DOI 10.1007/s00340-009-3856-6
Investigation of wavelength-dependent efficiency droop in InGaN
light-emitting diodes
J.-R. Chen· Y.-C. Wu · S.-C. Ling · T.-S. Ko · T.-C. Lu · H.-C. Kuo· Y.-K. Kuo · S.-C. Wang
Received: 29 May 2009 / Revised version: 15 October 2009 / Published online: 18 December 2009 © Springer-Verlag 2009
Abstract The physical mechanisms leading to the effi-ciency droop of InGaN/GaN light-emitting diodes (LEDs) are theoretically investigated. We first discuss the effect of Auger recombination loss on efficiency droop by taking dif-ferent Auger coefficients into account. It is found that the Auger recombination process plays a significant nonradia-tive part for carriers at typical LED operation currents when the Auger coefficient is on the order of 10−30cm6s−1. Fur-thermore, the InGaN/GaN multiple-quantum-well (MQW) LEDs with varied indium compositions in InGaN quan-tum wells are studied to analyze the wavelength-dependent efficiency droop. The simulation results show that the wavelength-dependent efficiency droop is caused by sev-eral different effects including non-uniform carrier distrib-ution, electron overflow, built-in electrostatic field induced by spontaneous and piezoelectric polarization, and Auger recombination loss. These internal physical mechanisms are the critical factors resulting in the wavelength-dependent ef-ficiency droop in InGaN/GaN MQW LEDs.
J.-R. Chen (
)· Y.-C. Wu · S.-C. Ling · T.-S. Ko · T.-C. Lu · H.-C. Kuo· S.-C. WangDepartment of Photonics & Institute of Electro-Optical Engineering, National Chiao Tung University, Hsinchu 300, Taiwan
e-mail:jrchen.eo95g@nctu.edu.tw
Fax: +886-3-5716631 T.-C. Lu
Institute of Lighting and Energy Photonics, National Chiao Tung University, Tainan 711, Taiwan
Y.-K. Kuo
Department of Physics, National Changhua University of Education, Changhua 500, Taiwan
1 Introduction
Nitride-based wide band-gap alloys have been widely used in light-emitting devices including light-emitting diodes (LEDs) [1], laser diodes (LDs) [2], and vertical-cavity surface-emitting lasers (VCSELs) [3]. Especially, GaN-based high-brightness LEDs would play an important role in future applications of solid-state lighting [4,5]. Although GaN-based high-brightness LEDs have been demonstrated by Nakamura et al. in 1995 [1] and the related ultravio-let, blue, and green LEDs have been commercialized in the following years, the efficiency of the state-of-the-art In-GaN/GaN LEDs should be further improved for many spe-cific applications which require high current injection op-eration. However, for InGaN/GaN multiple-quantum-well (MQW) LEDs, it is commonly observed that the efficiency decreases gradually with increasing injection current, which is the so-called efficiency droop phenomenon. The physical mechanism leading to an efficiency droop in InGaN/GaN LEDs is still controversial and different explanations have been proposed in recent years. Since the obvious efficiency droop is not observed in conventional GaAs-based LEDs, it is expected at an instant’s glance that the dislocation den-sity may be the main mechanism causing efficiency droop for GaN-based LEDs. Nevertheless, Schubert et al. have demonstrated that dislocation density reduces the overall ef-ficiency but does not affect the efef-ficiency droop [6]. Further-more, Auger recombination was proposed as the origin of the efficiency droop by Shen et al. [7] since they measured an Auger coefficient of InGaN ranging from 1.4× 10−30 to 2.0× 10−30 cm6s−1 by photoluminescence (PL) tech-nique. However, the opposite views on the importance of Auger loss were reported by Hader et al. based on fully mi-croscopic many-body models [8]. Recently, Delaney et al. have suggested again that Auger recombination is indeed
an important loss mechanism in wurtzite InGaN in 2009 [9]. Based on rigorous first-principle calculations, they in-dicated that the calculated Auger coefficients were in good agreement with the values reported by Shen et al. [7]. The calculated results by Hader et al. did not include a criti-cal second conduction band in their study, leading them to conclude that direct Auger losses are negligible in InGaN quantum wells (QWs) [8]. Moreover, experimental results show that the efficiency droop is unrelated to junction tem-perature according to the temtem-perature-dependent measure-ment [10] and the comparison of continuous-wave (CW) and pulse measurements [11]. Besides, it has been found that the efficiency droop is not observed from the power-dependent PL measurements when the carrier density is in the same range of the electrically injected one, which reveals that efficiency droop could be mainly related to the carrier injection, transport, and leakage processes [10]. Kim et al. have proposed that the polarization fields in the InGaN/GaN MQW region enable the escape of electrons from the MQW region and thus are the physical origin of the droop [10]. Xie et al. have also claimed that the efficiency droop is caused by severe electron leakage due to the heavy effective mass of holes and low hole injection efficiency [12]. Other mech-anisms including carrier delocalization from indium-rich re-gions [13], and piezoelectric polarization related to electron-hole separation [14], have been proposed as well. Moreover, another interesting study showed that the efficiency droop is more obvious for green InGaN LEDs and decreases toward shorter wavelengths. Based on these experimental results, some studies deduced that the wavelength-dependent effi-ciency droop might be caused by the current overflow from localized states or the increase of polarization with increas-ing indium composition in QWs [15–17].
