## Abstract

In this paper, the self-consistent solution of Schrödinger-Poisson equations was realized to estimate the radiative recombination coefficient and the lifetime of a single blue light InGaN/GaN quantum well (QW). The results revealed that the recombination rate was not in proportion to the total injected carriers, and thus the *Bnp* item was not an accurate method to analyze the recombination process. Carrier screening and band filling effects were also investigated, and an extended Shockley-Read-Hall coefficient *A*(*k _{t}*) with a statistical weight factor due to the carrier distributions in real and phase space of the QW was proposed to estimate the total nonradative current loss including carrier nonradiative recombination, leakage and spillover to explain the efficiency droop behaviors. Without consideration of the Auger recombination, the blue shift of the electroluminescence spectrum, light output power and efficiency droop curves as a function of injected current were all investigated and compared with the experimental data of a high brightness blue light InGaN/GaN multiple QWs light emitting diode to confirm the reliability of our theoretical hypothesis.

©2011 Optical Society of America

## 1. Introduction

Recently, there has been large progress in the blue and green light InGaN/GaN light-emitting diodes (LEDs), which are attracting great interest as back light unit, automotive headlights, and general illumination, because of their long lifetime, small size, high efficiency and low energy consumption [1,2]. In *c*-plane InGaN/GaN multiple quantum wells (MQWs) LEDs, the light emitting efficiency reaches its peak value at low current density and then rapidly decreases with the injection current. This phenomenon is known as “efficiency droop”, which is a severe problem to achieve high-power and high-efficiency LEDs for applications.

An essential problem in determining the origin of the droop is that it is difficult to measure the amount of carrier loss to individual channels. Based on the investigation of carrier loss mechanisms by means of *An + Bnp + Cn ^{2}p* (ABC model), Auger recombination was proposed to be responsible for the efficiency droop because Auger coefficient was estimated to be as large as 10

^{−30}~10

^{−29}cm

^{6}s

^{−1}[3–5]. However, some state-of-the-art theoretical calculations pointed out that Auger recombination in GaN materials were too small to be a convincing inducement of the droop [6,7]. Several other carrier loss mechanisms such as defect-related recombination [8–10], current leakage [11–14], carrier overflow and spillover [15–17], low hole injection efficiency [18] and piezoelectric polarization [19] have been proposed regardless of Auger process. Among these mechanisms, Hader proposed the droop should be derived from the density-activated defect recombination and gave an empirical expression

*en*(

_{w}*N-N*)

_{0}*to estimate the nonradiative current loss [8]. One*

^{2}/2τN_{0}*N-N*item represented the linear density dependence associated with defect recombination and the other represented the increasing number of available recombination centers above a certain carrier density

_{0}*N*. Therefore, an overall quadratic term for current loss could be gotten to elucidate the droop. In addition, Özgür used another phenomenological formula

_{0}*kJ*to calculate the leakage current density (

^{b}*J*–total current density,

*k*,

*b*–fit parameters), including the thermionic carrier emission from QWs, carrier overflow above QWs and defect-assisted carrier leakage [15]. Overall, to cause the efficiency droop, all the potential nonradiative current loss mechanisms need to exhibit superlinear rise with carrier density to exceed the increase of radiative recombination item at high current injection case. However, up to now, the detailed variations of carrier distributions in real and phase space of QWs with injected carrier density, such as carrier screening and band filling effects, have not been systematically analyzed or considered to be a potential explanation for the efficiency droop of InGaN/GaN MQWs LED chips.

In this paper, an extended carrier Shockley–Read-Hall (SRH) coefficient *A*(*k _{t}*) as a function of in-plane wave vector

*k*due to carrier screening and band filling effects was proposed instead of traditional ABC model as an alternative explanation for the droop behaviors in the framework of the self-consistent solution of Schrödinger-Poisson equations. The light output power and external quantum efficiency were derived and compared with an actual high brightness blue light InGaN/GaN MQWs LED, and we argue that this potential theoretical nonradiative current loss process can also reproduce the experimental observations of our LED chip regardless of Auger recombination.

_{t}## 2. Experiments

A high brightness blue light InGaN/GaN MQWs LED are deposited by metalorganic chemical vapor deposition on *c*-plane sapphire substrates. A 3-µm n-type GaN:Si layer was grown followed by the active region. The active region consists of five 3-nm InGaN MQWs with 17-nm GaN barriers. The nominal alloy composition of the InGaN layers was about 18% to produce a peak emission wavelength of 450 nm at 20 mA. At last, a 300-nm p-type GaN:Mg layer was deposited upon the active region. The light emitting properties were measured in an integrating sphere at room temperature in a quasi-pulse current (100 ms length of single pulse, and wait 20s for the samples to cool down before next measurement) to eliminate the heating effect. The peak shift of electroluminescence (EL), light output power and efficiency droop as a function of injected current were analyzed and compared with our theoretical analysis.

