## 1 Introduction  
Over the last two decades, there has been remarkable progress in the development of GaN-based light-emitting diodes (LEDs). Owing to its high efficiency and eco-friendliness, LED-based solid-state lighting has been rapidly replacing conventional light bulbs in general lighting and display applications [1; 2; 3; 4; 5; 6]. However, the high cost of LED illumination sources is still a hindering factor for increasing their market penetration rate in general lighting applications. One prominent way of lowering the production cost of LEDs is to utilize silicon-based technologies. The growth of InGaN/GaN layers on silicon substrates has attracted great interest because of the low substrate cost and the availability of large size wafers. High-performance GaN-based blue LEDs on silicon have been achieved by overcoming the technological difficulty in their growth caused by the large mismatch in the lattice constant and the thermal expansion coefficients between silicon and GaN [7; 8; 9; 10; 11]. Recently, one of the authors of this paper reported the experimental results of the comparison between the device characteristics of InGaN/GaN blue LEDs grown on silicon and sapphire substrates [12]. The efficiency droop of the LED sample on silicon substrates was found to be significantly lower than that of the identically grown LED on sapphire substrates. The reduced efficiency droop of the LED on silicon was attributed to its lower internal electric field [12]. It was shown that internal electric fields of InGaN multiple-quantum-wells (MQWs) on silicon was reduced by ~40% compared to that of the MQWs on sapphire. The polarization-induced internal electric field has been known to increase the electron leakage from MQWs to p-GaN, which could aggravate the efficiency droop problem [13; 14; 15]. However, evidence of the polarization-enhanced electron leakage was not observed clearly by either experiments or simulations. More specifically, the underlying mechanism for the reduced efficiency droop of the LED on silicon was not clearly identified although, in a previous study, the efficiency droop was shown to be caused by the internal electric field [12]. It is worth mentioning that as the internal field strength increases, the separation of electrons and holes in a quantum well (QW) increases and the carrier distribution in MQWs becomes increasingly inhomogeneous. This results in the reduction of the effective active volume, which contributes to the radiative recombination of electrons and holes. It has been reported that a reduction in the effective active volume of InGaN MQWs leads to a substantial increase in the efficiency droop [16; 17; 18; 19; 20]. Therefore, the influence of the internal electric field on the efficiency droop is expected to be explained by the reduced effective active volume due to the increasing internal field strength. In this paper, we use the rate equation model with the reduced effective active region to interpret the difference in the efficiency droops of the LED structures grown on silicon and sapphire substrates. The ratio of the effective active volume to the physical volume of QWs is determined by fitting the measured external quantum efficiency (EQE) curve of LEDs on silicon and sapphire with the rate equation model. Then, the differences in the degree of efficiency droop and the strength of internal electric fields between LEDs on silicon and sapphire are interpreted by the difference in the effective active volume of InGaN MQWs using modeling simulations. It is demonstrated that, as the strength of the internal electric fields increases, the effective active volume reduces and the efficiency droop increases, revealing the reason for the reduced efficiency droop of LEDs on silicon compared to the LED on sapphire. In Section 2 of this paper, the growth, fabrication, and characterization of InGaN/GaN LED structures on silicon and sapphire substrates are described.