Energy-Trapping Characteristics of Lateral Field Excited GdCOB Crystal Bulk Acoustic Wave Devices Based on Stepped Electrodes.

In this work, high-frequency forced vibrations of lateral field excitation (LFE) devices with stepped electrodes based on monoclinic crystals GdCOB are modeled, and the influence laws of the device parameters (the step number, size, and thickness of the stepped electrodes) on the energy-trapping effects of the device are revealed. The results show that the step number has a significant effect on the energy-trapping effect of the device: with the increase in the step number, the stronger energy-trapping effect of the device can be obtained; with the increase in the thickness difference of two layers of electrodes, the energy-trapping effect of the device becomes stronger; with the increase in the difference of the electrode radius, the energy-trapping effect of the device is enhanced gradually. The results of this work can provide an important theoretical basis for the design of stepped-electrode LFE resonators and sensors with high-quality factors based on monoclinic crystals.

Pure- and Pseudo-Lateral-Field-Excitation Characteristics of Relaxor Ferroelectric Single Crystal PMN-PT.

The relaxor ferroelectric single crystal (1-x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 (PMN-PT) has high piezoelectric constants, and thus has a good application prospect in the field of highly sensitive piezoelectric sensors. In this paper, for relaxor ferroelectric single crystal PMN-PT, the bulk acoustic wave characteristics on pure- and pseudo-lateral-field-excitation (pure- and pseudo-LFE) modes are investigated. LFE piezoelectric coupling coefficients and acoustic wave phase velocities for PMN-PT crystals in different cuts and electric field directions are calculated. On this basis, the optimal cuts of pure-LFE and pseudo-LFE modes of relaxor ferroelectric single crystal PMN-PT are obtained, namely, (zxt)45° and (zxtl)90°/90°, respectively. Finally, finite element simulations are carried out to verify the cuts of pure-LFE and pseudo-LFE modes. The simulation results show that the PMN-PT acoustic wave devices in pure-LFE mode have good energy-trapping effects. For PMN-PT acoustic wave devices in pseudo-LFE mode, when the device is in air, no obvious energy-trapping emerges; when the water (as a virtual electrode) is added to the surface of the crystal plate, an obvious resonance peak and the energy-trapping effect appears. Therefore, the PMN-PT pure-LFE device is suitable for gas-phase detections. While the PMN-PT pseudo-LFE device is suitable for liquid-phase detections. The above results verify the correctness of the cuts of the two modes. The research results provide an important basis for the development of highly sensitive LFE piezoelectric sensors based on relaxor ferroelectric single crystal PMN-PT.

Multimode Design and Piezoelectric Substrate Anisotropy Use to Improve Performance of Acoustic Liquid Sensors.

Using acoustic wave modes propagation in piezoelectric plates loaded with conductive liquids, peculiarities of the mode-liquid acoustoelectric interaction are studied. It is found that (i) in contrast to bulk and surface acoustic waves propagating in piezoelectric semiconductors, the acoustoelectric attenuation of the modes is not symmetric in respect to its maximum, (ii) a large increase in attenuation may be accompanied by a small decrease in phase velocity and vice versa, (iii) the peculiarities are valid for "pure" (without beam steering) and "not pure" (with beam steering) modes, as well as for modes of different orders and polarizations, and (iv) conductivity of test liquid increases electromagnetic leakage between input and output transducers, affecting results of the measurements. To decrease the leakage, the liquid should be localized between transducers, outside the zone over them. If so, the mode sensitivity may be as large as 8.6 dB/(S/m) for amplitude and 107°/(S/m) for phase. However, because of comparable cross-sensitivity towards viscosity and dielectric permittivity, modes with selective detection of liquid conductivity are not found.

An Analysis of the Thermal Behavior and Effects of Circular Quartz Crystal Resonators for Microbalance Applications.

The accurate calculation of vibration frequency is essential in design of circular quartz crystal resonators which are the core elements of high-precision microbalances used for testing and measurement. Currently, the prediction of thermal effects on frequency through an analytical analysis is still in its developmental stage, mainly due to the complexity of solving the 3-D equations with the consideration of asymmetric structure of resonators and electrodes along with material anisotropy. By using a scalar differential equation for vibrations with the eigen-displacement of thickness mode, the eigen-frequency of a plano-convex AT-cut circular quartz crystal plate with asymmetric electrodes is determined. Furthermore, the temperature effect in the scalar differential equation is successfully obtained by incorporating the incremental thermal field theory into the 1-D analysis. The theoretical results agree well with the experimental data. The combination of the thermal field and the thickness model for circular quartz crystal resonators can realize a full analysis of thermal properties in vital applications such as high-sensitivity microbalance sensors.