Although the specific physical mechanisms leading to ef-ficiency droop in InGaN/GaN LEDs are still debatable, two physical mechanisms are referred to as the most possible origins. One is Auger recombination [7,9,18–20]. It is gen-erally expected that the Auger recombination is not the issue for GaN materials based on the bandgap dependence in other materials [21,22]. Nevertheless, the Auger coefficient of the GaN materials is measured and estimated from 1× 10−34to 5× 10−28 cm6s−1 [7–9,21–24], which indicates that the Auger recombination is still a possible mechanism resulting in the efficiency droop. The other is the energy band profile of the whole active region, which is related to the quantum-well thickness, quantum-barrier thickness, bandgap energy, quantum confined Stark effect (QCSE), etc. Actually, the energy band profile strongly influences the carrier transport mechanism. Therefore, various band profile designs are pro-posed in order to reduce carrier overflow, eliminate QCSE, and inhibit efficiency droop [10–12,19,25,26]. Nowadays, most publications only indicate one possible mechanism, such as Auger recombination or carrier overflow. In this
study, we demonstrate that the efficiency droop in InGaN LEDs is caused by multiple factors. The effects of Auger loss and energy band profile on wavelength-dependent effi-ciency droop in InGaN LEDs will be systematically investi-gated by using an advanced device simulation software AP-SYS, which self-consistently combines QW band structure calculations by 6× 6 k · p theory, radiative and nonradia-tive carrier recombination, and the carrier drift and diffusion model [27]. The influence of Auger loss on the properties of efficiency droop is analyzed and the polarization-induced ef-fects, such as electron leakage current and electron-hole sep-aration in QW, are also systematically investigated. Further-more, the mutual influence between different physical mech-anisms on the efficiency droop is discussed in this study as well.
2 Theoretical method and device structure
In order to develop high-performance InGaN/GaN LEDs, systematic and compact theoretical modeling is a neces-sary approach to improve existing LED structures and un-derstand internal physical processes, which provides timely and efficient guidance toward the optimal structure design and device parameters. The self-consistent APSYS simula-tion program combines two-dimensional (2-D) simulasimula-tions of carrier transport, quantum-well effect, heat flux, etc. The carrier transport model includes drift and diffusion of elec-trons and holes in devices. Built-in polarization induced by spontaneous and piezoelectric polarization is considered at hetero-interfaces of nitride-related devices. In the quantum wells, self-consistent Poisson and Schrödinger equations are recomputed at every bias point for the states of quantum-well levels and carrier distributions.
