Vibrating Flexoelectric Micro-Beams as Angular Rate Sensors.

We studied flexoelectrically excited/detected bending vibrations in perpendicular directions of a micro-beam spinning about its axis. A set of one-dimensional equations was derived and used in a theoretical analysis. It is shown that the Coriolis effect associated with the spin produces an electrical output proportional to the angular rate of the spin when it is small. Thus, the beam can be used as a gyroscope for angular rate sensing. Compared to conventional piezoelectric beam gyroscopes, the flexoelectric beam proposed and analyzed has a simpler structure.

Effects of Semiconduction on Thickness-Extensional Modes of Piezoelectric Resonators.

We study thickness-extensional vibrations of a piezoelectric semiconductor plate for resonator application. A perturbation integral for the semiconduction-induced frequency shift is obtained, which shows that the first-order frequency shift represents a damping effect due to semiconduction. Numerical results for ZnO and AlN plates are presented.

Magnetically Induced Carrier Distribution in a Composite Rod of Piezoelectric Semiconductors and Piezomagnetics.

In this work, we study the behavior of a composite rod consisting of a piezoelectric semiconductor layer and two piezomagnetic layers under an applied axial magnetic field. Based on the phenomenological theories of piezoelectric semiconductors and piezomagnetics, a one-dimensional model is developed from which an analytical solution is obtained. The explicit expressions of the coupled fields and the numerical results show that an axially applied magnetic field produces extensional deformation through piezomagnetic coupling, the extension then produces polarization through piezoelectric coupling, and the polarization then causes the redistribution of mobile charges. Thus, the composite rod exhibits a coupling between the applied magnetic field and carrier distribution through combined piezomagnetic and piezoelectric effects. The results have potential applications in piezotronics when magnetic fields are relevant.

Trapped-Energy Thickness-Extensional Mode of a Partially Electroded ZnO Thin-Film Resonator.

Free and forced vibration analysis of ZnO thin-film resonator operating with trapped-energy thickness-extensional (TE) mode is performed based on the dispersion curves for both the unbounded fully electroded and unelectroded films. The thickness solutions to the free vibration problem consist of the mode branches of the dispersion curves for the electroded and the unelectroded regions, respectively. The mode branches in the unelectroded region that carry energy away from the vibration zone are neglected because this effective energy loss corresponds to a complex frequency in the frequency spectrum. Since the thickness solutions in each region have satisfied the differential equations and the boundary conditions on the major surfaces exactly, the substitution of the thickness solutions into the modified Hamilton variational principle derived by Tiersten gives an approximate continuity condition in the form of integral over the thickness of the resonator at the interface between the electroded and unelectroded regions. The stationary condition of the integral continuity condition leads to a system of homogeneous linear equations, which determines the frequency spectrum ranging from the first TE cutoff frequency of the fully electroded film to that of the unelectroded film where the conventional trapped-energy vibration occurs. Forced vibration analysis of ZnO thin film driven into TE mode by applying a voltage to the top and bottom electrodes is also performed, which further verifies the validity of the obtained results from the free vibration analysis.

Long thickness-extensional waves in thin film bulk acoustic wave filters affected by interdigital electrodes.

We studied free vibrations of thin-film bulk acoustic wave filters with interdigital electrodes theoretically using the scalar differential equations by Tiersten and Stevens. The filters are made from AlN or ZnO films on Si substrates with ground and driving electrodes. They operate with thickness-extensional modes. The basic vibration characteristics including resonant frequencies and mode shapes were obtained. Their dependence on various geometric parameters was examined. It was found that for properly design filters there exist trapped modes whose vibrations are strong in regions with a driving electrode and decay away from the electrode edges. These trapped modes are essentially long plate thickness-extensional modes modulated by the electrode fingers. The number of trapped modes is sensitive to the geometric parameters.

An in situ self-assembled Cu4I4-MOF-based mixed matrix membrane: a highly sensitive and selective naked-eye sensor for gaseous HCl.

The Cu4I4-MOF-based mixed matrix membranes (MMMs) can be readily prepared by one-step in situ self-assembly of a tri-armed oxadiazole-bridged ligand and CuI in polymer binder solution under ambient conditions. The visual and luminescent HCl(g) sensing detecting limits of the resulting Cu4I4-MOF-based MMMs are 3.2 and 1.6 ppb, respectively, and the response time is less than 1 min.

An estimate of the second-order in-plane acceleration sensitivity of a Y-cut quartz thickness-shear resonator.

