In this paper, Thickness-Lamé (TL) mode piezoelectrically-transduced silicon resonators are studied and demonstrated. It will be shown that unlike Planar-Lamé resonance modes, Thickness-Lamé modes could be efficiently excited using sputtered polycrystalline piezoelectric films such as Scandium Aluminum Nitride (ScAlN) due to the constructive contribution of both $d_{31}$ and $d_{33}$ piezoelectric coefficients in the coupling coefficient. Moreover, it is shown through finite element analysis and experimental results that the coupling coefficient improves with the order of the TL harmonic mode excited in a silicon slab. It is also confirmed that the quality-factor of TL resonators substantially enhances through utilization of properly-designed acoustic reflectors (i.e. acoustic isolation frames) around the tethered resonator block. The temperature coefficient of frequency is also modeled using finite-element eigen-frequency analysis. It is shown that the turnover temperature of TL resonators aligned to the [100] plane of a degenerately-doped n-type silicon substrate varies considerably as the mode shape transitions from a Thickness-Lamé to a Lateral-Extensional mode with the gradual increase of wavelength to thickness ratio. A record $Q$ of 23.2k is measured for a $\sim 185$ MHz fundamental TL resonator in vacuum ( $fxQ=4.3\times 10^{12}$ ) while quality factors of 12.6k ( $fxQ=4.6\times 10^{12}$ ) and 6k are also measured in vacuum for second- and third-harmonic TL resonators at 326 MHz and 555 MHz respectively. The combination of high turnover temperatures (>80 °C), high quality factor, and low motional resistance, promises the suitability of such resonators for extremely-stable oven-controlled oscillator applications.

中文翻译：

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厚度拉梅薄膜硅基压电谐振器

在本文中，对厚度-拉梅 (TL) 模式压电换能硅谐振器进行了研究和演示。将表明，与平面-Lamé 共振模式不同，由于两者的建设性贡献，可以使用溅射多晶压电薄膜（如氮化铝钪 (ScAlN)）有效地激发厚度-Lamé 模式。 $d_{31}$ 和 $d_{33}$ 耦合系数中的压电系数。此外，通过有限元分析和实验结果表明，耦合系数随着在硅板中激发的 TL 谐波模式的阶数而提高。还证实通过在系留谐振器块周围使用适当设计的声学反射器（即声学隔离框架），TL 谐振器的品质因数显着提高。频率的温度系数也使用有限元特征频率分析进行建模。结果表明，随着模式形状从厚度-拉梅转变为横向-扩展模式，与退化掺杂的 n 型硅衬底的 [100] 平面对齐的 TL 谐振器的翻转温度随着波长厚度比。 $Q$ 23.2k 的测量值是 $\sim 185$ 真空中 MHz 基波 TL 谐振器 ( $fxQ=4.3\times 10^{12}$ ) 而品质因数为 12.6k ( $fxQ=4.6\times 10^{12}$ ) 和 6k 也在真空中分别在 326 MHz 和 555 MHz 下测量二次和三次谐波 TL 谐振器。高周转温度 (>80 °C)、高品质因数和低运动电阻的组合保证了此类谐振器适用于极其稳定的恒温振荡器应用。