L C Shao

The nonlinearity cancellation phenomenon in micromechanical resonators

In this paper, we present comprehensive analysis of the nonlinearities in a micromechanical clamped-clamped beam resonator. A nonlinear model which incorporates both mechanical and electrostatic nonlinear effects is established for the resonator and verified by experimental results. Both the nonlinear model and experimental results show that the first-order cancellation between the mechanical and electrostatic nonlinear spring constants occurs at about 45 V dc polarization voltage for a 193 kHz resonator in vacuum pressure of 37.5 µTorr. Our study also reveals that the nonlinearity cancellation is helpful in optimizing the overall resonator performance. On top of improving the frequency stability of the resonator by reducing its amplitude-frequency coefficient to almost zero, the nonlinearity cancellation also boosts the critical vibration amplitude of the resonator (0.57 µm for the beam resonator with 2 µm nominal gap spacing), leading to better power handling capabilities. The results from the clamped-clamped beam resonator studied in this work can be easily generalized and applied to other types of resonators.

Nonlinearity in micromechanical free–free beam resonators: modeling and experimental verification

In this paper, we present a systematic characterization and modeling technique for the micromechanical free?free beam resonator to analyze its nonlinear vibration behavior. Different from the conventional FEM-based approach whose simulation accuracy is usually limited around 60?70%, the proposed modeling method is able to accurately identify both the mechanical and electrostatic nonlinear parameters from just a few preliminary experimental observations. The nonlinear equation of motion is then numerically solved, demonstrating both the spring hardening and softening effects in the system. The simulated nonlinear behavior of the resonator under different driving conditions is validated by comparing them with the experimental data. In addition, based on the verified nonlinear model, design guidelines such as the nonlinearity cancellation are also highlighted. Although this work focuses on the free?free beam resonators, the proposed modeling approach can be applied to any other electrostatically driven microresonator to reveal different intrinsic nonlinear properties of the device.

Nonlinear behavior of Lamé-mode SOI bulk resonator

In this paper, we report for the first time the detailed analysis of the nonlinear behavior of a Lame-mode SOI bulk resonator. The measured resonant frequency of the resonator was 6.35 MHz with a quality factor of 1.7 million in the ambient pressure of 0.02 Pa. We used the two-step semi-analytic approach to characterize the model parameters of the resonator and the nonlinear model of the resonator was verified by the experimental results. Our study shows that the Lame-mode bulk resonator has three orders of magnitude larger maximum energy storage capability than the flexural beam resonator, leading to improved overall phase noise performance of the resonator-based oscillator.

Nonlinearities in a high-Q SOI Lamé-mode bulk resonator

In this paper, a detailed study of the nonlinearities in a high-Q SOI Lame-mode bulk resonator is reported. The bulk resonator is designed to operate at 6.35 MHz with a quality factor of 1.7 million at air pressure below 100 Pa. As the vibration amplitude increases, the transmission curve of the resonator progressively bends to lower frequencies due to spring softening effects. The model parameters of the resonator are quantified based on some preliminary experimental results and verified by numerical calculation. Compared with a flexural-mode beam resonator, the Lame-mode bulk resonator is much less susceptible to nonlinear effects and thus can store three orders of magnitude more vibration energy before frequency hysteresis occurs. Closed-loop measurement further demonstrates that the ultra-high quality factor and superior power handling capability enable the Lame-mode based oscillator to achieve 42 dB lower phase noise than the flexural-mode one at 10 Hz offset from the carrier frequency.

Study of the nonlinearities in micromechanical clamped–clamped beam resonators using stroboscopic SEM

This paper presents a comprehensive study of the nonlinearities in micromechanical clamped–clamped beam resonators using a stroboscopic scanning electron microscopy (SEM) technique. Stroboscopic SEM allows direct imaging and measurement of the resonators momentary displacement, hence eliminating the uncertainties associated with the conventional characterization methods. Five different silicon-on-insulator (SOI) comb-drive clamped–clamped beam resonators with resonant frequencies ranging from 113 kHz to 239 kHz were designed, fabricated and tested to investigate how their nonlinearities are related to the device dimensions. Both the theoretical analysis and experimental results conclusively show that the critical vibration amplitude of the resonator is around 1% of the beam width in a vacuum and is relatively independent of the beam length. Furthermore, it is found that the maximum storable energy of the resonator can be significantly increased by increasing the beam width and/or reducing the beam length if there are no restrictions on these dimensions. On the other hand, if a specific resonant frequency needs to be maintained, the maximum storable energy can be improved by increasing both the beam width and length by the same factor. Such a study not only helps to reveal the intrinsic nonlinear properties of the micromechanical clamped–clamped beam resonators, but also provides useful design guidelines for engineers to optimize the overall device performance.

Characterization of SOI Lamé-mode square resonators

Characterization of Lame-mode square resonators with different straight-beam anchor lengths, structural layer thickness, and number of anchor support reveals that there is likely an optimal range of anchor designs that provide high quality factor (Q) above one million, along with low motional resistance. Shorter anchor length restricts resonator vibrations and motional resistance could be increased by 3.5 times compared to resonators with longer anchor length. Two-anchor support design is able to achieve higher Qpsilas but results in higher motional resistance compared to four-anchor support. When structural thickness is reduced from 25 mum to 10 mum, Q gets degraded but still maintained above one million.

Nonlinear behavior modeling of SOI micromechanical free-free beam resonators

Nonlinear behavior of a capacitively driven and sensed micromechanical free-free beam resonator is characterized, modeled and experimentally verified in this paper. Both the mechanical and electrostatic nonlinear effects are included in the resonator model. Instead of using the FEM tools which introduces uncertainties to the simulation process, an alternative semi-analytic method is proposed to identify the resonator parameters from just a few preliminary testing results. A 615kHz free-free beam resonator was designed, fabricated and studied. From the experimental results, it is observed that the nonlinear effects in the free-free beam always shift the resonant peak of the beam to a higher frequency under nonlinear vibration. In order to validate the proposed modeling approach, a nonlinear model was constructed based on the experimentally extracted parameters and numerically solved in MATLAB. The simulation results were compared with the experimental data, showing that the measured large-signal frequency domain response can be accurately reproduced by simulation. Although this work focused on the free-free beam resonator, the proposed modeling approach is not specific to flexural designs, but is valid for all types of electrostatic resonators. Such a method to predict nonlinear effects of microresonators will be especially useful for MEMS oscillator and filter applications.