physical mechanism Electron leakage is commonly regarded as having an important contribution to the remarkable efficiency loss, especially at high injected current Such leakage is caused by poor hole-injection efficiency,6an ineffective electron blocking layer (EBL),7,8non-capture of electrons by quantum wells (QWs),9and electron escape from the QWs.10Recently, ballistic and quasi-ballistic transport in electron leakage was investigated for InGaN/GaN LEDs with single double heterostructure (DH) active regions, producing a good agree-ment with experiagree-ments.11,12Ballistic transport in more complex and significant multiple quantum well (MQW) structures however, which are widely used in InGaN/GaN LEDs, has yet to be studied

In this study, we propose a ballistic and quasi-ballistic transport model for random period MQW structures of InGaN/GaN LEDs We build on the preliminary work of Ni et al.,11,12 addition-ally considering the potential drop in the active region, the In mole fraction of the quantum well, and the temperature The relationships between electron leakage and the period number of the quantum well, the polarization field, the aluminum (Al) mole fraction of EBL, and temperature are analyzed

in detail These dependencies can shed light on the previously reported experimental observations and yield the basic criteria for reducing electron leakage

THEORETICAL APPROACH FOR BALLISTIC TRANSPORT IN InGaN/GaN MQW LEDs

In InGaN/GaN LEDs, electrons across the active region may be captured by the QWs through phonon-mediated or impurity-mediated scattering events; only those electrons falling into QWs

a Authors to whom correspondence should be addressed Electronic mail: lzq@semi.ac.cn , spring@semi.ac.cn

045219-2 Huang et al. AIP Advances 6, 045219 (2016)

are able to participate in radiative recombination Unscattered electrons may escape the active re-gion, which is called ballistic or quasi-ballistic transport (Fig.1) Electron energies in the n-GaN are distributed in a Fermi-Dirac distribution with an additional kinetic energy (E-Ec) They cross the quantum wells (QWs) and quantum barriers (QBs) over the EBL to p-GaN, with or without energy exchange through interaction with phonons (in which case electron-LO-phonon scattering dominates).13,14Since the quasi-ballistic transport (scattering with more than one LO phonon in a single QW or QB) is negligible,12we just consider the electrons experiencing ballistic transport (no scattering in the QWs or QBs) and the quasi-ballistic transport (scattering with one LO phonon in a single QW or QB)

For ballistic or quasi-ballistic transport, the percentage of electrons leaking is given by12

P=

+∞

Elimitf(E) ∗ N (E) ∗ Q (E) dE

+∞

where f(E) and N(E) are the Fermi-Dirac distribution function and the conduction band density of states in the n-GaN, respectively We assume that the electrons traverse only in the direction normal

to the heterointerface, which would very slightly overestimate the escape probability, especially at high injected current At a given energy E, the probability of ballistic or quasi-ballistic transport Q(E) is proportional to exp(−t/τsc),11,12where t is the transit time and τscis the electron-LO-phonon scattering time We can consider Q(E) in the QW or QB under the two transport cases under a flatband approximation

i Ballistic case (no scattering)

Qiw

free(E)= exp( −Lw

ν E+ ∆Ec+ nwell

i ∗ }ωph
∗ τwell

sc )

Qib

free(E)= exp( −Lb

ν E+ ∆Ec+ nbarri

i ∗ }ωph
∗ τbarri

sc

where iwand ibdenote the ithQW and ithQB in MQWs starting from n-GaN, respectively; Qiw

free(E) and Qib

free(E) are the Q(E) without scattering; τwell

sc and τbarrisc are the electron-LO-phonon scattering time of the corresponding QW or QB; nwelli and nbarrii refer to the total phonons absorbed by the electron from n-GaN to the corresponding QW and QB (negative values represent the emission of phonons); Lwand Lbare the width of the QW and QB; ∆Ecis the conduction band offset between n-GaN and the QB or QW; and }ωph(≈ 88mev) is the LO phonon energy The velocity of electrons can be written as

where meis the effective electron mass

ii Quasi-ballistic transport with one phonon emission

FIG 1 Schematic illustration of ballistic or quasi-ballistic transport of electrons under the flatband approximation in InGaN /GaN LEDs.

