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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 22 — Oct. 24, 2011
  • pp: 21818–21831

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

Weng W. Chow  »View Author Affiliations


Optics Express, Vol. 19, Issue 22, pp. 21818-21831 (2011)
http://dx.doi.org/10.1364/OE.19.021818


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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

Considerable progress is being made in advancing InGaN light-emitting diodes (LEDs). However, there are still concerns involving performance limitations. An example is efficiency loss at high current density (efficiency droop) [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]

], which can limit use of LEDs in applications requiring intense illumination. Understanding and mitigating the efficiency droop mechanism is important. Several explanations have been proposed, including carrier leakage [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]

], Auger recombination [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]

], junction heating [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]

], carrier and defect delocalizations [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]

]. The assertions are much debated. For example, in the case of Auger scattering, discrepancy exists in the Auger coefficient estimation between experimental-curve fitting and microscopic calculations [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]

].

Discussions involving InGaN LED efficiency are commonly based on a rate equation for the total carrier density. The approach allows one to describe radiative and nonradiative carrier loss rates, where the latter typically includes ad-hoc terms for producing an efficiency droop. A particularly successful model, in terms of reproducing experimental efficiency versus injection current data, is the ABC model. [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]

] The model’s name derives from the three phenomenological constants (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.

It is known that the bandstructure in wurtzite quantum-well (QW) structures can change noticeably with carrier density because of screening of the quantum-confined Stark effect (QCSE) [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]

]. Incorporating these changes into the 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

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]

]. The additions include an algorithm for simplifying and extracting bandstructure information relevant to the dynamical equations. Detailed bandstructure properties are obtained from solving k · p and Poisson equations [13

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

]. Furthermore, since distinction between QW and barrier states is sometimes difficult in the presence of strong internal electric fields, extension is made to treat optical emission from these states on equal footing.

Section 2 describes the model, derivation of the working equations and calculation of input bandstructure properties. Section 3 demonstrates the application of the 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

The following Hamiltonian, adapted from quantum optics [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]

], is used in the derivation of spontaneous emission from QW and barrier transitions:
H= ∑i ɛie ai† ai+ ∑j ɛjh bj† bj+ ∑q h¯ Ωq cq† cq− ∑ i,j,q ℘ ij h¯ Ωq V ɛb ( ai bj cq†+ cq bj† ai†).
(1)
The summations are over QW and barrier states with subscript i(j) representing e,αe, k⊥ (h,αh, 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 ασ and in-plane momentum k⊥. A bulk state is specified by its charge σ and 3-dimensional carrier momentum k. In Eq. (1), ai, ai† ( bj, bj†) are electron (hole) annihilation and creation operators, cq, cq† are corresponding operators for the photons, ɛiσ is the carrier energy, Ωq is the photon frequency, ℘ij is the dipole matrix element, V is the active region volume and ɛb is the host permittivity. Using the Hiesenberg operator equations of motion and the above Hamiltonian, the carrier populations and polarizations evolve according to
d 〈 ai† ai〉 dt=cosi ∑ j,q ℘ ij Ωq h¯V ɛb [ 〈 cq bj† ai†〉− 〈 ai bj cq†〉],
(2)
d 〈 bj† bj〉 dt=i ∑ i,q ℘ ij Ωq h¯V ɛb [ 〈 cq bj† ai†〉− 〈 ai bj cq†〉],
(3)
d 〈 ai bj cq†〉 dt = −i ( Ωq− Ω ij) 〈 ai bj cq†〉 +i ℘ ij Ωq h¯V ɛb 〈 ( ai† ai+ bj† bj−1) cq† cq+ ai† bj† bj〉+…,
(4)
where Ω ij= ( ɛie+ ɛjh)/ h¯ is the transition frequency. Factorizing the operator products and truncating at the first level (Hartree-Fock approximation) give for Eq. (4)
d 〈 ai bj cq†〉 dt =i ( Ωq− Ω ij) 〈 ai bj cq†〉 −i ℘ ij Ωq h¯V ɛb [ ( 〈 ai† ai〉+ 〈 bj† bj〉−1) 〈 cq† cq〉+ 〈 ai† ai〉 〈 bj† bj〉].
(5)
For an LED, it is customary to assumed that cavity influence is sufficiently weak so that 〈 cq† cq〉≪1 and only the spontaneous emission contribution is kept. Additionally, polarization dephasing is introduced, where the dephasing (with coefficient 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
d 〈 ai† ai〉 dt=− 〈 ai† ai〉 ∑ j,q 2 Ωq h¯ ɛbVγ | ℘ ij|2 〈 bj† bj〉 [ 1+ ( Ω ij− Ωqγ)2] −1,
(6)
d 〈 bj† bj〉 dt=− 〈 bj† bj〉 ∑ i,q 2 Ωq h¯ ɛbVγ | ℘ ij|2 〈 ai† ai〉 [ 1+ ( Ω ij− Ωqγ)2] −1.
(7)
Converting the photon momentum summation into an integral, i.e.
∑q →2 V ( 2π)3 ∫0∞ dq4π q2,
(8)
where Ωq = 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
d 〈 ai† ai〉 dt=− 〈 ai† ai〉 ∑j nb h¯ ɛ0π c3 | ℘ ij|2 Ω ij3 〈 bj† bj〉,
(9)
d 〈 bj† bj〉 dt=− 〈 bj† bj〉 ∑i nb h¯ ɛ0π c3 | ℘ ij|2 Ω ij3 〈 ai† ai〉.
(10)

