## 3 Experiment and Results  
To verify the reliability of the simulation results aforementioned, we selected a device with a silicon top thickness of 4 \(\upmu\)m and a back cavity diameter of 1000 \(\upmu\)m, whose initial resonant frequency at room temperature closely matched the simulation results. The device was placed in a cascade probe station equipped with a heating function and the positive and negative electrodes of the device were directly connected to the E4990A impedance analyzer using probes. The probe stage was heated in increments of ten degrees and the changes in the resonant frequency were recorded. The variation in resonant frequency with temperature for this design was also calculated using FEM, and both results are compared in Figure 6. The comparison shows that the experimental results agree well with the simulation in both numerical values and trends, demonstrating the reliability of the simulation for the design of devices related to temperature drift. Four devices with a diameter of the back cavity of 1000 \(\upmu\)m were selected from the fabricated 8-inch wafer for further investigation. The heating and testing method for the devices was as described above. Figure 7 shows the change in resonant frequency with temperature for the four devices. The initial frequency differences among the four devices were mainly caused by processing errors and differences in residual stress. In the figure, all devices show a similar trend of resonant frequency change with temperature, and the results are summarized in Table 4. From room temperature to 80 \({}^{\circ}\)C, the frequency of the devices exhibited a slow upward trend, with a frequency drift ranging from 7.5% to 11% at 80 \({}^{\circ}\)C. As the temperature increased further to 200 \({}^{\circ}\)C, the slope of the frequency vs. temperature curve appeared to be nearly linear. A linear fit of the high-temperature range of the resonant frequency for the four devices revealed slopes between 0.186 and 0.19, and at 200 \({}^{\circ}\)C, the frequency drift exceeded 44% compared to the initial frequency. Figure 6: FEA and experimental result comparison of PMUT with 1000 \(\upmu\)m back cavity. The impedance phase angles of the four devices were analyzed, and the results are shown in Figure 8. The impedance phase angles of all four devices increased with increasing temperature, with the largest change observed in a device whose impedance phase increased from \(-18.56^{\circ}\) at room temperature to \(+17.45^{\circ}\) at 200 \({}^{\circ}\)C. The electromechanical coupling coefficient of the four devices was also analyzed using  
\[k_{t}^{2}=\frac{C_{m}}{C_{m}+C_{0}}=\frac{\pi^{2}}{4}\frac{f_{s}f_{p}-f_{s}}{f_ {p}}. \tag{1}\]  
for the electromechanical coupling coefficient, and the results are shown in Figure 9. Figure 9 indicates that the electromechanical coupling coefficient of the four PMUTs initially increases with temperature and then decreases. Compared to room temperature, the electromechanical coupling coefficient increased by 21.4% to 68.6%, reaching its maximum around 100 \({}^{\circ}\)C to 140 \({}^{\circ}\)C.  
To further explore the influence of temperature on PMUTs based on scandium-doped AlN SOI, the aforementioned tests were repeated with devices with a back cavity diameter of 600 \(\upmu\)m. The piezoelectric layer and the top silicon thickness were the same as those of the previous devices. The frequency and impedance phase changes with temperature are shown in Figure 10. The frequency change with temperature follows a similar trend, with a slow increase at lower temperatures followed by a roughly linear change at higher