## 2 Epitaxial Growth on InP-on-Insulator Substrate
### Assessment of Thermal Strain at Epitaxial Growth Temperature  
To grow a high-quality epitaxial layer using a III-V/Si integrated substrate, the total III-V layer thickness is critical [46]. This subsection describes the calculation model of the critical thickness of the InP layer on an InP-on-insulator substrate. Since the process temperature of metalorganic vapor phase epitaxy (MOVPE) for InP-based materials is about 600 \({}^{\circ}\)C, the difference of the CTEs between InP-based materials and SiO\({}_{2}\)/Si causes thermal strain. Thermal strain \(\varepsilon\) is given by the product of the difference between the CTEs \(\Delta\alpha\) and temperature change \(\Delta T\). The CTEs of Si and InP at 25 \({}^{\circ}\)C are 2.62 and 4.56 ppm/K, respectively [55]. For simplification, we disregard the SiO\({}_{2}\) layer because it is much thinner (typically 2 \(\upmu\)m) than the Si substrate. Since we apply pressure to the bonded wafer at 200 \({}^{\circ}\)C, we assume that the directly bonded InP membrane layer is free standing at 200 \({}^{\circ}\)C. On the basis of the above, epitaxial growth temperature (600 \({}^{\circ}\)C) induces a compressive strain of approximately 780 ppm to the InP membrane layer on Si. Next, the critical thickness was calculated based on the thermal strain applied at epitaxial growth temperature. When the total InP-based layer is thicker than the critical thickness, a dislocation should be generated in both the original InP membrane and regrown layer, resulting in degradation of photoluminescence (PL) and electroluminescence characteristics. There are two types of models to calculate critical thickness, mechanical equilibrium models and strain-energy-density equilibrium models. The thickness at which the dislocations appear in the actual strained crystal tends to be between the critical thicknesses calculated using these models [56]. Figure 2 compares the critical thickness versus strain calculated using one of each type of calculation model. The mechanical equilibrium model expresses the critical thickness \(h_{\rm c}\) as follows [57]:  
\[h_{\rm c}=\frac{b(1-\
u/4)}{4\pi f(1+\
u)}\left(\ln\frac{h_{\rm c}}{b}+1\right), \tag{1}\]  
where \(\
u\) is [PERSON]\({}^{\prime}\)s ratio for strained material, \(b\) is [PERSON]\({}^{\prime}\)s vector of dislocation, and \(f\) is strain. On the other hand, the strain-energy-density equilibrium model expresses the critical thickness as follows [54]:  
\[h_{\rm c}=\frac{\sqrt{6}b(2-\
u)}{8\sqrt{3/2}\pi f(1+\
u)}\left\{\ln\left( \frac{8\sqrt{3/2}ah_{\rm c}}{b}\right)-2\right\}, \tag{2}\]  
where \(a\) is a parameter that is related to the dislocation energy (typically \(a\sim 4\)). On the basis of the two models, the critical thickness of InP on a SiO\({}_{2}\)/Si substrate is calculated to  
Figure 1: Process flow of fabricating membrane photonic devices on Si using epitaxial growth technique on a III-V-on-insulator platform. be between approximately 200 nm (mechanical equilibrium model) and 430 nm (energy equilibrium model) for a thermal strain of about 800 ppm. Therefore, high-temperature MOVPE growth should not induce the formation of dislocation even though the III-V/Si hybrid substrate is used, unless the InP membrane thickness after the growth exceeds 200 or 430 nm. To experimentally evaluate the critical thickness, we fabricated III-V membranes that are bonded to SiO\({}_{2}\)/Si substrates [46]. Figure 3a shows the process flow. The fabrication began with the epitaxial growth of an InP buffer layer, InGaAs etch-stop layer (100 nm), and MQW double-heterostructure (DH) sandwiched between the InP layer on a 2-inch standard InP (001) substrate by MOVPE. The growth was undertaken at 630 \({}^{\circ}\)C and 6.7 kPa, and the precursors for In, Ga, As and P were trimethyl-indium (TMI), triethyl-gallium (TEG), arsine (AsH\({}_{3}\)) and phosphine (PH\({}_{3}\)), respectively. The DH included six-period strained InGaAsP MQWs (PL peak wavelength: 1520 nm).