Piezoelectric Nonlinearity Research Articles

Precise diffraction efficiency measurements of large-area greater-than-99%-efficient dielectric gratings at the Littrow angle

We have developed improved cavity-finesse methods for characterizing the diffraction efficiencies of large gratings at the Littrow angle. These methods include measuring cavity length with optical techniques, using a Michelson interferometer to calibrate piezoelectric transducer nonlinearities and angle-tuning procedures to confirm optimal alignment. We used these methods to characterize two 20 cm scale dielectric gratings. The values taken from across their surfaces collectively had means and standard deviations of micro=99.293% and sigma=0.164% and micro=99.084% and sigma=0.079%. The greatest efficiency observed at a single point on a grating was (99.577+/-0.002)%, which is also the most accurate measurement of the diffraction efficiency in the literature of which we are aware. These results prove that a high diffraction efficiency with low variation is achievable across large apertures for gratings.

Active damping of the forced vibration of a hinged beam with piezoelectric layers, geometrical and physical nonlinearities taken into account

The problem of forced vibration of a hinged beam with piezoelectric layers is solved. Issues of mechanical and electric excitation of vibration and the possibility of damping mechanically induced vibration by applying a voltage to the electrodes of the piezolayers are studied. The effect of the physically nonlinear behavior of the passive layers on the response of the sensor layer and entire structure and the effect of geometric nonlinearity on the behavior of the structure and sensor layer are analyzed. The interaction of physical and geometrical nonlinearities for transient and stationary processes is studied

Coupled Flexural-Torsional Nonlinear Vibrations of Piezoelectrically Actuated Microcantilevers With Application to Friction Force Microscopy

The problem of vibrations of microcantilevers has recently received considerable attention due to its application in several nanotechnological instruments, such as atomic force microscopy, nanomechanical cantilever sensors, and friction force microscopy. Along this line, this paper undertakes the problem of coupled flexural-torsional nonlinear vibrations of a piezoelectrically actuated microcantilever beam as a typical configuration utilized in these applications. The actuation and sensing are both facilitated through bonding a piezoelectric layer (here, ZnO) on the microcantilever surface. The beam is considered to have simultaneous flexural, torsional, and longitudinal vibrations. The piezoelectric properties combined with nonlinear geometry of the beam introduce both linear and nonlinear couplings between flexural vibration as well as longitudinal and torsional vibrations. Of particular interest is the inextensibility configuration, for which the governing equations reduce to coupled flexural-torsional nonlinear equations with piezoelectric nonlinearity appearing in quadratic form while inertia and stiffness nonlinearities as cubic. An experimental setup consisting of a commercial piezoelectric microcantilever installed on the stand of an ultramodern laser-based microsystem analyzer is designed and utilized to verify the theoretical developments. Both linear and nonlinear simulation results are compared to the experimental results and it is observed that nonlinear modeling response matches the experimental findings very closely. More specifically, the softening phenomenon in fundamental flexural frequency, which is due to nonlinearity of the system, is analytically and experimentally verified. It is also disclosed that the initial twisting in the microcantilever can influence the value of the flexural vibration resonance. The experimental results from a macroscale beam are utilized to demonstrate such twist-flexure coupling. This unique coupling effect may lead to the possibility of indirect measurement of small torsional vibration without the need for any angular displacement sensor. This observation could significantly extend the application of friction force microscopy to measure the friction of a surface indirectly.

Theoretical study of the effect of nonlinear piezoelectricity on the pressure coefficient of the light emission in (111)-orientedInGaAs∕GaAsquantum wells

