An Averaging Method for Phase Noise Studies of Quartz Phononic Frequency Combs.

A fast and accurate averaging method was derived and developed for the analysis and design of quartz phononic frequency combs. The phononic frequency combs were obtained from a pair of coupled nonlinear Duffing equations for quartz resonators by solving the equations in the time domain and performing a fast Fourier transformation (FFT) of the steady-state vibrations of the time series. Noise simulations were added to the drive frequency to study noise transfer characteristics between the drive signal and the resonances of the phononic frequency combs produced in 100-MHz quartz shear-mode resonators. Our new method averaged out the carrier frequency, thus allowing for a fast and efficient computation at parts per million accuracies of noise close to the carrier (  âˆ¼  10 Hz). The goal of our study was to develop methods and resonator requirements for engineering the properties of the phononic frequency combs for low-noise clock applications.

Design Considerations for Frequency Shifts in a Laterally Finite FBAR Sensor in Contact With the Newtonian Liquid.

Mode-coupled vibrations in a thickness-shear (TSh) mode and laterally finite film bulk acoustic resonator (FBAR) with one face in contact with Newtonian (linearly viscous and compressional) liquid are investigated. With boundary conditions and interface continuity conditions, exact dispersion curves in FBAR sensors contacting with two kinds of liquids are obtained, and they are compared with the dispersion curves in a bare sensor without liquid contact. Frequency spectra, describing mode couplings between the main TSh modal branch and undesirable modal branches, are calculated by employing weak boundary conditions at lateral free edges constructed based on the variational principle. Mode shapes of mechanical displacements in both the sensor and liquid layer are presented, and mode transformations are observed due to the liquid contact and lateral edge effect. The effect of liquid thickness on frequency spectra is also studied. Numerical results reveal that the generation of shear wave in the liquid layer results in the shifts of spectrum curves along the frequency axis and hence it is the main factor of frequency shifts of FBAR sensors. The compressional wave causes the shifts of spectrum curves along the lateral aspect ratio axis. Then for a given FBAR sensor, the liquid thickness changes could also cause frequency shifts. Therefore, desirable vibration modes should be chosen based on the frequency spectra to avoid strong mode couplings and to eliminate frequency shifts induced by the liquid thickness changes in real applications.

Lateral Size-Dependence in UHF Mode-Coupled ZnO FBARs to Suppress Undesirable Eigen-Modes and Weaken Mounting Effect.

Mode-coupled vibrations in an ultra-high frequency (UHF) ZnO thin film bulk acoustic resonator (FBAR) operating at thickness-extensional (TE) mode are studied by employing weak boundary conditions (WBCs), constructed based on Saint-Venant's principle and mixed variational principle in the piezoelectric theory. The frequency spectra, describing the lateral size-dependence of mode couplings between the main mode (TE) and undesirable eigen-modes, for clamped lateral edges are compared with the existing frequency spectra for free lateral edges to illustrate the boundary influence. The displacement and stress variations in FBAR volume are also presented to intuitionally understand and distinguish the difference of frequency spectra between these two different lateral edges, and then we discuss how to select outstanding lateral sizes to weaken the mounting effect. The frequency spectra predicted from our approximate WBCs are also compared with and agree well with those predicted by the finite element method (FEM) using COMSOL, which proves the correctness and accuracy of our theoretical method. These results indicate that the WBCs could have potentials in the valid predictions of lateral size-dependence of mode couplings in piezoelectric acoustic wave devices.

Simulation of Nonlinear Resonance, Amplitude-Frequency, and Harmonic Generation Effects in SAW and BAW Devices.

Nonlinearly coupled sets of piezoelectric field equations in the frequency domain were derived for the nonlinear propagation of finite-amplitude waves in piezoelectric bulk acoustic wave (BAW) and surface acoustic wave (SAW) devices. To verify their accuracy, we have embedded these sets of equations in the finite-element method (FEM) of COMSOL Multiphysics software and compared the FEM results with both the analytical and experimental results found in the published literature. The nonlinear frequency responses for both plano- and contoured-plate resonators of AT-cut quartz were investigated under various voltage drives, circuit resistances, and quality factors. The proposed equations with FEM have also been employed to study 33.3-MHz very-high-frequency (VHF) quartz resonators, showing that under different conditions of how well the third overtone mode (f3) matches the third harmonic (3f) and how the fractional frequency shift of the third overtone mode (f3) occurs as a function of the fundamental mode current. Furthermore, we have studied the nonlinear harmonic generation of an 840-MHz 128° Y-cut X-propagating (128° YX) LiNbO3 SAW resonator. The second-harmonic (H2) and third-harmonic (H3) modes were observed to occur, respectively, at two-time (2f) and three-time (3f) frequencies of fundamental frequency (f) when such resonators were driven with high power. The effects of substrate thickness, bottom surface conditions of the substrate, and different circuit connections on the H2 and H3 generations were simulated and compared with available measurements. Current proposed sets of equations are general and could be used for the study of nonlinear resonance, amplitude-frequency effect, and harmonic generation in any piezoelectric devices, provided that the necessary nonlinear material constants are known.

