Dejin Huang

The analysis of high frequency vibrations of layered anisotropic plates for FBAR applications

In this paper, the thickness-extension vibration of a layered piezoelectric plate is investigated. The vibration deformation consists of symmetric and asymmetric deformation. Thicknesses of the adhered layers will influence the frequency of the plate and the amplitude ratio between the layers significantly.

An analysis of vibrations of quartz crystal plates with nonlinear Mindlin plate equations

The nonlinear effects of material constants and initial stresses and strains in quartz crystal resonators is well known f on the frequency-temperature curves, drive-level dependency, acceleration sensitivity, and stress compensation. Consequently, accurate predictions on resonator behavior and their electrical circuit parameters require the use of nonlinear vibration equations and their solutions. The effectiveness of nonlinear analyses has been shown by a few researchers with the finite element and perturbation methods. The Mindlin plate theory, which has been used extensively for understanding plate modes and their coupling effects in plate vibrations analysis, is not enough in the study of the nonlinear behavior of quartz resonators. We have followed the Mindlin plate theory to derive the nonlinear equations with the inclusion of large displacements and higher order elastic constants. The coupling of vibration modes due to nonlinearity is clearly observed and it is quite different from linear cases that we are familiar with. We start from the equations of vibration for the thickness-shear mode to validate the solution techniques, which could be the perturbation method and the latest Homotopy Analytical Method (HAM). Then the methods are applied to the coupled equations of thickness-shear and flexural vibrations which are the two dominant modes of quartz crystal resonators of the thickness-shear type. These solutions, in the absence of the strong electrical field, can be used to study the frequency, deformation, and mode conversion in nonlinear vibrations. We hope the frequency spectra and spatial variations of the thickness-shear and flexural displacements from the accurate solutions of nonlinear equations will provide insights on the changes in each mode when compared with their linear vibrations. The further extension of nonlinear plate equations with the inclusion of piezoelectric effects will also provide useful examination of nonlinear behavior of quartz crystal resonators.

Nonlinear Mindlin plate equations for the thickness-shear vibrations of crystal plates

Mindlin plate equations have wide applications in engineering fields with piezoelectric crystal resonators in particular due to its accurate prediction of high frequency vibrations of plates in the vicinity of thickness-shear mode. As a linear theory based on the assumption of infinitesimal deformation, its applications have been limited mainly to vibration frequency analysis. Many important properties of quartz crystal resonators such as the electrical circuit parameters, and nonlinear phenomena such as the drive-level dependence (DLD), have to be studied with the consideration of higher-order material constants and subsequent inclusion of nonlinear strain components. This, in turn, implies that the consideration of large deformation related to the driving electrical field. The need of nonlinear theory for the analysis of high frequency vibrations of piezoelectric crystal plates have been noticed before, and research work concerning large electrical fields and deformation have been initiated for the calculation of electrical properties and DLD of quartz crystal resonators by many investigators recently. In this study, we start with higher-order constitutive relation which includes higher-order elastic constants. Then, nonlinear terms of strain components are considered in a compatible and systematic manner. By following Mindlins procedure in deriving the two-dimensional equations, nonlinear Mindlin plate equations for large deformation are obtained. The equations take the familiar form of Mindlin plate theory except the inclusion of nonlinear terms in the strain tensor.

The correction factors of mindlin plate theory with and without electrodes for SC-cut quartz crystal plates

Mindlin plate theory was first developed to provide accurate solutions for vibrations of thickness-shear mode, which has a much higher frequency than usual flexure vibrations. It has been widely used in the analysis of high frequency vibrations of quartz crystal plates, which are the core of resonators. The vibration frequency solutions obtained with Mindlin plate theory are proven much closer to the exact solutions. However, due to the truncation and approximation, the plate equations need to be corrected, as compared with the three-dimensional elasticity solutions. This has been done for the high-order Mindlin plate theory with and without electrodes for the AT-cut quartz crystal plates, and correction factors have been obtained though both natural and symmetric procedures. The correction factors could be used in the dispersion relationship and frequency spectrum in the analytical solutions, while the symmetric correction factors can be used in the finite element method implementation. Both correction schemes can provide improved and accurate results in the analysis of quartz crystal resonators. The electrodes are considered though its inertia effect as mass ratio known in resonator analysis.

