## 1 Introduction  
Internal quantum efficiency (IQE) in InGaN light-emitting diodes (LEDs) suffers a substantial decrease with increasing driven currents, leading to the so-called efficiency droop effect [1]. Until now, many different theories have been put forward to explain the origin of this experimental observation, such as Auger recombination [2; 3; 4], electron leakage [5; 6], lack of hole injection [7; 8; 9], and carrier delocalization [10; 11]. In addition, the efficiency droop in long wavelength InGaN LEDs becomes more severe as compared to the blue and the near-ultraviolet (NUV) LEDs [11]. Direct measurement of Auger electrons and clear correlation between the Auger current and droop suggest that the Auger process is the main cause of efficiency droop [4]. The conventional ABC model is widely employed to study carrier recombination dynamics in InGaN quantum wells (QWs). In this model, the efficiency equation can be expressed as \(\text{IQE}=\text{B}n^{2}/(\text{A}n+\text{B}n^{2}+\text{C}n^{3})\), where A, B, and C represent Shockley-Read-Hall (SRH), radiative recombination, and Auger coefficients respectively [12; 13; 14; 15]. The ABC model can explain non-thermal droop in InGaN-based LED structures in terms of Auger recombination, which agrees well with available observations [2; 13; 14]. However, the ABC model is criticized for oversimplified treatment of physical processes and some disagreements with a number of observations at high injection current densities [12; 15].  
IQE can be expressed in a first-order approximation as [15; 16]:  
\[\text{IQE}=V_{\text{active}}\text{B}n^{2}/(I/q) \tag{1}\]  
where \(V_{\text{active}}\) is physical volume of active region, B is the radiative recombination coefficient, \(n\) is carrier density within active region, \(I\) is current injected into active region, and \(q\) is elementary charge. In many studies carriers are assumed to be uniformly distributed within InGaN QW [12; 13; 14]. Nevertheless, carriers are actually localized in the potential minima of In-rich InGaN clusters [12; 17; 18; 19; 20], leading to a smaller effective active region volume (\(V_{\text{effective}}\)) than the physical \(V_{\text{active}}\). It has been estimated that the effective QW thickness of InGaN blue LEDs would be reduced to only \(\sim\)1.0 nm under a physical well thickness of 2.5-3.0 nm [17; 21; 22]. Consequently, carrier density in the numerical calculation is much smaller than real values, resulting in extracted coefficients of \(n^{3}\) (\(\sim\)10\({}^{-9}\) cm\({}^{6}\) s\({}^{-1}\)) [12], one order larger than reported Auger coefficients (\(\sim\)10\({}^{-30}\) cm\({}^{6}\) s\({}^{-1}\)) from the photoluminescence measurement [1] and the first principle calculation [2]. In our previous study, an effective volume ratio \(\beta\) was introduced as \(\beta=V_{\text{effective}}/V_{\text{active}}\) in the efficiency equation function to describe the effect of reduced effective active volume on the IQE droop [15]. In this study we present a comprehensive study of wavelength-dependent efficiency droop in InGaN LEDs with emission wavelengths centered from 400 nm to 525 nm by a reduced effective volume of the active region.