## 4 Discussion  
In this section, we quantify the ratio of the effective active volume as a result of the carrier distribution in Figure 6 and compare it with the result of rate equation modeling in Figure 4. The effective width of the radiative recombination \(d_{\text{eff}}\) is determined as the spatially integrated radiative recombination rate divided by half the peak radiative recombination rate:  
\[d_{\text{eff}}\ =\ \frac{\int R_{\text{rad}}(y)dy}{P_{\text{rad}}(\text{peak})/2} \tag{4}\]  
where \(R_{\text{rad}}(y)\) is the radiative recombination rate distribution and \(y\) is the vertical position across the QW layer. Using Equation (4), \(d_{\text{eff}}\) of the QW without an electric field, that of the QW on silicon, and that of the QW on sapphire were calculated to be 3.09, 2.82, and 2.45 nm, respectively. That is, \(d_{\text{eff}}\) of the QW without electric field was slightly larger than the physical width of the QW. \(d_{\text{eff}}\) values for silicon and sapphire are smaller than the QW width, and \(d_{\text{eff}}\) of the QW on silicon is 1.15 times larger than that on sapphire. The effective active volume of the QWs on silicon was estimated to be larger than that of the QWs on sapphire by \(\sim\)1.45 times in Figure 4b; therefore, the reduced radiative recombination width due to the separation of electrons and holes could only partly account for the reduced effective active volume. The remaining of the reduced effective active volume could be explained by the inhomogeneous distribution of carriers between QWs. To quantify the effective active volume in the MQW region, we define the effective number of QWs as  
\[N_{\text{eff}}\ =\ \frac{\sum\limits_{i\ =\ 1}^{N}\int R_{\text{rad}}^{i}(y)dy}{ \int R_{\text{rad}}^{1}(y)dy} \tag{5}\]  
where \(R_{\text{rad}}^{i}(y)\) is the radiative recombination rate distribution in the \(i\)-th QW and \(N\) is the number of QWs which is 5 in our LED samples. \(R_{\text{rad}}^{1}(y)\) is the radiative recombination rate distribution in the p-side QW that has the highest recombination rate between the QWs. For MQWs with perfectly uniform radiative recombination distribution, \(N_{\text{eff}}\) is 5. If the radiative recombination occurs only at the p-side QW, \(N_{\text{eff}}\) is 1. That is, \(N_{\text{eff}}\) varies from 1 to 5 depending on the radiative recombination distribution in the MQWs. \(N_{\text{eff}}\) was calculated to be 3.51 for the MQW without an internal field, 1.92 for the MQW on silicon, and 1.45 for the MQW on sapphire. The ratio of \(N_{\text{eff}}\) for the MQW on silicon and sapphire is 1.32, implying that the effective active MQW width of the LED on silicon is 1.32 times larger than that of the LED on sapphire. Recalling that the ratio of the effective active volume is \(\sim\)1.45, as shown in Figure 4, and the ratio of the effective QW width (\(d_{\text{eff}}\)) is 1.15, as shown in Figure 6a, the ratio of \(N_{\text{eff}}\) could explain the remaining \(\sim\)30% increase in the effective active volume for the LED on silicon. That is, the reduced effective active volume could be attributed to the combined effect of both the reduced active width of a single QW and the reduced effective number of QWs in MQW layers as a result of the internal electric field. In addition to the internal electric field, the fluctuation of the indium composition inside InGaN QWs may significantly influence the reduction in the effective active volume. However, we believe that the degree of indium fluctuation between the MQWs on silicon and sapphire is not much different, because the structures of MQWs on silicon and sapphire are identical and both MQW layers were grown under the same growth conditions. Therefore, the difference in the effective active volume of MQWs on silicon and sapphire is mainly attributed to the large difference in internal electric fields, which leads to the reduction in efficiency droop for Sample B.