We perform a theoretical analysis of the secondorder in-plane acceleration sensitivity of a Y-cut quartz thickness- shear mode resonator. The second-order nonlinear theory of elasticity for anisotropic crystals is used to determine the biasing fields in the resonator under in-plane acceleration. The acceleration-induced frequency shift is determined from a perturbation analysis based on the plate equations for small-amplitude vibrations superposed on a finite bias. We show that, whereas the first-order acceleration-induced frequency shift is zero for a structurally symmetric resonator under in-plane acceleration, the second-order frequency shift is nonzero and is quadratic in the acceleration. As the fourth-order nonlinear elastic constants of quartz have never been measured, we can only estimate the magnitude of the second-order frequency shift. For a particular case of interest, we find Δω/ω0 ~ 10(-18), 10(-16), and 10(-14) when the acceleration is 1, 10, and 100 g, respectively.

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.

Thickness-shear and thickness-twist modes in an AT-cut quartz acoustic wave filter.

We studied thickness-shear and thickness-twist vibrations of a monolithic, two-pole crystal filter made from a plate of AT-cut quartz. The scalar differential equations derived by Tiersten and Smythe for electroded and unelectroded quartz plates were employed which are valid for both the fundamental and the overtone modes. Exact solutions for the free vibration resonant frequencies and modes were obtained from the equations. For a structurally symmetric filter, the modes can be separated into symmetric and antisymmetric ones. Trapped modes with vibrations mainly under the electrodes were found. The effect of the distance between the two pairs of electrodes was examined.

Propagation of thickness-twist waves in elastic plates with periodically varying thickness and phononic crystals.

We study the propagation of thickness-twist (TT) waves in a crystal plate of AT-cut quartz with periodically varying, piecewise constant thickness. The scalar differential equation by Tiersten and Smythe is employed. The problem is found to be mathematically equivalent to the motion of an electron in a periodic potential field governed by Schrodinger's equation. An analytical solution is obtained. Numerical results show that the eigenvalue (frequency) spectrum of the waves has a band structure with allowed and forbidden bands. Therefore, for TT waves, plates with periodically varying thickness can be considered as phononic crystals. The effects of various parameters on the frequency spectrum are examined.

On the propagation of long thickness-stretch waves in piezoelectric plates.

We study the propagation of thickness-stretch waves in a piezoelectric plate of polarized ceramics with thickness poling or crystals of class 6 mm whose sixfold axis is along the plate thickness. For device applications we consider long waves with wavelengths much longer than the plate thickness. A system of two-dimensional equations in the literature governing thickness-stretch, extensional, and symmetric thickness-shear motions of the plate is further simplified. The equations obtained can be used to analyze piezoelectric plate acoustic wave devices operating with thickness-stretch modes.

Variational formulation of the Stevens-Tiersten equation and application in the analysis of rectangular trapped-energy quartz resonators.

The two-dimensional scalar differential equation for transversely varying thickness modes in quartz crystal resonators operating with thickness-shear modes is formulated into variational form for trapped-energy resonators with both electroded and unelectroded regions. A theoretical analysis of rectangular trapped-energy resonators of singly rotated quartz is performed using the Ritz method based on the variational formulation. Free vibration resonant frequencies and modes are obtained. The results show the existence of trapped modes under the electrodes. The effects of various geometric and physical parameters on the trapped modes are examined. It is also found that the classical frequency prediction given by Tiersten and Smythe from an approximate analysis using the scalar differential equation has an inaccuracy on the order of 100 ppm for the fundamental mode, significant in resonator design.

A piezoelectric energy harvester with increased bandwidth based on beam flexural vibrations in perpendicular directions.

We propose a new structure for piezoelectric energy harvesters. It consists of an elastic beam with two pairs of piezoelectric films operating with the fundamental flexural modes in perpendicular directions. A one-dimensional model is developed and used to analyze the proposed structure. The output power density is calculated and examined. Results show that, with simultaneous flexural motions in two perpendicular directions, the output power has two peaks close to each other resulting from the two fundamental flexural resonances. The distance between the two peaks can be adjusted through design to make the two peaks merge into one wide peak. Hence, the frequency bandwidth through which energy can be harvested is roughly doubled when compared with conventional beam bimorph energy harvesters operating with flexural motion in one direction and one resonance only.

Experimental measurement of the electroelastic effect in thickness-mode langasite resonators.