The physical model of the InGaN/GaN quantum wells is considered in such a way that the conduction bands are assumed to be decoupled from valence subbands and have isotropic parabolic bands due to the larger bandgap of nitride semiconductor [28–30]. This condition is different from a zinc-blende GaN structure, which has larger Kane matrix parameter and then causes significant coupling between con-duction and valence bands [31]. As for the calculation in va-lence bands, it includes the coupling of the heavy-hole (HH), the light-hole (LH), and the spin-orbit split-off bands, which are calculated by the 6× 6 Hamiltonian with envelop func-tion approximafunc-tion [32,33]. To obtain the numerical para-meters required for k· p calculations for the InGaN and Al-GaN materials, a linear interpolation between the parame-ters of the relevant binary semiconductors is utilized except for the unstrained bandgap energies. The material parame-ters of the binary semiconductors are taken from the paper by Vurgaftman and Meyer [34]. The unstrained InGaN and
AlGaN bandgap energies can be expressed as Eg(InxGa1−xN)
= x · Eg,InN+ (1 − x) · Eg,GaN− bInGaN· x · (1 − x), (1) Eg(AlxGa1−xN)
= x · Eg,AlN+ (1 − x) · Eg,GaN− bAlGaN· x · (1 − x), (2) where bInGaN and bAlGaNare the bandgap bowing parame-ters of InGaN and AlGaN, which are 1.4 and 0.7 eV, re-spectively [34]. The temperature-dependent bandgap ener-gies of the relevant binary semiconductors are calculated us-ing the commonly employed Varshni formula. The sponta-neous emission spectrum in active region can be expressed by [35]
rsp(E)= q2h
2m20εED(E)ρred(E)|M| 2fn c 1− fvm , (3)
where q is the free electron charge, h is the Planck con-stant,|M|2is the momentum matrix element in the strained quantum well, fcn and fvm are the Fermi functions for the conduction-band states and the valence band states respec-tively, D(E) is the optical mode density, and ρred(E)is the reduced density of states in each subband. The indices n and mdenote the electron states in the conduction band and the heavy-hole (light-hole) subband states in the valence band. The momentum matrix element is computed from the in-tegration of the envelope functions obtained from the k· p calculations [28]. The strain tensor εij in the well region is
comprised of εxx= εyy= a0− a a , (4) εzz= − 2C13 C33 εxx, (5) εxy= εyz= εzx= 0, (6)
where a0 and a are the lattice constants of the InGaN or AlGaN layers. C13 and C33 are the stiffness constants. The effects of calculated strain values will directly affect the quantum-well band structures and the interface charge den-sity in each heterojunction of the LED structures [28,36]. The optical mode density and reduced density of states can be expressed respectively as D(E)= εnrE 2 π23c3, (7) ρred(E)= mr π2d z , (8)
where nr is the index of refraction, c is the speed of light, dz
is the thickness of the quantum well, mris the reduced
effec-tive mass. To account for the broadening induced by intra-band scattering, the Lorentzian lineshape function is used in the expression of the spontaneous emission spectrum, which is given by Rsp= 1 π i,j rsp(E) Γ (Eij− E)2+ Γ2 dE, (9)
where Γ = /τ , which represents the broadening due to in-traband scattering relaxation time τ , and Eij is the
transi-tion energy from the ith conductransi-tion band to the j th valence band. To account for the broadening due to scattering, it is assumed that τ = 0.1 ps in the calculations [28–30]. The conduction-band offset ratio Ec/Egfor the InGaN/GaN
interface is assumed to be 0.6 based on the recent pub-lished literature [37–39]. The internal quantum efficiency, ηint, characterizing the device performance of a LED is de-fined as ηint= q J Rsp(V ) dV . (10)
Here J is the total current density in the device and V is the volume of the active region. The internal quantum effi-ciency is considered not only for the competition between the radiative and nonradiative recombination but also for the effect of carrier leakage from the active region.
The physical model of carrier transport is the traditional drift-diffusion model for semiconductors. The specific equa-tions can be expressed as
Jn( r )= qμnn(r) F ( r )+ qDn∇n(r), (11) Jp( r )= qμpp(r) F ( r )− qDp∇p(r), (12)
where n(r) and p(r) are the electron and hole concentra-tions, Jn(
r )and Jp(
r )are the current densities of elec-trons and holes,F ( r )is the electrostatic field, μn and μp
are the mobilities of electrons and holes. The diffusion con-stants Dnand Dp are replaced by mobilities using the
Ein-stein relation, D= μkBT /q. The equations used to describe
the semiconductor device behavior are the Poisson equation, ∇ ·ε0ε F ( r ) = qp(r)− n(r) + pD(r)− nA(r)± Nf (13) and current continuity equations for electrons and holes,
1 q∇· Jn( r )− Rn(r)+ Gn(r)= ∂n(r) ∂t , (14) 1 q∇ Jp( r )+ Rp(r)− Gp(r)= − ∂p(r) ∂t , (15)
Table 1 Net surface charge density at each interface of the In0.2Ga0.8N/GaN LED
Interface Built-in charge density
Al0.2Ga0.8N /GaN −8.8 × 1012cm−2 GaN/ Al0.2Ga0.8N +8.8 × 1012cm−2 In0.2Ga0.8N /GaN +2.09 × 1013cm−2 GaN/In0.2Ga0.8N −2.09 × 1013cm−2
where Gn(r)and Rn(r)are the generation rates and
recom-bination rates for electrons, Gp(r)and Rp(r)are the
gen-eration rates and recombination rates for holes, respectively. The recombination rate R accounts for radiative and non-radiative channels. The non-radiative carrier recombination is dependent on the spontaneous emission rate, Rsp, and the nonradiative recombination includes Shockley–Read–Hall (SRH), RSRH, and Auger recombination, RAug. The SRH re-combination is directly governed by the defect-related non-radiative SRH lifetime (τSRH). Defect density and nonra-diative lifetime depend on the substrate used and on the growth quality. In this study, we employ a common value of τSRH= 1 ns in our simulation [40–42]. The Auger re-combination rate is given by
RAug(r)= Cnn(r)+ Cpp(r) n(r)p(r)− n0(r)p0(r) , (16) where Cn and Cp are Auger coefficients, n0(r) and p0(r) are the equilibrium electron and hole concentration. The ef-fects of the Auger coefficients on the device performance will be discussed in the following sections.