High-Frequency Vibration Analysis of Piezoelectric Array Sensor under Lateral-Field-Excitation Based on Crystals with 3 m Point Group.

Based on Mindlin's first-order plate theory, the high-frequency vibrations of piezoelectric bulk acoustic wave array sensors under lateral-field-excitation based on crystals with 3 m point group are analyzed, and the spectral-frequency relationships are solved, based on which, the optimal length-thickness ratio of the piezoelectric crystal plate is determined. Then, the dynamic capacitance diagram is obtained by a forced vibration analysis of the piezoelectric crystal plate. The resonant mode conforming to good energy trapping is further obtained. The frequency interferences between different resonator units are calculated, and the influences of the spacing between two resonant units on the frequency interference with different electrode widths and spacings are analyzed. Finally, the safe spacings between resonator units are obtained. As the electrode spacing value of the left unit increases, the safe spacing d0 between the two resonator units decreases, and the frequency interference curve tends to zero faster. When the electrode spacings of two resonator units are equal, the safe distance is largest, and the frequency interference curve tends to zero slowest. The theoretical results are verified further by finite element method. The analysis model of high frequency vibrations of strongly coupled piezoelectric bulk acoustic array device based on LiTaO3 crystals with 3 m point group proposed in this paper can provide reliable theoretical guidance for size optimization designs of strongly coupled piezoelectric array sensors under lateral-field-excitation.

A Theoretical Model for Analyzing the Thickness-Shear Vibration of a Circular Quartz Crystal Plate With Multiple Concentric Ring Electrodes.

A dynamic model to analyze the thickness-shear vibration of a circular quartz crystal plate with multiple concentric ring electrodes on its upper and bottom surfaces is established with the aid of coordinate transformation. The theoretical solution is obtained, which can be written in a superposition form of Mathieu functions and modified Mathieu functions. The convergence of the solution is demonstrated, and the correctness is numerically validated via results from the finite element method (FEM). Subsequently, a systematic investigation is carried out to quantify the effect of the electrode size on the energy trapping phenomenon, i.e., the resonant frequency and mode shape, which reveals that the ring electrode has a great influence on the work performance of resonators. With the increase of the electrode inertia, i.e., the radius and mass ratio, new trapped modes emergence with the vibration mainly focused on the plate with partial electrodes. Besides, owing to the anisotropy, degenerated trapped modes have different resonant frequencies and the frequency discrepancy between them will become smaller for higher modes. Finally, the influence of multiple ring electrodes is investigated, and the qualitative analysis and quantitative results demonstrate that multiple ring electrodes will lead to a more uniform mass sensitivity compared with a single ring electrode. The outcome is widely applicable, which can provide theoretical guidance for the structural design and manufacturing of quartz resonators, as well as a thorough interpretation about the underlying physical mechanism.

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.

Two-Dimensional Plate Theory for the Analysis of Coupling Vibrations in Shear Mode FBARs.

The shear mode film bulk acoustic wave resonators (FBARs) have been developed rapidly in recent years due to the increased interest in biosensors. In this article, the mode coupling effects that are unavoidable in real devices are theoretically analyzed and discussed. Mindlin's approximation method of expanding the displacement fields with power series along the plate thickness direction is employed, and a system of 2-D plate equations is obtained. The accuracy of the plate theory is validated through the comparison of dispersion curves with the results calculated by 3-D exact theory. Free coupling vibrations of the cross-sectional plane of finite-sized shear mode FBAR plates are further studied with the derived 2-D theory. The results in this article show that the presented plate theory is accurate and efficient for the structural analyses of shear mode FBARs. Besides, the selection of aspect ratios can be used to avoid strong mode couplings, which is helpful for device designs.

The Design of a Frame-Like ZnO FBAR Sensor for Achieving Uniform Mass Sensitivity Distributions.

In this paper, an infinite circular ZnO thin film bulk acoustic resonator (FBAR) with a frame-like electrode operating at the thickness-extensional (TE) mode is studied. Two-dimensional scalar differential equations established for the problem in the Cartesian coordinate system are successfully solved by transforming them into normal Bessel equations and modified Bessel equations in the cylindrical coordinate system. Resonant frequencies and vibration distributions are obtained for this frame-like FBAR sensor. A nearly uniform mass sensitivity distribution in the active area is achieved by designing proper electrode size and mass ratio of the driving electrode to the ZnO film. Numerical results show that compared with the reported ring electrode FBAR sensor, the novel frame-like electrode FBAR can achieve a maximum optimization ratio (up to 97.90%) on the uniformity of the mass sensitivity distribution in the active area under the same structural parameters, which is also higher than the optimization ratio 77.63% obtained by the reported double-ring electrode design. Moreover, the mechanism to achieve a very uniform mass sensitivity distribution in the active area by the frame-like electrode is explained in detail according to dispersion curves. Namely, when the resonant frequency of the FBAR sensor is close to the cut-off frequency of the active region in the dispersion curve, the mass sensitivity distribution is nearly uniform. These conclusions provide a theoretical guidance for the design and optimization of ZnO FBAR mass sensors with high performance.