em(E)=

 Lw

0

exp *

,

−x

ν(E+ ∆Ec+ (nwell

i + 1) ∗ }ωph

)

∗ τwellsc +

ν(E+ ∆Ec+ (nwell

i + 1) ∗ }ωph

)

∗ τwellem

∗

exp * ,

−(Lw− x)

ν E+ ∆Ec+ nwell

i ∗ }ωph
∗ τwell

sc + -dx

Qib

em(E)=

 Lb

0 exp * ,

−x

ν E+ (nbarri

i + 1) ∗ }ωph
∗ τbarri

sc +

ν E+ (nbarri

i + 1) ∗ }ωph
∗ τbarri

em

∗

exp * ,

−(Lb− x)

ν E+ nbarri

i ∗ }ωph
∗ τbarri

sc +

where τwell

em and τbarri

em are the phonon emission times in QWs and QBs, respectively

iii Quasi-ballistic transport with one phonon absorption

Qiw

abs(E)=

 Lw

0

exp * ,

−x

ν(E+ ∆Ec+ (nwell

i − 1) ∗ }ωph

)

∗ τwellsc +

ν(E+ ∆Ec+ (nwell

i − 1) ∗ }ωph

)

∗ τwell abs

∗

exp * ,

−(Lw− x)

ν E+ ∆Ec+ nwell

i ∗ }ωph
∗ τwell

sc + -dx

Qib

abs(E)=

 Lb

0 exp * ,

−x

ν E+ (nbarri

i − 1) ∗ }ωph
∗ τbarri

sc +

ν E+ (nbarri

i − 1) ∗ }ωph
∗ τbarri

abs

∗

exp * ,

−(Lb−x)

ν E+ nbarri

i ∗ }ωph
∗ τbarri

sc +

where τwell

abs and τbarri

abs are the phonon absorption times in QWs and QBs, respectively

In accordance with above analysis, the total percentage of leakage electrons is described by

P=

3(2∗num−1)

j =1

+∞

Ejlimitf(E) ∗ N (E) ∗ Qj(E) dE

+∞

0 f(E) ∗ N (E) dE

Qj(E)=(num−1i=1 QiwQib

)

∗ Qiw

Qib= Qib free, Qib absor Qib

emQiw= Qiw

free, Qiw absor Qiw

em

Ejlimit= max{−nbarri

i ∗ }ωph, −nwell

num∗ }ωph+ φEBL− q ∗ V , 0}, (6) where ‘3(2∗num−1)’ is the number of possible transport processes and ‘num’ is the period number of MQWs in the active region Qj(E) gives the probability of the jthtransport process, which implies that electrons pass the last quantum barrier and EBL via thermionic emission due to the barrier layer being thinner than the electron mean free path Ejlimitis the lower energy limit of the integral for the

jthtransport process Only when the kinetic energy is above Ejlimitcan electrons flow over the EBL in the jthtransport process φEBLis the EBL barrier height and V is the potential drop across the active region

Although informative, the flatband approximation deviates significantly from the real band structure of InGaN/GaN LEDs In particular, the potential drop of the built-in field of the PN junction, the polarization field, and the applied voltage lead to band bending in QWs and QBs As

a result, electrons will be accelerated or decelerated depending on the direction of the net electric field, further affecting the probability of ballistic or quasi-ballistic transport

To verify the potential distribution in QWs and QBs, energy band simulation was performed via APSYS of crosslight In Fig 2, the conduction band at different drive voltages is illustrated Based on the simulated result, the turn-on voltage of the LED is around 2.8V As the applied voltage approaches 3V, the quasi-flatband condition is approached Further increasing injection makes the

045219-4 Huang et al. AIP Advances 6, 045219 (2016)

FIG 2 Energy band of the simulated InGaN /GaN LED at different applied voltages.