The internal electric field influences the band profile and carrier transport in devices and is affected by the charge distribution, including the electron and hole concentrations, dopant ions pD(r)and nA(r), and other fixed charges Nf
that are of special importance in nitride-based devices due to the effect of built-in polarization. The built-in polariza-tion induced by spontaneous and piezoelectric polarizapolariza-tions is known to influence the performance of nitride devices. In order to consider the built-in polarization within the inter-faces of nitride devices, the method developed by Fiorentini et al. is employed to estimate the built-in polarization, which is represented by fixed interface charges at each hetero-interface. They provided explicit rules to calculate the non-linear polarization for nitride alloys of arbitrary composi-tion [36]. For the InGaN/GaN LEDs under study, the net surface charges at all interfaces are calculated and listed in Table1. Although the interface charges can be obtained by this theoretical model, experimental investigations often find a weaker built-in polarization than that predicted by the-oretical calculation. It is mainly attributed to partial com-pensation of the built-in polarization by defect and interface charges [43]. Typical reported experimental values are some
20%, 50% or 80% smaller than the theoretically calculated values [44–46]. As a result, 50% of the theoretical polariza-tion values are used in our simulapolariza-tion from the average of the reported values.
A widely used empirical expression for modeling the mo-bility of electrons and holes is the Caughey–Thomas approx-imation, which is employed in our calculation and can be expressed as [47]
μ (N )= μmin+
μmax− μmin 1+ (N/Nref)α
, (17)
where μmin, μmax, Nref, and α are fitting parameters accord-ing to the experimental mobility measurements. We employ this carrier mobility model for binary GaN material in our calculation. The relative parameters are taken from Refs. [48,49]. As for ternary AlGaN, the analytical expressions for mobility as a function of doping density have been es-tablished by Monte Carlo simulation for various nitride al-loys [50]. The calculation of the carrier capture and escape from the quantum wells is considered in accordance with the model provided by Romero et al. [51]. As for the parameter of refractive index, the Adachi model is employed to calcu-late the refractive index values in each layer [52–54]. More description of the physical models utilized in the APSYS simulation program can be found in Refs. [22,41,55].
In this simulation, we first assume that the InGaN/GaN LEDs are grown on a 3-µm-thick n-type GaN layer. On top of this GaN layer, the MQW active region consists of five 3-nm-thick In0.2Ga0.8N QWs and 10-nm-thick GaN barri-ers. A 20-nm-thick p-type Al0.2Ga0.8N electronic blocking layer (EBL) is grown on top of the active region to reduce electron leakage into the p-type GaN layer [56–58]. Finally, a 0.2-µm-thick p-type GaN cap layer is grown to complete the preliminary LED structure. Moreover, in order to investi-gate the wavelength-dependent efficiency droop, the indium composition in the InxGa1−xN QW is changed from 10% to 30% to cover the emission wavelength from violet to green. The InGaN/GaN LED area is 300× 300 µm2. The detailed device structure and doping concentrations in each layer are described in Table2. The doping data in this table give the actual densities of free carriers.
