## 2 Epitaxial Growth on InP-on-Insulator Substrate
### Assessment of Thermal Strain at Epitaxial Growth Temperature
The epitaxial InP wafer and a 2-inch Si (001) wafer with a 2-\(\upmu\)m-thick thermal silicon dioxide (SiO\({}_{2}\)) layer (SiO\({}_{2}\)/Si substrate) were treated with oxygen plasma. This process grew the ultra-thin oxide layer, which makes the bonding surface very smooth and hydrophilic [58]. The wafers were then directly bonded at RT under atmospheric pressure. After that, post-annealing was carried out at 200 \({}^{\circ}\)C while pressure from 1.0 to 6.1 MPa was applied. To leave the InP-based membrane layers on the SiO\({}_{2}\) layer, the InP substrate side was mechanically polished and chemically etched using the InGaAs etch-stop layer [59]. Finally, the InGaAs etch-stop layer was chemically etched to obtain a 250-nm-thick MQW-DH on the SiO\({}_{2}\) layer. Figure 3b shows a typical PL intensity maps for the 250-nm-thick MQW-DH on the SiO\({}_{2}\)/Si substrate at 25 \({}^{\circ}\)C, measured with the excitation laser irradiation at a wavelength of 980 nm. The PL intensity distribution was very uniform (standard deviation: 1.31%), excluding the pe  
Figure 2: Strain dependence on calculated critical thickness of indium phosphide (InP) directly bonded on Si. Next, we prepared three MQW-DHs on SiO\({}_{2}\)/Si substrate with various III-V layer thicknesses and characterized their PL before and after 610 \({}^{\circ}\)C annealing (epitaxial regrowth temperature). The three epitaxial wafers included almost the same MQWs, but the total III-V layer thicknesses were varied from 250 to 850 nm. We annealed these wafers in an MOVPE reactor for 30 min under the PH\({}_{3}\)-atmosphere epitaxial-regrowth condition. The temperature was 610 \({}^{\circ}\)C and the pressure was 4.0 kPa. Figure 4a-c compares the PL spectra before (solid lines) and after (dashed lines) the MOVPE annealing for the samples with three different thicknesses, respectively. The median PL intensity for the wafer with a total thickness of 250 nm degraded only 8% after annealing. In contrast, for samples with total thicknesses of 500 and 850 nm which are thicker than the critical thicknesses calculated with the two models, degradation increased to 31 and 68%, respectively. To evaluate the degradation in the epitaxial wafers, we also evaluated the PL-intensity map with a PL microscope that can visualize dislocations and clocks. Figure 4d-f compares the PL-intensity distributions measured with PL microscope for these three samples. They clearly show that there were neither dark line nor dark spots in the 100 \(\times\) 100 \(\mu\)m\({}^{2}\) area for the 250-nm-thick sample, whereas dark lines were observed for the 500- and 850-nm-thick samples [46]. We also confirmed that a dark line did not appear when the total III-V thickness was 350 nm [60]. These results clearly show that the actual critical thickness for an InP-based membrane layer on a Si substrate at 610 \({}^{\circ}\)C should be between 350 and 500 nm; therefore, epitaxial regrowth using an InP-on-insulator substrate should not induce dislocation when the total III-V thickness is less than 350 nm. Figure 3: (**a**) Process flow of fabricating a multi-quantum well (MQW)-double-heterostructure (DH) on SiO\({}_{2}\)/Si substrate; (**b**) photoluminescence (PL) intensity map for 2-inch 250-nm-thick MQW-DH transferred to SiO\({}_{2}\)/Si substrate.