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.

Influence of Surface Conductivity on Dispersion Curves, Mode Shapes, Stress, and Potential for Lamb Waves Propagating in Piezoelectric Plate.

Lamb waves propagating in piezoelectric plate covered by thin-film electrodes with arbitrary conductivity are investigated. The structure consists of three parts, piezoelectric plate, top and bottom thin films with arbitrary conductivities, and top and bottom free space. The governing equations in each part are first solved, and then, continuous conditions are used to form the propagating matrix. Finally, the dispersion equations are obtained by boundary conditions. Calculating the dispersion equations with different conductivities, the dispersion curves in high conductivity and zero conductivity are compared. The wave mode shapes of different branches are investigated, including the distributions of mechanical displacements and stresses in the piezoelectric plate and distributions of electrical displacement and potential in both plate and free space. The relationships between frequency difference and wave mode shapes in zero and high conductivity are investigated. Furthermore, the dispersion curves of general conductivity are calculated. The attenuation of waves is compared in different branches and different conductivities. The wave mode shapes are also calculated and compared with the cases in zero and high conductivities.

Bulk Acoustic Wave Characteristics of Pseudo Lateral-Field-Excitation on LGT Single Crystal for Liquid Phase Sensing.

In the present study, pseudo lateral-field-excitation (LFE) bulk acoustic wave characteristics on LGT crystals are investigated to increase the sensitivity of LFE devices on the liquid characteristic variations. The cut orientation of LGT crystals for pseudo-LFE is investigated and verified experimentally. For an LFE device in the pseudo-LFE mode, the thickness shear mode wave is excited by the thickness field rather than the lateral field. The present work shows that when the (yxl) 13.8° LGT plate is excited by the electric field parallel to the crystallographic axis x, it operates in the pseudo-LFE mode. Moreover, characteristics of devices including the sensitivity and impedance are investigated. The present work shows that sensitivity of LFE devices to variation of the conductivity and permittivity of the aqueous solution are 9 and 3.2 times higher than those for AT-cut quartz crystal based devices, respectively. Furthermore, it has been found that the sensitivity of the LGT LFE sensor to liquid acoustic viscosity variations is 1.4 times higher than the one for the AT-cut quartz sensor. The results are a critical basis of designing high-performance liquid phase sensors by using pseudo-LFE devices.

Frequency interferences of two-unit quartz resonator arrays excited by lateral electric fields.

The coupled thickness-twist and face-shear vibrations of the lateral-field-excitation (LFE) two-unit resonator array are analyzed, and the frequency interference and shift of LFE resonator arrays are both studied. Different from most of the quartz resonator arrays based on thickness-field-excitation, the resonator array in this study operates with LFE generated by a pair of electrodes on the top surface of the resonator. By using Mindlin's first-order theory of piezoelectric plates, the electrically forced vibrations are analyzed. The effects of various structural parameters on the frequency interference and shift are examined; furthermore, the corresponding mechanisms are discussed. Varying trends of the frequency interference and shift with various structural parameters are verified by the finite element method. The results are crucial for avoiding frequency interferences between two adjacent units in parameter design of the resonator array operating on LFE.

Double-mode lateral-field-excitation bulk acoustic wave characteristics of Ca3TaGa3Si2O14 crystals.

The double-mode lateral-field-excitation (LFE) bulk acoustic wave characteristics of Ca3TaGa3Si2O14 (CTGS) crystals are investigated. It is found that LFE devices based on (yxl)-57° CTGS crystals can work on both pure-LFE and pseudo-LFE modes when the driving electric field direction is normal to the crystallographic x axis of the piezoelectric substrate. Several double-mode LFE bulk acoustic wave devices based on CTGS crystals are designed and tested. The experimental results conform to the theoretical prediction well. Being able to operate in pure-LFE and pseudo-LFE modes, the double-mode LFE sensors show high sensitivity to both mechanical and electrical property changes of analytes. The results provide the crystal cut for double-mode LFE sensors, which is a critical basis of designing high-performance chemical and biological sensors by using double-mode LFE devices.

