Microelectromechanical Oscillator Research Articles

Nonlinear Dynamics of a New Class of Micro-Electromechanical Oscillators—Open Problems

In this paper, we propose a new class of micro-electromechanical oscillators. Some investigations based on Melnikov’s approach are applied for identifying some chaotic possibilities. We demonstrate also some specialized modules for investigating the dynamics of these oscillators. This will be included as an integral part of a planned much more general Web-based application for scientific computing. It turns out that the theoretical apparatus for studying the circuit implementation (design, fabricating, etc.) of the considered differential model for large parameter values is extremely complex and requires a serious investigation. This is the reason to offer this model to the attention of specialists working in this scientific direction. Some open problems related to the use of existing computer algebraic systems for the study of this class of oscillators for large values of n,m and N are also posed. In general, the entire article is subordinated to this frank conversation with the readers with the sole purpose being the professional upgrading of the specialized modules provided for this purpose in subsequent licensed versions of CAS.

The analysis for the dynamic pull-in of a micro-electromechanical system

The pull-in behavior is an inherent property of the micro-electromechanical oscillator. The bisection method and an iterative method are introduced to find its pull-in voltage, and the main factors affecting the pull-in voltage are elucidated. The simple and efficient operability is demonstrated through theoretical analysis and result comparison.

Al0.68Sc0.32N Lamb wave resonators with electromechanical coupling coefficients near 10.28%

Aluminum scandium nitride (Al1−xScxN/AlScN) (x = 0.32) Lamb wave resonators (LWR) have been fabricated and tested to demonstrate electromechanical coupling coefficients (kt2) in excess of 10%. The resonators exhibited an average kt2 and unloaded quality factor (Qu) of 10.28% and 711, respectively, when calculated from the measured data. Applying the Butterworth Van-Dyke (BVD) model to the measured data enabled the extraction of the resonator's lumped element parameters to calculate the motional quality factor (Qm), which neglects the contributions of the electrical traces. For the best measured resonator response, results from the BVD model showed a Qm of 1184 and a resulting figure-of-merit (FOM = K2 · Qm) of 100. Comparing the response of similar AlScN and AlN resonators shows that the AlScN LWR has a significantly lower motional resistance (Rm), suggesting that AlScN has a strong potential for use in piezoelectric microelectromechanical oscillators.

Oven-Controlled MEMS Oscillator with Integrated Micro-Evaporation Trimming.

This study reports an oven-controlled microelectromechanical systems oscillator with integrated micro-evaporation trimming that achieves frequency stability over the industrial temperature range and permanent frequency trimming after vacuum packaging. The length-extensional-mode resonator is micro-oven controlled and doped degenerately with phosphorous to achieve a frequency instability of ±2.6 parts per million (ppm) in a temperature range of −40 to 85 °C. The micro-evaporators are bonded to the resonator, integrated face-to-face, and encapsulated in vacuum. During trimming, the micro-evaporators are heated electrically, and the aluminum layers on their surfaces are evaporated and deposited on the surface of the resonator that trims the resonant frequency of the resonator permanently. The impact of the frequency trimming on the temperature stability is very small. The temperature drift increases from ±2.6 ppm within the industrial temperature range before trimming to ±3.3 ppm after a permanent trimming of −426 ppm based on the local evaporation of Al. The trimming rate can be controlled by electric power. A resonator is coarse-trimmed by approximately −807 ppm with an evaporation power of 960 mW for 0.5 h, and fine-trimmed by approximately −815 ppm with an evaporation power of 456 mW for 1 h. Though the Q-factor decreases after trimming, a Q-factor of 304,240 is achieved after the trimming of −1442 ppm.

An on-chip fully electronic molecular clock based on sub-terahertz rotational spectroscopy

Mobile electronic devices require stable, portable and energy-efficient frequency references (or clocks). However, current approaches using quartz-crystal and microelectromechanical oscillators suffer from frequency drift. Recent advances in chip-scale atomic clocks, which probe the hyperfine transitions of evaporated alkali atoms, have led to devices that can overcome this issue, but their complex construction, cost and power consumption limit their broader deployment. Here we show that sub-terahertz rotational transitions of polar gaseous molecules can be used as frequency bases to create low-cost, low-power miniaturized clocks. We report two molecular clocks probing carbonyl sulfide (16O12C32S), which are based on laboratory-scale instruments and complementary metal–oxide–semiconductor chips. Compared with chip-scale atomic clocks, our approach is less sensitive to external influences and offers faster frequency error compensation, and, by eliminating the need for alkali metal evaporation, it offers faster start-up times and lower power consumption. Our work demonstrates the feasibility of monolithic integration of atomic-clock-grade frequency references in mainstream silicon-chip systems. The sub-terahertz rotational transitions of polar molecules can be used as frequency bases to create low-cost, low-power miniaturized clocks.

