Quality factor (Q) is an important property of micro- and nano-electromechanical (MEM/NEM) resonators that underlie timing references, frequency sources, atomic force microscopes, gyroscopes, and mass sensors. Various methods have been utilized to tune the effective quality factor of MEM/NEM resonators, including external proportional feedback control, optical pumping, mechanical pumping, thermal-piezoresistive pumping, and parametric pumping. This work reviews these mechanisms and compares the effective Q tuning using a position-proportional and a velocity-proportional force expression. We further clarify the relationship between the mechanical Q, the effective Q, and the thermomechanical noise of a resonator. We finally show that parametric pumping and thermal-piezoresistive pumping enhance the effective Q of a micromechanical resonator by experimentally studying the thermomechanical noise spectrum of a device subjected to both techniques.

Effective quality factor tuning mechanisms in micromechanical resonators

In this paper, we report Q-factor over 1 million on both n = 2 wineglass modes, and high-frequency symmetry (Af/f ) of 132 ppm on wafer-level microglassblown 3-D fused silica wineglass resonators at a compact size of 7-mm diameter and center frequency of 105 kHz. In addition, we demonstrate for the first time, out-of-plane capacitive transduction on microelectromechanical systems wineglass resonators. High Q-factor is enabled by a high aspect ratio, self-aligned glassblown stem structure, careful surface treatment of the perimeter area, and low internal loss fused silica material. Electrostatic transduction is enabled by detecting the spatial deformation of the 3-D wineglass structure using a new out-of-plane electrode architecture. Out-of-plane electrode architecture enables the use of sacrificial layers to define the capacitive gaps and 10 μm capacitive gaps have been demonstrated on a 7-mm shell, resulting in over 9 pF of active capacitance within the device. Microglassblowing may enable batch-fabrication of high-performance fused silica wineglass gyroscopes at a significantly lower cost than their precision-machined macroscale counterparts.

Demonstration of 1 million Q -factor on microglassblown wineglass resonators with out-of-plane electrostatic transduction

This paper reports a high-performance vibrating ring gyroscope fabricated in (111) oriented single-crystal silicon (SCS). High-performance microgyroscopes are needed in many applications, including inertial navigation, control, and defense/avionics/space. The ring gyroscope provides a number of advantages, including excellent mode matching, high-resolution, low zero-rate output, and long-term stability. In this paper, a SCS vibrating ring gyroscope with high aspect ratio silicon on glass structure was designed, fabricated and tested. The ring is 2.7mm in diameter and 150μm thick. The gyro has the following measured performance: high Q (12000), good nonlinearity (0.02%), large sensitivity (132 mV/°/sec), low output noise (10.4°/hr/Hz) and high resolution (7.2°/hr). The maximum bias shift is less than ±1°/sec over 10 hours without thermal control.

A single-crystal silicon vibrating ring gyroscope

Parametric amplification, resulting from intentionally varying a parameter in a resonator at twice its resonant frequency, has been successfully employed to increase the sensitivity of many micro- and nano-scale sensors. Here, we introduce the concept of self-induced parametric amplification, which arises naturally from nonlinear elastic coupling between the degenerate vibration modes in a micromechanical disk-resonator, and is not externally applied. The device functions as a gyroscope wherein angular rotation is detected from Coriolis coupling of elastic vibration energy from a driven vibration mode into a second degenerate sensing mode. While nonlinear elasticity in silicon resonators is extremely weak, in this high quality-factor device, ppm-level nonlinear elastic effects result in an order-of-magnitude increase in the observed sensitivity to Coriolis force relative to linear theory. Perfect degeneracy of the primary and secondary vibration modes is achieved through electrostatic frequency tuning, which also enables the phase and frequency of the parametric coupling to be varied, and we show that the resulting phase and frequency dependence of the amplification follow the theory of parametric resonance. We expect that this phenomenon will be useful for both fundamental studies of dynamic systems with low dissipation and for increasing signal-to-noise ratio in practical applications such as gyroscopes.

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Self-induced parametric amplification arising from nonlinear elastic coupling in a micromechanical resonating disk gyroscope

We demonstrate, for the first time, sub-1 Hz frequency symmetry in micro-glassblown wineglass resonators with integrated electrode structures. A new fabrication process based on deep glass dry etching was developed to fabricate micro-wineglasses with self-aligned stem structures and integrated electrodes. The wineglass modes were identified by electrostatic excitation and mapping the velocity of motion along the perimeter using laser Doppler interferometry. A frequency split (Δf) of 0.15 and 0.2 Hz was demonstrated for n=2 and n=3 wineglass modes, respectively. To verify the repeatability of the results, a total of five devices were tested, three out of five devices showed . Frequency split stayed below 1 Hz for dc bias voltages up to 100 V, confirming that the low frequency split is attributed to high structural symmetry and not to capacitive tuning. High structural symmetry and atomically smooth surfaces (0.23 nm Sa) of the resonators may enable new classes of high performance 3-D MEMS devices, such as rate-integrating MEMS gyroscopes.

