## Modeling excitation-dependent bandstructure effects on InGaN light-emitting diode efficiency |

Optics Express, Vol. 19, Issue 22, pp. 21818-21831 (2011)

http://dx.doi.org/10.1364/OE.19.021818

Acrobat PDF (1315 KB)

### Abstract

Bandstructure properties in wurtzite quantum wells can change appreciably with changing carrier density because of screening of quantum-confined Stark effect. An approach for incorporating these changes in an InGaN light-emitting-diode model is described. Bandstructure is computed for different carrier densities by solving Poisson and k·p equations in the envelop approximation. The information is used as input in a dynamical model for populations in momentum-resolved electron and hole states. Application of the approach is illustrated by modeling device internal quantum efficiency as a function of excitation.

© 2011 OSA

## 1. Introduction

1. M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Technol. **3**, 160–175 (2007). [CrossRef]

2. M. H. Kim, M. F. Schubert, Q. Dai, J. K. Kim, E. F. Schubert, J. Piprek, and Y. Park, “Origin of efficiency droop in GaN-based light-emitting diodes,” Appl. Phys. Lett. **91**, 183507–183510 (2007). [CrossRef]

3. Y. C. Shen, G. O. Müller, S. Watanabe, N. F. Gardner, A. Munkholm, and M. R. Krames, “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett. **91**, 141101–141101 (2007). [CrossRef]

4. A. A. Efremov, N. I. Bochkareva, R. I. Gorbunov, D. A. Larinvovich, Yu. T. Rebane, D. V. Tarkhin, and Yu. G. Shreter, “Effect of the joule heating on the quantum efficiency and choice of thermal conditions for high-power blue InGaN/GaN LEDs,” Semiconductors **40**, 605–610 (2006). [CrossRef]

5. S. F. Chichibu, T. Azuhata, M. Sugiyama, T. Kitamura, Y. Ishida, H. Okumurac, H. Nakanishi, T. Sota, and T. Mukai, “Optical and structural studies in InGaN quantum well structure laser diodes,” J. Vac. Sci. Technol. B **19**, 2177–2183 (2001). [CrossRef]

6. I. A. Pope, P. M. Smowton, P. Blood, and J. D. Thompson, “Carrier leakage in InGaN quantum well light-emitting diodes emitting at 480nm,” Appl. Phys. Lett. **82**, 2755–2757 (2003). [CrossRef]

3. Y. C. Shen, G. O. Müller, S. Watanabe, N. F. Gardner, A. Munkholm, and M. R. Krames, “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett. **91**, 141101–141101 (2007). [CrossRef]

7. H.-Y Ryu, H.-S. Kim, and J.-I. Shim, “Rate equation analysis of efficiency droop in InGaN light-emitting diodes,” Appl. Phys. Lett. **95**, 081114–081117 (2009). [CrossRef]

9. K. T. Dellaney, P. Rinke, and C. G. Van de Walle, “Auger recombination rates in nitrides from first principles,” Appl. Phys. Lett. **94**, 191109–191111 (2009). [CrossRef]

*ABC*model. [3

3. Y. C. Shen, G. O. Müller, S. Watanabe, N. F. Gardner, A. Munkholm, and M. R. Krames, “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett. **91**, 141101–141101 (2007). [CrossRef]

7. H.-Y Ryu, H.-S. Kim, and J.-I. Shim, “Rate equation analysis of efficiency droop in InGaN light-emitting diodes,” Appl. Phys. Lett. **95**, 081114–081117 (2009). [CrossRef]

*A*,

*B*and

*C*) introduced to account for Shockley-Read-Hall (SRH), radiative-recombination and Auger-scattering carrier losses, respectively. Bandstructure effects enter indirectly via these coefficients.

10. A. Bykhovshi, B. Gelmonst, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. **74**, 6734–6739 (1993). [CrossRef]

11. J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlGaN quantum wells,” Phys. Rev. B **57**, R9435–R9438 (1998). [CrossRef]

*ABC*model is challenging, without compromising the attractiveness of having only three fitting parameters, each with direct correspondence to a physical mechanism. This paper considers an alternative that allows direct input of bandstructure properties, in particular, the band energy dispersions, confinement energies and optical transition matrix elements, as well as their carrier-density dependences arising from screening of piezoelectric and spontaneous polarization fields. The model has the further advantage of providing a consistent treatment of spontaneous emission, carrier capture and leakage, and nonequilibrium effects. Thus, the fitting parameter,

*B*is eliminated and effects, such as plasma heating, are taken in account within an effective relaxation rate approximation for carrier-carrier and carrier-phonon scattering. All this is accomplished by extending a previously reported non-equilibrium LED model that is based on dynamical equations for electron and hole occupations in each momentum (