3 Simulation results and discussion
In first instance we will discuss the effects of Auger recom-bination on efficiency droop in blue In0.2Ga0.8N/GaN LEDs. As mentioned in Sect.1, the Auger coefficients of GaN ma-terials are reported to range from 10−34 to 10−28 cm6s−1 based on different theoretical and experimental estimations. Recently, the Auger coefficient in quasi-bulk InGaN layers
Table 2 Layer structure and room-temperature physical parameters of the GaN/AlGaN quantum-well laser under study (d, layer thickness; Ndop, doped carrier density; n, refractive index at wavelength 450 nm). The doped carrier density, Ndop, represents the actual density of free carriers
Parameter (unit) d(nm) Ndop(1/cm3) n
p-GaN (contact layer) 200 1.2× 1018 2.454
p-Al0.25Ga0.75N (blocking layer) 20 4× 1017 2.425
n-GaN (barrier layer) 10 2× 1017 2.454
i-Al0.2Ga0.8N (quantum well) 3 3.2393
n-GaN (barrier layer) 10 2× 1017 2.454
i-In0.2Ga0.8N (quantum well) 3 3.2393
n-GaN (barrier layer) 10 2× 1017 2.454
i-In0.2Ga0.8N (quantum well) 3 3.2393
n-GaN (barrier layer) 10 2× 1017 2.454
i-In0.2Ga0.8N (quantum well) 3 3.2393
n-GaN (barrier layer) 10 2× 1017 2.454
i-In0.2Ga0.8N (quantum well) 3 3.2393
n-GaN (barrier layer) 10 2× 1017 2.454
n-GaN 500 5× 1018 2.454
n-GaN 2500 5× 1018 2.454
sapphire(substrate) 100000
Fig. 1 Normalized quantum efficiency of the In0.2Ga0.8N/GaN LED as a function of input current when the calculations take different Auger coefficients into account
was measured from 1.4× 10−30to 2.0× 10−30cm6s−1 us-ing a photoluminescence technique [7] and the reported re-sults by Delaney et al. based on rigorous first-principle cal-culations are also in agreement with these values [9]. There-fore, the Auger coefficient of this magnitude will play a significant nonradiative part for carriers at typical LED op-erating currents. Figure1 shows the calculated normalized quantum efficiency of the In0.2Ga0.8N/GaN LED as a func-tion of input current when the calculafunc-tions take different Auger coefficients into account. It is evident that the effi-ciency droop becomes more obvious with Auger coefficient increasing from 3× 10−34 to 3× 10−30 cm6s−1. The re-duction of the efficiency is nearly 50% with increasing input current to become about 600 mA as using an Auger
coeffi-cient of 3× 10−30cm6s−1. Originally, the efficiency should increase with input current for the ideal case. However, the Auger recombination loss will compete with radiative re-combination as the input current increases and thus the ef-ficiency curves first go up to peak values and then gradu-ally decrease. Therefore, the degree of efficiency droop is changed with employing different Auger coefficients in the calculations. Moreover, the peak efficiency value shifts to-ward lower input current when the larger Auger coefficient is employed in the calculation, which means that the Auger recombination could still be a carrier loss process for the LED under low current injection if the Auger coefficient is really on the order of 10−30 cm6s−1. To further compare the effects of Auger recombination loss at different injection currents, we plotted the Auger recombination rate within the active region of the In0.2Ga0.8N/GaN LED using Auger co-efficients of 3× 10−30 and 3× 10−34 cm6s−1 at 30 and 480 mA, respectively, in Fig.2. The reason for the choice of these current values is to compare the effects of Auger re-combination rate under low and high current injection, and thus analyze how this mechanism influences the efficiency droop. The left-hand side of the figure is the n-side of the device. In Fig.2, the Auger recombination rate calculated using 3× 10−30cm6s−1is much higher than that calculated using 3× 10−34 cm6s−1 even though the input current is 30 mA. Furthermore, the highest Auger recombination rate is always observed in the QW closest to p-side. According to this condition, it is expected that a large number of carri-ers are within this QW since the Auger recombination rate is in proportion to n3, where n is the carrier concentration. It is found that this condition is caused by a non-uniform hole distribution within the MQW region, resulting from a
Fig. 2 Auger recombination rate within the active region of the In0.2Ga0.8N/GaN LED using Auger coefficients of 3× 10−30 and 3× 10−34cm6s−1at 30 and 480 mA, respectively
large hole effective mass and low hole injection due to rel-atively low hole concentration, which is in agreement with experimental investigations [12,59–61]. We will further dis-cuss this issue in the later analysis. According to the calcu-lation results shown in Figs. 1 and2, the effect of Auger recombination loss combining with serious inhomogeneous distribution of injected carriers plays a significant role in the efficiency droop of InGaN/GaN LEDs when the calculation employs an Auger coefficient of the order of 10−30cm6s−1 [7,9].