A Fully Integrated Quartz MEMS VHF TCXO.

We report on a 32-MHz quartz temperature compensated crystal oscillator (TCXO) fully integrated with commercial CMOS electronics and vacuum packaged at wafer level using a low-temperature MEMS-after quartz process. The novel quartz resonator design provides for stress isolation from the CMOS substrate, thereby yielding classical AT-cut f/T profiles and low hysteresis which can be compensated to < ±0.2 parts per million over temperature using on-chip third-order compensation circuitry. The TCXO operates at low power of 2.5 mW and can be thinned to as part of the wafer-level eutectic encapsulation. Full integration with large state-of-the-art CMOS wafers is possible using carrier wafer techniques.

On the acceleration sensitivity and its active reduction by edge electrodes in AT-cut quartz resonators.

Incremental piezoelectric equations for small vibrations superposed on initial deformations are presented. The equations are implemented in COMSOL finite element models (FEA). Equations are validated by comparing the results for the force sensitivity coefficient Kf of a circular quartz plate subjected to a pair of diametrical forces with measured data. The model results show a consistent trend with the experimental results, and the relative difference between our FEA results and Ballato's measured result is about 13%. A detailed study of the acceleration sensitivity of a rectangular AT-cut quartz plate is presented. The plate resonator is fixed along one edge as a cantilever. For AT-cut quartz resonators with the crystal digonal X-axis perpendicular to plate X-axis, the in-plane acceleration sensitivity is found to be negligible compared with the out-of-plane (Y-axis) acceleration sensitivity. For AT-cut quartz resonators with the crystal digonal X-axis parallel to plate X-axis, the Y-axis acceleration sensitivity is found to be rectified, that is the fractional change in frequency is positive with respect to both positive and negative Y-axis accelerations. The Y-axis acceleration sensitivity is small in comparison with the in-plane acceleration sensitivity for small body forces. However, for large body forces, the Y-axis acceleration sensitivity dominates because it increases nonlinearly with the Y-axis acceleration. The resonator rectified acceleration sensitivity is confirmed by phase noise measurements. For reduced acceleration sensitivity, two pairs of electrodes along the plate edges reduce the bending of the plate resonator and subsequently reduce acceleration sensitivity. We present a new method using these edge electrodes in which a dc bias field is employed to control the resonant frequency of resonator subjected to g body forces. A dc bias field with an appropriate dc bias voltage could potentially yield a reduction of acceleration sensitivity in Y-axis direction of about two orders of magnitude.

Theory and experimental verifications of the resonator Q and equivalent electrical parameters due to viscoelastic and mounting supports losses.

A novel analytical/numerical method for calculating the resonator Q and its equivalent electrical parameters due to viscoelastic, conductivity, and mounting supports losses is presented. The method presented will be quite useful for designing new resonators and reducing the time and costs of prototyping. There was also a necessity for better and more realistic modeling of the resonators because of miniaturization and the rapid advances in the frequency ranges of telecommunication. We present new 3-D finite elements models of quartz resonators with viscoelasticity, conductivity, and mounting support losses. The losses at the mounting supports were modeled by perfectly matched layers (PMLs). A previously published theory for dissipative anisotropic piezoelectric solids was formulated in a weak form for finite element (FE) applications. PMLs were placed at the base of the mounting supports to simulate the energy losses to a semi-infinite base substrate. FE simulations were carried out for free vibrations and forced vibrations of quartz tuning fork and AT-cut resonators. Results for quartz tuning fork and thickness shear AT-cut resonators were presented and compared with experimental data. Results for the resonator Q and the equivalent electrical parameters were compared with their measured values. Good equivalences were found. Results for both low- and high-Q AT-cut quartz resonators compared well with their experimental values. A method for estimating the Q directly from the frequency spectrum obtained for free vibrations was also presented. An important determinant of the quality factor Q of a quartz resonator is the loss of energy from the electrode area to the base via the mountings. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base substrate. The base affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a base. Hence, the base is an energy sink and the Q will be affected by the shape and size of this base. A lower-bound Q will be obtained if the base is a semi-infinite base because it will absorb all acoustical energies radiated from the resonator.

Conceptual design of a high-Q, 3.4-GHz thin film quartz resonator.

Theoretical analyses and designs of high-Q, quartz thin film resonators are presented. The resonators operate at an ultra-high frequency of 3.4 GHz for application to high-frequency timing devices such as cesium chip-scale atomic clocks. The frequency spectra for the 3.4-GHz thin film quartz resonators, which serve as design aids in selecting the resonator dimensions/configurations for simple electrodes, and ring electrode mesa designs are presented here for the first time. The thin film aluminum electrodes are found to play a major role in the resonators because the electrodes are only one third the thickness and mass of the active areas of the plate resonator. Hence, in addition to the material properties of quartz, the elastic, viscoelastic, and thermal properties of the electrodes are included in the models. The frequency-temperature behavior is obtained for the best resonator designs. To improve the frequency-temperature behavior of the resonators, new quartz cuts are proposed to compensate for the thermal stresses caused by the aluminum electrodes and the mounting supports. Frequency response analyses are performed to determine the Q-factor, motional resistance, capacitance ratio, and other figures of merit. The resonators have Q's of about 3800, resistance of about 1300 to 1400 ohms, and capacitance ratios of 1100 to 2800.

