## 3 Results and Discussion
### Microscale Structural Properties
The sudden drop of the Si component at a 1 \(\upmu\)m depth may have been caused by parameter alternation during the ultra-high-vacuum chemical-vapor epitaxy, which helps to locate the Ge\({}_{\mathrm{x}}\)Si\({}_{\mathrm{1-x}}\)/Ge\({}_{\mathrm{0.2}}\)Si\({}_{\mathrm{0.8}}\) interface. Both elements decreased to zero because the EDS sampling point entered the epoxy resin layer of the sample cross-section. The EDS data did not show the signal from the \(\varepsilon\)-Si layer, which should be because the \(\varepsilon\)-Si layer thickness (\(\sim\)10 nm) was less than the EDS sampling step length. Figure 4a shows the wavenumber image of the Si-Si band (\(\sim\)520 cm\({}^{-1}\)) that was obtained by Raman mapping at a 25 \(\upmu\)m \(\times\) 24 \(\upmu\)m region near the multilayer interfaces on the sample cross-section. The step lengths for Raman mapping along the parallel and vertical direction to the interfaces were 1.2 \(\upmu\)m and 0.2 \(\upmu\)m, respectively. Figure 4b shows the mean values of the wavenumbers at different locations along the depth direction based on Figure 4a. Figure 4a,b shows that the wavenumber at \(\sim\)520 cm\({}^{-1}\) changes slightly inside the Si substrate, acutely in the buffer layers, and continuously at the interfaces. When the scanning spot originated from the Si substrate into the buffer layers, the wavenumber showed a sharp and linear decrease with a speed of \(-\)4.97 \(\pm\) 0.07 cm\({}^{-1}\)/\(\upmu\)m. However, this rapidly decreasing trend stopped within a region of \(\sim\)1 \(\upmu\)m next to the epoxy layer (compared with the top surface of the strained-silicon wafer), where the wavenumber decreased gently with a speed of \(-\)0.83 \(\pm\) 0.28 cm\({}^{-1}\)/\(\upmu\)m. Hence, from the wavenumber image, the thickness of the rapidly decreased wavenumber region is \(\sim\)2.6-2.8 \(\upmu\)m, and that of the gently decreased wavenumber region is \(\sim\)0.8-1 \(\upmu\)m. The former region is the Ge\({}_{\mathrm{x}}\)Si\({}_{\mathrm{1-x}}\) layer and the latter is the Ge\({}_{\mathrm{0.2}}\)Si\({}_{\mathrm{0.8}}\) layer. The strain distribution can be achieved by using the wavenumber results, as discussed below. Figure 5a shows the Raman intensity image of the same region in Figure 4a. Figure 5b shows the mean values at different locations along the depth direction based on Figure 5a. In Figure 5, when the sampling spot for Raman mapping from the Si substrate crossed the buffer layer, the Raman intensity of the Si-Si band decreased first and then increased. The Raman-intensity-decreased region in Figure 5 is almost the same as that of the wavenumber rapid-decreased region in Figure 4, so the Raman intensity-increased region to the wavenumber gently-decreased region. This result indicates that the quantity of elemental Si in the monocrystalline stage decreased with a continuous enhancement of elemental Ge in the graded-buffer (Ge\({}_{\mathrm{x}}\)Si\({}_{\mathrm{1-x}}\)) layer, and then improved gradually  
Figure 4: (a) Si-Si band wavenumber image obtained by Raman mapping on the sample cross-section, (b) wavenumber distribution along the depth direction. inside the relaxed-buffer (Ge\({}_{0.2}\)Si\({}_{0.8}\)) layer even though the elemental content of the materials did not change any more. Figure 6a shows the full-width-at-half-maximum (FWHM) image of the same region of Figure 4a. Figure 6b gives the mean values at different locations along the depth direction based on Figure 6a. In Figure 6, the FWHM of the Si-Si band broadened first and then narrowed when the sampling spot of Raman mapping from the Si substrate crossed the buffer layer. The FWHM broadened region in Figure 6 is different from (smaller than) the wavenumber rapid-decreased region in Figure 4 and the Raman-intensity-decreased region in Figure 5. Therefore, with a continuous enhancement of elemental Ge, the average lattice quality of the Si element in the monocrystalline stage in the graded-buffer (Ge\({}_{\mathrm{x}}\)Si\({}_{1-\mathrm{x}}\)) layer decreased to an extreme and then improved continuously.