band gradually shift upward due to voltage saturation The net potential drop at different applied voltages within QWs and QBs, expressed as ∆Vwell and ∆Vbarrier, are exhibited in Fig.3 At the working voltage, the build-in field of the PN junction is compensated by the applied voltage Thus, the net potential drop in QWs and QBs, which is primarily driven by the polarization charge, have almost the same absolute values with opposite directions It is therefore reasonable to approximate that the ∆Vwelland ∆Vbarrierare constant within the entire active region and that ∆Vwellequals to

∆Vbarrier Due to the opposite directions of the net potential drop in QWs and QBs, electrons are accelerated within QWs and decelerated within QBs Because

∆Vwell= ∆Vbarrier= ∆V, (7) the average increment of velocity in QWs and QBs, in contrast to the flatband approximation, is

¯

∆ν= ¯∆νwell= ¯∆νbarrier=1

2 ∗

 2 ∗ q ∗ ∆V

me

Hence, the average increment of velocity ¯∆νmust be added to ν(E) in equations (2)-(5) to account for the potential drop in the active region

Except for the last barrier, the QB and QW are generally deposited in pairs The direction

of potential in the last barrier however is opposite of the other barriers, as exhibited in Fig 1 Thus we classify the net voltage drop in MQWs into two parts: the last barrier and the remainder For calculation convenience, we take the net voltage drop in the MQWs except the last barrier as determining the velocity variation of electrons, while the net potential drop in the last barrier, named

FIG 3 Potential drop in QWs and QBs at di fferent applied voltages based on the band simulations presented in Fig 2

Vb, is taken as equivalent to a reduction of the effective height of the EBL It is therefore necessary

to correct equation (6) as

Ejlimit= max{−nbarri

i ∗ }ωph, −nwell

num∗ }ωph+ φEBL−q ∗ Vb,0} (9)

DISCUSSION

To date, research on carrier leakage of InGaN/GaN LEDs are scattered in experiment and theory,6,8,11,15 limiting their application Here, on our ballistic transport model, we systematically analyze the properties of InGaN/GaN LEDs electron leakage

To eliminate carrier leakage, an AlGaN layer is typically inserted between MQWs and p-GaN

as an EBL The Al fraction in the AlGaN layer is a crucial parameter in EBL design The percent electron leakage (electron leakage current divided by the total current) as a function of φEBLunder the flatband approximation is calculated and shown in Fig 4 In the quantum wells, we used a phonon scattering time (τwell

sc ), phonon emission time (τwell

em), and phonon absorption time (τwell

abs) of 0.009 ps, 0.01 ps, and 0.1 ps, respectively, for room temperature calculations In the quantum bar-riers, we used a phonon scattering time (τbarrisc ), phonon emission time (τbarriem ), and phonon absorption time (τbarriabs ) of 0.05 ps, 0.055 ps, and 0.55 ps, respectively.16As shown in Fig 4, our calculated results for one QW (i.e., a single DH structure) are identical to that of Ni et al.12,17under the same conditions the simulated result of the MQW structure with three periods of 3-nm-thick QWs and 7.5-nm-thick QBs also has been presented in Fig.4 In these two structures, the percent leakage current decreases monotonically with increasing Al content in the EBL, confirming that EBL design

is an effective means of suppressing carrier leakage

Compared to the other QBs, the last barrier has an opposite potential drop As we know, the direction of piezoelectric polarization in the EBL (AlGaN) is opposite to that of a QW (InGaN); unlike other barriers, the last barrier has its deposited upper layer of EBL Therefore, when the applied voltage is above a certain value under working conditions, the net potential drop in the last barrier will have the opposite sign Though the compensation of Vbfor the electron blocking barrier can lead to high electron leakage, increasing the Al content in the EBL may effectively inhibit this

effect, as seen in Fig.5 However, the potential drop Vbis not only relevant to the width of the last barrier, but also to the aluminum composition in EBL As a result, incrementally increasing the Al fraction in the EBL is not going to reduce overflow current unless further details are investigated The acceleration or deceleration of electrons in the electric field of the active region has been analyzed above We note that the built-in potential of the PN junction is compensated by the applied voltage, and that the applied voltage saturates with increasing current at high injection via the Shockley equation (SE) Hence, the polarization field dominates the band bending in QWs and QBs under operating conditions From Fig 2, we obtain the potential drop ∆V ≈ 0.14V, of the

FIG 4 Calculated ratio of electron leakage current to total current as a function of EBL barrier height (φ EBL ) under the flatband approximation for two di fferent designed active region structures.

