Silicon is widely used as the device material for many micro resonators applied in timing and frequency referencing. One key disadvantage of silicon resonators compared to quartz resonators is their high thermal sensitivities. Doping silicon is a promising approach for temperature stability. Doped resonators operating at large deformation and finite strain amplitude often go to nonlinear regimes; therefore, nonlinear dynamics must be considered for adequately predicting the system behavior. In this article, the nonlinear vibration analysis is given for rectangular resonators operating in the Lamé mode, incorporating both the second- and third-order elastic constant (SOEC and TOEC) components. This article presents an analytic demonstration for the linear and nonlinear lumped mass system equivalent spring constants explicitly in terms of SOEC and TOEC. We show that, for a rectangular resonator in the Lamé mode, the first-order nonlinear spring constant would be an explicit expression in terms of TOEC components, which, for a square resonator, will be nullified. We show that there exist optimal doping levels where the anharmonic stiffness coefficient is minimized, implying the most dynamic stable vibrations. Furthermore, this article shows that there exists a tradeoff between dynamical and temperature–frequency stability in terms of the doping level.

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掺杂硅微谐振器的非线性动力学和温度稳定性建模

硅被广泛用作许多用于定时和频率基准的微型谐振器的器件材料。与石英谐振器相比，硅谐振器的一个关键缺点是它们的高热敏性。掺杂硅是提高温度稳定性的一种有前途的方法。在较大的变形和有限的应变幅度下工作的掺杂谐振器通常会进入非线性状态。因此，必须考虑非线性动力学以充分预测系统行为。在本文中，给出了在Lamé模式下工作的矩形谐振器的非线性振动分析，其中包含了二阶和三阶弹性常数（SOEC和TOEC）。本文以SOEC和TOEC的形式给出了线性和非线性集总质量系统等效弹簧常数的解析演示。我们表明，对于处于Lamé模式的矩形谐振器，一阶非线性弹簧常数将是根据TOEC分量的一个明确表达式，对于方形谐振器，它将被作废。我们表明存在最佳掺杂水平，其中非谐刚度系数最小，这意味着动态最稳定的振动。此外，本文表明，就掺杂水平而言，动态稳定性和温度-频率稳定性之间存在折衷。对于方谐振器，将被取消。我们表明存在最佳掺杂水平，其中非谐刚度系数最小，这意味着动态最稳定的振动。此外，本文表明，就掺杂水平而言，动态稳定性和温度-频率稳定性之间存在折衷。对于方谐振器，将被取消。我们表明存在最佳掺杂水平，其中非谐刚度系数最小，这意味着动态最稳定的振动。此外，本文表明，就掺杂水平而言，动态稳定性和温度-频率稳定性之间存在折衷。