## 3 Results  
### Rate Equation Modeling and Effective Active Volume  
In the \(ABC\) rate equation model, the relation between the current, \(I\) and carrier density, \(N\) can be written as [15; 16; 17; 18]  
\[I\ =\ qV_{\text{eff}}[AN+BN^{2}+CN^{3}], \tag{1}\]  
where \(q\) is the elementary charge, and \(V_{\text{eff}}\) is the effective active volume of QWs. The parameters, \(A\), \(B\), and \(C\) represent the coefficient of Shockley-Read-Hall (SRH) recombination, bimolecular radiative recombination, and Auger recombination, respectively. \(V_{\text{eff}}\) is the active volume that actually contributes to the light emission in QWs, which is believed to be smaller than the physical volume of QWs, \(V_{0}\). Here, we introduce a scaling parameter, \(k\), which represents the ratio of the effective active volume (\(V_{\text{eff}}\)) to the physical active volume (\(V_{0}\)), is defined as  
\[k\equiv V_{\text{eff}}/V_{0} \tag{2}\]  
The IQE of LEDs is expressed as  
\[\eta\ =\ \frac{BN^{2}}{AN+BN^{2}+CN^{3}} \tag{3}\]  
Using Equations (1)-(3), the IQE versus current relation (IQE curve) can be obtained. We perform the fitting of the measured EQE curves of the LEDs on silicon and sapphire using Equations (1)-(3). According to References [25; 26], for a given EQE versus current data, the IQE fit curve could be uniquely determined for any set of coefficients \(A\), \(B\), \(C\), and \(k\) parameter. That is, the IQE fit curves are identical and hence the same peak IQE is obtained irrespective of the choice of \(A\), \(B\), \(C\), and \(k\) parameters. Therefore, the uncertainty of the fit curves and IQE values for different sets of fit parameters could be negligible. Here, \(k\) and \(A\) are used as fitting parameters for given \(B\) and \(C\) values. The \(B\) coefficient is fixed at \(1\times 10^{-11}\) cm\({}^{3}\)/s, while the \(C\) coefficient is varied from \(1\times 10^{-31}\) to \(1\times 10^{-30}\) cm\({}^{6}\)/s considering the uncertainty of the reported \(C\) coefficient values [27; 28; 29; 30; 31]. Because the LEDs on silicon and sapphire have the same active layer structures and indium composition in the QWs, the \(B\) and \(C\) coefficients are believed to be the same. However, \(k\) and \(A\) can be different because of the difference in the crystal quality and effective active volume in MQWs. Figure 3 shows semi-log plots of the IQE fitting curves for LEDs on silicon and sapphire when \(C\) is \(5\times 10^{-31}\) cm\({}^{6}\)/s. The IQE curves for both samples fit reasonably well with the measured EQE data. The deviation of measured data and fit curves at low currents (\(<\)10 mA) are due to the measurement  
Figure 2: External quantum efficiency (EQE) versus current relation of the LED on sapphire (Sample A, blue line) and silicon (Sample B, red line) substrates. uncertainty of low light output power. The nearly perfect agreement at high currents (\(>\)100 mA) implies that the leakage current can be negligible in our samples. The best fits were obtained when \(k\) and \(A\) were 0.085 and \(6.0\times 10^{6}\) s\({}^{-1}\), respectively, for Sample A and 0.125 and \(1.15\times 10^{7}\) s\({}^{-1}\), respectively, for Sample B. The higher value of the coefficient \(A\) for Sample B indicates that it had a higher nonradiative recombination rate owing to the poorer crystal quality. The peak IQEs were obtained to be 0.743 and 0.676 for Sample A and Sample B, respectively. By comparing the measured EQE and the fitted IQE values, the light extraction efficiencies (LEEs) were obtained to be 0.787 and 0.779 for the LED on sapphire and silicon, respectively. The almost equal LEE values are reasonable considering that the chip and package structures of the two LED samples are identical. Because the \(k\) value obtained from the IQE fitting may vary for different \(C\) coefficients, we performed the fitting with \(C\) varying from \(1\times 10^{-31}\) to \(1\times 10^{-30}\) cm\({}^{6}\)/s. For all of the \(C\) coefficient values, almost the same fitting results as in Figure 3 were obtained.