## 3 Results
### Simulation of Carrier and Recombination Rate Distribution  
The difference in the effective active volume between the two samples could be related to their difference in the internal electric field of the QWs. In the previous study, based on reverse-biased electro-reflectance measurements, the internal electric fields of the InGaN/GaN QWs on sapphire and silicon substrates were determined to be 1.50 MV/cm and 0.92 MW/cm, respectively [12]. Figure 5 shows the simulated energy band diagram and electron/hole concentration distribution in a QW corresponding to the QW of the sapphire and silicon substrates. For the simulation, we employed the simulation software, APSYS [32]. APSYS solves the QW band structures, the drift, and diffusion equations, and the carrier recombination rates self-consistently, and it has been widely employed for simulating the device characteristics of LED structures. The internal electric field in the QW separates the spatial distribution of electron and hole from each other. Therefore, electrons and holes are more separated in the QW on sapphire substrates, which have larger internal electric fields, as indicated in Figure 5. The differences of the peak carrier concentration between electrons and holes are 1.98 nm for the QW on sapphire and 1.40 nm for the QW on silicon.  
The larger is the separation of the electrons and holes; the smaller is the overlapped region of the electron and hole concentration. This means that the carrier recombination region can be reduced as the internal electric field increases. Figure 6a shows the simulated distribution of radiative recombination rates for the QW on silicon and sapphire along with the recombination rate of the QW without a polarization-induced electric field. The radiative recombination rate was also simulated using the APSYS program. In Figure 6a, the calculated radiative recombination rate was normalized to the peak for each case. With increasing internal electric field, the peak position shifts, and the recombination distribution narrows.  
Figure 6b depicts the distribution of the radiative recombination rate in five QWs for three cases of the internal electric field: without polarization, MQWs on silicon, and MQWs on sapphire. As the internal electric field increases, the radiative recombination rate in the MQWs becomes increasingly inhomogeneous, which results mainly from the restriction of hole transport from the p-side to the n-side QWs with increasing internal electric field [33; 34; 35]. For the QW without the internal electric field, the radiative recombination rate is reasonably uniform for five QWs. By contrast, for the QW on sapphire, the radiative recombination mostly occurs at the QW adjacent to the p-side layers. For the QW on silicon, although the radiative recombination rate decreases significantly from the p-side to the n-side QWs, the radiative recombination distribution of the MQWs on silicon is more homogeneous than that of the MQWs on sapphire. Therefore, the inhomogeneous recombination distribution in the MQWs could also partly explain the larger effective active volume of Sample B compared with that of Sample A.  
Figure 5: Simulated energy band diagram and electron/hole concentration profile in an InGaN/GaN QW. (a) QW on sapphire substrates with internal electric field of 1.50 MV/cm. (b) QW on silicon substrates with an internal electric field of 0.92 MV/cm. The differences in the position of the peak carrier concentration are indicated as dotted arrows.  
Figure 6: (a) Normalized radiative recombination rate profile for a QW without an internal electric field (black line), a QW on silicon (red line), and a QW on sapphire (blue line). (b) Radiative recombination rate profile for InGaN/GaN MQWs without an internal electric field (black line), MQWs on silicon (red line), and MQWs on sapphire (blue line).