Computational and quasi-analytical models for non-linear vibrations of resonant MEMS and NEMS sensors

International audienceLarge-amplitude non-linear vibrations of micro- and nano-electromechanical resonant sensors around their primary resonance are investigated. A comprehensive multiphysics model based on the Galerkin decomposition method coupled with the averaging method is developed in the case of electrostatically actuated clamped-clamped resonators. The model is purely analytical and includes the main sources of non-linearities as well as fringing field effects. The influence of the higher modes and the validation of the model is demonstrated with respect to the shooting method as well as the harmonic balance coupled with the asymptotic numerical method. This model allows designers to investigate the sensitivity variation of resonant sensors in the non-linear regime with respect to the electrostatic forcing

Capteurs résonants M/NEMS et phénomènes non linéaires

Accessible via http://www.bruit.fr/flipbook/AT57/index.html#/10/Les capteurs résonants de type M/NEMS jouent et joueront un rôle essentiel dans les nouvelles technologies. Cependant leur comportement est souvent fortement non linéaire ce qui est préjudiciable à la précision de la mesure exigée. Les résonateurs M/NEMS analysés ont des comportements complexes combinant raidissements, assouplissements, instabilités latérales car régis par des larges déflexions, des excitations paramétriques, des couplages géométrique et électrique. Ces comportements nécessitent une conception soigneuse qui doit s'appuyer sur des modèles les plus simples possibles mais tout en gardant leur pertinence pour modéliser au mieux les différents phénomènes physiques en jeu

Pull-In Retarding in Nonlinear Nanoelectromechanical Resonators Under Superharmonic Excitation

International audienceIn order to compensate for the loss of performance when scaling resonant sensors down to NEMS, a complete analytical model, including all main sources of nonlinearities, is presented as a predictive tool for the dynamic behavior of clamped-clamped nanoresonators electrostatically actuated. The nonlinear dynamics of such NEMS under superharmonic resonance of an order half their fundamental natural frequencies is investigated. It is shown that the critical amplitude has the same dependence on the quality factor Q and the thickness h as the case of the primary resonance. Finally, a way to retard the pull-in by decreasing the AC voltage is proposed in order to enhance the performance of NEMS resonators

Nonlinear phenomena in nanomechanical resonators: mechanical behaviors and physical limitations

International audienceIn order to overcome the loss of performances issue when scaling resonant sensors down to NEMS, it proves extremely useful to study the behavior of resonators up to large displacements and hence high nonlinearities. A comprehensive nonlinear multiphysics model based on the Euler-Bernoulli equation which includes both mechanical and electrostatic nonlinearities in the case of a capacitive doubly clamped beam is presented. This purely analytical model captures all the nonlinear phenomena present in NEMS resonators electrostatically actuated including bistability, multistability which can lead to several physical limitations such as noise mixing, frequency stability deterioration as well as dynamic pull-in. Moreover, close-form expressions of the critical amplitudes and pull-in domain initiation amplitude are provided which can potentially serve for NEMS designers as quick design rules

Mixed behavior identification in nonlinear nanomechanical resonators

International audienceThe small size of NEMS resonators combined with their physical attributes make NEMS quite attractive and suitable for a wide range of technological applications such as ultrasensitive force and mass sensing, narrow band filtering, and time keeping. However, at this size regime, nonlinearities occur sooner which reduce the NEMS dynamic range [1]. Consequently, a nonlinear multiphysics model is needed as a tool of performances optimization in resonant sensors for MEMS and NEMS designers. To this end, the nonlinear behavior of resonators remains yet to be explored, and numerous models have been presented. Some models are purely analytical [2, 3] but they include coarse assumptions concerning nonlinearities. Other models [4, 5] are more complicated and use numerical simulations which make them less interesting for MEMS designers. In the present paper, a compact and analytical model including the main sources of nonlinearities (mechanical and electrostatic) is presented and validated thanks to the fabrication of a resonant accelerometer and the characterization of its sensing element, an electrostatically doubly clamped beam. A perturbation technique, the averaging method [6], has been used to obtain two first-order-nonlinear-ordinary-differential equations which describe the amplitude and phase modulation of the response and allows the computation of its stability. This analytical approach proves to be a powerful and quick tool for the sensor design, enabling the description of the competition between the hardening and the softening behavior (Figure 1), and thus the capture of all possible dynamic behaviors, particularly, the mixed behavior characterized by four bifurcation points. Furthermore, the model may be used as a tool to enhance the dynamic range of the resonator, i.e. its detectability. On the way from MEMS to NEMS, a "small" MEMS resonant accelerometer [1] shown in Figure 2a has been fabricated. The sensor structure has not been designed to display high inertial performances, but rather is a way to validate process [7], characterization and model choices. In order to electrically characterize its sensitive part (the resonator described in Figure 2b), the device was placed in a vacuum chamber (down to 1 mTorr), and the 2-port electrical measurements were performed at room temperature using a low noise lock-in amplifier (Signal Recovery 7280). The drive voltage is V ac = 0.5V and the beam is polarized with V dc = 10V. This ensures a mixed behavior as shown in Figure 3. The quality factor obtained with this polarization voltage and in a linear regime is 4000. The critical amplitude [7] is then A c = 53nm, i.e. V c = 25µV. The peak obtained is then far beyond A c , up to 75% of the gap. The mixed behavior is fully identified with its four bifurcation points using a sweep up frequency to capture the points P and 2 as well as a sweep down frequency for points 1 and 3. Moreover, we experimentally track the point P (mixed behavior initiation point) while varying the drive and the bias voltage and we show that this bifurcation point is fixed by the design parameters. Then, the mixed behavior can be retarded and avoided by designing resonators for which the P point amplitude is close to the gap. Further measurements are under work to validate the dynamic range enhancement based on the compensation of the nonlinearities shown by the model

