The chaotic behavior of MEMS structures constitutes an exotic field exhibiting a consid-erable potential. It is important to emphasize that, up to date, no experimental results of chaotic motion were reported on simple structures like cc-beams. Chaotic behavior has been only experimentally observed for relatively large and complex structures using non-overlapping comb fingers as driver electrodes working in the kHz range. In this chapter, the possibility of obtaining chaotic behavior with a MEMS simple resonator (cc-beam and cantilever) placed in-plane has been evaluated from two approaches. The first approach was oriented to explore the one-well chaotic behavior of simple and in-plane oscillators (cc-beams or cantilevers) as culmination of period doubling route. An exhaustive revision of the works found in the literature indicates the unfeasibility of the in-well chaotic behavior since the extremely narrow range of the actuations voltages that disables chaotic performance in real systems. In addition, numerical simulations of the Duffing system (performed in section 2.3.2) shown the need of great excitation amplitudes, and a weakness of the chaotic response in comparison with the homoclinic chaos. These results discourage definitively
the practical exploitation of in-well chaos using the proposed approach. In the second ap-proach, the reproduction of the well known 2WP Duffing chaotic attractor with a beam MEMS resonator has been analyzed. This chaotic behavior is found to be robust, broad banded, stationary and provided by a wide range of parameters. In fact, two-well Duffing chaos is the most widely used among most chaotic oscillators references. In any case, it is important to mention inconsistencies found in some previous works ([26], [27], [116]). As we remark in the first section, in a cc- beam resonator the effective mass, linear and non-linear stiffness and the capacitance for zero displacement parameters are not independent.
Solving the system of equations to size the microresonator considered in [26] reveals a set of values for these parameters that are fully incompatible. The given capacitance for zero displacement takes an unusually high value ofC0 = 0.94pF for a simple cc-beam resonator (C0×s = 1.875×1018F m and s = 2µm). Assuming a Young modulus of E = 160 GPa for the silicon cc-beam resonator and by the equations intended fork1, k3 and C0 the mi-croresonator dimensions are found to be: w = 506nm, l = 344µm and t = 616µm. The aspect ratioth/w obtained from such values (∼103) is absolutely unfeasible for any MEMS fabrication technology. In fact, the maximumth/wratio reported up to now is less than 100 [117]. In addition, a mass density value of 122kg/m3 is also derived from last parameters, which is an unrealistic value for common MEMS materials.
To summarize, taking a first approximation of a simple lumped model to easily get the essential nonlinear performance of beam-shaped resonators, both the technological (mate-rial and aspect ratios) and electrical (voltage values) requirements for the generation of a double-well potential distribution (which enables its operation as cross-well based chaotic generator in an extensive and robust way) have been determined. In contrast to typi-cal applications (sensors, RF oscillators, etc.), a relatively large gap is required making the capacitive readout a key issue. In this way, the resonator width minimization becomes mandatory if high frequency operation is desired, since the required bias voltage value scales linearly with the resonance frequency. These results are given for some CMOS technolo-gies (which have been reported in literature to be used for MEMS resonator fabrication) to illustrate the order of magnitude of the polarization voltages depending on the required operating frequency and the approximated real resonator dimensions.
Chapter 5
Nonlinear macro-model for cc-beam microresonators
In chapter 4, the design and operating conditions for cross-well motion (based on bistabil-ity) have been reported and numerically reproduced to obtain rich and sustained chaotic behavior. Bistability (two-well potential distribution) can be seen as the first step to attain the chaotic behavior in MEMS resonators, having also some other applications, like thresh-old switches, memory cells, relays, valves etc. [118]. Up to now, all the analytic procedure relies on the parallel plate approximation, namely the conception of that all the beam is sliding as a solid without deformation and that it keeps always a perfect parallel orienta-tion with respect to the electrodes. However, bistable behavior requires relative large beam displacements making the resonator to electrode parallel plate approximation ([21], [26]) inaccurate for its design, especially in narrow beams. In addition, the fringing field effects and residual stress (second order nonlinear effects) may modify significantly the resonator-electrode capacitance and the resonance frequency respectively [119] and, in consequence, must be taken into account for a more realistic and reliable modeling.
In this chapter a nonlinear compact model with accurate near-real resonator deflection profile based on finite difference method is developed. The finite difference method provides a good accuracy with low computational cost. The model includes the mechanical stiffness cubic nonlinearity term, the intrinsic electrostatic nonlinearities, the fringing field contribu-tions and the residual fabrication stress [114], [120]. Since only time-derivatives equacontribu-tions are used, the model can be embedded within common electrical simulators for system level
simulations into the IC design flow (see appendix B for more details). Finally, the model has been applied to predict the design parameters and biasing conditions to achieve two-well potential distribution (or bistable behavior) in a narrow cc-beam resonator reported in references [109], [121]. Moreover, the accuracy of the model has been validated through extensive FEM simulations and experimental data.