Energy band diagram of LED A at I=200mA. 

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... C n and C p are the Auger recombination coefficients for electrons and holes, respectively. The detail band parameters for the III-nitride alloys were obtained from Ref. [ 33]. Furthermore, the dislocation density is set to be 1×109 cm -2 , and the corresponding Shockley-Read-Hall lifetime and internal loss, are set to be 10 ns and 1600 m -1 , respectively. With respect to the simulation parameters, the LED was assumed to be grown on the c-plane of the sapphire substrate, on which is a GaN nucleation layer, a 2 μ m-thick undoped GaN layer and a 2.5 μ m-thick n-type (5×10 18 cm -3 ) GaN cladding layer in that order. The active region comprises five periods of undoped 3 nm-thick In 0.30 Ga 0.70 N/ 12 nm-thick GaN multiple quantum wells (QW, λ = 530 nm). On top of the active region is a 15 nm-thick p-type (1×10 18 cm -3 ) Al 0.15 Ga 0.85 N electron blocking layer (EBL) and a 200 nm-thick p-type (1×10 18 cm -3 ) GaN cap layer. Four structures were considered to elucidate the dependence of the efficiency droop on the location of high indium layer in staggered InGaN QWs, as displayed in Fig. 1(a). In this investigation, the thickness of high indium layer is kept to be 1 nm with an indium content of 40%. The high indium layer is placed on the left side (LED B), in the middle (LED C), and on the right side (LED D) of the In 0.30 Ga 0.70 N QW. LED A is the control sample (without the high indium layer). The chip size of all LEDs is set to be 300×300 μ m 2 . Figure 1 (b) plots the simulated light-output power against current ( L-I ) and current against voltage ( I-V ) characteristics for four LEDs. At first glance, the I-V properties of the four LEDs seem to be almost identical, with a turn-on voltage of 3.0 V and a series resistance of 18.4 Ω . In contrast, the light-output power of all LEDs depends strongly on the location of the high indium layer in the staggered QW. Clearly, LED D exhibits the largest light-output power under high current injection ( I > 50 mA). The light-output power of LED D is by 9.4% higher than that of the control sample (LED A) at I = 20 mA and 41.5% higher at I = 200 mA, revealing a considerable reduction of the efficiency-droop. The light-output power of the LED declines as the location of the high indium layer is changed from the right side to the left side of the QW. Notably, the light-output power of LED B is lower than that of LED A throughout the range of current injection. This observation reveals that the location of the high indium layer in the staggered QW is extremely important in realizing high-power InGaN-based green LEDs. Figure 2 (a) plots the internal quantum efficiency (IQE) as a function of injected current of the four LEDs. The IQE of the LEDs other than LED D exhibits the typical efficiency-droop effect. For example, the IQE of LED A reaches its peak of 29.5% at I = 3 mA, and decreases rapidly to 17.8% at I = 200 mA. The degree of efficiency-droop, defined as [(IQEpeak - IQE200mA)/ IQEpeak] × 100%, is 39.6% for LED A and 42.4% for LED B, respectively. It is seen that LED B is similar to that of LED A. However, the efficiency-droop in LED C is highly significant, at 51.2%, since the IQE peak of LED C (44.9% at I = 1mA) is considerably larger than that of LED A. Importantly, the degree of efficiency droop calculated for LED D is only 11.3%. Figure 2(b) plots the electrostatic-field in the active region in four LEDs at I = 200 mA. For all LEDs, the electrostatic-field in the last QW is weaker than in the other QWs, suggesting the accumulation of many injected carriers, which screen the polarization charges at the InGaN/GaN interface. LED A has a much weaker electrostatic field than the other LEDs in the active regions, purely because the QW in LED A contains less indium than those in the other LEDs. Moreover, the incorporation of high indium layer alters the distribution of the electrostatic-field in the QW, as presented in the inset in Fig. 2(b). Accordingly, injected holes in LED D are likely transited across the QW because the intensity of the electrostatic-field on the right-side of the QW is evidently larger than in the other regions of the QW. Consequently, more holes can be injected into the active region than elsewhere. In contrast, most injected holes in LED B accumulate in the last QW. Because of more injected holes will concentrate in the low energy layer of LED B, the distribution of injected holes is less uniform in the QW and the induction of the