Writing explicitly for QW populations and adding phenomenologically SRH carrier loss and relaxation contributions from carrier-carrier and carrier-phonon scattering, gives
d n σ, ασ, k⊥ dt = − n σ, nσ, k⊥ ∑ α σ′ b ασ, α σ′, k⊥ n σ′, α σ′, k⊥−A n σ, nσ, k⊥ − γ c−c [ n σ, nσ, k⊥−f ( ɛ σ, k⊥, μσ,T)] − γ c−p [ n σ, nσ, k⊥−f ( ɛ σ, k⊥, μσL, TL)],
(11)
where σ,σ′ is e,h or h,e. In Eq. (11), A is the SRH coefficient for QW states, γc−c and γc−p are the effective carrier-carrier and carrier-phonon collision rates, respectively, and
b ασ, α σ′, k⊥= 1 h¯ ɛbπ c3 | ℘ αα, α σ′, k⊥|2 Ω ασ, α σ′, k⊥3,
(12)
where ℘ασ,ασ′,k⊥ and Ωασ,ασ′,k⊥ are the QW dipole matrix element and transition energy. Similarly, for the barrier populations,
d n σ,kb dt = − bk n e,kb n h,kb+ J e Nσpf ( ɛ σ,kb, μσp, Tp) ( 1− n σ,kb)− Ab n σ,k − γ c−c [ n σ,kb−f ( ɛ σ,kb, μσ,T)] − γ c−p [ n σ,kb−f ( ɛ σ,kb, μσL, TL)],
(13)
where
bk= 1 h¯ ɛbπ c3 | ℘k|2 Ωk3,
(14)
℘k and Ωk are the barrier dipole matrix element and transition energy. In Eq. (13), Ab is the SRH coefficient for barrier states and there is a pump contribution, where J is the current density, e is the electron charge, Nσp= Σkf ( ɛ σ,kb, μσp, Tp) and f ( ɛ σ,kb, μσp, Tp), the injected carrier distribution, is a Fermi-Dirac function with chemical potential μσp and temperature Tp. For the asymptotic Fermi-Dirac distributions approached via carrier-carrier collisions, the chemical potential μσ and plasma temperature T are determined by conservation of carrier density and energy. In the case of carrier-phonon collisions, the chemical potential μσL is determined by conservation of carrier density and the lattice temperature TL is an input quantity. Total carrier density and energy are computed by converting the sum over states to integrals, i.e.,
∑ k⊥ → S ( 2π)22 ∫0∞ d k⊥ 2π k⊥ and  ∑k → hS ( 2π)32 ∫0∞ dk4π k2,
(15)
where 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

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]

]. Many-body effects [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]

, 18

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).

] are neglected in Eqs (11) and (13), as in recent studies involving bandstructure effects, such as from increasing QW envelope function overlap [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]

], on InGaN LED performance. While behaviorial trends from bandstructure engineering many be described without many-body effects, some caution is advisable when using the quantitative results. One reason is that Coulomb enhancement has been shown to change optical gain and absorption in QWs and bulk structures by as much as a factor of two [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]

]. Equally important, corresponding changes in SRH and Auger contributions will have to be taken into account. These concerns were addressed in a theory with first-principles treatment of Auger scattering and optical emission [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]

]. However, the simulations are very time demanding, and carrier transport, current leakage, as well as nonequilibrium effects are not modeled. Perhaps, a good compromise is the extension of the model described in this paper to include many-body effects at the level of the screened Hartree-Fock approximation [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]

].

