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 times 10⁶/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.     

Frequency-temperature behavior of flexural quartz resonators by means of Timoshenko's model

The frequency of a flexural resonator and its frequency-temperature behavior usually are computed by Bernoulli's classical approximation. This approach is valid for beams with a large length-over-thickness-ratio. For shorter beams, the effects of shear stress and rotary inertia may play a significant role for temperature-compensated resonators. These effects have been taken into account for isotropic beams. The aim of this paper is to discuss the extension of the shear coefficient in the case of an anisotropic material and to compute the frequency-temperature characteristic of an (XYt)theta cut resonator when the shear stress and the rotary inertia have been taken into account. Comparisons between the classical approximation and this treatment are given for quartz. Furthermore, the numerical predictions obtained by means of different sets of data available for thermal sensitivities of elastic coefficients are compared.     

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.     

An automated system for the precision measurement of the long-term frequency stability of quartz resonators

An automated system for the precision measurement of long-term frequency stability, designed for testing quartz resonators during production, is developed. During such tests, the point of inflection of the frequency-temperature characteristic is determined and possible “aging” of the resonator frequency during use is predicted.     

The thermal effect of metal electrodes on thickness-shear vibrations of crystal plates

The effectiveness of theoretical approaches based on the initial thermal field on vibrations elastic solids has been demonstrated by a series of analyses concerning the thermal behavior of crystal resonators with results in good agreement with measurements. These equations have been implemented in a finite-element method in addition to simple analytical solutions we are familiar with. Starting with an infinite plate under a temperature increase in both crystal and metal layers, the frequency equation based on incremental thermal field formulation is obtained and solved analytically for thickness-shear vibrations with electrodes of different metals and configuration. Numerical examples show that the thermal effect of thicker electrodes on frequency-temperature relations of thickness-shear vibrations of AT-cut quartz crystal plates are generally positive and should be considered in the design process.     

Bandpass filters for 8 GHz using solidly mounted bulk acoustic wave resonators

Frequency shift, design, and fabrication issues have been investigated for the realization of 8 GHz bandpass filters based on AlN thin film bulk acoustic wave resonators. Fabrication includes well-textured AlN thin films on Pt (111) electrodes and SiO/sub 2//AlN Bragg gratings for the solidly mounted resonators. The chosen ladder filter design requires the tuning of the shunt resonators with respect to the series one. For this purpose, mass loading of the shunt resonators with aluminum (Al) and SiO/sub 2/ were studied. Design simulations showed that the channel bandwidth can be doubled by shifting more than the difference of resonance and antiresonance frequency. Bandpass filters at 8 GHz were successfully fabricated with -5.5 dB insertion loss, -26 dB out-of-band rejection, 99 MHz (1.2%) /spl plusmn/0.2 dB channel bandwidth, and 224 MHz (2.8%) 3 dB bandwidth. The group delay variations within any 30 MHz channel inside the channel bandwidth amounts to <0.2 ns. Comparisons with simulation calculations and single resonator characteristics show that each /spl pi/-section includes a parasitic series resistance and inductance.     

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.     

The frequency-temperature analysis equations of piezoelectric plates with Lee plate theory

Frequency-temperature analysis theory of crystal plates based on the incremental field theory (IFT) and the Mindlin plate theory have been widely used in the vibration analysis of crystal resonators subject to temperature change. As one of the two major plate theories for the resonator analysis, the Lee plate theory based on the trigonometric series expansion of displacements has been extensively used for both analytical and numerical analyses of quartz resonators, and efforts have also been made in correcting and perfecting the theory for much broader applications. In this paper, the earlier frequency-temperature analysis equations based on the IFT is further extended to the trigonometric series expansion in a systematic manner. By incorporating the frequency-temperature analysis into the Lee plate theory, the complete analysis of crystal resonators can be made in a consistent way. The thickness-shear and flexural vibration equations have been compared with the Mindlin plate theory to demonstrate the similarity and consistency.     