Although the specific Auger coefficient of GaN materials may be still debatable, we employ the Auger coefficient of 3×10−30cm6s−1based on the latest report in the following investigation [7,9]. We further focus our study on the inter-esting issue of the wavelength-dependent efficiency droop in InGaN LEDs. The LED structures are the same as pre-viously except for the indium composition in QWs which is varied from 10% to 30% to cover the emission wave-length from near ultraviolet to green. Figure3shows the cal-culated normalized quantum efficiency of the InGaN/GaN LEDs with different indium compositions in QWs as a func-tion of input current. It can be observed that the efficiency droop appears to be a strong function of the indium compo-sition in the active region, which has been demonstrated in
Fig. 3 Normalized quantum efficiency of the InGaN/GaN LEDs with different indium compositions in QWs as a function of input current
experimental results [15–17]. In Fig.3, the efficiency of the InxGa1−xN/GaN LED decreases from 62% to 35% with in-creasing indium composition in QWs at the injection current of 600 mA. Additionally, the peak efficiency value shifts toward lower input current with increasing indium compo-sition. These calculated results represent that other physi-cal mechanisms will participate in the efficiency droop in addition to the effect of Auger recombination. Practically, there are three possible effects governing the wavelength-dependent efficiency droop in InGaN/GaN LEDs. The first one is the carrier localization originating from the lateral po-tential fluctuations due to inhomogeneous indium incorpo-ration. Although localization effects improve the radiative efficiency due to the prevention of carriers from nonradia-tive centers [62], the increased input current may give rise to carriers escaping from the localized states, thus lowering the radiative efficiency owing to the enhanced nonradiative re-combination [15–17]. Additionally, with increasing indium composition, the effects of strain and indium phase separa-tion will deteriorate the crystalline quality, which may be an issue for efficiency droop. Nevertheless, Yang et al. demon-strated that crystal quality plays a minor role in light emis-sion and is not responsible for the efficiency droop behav-iors in InGaN/GaN green LEDs [17]. Although the indium inhomogeneity is experimentally observed in InGaN QWs and may be the origin of the wavelength-dependent effi-ciency droop, it is difficult to quantitatively take this effect into account in our calculation. The second one is that the QW energy band profile is affected by the indium compo-sition and the spontaneous and piezoelectric polarizations in QWs. Since the lattice mismatch between GaN barrier and InGaN QW increases with indium composition, the cor-respondingly increased piezoelectric polarization enhances the tilt of QW potential. The third one is that the Auger co-efficient could depend on the indium concentration of the In-GaN alloy as reported by Delaney et al. from first-principle
Fig. 4 Vertical profiles of (a)–(f) electron and hole concentration distribution, conduction-band edge, and (g)–(l) radiative recombination rate in the active regions of the LED structures with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N QWs
calculation [9]. That would induce the additional possibil-ity of the wavelength-dependent efficiency droop. Neverthe-less, the experimental measurements of the indium-content dependent Auger coefficients are hardly reported in the lit-erature. Although the process of carriers escaping from the localized states with increasing input current is not consid-ered in our simulation due to the difficulty in quantitatively modeling the randomly position-dependent indium distrib-ution, the wavelength-dependent efficiency droop can still be observed, as shown in Fig. 3. Therefore, we will fur-ther discuss the possible physical mechanisms leading to the wavelength-dependent efficiency droop in InGaN/GaN LEDs.