Forced large amplitude periodic vibrations of non-linear Mathieu resonators for microgyroscope applications

International audienceThis paper describes a comprehensive non-linear multiphysics model based on the Euler-Bernoulli beam equation that remains valid up to large displacements in the case of electrostatically actuated Mathieu resonators. This purely analytical model takes into account the fringing field effects and is used to track the periodic motions of the sensing parts in resonant microgyroscopes. Several parametric analyses are presented in order to investigate the effect of the proof mass frequency on the bifurcation topology. The model shows that the optimal sensitivity is reached for resonant microgyroscopes designed with sensing frequency four times faster than the actuation one

High Order Nonlinearities and Mixed Behavior in Micromechanical Resonators

International audienceThis paper investigates the sensitivity of the third order nonlinearity cancellation to the mixed (hardening and softening) behavior in electrostatically actuated micromechanical resonators under primary resonance at large amplitudes compared to the gap. We demonstrate the dominance of the mixed behavior due to the quintic nonlinearities, beyond the critical amplitude when the third order mechanical and electrostatic nonlinearities are balanced. We also report the experimentalobservation of a strange attraction which can lead to a chaotic resonator

Pull-in retarding in nonlinear Mathieu NEMS resonator under superharmonic excitation

International audienceIn order to compensate the loss of performance when scaling resonant sensors down to NEMS, a complete analytical model including all main sources of non linearities is presented as a predictive tool for the dynamic behaviour of Mathieu NEMS resonators. The nonlinear dynamics of such resonators under superharmonic resonance of order half their fundamental natural frequency is investigated. The device consists of a clamped-clamped nanobeam subject to viscous damping and actuated by an electric load $v (t) =V_{ac}\cos[\tilde{Ω}t]−V_{dc}$ where $V_{dc}$ is the DC polarization voltage, $V_{ac}$ is the amplitude of the applied AC voltage, and $\tilde{Ω}$ is the excitation frequency. The particularity of this device is the use of two electrodes with different gap thickness: a first electrode for the actuation and a second one for the transduction. The transverse deflection of the nanobeam $\tilde{w}( x,t)$ is governed by the nonlinear Euler-Bernoulli equation for thin beams. The fringing field effects are introduced into the model since they can have a significant influence. A Galerkin discretization procedure with basis functions satisfying the boundary conditions of a clamped-clamped nanobeam permits the transformation of the nonlinear partial differential equations of motion into a finite system of nonlinear Mathieu equations. Since we are interested in the response of the resonator at resonance when the first mode is dominant, only the first equation is considered. A perturbation technique is then used in order to obtain two first order non-linear ordinary-differential equations which describe the amplitude and phase modulation of the response and permit a stability analysis. It is shown that the critical amplitude has the same dependence on the quality factor Q and the vibrating width h as in the case of the primary resonance. Finally, a way to shift up the pull-in amplitude by decreasing the AC voltage is proposed in order to enhance the performances of NEMS resonators

Compact and explicit physical model for lateral metal-oxide-semiconductor field-effect transistor with nanoelectromechanical system based resonant gate

International audienceWe propose a simple analytical model of a metal-oxide-semiconductor field-effect transistor with a lateral resonant gate based on the coupled electromechanical equations, which are self-consistently solved in time. All charge densities according to the mechanical oscillations are evaluated. The only input parameters are the physical characteristics of the device. No extra mathematical parameters are used to fit the experimental results. Theoretical results are in good agreement with the experimental data in static and dynamic operation. Our model is comprehensive and may be suitable for any electromechanical device based on the field-effect transduction