Impact of excitation waveform on the frequency stability of electrostatically-actuated micro-electromechanical oscillators

In this paper, the frequency stability of high-Q electrostatically-actuated MEMS oscillators with cubic restoring forces, and its relation with the amplitude, the phase and the shape of the excitation waveform, is studied. The influence on close-to-the carrier frequency noise of additive processes (such as thermomechanical noise) or parametric processes (bias voltage fluctuations, feedback phase fluctuations, feedback level fluctuations) is taken into account. It is shown that the optimal operating conditions of electrostatically-actuated MEMS oscillators are highly waveform-dependent, a factor that is largely overlooked in the existing literature. This simulation-based study covers the cases of harmonic and pulsed excitation of a parallel-plate capacitive MEMS resonator.

Direct observation of coherent energy transfer in nonlinear micromechanical oscillators

Energy dissipation is an unavoidable phenomenon of physical systems that are directly coupled to an external environmental bath. In an oscillatory system, it leads to the decay of the oscillation amplitude. In situations where stable oscillations are required, the energy dissipated by the vibrations is usually compensated by replenishment from external energy sources. Consequently, if the external energy supply is removed, the amplitude of oscillations start to decay immediately, since there is no means to restitute the energy dissipated. Here, we demonstrate a novel dissipation engineering strategy that can support stable oscillations without supplying external energy to compensate losses. The fundamental intrinsic mechanism of resonant mode coupling is used to redistribute and store mechanical energy among vibrational modes and coherently transfer it back to the principal mode when the external excitation is off. To experimentally demonstrate this phenomenon, we exploit the nonlinear dynamic response of microelectromechanical oscillators to couple two different vibrational modes through an internal resonance.

Anomalous Resonance Frequency Shift of a Microelectromechanical Oscillator in Superfluid $$^3$$He-B

A superfluid \(^3\)He film with a thickness of 1.25 \(\upmu \)m was studied using a microelectromechanical oscillator at various pressures of 9.2, 18.2, 25.2, and 28.6 bars. The oscillator was driven in the linear damping regime where the damping coefficient is independent of the velocity of the oscillator. The resonance frequency shows weak temperature and pressure dependences in the low temperature limit. An inertia coefficient of \(\approx \)0.1 was obtained in the ballistic regime. When the temperature rose from the lowest temperature, the resonance frequency of the resonator exhibited an unusual behavior, a rapid increase beyond the intrinsic value as temperature increases, for 9.2 and 18.2 bars.

Critical Velocity in the Presence of Surface Bound States in Superfluid He3−B

A microelectromechanical oscillator with a gap of 1.25 μm was immersed in superfluid ^{3}He-B and cooled below 250 μK at various pressures. Mechanical resonances of its shear motion were measured at various levels of driving force. The oscillator enters into a nonlinear regime above a certain threshold velocity. The damping increases rapidly in the nonlinear region and eventually prevents the velocity of the oscillator from increasing beyond the critical velocity which is much lower than the Landau critical velocity. We propose that this peculiar nonlinear behavior stems from the escape of quasiparticles from the surface bound states into the bulk fluid.

A Fully Integrated Oven Controlled Microelectromechanical Oscillator—Part II: Characterization and Measurement

This paper, the second of two parts, reports the measurement and characterization of a fully integrated oven controlled microelectromechanical oscillator (OCMO). The OCMO takes advantage of high thermal isolation and monolithic integration of both aluminum nitride (AlN) micromechanical resonators and electronic circuitry to thermally stabilize or ovenize all the components that comprise an oscillator. Operation at microscale sizes allows implementation of high thermal resistance platform supports that enable thermal stabilization at very low-power levels when compared with the state-of-the-art oven controlled crystal oscillators. A prototype OCMO has been demonstrated with a measured temperature stability of -1.2 ppb/°C, over the commercial temperature range while using tens of milliwatts of supply power and with a volume of 2.3 mm <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (not including the printed circuit board-based thermal control loop). In addition, due to its small thermal time constant, the thermal compensation loop can maintain stability during fast thermal transients (>10 °C/min). This new technology has resulted in a new paradigm in terms of power, size, and warm up time for high thermal stability oscillators.

