J. E. Huber

A constitutive model for ferroelectric polycrystals

Abstract A constitutive model is developed for the non-linear switching of ferroelectric polycrystals under a combination of mechanical stress and electric field. It is envisaged that the polycrystal consists of a set of bonded crystals and that each crystal comprises a set of distinct crystal variants. Within each crystal the switching event, which converts one crystal variant into another, gives rise to a progressive change in remanent strain and polarisation and to a change in the average linear electromechanical properties. It is further assumed that switching is resisted by the dissipative motion of domain walls. The constitutive model for the progressive switching of each crystal draws upon elastic–plastic crystal plasticity theory, and a prescription is given for the tangent moduli of the crystal, for any assumed set of potentially active transformation systems. A self-consistent analysis is used to estimate the macroscopic response of tetragonal crystals (representative of lead titanate) under a variety of loading paths. Also, the evolution of the switching surface in stress-electric field space is calculated. Many of the qualitative features of ferroelectric switching, such as butterfly hysteresis loops, are predicted by the analysis.

Multi-axial electrical switching of a ferroelectric: theory versus experiment

Abstract Samples of the polycrystalline ferroelectric ceramic PZT-5H were poled by applying an electric field at room temperature. Subsequently, an electric field was applied to the samples at a range of angles to the poling direction. The measured non-linear responses in electric displacement are used to construct “yield surfaces” in electric field space corresponding to the onset of ferroelectric switching. The results are compared with predictions from three models: (i) a previous self-consistent polycrystal calculation with rate-independent, non-hardening crystal plasticity; (ii) a simplified crystal plasticity model with viscoplastic (rate-dependent) behaviour and a sufficient number of transformation systems to reproduce the polycrystalline behaviour; (iii) a phenomenological model based on rate-independent flow theory, using kinematic hardening and a quadratic yield surface in electric field and stress space. The experiments suggest that the self-consistent crystal plasticity formulation is most able to reproduce the multi-axial electrical response and yield surface of the polycrystal. The phenomenological model is able to reproduce the uniaxial response accurately, but gives relatively poor performance for multi-axial loading paths, in its present form. A tolerable compromise in multi-axial modelling is the simplified crystal plasticity approach. This is able to reproduce multi-axial constitutive behaviour with reasonable accuracy, whilst offering computational simplicity and speed similar to that of the phenomenological model.

THE SIMULATION OF SWITCHING IN POLYCRYSTALLINE FERROELECTRIC CERAMICS

A polarization switching model for polycrystalline ferroelectric ceramics has been developed. It is assumed that a single ferroelectric crystallite in a ceramic, which is subjected to an electric field and/or a stress, undergoes a complete polarization change and a corresponding strain change if the resulting reduction in potential energy exceeds a critical value per unit volume of switching material. The crystallite’s switch causes a change in the interaction of its field and stress with the surrounding crystallites, which is modeled by the Eshelby inclusion method to provide a mean field estimate of the effect. Thus the model accounts for the effects of the mean electric and stress fields arising from the constraints presented by surrounding crystallites as well as the externally applied mechanical and electrical loads. The switching response of the ceramic polycrystal is obtained by averaging over the behavior of a large number of randomly oriented crystallites. The model, along with the linear dielect...

The selection of sensors

Abstract A systematic method is developed to select the most appropriate sensor for a particular application. A wide range of candidate sensors exist, and many are based on coupled electrical and mechanical phenomena, such as the piezoelectric, magnetostrictive and the pyro-electric effects. Performance charts for sensors are constructed from suppliers data for commercially available devices. The selection of an appropriate sensor is based on matching the operating characteristics of sensors to the requirements of an application. The final selection is aided by additional considerations such as cost, and impedance matching. Case studies illustrate the selection procedure.

