Waves In Piezoelectric Crystals Research Articles

Second gradient continuum model for anisotropic elastic and piezoelectric structures calibrated based on phonon dispersion relations

In this paper, we present a variant of the second gradient continuum model that allows description of the spatial dispersion effects of high-frequency waves in anisotropic crystals. The considered model takes into account the strain and inertia gradient effects and the classical electro-mechanical coupling. The simplified constitutive equations of the model contain seven and four length scale parameters for piezoelectric hexagonal crystals and elastic cubic crystals, respectively. It is shown that all except one length scale parameter can be identified based on the fitting of the continuum model to the lattice dynamics calculations for the dispersion relations of the bulk acoustic phonons. Examples of the identification are presented for hexagonal piezoelectric ZnO and AlN and for cubic diamond and Si crystals. Using a calibrated model, we then obtain the dispersion relations for the high-frequency Bleustein–Gulyaev wave in piezoelectric crystals and for the shear horizontal surface wave in elastic crystals. The last one is predicted by the gradient models only for the high frequencies, while at low frequencies this wave degenerates to the bulk shear wave. Examples of calculations are also provided for the Rayleigh wave propagating through the homogeneous piezoelectric half-space and layered ZnO(AlN)/diamond/Si structures. It is shown that this kind of surface wave can be used to identify the last unknown length scale parameter of the model.

Controlling Acoustic Wave Energy Fluxes in Piezoelectric Crystals

Abstract—The article investigates the possibility of controlling the energy fluxes of surface (SAW) and normal (NAW) acoustic waves in piezoelectric crystals without changing the orientation of the crystals. It is shown that the angle Ψ between the direction of the energy flux and the wave normal of the SAW changes jumpwise during surface metallization, and the size of the jump depends on the value of the electromechanical coupling coefficient wave and its anisotropy in the propagation plane. For NAW, the angle Ψ additionally depends on the mode number, plate thickness, and acoustic wavelength. Therefore, the radiation of modes with different numbers using a periodic interdigital transducer (IDT) at different frequencies is fan-shaped.

Stable perfectly-matched-layer boundary conditions for finite-difference time-domain simulation of acoustic waves in piezoelectric crystals

Perfectly-matched-layer (PML) boundary conditions are derived for finite-difference time-domain analysis of acoustic waves within piezoelectric crystals. The robustness and effectiveness of the derived boundary conditions are demonstrated by simulating acoustic wave propagation in the bismuth germanate material system—a system in which simple absorbing boundary conditions cause instabilities. An investigation into the stability and effectiveness of the PML is then presented in terms of the PML thickness and absorption profile. A range of optimised absorption profiles were determined by finding the maximum permissible absorption within the stability limit of the system. In the optimised case, the form of the absorption profile had little influence on the effectiveness of the PML. However, in the unoptimised case the linearly increasing absorption profile was found to be the most effective.

Resonant generation of surface acoustic waves between moving and stationary piezoelectric crystals

The propagation of surface acoustic waves in a system composed of two piezoelectric crystals moving with respect to each other and separated by a vacuum gap is considered. The waves are localized on different sides of the gap and coupled only through the electrostatic interaction. It is shown that when the velocity of the relative motion of crystals is close to some value, there occurs a wave instability resulting in a resonant generation of these surface waves. The rate of growth of Bleustein-Gulyaev waves in piezoelectric crystals of 6mm symmetry class is determined analytically.

The role of electromagnetic waves in the reflection of acoustic waves in piezoelectric crystals

The role of electromagnetic effects in the reflection and transmission of acoustic waves in piezoelectric materials is discussed. Attention is focused on the analysis of a number of situations where the coefficients of plane wave conversion derived on the basis of the electromagnetic theory differ significantly from the coefficients calculated within the frame of the quasi-electrostatic approximation. This effect occurs at quasi-normal incidence for angles of the order of v a/ v el, where v a and v el are the typical velocities of the sound and electromagnetic waves, respectively. For instance, according to the electromagnetic description, the acoustic wave must suffer total reflection while the quasi-electrostatic approximation predicts almost total transmission. The possibility of experimental observation of the above effects is also discussed.

Non-piezoactivity in piezoacoustics: basic properties and topological features

The existence conditions of zero electric fields E and zero electric displacements D are studied for bulk acoustic waves in piezoelectric crystals. General equations are derived for lines of zero electric fields, E(m)=0, and for specific points m0 of vanishing electric displacements, D(m0)=0, on the unit sphere of propagation directions m2=1. The obtained equations are solved for a series of examples of particular crystal symmetry. It is shown that the vectors Dα(m) being generally orthogonal to the wave normal m are characterized by definite orientational singularities in the vicinity of m0 and can be described by the Poincare indices n=0, ±1 or ±2. The algebraic expressions for the indices n are found both for unrestricted anisotropy and for a series of particular cases.

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. I. Theory.