The electroelastic effect describes the shift in resonant frequency that a resonator experiences as a result of the application of a dc electrical field. We report on experimental measurements of the electroelastic effect observed in fourteen plano-plano configuration thickness-mode langasite (La3Ga5SiO14) resonators. The orientations of the fourteen samples provide a sufficient data set to extract all eight of the third-order piezoelectric constants of this material. The role of this type of measurement in determining third-order piezoelectric constants is discussed. We compare the experimentally observed behavior to that predicted when using the langasite material constants currently available in the literature.

A piezoelectric spring-mass system as a low-frequency energy harvester.

We propose a new structure consisting of a piezoelectric spring-mass system as a low-frequency piezoelectric energy harvester. A theoretical model is developed for the system from the theory of piezoelectricity. An analysis is performed to demonstrate the low-frequency nature of the system. Other basic characteristics of the energy harvester, including the output power, voltage, and efficiency, are also calculated and examined.

Overtone frequency spectra for x3-dependent modes in AT-cut quartz resonators.

We study straight-crested waves and vibration modes with spatial variations along the x3 direction only in an AT-cut quartz plate resonator. The equations of anisotropic elasticity are used. Dispersion relations for face-shear and thickness-twist waves in unbounded plates are plotted. Frequency spectra are obtained for face-shear and thickness-twist vibrations of finite plates in which these modes are coupled by boundary conditions. Most importantly, our analysis produces the frequency spectra for overtone modes which do not seem to have been obtained before for x3-dependent modes. Numerical results for third- and fifth-overtone AT-cut quartz resonators are presented, showing that higher-order overtone modes are associated with more mode couplings.

Effects of air resistance on AT-cut quartz thickness-shear resonators.

We study theoretically the effects of air resistance on an AT-cut quartz plate thickness-shear mode resonator. Mindlin's two-dimensional equations for coupled thickness-shear and flexural motions of piezoelectric plates are employed for the crystal resonator. The equations of a Newtonian fluid and the equations of linear acoustics are used for the shear and compressive waves in the air surrounding the resonator, respectively. Solutions for free and electrically forced vibrations are obtained. The impedance of the resonator is calculated. The effects of air resistance are examined. It is found that air viscosity causes a relative frequency shift of the order of ppm. When the material quality factor of quartz Q = 10(5), the air viscosity and compressibility both have significant effects on resonator impedance. For resonators with larger aspect ratios the effects of air resistance are weaker, and the effect of air compressibility is weaker than air viscosity.

Thickness-shear modes of an elliptical, contoured AT-cut quartz resonator.

We study free vibrations of an elliptical crystal resonator of AT-cut quartz with an optimal ratio between the semi-major and semi-minor axes as defined by Mindlin. The resonator is contoured with a quadratic thickness variation. The scalar equation for thickness-shear modes in an AT-cut quartz plate by Tiersten and Smythe is used. Analytical solutions for the frequencies and modes to the scalar equation are obtained using a power series expansion that converges rapidly. The frequencies and modes are exact in the sense that they can satisfy the scalar differential equation and the free edge condition to any desired accuracy. They are simple and can be used conveniently for further studies on other effects on frequencies and modes of contoured resonators.

Effects of interface bonding on acoustic wave generation in an elastic body by surface-mounted piezoelectric transducers.

We study the effects of interface bonding on acoustic wave generation in an elastic body using surface-mounted piezoelectric transducers driven electrically. A theoretical analysis is performed based on a physical model of a piezoelectric layer on an elastic substrate. The transducer-substrate interface is described by the shear-slip model, representing a viscoelastic interface. Different from the results in the literature on free vibrations of structures with weak interfaces, this paper presents an electrically forced vibration analysis. An analytical solution for the generated acoustic wave is obtained and used to calculate its energy flux and the efficiency of the transduction. The effects of the interface parameters are examined. It is found that the interface bonding affects the performance of the transducer in multiple ways, some of which may be exploitable in designs for better transducer performance. In particular, optimal transduction is not necessarily associated with a perfectly bonded interface.

Quasi-thickness-shear waves in thin-film piezoelectric resonators of ZnO and AlN with tilted C-axis.

We study the propagation of Lamb waves in a piezoelectric plate of crystals of class 6mm with tilted c-axis. Dispersion relations for waves in ZnO and AlN plates with a few specific tilting angles of the c-axis are obtained. Displacement amplitude and phase angle distributions along the plate thickness are also calculated. It is shown that for certain values of the tilting angle of the c-axis, there exist long waves with a dominating thickness-shear component. The results further support the possibility of thin-film resonators operating with thickness-shear modes.