Bandstructure information enters directly into Eqs (11) and (13) via the dipole matrix elements ℘ασ,ασ′,k⊥, ℘k and carrier energies ɛσ,k⊥, ɛ σ,kb. From k · p theory, the QW electron and hole eigenfunctions are [18

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).

]
〈 r | ϕ σ, ασ, k⊥〉= e i k⊥⋅ r⊥ ∑ mσ ∑ βσ A βσ, ασ, k⊥ u mσ, βσ (z) 〈 r | mσ〉,
(16)
where |mσ〉 is a bulk electron or hole state, umσ,βσ (z) is the βσ-th envelope function associated with the mσ bulk state, Aβσ,ασ,k⊥ is the amplitude of the βσ-th envelope function contributing to the ασ-th subband at momentum 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
| ℘ αe, αh, k⊥|2 ≡ | 〈 ϕ e, αe, k⊥|ex | ϕ h, αh, k⊥〉|2 = | ℘ bulk|2 ξ αe, αh, k⊥,
(17)
where
ξ αe, αh, k⊥ = 14 | ∑ βe ∑ βh ∑ me ∑ mh A βe, αe, k⊥ A βh, αh, k⊥ × ∫ −∞∞ dz u me, βe (z) u mh, βe (z)|2,
(18)
and the bulk dipole matrix element in the absence of an electric field is given by
| ℘ bulk|2= h¯2 2 m0 ɛg ( m0 me−1) ( 1+ Δ1+ Δ2 ɛg),
(19)
ɛg is the bulk material bandgap energy, m0 and me are the bare and effective electron masses, Δ1 and Δ2 are energy splittings associated with the bulk hole states. An iterative solution of the k · p and Poisson equations [13

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

] is used to obtain the energies ɛσ,ασ,k⊥ and ɛ σ,kb and the overlap integral ξαe,αh,k⊥. For these calculations we use the the bulk wurtzite material parameters listed in Refs. [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

With the present model, it is necessary to solve the bandstructure and population problems self consistently. Simultaneous solution of both problems is very challenging and perhaps unnecessary. The approach used in this paper is to first take care of the bandstructure part by iteratively solving the k · p and Poisson equations for a range of carrier densities. Bandstructure information needed for the population part are ɛσ,ασ,k⊥, ɛ σ,kb and ξαe,αh,k⊥ versus total QW carrier density, nσ qw= S −1 ∑ ασ,, k⊥ n σ, ασ, k⊥, where the nσ qw dependences are from screening of the QW electric field.

To facilitate the solution of the dynamical population equations, the carrier states are grouped into two categories: those belonging to the QWs and those belonging to the barriers. The QW states are treated using Eq. (11) and the barrier states are treated collectively with Eq. (13). With a high internal electric field, the distinction between QW and barrier states may be ambiguous. In this paper, the choice is made by calculating ∫QWdz |umσ,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.

The second step involves numerically solving Eqs. (11) and (13) with the bandstructure quantities updated at each time step according to the instantaneous value of nσ qw. When steady state is reached, IQE is obtained from dividing the rate of carrier (electron or hole) loss via spontaneous emission by the rate of carrier injection:
IQE= e JS ( ∑ αe, αh, k⊥ b αe, αh, k⊥ n e, αe, k⊥ n h, αh, k⊥+ ∑k bk n e,kb n h,kb).
(20)

Computed IQE versus current density curves for different values of SRH coefficients in the QWs are plotted in Fig. 1. Each curve shows an initial sharp increase in IQE with injection current, with emission occurring the instant there is an injected current. Quite interesting, especially because Auger carrier loss is not included in the model, is the appearance of efficiency droop for A/Ab ≳ 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

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

]. The calculations are performed assuming an active region consisting five 4nm In0.2Ga0.8N 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 Ab = 107s−1, TL = 300K, γc−c = 5 × 1013s−1 and γc−p = 1013s−1. Effects arising from doping profile, presence of carrier blocking layers and interface irregularities are ignored [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]

]. These may be treated with the present model as details involving the growth-direction active-medium configuration. Not so straightforward is the description of effects arising from inhomogeneities within the QW plane, such as current crowding and carrier localization [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]

]. To incorporate such in-plane spatial inhomogeneities, one could divide the active region into different domains. However, nonuniform current spreading will likely require the full (3-d) solution of Poisson equation [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]

]. To use the results in a microscopic (rather than 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

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]

].

