Saurabh A. Chandorkar

Real-Time Temperature Compensation of MEMS Oscillators Using an Integrated Micro-Oven and a Phase-Locked Loop

We present a new temperature compensation system for microresonator-based frequency references. It consists of a phase-locked loop (PLL) whose inputs are derived from two microresonators with different temperature coefficients of frequency. The resonators are suspended within an encapsulated cavity and are heated to a constant temperature by the PLL controller, thereby achieving active temperature compensation. We show repeated real-time measurements of three 1.2-MHz prototypes that achieve a frequency stability of ± 1 ppm from -20°C to +80°C, as well as a technique to reduce steady-state frequency errors to ±0.05 ppm using multipoint calibration.

Temperature-Insensitive Composite Micromechanical Resonators

Utilizing silicon and silicon dioxides opposing temperature coefficients of Youngs modulus, composite resonators with zero linear temperature coefficient of frequency are fabricated and characterized. The resulting resonators have a quadratic temperature coefficient of frequency of approximately -20 ppb/degC2 and a tunable turnover temperature in the -55degC to 125degC range. Reduction of the temperature dependence of frequency is shown in flexural-mode resonators (700 kHz-1.3 MHz) and extensional-mode ring resonators (20 MHz). The linear temperature coefficient of Youngs modulus of silicon dioxide is extracted from measurements to be +179 ppm/degC. The composite resonators are fabricated and packaged in a CMOS-compatible wafer-scale hermetic encapsulation process. The long-term stability of the resonators is monitored for longer than six months. Although most devices exhibit less than 2 ppm frequency drift, there is evidence of dielectric charging in the silicon dioxide.

Temperature-compensated high-stability silicon resonators

Composite micromechanical resonators were encapsulated in a hermetic environment using a wafer-scale encapsulation process compatible with complementary metal-oxide semiconductor processing. The resonator structure is comprised of single crystal silicon with a silicon dioxide coating and shows a frequency-temperature sensitivity that is comparable to uncompensated quartz crystal resonators. A frequency variation of less than 200ppm is achieved over a −40–125°C temperature range. The resonator exhibits a quadratic temperature behavior with a turnover temperature at which the frequency becomes insensitive to small temperature changes. The turnover temperature can be controlled for use in high precision frequency references.

Limits of quality factor in bulk-mode micromechanical resonators

In this paper we present the dominant energy loss mechanisms and quality factor (Q) limits in bulk mode micromechanical resonators. We demonstrate that in resonators with an appropriately designed stem connection to anchor the maximum achievable Q limit is set by either Thermoelastic dissipation (TED) or the Akhieser effect (AKE). Furthermore, we suggest a choice of materials for achieving maximum Qs in micromechanical resonators. It is established here that silicon resonators can theoretically achieve higher Qs than quartz and we predict that by using alternative materials, such as silicon carbide, it is possible to surpass the Q of quartz by more than an order of magnitude.

Impact of geometry on thermoelastic dissipation in micromechanical resonant beams

Thermoelastic dissipation (TED) is analyzed for complex geometries of micromechanical resonators, demonstrating the impact of resonator design (i.e., slots machined into flexural beams) on TED-limited quality factor. Zener first described TED for simple beams in 1937. This work extends beyond simple beams into arbitrary geometries, verifying simulations that completely capture the coupled physics that occur. Novel geometries of slots engineered at specific locations within the flexural resonator beams are utilized. These slots drastically affect the thermal-mechanical coupling and have an impact on the quality factor, providing resonators with quality factors higher than those predicted by simple Zener theory. The ideal location for maximum impact of slots is determined to be in regions of high strain. We have demonstrated the ability to predict and control the quality factor of micromechanical resonators limited by thermoelastic dissipation. This enables tuning of the quality factor by structure design without the need to scale its size, thus allowing for enhanced design optimization

Thermal Isolation of Encapsulated MEMS Resonators

This paper presents an in-chip thermal-isolation technique for a micro-ovenized microelectromechanical-system resonator. Resonators with a microoven can be used for high-precision feedback control of temperature to compensate for the temperature dependence of resonator frequency over a wide temperature range. However, ovenization requires power consumption for heating, and the thermal time constant must be minimized for effective temperature control. This paper demonstrates an efficient local-thermal-isolation mechanism, which can reduce the power requirement to a few milliwatts and the thermal time constant to a few milliseconds. In this method, the mechanical suspension of the resonator is modified to provide thermal isolation and include an integrated resistive heater. This combination provides mechanical suspension, electrical heating, and thermal isolation in a compact structure that requires low heating power and has a small thermal time constant. A power consumption of approximately 12 mW for a 125degC temperature rise and a thermal time constant ranging from 7 to 10 ms is reported in this paper, which is orders of magnitude lower than that of commercially available ovenized quartz resonators. A CMOS-compatible wafer-scale encapsulation process is used to fabricate this device, and the thermal-isolation design is achieved without any modification to the existing resonator fabrication process.

Energy loss in MEMS resonators and the impact on inertial and RF devices

In this paper, we review the current understanding of energy loss mechanisms in micromachined (MEMS and NEMS) devices. We describe the importance of high quality factor (Q) to the performance of MEMS gyros and MEMS resonators used in radio-frequency applications.

Using the temperature dependence of resonator quality factor as a thermometer

Silicon micromechanical resonators have been designed to have a quality factor (Q) that is a strong function of temperature. This is an ideal sensor for the temperature of the resonator—it is instantaneous, consumes no power, and indicates the temperature of the resonator structure with high sensitivity. The authors present a practical implementation of an oscillator system using these resonators with a temperature resolution of better than 0.002°C. The Q(T) signal is uniquely suited for implementing feedback control of the resonator temperature, thereby stabilizing the frequency silicon micromechanical resonators and enabling their use in high-stability frequency reference applications.

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.

Multimode thermoelastic dissipation

In this paper, we investigate thermoelastic dissipation (TED) in systems whose thermal response is characterized by multiple time constants. Zener [Phys. Rev. 52, 230 (1937)] analyzed TED in a cantilever with the assumption that heat transfer is one dimensional. He showed that a single thermal mode was dominant and arrived at a formula for quantifying the quality factor of a resonating cantilever. In this paper, we present a formulation of thermoelastic damping based on entropy generation that accounts for heat transfer in three dimensions and still enables analytical closed form solutions for energy loss estimation in a variety of resonating structures. We apply this solution technique for estimation of quality factor in bulk mode, torsional, and flexural resonators. We show that the thermoelastic damping limited quality factor in bulk mode resonators with resonator frequency much larger than the eigenfrequencies of the dominant thermal modes is inversely proportional to the frequency of the resonator un...