Eldwin J. Ng

Quantum Limit of Quality Factor in Silicon Micro and Nano Mechanical Resonators

Micromechanical resonators are promising replacements for quartz crystals for timing and frequency references owing to potential for compactness, integrability with CMOS fabrication processes, low cost, and low power consumption. To be used in high performance reference application, resonators should obtain a high quality factor. The limit of the quality factor achieved by a resonator is set by the material properties, geometry and operating condition. Some recent resonators properly designed for exploiting bulk-acoustic resonance have been demonstrated to operate close to the quantum mechanical limit for the quality factor and frequency product (Q-f). Here, we describe the physics that gives rise to the quantum limit to the Q-f product, explain design strategies for minimizing other dissipation sources, and present new results from several different resonators that approach the limit.

Epitaxially-encapsulated polysilicon disk resonator gyroscope

We present a 0.6 mm diameter, 20 μm thick epitaxially-sealed polysilicon disk resonator gyro (DRG). High Q (50,000) combined with electrostatic mode-matching and closed-loop quadrature null performed by dedicated electrode sets enables a scale-factor of 0.286 mV/(°/s) and Angle Random Walk (ARW) of 0.006 (°/s)/√Hz. Without precise control of temperature, the minimum Allan deviation is 3.29 °/hr.

Temperature Dependence of the Elastic Constants of Doped Silicon

Resonators fabricated in heavily doped silicon have been noted to have a reduced frequency-temperature dependence compared with lightly doped silicon. The resonant frequency of silicon microelectromechanical systems (MEMS) resonators is largely governed by the materials elastic properties, which are known to depend on doping. In this paper, a suite of different types and orientations of resonators were used to extract the first- and second-order temperature dependences of the elastic constants of p-doped silicon up to 1.7e20 cm-3, and n-doped up to 6.6e19 cm-3 . It is shown that these temperature-dependent elastic constants may be used in finite element analysis to predict the frequency-temperature dependence of similarly doped silicon resonators.

Mode-Matching of Wineglass Mode Disk Resonator Gyroscope in (100) Single Crystal Silicon

In this paper, we present four design methods to overcome (100) silicon crystalline anisotropy and achieve mode-matching in wineglass-mode disk resonator gyroscope (DRG). These methods were validated through experimental characterization of more than 145 different devices that arose from simulations. With the proposed methods, the frequency split of the 250-kHz DRG wineglass modes in (100) silicon was reduced from >10 kHz to as low as 96 Hz (<;0.04% of 250-kHz resonant frequency) without any electrostatic tuning. Perfect mode-matching is then achieved using electrostatic tuning. Mode-matching was maintained within ±10 Hz over a temperature range from -20 °C to 80 °C. The temperature dependence of quality factor is also discussed in this paper. These results allow for the development of high-performance miniature DRGs tuned for degenerate wineglass mode operation from high-quality crystalline silicon material.

100K Q-factor toroidal ring gyroscope implemented in wafer-level epitaxial silicon encapsulation process

This paper reports a new type of degenerate mode gyroscope with measured Q-factor of > 100,000 on both modes at a compact size of 1760 μm diameter. The toroidal ring gyroscope consists of an outer anchor ring, concentric rings nested inside the anchor ring and an electrode assembly at the inner core. Current implementation uses n = 3 wineglass mode, which is inherently robust to fabrication asymmetries. Devices were fabricated using high-temperature, ultra-clean epitaxial silicon encapsulation (EpiSeal) process. Over the 4 devices tested, lowest as fabricated frequency split was found to be 8.5 Hz (122 ppm) with a mean of 21 Hz (Δf/f = 300 ppm). Further electrostatic tuning brought the frequency split below 100 mHz (<; 2 ppm). Whole angle mechanization and pattern angle was demonstrated using a high speed DSP control system. Characterization of the gyro performance using force-rebalance mechanization revealed ARW of 0.047°/√hr and an in-run bias stability of 0.65 deg/hr. Due to the high Q-factor and robust support structure, the device can potentially be instrumented in whole angle mechanization for applications which require high rate sensitivity and robustness to g-forces.

