# Thermomechanical Stresses in Silicon Chips for
## 2 Materials and Methods
### Determination of Stress from Raman Spectra in Si  
The frequencies of the Raman modes are affected by thermomechanical strains or stresses. Raman measurements are used to quantify the local residual stress of Si, using the secular equation of Equation (1):  
\[\begin{bmatrix}p\epsilon_{11}+q(\epsilon_{22}+\epsilon_{33})-\lambda&2r \epsilon_{12}&2r\epsilon_{13}\\ 2r\epsilon_{12}&p\epsilon_{22}+q(\epsilon_{33}+\epsilon_{11})-\lambda&2r \epsilon_{23}\\ 2r\epsilon_{13}&2r\epsilon_{23}&p\epsilon_{33}+q(\epsilon_{11}+\epsilon_{22}) -\lambda\end{bmatrix}=0 \tag{1}\]  
where \(p\), \(q\) and \(r\) represent the deformation potentials, which are material constants, and \(\epsilon_{ij}\) are the strain phonon tensor components along each direction [23, 24], related to the strain tensor (\(\sigma\)) through the Hooke's law \(\epsilon_{ij}=S_{ij}\sigma\). \(S_{ij}\) are the elastic compliance tensor elements of silicon, reported in Table 2.  
From the eigenvalues \(\lambda_{j}\) (\(j\) = 1,2,3), it is possible to calculate the difference in Raman frequencies for each mode in the presence (\(\omega_{j}\)) and in the absence (\(\omega_{j0}\)) of stress: \(\lambda_{j}=\omega_{j}^{2}-\omega_{j0}^{2}\) or  
\[\Delta\omega_{j}=\omega_{j}-\omega_{0}\approx\frac{\lambda_{j}}{2\omega_{j0}} \tag{2}\]  
\begin{table}
\begin{tabular}{c c c c} \hline \(\mathbf{S_{11}}\) [\(\mathbf{\mathrm{Pa}}^{-1}\)] & \(\mathbf{S_{12}}\)[\(\mathbf{\mathrm{Pa}}^{-1}\)] & \(\mathbf{p}\)[\(s^{-2}\)] & \(\mathbf{q}\)[\(s^{-2}\)] \\ \hline
7.68\(\cdot\)10\({}^{-12}\) & \(-\)2.14\(\cdot\)10\({}^{-12}\) & \(-\)1.85\(\cdot\omega_{0}\)\({}^{2}\) & \(-\)2.31\(\cdot\omega_{0}\)\({}^{2}\) \\ \hline \end{tabular}
\end{table}
Table 2: Elastic compliance constants and phonon deformation potential values for silicon [3, 9, 11, 25].  
Figure 4: Schematic subdivision of the surface of silicon chips (**a**). The Raman signal is acquired at the center position of each square. The diagonal points [1, 1] to [11, 11] are colored yellow in panel (**b**), while the positions [1, 1] and [6, 6] are colored yellow in panel (**c**).  
Given the values of \(S_{11}\), \(S_{12}\), \(p\), and \(q\), it is possible to solve the secular matrix of Equation (1). The biaxial stress along the [100] and [010] directions (\(xy\) plane, see Figure 1), for back scattering from a (001) surface, will be  
\[\Delta\omega_{3}=\frac{\lambda_{3}}{2\omega_{0}}=\frac{1}{2\omega_{0}}[pS_{12}+ q(S_{11}+S_{12})]\big{(}\sigma_{xx}+\sigma_{yy}\big{)} \tag{3}\]  
Consequently, the biaxial stress values can be calculated using Equation (4).  
\[\big{(}\sigma_{xx}+\sigma_{yy}\big{)}=-2.26\times 10^{5}\frac{\Delta\omega_{j}}{ \omega_{0}} \tag{4}\]  
In general, a Raman frequency value higher than the stress-free frequency indicates compressive stress in the sample, while a Raman frequency value lower than the stress-free frequency indicates tensile stress [24].  
The evaluation of the stress induced in the chips by the assembly process can be performed considering the unmounted chip as a reference (providing the value of \(\omega_{0}\)) and measuring the position of the Raman shift of the assembled samples.