Zhao Zinan, Wang Bin, Qian Zhenghua, Yang Jiashi
IEEE Trans Ultrason Ferroelectr Freq Control. 2018 Sep;65(9):1669-1679. doi: 10.1109/TUFFC.2018.2845386. Epub 2018 Jun 8.
Free and forced vibration analysis of ZnO thin-film resonator operating with trapped-energy thickness-extensional (TE) mode is performed based on the dispersion curves for both the unbounded fully electroded and unelectroded films. The thickness solutions to the free vibration problem consist of the mode branches of the dispersion curves for the electroded and the unelectroded regions, respectively. The mode branches in the unelectroded region that carry energy away from the vibration zone are neglected because this effective energy loss corresponds to a complex frequency in the frequency spectrum. Since the thickness solutions in each region have satisfied the differential equations and the boundary conditions on the major surfaces exactly, the substitution of the thickness solutions into the modified Hamilton variational principle derived by Tiersten gives an approximate continuity condition in the form of integral over the thickness of the resonator at the interface between the electroded and unelectroded regions. The stationary condition of the integral continuity condition leads to a system of homogeneous linear equations, which determines the frequency spectrum ranging from the first TE cutoff frequency of the fully electroded film to that of the unelectroded film where the conventional trapped-energy vibration occurs. Forced vibration analysis of ZnO thin film driven into TE mode by applying a voltage to the top and bottom electrodes is also performed, which further verifies the validity of the obtained results from the free vibration analysis.
基于无界全电极化薄膜和无电极化薄膜的色散曲线，对以俘获能量厚度伸缩（TE）模式工作的ZnO薄膜谐振器进行了自由振动和受迫振动分析。自由振动问题的厚度解分别由电极化区域和无电极化区域色散曲线的模式分支组成。无电极化区域中携带能量离开振动区的模式分支被忽略，因为这种有效的能量损失对应于频谱中的复频率。由于每个区域的厚度解都精确满足了微分方程和主表面上的边界条件，将厚度解代入Tiersten推导的修正哈密顿变分原理，得到了一个以电极化区域和无电极化区域界面处谐振器厚度积分形式表示的近似连续性条件。积分连续性条件的驻值条件导致一个齐次线性方程组，该方程组确定了从全电极化薄膜的第一TE截止频率到发生传统俘获能量振动的无电极化薄膜截止频率范围内的频谱。还对通过向顶部和底部电极施加电压驱动进入TE模式的ZnO薄膜进行了受迫振动分析，这进一步验证了自由振动分析结果的有效性。