*k*) state [12

12. W. W. Chow, M. H. Crawford, J. Y. Tsao, and M. Kneissl, “Internal efficiency of InGaN light-emitting diodes: beyond a quasiequilibrium model,” Appl. Phys. Lett. **97**, 121105–121107 (2010). [CrossRef]

*k*·

*p*and Poisson equations [13

13. S. L. Chuang and C. S. Chang, “*k* · *p* method for strained wurtzite semiconductors,” Phys. Rev. B **54**, 2491–2504 (1996). [CrossRef]

*k*–resolved model by calculating internal quantum efficiency (IQE) as a function of injection current for a multi-QW InGaN LED. Results are presented to illustrate the role of excitation dependences of band-structure. An example involves a higher SRH coefficient in QWs than barriers combining with screening of QCSE to produce an efficiency droop in certain LED configurations. Section 4 explains the role of the bandstructure by discussing the changes in QW confinement energies and envelope function overlap with increasing excitation. The section also describes the incorporation of Auger carrier loss into the model. With the

*k*–resolved model, any increase in plasma temperature or carrier leakage resulting from the Auger scattering is taken into account. Section 5 summarizes the paper.

## 2. Theory

14. E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE **51**, 89–109 (1963). [CrossRef]

*i*(

*j*) representing

*e*,

*α*,

_{e}*k*

_{⊥}(

*h*,

*α*,

_{h}*k*

_{⊥}) for QW states and

*e*,

*k*(

*h*,

*k*) for barrier states. In this notation, each QW state is denoted by its charge

*σ*, subband

*α*and in-plane momentum

_{σ}*k*

_{⊥}. A bulk state is specified by its charge

*σ*and 3-dimensional carrier momentum

*k*. In Eq. (1),

*a*,

_{i}*c*,

_{q}*is the photon frequency,*

_{q}*℘*is the dipole matrix element,

_{ij}*V*is the active region volume and

*ɛ*is the host permittivity. Using the Hiesenberg operator equations of motion and the above Hamiltonian, the carrier populations and polarizations evolve according to where

_{b}*γ*) is assumed to be considerably faster than the population changes. This allows integration of Eq. (5). The result is used to eliminate the polarization in Eqs. (2) and (3), giving Converting the photon momentum summation into an integral, i.e. where Ω

*=*

_{q}*qc*and

*c*is the speed of light in the semiconductor, the right-hand sides of Eqs. (6) and (7) may be integrated to give

*σ*,

*σ*′ is

*e*,

*h*or

*h*,

*e*. In Eq. (11),

*A*is the SRH coefficient for QW states,

*γ*

_{c−c}and

*γ*

_{c−p}are the effective carrier-carrier and carrier-phonon collision rates, respectively, and where

*℘*

_{ασ,ασ′,k⊥}and Ω

_{ασ,ασ′,k⊥}are the QW dipole matrix element and transition energy. Similarly, for the barrier populations,

*℘*and Ω

_{k}*are the barrier dipole matrix element and transition energy. In Eq. (13),*

_{k}*A*is the SRH coefficient for barrier states and there is a pump contribution, where

_{b}*J*is the current density,

*e*is the electron charge,

*T*. For the asymptotic Fermi-Dirac distributions approached via carrier-carrier collisions, the chemical potential

_{p}*μ*and plasma temperature

_{σ}*T*are determined by conservation of carrier density and energy. In the case of carrier-phonon collisions, the chemical potential

*T*is an input quantity. Total carrier density and energy are computed by converting the sum over states to integrals, i.e., where

_{L}*S*and

*h*are the surface area and thickness of the active active region consisting of all QW and barrier layers. Further details involving implementation and comparison with results from quantum-kinetic calculations are reported elsewhere [15

15. W. W. Chow, H. C. Schneider, S. W. Koch, C. H. Chang, L. Chrostowski, and C. J. Chang-Hasnain, “Nonequilibrium model for semiconductor laser modulation response,” IEEE J. Quantum Electron. **38**, 402–409 (2002). [CrossRef]

16. I. Waldmueller, W. W. Chow, M. C. Wanke, and E. W. Young, “Non-equilibrium many-body theory of intersub-band lasers,” IEEE J. Quantum Electron. **42**, 292–301 (2006). [CrossRef]

17. W. W. Chow, A. F. Wright, A. Girndt, F. Jahnke, and S. W. Koch, “Microscopic theory of gain for an In-GaN/AlGaN quantum well laser,” Appl. Phys. Lett. **71**, 2608–2610 (1997). [CrossRef]

19. H. Zhao, G. Liu, J. Zhang, J. Poplawsky, V. Dierolf, and N. Tansu, “Approaches for high internal quantum efficiency green InGaN light-emitting diodes with large overlap quantum wells,” Opt. Express **19**, A991–A1007 (2011). [CrossRef] [PubMed]

20. W. W. Chow, A. Knorr, and S. W. Koch, “Theory of laser gain in group-III nitrides,” Appl. Phys. Lett. **67**, 754–756 (1995). [CrossRef]