We investigate the effect of second-order piezoelectricity on the pressure coefficient of the light emission $(d{E}_{E}∕dP)$ in (111)-oriented $\mathrm{In}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ quantum wells (QWs). In the framework of continuum theory of elasticity and piezoelectricity, we propose the analytic model of pressure tuning of the built-in electric field in a (111)-oriented zinc-blende QW, which takes into account the second-order (nonlinear) piezoelectric effect. Calculations performed using this model reveal that changes of the built-in electric field with pressure in (111)-oriented $\mathrm{In}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QWs are significantly enlarged by the effect of nonlinear piezoelectricity, in comparison to the case when linear piezoelectricity is used. Next, we discuss the influence of the effects of nonlinear piezoelectricity and nonlinear elasticity on $d{E}_{E}∕dP$ in (111)-oriented $\mathrm{In}\mathrm{Ga}\mathrm{As}∕\mathrm{Ga}\mathrm{As}$ QWs as a function of In concentration and the QW width. We show that the effect of nonlinear piezoelectricity increases $d{E}_{E}∕dP$ in In-poor QWs and decreases $d{E}_{E}∕dP$ in In-rich QWs. The effect of nonlinear elasticity reduces $d{E}_{E}∕dP$ in the whole range of QW compositions. Contribution to $d{E}_{E}∕dP$ originating from the effect of nonlinear piezoelectricity depends more significantly on the QW width than the contribution coming from the effect of nonlinear elasticity.

Micromechanics of Ferroelectrics: From Domain Walls to Piezoceramic Devices

A micromechanical approach utilizing the features of thermally activated domain wall motion is applied to simulate the structural changes in ferroelectric ceramics for the basic processes employed in piezoelectric devices. The approach combines an Arrhenius equation (with the temperature insensitive pre-exponential factor) for the domain wall mobility and a time-dependent constitutive model, using the volume fractions of domains with representative polarization directions as structural (internal) variables. Computer simulation of the piezoceramic poling process in a sheet actuator with interdigitated electrodes shows the formation of the inhomogeneous poling state and peculiarities of the nonlinear longitudinal piezoelectric effect. **Permanent address: Institute of Crystallography RAS, Moscow, Russia.

Experimental Verification of Models for Microfabricated Piezoelectric Vibration Energy Harvesters

Experiments have been performed to verify power-optimized modal models of piezoelectric vibration harvesters for microelectromechanical systems. Such harvesters can power a variety of sensors, and there have been recent national workshops dedicated to harvesting. Detailed experimental results, including displacement histories and electrical output, are provided over a range of frequencies and electrical loadings to compare with (optimized) modal models. The harvester geometry considered is that of a symmetric bimorph macroscale cantilever. Although some experimental work for cantilevered bimorph harvesters has been published, key testing and/or device parameters needed for model verification are missing and/or data at and near power optima (the most interesting operating points) are not provided. Therefore, a detailed set of experiments was performed using power-optimized modeling results to guide the test matrix. Over the broad range of parameters tested, the models accurately predicted all trends and device performance away from device resonances (resonance and antiresonance frequencies). Near the resonance frequencies, the model consistently underpredicts electrical performance, which is satisfactorily attributed (and experimentally supported) to the well-known piezoelectric coupling nonlinearity in the large-strain region. The data presented herein can serve as benchmark data to verify other modeling efforts. The verified models have been used to optimally design microelectromechanical system harvesters for commercial aircraft and microfabrication is ongoing.

Nanoscale characterization of polycrystalline ferroelectric materials for piezoelectric applications

In this work, the nanoscale electromechanical properties of several important piezoelectric materials [as exemplified by PbZrxTi1−xO3 (PZT)] suitable for both bulk actuator and microelectromechanical system (MEMS) applications are reported. The investigations are performed by the piezoresponse force microscopy (PFM) that is currently the most suitable tool for both ferroelectric domain imaging and local piezoelectric studies. The local piezoresponse of individual grains is measured in PZT films and compared with average piezoelectric behavior. Frequency dependencies of local piezoelectric coefficients are presented and analyzed. The results on local piezoelectric nonlinearity, as well as on nanoscale fatigue and aging are briefly discussed. These measurements demonstrate that PFM is promising for studying local piezoelectric phenomena in polycrystalline ferroelectrics where defects and other inhomogeneities are essential for the interpretation of macroscopic piezoelectric properties. Finally, local electromechanical properties of polycrystalline relaxors (PMN-PT, PLZT, doped BaTiO3) are briefly outlined.