045219-6 Huang et al. AIP Advances 6, 045219 (2016)

FIG 5 Calculated ratio of overflow electron current to total current vs the net voltage drop in the last barrier V b with varying aluminum content in the EBL.

same order of magnitude as the reported polarization potential18,19the average increment of velocity calculated in QWs and QBs is ¯∆ν= 2.965 × 105m, near the electron saturation velocity of GaN (∼3 × 105m/s).20Ignoring the velocity saturation effect, we display the dependence of the percent electron leakage current on the potential drop ∆V in Fig 6(a) Similar to previous reports, we find that the polarization field accounts for the promotion of ballistic or quasi-ballistic transport, or carrier leakage In some publications,21,22the dependence of luminescence efficiency on In compo-sition of the quantum well was observed, and the difference in polarization field was hypothesized

to be the main cause Nevertheless, as showed in Fig.6(a), with the same potential drop, the calcu-lated electron leakage current increases monotonically with increasing In mole fraction in QWs, resulting in more luminescence efficiency loss That indicates that the In composition dependence of electron velocity in QWs with different conduction band offsets also contributes to the luminescence difference That suggests that the electron leakage resulting from ballistic transport could explain the efficiency difference between LEDs of different wavelengths To better evaluate the effect of the velocity increment due to the polarization field on electron leakage current, we introduce a concept

of gain ratio, a relative scale of electron leakage current, defined as

Gain Ratio= (electron leakage current / total current)∆V

(electron leakage current /total current)∆V =0 (10) The gain ratio as a function of ∆V is presented in Fig.6(b) It demonstrates that the polarization field plays a more important role in electron leakage for LEDs with lower indium compositions For this reason, enough attention should be paid to the polarization effect on short wavelength LEDs

FIG 6 (a) Calculated ratio of electron leakage current to total current and (b) calculated gain ratio vs potential drop ∆V in QBs and QWs with varying In mole fraction.

despite their low field intensity, especially where carrier leakage is concerned However, neglecting the velocity saturation effect we may overestimate the electron overflow in InGaN/GaN LEDs, and for MQWs with high In compositions, even the possibility of negative differential mobility should

be considered

The luminescence of LEDs is strongly dependent on the number of quantum wells in MQW structures because they affect the injection efficiency and distribution of carriers Optimization of the period number of MQW is therefore necessary for improving InGaN/GaN LEDs’ performance

In Fig 7, we compare the electron leakage in structures with different periods of MQWs and Al concentrations in the EBL Without an EBL (no Al in the EBL), the ratio of leakage current drops dramatically as the period number of MQWs increases in our calculation Furthermore, the lower the concentration of aluminum in the EBL, the higher the leakage current As the aluminum concen-tration approaches 8%, the ratio of leakage current is nearly invariant with increasing MQW period number Thus, with low aluminum content in the EBL, the electron leakage is controlled by both the EBL barrier height φEBLand the MQW period number For structures with a high-Al-content EBL, however, the influence of the period number on leakage is not as significant Hence, when EBLs with low aluminum content are required for enhancement of hole injection, it is optimal to moderately increase the quantity of QWs; when QWs are sparse and uncontrolled, increasing the Al content of the EBL is ideal

While superficially it appears that the electron leakage current in high-period-number MQWs

is unremarkable (Fig.7), we must stress that the phonon scattering time (τbarrisc = 0.05ps) used in our simulation is relatively short compared to experimentally measured values (approximately 5ps),23 which would significantly improve the electron leakage current However, for comparison purposes, the phonon scattering time illustrated in Fig.4is used in this paper unless otherwise specified The temperature in the active region of InGaN/GaN LEDs varies with working current and circumstance The performance of LEDs at various temperatures varies considerably Both the temperature dependence of the electron-LO-phonon scattering time and the band offset (φEBLand