21. W. W. Chow, A. F. Wright, and J. S. Nelson, “Theoretical study of room temperate optical gain in GaN strained quantum wells,” Appl. Phys. Lett. **68**, 296–298 (1996). [CrossRef]

8. J. Hader, J. V. Moloney, B. Pasenow, S. W. Koch, M. Sabathil, N. Linder, and S. Lutgen, “On the important of radiative and Auger losses in GaN-based quantum wells,” Appl. Phys. Lett. **92**, 261103–261105 (2008). [CrossRef]

15. W. W. Chow, H. C. Schneider, S. W. Koch, C. H. Chang, L. Chrostowski, and C. J. Chang-Hasnain, “Nonequilibrium model for semiconductor laser modulation response,” IEEE J. Quantum Electron. **38**, 402–409 (2002). [CrossRef]

22. S.-H. Park, D. Ahn, J. Park, and T -T. Lee, “Optical properties of staggered InGaN/InGaN/GaN quantum-well structures with Ga- and N-Faces,” Jpn. J. Appl. Phys. **50**, 072101–07214 (2011). [CrossRef]

*℘*

_{ασ,ασ′,k⊥},

*℘*and carrier energies

_{k}*ɛ*

_{σ,k⊥},

*k*·

*p*theory, the QW electron and hole eigenfunctions are [18] where |

*m*〉 is a bulk electron or hole state,

_{σ}*u*

_{mσ,βσ}(

*z*) is the

*β*-th envelope function associated with the

_{σ}*m*bulk state,

_{σ}*A*

_{βσ,ασ,k⊥}is the amplitude of the

*β*-th envelope function contributing to the

_{σ}*α*-th subband at momentum

_{σ}*k*

_{⊥},

*z*is position in the growth direction and

*r*

_{⊥}is position in the QW plane Using Eq. (16), the square of the dipole matrix element may then be written as where and the bulk dipole matrix element in the absence of an electric field is given by

*ɛ*is the bulk material bandgap energy,

_{g}*m*

_{0}and

*m*are the bare and effective electron masses, Δ

_{e}_{1}and Δ

_{2}are energy splittings associated with the bulk hole states. An iterative solution of the

*k*·

*p*and Poisson equations [13

13. S. L. Chuang and C. S. Chang, “*k* · *p* method for strained wurtzite semiconductors,” Phys. Rev. B **54**, 2491–2504 (1996). [CrossRef]

*ɛ*

_{σ,ασ,k⊥}and

*ξ*

_{αe,αh,k⊥}. For these calculations we use the the bulk wurtzite material parameters listed in Refs. [23

23. S. J. Jenkins, G. P. Srivastava, and J. C. Inkson, “Simple approach to self-energy corrections in semiconductors and insulators,” Phys. Rev. B **48**, 4388–4397 (1993). [CrossRef]

26. O. Ambacher, “Growth and applications of Group III-nitrides,” J. Phys. D: Appl. Phys. **31**, 2653–2710 (1998). [CrossRef]

## 3. Results

*k*·

*p*and Poisson equations for a range of carrier densities. Bandstructure information needed for the population part are

*ɛ*

_{σ,ασ,k⊥},

*ξ*

_{αe,αh,k⊥}versus total QW carrier density,

*dz |*

_{QW}*u*

_{mσ,βσ}(

*z*)|

^{2}, where integral is performed over the QWs. The states where the integral is greater than a half are grouped as QW states and the rest as barrier states. For the problem being addressed, which is the excitation dependence of IQE, the distinction is only important because only QW transitions are affected by QCSE. For the barrier transitions, the dipole matrix element in the presence of an internal electric field is approximated by an average, where each transition is weighted according to the occupations of the participating states. When solving the population equations, grouping the barrier states appreciably reduces numerical demand, which remains substantial because one is still keeping track of a large number of

*k*-states.

*A*/

*A*≳ 1. A larger SRH coefficient in QW than barrier is possible in present experimental devices, based on the roughly three times higher defect density in QWs than in barriers in LEDs measured at Sandia [27]. The calculations are performed assuming an active region consisting five 4nm In

_{b}_{0.2}Ga

_{0.8}N QWs separated by 6nm GaN barriers and bounded by 20nm GaN layers. Electric field in the QWs is determined from the sum of piezoelectric and spontaneous polarization fields. The electric fields in the barriers are from spontaneous polarization. Screening of these fields are determined semiclassically according to Poisson equation, and electron and hole envelope functions. Input parameters are

*A*= 10

_{b}^{7}

*s*

^{−1},

*T*= 300

_{L}*K*,

*γ*

_{c}_{−}

*= 5 × 10*

_{c}^{13}

*s*

^{−1}and

*γ*

_{c}_{−}

*= 10*

_{p}^{13}

*s*

^{−1}. Effects arising from doping profile, presence of carrier blocking layers and interface irregularities are ignored [2

