# High-Frequency and Spectrum-Clean Shear-Horizontal Acoustic Wave Resonators with AlN Overlay
## 4 Propagation Characteristics of SH Wave in AlN/LiNbO\({}_{3}\)/Si
### Improvement of TCF  
In addition, without a careful design of the substrate, the layered SH potentially has spurious responses in the out-of-band frequencies (Lamb modes) and near-band frequencies. This paper focuses on the design of layered substrate not only to optimize its narrow-band characteristics--\(f\), \(Q\), \(k_{\text{eff}}\)\({}^{2}\), temperature coefficient of frequency (TCF)--but also for eliminating the out-of-band spurious responses. The _TCF_ performance measuring the thermal stability of a resonator is set by the temperature dependence of phase velocity and the thermal-expansion coefficient of the wave along the propagation direction. The first-order _TCF_'s for the series resonance (\(TCF_{\text{s,1 st}}\)) and parallel resonance (\(TCF_{\text{p,1 st}}\)) are calculated as  
\[TCF_{\text{s,1 st}}=\frac{1}{f_{s}}\frac{\partial f_{s}}{T}=\frac{1}{v_{p,SC}} \frac{\partial v_{p,SC}}{T}-\alpha_{x}, \tag{3}\]  
\[TCF_{\text{p,1 st}}=\frac{1}{f_{p}}\frac{\partial f_{p}}{T}=\frac{1}{v_{p,OC}} \frac{\partial v_{p,OC}}{T}-\alpha_{x}, \tag{4}\]  
where \(v_{\text{p,SC}}\) and \(v_{\text{p,OC}}\) refer to the phase velocities under short-circuited (SC) and open-circuited (OC) grating BCs, shown in the inset of FIG 10. From the coupling-of-modes (COM) theory, these BCs corresponds to the \(f_{s}\) and \(f_{p}\), respectively. Their temperature dependence \(\partial/\partial T\) is calculated from the temperature coefficients of stiffness constants, temperature coefficients of piezoelectric constants, and temperature coefficients of permittivity of LiNbO\({}_{3}\), AlN, Si, and Al listed in Table 3. The \(\alpha_{x}\) corresponds to the thermal-expansion coefficient of the substrate in the wave-propagation direction \(x\), also listed in Table 3. Since the Si substrate is much thicker than the LiNbO\({}_{3}\) and AlN, the effect of thermal expansion is limited by the clamping substrate Si, and its \(\alpha_{11}\) of 2.6 ppm/\({}^{\circ}\)C from literature [19] is used herein for the derivation. Figure 10 shows the calculated \(TCF_{\text{s,1 st}}\) and \(TCF_{\text{p,1 st}}\) by varying the AlN thickness for the nonleaky SH waves propagating in AlN/LiNbO\({}_{3}\)/Si with the 0.25\(\lambda\)- thick 15\({}^{\circ}\) YX-LiNbO\({}_{3}\) optimized from the previous analysis. Although AlN also becomes softer \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Parameter** & **Symbol** & **LiNbO\({}_{3}\)[21]** & **AIN [15]** & **Si [16]** & **AI [15]** & **Units** \\ \hline \multirow{8}{*}{Temperature coefficients of stiffness constants (1 st order)} & \(\bar{T}_{C11}\) & \(-174\) & \(-37\) & & \(-590\) & \\ \cline{2-6}  & \(\bar{T}_{C12}\) & \(-252\) & \(-1.8\) & & \(-80\) & \\ \cline{2-6}  & \(\bar{T}_{C13}\) & \(-159\) & \(-1.8\) & & \(-80\) & \\ \cline{2-6}  & \(\bar{T}_{C14}\) & \(-214\) & \(-\) & \(-63\) & \(-\) & \((10^{-6}\)\(1/^{\circ}\)C) \\ \cline{2-6}  & \(\bar{T}_{C33}\) & \(-153\) & \(-65\) & & \(-590\) & \\ \cline{2-6}  & \(\bar{T}_{C44}\) & \(-204\) & \(-50\) & & \(-520\) & \\ \cline{2-6}  & \(\bar{T}_{C66}\) & \(-143\) & \(-57\) & & \(-520\) & \\ \hline \multirow{8}{*}{Temperature coefficients of piezoelectric constants} & \(\bar{T}_{C15}\) & \(147\) & & & & \\ \cline{2-6}  & \(\bar{T}_{C22}\) & \(79\) & & & & \\ \cline{2-6}  & \(\bar{T}_{C31}\) & \(221\) & & & & \\ \cline{1-1} \cline{2-6}  & \(\bar{T}_{C33}\) & \(887\) & & & & \\ \hline Temperature coefficients of permittivity & \(\bar{T}_{C11}\) & \(323\) & & & & \\ \cline{1-1} \cline{2-6}  & \(\bar{T}_{C33}\) & \(627\) & & & & \\ \hline Thermal-expansion coefficients & \(\alpha_{11}\) & \(15.4\) & \(5.27\) & \(2.6\) & \(18\) & \\ \cline{1-1} \cline{2-6}  & \(\bar{\alpha}_{33}\) & \(7.5\) & \(4.15\) & \(2.6\) & \(18\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Temperature coefficients of material constants of LiNbO\({}_{3}\), AlN, Si, and Al used in the simulations [15; 16; 21].