Square-extensional Research Articles

Mechanically Coupled Single-Crystal Silicon MEMS Resonators for TCF Manipulation

This paper presents mechanically coupled single-crystal silicon (SCS) microelectromechanical system (MEMS) resonators for temperature coefficient of frequency (TCF) manipulation. The reported mechanically coupled square extensional (SE) mode resonator consists of two square plates coupled with a beam or a third square plate purposely vibrating in another bulk mode. The SE mode resonators and the coupled structure can be considered as two kinds of individual resonators with different TCFs. Within the specific dimensions of the coupled resonator, the resonant frequencies of the two components are assumed to be equal. Hence, the TCF of the mechanically coupled resonator could be manipulated by adjusting the sizes of the coupling parts. The design and fabrication of coupled resonators have been completed in a degenerate-doped (100) SCS wafer. Measurement results illustrate that the TCFs of the coupled resonators can be purposely manipulated by adjusting the volume and equivalent spring constant ratios of the individual resonators. The minimal measured frequency shift is roughly <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\pm$</tex-math> </inline-formula> 10 ppm for an 18 MHz coupled SE mode resonator, over the entire industrial temperature range of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$-$</tex-math> </inline-formula> 40 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> C to 85 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{\circ}$</tex-math> </inline-formula> C. This work highlights a novel and effective way to manipulate the TCFs of MEMS resonators via mechanical coupling. 2022-0197

Temperature Compensated Bulk-Mode Capacitive MEMS Resonators With ±16 ppm Temperature Stability Over Industrial Temperature Ranges

This letter reports temperature compensated single crystal silicon (SCS) bulk-mode capacitive microelectromechanical system (MEMS) resonators with high temperature stability of less than ±20 ppm over industrial temperature range. Degenerate doping is adopted to change the temperature coefficient of frequency ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TCF$ </tex-math></inline-formula> ) and adjusting of crystal orientation is implemented to tune the turnover points for the MEMS resonators, thanks to the different temperature coefficient of elastic constants ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TCE$ </tex-math></inline-formula> ) of the resonators along different crystal orientation. The adjustment effects are verified by using finite element method (FEM) simulations as well as measurement results. Two bulk-modes, namely length-extensional (LE) and square-extensional (SE) mode resonators along various crystal orientations in a degenerate-doped (100) SCS wafer are designed and fabricated. The measurement frequency shifts are approximately ±20 ppm for the LE mode resonator and ±16 ppm for the SE mode resonator as the resonators are placed along 22.5° from the <110> direction over industrial temperature range of −40°C to 85 °C, respectively. It highlights an effective way to reduce the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$TCF$ </tex-math></inline-formula> via degenerate doping and precisely adjust the turnover point by crystal orientation tuning for MEMS resonators. [2022-0046]

Piezoelectric-on-Silicon Square Wine-Glass Mode Resonator for Enhanced Electrical Characterization in Water

We describe a piezoelectric square wine-glass (SWG) mode micromechanical resonator as a unique device topology for liquid phase sensing applications that offers enhanced quality (Q)-factor in water over other common contour modes. We show a Q-factor in water of 377 for 14.08-MHz SWG mode resonator (mean of multiple measurements), which is one of the highest Q-factors among piezoelectric resonators tested when covered in a fluid. We show that the Q-factor in water of the SWG mode increases much more significantly with a reduction in resonator size relative to the square-extensional (SE) mode, such that the Q-factor the SWG mode can be almost twice of the SE mode. We present an analytical model for the mode shape of the proposed SWG resonators and their effective electromechanical coupling factor $({k}_{\text {eff}}^{{2}})$ . Due to the enhanced Q-factor in water, reasonable ${k}_{\text {eff}}^{{2}}$ , and feedthrough cancelation (using fully differential transduction with the fluid loaded from the backside of the device), we show that a strong resonant peak of 29.8 dB is obtainable in water.