Stability of Silicon Microelectromechanical Systems Resonant Thermometers

The frequency stability of single-crystal silicon microelectromechanical systems resonators encapsulated with epitaxial polysilicon (epi-seal) is investigated. As silicon resonators have significant temperature dependence, the inherent frequency stability of the resonators is masked by temperature-induced noise. Using two resonators side-by-side and assuming identical temperatures and fluctuations, temperature effects are eliminated, resulting in the two resonators tracking each other within ±10 ppb, or 3 × 10-4 °C, over a month. Power and thermal cycling the resonators produced no observable effects on the resonant frequency. This result indicates that silicon resonators make excellent on-chip thermometers, or high stability timing references if temperature is compensated well.

Encapsulated high frequency (235 kHz), high-Q (100 k) disk resonator gyroscope with electrostatic parametric pump

In this paper, we explore the effects of electrostatic parametric amplification on a high quality factor (Q > 100 000) encapsulated disk resonator gyroscope (DRG), fabricated in 〈100〉 silicon. The DRG was operated in the n = 2 degenerate wineglass mode at 235 kHz, and electrostatically tuned so that the frequency split between the two degenerate modes was less than 100 mHz. A parametric pump at twice the resonant frequency is applied to the sense axis of the DRG, resulting in a maximum scale factor of 156.6 μV/(°/s), an 8.8× improvement over the non-amplified performance. When operated with a parametric gain of 5.4, a minimum angle random walk of 0.034°/√h and bias instability of 1.15°/h are achieved, representing an improvement by a factor of 4.3× and 1.5×, respectively.

Lorentz force magnetometer using a micromechanical oscillator

This paper presents a Lorentz force magnetometer employing a micromechanical oscillator. The oscillator, actuated by both electrostatic force and Lorentz force, is based on a 370 μm by 230 μm silicon micromechanical resonator with quality factor (Q) of 13 000. This field-sensitive micromechanical oscillator eliminates the need for an external electronic oscillator and improves magnetometers stability over temperature. The resonator uses no magnetic materials and is encapsulated using an epitaxial polysilicon layer in a process that is fully compatible with complementary metal-oxide-semiconductor manufacturing. The sensor has a magnetic field resolution of 128 nT/rt-Hz with 2.1 mA bias current.

Fatigue Experiments on Single Crystal Silicon in an Oxygen-Free Environment

The fatigue lifetime of single crystal silicon (SCS) was characterized in an environment free of oxygen, humidity, and organics. Long-term (> 1010 Hz) fatigue experiments performed with smooth-walled SCS devices showed no signs of fatigue damage up to 7.5 GPa. In contrast, experiments using SCS devices with a silicon dioxide (SiO2) coating and rough sidewalls due to scalloping from deep reactive ion etching exhibited fatigue drift at 2.7 GPa and suffered from short-term (<; 1010 Hz) fatigue failure at stress levels >3 GPa. In these SCS-SiO2 experiments, the initiation of fracture occurs in the SiO2 layer. It is concluded that fatigue in this case is likely attributed to a subcritical cracking mechanism; not reaction-layer nor dislocation related. A cross-comparison with other works from literature is developed to show that packaging a pristine device in an inert environment is necessary in order to operate devices at high-stress levels.

A Unified Epi-Seal Process for Fabrication of High-Stability Microelectromechanical Devices

This paper presents a thin-film wafer-level encapsulation process based on an epitaxial deposition seal that incorporates both narrow and wide lateral transduction gaps (0.7-50 μm), both in-plane and out-of-plane electrodes, and does not require release etch-holes in the device layer. Resonant structures fabricated in this process demonstrate high-quality factors ( f × Q products of up to 2.27e + 13 Hz) and exceptional stability (±18 ppb over one month) with no obvious aging trends. Studies on cavity pressure indicate that vacuum levels better than 0.1 Pa can be achieved after final encapsulation, thus reducing gas damping for high surface-to-volume devices. The vast diversity of functioning devices built in this process demonstrates the potential for combinations of high-performance MEMS devices in a single process and/or single chip.