∆Ec) make the ballistic or quasi-ballistic transport sensitive to temperature The correlation between electron-LO-phonon scattering time and temperature can be simply expressed as.24

where T is the average temperature of MQWs,τ∞is the scattering time at the temperature of infinite, and the characteristic temperature is defined as

Θ=}wph

kB

FIG 7 Calculated ratio of electron leakage current to total current vs the period number of MQWs of InGaN /GaN LEDs with different EBLs.

045219-8 Huang et al. AIP Advances 6, 045219 (2016)

FIG 8 Calculated ratio of electron leakage current to total current vs the temperature of MQWs.

where kBis the Boltzmann constant For LO-phonon scattering in GaN or InGaN, Θ ≈ 1020K The relationship between temperature and band gap (or the band offset) is not discussed here, as they are addressed elsewhere.25

The percentage of electron leakage current at temperatures ranging from 1K to 500K is pre-sented in Fig.8 There is a maximum overflow current at approximately 200K, below which the ratio of leakage current is almost constant From that point to higher temperatures, the ratio signifi-cantly decreases Our model explains this behavior At low temperatures, the impact of temperature

on the band offset is dominant,26 and increasing temperature leads to low φEBL; at high temper-atures, the scattering time becomes shorter and the strong scattering hinders ballistic transport.24 Recently, an anomalous temperature-dependence of electroluminescence intensity in GaN-based LEDs was reported.27–30It was found that when temperature is decreased to around 200 K, the EL intensity rises, as expected from the improved quantum efficiency However with a further decrease

in temperature, the EL intensity falls dramatically That phenomenon was explained by reduced carrier capture by QWs at low temperature,27–29 or by the low ionization rate of magnesium (Mg)

in p-GaN at low temperature.30 In our simulation, Fig 8 reveals that the electron non-capture is much stronger at low temperature due to the weaker electron-LO-phonon scattering, which leads to the weak EL intensity below 200k, but it does not grow worse with a further fall in temperature, indicating that the weak EL intensity is nearly temperature-independent over the low temperature range (below 200 K) The measured EL intensity however further decreases at lower temperatures Given these considerations, the Mg freeze-out in p-GaN at low temperature must also be considered for this intensity decrease Therefore, both proposed mechanisms are responsible for the anomalous temperature-dependence of electroluminescence intensity at low temperature

CONCLUSIONS

In this work, an analytic model of ballistic and quasi-ballistic transport for InGaN/GaN MQW LEDs is proposed The dependence of electron overflow on the polarization field, Al mole fraction

in the EBL, and temperature were investigated, providing new design rules for InGaN/GaN LEDs Several lingering issues however, including the relationship between the potential drop in the last barrier and the Al mole fraction in the EBL, and he influence of electron velocity saturation in MQWs on ballistic transport, should be further investigated

ACKNOWLEDGEMENTS

This work was supported by the National High Technology Program of China (Grant No.2013A A03A101) and the National Natural Science Foundation of China (Grant No.61306051 and No.61306050)

1 J Cho, E F Schubert, and J K Kim, Laser & Photonics Reviews 7(3), 408–421 (2013).

2 J Piprek, physica status solidi (a) 207(10), 2217–2225 (2010).

3 X A Cao and S D Arthur, Applied Physics Letters 85(18), 3971 (2004).

4 S F Chichibu, A Uedono, T Onuma, B A Haskell, A Chakraborty, T Koyama, P T Fini, S Keller, S P Denbaars, J.

S Speck, U K Mishra, S Nakamura, S Yamaguchi, S Kamiyama, H Amano, I Akasaki, J Han, and T Sota, Nat Mater 5(10), 810–816 (2006).

