## 2 Device and Methodology
Before the second backside etching, the wafer was treated with hexamethyldisilazane (HDMS) to enhance the adhesion of the photoresist. A double-sided mask aligner (SUSS, MA/BA8 Gen3) was used for lithography, followed by the precise patterning and etching of the cavities. The crystal structure of the Al\({}_{0.8}\)Sc\({}_{0.2}\)N film was analyzed using the X-ray diffraction (XRD) technique. The rocking curve of the (002) peak for Al\({}_{0.8}\)Sc\({}_{0.2}\)N is shown in Figure 3. The full width at half maximum (FWHM) of the peak was measured to be 1.738\({}^{\circ}\), indicating that the Al\({}_{0.8}\)Sc\({}_{0.2}\)N film deposited on the Mo substrate exhibits excellent piezoelectric properties. Figure 3: XRD result of AlScN thin film. Energy-dispersive X-ray spectroscopy (EDS) was used to characterize the composition of Al\({}_{0.8}\)Sc\({}_{0.2}\)N thin films deposited with 20% Sc-doped Al-Sc alloy targets. The EDS results are summarized in Table 1. The atomic percentage of Sc is 9. 26%, which shows good deposition composition. The device dimensions and core material parameters used in this study are given in Table 2[24]. The impact of varying structural parameters on the temperature characteristics of SOI-based PMUTs was analyzed using finite element analysis (FEA). The FEA focused on three key parameters: the diameter of the back cavity and the thickness of the top silicon layer. These parameters are also critical for determining the final resonant frequency of the device. The material properties used in the FEA are listed in Table 3, including the mechanical and electrical properties of the Molybdenum electrodes, piezoelectric layer, and silicon, which are obtained from the Comsol 5.6 Multiphysics Materials database. The properties of Al\({}_{0.8}\)Sc\({}_{0.2}\)N were obtained from previous work [25]. The FEA results of the effect of temperature on the frequency of the device under varying diameters of the back cavity are presented in Figure 4. When the diameter of \begin{table}
\begin{tabular}{c c c c} \hline
**Element** & **wt\%** & **wt\% Sigma** & **Atomic Percentage (\%)** \\ \hline N & 33.35 & 0.38 & 52.02 \\ O & 0.32 & 0.20 & 0.43 \\ Al & 47.28 & 0.34 & 38.28 \\ Sc & 19.06 & 0.30 & 9.26 \\ \hline Total & 100.00 & – & 100.00 \\ \hline \end{tabular}
\end{table} 
Table 1: Elemental composition of Al\({}_{0.8}\)Sc\({}_{0.2}\)N thin film. \begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Dimension** \\ \hline Top Mo electrode diameter (\(\upmu\)m) & 468/780 \\ Membrane diameter (\(\upmu\)m) & 600/1000 \\ Top Mo electrode thickness (\(\upmu\)m) & 0.1 \\ ScAIN piezoelectric layer thickness (\(\upmu\)m) & 1 \\ Bottom Mo electrode thickness (\(\upmu\)m) & 0.2 \\ Seed layer thickness (\(\upmu\)m) & 0.05 \\ Si structural layer thickness (\(\upmu\)m) & 4 \\ \hline \end{tabular}
\end{table} 
Table 2: Dimensions of the proposed PMUTs. \begin{table}
\begin{tabular}{c c c c} \hline
**Property** & **Mo** & **ScAIN** & **Si** \\ \hline Dielectric permittivity & & 13.7 & \\ Density (kg/m\({}^{3}\)) & 10,200 & 3560 & 2320 \\ Young’s Modulus (GPa) & 312 & 230 & 130 \\ C\({}_{11}\) = C\({}_{12}\) (GPa) & & 325 & \\ C\({}_{33}\) (GPa) & & 279 & \\ C\({}_{13}\) (GPa) & & 131 & \\ C\({}_{44}\) = C\({}_{55}\) (GPa) & & 99 & \\ C\({}_{66}\) (GPa) & & 94 & \\ d\({}_{31}\) (pm/V) & & \(-4\) & \\ d\({}_{33}\) (pm/V) & & 9.9 & \\ \hline \end{tabular}
\end{table} 
Table 3: Properties of the materials for PMUT FEM. the back cavity diameter is 600 \(\upmu\)m, the resonant frequency of the device changes from 177.08 kHz at room temperature to 189.79 kHz, corresponding to a variation of 7.17%. As the diameter of the back cavity increases, the frequency shift rate increases, reaching 12.6% at 700 \(\upmu\)m, 20.4% at 800 \(\upmu\)m, 30.6% at 900 \(\upmu\)m, and ultimately 43.1% at 1000 \(\upmu\)m. The back cavity primarily affects the stiffness of the device, influencing its temperature stability.