In order to understand the internal physical mechanisms which result in the wavelength-dependent efficiency droop,
the vertical profiles of electron and hole concentration dis-tribution, conduction-band edge, and radiative recombina-tion rate in the active regions of the LED structures with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N QWs are plotted in Fig.4, respectively. The gray areas in Figs.4(g)–4(l) rep-resent the positions of QWs. We investigate these profiles under low injection (30 mA) and high injection (480 mA), respectively, to analyze the possible radiative and nonradia-tive processes for a wavelength-dependent efficiency droop. In the case of the In0.1Ga0.9N/GaN LED, the relatively uni-form electron and hole concentration distributions can be observed whatever the injection current is; it may be 30 or 480 mA due to the shallow QWs, as shown in Figs. 4(a) and4(d). In this condition, all the quantum wells can ef-fectively participate in the radiative recombination process
when the input current is 30 mA, as shown in Fig. 4(g). On the contrary, when the indium composition in the QWs increases, the non-uniform hole concentration distribution is easily observed at low current injection [Figs.4(b) and 4(c)]. The hole concentration in the QW near the p-side is larger than that in the QW adjacent to the n-side due to the large effective mass of the holes and the deep QWs for the In0.2Ga0.8N/GaN and In0.3Ga0.7N/GaN LEDs [59–61]. However, under the same injection current the electrons can relatively easily transport through the MQW region. In addi-tion to the effect of barrier height on the non-uniform carrier distribution, the barrier height created by Al0.2Ga0.8N elec-tronic blocking layer is substantially reduced by the high density of positive polarization charges at the interface be-tween the GaN barrier layer and the Al0.2Ga0.8N electronic blocking layer. This condition will lead to the severe ac-cumulation of electrons at this interface due to the attrac-tion of positive charges by Coulomb force, causing strong band bending at this interface, as indicated in Figs. 4(a)– 4(f). Consequently, a higher electron concentration at the GaN/Al0.2Ga0.8N interface and in the QW closest to the p-side is observed. Therefore, because of the non-uniform hole distribution and the electron accumulation, most of the ra-diative recombination occurs only within the first QW close to the p-side when the indium composition in the QWs in-creases, as shown in Figs.4(h) and4(i). This phenomenon is in agreement with experimental observation [61]. Addi-tionally, the effect of band bending at the GaN/Al0.2Ga0.8N interface will reduce the effective energy barrier height and thus significantly enhances the escape of electrons from the active region [10,63,64]. Furthermore, when the input cur-rent is 480 mA, the electron and hole concentration distrib-utions in the active region of the In0.1Ga0.9N/GaN LED are relatively uniform across all the QWs, which correspond-ingly leads to the similar magnitude of radiative recombina-tion rate in each QW, as shown in Fig.4(j). Nevertheless, with increasing indium composition in the QWs the non-uniform carrier distribution and radiative recombination rate within the MQWs can still be observed when the injection current is 480 mA [Figs.4(f) and4(l)]. The deep QWs and large effective mass of holes cause the significant increase of hole concentration in the QW nearest the p-layer under this high current injection. Combining the effect of electron ac-cumulation, the QW closest to p-type layer still dominates the radiative recombination process, as shown in Fig.4(l). Therefore, to enhance all the QWs participating in the radia-tive recombination process, a more effecradia-tive hole transport within the MQW region is required, especially for the high-indium-content MQWs.
In addition to the effect of increased barrier height with indium composition in QWs, the piezoelectric polarization induced by lattice mismatch between GaN barrier layer
Fig. 5 Calculated interface charge density at the interface between the InGaN QW and the GaN barrier layer as a function of the indium com-position
Fig. 6 Internal electric field in the QWs of the LED structures with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N QWs, respectively, as a function of input current
and InGaN QW will also increase with indium composi-tion. Figure 5 shows the calculated interface charge den-sity at the interface between the InGaN QW and the GaN barrier layer as a function of the indium composition. The interface charge density of the In0.3Ga0.7N/GaN interface is nearly three and a half times larger than that of the In0.1Ga0.9N/GaN interface. In this situation, the built-in po-larization causes a strong deformation of the band struc-ture in In0.3Ga0.7N/GaN QWs accompanied by an elec-trostatic field. Therefore, the separation of electrons and holes in In0.3Ga0.7N/GaN QWs becomes more obvious than that in In0.1Ga0.9N/GaN QWs, as shown in Figs.4(d) and 4(f). Although the deep QWs can confine higher carrier concentrations, the non-uniform carrier distribution and se-vere electron–hole separation with increasing indium com-position result in low radiative recombination rate in the
Fig. 8 Auger recombination rate within the active region of InGaN/GaN LEDs with different indium compositions under the injection current of 30 and 480 mA, respectively
Fig. 7 Vertical electron current density profiles within the active re-gions of the LEDs with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N MQW, respectively, at the input current of 480 mA
In0.3Ga0.7N/GaN QWs, as shown in Fig.4(l). Figure6plots the internal electric field in the QWs of the LED structures with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N QWs, re-spectively, as a function of input current. The electric field in these three LEDs decreases with increasing input current due to the screening effect. Nevertheless, the increased in-put current still cannot completely screen the large built-in polarization, especially for In0.3Ga0.7N/GaN QWs [63]. Figure 7 shows the vertical electron current density pro-files within the active regions of the LED structures with In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N QWs,
respec-tively, at the input current of 480 mA. The positions of the QWs are marked with gray areas. The electron current is injected from n-type layers into quantum wells and recom-bines with holes in QWs. Therefore, the electron current density is consumed after passing each QW region. Elec-tron current which overflows to the p-type region is viewed as the leakage current. In Fig.7, the electron leakage cur-rent is easily observed under this high driving curcur-rent for the three LED structures. However, the In0.3Ga0.7N LED re-veals a more serious electron leakage current than the other two LEDs. Although the In0.3Ga0.7N/GaN LED provides deep QWs in the active region, the relatively light effective mass of electrons, the attraction of positive charges at the GaN/Al0.2Ga0.8N interface [63], the large built-in polariza-tion [10,14], and the non-uniform hole distribution [12] lead to higher electron leakage current.
