## 4 Results and Discussion  
Measurement results of IQE versus injection current density for the three InGaN LEDs are plotted as dots in Figure 1a-c, respectively. Green LEDs present a more severe IQE droop, as well as a lower onset-droop current density, as compared with NUV and blue LEDs. By fitting experimental data through Equation (3), the values of parameters are extracted, as shown in Figure 2. Without introducing a large value of Auger coefficient, accurate fitting results can be achieved. Extracted Auger coefficients for these three LEDs are in the range of 0.8-2 \(\times\) 10\({}^{-30}\) cm\({}^{6}\) s\({}^{-1}\), which agree well with expectations from theoretical calculations [2; 25] and experimental measurements [1]. Figure 1: Fitting results to experimental internal quantum efficiency (IQE) versus current for InGaN (**a**) near-ultraviolet (NUV), (**b**) blue, and (**c**) green LEDs. Droop ratio is defined as (Max_IQE-IQE)/Max_IQE, where Max_IQE represents peak IQE. The QW In composition-dependent onset-droop current density and droop ratio at 100 A/cm\({}^{2}\) are shown in Figure 2. With increasing emission wavelength, the onset-droop shifts towards a lower current density. Meanwhile, the droop ratio increases. As the QW indium content increases from 6 to 30%, onset-droop current density reduces from 20 to 1.8 A/cm\({}^{2}\), while the droop ratio rises from 10 to 60%. Wavelength-dependent efficiency droop behavior has previously been investigated in literature [11; 18; 26]. The Auger recombination, piezoelectric field, and carrier leakage in InGaN green LEDs are claimed as the main mechanisms for such droop behavior. In this study, the wavelength-dependent efficiency droop behavior can be well explained by a reduced effective active region volume. Figure 3a,b presents QW In content-dependent \(\beta\) and carrier density at various current densities. It is worth pointing out that \(\beta\) decreases nearly linearly with QW In content. In addition, effective thicknesses for the three InGaN LEDs are noted in the figures. Under a physical well thickness of 3.0 nm, the effective thickness for NUV, blue, and green LEDs is estimated to be only 2.07, 1.29, and 0.24 nm respectively. Our extracted effective thickness of InGaN blue LEDs is consistent with a value of 1.0 nm, as reported by [PERSON] et al., suggesting accuracy in our calculation [1]. Such a significantly reduced effective active region volume is ascribed to inhomogeneous carrier distributions in localized centers, as well as less overlap of electron-hole wave-functions by large piezoelectric fields. The shrink in effective volume of InGaN QW can be evidenced by the experimental confocal micro-photoluminescence [18; 20], near-field scanning optical microscopy [17; 27], and atom probe tomography [28]. In addition, spatial fluctuations in indium composition have been theoretically modeled to investigate the carrier localization effect [18] and device performance [19]. Moreover, carrier density increases with QW In composition at a given current density, opposite to the trend of \(\beta\). For InGaN LEDs with a long emission wavelength, carriers accumulate strongly in the deeply In-rich InGaN clusters by a high degree of localization effect [20; 29], leading to a further reduced effective active region volume as well as a larger carrier density. Figure 2: QW In composition-dependent onset-droop current density and droop ratio at 100 A/cm\({}^{2}\). Figure 4a-c presents the peak wavelength blue-shift and full width at half maximum (FWHM) broadening of emission wavelength with injection current density for the three LEDs. The FWHM broadens from 10 to 30 nm as the emission wavelength increases from 400 to 525 nm, with a current density varied from 1 to 100 A/cm\({}^{2}\). The wider FWHM for InGaN LEDs with a long emission wavelength is attributed to stronger spatial fluctuations of indium composition within the InGaN QW.