A Lorentz force magnetometer based on a piezoelectric-on-silicon square-extensional mode micromechanical resonator

In this letter, we present a Lorentz force magnetic field sensor based on a thin-film piezoelectric-on-silicon (TPoS) CMOS-compatible resonator for the detection of an out-of-plane (perpendicular to the plane of fabrication) magnetic field. We here exploit the fundamental breathing mode of vibration in a suspended square plate, which is commonly referred to as the square-extensional (SE) mode. The symmetric stress profile of the SE mode avails stresses in the two orthogonal in-plane axes to be effectively coupled into a charge output through the piezoelectric transducers. This in turn enhances the output motional current from the device, which effectively determines the responsivity of the device. In this context, the responsivity has been defined as a ratio of output motional current to the external magnetic field, which has been further normalized against the input reference current of the device. The reported device has recorded a responsivity of 6950 ppm/T (μA/A.T) at a resonant frequency of 5.28 MHz and a reasonable mechanical quality (Q) factor of 1056 in air.

Modelling of resonant MEMS magnetic field sensor with electromagnetic induction sensing

This paper presents an analytical model of resonant MEMS magnetic field sensor with electromagnetic induction sensing. The resonant structure vibrates in square extensional (SE) mode. By analyzing the vibration amplitude and quality factor of the resonant structure, the magnetic field sensitivity as a function of device structure parameters and encapsulation pressure is established. The developed analytical model has been verified by comparing calculated results with experiment results and the deviation between them is only 10.25%, which shows the feasibility of the proposed device model. The model can provide theoretical guidance for further design optimization of the sensor. Moreover, a quantitative study of the magnetic field sensitivity is conducted with respect to the structure parameters and encapsulation pressure based on the proposed model.

Dependence of temperature coefficient of frequency (TCf) on crystallography and eigenmode in N-doped silicon contour mode micromechanical resonators

This paper reports the effects of crystal orientations on the temperature coefficient of frequency (TCf) of single crystal silicon square-plate micromechanical resonators vibrating in two distinct contour modes: Lamé mode and square extensional (SE). For the Lamé mode, the same TCf was found over several devices aligned to the 〈110〉 direction, while much greater variation in the TCf was observed among the devices aligned against the 〈100〉 direction. For the SE mode, the devices in both 〈100〉 and 〈110〉 orientations exhibit similar TCf values for varying doping levels. The sensitivity of TCf to doping concentration is also investigated. The TCf of Lamé 〈100〉 device is more easily influenced by n-type doping concentration than SE mode devices in both orientations while the Lamé 〈100〉 device is almost immune to doping variation. Devices with different dimensions are tested, and the TCf values are proved to be free of size scaling. Quantitative study based on free carrier contribution on elastic constants is performed and supports our observations. Close agreement among experiments, theoretical predictions and simulations is achieved.

Does greater piezo-resistive transduction give rise to higher anchor loss in a square-extensional mode micromechanical resonator?

The square-extensional (SE) mode resonator was used as the timing element in one of the first demonstrations of a MEMS oscillator meeting the GSM specification for phase noise. Crucial to its high Q was the use of T-shaped tethers in the anchoring design, which could also be used as piezo-resistors that could provide higher transduction. In this work, we examine the relationship between Q and corresponding piezo-resistive transduction efficiency, affected by changes in the compliance of the T-shaped tether anchoring design. Does a high Q necessarily mean a lower piezo-resistive transduction factor as an unavoidable performance tradeoff? Our measurement results, corroborated by finite element simulations, suggest that Q and the related piezo-resistive transduction factor are not opposed to each order in the SE mode resonator. In fact, the two indicative metrics of performance exhibit similar trends to the other with respect to changing the compliance of the anchoring design.

Nonlinear excitation of a rotational mode in a piezoelectrically excited square-extensional mode resonator

We present an experimental study of the nonlinear behavior of a square-extensional (SE) mode microelectromechanical resonator, actuated with a piezoelectric AlN thin film. The acoustic vibration fields of the device are characterized using laser probing. A nonlinear vibration behavior of the SE mode is observed above a drive power level of −10 dBm such that the vibration amplitude of the SE mode saturates and a rotational in-plane vibration mode is excited at a significantly lower frequency (0.725 MHz) than the SE mode (16.670 MHz). Interestingly, the measured ∼10 nm saturation amplitude of the SE mode is more than a decade below the amplitude value at which mechanical or electromechanical nonlinearities are estimated to become significant.