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Achieving Sub-Hz Frequency Symmetry in Micro-Glassblown Wineglass Resonators

We present a 2-mm diameter, 35-μm-thick disk resonator gyro (DRG) fabricated in silicon with integrated 0.35-μm CMOS analog front-end circuits. The device is fabricated in the commercial InvenSense Fabrication MEMSCMOS integrated platform, which incorporates a wafer-level vacuum seal, yielding a quality factor (Q) of 2800 at the DRGs 78-kHz resonant frequency. After performing electrostatic tuning to enable mode-matched operation, this DRG achieves a 55 μV/°/s sensitivity. Resonator vibration in the sense and drive axes is sensed using capacitive transduction, and amplified using a lownoise, on-chip integrated circuit. This allows the DRG to achieve Brownian noise-limited performance. The angle random walk is measured to be 0.008°/s/√(Hz) and the bias instability is 20°/h.

Silicon MEMS Disk Resonator Gyroscope With an Integrated CMOS Analog Front-End

We present a demonstration of a whole-angle mode operation of a 0.6 mm single-crystal silicon disk resonator gyroscope (DRG). This device has a Q factor of ~80,000 and a resonant frequency of ~250 kHz and is fabricated in the epi-seal process. Discrete-time control algorithms for rate-integrating gyro operation were implemented based on Lynch’s algorithm. Despite the fact that this DRG is over 5 orders of magnitude smaller than the 58 mm HRG, the device’s error sources are shown to be accurately modeled by the basic error models developed by Lynch.

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Disk resonator gyroscope with whole-angle mode operation

The hemispherical resonator gyro (HRG) is low loss and high stability, spurring recent interest in micro-scale hemispherical resonators. To achieve mode-matching and high-Q performance in a hemispherical resonator, geometric symmetry in combination with low thermoelastic damping structural material are critical. In this work, we describe the development of millimeter scale 3D hemispherical shell resonators fabricated from polycrystalline diamond, a material with low thermoelastic damping and very high stiffness. The relation between the fourth harmonic (4θ) in a Fourier analysis of the resonator's radius r(θ) and frequency mismatch (Δf) of the 2θ elliptical vibration modes of the shell resonator is demonstrated.

Micromachined polycrystalline diamond hemispherical shell resonators

We present the development of millimeter scale 3D hemispherical shell resonators fabricated from the polycrystalline diamond, a material with low thermoelastic damping and very high stiffness. These hemispherical wineglass resonators with 1.1 mm diameter are fabricated through a combination of micro-electro discharge machining (EDM) and silicon micromachining techniques. Using piezoelectric and electrostatic excitation and optical vibration measurement, the elliptical wineglass vibration mode is determined to be at 18.321 kHz, with the two degenerate wineglass modes having a relative frequency mismatch of 0.03%. A study on the effect of the size and misalignment of the anchor and resonator’s radius variation on both the average frequency and frequency mismatch of the 2θ elliptical vibration modes is carried out. It is shown that the absolute frequency of a wineglass resonator will increase with the anchor size. It is also demonstrated that the fourth harmonic of radius variation is linearly related to the frequency mismatch. (Some figures may appear in colour only in the online journal)

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Hemispherical wineglass resonators fabricated from the microcrystalline diamond

Toward the objective of direct angle measurement using a rate integrating gyroscope (RIG) without a minimum rate threshold and with performance limited only by electrical and mechanical thermal noise, in this paper, we present the implementation of a generalized electronic feedback method for the compensation of MEMS gyroscope damping asymmetry (anisodamping) and stiffness asymmetry (anisoelasticity) on a stand-alone digital signal processing platform. Using the new method, the precession angle-dependent bias error and minimum rate threshold, two issues identified by Lynch for a MEMS RIG [1] due to anisodamping are overcome. To minimize angledependent bias, we augment the electronic feedback force of the amplitude regulator with a non-unity gain output distribution matrix selected to correct for anisodamping. Using this method, we have decreased the angle-dependent bias error by a factor of 12, resulting in a minimum rate threshold of ~3 °/s. To further improve RIG performance, an electronically induced self-precession rate is incorporated and successfully demonstrated to lower the rate threshold.

Parsa Taheri-Tehrani

Papers

Silicon MEMS Disk Resonator Gyroscope With an Integrated CMOS Analog Front-End

Disk resonator gyroscope with whole-angle mode operation

Micromachined polycrystalline diamond hemispherical shell resonators

Hemispherical wineglass resonators fabricated from the microcrystalline diamond

Micromechanical Rate Integrating Gyroscope With Angle-Dependent Bias Compensation Using a Self-Precession Method