# High-Frequency and Spectrum-Clean Shear-Horizontal Acoustic Wave Resonators with AlN Overlay
## 2 Substrate Leakage  
Figure 1 depicts the FEM-simulated mode shapes of the SH wave propagating in LiNbO\({}_{3}\) substrate, LiNbO\({}_{3}\)/Si layered substrate, and AlN/LiNbO\({}_{3}\)/Si layered substrate. Perfect-matched-layer (PML) physics is assigned to bottom layers of the 3D unit cell models to simulate the substrate that is too thick, comparing it to wavelengths to generate wave reflections from its bottom. The periodic conditions are applied to both \(x\) (perpendicular to IDT fingers) and \(y\) (in parallel with IDT fingers) directions so that the basic semi-infinite plane condition is assumed for the wave propagation. The material con \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbol** & **Definition** \\ \hline \(Q\) & Quality Factor \\ \hline \({k_{\text{eff}}}^{2}\) & Electromechanical Coupling Coefficient \\ \hline \(\lambda\) & Acoustic Wavelength \\ \hline \(f_{\text{s}}\) & Series-frequency \\ \hline \(u_{\text{x}}\) & \(x\)-direction displacement \\ \hline \(u_{\text{y}}\) & \(y\)-direction displacement \\ \hline \(u_{\text{z}}\) & \(z\)-direction displacement \\ \hline \({h_{\text{LiNbO}}}\) & The thickness of LiNbO\({}_{3}\) \\ \hline \({h_{\text{AlN}}}\) & The thickness of AlN \\ \hline \({\text{S}_{0}}\) mode & The lowest symmetric mode \\ \hline \(v_{\text{p,o}}\) & Open phase velocity \\ \hline \(TCF\) & Temperature coefficients of frequency \\ \hline \(TCV\) & Temperature coefficients of velocity \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Symbols used in this paper. stants are from [14; 15; 16; 17] and are listed in Table 2. In the structures shown, \(h_{\text{LINDO3}}/\lambda=0.25\), \(h_{\text{AIN}}/\lambda=0.2\) are assumed; 8\(\%\lambda\)-thick aluminum (Al) is used as IDT electrodes here and throughout the paper. As shown in Figure 1a, the shear component in _YX_-LiNbO\({}_{3}\) substrate still concentrates at the surface, but strong leakage results from coupling to the slow shear bulk acoustic wave and mechanically propagates down obliquely, as shown in \(u_{z}\) and \(u_{x}\). The lower-case coordinate \(x\) is the propagation direction proportional to IDT, \(y\) is the transversal direction parallel to IDT, and \(z\) the direction down into substrate. By bonding the high-velocity Si \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Symbol** & **LiNbO\({}_{3}\)[14]** & **AIN [15]** & **Si [16]** & **Al [16]** & **Units** \\ \hline \multirow{8}{*}{Stiffness constants} & \(C_{11}\)\({}^{\text{E}}\) & 198.39 & 345 & & 111 & \\ \cline{2-6}  & \(C_{12}\)\({}^{\text{E}}\) & 54.72 & 125 & & 59 & \\ \cline{2-6}  & \(C_{13}\)\({}^{\text{E}}\) & 65.13 & 120 & & 59 & \\ \cline{2-6}  & \(C_{14}\)\({}^{\text{E}}\) & 7.88 & – & \multirow{2}{*}{170} & \multirow{2}{*}{(GPa)} & \multirow{2}{*}{(GPa)} \\ \cline{2-6}  & \(C_{33}\)\({}^{\text{E}}\) & 227.90 & 395 & & 111 & \\ \cline{2-6}  & \(C_{44}\)\({}^{\text{E}}\) & 59.65 & 118 & & 26 & \\ \cline{2-6}  & \(C_{66}\)\({}^{\text{E}}\) & 71.84 & 110 & & 26 & \\ \hline \multirow{2}{*}{Density} & \(\rho\) & 4628 & 3260 & 2329 & 2700 & (kg/m\({}^{3}\)) \\ \cline{2-6}  & \(e_{15}\) & 3.69 & –0.48 & – & – & \\ \cline{2-6}  & \(e_{22}\) & 2.42 & – & – & – & \\ \cline{2-6}  & \(e_{31}\) & 0.30 & –0.58 & – & – & \\ \cline{2-6}  & \(e_{33}\) & 1.77 & 1.55 & – & – & \\ \hline \multirow{2}{*}{Dielectric constants} & \(\varepsilon_{11}\)\({}^{\text{S}}\) & 45.6 & 8.0 & 11.7 & 1 & \\ \cline{2-6}  & \(\varepsilon_{33}\)\({}^{\text{S}}\) & 26.3 & 9.5 & 11.7 & 1 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: LiNbO\({}_{3}\), AlN, Si, and Al material constants used in the simulations [14; 15; 16]. Figure 1: Comparison of simulated displacement amplitudes of (**a**) LiNbO\({}_{3}\) substrate, (**b**) LiNbO\({}_{3}\)/Si, and (**c**) AlN\(/\) LiNbO\({}_{3}\)/Si for the SH wave. The insets are the mode shapes at antiresonances simulated with periodic structures using FEM simulation (\(h_{\text{LINDO3}}/\lambda=0.25\), \(h_{\text{AIN}}/\lambda=0.2\), \(h_{\text{IDT,Al}}/\lambda=0.08\) are used in the plots).