Free and forced vibrations of SC-cut quartz crystal rectangular plates with the first-order Mindlin plate equations.

Mindlin plate theory was used to provide accurate solutions to thickness-shear vibrations of plates, which have a much higher frequency than usual flexural vibrations and are the functioning modes of quartz crystal resonators. The vibration frequency solutions obtained with the Mindlin plate theory are proven being accurate along with mode shapes. In this paper, straight-crested wave solutions of free and forced vibrations of doubly rotated SC-cut of quartz crystal plates of rectangular shapes with four free edges are obtained with validated Mindlin plate equations. A procedure has been established for the calculation of dispersion relations, frequency spectra, mode shapes, and capacitance ratios of forced vibrations needed in resonator design.

Analysis of coupled fast-shear and extensional vibrations of a LiTaO3 crystal plate with a ferroelectric inversion layer.

The resonance vibrations of LiTaO3 fast-shear overtone mode resonators with a ferroelectric inversion layer are analyzed. In addition to the fast-shear mode, the coupled extensional mode is considered. Different from most of the LiTaO3 resonators studied in the literature that are based on the slow-shear mode, the resonator in this paper operates with the fast-shear overtone mode. Results show that the capacitance ratio assumes maxima at two resonances, which are identified to be the second overtone modes of fast-shear and extension, respectively. It is found that the thickness of the inversion layer has obvious influences on the capacitance ratio of fast-shear and extensional modes. This condition may provide a simple method to adjust capacitance ratios of piezoelectric resonators. The influence mechanisms are also discussed. Besides, the effect of the cut angle of the crystal on the mode shape of vibrations is also investigated. The results can be used as important basis of parameters designs of LiTaO3 resonators operating on the fast-shear overtone mode.

Forced vibrations of SC-cut quartz crystal rectangular plates with partial electrodes by the Lee plate equations.

Lee plate equations for high frequency vibrations of piezoelectric plates have been established and perfected over decades with the sole objective of obtaining accurate predictions of frequency and mode shapes to aid the analysis and design of quartz crystal resonators. The latest improvement includes extra terms related to derivatives of the flexural displacement to provide much accurate solutions for vibrations of the thickness-shear mode, which is the functioning mode of resonators and has much higher frequency than the flexural mode. The improved Lee plate equations have been used in the analysis of high frequency vibrations of quartz crystal plates as an essential step for analysis of AT- and SC-cut quartz crystal resonators after validations with fully electrode quartz crystal piezoelectric plates. In this study, closed-form solutions of free and forced vibrations of SC-cut quartz plates with partial electrodes are obtained. A procedure has been established for the calculation of dispersion relations, frequency spectra, selected vibration modes, and capacitance ratios of forced vibrations. The vibration solutions obtained with the first-order Lee plate equations are proven to be close to solutions from the Mindlin plate equations. It is now clear that both the Mindlin and Lee plate equations can be used in the analysis and design of quartz crystal resonators.

Resonances and energy trapping in AT-cut quartz resonators operating with fast shear modes driven by lateral electric fields produced by surface electrodes.

We analyze coupled thickness-shear and extensional vibrations of a piezoelectric resonator of AT-cut quartz. Different from most of the AT-cut quartz resonators studied in the literature which are based on the slow shear mode excited by a thickness electric field, the resonator in this paper operates with the fast shear mode driven by a lateral electric field produced by a pair of electrodes on the top surface of the resonator. Mindlin's first-order theory of piezoelectric plates is used. Dispersion relations of the relevant waves in unelectroded and electroded plates are presented and compared. The motional capacitance, resonant frequencies and mode shapes near resonances are obtained from an electrically forced vibration analysis. Trapped modes without vibration near the resonator edges are identified. The effects of various structural parameters on energy trapping are examined and the mechanisms are discussed. The results can provide important bases for the parameters design of new resonators operating with the fast shear mode with new excitation schemes.

Investigation of quasi lateral-field-excitation on (yxl)-17° LiNbO3 single crystal.

Quasi lateral-field-excitation (LFE) on LiNbO3 crystal is investigated both theoretically and experimentally. It is found that when the driving electric field direction is parallel to the crystallographic X-axis of the piezoelectric substrate, (yxl)-17° LiNbO3 LFE bulk acoustic wave devices work on quasi-LFE mode. The experimental results agreed with the theoretical prediction well. The results provide the cut of LiNbO3 crystal for quasi-LFE bulk acoustic wave devices, which is important for designing high performance LFE sensors on LiNbO3 substrates.

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 AT-cut 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.