## 3 Theoretical Framework
### Premises
In the case of transversely isotropic medium with the vertical axis of symmetry \(Oz\), the compliance matrix \(S\) written in terms of the Young modulus \(E_{x}=E_{y}\) and \(E_{z}\), shear modulus \(\mu_{z}\), and Poisson's ratios \(\
u_{x}=\
u_{y}\) and \(\
u_{z}\) takes the form  
\[S=\left( \begin{array}{cc}S_{1}&0\\ 0&S_{2}\end{array} \right)\]  
\[S_{1}=\left( \begin{array}{ccc}1/E_{x}&-\
u_{x}/E_{x}&-\
u_{z}/E_{z}\\ -\
u_{x}/E_{x}&1/E_{x}&-\
u_{z}/E_{z}\\ -\
u_{z}/E_{z}&-\
u_{z}/E_{z}&1/E_{z}\end{array} \right)\]  
\[S_{2}=\text{diag}(1/\mu_{z},1/\mu_{z},1/\mu_{x}),\quad\mu_{x}=E_{x}/(2(1+\
u_{ x}))\] \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline  & \(\mathbf{C_{11}}\) & \(\mathbf{C_{33}}\) & \(\mathbf{C_{13}}\) & \(\mathbf{C_{44}}\) & \(\mathbf{C_{66}}\) & \(\rho\) & \(\mathbf{E_{xy}}\) & \(\mathbf{E_{z}}\) & \(\mathbf{\mu_{z}}\) & \(\mathbf{\
u_{xy}}\) & \(\mathbf{\
u_{z}}\) \\ \hline CrPt & 315 & 315 & 193 & 61 & 61 & 21.5 & & & & \\ \hline Si[111] & 195 & 205 & 44.7 & 60.7 & 70.3 & 2.33 & 175 & 189 & 60.7 & 0.241 & 0.179 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Unmodified input material constants for the coating CrPt and substrate Si[111]. Elastic moduli \(C_{ij}\) [GPa], density \(\rho\) [\(10^{3}\) kg/m\({}^{3}\)], thickness \(h\) [nm], and the corresponding engineering constants: [PERSON]’s moduli \(E_{x}\), \(E_{z}\) and shear modulus \(\mu_{z}\) [GPa]; [PERSON]’s ratios \(\
u_{x}\) and \(\
u_{z}\). Figure 5: (**a**) Electronic microscope picture of the cross-section of the cleaved specimen. Here, NWs are embedded into HSQ matrix, Cr/Pt metal electrode is deposited on the top. (**b**) Coordinate system for a multilayered elastic half-space used to model SAWs; dashed line is for a further subdivision in accordance with vertical inhomogeneity of the HSQ-NW layer. Accordingly,  
\[C=\left( \begin{array}{cc}C_{1}&0\\ 0&C_{2}\end{array} \right),\quad C_{1}=S_{1}^{-1},\quad C_{2}=S_{2}^{-1}\]  
A three-layer elastic anisotropic half-space is a basic model for the SAW simulation. However, since some samples present straight and homogeneously grown NWs (e.g., Figure 1), in other samples, the NWs are of _\"pyramidal\"_ shape (Figure 2) in the sense where the straight NWs present a large base resulting from the parasitic layer grown at their bottom due to the lower temperature. It means that the elastic properties and density of the HSQ-NW layer vary with the depth, and it should be modeled by a vertically inhomogeneous (layered or functionally gradient) elastic medium. Nevertheless, at the first stage, such a core layer can also be roughly modeled by a homogeneous transversely isotropic material. And the numerical analysis shows that such an approximation provides quite reasonable results. However, judging by the pictures of _\"pyramidal\"_ NWs, it is more precise to divide such a layer in two sublayers with different material properties (e.g., sublayers 2.1 and 2.2 separated in Figure 5b by the horizontal dashed line). Thus, the wave processes in the samples under study are simulated on the basis of solutions to the elastodynamic boundary value problems (BVP) for multilayered anisotropic half-spaces.