electrostatic field therein tends to prevent the transit of injected holes into the active region [34]. Hence, of all LEDs, LED B has the lowest IQE over the whole range of injected currents, as shown in Fig. 2(a). Figure 3 plots the calculated band diagram of LED A at I = 200 mA. In this figure, gray regions represent the QWs. Clearly, the energy of the conduction band on the n -side substantially exceeds that on the p -side. Additionally, the Al 0.15 Ga 0.85 N EBL is inefficient in preventing the leakage of injected electrons, and so most of the injected electrons are accumulated in the last QW. Similarly, most of the injected holes are also accumulated in the last QW because they have a relatively large effective mass. As a result, the last well is alone responsible for nearly all of the radiative recombination. The calculated band diagrams of the other LEDs reveal similar behaviors. However, the band diagram on the individual QW is completely different. Figure 4(a) magnifies the valence band of the first QW in all LEDs. The energy offset between the quasi-Fermi level (red dashed line) and the apex of the valence band is mainly determined by the location of high indium layer in the QW. As the high indium layer is moved from the right side to the left side of the QW, the energy offset slowly increases. Consequently, the population of injected holes in LED D in the first QW is expected to exceed that in the other LEDs, as LED D has the smallest energy offset of -0.2 eV. The energy offsets of LED C and LED B are 0.1 eV and 0.74 eV, respectively. As previously discussed with reference to Fig. 2(b), LED D has the smallest energy offset mainly because of the strong electrostatic field inherent in the right-side of its QW, which pushes the injected holes to the n-side, reducing the energy offset. In contrast, when the high indium layer is placed in the left-side of the QW (LED B), the induced electrostatic-field in the QW inhibits the transit of injected holes across the active region, yielding an extraordinarily large energy offset of 0.74 eV. Similarly, Fig. 4(b) enlarges the conduction band of the last QW in all LEDs. Owing to the relative small effective mass of injected electrons, the energy offsets (between the quasi-Fermi level and the bottom of conduction band) for all LEDs are almost identical to each other, and are dominated by the overall energy band profile. Restated, the incorporation of the high indium layer in the staggered QW barely influences the energy offset in the conduction band. Figure 5 plots the distribution of electron and hole concentrations at I = 200 mA in all LEDs. Apparently, for all LEDs, the hole concentration in the last QW closest to the p-side is larger than that in the QW adjacent to the n-side, mainly because of the large effective mass of the injected holes. However, LED D evidently has a more uniform distribution and higher concentration of injected holes than the other LEDs, especially in the first four QWs. These results arise mainly from the fact that a large electrostatic field is induced in the right side of the QW by the incorporation of the high indium layer, which, in turn, causes the injected holes to be transported across the active region. Notably, as previously discussion with reference to Fig.4 (b), the distributions of the electron concentration in active regions in all LEDs are almost identical. Figure 6 plots radiative (red dashed line) and Auger (blue line) recombination rates in the active regions in all LEDs at I = 200 mA. The radiative recombination is dominated by the last QW in every LED. However, the first four QWs of LED D can also participate in the radiative recombination process. Furthermore, as the accumulation of injected holes in the last QW of LED D is reduced, the Auger recombination process and the efficiency-droop effect are quenched. In conclusion, when the high indium layer was appropriately placed in the InGaN staggered QW, the transport efficiency of injected holes was increased, because the localized electrostatic-field with high intensity pushed injected holes across the active region, increasing the rate of radiative recombination. Most importantly, the accumulation of injected holes in the last QW was correspondingly reduced, reducing the Auger recombination rate and the efficiency-droop in the LED. Y.-J. Lee gratefully acknowledge the financial support from the National Science Council of Republic of China (ROC) in Taiwan under contract No. NSC–100–2112–M–003–006–MY3 and from the National Taiwan Normal University under ...