2. M. H. Kim, M. F. Schubert, Q. Dai, J. K. Kim, E. F. Schubert, J. Piprek, and Y. Park, “Origin of efficiency droop in GaN-based light-emitting diodes,” Appl. Phys. Lett. **91**, 183507–183510 (2007). [CrossRef]

28. S. Choi, H. J. Kim, S.-S. Kim, J. Liu, J. Kim, J.-H. Ryou, R. D. Dupuis, A. M. Fishcer, and F. A. Ponce, “Improvement of peak quantum efficiency and efficiency droop in III-nitride visible light-emitting diodes with an InAlN electron-blocking layer,” Appl. Phys. Lett. **96**, 221105–221107 (2010). [CrossRef]

6. I. A. Pope, P. M. Smowton, P. Blood, and J. D. Thompson, “Carrier leakage in InGaN quantum well light-emitting diodes emitting at 480nm,” Appl. Phys. Lett. **82**, 2755–2757 (2003). [CrossRef]

29. J. Hader, J. V. Moloney, and S. W. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. **96**, 221106–221108 (2010). [CrossRef]

30. Y. Y. Kudryk and A. V. Zinovchuk, “Efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes with nonuniform current spreading,” Semicond. Sci. Technol. **26**, 095007–095011 (2011). [CrossRef]

*ABC*) treatment will lead to a substantially more complicated numerial model, similar to the ones used in modeling spatio-temporal behavior in semiconductor lasers [31

31. C. Z. Ning, J. V. Moloney, A. Egan, and R. A. Indik, “A first-principles fully space-time resolved model of a semiconductor laser,” Quantum Semiclassical Opt. **9**, 681–691 (1997). [CrossRef]

29. J. Hader, J. V. Moloney, and S. W. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. **96**, 221106–221108 (2010). [CrossRef]

*ABC*model to distinguish between QW and barrier carrier densities,

*N*and

_{σ}*σ*=

*e*or

*h*. 3-d (volume) densities are used to connect with the

*ABC*model, especially in terms of the SRH and spontaneous emission coefficients. Equations (21) and (22) are coupled by assuming that intraband collisions are sufficiently rapid so that QW and barrier populations are in equilibrium at temperature

*T*. Defining a total 2-d carrier density,

*N*is the number of QWs in the structure,

_{qw}*h*is the width of individual QWs,

*k*is Boltzmann constant and Δ

_{B}*is the averaged QW confinement energy. The steady state solution to Eq. (23) gives the internal quantum efficiency, where*

_{σ}*A*/

*A*=

_{b}*N*/

_{qw}h*h*is assumed to simplify the above expressions.

_{b}*, Δ*

_{e}*and the QW*

_{h}*B*coefficient as functions of total carrier density,

*N*

_{2d}. The information is extracted from the same bandstructure calculations performed for the more comprehensive

*k*–resolved model, with the exception that only the zone center (

*k*

_{⊥}=

*k*= 0) values are used. Confinement energies are approximated by

*indicates an average over QW states. Based on Eqs. (12), (14) and (17), the assumption*

_{QW}*B*= 〈

*ξ*

_{αe,αh,0}〉

_{QW}*ηB*is made, where 〈

_{b}*ξ*

_{αe,αh,0}〉

*is the average envelope function overlap of the allowed QW transitions and*

_{QW}*η*is introduced to account for the difference in QW and barrier densities of states. This difference is automatically taken care of in the

*k*–resolved model based on Eqs. (11) and (13). Δ

*, Δ*

_{e}*and 〈*

_{h}*ξ*

_{αe,αh,0}〉

*versus carrier density*

_{QW}*N*

_{2}

*are plotted in Fig. 2. The sheet (2-d) density*

_{d}*N*

_{2}

*is for a heterostructure consisting 5 QWs and 6 barrier layers that totals 84nm in width.*

_{d}*η*. Input parameters are

*T*= 300

*K*,

*N*/

_{qw}h*h*= 0.16 and

_{b}*ABC*model, where carrier dependences of confinement energies and QW bimolecular radiative coefficient are taken into account. They also indicate that the appearance of droop is insensitive to the fitting parameter

*η*, which affects only the IQE recovery arising from increase in QW emission.

*J*= 150

*A*/

*cm*

^{2}, the fits indicate elevated plasma temperatures of

*T*> 360

*K*for carrier-phonon scattering rate

*γ*

_{c}_{−}

*= 10*

_{p}^{13}

*s*

^{−1}and

*T*> 600

*K*for

*γ*

_{c}_{−}

*= 10*

_{p}^{12}

*s*

^{−1}.