Fig. 1 Internal quantum efficiency versus current density for different A/Ab. The curves are computed using the k–resolved model described by Eqs. (11) and (13) for a LED with a In0.2Ga0.8N/GaN multi-QW active region (see Fig. 4).

Defect recombination has been proposed as a mechanism for efficiency droop [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]

]. The modeling was based on a rigorous microscopic treatment of the radiative process and a postulated nonlinear density-activated defect recombination current density. The IQE droop shown in Fig. 1 is also from a nonlinear total carrier density dependence of defect recombination, where the nonlinearity arises from bandstructure changes caused by screening of QCSE. These band-structure changes increase the fraction of injected carriers populating the higher SRH-loss, QW states relative to the lower SRH-loss, barrier states. Since the mechanism relates directly to defect densities in the QW and barrier layers, a more direct connection to measurable parameters is made.

For further insight, it is more effective to use a less comprehensive model that ignores carrier leakage and nonequilibrium effects, so as to isolate the bandstructure effects. Such a model is possible by extending the ABC model to distinguish between QW and barrier carrier densities, Nσ and Nσb, respectively. The following phenomenological (and less rigorous than Eqs. (11) and (13)] rate equations may be written:
d Nσ dt=−B Ne Nh−A Nσ,
(21)
d Nσb dt=− Bb Neb Nhb− Ab Nσb+ J e hb,
(22)
where 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 2d= N qwh Nσ+ hb Nσb allows combining these equations to give
d N 2d dt=−β N 2d2− Ab N 2d+ Je,
(23)
where Nqw is the number of QWs in the structure, h is the width of individual QWs,
β= h hb N qwB+ Bbexp ( Δe+ Δh kBT) [ 1+exp ( Δe kBT)] [ 1+exp ( Δh kBT)],
(24)
kB is Boltzmann constant and Δσ is the averaged QW confinement energy. The steady state solution to Eq. (23) gives the internal quantum efficiency,
IQE= β N 2d2 J/e=1−2 J0J [ J J0+1−1],
(25)
where J0=e γb2 ( 4β) −1 and A/Ab = Nqwh/hb is assumed to simplify the above expressions.

Bandstructure input to Eq. (25) are the confinement energies Δe, Δh and the QW B coefficient as functions of total carrier density, N2d. 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 Δσ= 〈 ɛ σ, ασ,0〉 QW− ɛ σ,0b, where 〈〉QW indicates an average over QW states. Based on Eqs. (12), (14) and (17), the assumption B = 〈ξαe,αh,0〉QW ηBb is made, where 〈ξαe,αh,0〉QW is the average envelope function overlap of the allowed QW transitions and 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). Δe, Δh and 〈ξαe,αh,0〉QW versus carrier density N2d are plotted in Fig. 2. The sheet (2-d) density N2d is for a heterostructure consisting 5 QWs and 6 barrier layers that totals 84nm in width.

Fig. 2 Average QW confinement energies (left axis) and electron-hole wavefunction overlap (right axis) versus carrier density. The curves are extracted from solving k · p and Poisson equations. A negative average hole confinement energy is possible because of the tilt in QW confinement potential and the presence of states in the outer barrier regions cladding the QWs, as shown in Fig. 4(a).

Figure 3 shows IQE versus current density computed with Eq. (25) and for different η. Input parameters are T = 300K, Nqwh/hb = 0.16 and Ab2/ Bb=1.1× 10 24c m −3 s −1. All the curves depict efficiency droop from the extended 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.

Fig. 3 Internal quantum efficiency versus current density computed using Eq. (25) from an extended ABC model, which isolates bandstructure effects. The curves are for different η, a free parameter accounting for difference between QW and barrier bimolecular radiative recombination coefficients (B and Bb, respectively) because of differences in densities of state.

While the above exercise reveals that bandstructure changes play a role in IQE droop, differences between Figs. 1 and 2 suggest that there is also influence from other contributions in present experiments. That experimental results are in closer agreement with Fig. 1 indicates the importance of these contributions in present LEDs. They include energy dispersions, carrier leakage and nonequilibrium carrier effects, such as an incomplete transfer of the carrier population from barrier to QW states because of finite intraband collision rates. The presence of nonequilibrium effects is verified from least-squares fits of computed carrier populations to Fermi-Dirac distributions. For J = 150A/cm2, the fits indicate elevated plasma temperatures of T > 360K for carrier-phonon scattering rate γc−p = 1013s−1 and T > 600K for γc−p = 1012s−1.