After some necessary recalls on the nonlinear theory of thermoelectroelasticity in piezoelectric crystals, asserting the need of constitutive equations which derive from a rotationally invariant energy function, this paper presents the governing equations for a small vibration superimposed on a bias originated by a slow and homogeneous temperature variation from a well-defined reference state. Thereafter, the authors define the effective coefficients appearing in the linearized incremental dynamic balance equations for linear momentum and electrical charge in Lagrange configuration, not omitting associated boundary conditions. The main features of these coefficients are discussed and explicit relations with more conventionally defined coefficients are given. Determination of numerical values of the proposed effective coefficients and examples of their use in the higher order modeling of static frequency-temperature characteristics of either bulk acoustic wave or surface acoustic wave devices are given in a companion paper.

Lagrangian effective material constants for the modeling of thermal behavior of acoustic waves in piezoelectric crystals. II. Applications and numerical values for quartz.

This paper presents a set of numerical values of temperature derivatives of Lagrangian effective elastic coefficients suitable for the modeling of small vibrations in quartz devices submitted to slow and homogeneous temperature variations. After a short description of the proper writing of vibration problems in practical applications with the help of this kind of coefficient, we determine the proposed set of numerical values from the frequency-temperature characteristics of an heterogeneous set of bulk and surface acoustic wave devices.

Real-space field of surface sources and the problem of fast leaky wave generation in a piezoelectric half-space

The total solution of the boundary problem for a piezoelectric half-space in the case of oscillating electric and mechanical surface sources is presented as a function of coordinates in terms of a Fourier integral over the real axis of acoustic wave slowness along the surface. In order to investigate the possible leaky wave contributions to the total field, the surface impedance matrix and the Fourier transform of the Green function are uniquely defined for complex values of slowness, using the procedure of analytic continuation from the real axis of slowness and appropriate energy relations. A simple expression for the calculation of plane partial wave group velocities is also proposed. This approach allows the problem to be automatically confined to leaky waves that can be generated by surface sources. The main calculations refer to the new fast leaky waves, which have high phase velocities not far from the velocities of the fastest bulk wave in piezoelectric crystals, and are thus potentially attractive for high frequency surface acoustic wave devices. In the case of lithium tetraborate with Euler angles (0°, 47.3°, 90°), having assumed a simple distribution of surface sources, the total field is exactly calculated in the vicinity of the sources both at the surface and in the bulk region. The asymptotic field far from the sources is also estimated: it is shown that fast leaky waves, ordinary surface acoustic waves, and so-called limiting bulk waves are present, and all contributions in relation to the distance from the sources are compared. It is shown that the fast leaky waves may have different ratios between the magnitudes of components, when propagating in opposite directions. Both free and metallized crystal surfaces are considered. In contrast to the free surface case, it is found that in the case of a metallized surface the new leaky waves give a good approximation of the total field near the sources. This different behavior should be taken into account in practical applications. A comparison with the well-known case of lithium niobate (0°, −49°, 0°), in which the slow leaky waves can be generated, is made. Previously published numerical and experimental results are discussed. Some experimental results are reinterpreted and explained by applying the present approach to a model of the experimental situation considered.

Acoustic waves effects on raman spectra of piezoelectric crystals

Abstract A technique is proposed to observe effects of acoustic waves in piezoelectric crystals by differential Raman spectroscopy. Anisotropic deformations induced by acoustic waves are shown to change crystal symmetry and result in redistribution of lines’ intensity and new Raman lines appearance. Such effects in KDP and α-quartz crystals are presented as examples.

Observation of nonlinear accompanying electric potential wave in piezoelectric crystals

The theory of three-field acoustic interactions in nonlinear piezoelectric crystals predicts that the second-harmonic acoustic wave is accompanied by the second-harmonic electric potential wave due to the electroacoustic nonlinearities of the media. An experiment to demonstrate the phenomenon carried out for crystal LiNbO3 is described in this letter. Ultrasonic waves excited by a 30 MHz quartz transducer propagate along the z direction of LiNbO3 crystal. The nonlinear electric potential wave accompanying with the acoustic wave is directly detected by use of the capacitive detector. To the author’s knowledge, this kind of observation has not been made before.

Phase conjugate reflection and amplification of a bulk acoustic wave in piezoelectric crystals

A general theory of phase conjugation of bulk acoustic waves in piezoelectric crystals is proposed. The formulae obtained for the ratio of amplitudes in two cases of special boundary conditions allow a quantitative prediction for the so-called 'space resonance'. The theory is in good agreement with the experiments.

Nonlinear properties of bulk and surface acoustic waves in piezoelectric crystals

Abstract Nonlinear properties of bulk and surface acoustic waves are described. Two classes of phenomena are distinguished: the nonlinear propagation of a finite amplitude wave and the nonlinear coupling of a small amplitude wave with the surrounding medium. Studies of the first one are harmonic generation, amplitude-frequency effect, intermodulation and convolution, mainly for signal processing applications. The coupling with external, thermal, mechanical and electrical perturbations are analyzed for stable oscillators or sensor applications.