## 3. Theoretical model and simulations

The self-consistent solution of Schrödinger-Poisson equations method was introduced in our simulation [20,21]. Single InGaN/GaN QW model with the same structure parameters as actual LED mentioned above was proposed to analyze the detailed carrier occupation states in real and phase space because David revealed only the QW nearest the p-type layer emitted light under electrical pumping, regardless of the actual number of QWs [22]. The spontaneous and piezoelectric polarization fields were all taken into consideration [23].

Figure 1(a)
shows the energy band profiles of the lowest conduct band and the uppermost valence band under an injected carrier density of 5 × 10^{18} cm^{−3}. Spontaneous and piezoelectric polarization fields were set to be −0.0278 C/m^{2} and 0.0032 C/m^{2} for blue light emitting. Energy levels and wave functions of two lowest conduct subbands (*e1* and *e2*) and the upmost valence subband (*hh1*) are showed in this figure. Figure 1(b) shows the occupancy percentages of the electron and hole carriers distributed in *e1* and *hh1* subbands as a function of total carrier densities, considering that *e1* and *hh1* are the actual subbands contributed to the EL emission. *n1*, *p1* are electron and hole carrier densities occupied in *e1* and *hh1* and *n*, *p* are the total carrier densities. We can confirm that the electrons almost occupy the *e1* energy level even at very high injection case because of the large energy difference between *e1* and *e2* (about 0.28 eV). However, due to the three adjacent valence bands mixing effect, the percentage of the hole occupied *hh1* energy level is just about 81% and decreases sharply when total hole density goes up to 1 × 10^{17} cm^{−3}. This phenomenon illuminates that the magnitude of hole contributed to the radiative recombination transition (*e1*-*hh1*) does not linearly increase with the total injected carriers, so that the expression *Bnp* in usual ABC model is not an accurate method to calculate the radiative recombination rates. Therefore, a more accurate calculation of the spontaneous emission rate can be achieved in our simulation.

In order to analyze the carrier nonradiative loss mechanisms based on our single QW model, the carrier screening and band filling effects induced by the carrier real and phase space distributions were also analyzed in our simulation. Figure 2(a)
shows the normalized electron distribution in phase space as a function of energy state *E*(*k _{t}*) of

*e1*subband. It is clearly to see that most of electrons are collected by the energy band edge of

*e1*within a small value of wave vector

*k*at the carrier density blow 1 × 10

_{t}^{17}cm

^{−3}. As injected carrier density increases, the Brillouin zone center (marked as

*E*(

*k*) in figure) shifts from 3.044 eV to 3.065 eV due to the carrier screening field estimated by Poisson equation, and much more electrons are collected by higher energy states with a relative larger

_{t}= 0*k*. In detail, the energy value of the most probable electron distribution shifts from 3.057 eV to 3.152 eV when the total injected carrier density increases from 1 × 10

_{t}^{16}cm

^{−3}to 5 × 10

^{18}cm

^{−3}. In other words, a total shift of 95 meV can be obtained due to the carrier screening and band filling effects.

Figure 2(b) illustrates the main possible carrier loss channels in the InGaN/GaN QW LEDs. Besides radiative recombination process, four main nonradiative current loss mechanisms could be involved in this model: defect related nonradiative recombination centers in QW [8–10], defect states in barriers as carrier tunneling channels [11–14], carrier thermal spillover from QW [15–17] and threading dislocations related V-shaped hexagonal pits as nonradiative recombination or current leakage channels [9,24,25]. Under high carrier injection conditions, the carriers occupied in high energy states should have relative larger possibilities to loss through these nonradiative loss channels. Therefore, taken advantage of the single QW model and self-consistent solution, an extended SRH coefficient *A*(*k _{t}*) =

*A*((

_{0}exp*E*(

*k*)

_{t}*-E*(

*k*))

_{t}= 0*/k*) with a statistical weight factor

_{B}T*exp*((

*E*(

*k*)

_{t}*-E*(

*k*))

_{t}= 0*/k*) due to carrier screening and band filling effects instead of a constant

_{B}T*A*used in usual ABC model, was introduced to describe a superlinear rise of nonradiative loss without consideration of Auger recombination.

*k*is Boltzmann's constant and

_{B}*T*is room temperature.

*A*is initial SRH coefficient at very low carrier injection which can be determined by our experimental fitting.

_{0}*E*(

*k*) represents the energy of carrier with wave vector

_{t}*k*. At last, the magnitude of total nonradiative current loss based on our hypothesis can be calculate using formula

_{t}*q*is electronic charge,

_{e}*L*is effective width of the active region and we take 18 nm in our simulation,

*f*(

_{c}*k*) is Fermi-Dirac distribution function for the first conduct subband

_{t}*e1*.