A Fully Integrated Oven Controlled Microelectromechanical Oscillator—Part I: Design and Fabrication

This paper, the first of two parts, reports the design and fabrication of a fully integrated oven controlled microelectromechanical oscillator (OCMO). This paper begins by describing the limits on oscillator frequency stability imposed by the thermal drift and electronic properties (Q, resistance) of both the resonant tank circuit and feedback electronics required to form an electronic oscillator. An OCMO is presented that takes advantage of high thermal isolation and monolithic integration of both micromechanical resonators and electronic circuitry to thermally stabilize or ovenize all the components that comprise an oscillator. This was achieved by developing a processing technique where both silicon-on-insulator complementary metal-oxide-semiconductor (CMOS) circuitry and piezoelectric aluminum nitride, AlN, micromechanical resonators are placed on a suspended platform within a standard CMOS integrated circuit. Operation at microscale sizes achieves high thermal resistances (~10 °C/mW), and hence thermal stabilization of the oscillators at very low-power levels when compared with the state-of-the-art ovenized crystal oscillators, OCXO. A constant resistance feedback circuit is presented that incorporates on platform resistive heaters and temperature sensors to both measure and stabilize the platform temperature. The limits on temperature stability of the OCMO platform and oscillator frequency imposed by the gain of the constant resistance feedback loop, placement of the heater and temperature sensing resistors, as well as platform radiative and convective heat losses are investigated.

Predicting the closed-loop stability and oscillation amplitude of nonlinear parametrically amplified oscillators

This work investigates the closed-loop operation of microelectromechanical oscillators in the presence of both cubic (Duffing) nonlinearities and parametric amplification. We present a theoretical model for this system that enables us to predict oscillation amplitude and instability and experimentally verify it using a silicon disk resonator with a quality factor (Q) of 85 000 and a natural frequency of 251 kHz. We determine that, contrary to previous understanding gained from analyzing the open-loop system, the presence of cubic nonlinearities does not limit the maximum stable oscillation amplitude if the resonator is operated in a closed loop. In addition, the stability and amplitude behavior predicted by our theoretical model are independent of the presence or severity of cubic nonlinearities, or on drive amplitude.

Analysis of resonant pull-in of micro-electromechanical oscillators

In this paper, the equations governing the pull-in of electrostatic (micro-electromechanical systems MEMS) oscillators are established and analyzed. This phenomenon defines the maximal oscillation amplitude that can be obtained without incurring instability and, hence, an upper limit to the performance of a given device. The proposed approach makes it possible to accurately predict pull-in behavior from the purely resonant case, in which the electrostatic bias is very small, to the static case. The method is first exposed in the case of a parallel-plate resonator and the influence of the excitation waveform on the resonant pull-in characteristics is assessed. It is then extended to the more complex case of clamped–clamped and cantilever beams. The results are validated by comparison with transient simulations.

Synchronization in a coupled architecture of microelectromechanical oscillators

There has been much recent interest in engineering the phenomenon of synchronization in coupled micro-/nano-scale oscillators for applications ranging from precision time and frequency references to new approaches to information processing. This paper presents descriptive modelling detail and further experimental validation of the phenomenon of mutual synchronization in coupled MEMS oscillators building upon recent experimental validation of this concept by the present authors. In particular, the underlying dependence of the observation of synchronization on system parameters is studied through numerical and analytical modelling while considering essential nonlinearities in both the resonator and circuit domain. Experimental results demonstrating synchronized response are elaborated based on the realization of electrically coupled MEMS resonator based square-wave oscillators. The experimental results on frequency entrainment are found to be in general agreement with results obtained through analytical modeling and numerical simulation. The concept presented here is scalable and could be used to investigate the dynamics of large-arrays of coupled MEMS oscillators.

