Nonlinear Limits for Single-Crystal Silicon Microresonators

Abstract

Abstract-Nonlinear effects in single-crystal silicon microresonators are analyzed with the focus on mechanical nonlinearities. The bulk acoustic wave (BAW) resonators are shown to have orders-of-magnitude higher energy storage capability than flexural beam resonators. The bifurcation point for the silicon BAW resonators is measured and the maximum vibration amplitude is shown to approach the intrinsic material limit. The importance of nonlinearities in setting the limit for vibration energy storage is demonstrated in oscillator applications. The phase noise calculated for silicon microresonator-based oscillators is compared to the conventional macroscopic quartz-based oscillators, and it is shown that the higher energy density attainable with the silicon resonators can partially compensate for the small microresonator size. Scaling law for microresonator phase noise is developed.[1246]

Addressing geometric nonlinearities with cantilever microelectromechanical systems: Beyond the Duffing model

“…This non-linearity is of pure geometrical origin [26,27], and in its most general form it will contain other terms in the dynamics equation in addition to the cubic restoring force, with a straightforward second-order force F non−lin. ∝ x 2 [28,29], and less intuitively inertia non-linear terms [30][31][32]. To put it in crude words, the Duffing equation is, even for these perfectly elastic devices, only a convenient model describing correctly the measurements.…”

Large amplitude dynamics of micro-/nanomechanical resonators actuated with electrostatic pulses

“…4 When 0 = δ , up to three oscillation states may be observed. This corresponds to the well-known 'critical amplitude' phenomenon associated with the Duffing pendulum [9]. pi V ≈29.63V, γ ≈0.719.…”

“…In order to maximize the signal-to-noise ratio (SNR) and, thus, to relax the constraints on the electronic design, the detected signal must be as large as possible, which means that, for a given set of structural parameters and a given bias voltage, the oscillation amplitude of the resonant beam must also be as large as possible. This raises questions concerning what oscillation amplitude can be sustained without incurring mechanical [9][10] or electrostatic [11][12] instability. We show in this paper that, in spite of the nonlinearities, the proposed feedback scheme ensures the stability of the motion of the beam much beyond the critical Duffing amplitude.…”

Nanoelectromechanical Systems

“…15,30 We note that for a cantilever, a similar formulation based upon a single dominant effect cannot be made. 31 The onset of nonlinearity based upon the aforementioned criterion in doubly clamped beams is displayed in Table I for the representative devices. With these limits, the DR within the operation bandwidth ⌬f becomes DR = 10 log ͫ ͗x c ͘ 2 / ͵ 2 ⌬f S X ͑eff͒ ͑ ͒d ͬ .…”