On the Propagation of Elastic Surface Waves in Piezoelectric Crystals

On the Propagation of Elastic Surface Waves in Piezoelectric Crystals

Functional transformations of signals in the nonlinear interaction of volume and surface elastic waves in piezoelectric crystals

Functional transformations of signals in the nonlinear interaction of volume and surface elastic waves in piezoelectric crystals

Integral formalism for surface waves in piezoelectric crystals. Existence considerations

An integral formalism for surface waves in piezoelectric half-infinite solids valid up to the critical velocity is developed. Various boundary conditions are considered and, in particular, the problem of which boundary conditions allow surface-wave solutions for velocities below the limiting velocity vL is discussed in detail. It is proved that (a) with a mechanically free surface and zero dielectric constant for adjoining medium, at most one solution is possible for v<vL, (b) with a mechanically clamped surface and zero dielectric constant for adjoining medium, no solution is possible for v<vL, (c) with a mechanically clamped surface and an infinitely conducting adjoining medium, no solution is possible for v<vL, and (d) with a mechanically free surface and an infinitely conducting adjoining medium, at least one and at most two solutions are possible for v<vL. When two solutions are possible, one solution is of the Bluestein-Gulyaev type.

Elastic-Wave Formulation for Electroelastic Waves in Unbounded Piezoelectric Crystals

The equations describing the elastic behavior of plane-wave disturbances in an infinite piezoelectric crystal are reduced in form to those of a purely elastic medium in the quasistatic approximation. The internal energy of the piezoelectric solid is expressed as one elastic-type term, and an alternative expression for the elastic energy flux yields the combined elastic and electric energy flux on replacing the elastic constant ${c}^{E}$ with the stiffened ${c}^{\mathrm{kD}}$ without introducing $\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\nabla}}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}=\stackrel{\ensuremath{\rightarrow}}{\mathrm{D}}$ and $\stackrel{\ensuremath{\rightarrow}}{\mathrm{P}}=\stackrel{\ensuremath{\rightarrow}}{\mathrm{E}}\ifmmode\times\else\texttimes\fi{}\stackrel{\ensuremath{\rightarrow}}{\mathrm{H}}$. Similarly, the electric displacement is expressed in terms of the strain variables alone by means of modified piezoelectric ${e}^{\mathrm{kD}}$. Properties of this formalism and of the usual mechanical-electrical formulation are the same, and the two formulations are discussed in relation to each other. The elastic-propagation properties of piezoelectrics are describable by a ray or wave surface, which exists in terms of the ${c}^{\mathrm{kD}}$, and the techniques of variational elasticity carry over to piezoelectrics. The positive definiteness of the ${c}^{\mathrm{kD}}$ is asserted for an arbitrary direction to realize physical stability and, whereas no new limiting restrictions among material constants obtain, their use in Rayleigh-Ritz procedures assures its monotonic convergence. The symmetry and transformation properties of the ${c}^{\mathrm{kD}}$ show that, whereas their symmetry is lower than that of the ${c}^{E}$, their centrosymmetric nature requires only the centrosymmetric crystal groups to describe the elastic properties of piezoelectric crystals. Various stiffened-elastic properties are numerically evaluated for an arbitrary nonpure mode and symmetry-related directions for $\ensuremath{\alpha}$-quartz (class 32), LiNb${\mathrm{O}}_{3}$ ($3m$), CdS ($6mm$), ${\mathrm{Ba}}_{2}$Na${\mathrm{Nb}}_{5}$${\mathrm{O}}_{15}$ ($2mm$), ${\mathrm{Bi}}_{12}$Ge${\mathrm{O}}_{12}$ (23), and GaAs ($43m$). The work offers a simplified approach to characterizing bulk and surface elastic-wave properties of electroelastic waves in piezoelectric crystals and of modally analyzing particular classes of piezoelectric structures.

Nonlinear Acoustical Parameters of Piezoelectric Crystals

The wave equation for elastic waves in piezoelectric crystals is found, including the third-order terms in the expansion of the thermodynamic potential in the strain and electric field. The effective nonlinear parameters for a general crystal are obtained. In the general case these parameters depend on the second-order and third-order coefficients: nonlinear piezoelectric, electrostrictive, and linear electro-optic. These coefficients are considered for piezoelectric and nonpiezoelectric crystals. Some estimates are given of the values of the nonlinear coefficients.

Nonlinear Acoustical Parameters of Piezoelectric Crystals

The wave equation for elastic waves in piezoelectric crystals is found, including the third-order terms in the expansion of the thermodynamic potential in the strain and electric field. The effective nonlinear acoustical parameters for a general crystal are obtained. In the general case these parameters depend on the second-order and third-order coefficients: nonlinear piezoelectric, electrostrictive, and linear electrooptic. These coefficients are considered for piezoelectric and nonpiezoelectric crystals. The properties of the nonlinear piezoelectric tensor are discussed giving some estimate of the value of the nonlinear coefficients. The dependence of sound velocity on the external fields (strain and electric field) and wave interaction is discussed. [Research carried out while visiting at the University of California, Berkeley 94720.]

Theory of parametric amplification of surface waves

Conditions for net parametric gain of Rayleigh waves in piezo-electric crystals have been obtained. The waves are coupled through their interaction with conduction electrons. For CdS the parametric amplification of Rayleigh waves propagating in the basal plane of the crystal is found to be larger than that of bulk transverse waves.