*N*in the case of the QW and by the total barrier width

_{qw}h*h*in the case of the barrier. When performing the bandstructure calculation, quasiequilibrium condition is assumed to determine the QW and barrier bulk densities used in the solution of Poisson equation. This is an inconsistency that is acceptable provided the dynamical solution does not produce carrier distributions deviating too far from quasiequilibrium distributions. Even though the current density versus carrier density relationship depends on the input to the dynamical problem, and therefore, different for the different curves in Fig. 1, some insight into the connection between bandstructure and IQE excitation dependence may be obtained by examining Figs. 1 and 2 together. The onset of droop in the curves in Fig. 1 occurs around 45

_{b}*A*/

*cm*

^{2}, which corresponds to

*N*

_{2d}around 10

^{13}

*cm*

^{−2}or a 3-d QW carrier density of 1× to 1.2 × 10

^{18}

*cm*

^{−3}. At these densities, the QCSE is essentially unscreened. At the start of IQE recovery which occurs over the range of 60 to 120

*A*/

*cm*

^{2}, the corresponding carrier densities are 2.8 × 10

^{13}<

*N*

_{2d}< 3.0 × 10

^{13}

*cm*

^{−2}or 3-d QW carrier density of 8.5× to 9 × 10

^{18}

*cm*

^{−2}. According to Fig. 2, these are densities where wavefunction overlap is no longer negligible. Between the IQE peak and recovery,

*N*

_{2}

*changes from approximately 10*

_{d}^{13}to 3.0 × 10

^{13}

*cm*

^{−2}. Within that carrier density range, Fig. 2 shows significant increase in QW-barrier electron and hole energy separations.

## 4. Discussion of results

*k*

_{⊥}=

*k*= 0) for four different carrier densities. For clarity, the curves are separated vertically according to their associated energies. The black lines plot the electron and hole confinement potentials, while the red and blue curves indicate the QW and barrier states, respectively.

*N*

_{2d}= 2.3 × 10

^{13}

*cm*

^{−2}, Fig. 4(a) depicts confinement potentials differing appreciably from the flat-band situation [see Fig. 4(d)]. A result is small energy separation between QW and barrier states, leading to comparable QW and barrier populations, especially for the holes. Optical emission from barrier transitions occur via the contribution

_{αe,αh,k⊥}

*b*

_{αe,αh,k⊥}

*n*

_{e,αe,k⊥}

*n*

_{h,αh,k⊥}is negligible, even though the product

*n*

_{e,αe,k⊥}

*n*

_{h,αh,k⊥}may be appreciable. This is because QCSE spatially separates electrons and holes in the QWs, resulting in very small dipole matrix elements for QW transitions.

*N*

_{2d}= 3.4 × 10

^{13}

*cm*

^{−2}, increased screening of QCSE leads to higher energy separation between QW and barrier states as shown in Fig. 4(b). This causes the barrier populations to decrease relative to those of the QW. However, the QCSE is still sufficient to suppress the dipole matrix element. Moreover, for

*A*/

*A*> 1, a larger fraction of the injected carriers are populating the lower-lying and higher-loss QW states. The net result is reduced IQE because the smaller increase in

_{b}_{αe,αh,k⊥}

*b*

_{αe,αh,k⊥}

*n*

_{e,αe,k⊥}

*n*

_{h,αh,k⊥}. Important to the appearance of droop is a lag between the increase in confinement energies and the increase in QW dipole matrix element, as illustrated in Fig. 2 within the region 2.5 × 10

^{13}

*cm*

^{−2}<

*N*

_{2d}< 5 × 10

^{13}

*cm*

^{−2}.

*N*

_{2}

*= 6.8 × 10*

_{d}^{13}

*cm*

^{−2}. An appreciable QW emission leads to a reversal of the IQE droop as shown in Figs. 1 and 3. Lastly, Fig. 4(d) shows the asymptotic flat-band case, both for reference and as a guide for assigning QW and barrier states. Note that some ambiguity remains, especially with the

*n*= 2 subbands, which lie mostly in the triangular barrier regions of the confinement potentials at finite carrier densities.

*ɛ*

_{e,αe,0}〉

*+ 〈*

_{QW}*ɛ*

_{h,αh,0}〉

*and*

_{QW}12. W. W. Chow, M. H. Crawford, J. Y. Tsao, and M. Kneissl, “Internal efficiency of InGaN light-emitting diodes: beyond a quasiequilibrium model,” Appl. Phys. Lett. **97**, 121105–121107 (2010). [CrossRef]

*A*/

*A*= 0.5, 2 and 4, with Auger coefficient

_{b}*C*= 0, 10

^{−32}, 5 × 10

^{−32}and 10

^{−31}

*cm*

^{6}

*s*

^{−1}(dotted, dashed, dot-dashed and solid curves, respectively). For clarity, the

*A*/

*A*= 1 case in Fig. 1 is omitted. The curves show the prolonging of the efficiency droop by Auger carrier loss. More importantly, the necessary Auger coefficient is shown to be

_{b}*C*< 10

^{−31}

*cm*

^{6}

*s*

^{−1}which is smaller than that used in

*ABC*models and are within the range predicted by microscopic calculation [9

9. K. T. Dellaney, P. Rinke, and C. G. Van de Walle, “Auger recombination rates in nitrides from first principles,” Appl. Phys. Lett. **94**, 191109–191111 (2009). [CrossRef]

## 5. Summary

*ABC*model.