With the present approach, the dynamical solution gives the carrier densities in QW and barrier states. The conversion to bulk (3-d) density is via division by the total QW layer width Nqwh in the case of the QW and by the total barrier width hb 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 45A/cm2, which corresponds to N2d around 1013 cm−2 or a 3-d QW carrier density of 1× to 1.2 × 1018 cm−3. At these densities, the QCSE is essentially unscreened. At the start of IQE recovery which occurs over the range of 60 to 120A/cm2, the corresponding carrier densities are 2.8 × 1013 < N2d < 3.0 × 1013 cm−2 or 3-d QW carrier density of 8.5× to 9 × 1018 cm−2. According to Fig. 2, these are densities where wavefunction overlap is no longer negligible. Between the IQE peak and recovery, N2d changes from approximately 1013 to 3.0 × 1013 cm−2. Within that carrier density range, Fig. 2 shows significant increase in QW-barrier electron and hole energy separations.

4. Discussion of results

Further insight into the droop behavior described in the previous section is possible from closer examination of the bandstructure changes with excitation. Figure 4 shows the absolute square of electron and hole envelope functions at zone center (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.

Fig. 4 Absolute square of envelope functions for electrons and holes for carrier densities, N2d = (a) 2.25×, (b) 3.47× and (c) 6.89 × 1013 cm−2. Figure 4(d) is the flat-band limit. Each curve is displaced according to its bandedge energy for clarity. Envelope functions belonging to QW and barrier states are indicated by red and blue curves, respectively. The black lines plot the confinement potentials. The x-axis is along the growth direction.

Starting at a carrier density of N2d = 2.3 × 1013 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 ∑k bk n e,kb n h,kb, as soon as the product of electron and hole populations, n e,kb n h,kb becomes nonzero. In contrast, the QW contribution Σαe,αh,k⊥ bαe,αh,k⊥ ne,αe,k⊥ nh,αh,k⊥ is negligible, even though the product ne,αe,k⊥ nh,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.

At a higher carrier density of N2d = 3.4 × 1013 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/Ab > 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 ∑k bk n e,kb n h,kb with increasing excitation that is not compensated by a corresponding increase in Σαe,αh,k⊥ bαe,αh,k⊥ ne,αe,k⊥ nh,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 × 1013 cm−2 < N2d < 5 × 1013 cm−2.

For QW emission to increase, a high carrier density is necessary to sufficiently screen the QW electric field. That is the case for Fig. 4(c), where N2d = 6.8 × 1013 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.

Some questions remain. For example, one might expect a significant red shift of emission energy when optical transitions changes from barrier dominated to QW dominated. This need not be the case because of the energy level shifts associated with the QCSE and Franz-Keldysh effects [32

L. V. Keldysh, “Behaviour of non-metallic crystals in strong electric fields,” Sov. Phys. JETP 6, 763–770 (1958).

]. The curves in Fig. 5 show the carrier density dependences of the average QW and barrier bandedges, 〈ɛe,αe,0〉QW + 〈ɛh,αh,0〉QW and ɛ e,0b+ ɛ h,0b, respectively. To a good approximation, emission energy is centered around the lower curve, which means that except for slight deviations around the cross-over region, the emission energy is blue shifted with increasing excitation. Furthermore, it is always below the zero-field barrier bandgap.

Fig. 5 Average QW and barrier bandedge energies (solid and dashed curves, respectively) versus carrier density. Optical emission should be centered approximately at the lower curve. The upper and lower dotted lines indicate the strained-InGaN and unstrained-GaN bulk bandgap energies.

Another question concerns the curves depicting IQE recovery at current densities lower than observed in present experiments. This discrepancy suggests the presence of other loss mechanisms, such as Auger carrier loss. To illustrate the effect of Auger scattering, Auger carrier loss is incorporated into Eqs. (11) and (13), as described in Ref. [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]

]. The results are shown in Fig. 6 for A/Ab = 0.5, 2 and 4, with Auger coefficient C = 0, 10−32, 5 × 10−32 and 10−31 cm6s−1 (dotted, dashed, dot-dashed and solid curves, respectively). For clarity, the A/Ab = 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 C < 10−31 cm6s−1 which is smaller than that used in ABC models and are within the range predicted by microscopic calculation [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]

].