5 J Iveland, L Martinelli, J Peretti, J S Speck, and C Weisbuch, Physical Review Letters 110(17), (2013).

6 G B Lin, D Meyaard, J Cho, E Fred Schubert, H Shim, and C Sone, Applied Physics Letters 100(16), 161106 (2012).

7 S H Han, D Y Lee, S J Lee, C Y Cho, M K Kwon, S P Lee, D Y Noh, D J Kim, Y C Kim, and S J Park, Applied Physics Letters 94(23), 231123 (2009).

8 K J Vampola, M Iza, S Keller, S P DenBaars, and S Nakamura, Applied Physics Letters 94(6), 061116 (2009).

9 M F Schubert and E F Schubert, Applied Physics Letters 96(13), 131102 (2010).

10 M F Schubert, J Xu, Q Dai, F W Mont, J K Kim, and E F Schubert, Applied Physics Letters 94(8), 081114 (2009).

11 X Ni, X Li, J Lee, S Liu, V Avrutin, U Özgür, H Morkoç, A Matulionis, T Paskova, G Mulholland, and K R Evans, Applied Physics Letters 97(3), 031110 (2010).

12 X Ni, X Li, J Lee, S Liu, V Avrutin, U Özgür, H Morkoç, and A Matulionis, Journal of Applied Physics 108(3), 033112 (2010).

13 J L A Matulionis, I Matulioniene, and M Ramonas, PHYSICAL REVIEW B 68, 035338 (2003).

14 H Ye, G W Wicks, and P M Fauchet, Applied Physics Letters 74(5), 711 (1999).

15 B J Ahn, T S Kim, Y Dong, M T Hong, J H Song, J H Song, H K Yuh, S C Choi, D K Bae, and Y Moon, Applied Physics Letters 100(3), 031905 (2012).

16 I M J Liberis, A Matulionis, M Ramonas, and L F Eastman, Advanced Semiconductor Materials and Devices Research: III-Nitrides and Sic (2009).

17 X Ni, X Li, J Lee, S Liu, V Avrutin, Ü Özgür, H Morkoç, A Matulionis, T Paskova, G Mulholland, and K R Evans, physica status solidi (RRL) - Rapid Research Letters 4(8-9), 194–196 (2010).

18 V Fiorentini, F Bernardini, and O Ambacher, Applied Physics Letters 80(7), 1204 (2002).

19 S S Tetsuya Takeuchi, Maki Katsuragawa, Miho Komori, Hideo Takeuchi, Hiroshi Amano, and Isamu Akasaki, Jpn J Appl Phys 36, L382–L385 (1997).

20 J Khurgin, Y J Ding, and D Jena, Applied Physics Letters 91(25), 252104 (2007).

21 G Chen, M Craven, A Kim, A Munkholm, S Watanabe, M Camras, W Götz, and F Steranka, physica status solidi (a) 205(5), 1086–1092 (2008).

22 M S Shuji Nakamura, Naruhito Iwasa, and Shinichi Nacahama, Jpn J Appl Phys 34, L797–L799 (1995).

23 K T Tsen, D K Ferry, A Botchkarev, B Sverdlov, A Salvador, and H Morkoc, Applied Physics Letters 72(17), 2132 (1998).

24 H Morkoc, Handbook of Nitride Semiconductors and Devices, volume 2: Electronic and Optical Processes in Nitrides (2008), p 227.

25 J M W K Karch and F Bechstedt, PHYSICAL REVIEW B 57(12), 7043–7049 (1998).

26 J N Kolniík, I S H Oˇguzman, K F Brennan, R Wang, P P Ruden, and Y Wang, Journal of Applied Physics 78(2), 1033 (1995).

27 X A Cao, S F LeBoeuf, L B Rowland, C H Yan, and H Liu, Applied Physics Letters 82(21), 3614 (2003).

28 A Hori, D Yasunaga, A Satake, and K Fujiwara, Applied Physics Letters 79(22), 3723 (2001).

29 A Hori, D Yasunaga, A Satake, and K Fujiwara, Journal of Applied Physics 93(6), 3152 (2003).

30 C M Lee, C C Chuo, J F Dai, X F Zheng, and J I Chyi, Journal of Applied Physics 89(11), 6554 (2001).