*k*-resolved LED model described in this paper can provide a more precise estimation of their relative strengths than the commonly used

*ABC*model [3

**91**, 141101–141101 (2007). [CrossRef]

8. J. Hader, J. V. Moloney, B. Pasenow, S. W. Koch, M. Sabathil, N. Linder, and S. Lutgen, “On the important of radiative and Auger losses in GaN-based quantum wells,” Appl. Phys. Lett. **92**, 261103–261105 (2008). [CrossRef]

## Acknowledgment

## References and links

1. | M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Technol. |

2. | M. H. Kim, M. F. Schubert, Q. Dai, J. K. Kim, E. F. Schubert, J. Piprek, and Y. Park, “Origin of efficiency droop in GaN-based light-emitting diodes,” Appl. Phys. Lett. |

3. | Y. C. Shen, G. O. Müller, S. Watanabe, N. F. Gardner, A. Munkholm, and M. R. Krames, “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett. |

4. | A. A. Efremov, N. I. Bochkareva, R. I. Gorbunov, D. A. Larinvovich, Yu. T. Rebane, D. V. Tarkhin, and Yu. G. Shreter, “Effect of the joule heating on the quantum efficiency and choice of thermal conditions for high-power blue InGaN/GaN LEDs,” Semiconductors |

5. | S. F. Chichibu, T. Azuhata, M. Sugiyama, T. Kitamura, Y. Ishida, H. Okumurac, H. Nakanishi, T. Sota, and T. Mukai, “Optical and structural studies in InGaN quantum well structure laser diodes,” J. Vac. Sci. Technol. B |

6. | I. A. Pope, P. M. Smowton, P. Blood, and J. D. Thompson, “Carrier leakage in InGaN quantum well light-emitting diodes emitting at 480nm,” Appl. Phys. Lett. |

7. | H.-Y Ryu, H.-S. Kim, and J.-I. Shim, “Rate equation analysis of efficiency droop in InGaN light-emitting diodes,” Appl. Phys. Lett. |

8. | J. Hader, J. V. Moloney, B. Pasenow, S. W. Koch, M. Sabathil, N. Linder, and S. Lutgen, “On the important of radiative and Auger losses in GaN-based quantum wells,” Appl. Phys. Lett. |

9. | K. T. Dellaney, P. Rinke, and C. G. Van de Walle, “Auger recombination rates in nitrides from first principles,” Appl. Phys. Lett. |

10. | A. Bykhovshi, B. Gelmonst, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys. |

11. | J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlGaN quantum wells,” Phys. Rev. B |

12. | W. W. Chow, M. H. Crawford, J. Y. Tsao, and M. Kneissl, “Internal efficiency of InGaN light-emitting diodes: beyond a quasiequilibrium model,” Appl. Phys. Lett. |

13. | S. L. Chuang and C. S. Chang, “ |

14. | E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE |

15. | W. W. Chow, H. C. Schneider, S. W. Koch, C. H. Chang, L. Chrostowski, and C. J. Chang-Hasnain, “Nonequilibrium model for semiconductor laser modulation response,” IEEE J. Quantum Electron. |

16. | I. Waldmueller, W. W. Chow, M. C. Wanke, and E. W. Young, “Non-equilibrium many-body theory of intersub-band lasers,” IEEE J. Quantum Electron. |

17. | W. W. Chow, A. F. Wright, A. Girndt, F. Jahnke, and S. W. Koch, “Microscopic theory of gain for an In-GaN/AlGaN quantum well laser,” Appl. Phys. Lett. |

18. | W. W. Chow and S. W. Koch, |

19. | H. Zhao, G. Liu, J. Zhang, J. Poplawsky, V. Dierolf, and N. Tansu, “Approaches for high internal quantum efficiency green InGaN light-emitting diodes with large overlap quantum wells,” Opt. Express |

20. | W. W. Chow, A. Knorr, and S. W. Koch, “Theory of laser gain in group-III nitrides,” Appl. Phys. Lett. |

21. | W. W. Chow, A. F. Wright, and J. S. Nelson, “Theoretical study of room temperate optical gain in GaN strained quantum wells,” Appl. Phys. Lett. |

22. | S.-H. Park, D. Ahn, J. Park, and T -T. Lee, “Optical properties of staggered InGaN/InGaN/GaN quantum-well structures with Ga- and N-Faces,” Jpn. J. Appl. Phys. |