Position Controller for Piezoelectric Actuator

This paper presents a computer-based tracking control approach for a piezoelectric actuator based on incorporating a feedforward loop with a PID feedback controller. The purpose of this paper is to improve the hysteresis characteristics of a stack type piezoelectric actuator using the hysteresis nonlinearity compensator. The system proposed in this study prints by spraying the molten metal, and consists of a nozzle, heating furnace, operating actuator, and an XYZ 3-axis stage. As an operating system, the piezoelectric(PZT) method has very valuable uses. The PZT actuator, however, has a hysteresis nonlinearity due to the ferroelectric characteristics of the PZT element. This causes problems in the system position control characteristics and deteriorates the performance of the system. This study proposed a inverse hysteresis model, a mathematic modeling method that can express the geometric relationship between voltage and displacement, in order to reduce the hysteresis of the PZT actuator. In addition, system identification and PID control methods were examined. Also, it was confirmed that the proposed control strategy gives good tracking performance.

Nonlinear piezoelectric resonance: A theoretically rigorous approach to constant I−V measurements

Methods for piezoelectric characterization include the standard resonance test. At higher powers, however, the material’s inherent nonlinearity acts to significantly affect the expected resonance response. High-power resonance methods have previously been developed to describe piezoelectric nonlinearity. In this article we specify and describe the approximations adopted in the current theory and propose a more rigorous theory derived from fundamental principles. We first use thermodynamics to derive the form of the constitutive equations. In particular, we propose that the envelope rather than instantaneous values of stress, strain, and electric field must appear in these equations to yield a type of nonlinearity which, nonetheless, yields a sinusoidal current for sinusoid applied voltage. An alternative approach is set out describing the highly nonlinear experimental data by fitting just one adjustable material parameter to the entire impedance response measured around resonance. Theoretical descriptions for conventional constant-voltage and modified constant-current excitation are developed and compared with experiments for soft and hard compositions of a piezoelectric ceramic (PZT-5H and PZT-4D). The theory is able to match whole families of constant-voltage and constant-current curves with only one adjustable parameter. These models may be used to characterize the high-power properties of piezoelectric materials.

Nanoscale Characterization of Ferroelectric Materials for Piezoelectric Applications

In this work, the nanoscale electromechanical properties of Pb-based perovskites that are currently the main candidates for piezoelectric applications are investigated by Piezoresponse Force Microscopy (PFM). Local response is compared with that studied via conventional technique. The effect of local polarization switching by mechanical force is demonstrated. This effect may limit the functionality of the films in piezoelectric applications. Local piezoelectric nonlinearity, as well as the nanoscale degradation are also studied by PFM. These measurements demonstrate that the defects may act as pinning centers for domain walls and thus influence nanoscale properties. Finally, local properties of ferroelectric relaxors are presented.

Ferroelectric thin films: Review of materials, properties, and applications

An overview of the state of art in ferroelectric thin films is presented. First, we review applications: microsystems’ applications, applications in high frequency electronics, and memories based on ferroelectric materials. The second section deals with materials, structure (domains, in particular), and size effects. Properties of thin films that are important for applications are then addressed: polarization reversal and properties related to the reliability of ferroelectric memories, piezoelectric nonlinearity of ferroelectric films which is relevant to microsystems’ applications, and permittivity and loss in ferroelectric films—important in all applications and essential in high frequency devices. In the context of properties we also discuss nanoscale probing of ferroelectrics. Finally, we comment on two important emerging topics: multiferroic materials and ferroelectric one-dimensional nanostructures.

Piezoelectric nonlinearity in ferroelectric thin films

The piezoelectric properties of {001} oriented 0.5Pb(Yb1∕2Nb1∕2)O3–0.5PbTiO3 films were studied. It was found that the films show Rayleigh-like behavior in that the magnitude of the piezoelectric coefficient depends linearly on the amplitude of the exciting ac electric field used for the measurement. The reversible and irreversible Rayleigh piezoelectric coefficients decreased logarithmically with frequency. The frequency dependence of the Rayleigh coefficients measured under applied bias was identical for the dielectric and piezoelectric responses, suggesting that the same interface motion is responsible for both phenomena. It is demonstrated that dynamic poling of the material by the external ac field self-consistently explains both the Rayleigh-like response and the higher harmonics in the strain response.