An analysis of thickness-shear vibrations of doubly-rotated quartz crystal plates with the corrected first-order mindlin plate equations

The Mindlin plate equations have been widely used in the analysis of high-frequency vibrations of quartz crystal resonators with accurate solutions, as demonstrated by the design procedure based on analytical results in terms of frequency, mode shapes, and optimal parameters for the ATcut quartz crystal plate, which is the core element in a resonator structure. Earlier studies have been focused on the AT-cut (which is one type of rotated Y-cut) quartz crystal plates because it is widely produced and has relatively simple couplings of vibration modes at thickness-shear frequencies of the fundamental and overtone modes. The simplified equations through the truncation, correction, and modification of the Mindlin plate equations have been widely accepted for practical applications, and further efforts to expand their applications to similar problems of other material types, such as doubly-rotated quartz crystals, with the SC-cut being a typical and popular one, are also naturally expected. We have found out that the Mindlin plate theory can be truncated and corrected for the SC-cut quartz crystal plates in a manner similar to the AT-cut plates. The analytical results show that the corrected Mindlin plate equations are equally accurate and convenient for obtaining essential design parameters of resonators for the thickness-shear vibrations of SC-cut quartz crystal plates.

The fifth-order overtone vibrations of quartz crystal plates with higher-order Mindlin plate equations

As demands for high frequency quartz crystal resonators rise, we are prompted to design and make overtone devices with the same material, process, and technology. Indeed, high-order overtone resonators have been widely utilized in many applications and further demands in precision products are also growing. The design of overtone resonators and further improvement of the existing ones to meet precision requirements are largely based on empirical approaches, but we found that the technique can be polished with theoretical and analytical efforts as we examine the applications of the Mindlin plate theory in the design and analysis of the fundamental type. Through the extensive improvements of the Mindlin plate theory, we can now analyze the vibration mode couplings, electrode effect, optimal sizes, and thermal behavior, among others. These essential analytical procedures have been implemented in finite element analysis tools with more advanced features such as the nonlinear behavior prediction and eventually circuit parameter extraction. Since it has been proven that the Mindlin plate equations can be used for the vibration analysis of plates at the higher-order overtone modes with accurate prediction of frequency and dispersion relations in the vicinity of cut-off frequencies, we extended the equations to the third-order for the modal behavior and frequency spectra. The results show that earlier knowledge on the proper selection of the sizes of electrode can be proven from our analysis. In addition, the spatial variation and end effects of displacements, particularly of the working mode, can be used in the optimal selection of resonator configuration. The design changes can be used as a way to improve the resonator performance, which has been increasingly degenerating for higher-order overtone types, to meet more stringent requirements. We now extend the plate equations to the fifth-order so the design principle and guidelines can be summarized from more analytical results of overtone vibrations. These predictions on frequency, deformation, and electrode effects from studies with successive orders of equations can be used for resonator design at higher overtone frequencies.

Correction Factors for Mindlin Higher-order Plate Theory with the Consideration of Electrodes

Mindlin plate theory has been the choice for the analysis of high frequency vibrations of piezoelectric quartz crystal resonators, which utilize electroded crystal plates vibrating at the thickness-shear and overtone modes to achieve frequency generation functions and applications. For this purpose, Mindlin plate equations have been simplified, modified, and corrected to accommodate the actual geometry, structural complications, practical boundary conditions, and various bias fields such as temperature change and initial stresses. Accordingly, the correction of Mindlin plate theory should also include complications through factors which adjust the thickness-shear frequencies to the exact three-dimensional solutions from earlier studies with the first-order plate equations. In recent studies, correction factors for equations up to the third-order theory have been introduced and they have the capability to make the cut-off frequencies for the thickness-shear modes accurate up to much higher-orders and improving the accuracy of the extensional mode groups also. These correction factors are derived based on the general principle of comparison and matching of exact frequencies of interested vibration modes in the coupled two-dimensional equations, and the inclusion of structural complications such as electrodes is an natural extension of the derivation with practical importance. In this study, we consider the effect of thin electrodes through their mass loading formulation by adding the mass ratio in the inertia terms as demonstrated by Mindlin and others. Consequently, a set of frequency equations for both thickness-shear and extensional modes are established with the inclusion of mass ratio of electrodes. These equations are solved numerically for mass ratios to obtain the corresponding correction factors which have been modified by the presence of electrodes. These correction factors can be utilized in the improvement of applications of the higher-order plate equations with the finite element implementation, which remains as one of the reliable tool in the precise analysis and design of quartz crystal resonators. The derivation of these correction factors can be further incorporated into nonlinear equations of piezoelectric solids.