23. | S. J. Jenkins, G. P. Srivastava, and J. C. Inkson, “Simple approach to self-energy corrections in semiconductors and insulators,” Phys. Rev. B |

24. | A. F. Wright and J. S. Nelson, “Consistent structural properties for AlN, GaN, and InN,” Phys. Rev. B |

25. | S. H. Wei and A. Zunger, “Valence band splittings and band offsets of AlN, GaN, and InN,” Appl. Phys. Lett. |

26. | O. Ambacher, “Growth and applications of Group III-nitrides,” J. Phys. D: Appl. Phys. |

27. | A. Armstrong, Sandia National Laboratories, Albuquerque, NM 87185 (personal communication, 2010). |

28. | S. Choi, H. J. Kim, S.-S. Kim, J. Liu, J. Kim, J.-H. Ryou, R. D. Dupuis, A. M. Fishcer, and F. A. Ponce, “Improvement of peak quantum efficiency and efficiency droop in III-nitride visible light-emitting diodes with an InAlN electron-blocking layer,” Appl. Phys. Lett. |

29. | J. Hader, J. V. Moloney, and S. W. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett. |

30. | Y. Y. Kudryk and A. V. Zinovchuk, “Efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes with nonuniform current spreading,” Semicond. Sci. Technol. |

31. | C. Z. Ning, J. V. Moloney, A. Egan, and R. A. Indik, “A first-principles fully space-time resolved model of a semiconductor laser,” Quantum Semiclassical Opt. |

32. | L. V. Keldysh, “Behaviour of non-metallic crystals in strong electric fields,” Sov. Phys. JETP |

**OCIS Codes**

(230.3670) Optical devices : Light-emitting diodes

(230.5590) Optical devices : Quantum-well, -wire and -dot devices

(250.5590) Optoelectronics : Quantum-well, -wire and -dot devices

**ToC Category:**

Optical Devices

**History**

Original Manuscript: August 16, 2011

Revised Manuscript: September 22, 2011

Manuscript Accepted: September 22, 2011

Published: October 20, 2011

**Citation**

Weng W. Chow, "Modeling excitation-dependent bandstructure effects on InGaN light-emitting diode efficiency," Opt. Express **19**, 21818-21831 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-22-21818