In situ calibration of a Michelson type, speckle-shearing interferometer: Wobbling mirror effect

In situ calibration of piezoelectric transducers in a phase stepping, Michelson type, speckle-shearing interferometer can be affected by mirror wobbling introduced by the transducers. When this happens, a modulation of the curves employed for calibration can be readily detected in imaging systems with large apertures. A calibration method that takes into account the modulation of these curves is, therefore, essential to insure the required regularity of the phase steps. A computer model of speckle-shearing interferometry is used to deal with piezoelectric non-linearities and mirror wobbling in the interferometer, and to show how the calibration curves are affected when both phenomena are present. Computer simulated and experimental results validate the calibration technique advanced here.

Piezoelectric nonlinearity due to motion of 180° domain walls in ferroelectric materials at subcoercive fields: A dynamic poling model

A mechanism is described whereby 180° domain wall motion can contribute to the electrically induced strain in a ferroelectric material at subcoercive ac electric fields. The field-dependent, largely reversible motion of ferroelectric, 180° domain walls due to an applied ac electric field is considered. The Rayleigh law is modified to describe the piezoelectric response of the system. This results in both a linear dependence of the piezoelectric coefficient on the amplitude of the applied electric field and the creation of a second order harmonic of strain which adds to the electrostrictive response. The model was experimentally confirmed in Pb(Zr1−xTix)O3 thin films.

Piezoelectric response of compressive loaded multilayer ceramic actuator

The non-linear piezoelectricity of multilayer ceramic actuators (MCA) using PMN–PZT ceramics was investigated under the combined loading of electric field and compressive stress. These piezoelectric actuators present the non-linear ferroelectric behaviors. When electric field and compressive stress are simultaneously applied to the ceramics, the ferroelectricity is expected to be different from that of ceramics induced by only one quantity, electric field or mechanical load. This property difference was estimated via piezoelectric equation. Unlike the case of application of only one inducing quantity, electric field or mechanical stress, the piezoelectric and dielectric behavior was found to be strongly dependent upon variation in dielectric permittivity under the mechanical loading.

Contributions to the Piezoelectric Effect in Ferroelectric Single Crystals and Ceramics

The piezoelectric effect in ferroelectric single crystals and ceramics is investigated considering intrinsic (lattice), and extrinsic (originating mainly from displacement of domain walls) contributions. The focus of the study of intrinsic properties is on piezoelectric anisotropy, which was examined using the Landau–Ginsburg–Devonshire phenomenological theory. It is shown that the enhanced piezoelectric response along nonpolar directions, observed in many perovskite systems, is a consequence of the flattening of the Gibbs free energy profile. This flattening is common for temperature‐, composition‐, and external field‐induced enhancement of the piezoelectric properties along nonpolar axes. A brief review of recent advances in understanding the origins of the piezoelectric nonlinearity, hysteresis, and frequency dispersion is also given.