Fig. 6 Internal quantum efficiency versus current density showing the influence of Auger carrier loss for different QW SRH coefficients. The Auger coefficients are C = 0, 10−32, 5 ×10−32 and 10−31 cm6s−1 (dotted, dashed, dot-dashed and solid curves, respectively).

5. Summary

This paper describes an approach to modeling InGaN LEDs that involves the self-consistent solution of bandstructure and carrier population problems. The motivation is to provide direct input of bandstructure properties, in particular, their carrier-density dependences arising from screening of piezoelectric and spontaneous polarization fields. Other advantages include consistent treatment of spontaneous emission, carrier capture and leakage and nonequilibrium effects, as well as description of optical emission from quantum-well and barrier transitions on equal footing.

Application of the model is demonstrated with two examples, that are chosen to illustrate of the role of bandstructure changes on LED efficiency. The first example shows that higher defect recombination in QWs than barriers, when combined with bandstructure changes from screening of QCSE, can give rise to an efficiency droop. By casting the conditions in terms of defect densities in QW versus barrier layers, the model provides direct connection to measurable device properties. Within this model, LED efficiency would increase again for excitation at which the QCSE is sufficiently screened. The second example describes the role of Auger carrier loss in maintaining efficiency droop to high current densities, as observed in present experiments. By also including the effects of bandstructure changes, carrier capture and leakage, and plasma heating, one finds the necessary Auger coefficient to be in closer agreement with microscopic calculations than estimates from experimental curve fitting using the ABC model.

Lastly, the paper does not make use of any mechanism for the efficiency droop that is not already proposed in the literature. Rather, its goal is to introduce an approach for systematically incorporating potential contributions, both intrinsic and extrinsic, to produce a comprehensive model based on microscopic physics. It is possible that the differences in observed droop behavior (involving different LED emitting wavelengths, polar versus non polar substrates, with or without electron blocking layers, etc.) arise from differences in the relative importance of various mechanisms. The 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

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]

] and is easier to implement than a first-principles, many-body approach [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

This work is performed at Sandia’s Solid-State Lighting Science Center, an Energy Frontier Research Center (EFRC) funded by the U. S. Department of Energy, Office of Science, Office of Basic Energy Sciences. The author thanks A. Armstrong, M. Crawford, S. W. Koch, P. Smowton and J. Tsao for helpful discussions.

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. 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]

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]

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]

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]

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]

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]

13.

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

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]

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]

18.

W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).

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]

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]

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]

24.

A. F. Wright and J. S. Nelson, “Consistent structural properties for AlN, GaN, and InN,” Phys. Rev. B 51, 7866–7869 (1995). [CrossRef]

25.

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]

26.

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

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. 96, 221105–221107 (2010). [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]

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]

32.

L. V. Keldysh, “Behaviour of non-metallic crystals in strong electric fields,” Sov. Phys. JETP 6, 763–770 (1958).

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

  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,” Semiconductors40, 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. B19, 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]
  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]
  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]
  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]
  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. B57, R9435–R9438 (1998). [CrossRef]
  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]
  13. S. L. Chuang and C. S. Chang, “k · p method for strained wurtzite semiconductors,” Phys. Rev. B54, 2491–2504 (1996). [CrossRef]
  14. E. Jaynes and F. Cummings, “Comparison of quantum and semiclassical radiation theories with application to the beam maser,” Proc. IEEE51, 89–109 (1963). [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]
  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]
  18. W. W. Chow and S. W. Koch, Semiconductor-Laser Fundamentals: Physics of the Gain Materials (Springer, 1999).
  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. Express19, 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]
  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]
  23. 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]
  24. A. F. Wright and J. S. Nelson, “Consistent structural properties for AlN, GaN, and InN,” Phys. Rev. B51, 7866–7869 (1995). [CrossRef]
  25. 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]
  26. O. Ambacher, “Growth and applications of Group III-nitrides,” J. Phys. D: Appl. Phys.31, 2653–2710 (1998). [CrossRef]
  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.96, 221105–221107 (2010). [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]
  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]
  32. L. V. Keldysh, “Behaviour of non-metallic crystals in strong electric fields,” Sov. Phys. JETP6, 763–770 (1958).

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