Compact, high-Q, zero temperature coefficient, TE011 sapphire-rutile microwave distributed Bragg reflector resonators

Some novel new resonator designs based on the distributed Bragg reflector are presented. The resonators implement a TE₀₁₁ resonance in a cylindrical sapphire dielectric, which is confined by the addition of rutile and sapphire dielectric reflectors at the end faces. Finite element calculations are utilized to optimize the dimensions to obtain the highest Q-factors and zero frequency-temperature coefficient for a resonator operating near 0°C. We show that a Q-factor of 70,000 and 65,000 can be achieved with and without the condition of zero frequency-temperature coefficients, respectively     

Characteristics of a Lagrangian, high-frequency plate element for the static temperature behavior of low-frequency quartz resonators

Finite-element matrix equations based on the Lagrangian, first-order, incremental plate equations of motion superposed on homogeneous thermal strains were formulated using virtual work principles. A program for an isoparametric, four-node quadrilateral element was written and applied to the study of the frequency-temperature (FT) behavior of flexure-mode quartz resonators. The lumped-mass and consistent-mass matrices were found to yield practically the same FT curves. For simple prismatic resonators, two schemes, reduced/selective integration and incompatible modes, produced relatively similar FT curves. The incompatible modes scheme yielded better results for resonators of more complex shapes, such as the tuning fork. It is concluded that the six-degree-of-freedom per node element is needed for the analysis of the FT behavior of a fully anisotropic flexure-mode resonator.     

Micromachined thin film plate acoustic resonators utilizing the lowest order symmetric lamb wave mode

Thin film integrated circuits compatible resonant structures using the lowest order symmetric Lamb wave propagating in thin aluminum nitride (AlN) film membranes have been studied. The 2-mum thick, highly c-oriented AlN piezoelectric films have been grown on silicon by pulsed, direct-current magnetron reactive sputter deposition. The films were deposited at room temperature and had typical full-width, half-maximum value of the rocking curve of about 2 degrees. Thin film plate acoustic resonators were designed and micromachined using low resolution photolithography and deep silicon etching. Plate waves, having a 12-mum wavelength, were excited by means of both interdigital (IDT) and longitudinal wave transducers using lateral field excitation (LW-LFE), and reflected by periodical aluminum-strip gratings deposited on top of the membrane. The existence of a frequency stopband and strong grating reflectivity have been theoretically predicted and experimentally observed. One-port resonator designs having varying cavity lengths and transducer topology were fabricated and characterized. A quality factor exceeding 3000 has been demonstrated at frequencies of about 885 MHz. The IDT based film plate acoustic resonators (FPAR) technology proved to be preferable when lower costs and higher Qs are pursued. The LW-LFE-based FPAR technology offers higher excitation efficiency at costs comparable to that of the thin film bulk acoustic wave resonator (FBAR) technology     

A Frequency-Lock System for Improved Quartz Crystal Oscillator Performance

The intrinsic noise of the best quartz crystal resonators is significantly less than the noise observed in oscillators employing these resonators Several problem areas common to traditional designs are pointed out and a new approach is suggested for their solution. Two circuits are described which frequency lock a spectrally pure quartz crystal oscillator to an independent quartz crystal resonator. The performance of the composite system is predicted based on the measured performance of its components.     

Analysis of aging of piezoelectric crystal resonators

Aging of piezoelectric (quartz crystal) resonator has been identified as one of the most important quality control problems of quartz crystal products. Aging is defined as frequency change with time. Aging in quartz resonators can be due to several sources: mass transfer due to contamination inside the resonator enclosure, stress-strain in the resonator blank, quartz defect, etc. In this study, the stress-strain effect, which has been believed as a dominant factor contributing to aging, is studied. The stress-strain effect is caused mainly by the long-term viscoelastic properties of bonding adhesive that attach quartz crystal plate to the ceramic base package. With the available accelerating testing method under elevated temperatures, the stress-strain induced aging in the quartz crystal resonators can be investigated. Because of the miniaturized size of the resonator, a digital image analysis method called image intensity matching technique (IIMT) is applied to obtain deformation patterns in the quartz blank due to thermal load. Our preliminary results showed that the unsymmetric thermal deformations may be a dominant contributing factor to aging. For simulation purposes, finite-element analysis is used to investigate the deformation patterns (i.e., stress-strain distributions) and corresponding natural frequency shift in the piezoelectric resonators. The viscoelastic behavior of mounting adhesives is incorporated into the analysis to show the dominant effect of long-term behavior of stress-strain developed in the crystal resonators. Also, some geometrical aspects-such as uneven mounting supports due to distances, volumes and heights of the adhesives-are simulated in the model.     