An analysis of nonlinear vibrations of coupled thickness-shear and flexural modes of quartz crystal plates with the homotopy analysis method

We investigated the nonlinear vibrations of the coupled thickness-shear and flexural modes of quartz crystal plates with the nonlinear Mindlin plate equations, taking into consideration the kinematic and material nonlinearities. The nonlinear Mindlin plate equations for strongly coupled thickness- shear and flexural modes have been established by following Mindlin with the nonlinear constitutive relations and approximation procedures. Based on the long thickness-shear wave approximation and aided by corresponding linear solutions, the nonlinear equation of thickness-shear vibrations of quartz crystal plate has been solved by the combination of the Galerkin and homotopy analysis methods. The amplitudefrequency relation we obtained showed that the nonlinear frequency of thickness-shear vibrations depends on the vibration amplitude, thickness, and length of plate, which is significantly different from the linear case. Numerical results from this study also indicated that neither kinematic nor material nonlinearities are the main factors in frequency shifts and performance fluctuation of the quartz crystal resonators we have observed. These efforts will result in applicable solution techniques for further studies of nonlinear effects of quartz plates under bias fields for the precise analysis and design of quartz crystal resonators.

The analysis of film acoustic wave resonators with the consideration of film piezoelectric properties

The vibration frequency analysis of film bulk acoustic resonators (FBAR) is based on the assumption of layered infinite plates vibrating at a working mode, which can be the thickness-extension or thickness-shear depending on the choice of the mode. A transcendental equation is used to determine the vibration frequency with given materials and plate thicknesses. Similar to the analysis and design of quartz crystal resonators of thickness-shear type, frequency equations and displacements in films can be used for the calculation of resonator properties which are important for improvement and modeling. By expanding the formulation to include the piezoelectric effect, we shall also be able to obtain the electrical field as a vital addition to mechanical solutions. Of course, the piezoelectric effect will also be included in the solutions of frequency and displacements. The solutions can be used to calculate the electrical circuit parameters of a resonator. We study vibrations of layered FBAR structures for both thickness-extension and thickness-shear modes and the solutions also include the electrical field under an alternating voltage. With these equations, solutions, and further formulations on the electrical circuit properties of FBAR, we can establish a systematic procedure for the analysis and design, thus completing the currently empirical methodology in resonator development. These one-dimensional formulation based on the infinite plate assumption can be further improved through the consideration of finite plates and numerical solutions based on the commonly used finite element analysis. These studies will be the basis for the formulation and calculation of electrical circuit parameters that are highly demanded as FBAR technology is expanding quickly to other applications. The accurate analysis and resonator property extension will contribute to the sophistication of FBAR technology with improved design procedure and performance.

An analysis of quartz crystal resonators with mindlin plate theory based parallel finite element method on a Linux cluster

Finite element method has been applied to quartz crystal resonator engineers due to its long history in research based on Mindlin plate and three-dimensional approaches for piezoelectric plates with considerations of complication factors such as electrodes, thermal effect, mounting, and packaging, among others. The earlier efforts have been dampened, however, by the fact that the high vibration frequency in the thickness-shear mode has caused significant increase of the problem size in terms of number of equations, or total degree of freedom, in the linear system resulted. It is typical that an accurate analysis of quartz crystal resonator vibrating at the fundamental thickness-shear mode may require solving a linear system around one million for both free and forced vibrations. What we can get from the analysis are the frequency spectra, which are the relationship between frequencies and geometry, and mode shapes of the resonator structure with mountings. This, not surprisingly, is beyond the computing capabilities for many industrial engineers who do not have access to resources like supercomputers widely available to academic researchers. On the other hand, the finite element method is an excellent tool for crystal resonator analysis we should utilize for quick and precise prototyping process. In order to overcome the challenge of computing power, the finite element program development has been taking steps through employing Mindlin plate theory, efficient eigenvalue solvers, and sparse matrix handling algorithms, as ways to aggressively improve the efficiency and reduce stringent requirements on hardware. In this continuing research, we have replaced our earlier version of the finite element program with ARPACK as the eigenvalue solver, added sparse matrix handling functions, and implemented parallel computing capability on a cost-effective Linux cluster. The computing capability has been improved with the utilization of multiple processors significantly while the cost of infrastructure and operations is within the reach of the frequency control industry. We shall introduce the core improvements on algorithms, parallel features and implementation, hardware requirements, and computing power gain based on our current program configuration. The program has been developed in partnership with major industry players to demonstrate the effectiveness of supercomputing and finite element method as an important design tool for resonator design and improvement.