Effects of electromagnetic radiation on the Q of quartz resonators.

The quartz resonator Q with aluminum electrodes was studied with respect to its fundamental thickness shear mode frequency and its viscoelastic, viscopiezoelectric, and viscopiezoelectromagnetic behaviors. The governing equations for viscoelasticity, viscopiezoelectricity, and viscopiezoelectromagnetism were implemented for an AT-cut quartz resonator. To simulate the radiation conditions at infinity for the viscopiezoelectromagnetic model, perfectly matched layers over a surface enclosing the resonator were implemented to absorb all incident electromagnetic radiation. The shape of the radiation spectrum of a 5.6 MHz AT-cut quartz resonator was found to compare relatively well the measured results by Campbell and Weber. The mesa-plate resonator was studied for a frequency range of 1.4 GHz to 3.4 GHz. The resonator Q was determined to be influenced predominantly by the quartz viscoelasticity; however at frequencies greater than 2.3 GHz, the quartz electromagnetic radiation had an increasingly significant effect on the resonator Q. At 3.4 GHz, the electromagnetic radiation accounted for about 14% of the loss in resonator Q. At frequencies less than 2 GHz, the calculated resonator Q compared well with the intrinsic Q(x) provided by the formula Q(x) = 16 x 10(6)/f where f was in MHz. At frequencies higher than 2.3 GHz, the aluminum electrodes had significant effects on the resonator Q. At 3.4 GHz, the electromagnetic radiation loss in the electrodes was an order of magnitude greater than their viscoelastic loss; hence, the vibrating aluminum electrodes became an efficient emitter of electromagnetic waves. The effects of electrical resistance in both the electrodes and quartz were determined to be negligible.

Love wave propagation in piezoelectric layered structure with dissipation.

We investigate analytically the effect of the viscous dissipation of piezoelectric material on the dispersive and attenuated characteristics of Love wave propagation in a layered structure, which involves a thin piezoelectric layer bonded perfectly to an unbounded elastic substrate. The effects of the viscous coefficient on the phase velocity of Love waves and attenuation are presented and discussed in detail. The analytical method and the results can be useful for the design of the resonators and sensors.

Dedicated finite elements for electrode thin films on quartz resonators.

The accuracy of the finite element analysis for thickness shear quartz resonators is a function of the mesh resolution; the finer the mesh resolution, the more accurate the finite element solution. A certain minimum number of elements are required in each direction for the solution to converge. This places a high demand on memory for computation, and often the available memory is insufficient. Typically the thickness of the electrode films is very small compared with the thickness of the resonator itself; as a result, electrode elements have very poor aspect ratios, and this is detrimental to the accuracy of the result. In this paper, we propose special methods to model the electrodes at the crystal interface of an AT cut crystal. This reduces the overall problem size and eliminates electrode elements having poor aspect ratios. First, experimental data are presented to demonstrate the effects of electrode film boundary conditions on the frequency-temperature curves of an AT cut plate. Finite element analysis is performed on a mesh representing the resonator, and the results are compared for testing the accuracy of the analysis itself and thus validating the results of analysis. Approximations such as lumping and Guyan reduction are then used to model the electrode thin films at the electrode interface and their results are studied. In addition, a new approximation called merging is proposed to model electrodes at the electrode interface.

Estimation of quartz resonator Q and other figures of merit by an energy sink method.

An important determinant of the quality factor Q of a quartz resonator is the loss of energy from the electrode area to the base via the mountings. The acoustical characteristics of the plate resonator are changed when the plate is mounted onto a base substrate. The base substrate affects the frequency spectra of the plate resonator. A resonator with a high Q may not have a similarly high Q when mounted on a base. Hence, the base is an energy sink and the Q will be affected by the shape and size of this base. A lower bound Q will be obtained if the base is a semi-infinite base since it will absorb all acoustical energies radiated from the resonator. A scaled boundary finite element method is employed to model a semi-infinite base. The frequency spectra of the quartz resonator with and without the base are presented. In addition to the loss of energy via the base, there are other factors which affect the resonator Q, such as, for example, material dissipation, and damping at the interfaces of quartz and electrodes. The energy dissipation due to material damping increases with the resonant frequency and the reduction of resonator size; hence material damping becomes important in the current and future miniaturized resonators operating at very high frequencies. An energy sink model along with material dissipation would provide realistic Q, motional capacitance, motional resistance, and other figures of merit useful for designing resonators. The model could be used for evaluating resonator and mountings designs of microelectromechanical systems and miniaturized devices. The effect of the mountings, and plate and electrode geometries on the resonator Q and other electrical parameters are presented for AT-cut quartz resonators. Model results from the energy sink method were compared with experimental results and were found to be good.