Frequency-temperature behavior of spurious vibrations of rectangular AT-cut quartz plates

We evaluate the frequency-temperature behavior of spurious modes in a rectangular AT-cut quartz plate resonator based on three-dimensional linear equations. The elastic constants and three geometrical dimensions of the resonator are defined in terms of cubic polynomials of a temperature change. Assuming that the resonator holds its rectangular plate shape irrespective of temperature, we can determine the relationship between frequency and the dimensions of the resonator for a given temperature using the previous technique. We compare the calculated results with our own experimental data, and show that agreement between the calculated and observed data is excellent.     

Experimental verification of stress compensation in the SBTC-cut

The authors have demonstrated experimental verification of the stress compensation feature for the fast thickness shear mode of vibration of stress-compensated for B-mode and temperature-compensated for C-mode (SBTC)-cut quartz resonators. For the resonator design used in the cylindrical probe structure, the motional resistance for the B-mode of vibration was approximately 12% of that of the C-mode. The relatively large motional resistance for the C-mode of vibration of the SBTC-cut was found to be largely due to the lower piezoelectric coupling for the thickness excitation of this mode. In addition the proximity of the third overtone of the A-mode to the fifth overtone of the C-mode also contributed to the increase in the motional resistance. The authors have obtained experimental data on the temperature dependence of the planar stress coefficient and pressure dependence of the frequency-temperature characteristic for both the thickness-shear modes of the SBTC-cut. It is noted that such a doubly rotated cut can have applications in the design of either stable frequency sources or sensors for pressure and temperature measurements.     

Investigations on LGS and LGT crystals to realize BAW resonators

The LGS family are promising materials for the design of high quality bulk acoustic wave resonators. We have manufactured many plano-convex 10 MHz 5th overtone Y-cut resonators using langasite (LGS, La₃Ga₅SiO₁₄) and langatate (LGT, La₃Ga_5.5Ta_0.5O₁₄) crystals. We observed that the quality factor strongly depends on the polishing method, the supplier of the material, and on the energy trapping. For quartz crystals, we have found that resulting IR spectra exhibit absorption peaks more or less deep, linked to defects. These predominant criteria are not surprising, but they have to be defined in manner similar to that used for quartz crystal. A satisfying machining and polishing method has been first applied to elaborate high Q resonators, and a comparison between samples of LGS and LGT materials from different suppliers is established. In addition, LGT resonators are characterized by their motional parameters and frequency-temperature curves. Nevertheless, one of the main results is that the measured Q × f product is not the expected one. We present results of Q-factor versus radius of curvature: it appears that an optimization should be performed and that this last one cannot be directly transposed from that of quartz crystal resonator. Currently, the best resonator that we have made has a Q × f product of 1.4 × 10¹³ on its 5th overtone (1.7 × 10¹³ on its 9th overtone). This result is slightly higher than the similar parameter obtained on a state-of-the-art SC-cut quartz crystal resonator working at the same frequency.     

Beam-supported AlN thin film bulk acoustic resonators

A novel, suspended thin film bulk acoustic wave resonator (SFBAR) has been fabricated from an aluminum nitride film sputtered directly on a (100) silicon substrate. The suspended membrane design uses thin beams to support, as well as electrically connect, the resonator and has been fabricated using both thin film processing and bulk silicon micromachining. The quality factor and the effective electromechanical coupling coefficient were characterized as a function of the number and the length of the support beams. The length of the support beams was found to affect neither the quality factor at resonance nor the effective electromechanical coupling factor. However, longer support beams did facilitate better frequency pair response. Device performance varied with the number of support beams: 70% of the resonators tested showed a higher figure of merit with eight support beams than with four support beams.     

Analysis of the rutile-ring method of frequency-temperature compensating a high-Q whispering gallery sapphire resonator

The rutile-ring method of dielectrically frequency-temperature compensating a high-Q whispering gallery (WG) sapphire resonator is presented. Two and three-dimensional finite element (FE) analysis has been implemented to design and analyze the performance of such resonators, with excellent agreement between theory and experiment. A high-Q factor of 30 million at 13 GHz, and compensation temperature of 56 K was obtained. It is shown the frequency-temperature compensation can occur either because the rutile adds a small perturbation to the sapphire resonator or because of a mode interaction with a resonant mode in the rutile. The characteristics of both of these methods are described, and it is shown that for high frequency stability, it is best to compensate perturbatively