## 4. Results and discussion

Based on the carrier distributions in real and phase space of single InGaN/GaN QW model, the variations of radiative recombination lifetime *τ _{r}* and coefficient

*B*as a function of carrier density were investigated and results are shown in Fig. 3(a) . The lifetime

*τ*is about 100 ns at low carrier density, but goes sharply to less than 20 ns at the carrier density above 1 × 10

_{r}^{18}cm

^{−3}owning to the decrease of internal field induced by carrier screening effect. Then the increasing rate of

*τ*becomes lower and reaches a limit value of about 6 ns. On the other hand, the radiative recombination coefficient

_{r}*B*increases slowly and reaches its maximum value of 5.12 × 10

^{−11}cm

^{−3}s

^{−1}at carrier density of 1.33 × 10

^{18}cm

^{−3}. Afterward, the radiative coefficient

*B*begins to decrease with carrier density due to the evident carrier band-filling effect. Overall, through the analysis of these radiative recombination parameters, a more meticulous radiative recombination rate can be obtained than ABC model to reveal the origin of efficiency droop.

To keep consistent with experiment data, 5 × 10^{6} s^{−1} was fixed as the value of initial SRH coefficient *A _{0}*, and the relationship between the total nonradiative carrier loss lifetime

*τ*and the injected carrier density was simulated and results are shown in Fig. 3(b). The lifetime

_{nonr}*τ*keeps almost constant at 100 ns until carrier density reaches 1 × 10

_{nonr}^{18}cm

^{−3}. Then

*τ*exhibits a superlinear decrease and gets about 7 ns at a high carrier density 5 × 10

_{nonr}^{18}cm

^{−3}, which is almost equal to the radiative recombination lifetime, indicating that electrons occupied the high energy states have much larger probability to be captured by nonradiative recombination centers or leak from QW.

In order to verify the carrier screening and band filling effects simulated by our single QW model, a comparison of the blue shift of EL spectrums with current between our simulation and the actual InGaN/GaN MQWs LED was investigated because the blue shift of EL spectrum was considered to be the first preferred phenomenon induced by carrier distribution in real and phase space. Figure 4(a)
shows the comparison results of EL peak blue shift. Using the initial SRH coefficient *A _{0}* of 5 × 10

^{6}s

^{−1}, our theoretical analysis fits very well with the experiment results with current less than 300 mA, indicating the existence of the non-ignorable carrier screening and band filling effects prefigured by simulation. The little deviation of our simulation result from experimental data with current above 300 mA may be related to the electron overflow or the Fermi level pinning effect not involved in our calculation.

At last, the theoretical and experimental light output power and efficiency droop behaviors as a function of injected current were investigated and showed in Figs. 4(b) and 4(c). The theoretical output power also fits very well with experimental data unless at very high injection case. To fit the efficiency droop curve, a presumed light extraction efficiency *η _{LEE}* is fixed at 50% in our simulation. For comparison, usual ABC model was also preformed to fit the droop curve with parameters

*A*= 1.3 × 10

^{7}s

^{−1}and

*C*= 1 × 10

^{29}cm

^{6}s

^{−1}, but the values of coefficient

*B*is still dependent on our theoretical calculation. We can see that using 5 × 10

^{6}s

^{−1}as

*A*in extended SRH coefficient

_{0}*A*(

*k*) and 1.3 × 10

_{t}^{7}s

^{−1}as

*A*in ABC model both fit very well with the experimental data before the onset of droop at low current densities, once more indicating the reliability of our theoretical model. The little difference between the two SRH coefficients should due to the variation of the carrier filling state with current in our theoretical simulation. After reaching the efficiency maximum, the simulated efficiency began to decrease monotonically and also fit very well with actual data. However, at very high current density (above 200 A/cm

^{2}), both theoretical fittings shows a little deviation from the experiment and carrier noncapture mechanism such as electron overflow above active layer should be answerable to it.

## 5. Summary

Based on the single blue light InGaN/GaN QW model and self-consistent solution of Schrödinger-Poisson equations, the significant carrier distributions in real and phase space of QW should be essential factors for the exploration of radiative and nonradiative process of GaN-based LEDs. The superlinear increase of carrier nonradiative recombination, leakage and spillover loss mechanisms with injected carrier, were considered to be the main reason for the efficiency droop, and the extended SRH coefficient *A*(*k _{t}*) was proved to be an alternative method to analyze the nonradiative current loss properties of LEDs. At last, referring to our analysis, we can get the conclusion that, reducing nonradiative recombination centers, threading dislocations or defect states at barriers, as well as using wide thickness QWs or double-heterostructure as active layers to avoid carrier high energy state filling, are effective methods to relieve the severe efficiency droop phenomenon in GaN-based LEDs.

## Acknowledgments

This work is supported by the National Natural Science Foundation of China under Grant Nos. 61076013, 60776042 and 60990313, the National High Technology Program of China under Grant No. 2007AA03Z403.

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