Nonlinear dynamics of a microelectromechanical oscillator with delayed feedback

We study the dynamics of a nonlinear electromechanical oscillator with delayed feedback. Compared to their linear counterparts, we find that the dynamics is dramatically different. The well-known Barkhausen stability criterion ceases to exist, and two modes of operation emerge: one characterized by hysteresis in combination with a bistable frequency and amplitude; the other, by self-stabilization of the oscillation frequency and amplitude. The observed features are captured by a model based on a Duffing equation with delayed force feedback. Nonlinear oscillators with delayed force feedback are exemplary for a large class of dynamic systems.

Metal microelectromechanical oscillator exhibiting ultra-high water vapor resolution

Water vapor sensing characterization of a metal resonator fabricated with an industrial 0.35 μm CMOS technology is reported. The resonator frequency is ∼13.2 MHz and exhibits a sensitivity magnitude of ∼3.5 kHz per %RH without requiring any additional hygroscopic coating layer. An on-chip integrated oscillator circuit enables an unprecedented resolution of 0.005 %RH.

NEW RESULTS FOR THE CASIMIR INTERACTION: SAMPLE CHARACTERIZATION AND LOW TEMPERATURE MEASUREMENTS

We describe our latest results in the separation dependence of the Casimir interaction in the sphere-plane geometry for two Au-coated surfaces. All results are obtained by measuring the change in the resonant frequency of a sensitive microelectromechanical torsional oscillator as the separation between the sphere and the plane is changed. By means of the proximity force approximation, the change in resonant frequency yields the Casimir pressure between two parallel plates at the same separation. We present results for a new sample at room temperature, where the dielectric function has been measured in the 190-825 nm range. We show that the results of the Casimir force in this sample and in previous samples are virtually indistinguishable. Furthermore, the observed differences between measured and tabulated optical properties data do not show any effect on the calculation of the Casimir interaction. We also present results of the measurement of the Casimir force between a sphere and a plane at 300, 77, 4.2 and 2.1 K. While low temperature results are noisier than room temperature ones, precluding a direct exclusion of either the Drude or the plasma model, the average of the measurements coincide at all temperatures.

1.05-GHz CMOS oscillator based on lateral- field-excited piezoelectric AlN contour- mode MEMS resonators

This paper reports on the first demonstration of a 1.05-GHz microelectromechanical (MEMS) oscillator based on lateral-field-excited (LFE) piezoelectric AlN contourmode resonators. The oscillator shows a phase noise level of -81 dBc/Hz at 1-kHz offset frequency and a phase noise floor of -146 dBc/Hz, which satisfies the global system for mobile communications (GSM) requirements for ultra-high frequency (UHF) local oscillators (LO). The circuit was fabricated in the AMI semiconductor (AMIS) 0.5-microm complementary metaloxide- semiconductor (CMOS) process, with the oscillator core consuming only 3.5 mW DC power. The device overall performance has the best figure-of-merit (FoM) when compared with other gigahertz oscillators that are based on film bulk acoustic resonator (FBAR), surface acoustic wave (SAW), and CMOS on-chip inductor and capacitor (CMOS LC) technologies. A simple 2-mask process was used to fabricate the LFE AlN resonators operating between 843 MHz and 1.64 GHz with simultaneously high Q (up to 2,200) and kt 2 (up to 1.2%). This process further relaxes manufacturing tolerances and improves yield. All these advantages make these devices suitable for post-CMOS integrated on-chip direct gigahertz frequency synthesis in reconfigurable multiband wireless communications.

Dynamical mechanism of intrinsic localized modes in microelectromechanical oscillator arrays

Experimental evidence of intrinsic localized modes (ILMs) in microelectromechanical oscillator arrays has been reported recently. In this paper, we carry out a detailed analysis of a new mechanism for ILMs in typical experimental settings; that is, spatiotemporal chaos is ubiquitous and it provides a natural platform for actual realization of various ILMs through frequency control. We find that unstable periodic orbits associated with ILMs are pivotal for spatiotemporal chaos to arise and these orbits are the keys to stabilizing ILMs by frequency modulation.

Limit Cycles in a MEMS Oscillator

In this paper, we apply methods of nonlinear dynamics to study the behavior of a microelectromechanical oscillator. We show how the analysis predicts the appearance of a devil's staircase-like relationship between frequencies, but also show how the output frequency of the oscillator, and hence the devil's staircase, may not be uniquely determined. Both of these features - the sequence of steps corresponding to different output frequencies and the fact that the output frequency may not be uniquely defined for a given set of parameters - have a direct bearing on the ease with which a designer can produce an useful oscillator of this form.