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### References

- M. R. Krames, O. B. Shchekin, R. Mueller-Mach, G. O. Mueller, L. Zhou, G. Harbers, and M. G. Craford, “Status and future of high-power light-emitting diodes for solid-state lighting,” J. Display Technol.3, 160–175 (2007). [CrossRef]
- M. H. Kim, M. F. Schubert, Q. Dai, J. K. Kim, E. F. Schubert, J. Piprek, and Y. Park, “Origin of efficiency droop in GaN-based light-emitting diodes,” Appl. Phys. Lett.91, 183507–183510 (2007). [CrossRef]
- Y. C. Shen, G. O. Müller, S. Watanabe, N. F. Gardner, A. Munkholm, and M. R. Krames, “Auger recombination in InGaN measured by photoluminescence,” Appl. Phys. Lett.91, 141101–141101 (2007). [CrossRef]
- A. A. Efremov, N. I. Bochkareva, R. I. Gorbunov, D. A. Larinvovich, Yu. T. Rebane, D. V. Tarkhin, and Yu. G. Shreter, “Effect of the joule heating on the quantum efficiency and choice of thermal conditions for high-power blue InGaN/GaN LEDs,” Semiconductors40, 605–610 (2006). [CrossRef]
- S. F. Chichibu, T. Azuhata, M. Sugiyama, T. Kitamura, Y. Ishida, H. Okumurac, H. Nakanishi, T. Sota, and T. Mukai, “Optical and structural studies in InGaN quantum well structure laser diodes,” J. Vac. Sci. Technol. B19, 2177–2183 (2001). [CrossRef]
- I. A. Pope, P. M. Smowton, P. Blood, and J. D. Thompson, “Carrier leakage in InGaN quantum well light-emitting diodes emitting at 480nm,” Appl. Phys. Lett.82, 2755–2757 (2003). [CrossRef]
- H.-Y Ryu, H.-S. Kim, and J.-I. Shim, “Rate equation analysis of efficiency droop in InGaN light-emitting diodes,” Appl. Phys. Lett.95, 081114–081117 (2009). [CrossRef]
- J. Hader, J. V. Moloney, B. Pasenow, S. W. Koch, M. Sabathil, N. Linder, and S. Lutgen, “On the important of radiative and Auger losses in GaN-based quantum wells,” Appl. Phys. Lett.92, 261103–261105 (2008). [CrossRef]
- K. T. Dellaney, P. Rinke, and C. G. Van de Walle, “Auger recombination rates in nitrides from first principles,” Appl. Phys. Lett.94, 191109–191111 (2009). [CrossRef]
- A. Bykhovshi, B. Gelmonst, and M. Shur, “The influence of the strain-induced electric field on the charge distribution in GaN-AlN-GaN structure,” J. Appl. Phys.74, 6734–6739 (1993). [CrossRef]
- J. S. Im, H. Kollmer, J. Off, A. Sohmer, F. Scholz, and A. Hangleiter, “Reduction of oscillator strength due to piezoelectric fields in GaN/AlGaN quantum wells,” Phys. Rev. B57, R9435–R9438 (1998). [CrossRef]
- W. W. Chow, M. H. Crawford, J. Y. Tsao, and M. Kneissl, “Internal efficiency of InGaN light-emitting diodes: beyond a quasiequilibrium model,” Appl. Phys. Lett.97, 121105–121107 (2010). [CrossRef]
- S. L. Chuang and C. S. Chang, “k · p method for strained wurtzite semiconductors,” Phys. Rev. B54, 2491–2504 (1996). [CrossRef]
- E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE51, 89–109 (1963). [CrossRef]
- W. W. Chow, H. C. Schneider, S. W. Koch, C. H. Chang, L. Chrostowski, and C. J. Chang-Hasnain, “Nonequilibrium model for semiconductor laser modulation response,” IEEE J. Quantum Electron.38, 402–409 (2002). [CrossRef]
- I. Waldmueller, W. W. Chow, M. C. Wanke, and E. W. Young, “Non-equilibrium many-body theory of intersub-band lasers,” IEEE J. Quantum Electron.42, 292–301 (2006). [CrossRef]
- W. W. Chow, A. F. Wright, A. Girndt, F. Jahnke, and S. W. Koch, “Microscopic theory of gain for an In-GaN/AlGaN quantum well laser,” Appl. Phys. Lett.71, 2608–2610 (1997). [CrossRef]
- W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).
- H. Zhao, G. Liu, J. Zhang, J. Poplawsky, V. Dierolf, and N. Tansu, “Approaches for high internal quantum efficiency green InGaN light-emitting diodes with large overlap quantum wells,” Opt. Express19, A991–A1007 (2011). [CrossRef] [PubMed]
- W. W. Chow, A. Knorr, and S. W. Koch, “Theory of laser gain in group-III nitrides,” Appl. Phys. Lett.67, 754–756 (1995). [CrossRef]
- W. W. Chow, A. F. Wright, and J. S. Nelson, “Theoretical study of room temperate optical gain in GaN strained quantum wells,” Appl. Phys. Lett.68, 296–298 (1996). [CrossRef]
- S.-H. Park, D. Ahn, J. Park, and T -T. Lee, “Optical properties of staggered InGaN/InGaN/GaN quantum-well structures with Ga- and N-Faces,” Jpn. J. Appl. Phys.50, 072101–07214 (2011). [CrossRef]
- S. J. Jenkins, G. P. Srivastava, and J. C. Inkson, “Simple approach to self-energy corrections in semiconductors and insulators,” Phys. Rev. B48, 4388–4397 (1993). [CrossRef]
- A. F. Wright and J. S. Nelson, “Consistent structural properties for AlN, GaN, and InN,” Phys. Rev. B51, 7866–7869 (1995). [CrossRef]
- S. H. Wei and A. Zunger, “Valence band splittings and band offsets of AlN, GaN, and InN,” Appl. Phys. Lett.69, 2719–2711 (1996). [CrossRef]
- O. Ambacher, “Growth and applications of Group III-nitrides,” J. Phys. D: Appl. Phys.31, 2653–2710 (1998). [CrossRef]
- A. Armstrong, Sandia National Laboratories, Albuquerque, NM 87185 (personal communication, 2010).
- S. Choi, H. J. Kim, S.-S. Kim, J. Liu, J. Kim, J.-H. Ryou, R. D. Dupuis, A. M. Fishcer, and F. A. Ponce, “Improvement of peak quantum efficiency and efficiency droop in III-nitride visible light-emitting diodes with an InAlN electron-blocking layer,” Appl. Phys. Lett.96, 221105–221107 (2010). [CrossRef]
- J. Hader, J. V. Moloney, and S. W. Koch, “Density-activated defect recombination as a possible explanation for the efficiency droop in GaN-based diodes,” Appl. Phys. Lett.96, 221106–221108 (2010). [CrossRef]
- Y. Y. Kudryk and A. V. Zinovchuk, “Efficiency droop in InGaN/GaN multiple quantum well light-emitting diodes with nonuniform current spreading,” Semicond. Sci. Technol.26, 095007–095011 (2011). [CrossRef]
- C. Z. Ning, J. V. Moloney, A. Egan, and R. A. Indik, “A first-principles fully space-time resolved model of a semiconductor laser,” Quantum Semiclassical Opt.9, 681–691 (1997). [CrossRef]
- L. V. Keldysh, “Behaviour of non-metallic crystals in strong electric fields,” Sov. Phys. JETP6, 763–770 (1958).

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