As our previous discussion, the Auger recombination process really plays an important role if the Auger recom-bination coefficient of the GaN materials is on the order of 10−30 cm6s−1. Therefore, we will further discuss the wavelength-dependent Auger recombination process. Fig-ure8shows the Auger recombination rate within the active region of InGaN/GaN LEDs with different indium compo-sitions under the injection current of 30 and 480 mA, re-spectively. When the input current is 30 mA, the total Auger recombination rate in the QWs of In0.1Ga0.9N, In0.2Ga0.8N, and In0.3Ga0.7N LEDs, respectively, is comparable. How-ever, the Auger recombination rate of the In0.3Ga0.7N LED is obviously larger than that of the In0.1Ga0.9N LED as the input current is 480 mA. The higher Auger recombination
rate could be one of the mechanisms leading to the severe efficiency droop for the In0.3Ga0.7N/GaN LED. Moreover, the highest Auger recombination rate is still within the first QW close to the p-side for the In0.3Ga0.7N LED, which is attributed to the severely non-uniform carrier distribution within the MQW region, as discussed previously. As a re-sult, the Auger recombination process may be still one of the possible issues leading to the wavelength-dependent ef-ficiency droop in InGaN/GaN LEDs.
4 Conclusion
Theoretical simulation has been used to investigate the pos-sible physical mechanisms resulting in the efficiency droop in InGaN/GaN LEDs. The calculated results indicate that the effect of Auger recombination loss plays an important role in the efficiency droop of InGaN/GaN LEDs if the Auger co-efficient of GaN materials is on the order of 10−30cm6s−1. Furthermore, the wavelength-dependent efficiency droop is discussed by using different indium compositions in the In-GaN/GaN QWs. According to the simulation results, we de-duce that there are four internal physical mechanisms lead-ing to the wavelength-dependent efficiency droop. First, the deep MQWs employed by increasing the indium composi-tion in InGaN/GaN MQWs cause the severely non-uniform carrier distribution, which induces that only the QW close to p-side might effectively participate in the radiative recom-bination. Second, the built-in polarization increases with in-creasing indium composition due to the lattice-mismatch-induced piezoelectric polarization, which enhances the sep-aration of electrons and holes in QWs and thus reduces the radiative recombination. Third, the effect of electron over-flow is more obvious with increasing indium composition in QWs, which originates from the non-uniform carriers distri-bution and increased built-in polarization in high-indium-content MQWs. Fourth, the Auger recombination loss is higher with increasing indium composition due to the ef-fect of carrier accumulation in the QW closest to p-side. Although the effect of carrier localization may be still a possible mechanism of efficiency droop from the viewpoint of practical issue in InGaN materials, the condition of the wavelength-dependent efficiency droop can be observed in the calculation, which means that additional factors sum-marized as the four effects could result in the wavelength-dependent efficiency droop in InGaN/GaN LEDs.
Acknowledgements This work was supported in part by the MOE ATU program and in part by the National Science Council of the Re-public of China under Contracts NSC 98-3114-M-009-001, NSC 2221-E-009-094-MY3, NSC 98-2221-E-009-016-MY3, and NSC 96-2112-M-018-007-MY3.
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