Driving Forces for Interfacial Fatigue Crack Growth by Piezoelectric Actuator

A new experimental technique for accelerated fatigue testing has recently been developed (Du, T.B., Liu, M., Seghi, S., Hsia, K.J., Economy, J. and Shang, J.K. 2001. “Piezoelectric Actuation of Crack Growth along Polymer-Metal Interfaces in Adhesive Bonds,’’ Journal of Material Research, 16(10):2885-2892). Using a piezoelectric actuator, cyclic loading can be applied at frequencies up to 20 kHz, several orders of magnitude higher than that achieved by a conventional mechanical cyclic loading technique. Moreover, the new technique using piezoelectric actuators directly addresses the debonding problem in piezoelectric multilayered smart structures. However, the threshold energy release rate for interfacial crack propagation, evaluated based on plane strain and linear piezoelectricity assumptions in (Du, T.B. 2001. “Durability of Polymer/Metal Interfaces under Cyclic Loading,” PhD Thesis, University of Illinois at Urbana-Champaign, Urbana, IL, May), seems to be almost an order of magnitude lower than that measured by the conventional mechanical cyclic loading. In this article, we investigate the origin of such discrepancies, and find that the driving force provided by piezoelectric actuator is intrinsically three-dimensional in nature. To account for this effect, we develop both a plane strain model and a modified plane strain analytical model that takes into account the effects in the third dimension to evaluate the energy release rate. The results show that the plane strain solution underestimates the driving force. We also study the effect of nonlinear piezoelectricity on crack driving forces by performing detailed finite element simulations. The results show that the nonlinear piezoelectric effect is another important factor that contributes to the discrepancy between the results from the piezoelectric loading and that from the conventional mechanical loading.

Nonlinear local piezoelectric deformation in ferroelectric thin films studied by scanning force microscopy

Local piezoelectric deformation is investigated in (Pb,La)TiO3 (PLT) and Pb(Zr,Ti)O3 (PZT) thin films via scanning force microscopy (SFM) as a function of the ac voltage Vac applied between the conducting tip and the bottom electrode. Thus obtained voltage dependence of the effective piezoelectric coefficient (local piezoelectric nonlinearity) is compared with the corresponding macroscopic piezoelectric behavior determined by laser interferometry. As expected, the local piezoresponse of PLT films measured inside uniformly polarized areas (ferroelectric domains) remains almost linear with increasing Vac until the driving voltage becomes comparable with the coercive one. The corresponding macroscopic response is substantially nonlinear, suggesting significant contribution from the motion of 90° domain walls. On the contrary, in PZT films the local piezoelectric behavior is strongly nonlinear, whereas the macroscopic piezoelectric coefficient is almost field independent. Moreover, depending on the polarity of probed as-grown domains, the local piezocoefficient of PZT films is found to display either enhancement or reduction with increasing ac voltage. The “positive” domains (i.e., domains having polarization vector pointing to the film-free surface) are often unstable with increasing Vac and switch into the opposite polarization state under an ac voltage several times smaller than that required for global polarization reversal. This effect is explained by the presence of charged domain boundaries below the surface and their local depinning induced by external field. It is envisaged that SFM can be used not only for polarization mapping of ferroelectric surfaces but also as a probe for studying local polarization profiles beneath the SFM tip.

Piezoelectric field-dependent optical nonlinearities induced by interband transition in InGaN/GaN quantum well

The binding energy of GaN/InGaN quantum well (QW) has been studied by taking the strain-induced piezoelectric and spontaneous polarization effects into account. The variational calculations are presented for the ground exciton state in the quantum wells, and the third-order susceptibilities, as functions of In content x , well width w , and pump photon energy ℏ ω , have also been analyzed. In addition, the results are compared with the data of previous work and some conclusions are given.

Extrinsic contribution to the non-linearity in a PZT disc

Non-linear increases in elastic, piezoelectric (direct and reverse) and dielectric coefficients have been measured under a high electrical field or under high mechanical stress. The permittivity and reverse piezoelectric coefficient can be measured by applying a high voltage at a low frequency, while the elastic compliance and direct piezoelectric coefficient can be measured at the first radial resonance frequency in order to apply a high stress. The non-linear behaviour has been analysed at the radial resonance of a disc.In all the materials tested, the results show that there is a close relation between the non-linear increments of the different coefficients. An empirical model has been proposed in order to describe and understand these relations. It is assumed that either the strain or the electrical displacement is produced by intrinsic and extrinsic processes, but only the latter, which consist mainly in the motion of domain walls, contribute to the non-linearity. The model enables us to find the domain wall contribution to elastic, piezoelectric and dielectric non-linearities, and allows us to compare the amplitudes of the fields and stresses that produce the same displacement of domain walls.