## 3 Results and Discussion
### Distribution of Residual Stress with Depth  
The residual stress inside the samples was analyzed, using MRS, which was regarded as an effective method for residual stress analysis of any semiconductor material, such as silicon. The basic Raman-mechanical model for crystal silicon (C-Si) follows the secular equation of the lattice dynamics [21; 27]. According to existing investigations, the Raman-mechanical relationship of C-Si depends on the crystal plane and the stress state for measurement. In this work, the UV Raman tests were performed on the surface of the strained-silicon film, whose crystal plane was (100). Figure 9a shows a typical spectrum from the UV Raman tests. The 325-nm UV laser can only penetrate the crystal silicon to a depth of \(\sim\)10 nm, which is similar to the thickness of the strained silicon. Hence, the wavenumber detected by the 325-nm laser achieves a mean wavenumber of 511.98 cm\({}^{-1}\) based on results from 66 different sampling spots, which originated solely from the \(\epsilon\)-Si layer and were characterized by the average strain and stress of this layer. The residual stress in the \(\epsilon\)-Si layer can be treated as a biaxial stress state, with stress components \(\sigma_{x}=\sigma_{y}=\sigma\) and strain components \(\varepsilon_{x}=\varepsilon_{y}=\varepsilon\), where \(x\) and \(y\) denote the in-plane directions. According to the Raman-mechanical relationship of C-Si with the (100) crystal plane [28],  
\[ \begin{array}{l}\varepsilon=-1.204\times 10^{-3}\Delta w\\ \sigma=-217.28\Delta w\ (\text{MPa})\end{array} . \tag{1}\]  
According to Equation (1), \(\varepsilon=-1.204\times 10^{-3}\times(511.98-519.78)=9.39\%\epsilon\approx 1\%\) and \(\sigma=-217.28\times(511.98-519.78)\approx 1.69\) GPa, where wavenumber 519.78 cm\({}^{-1}\) was calibrated on a silicon wafer of the same production batch with the \(\epsilon\)-Si/GeSi/C-Si samples and on the same day of the measurements above. The results showed that the real in-plane strain in the \(\epsilon\)-Si layer was \(\sim\)1%. For the Raman measurement on the cross-section with a (110) crystal plane, the applicable Raman-mechanical relationship of (110) C-Si was given as Equation (2), where the residual stress is treated as a uniaxial stress state, because the normal component on the cross-section surface (denoted as \(\sigma_{x}\)) was released during the sample preparation, \(\sigma_{x}=0\), \(\sigma_{y}=\sigma\) and strain components \(\varepsilon_{x}=\varepsilon_{y}=\varepsilon\), where \(x\) and \(y\) denote the in-plane directions [28]. \[ \begin{array}{l}\varepsilon=-2.58\times 10^{-3}\Delta w\\ \sigma=-434.58\Delta w\ (\text{MPa})\end{array} . \tag{2}\]  
The cross-sectional stress analyses for the Raman data from the sampling spot inside the C-Si layer can use the Raman-mechanical relationship shown as Equation (2), whereas those from the GeSi  
Figure 9: Raman spectra of (**a**) strained silicon (surface) and (**b**) germanium-silicon (cross-section). layers cannot use Equation (2) directly. Figure 9b shows a typical Raman spectrum of the germanium silicon alloy, which includes three major peaks: A Si-Si band at \(\sim\)500 cm\({}^{-1}\), a Ge-Si band at \(\sim\)400 cm\({}^{-1}\) and a Ge-Ge band at \(\sim\)300 cm\({}^{-1}\). If elemental Si dominated the Ge-Si alloy, just like the buffer layers in this work, the Raman peak of the Si-Si band would be strong and sufficiently narrow (shown in Figures 5 and 6) for mechanical measurement. However, Equation (2) is not applicable directly for the Si-Si band of germanium silicon alloy. As shown in Figure 4, the Raman wavenumber decreases sharply inside the GeSi layers, whose increment is up to 14 cm\({}^{-1}\), denoting more than 6 GPa by using Equation (2). This is inconsistent with the reality of C-Si or GeSi strength. In fact, due to the influence of phonon confinement, the change in the wavenumber of the Si-Si band in the Raman spectrum of germanium silicon is not only induced by the strain, but also by the influence of phonon confinement.