A Review of Domain Modelling and Domain Imaging Techniques in Ferroelectric Crystals

The present paper reviews models of domain structure in ferroelectric crystals, thin films and bulk materials. Common crystal structures in ferroelectric materials are described and the theory of compatible domain patterns is introduced. Applications to multi-rank laminates are presented. Alternative models employing phase-field and related techniques are reviewed. The paper then presents methods of observing ferroelectric domain structure, including optical, polarized light, scanning electron microscopy, X-ray and neutron diffraction, atomic force microscopy and piezo-force microscopy. Use of more than one technique for unambiguous identification of the domain structure is also described.

An evaluation of switching criteria for ferroelectrics under stress and electric field

Abstract The multi-axial responses of barium titanate (BaTiO3) and hard lead zirconate titanate (PZT-4D) are measured for stress and electric field loadings, and are compared to the response of soft lead zirconate titanate (PZT-5H) taken from a previous study. First, poled ferroelectric specimens are subjected to an electric field at an angle to the original poling direction. Second, unpoled ferroelectric specimens are loaded by a uniaxial compressive stress and a parallel, proportional electric field. The switching surfaces of BaTiO3 and PZT-4D are constructed from the experimental measurements, and compared with existing data for PZT-5H. The measured responses are then used to evaluate the accuracy of existing micromechanical and phenomenological models of ferroelectric switching.

A hexadomain vortex in tetragonal ferroelectrics

We study internal flux closures in tetragonal ferroelectric crystals using a phase-field model and demonstrate the existence of a stable polarization vortex of six domains. This hexadomain vortex exhibits a threefold rotation axis and appears as a minimum energy equilibrium state over a narrow range of aspect ratios in freestanding cuboidal nanodots. Square polarization vortices of four domains represent lower energy solutions for a wide range of aspect ratios. The hexadomain vortex offers significant potential in memory storage applications as it has eight stable states.

Scale effects and the formation of polarization vortices in tetragonal ferroelectrics

Vortices consisting of 90° quadrant domains are rarely observed in ferroelectrics. Although experiments show polarization flux closures with stripe domains, it is as yet unclear why pure single vortices are not commonly observed. Here, we model and explore the energy of polarization patterns with vortex and stripe domains, formed on the square cross-section of a barium titanate nanowire. Using phase-field simulations, we calculate the associated energy of polarization patterns as a function of nanowire width. Further, we demonstrate the effects of surface energy and electrical boundary conditions on equilibrium polarization patterns. The minimum energy equilibrium polarization pattern for each combination of surface energy and nanowire width is mapped for both open- and short-circuit boundary conditions. The results indicate a narrow range of conditions where single vortices are energetically favorable: nanowire widths less than about 30 nm, open-circuit boundary condition, and surface energy of less than...

Domain evolution processes during poling of a near-morphotropic Pb(Zr, Ti)O3 ceramic

Domain wall motion during the poling of near-morphotropic Pb(Zr,Ti)O3 PZT was observed using Piezoresponse Force Microscopy (PFM). Poling was conducted on bulk polycrystalline PZT in a series of steps, interrupted by vertical PFM scans, which were used to identify the domain evolution processes. The mechanisms of evolution in complex domain patterns such as herringbone and checkerboard structures are revealed. Of interest, in the case of a herringbone pattern consisting of two sets of lamellae angled to each other, one set of lamellae expands and is observed to overwrite the other, transforming the herringbone structure into a single lamination. Also, lengthening without broadening, and simultaneous lengthening and broadening of lamellar domain bands in checkerboard structures are observed. The observations show that 180° and non-180° domain switching can occur simultaneously in complex domain patterns. Methods are developed for identifying the polarization directions of the individual domains in near-mor...

Multiaxial response of hard and soft ferroelectrics under stress and electric field

Samples of soft PZT-5H, hard PZT-4D and Barium Titanate were subjected to multi-axial loading in stress and electric field space. The loading paths were: (1) Poling with electric field, followed by repolarizing with electric field at an angle to the original poling direction. (2) Proportional loading with electric field and coaxial compressive stress, the proportions of stress and electric field being varied between tests. In case (1) the poled material was cut to produce faces angled to the original poling direction. The measured material responses are reported and initial switching surfaces are calculated based on an offset from linear response in electric displacement. The measurements are used to assess the features required in micromechanical or phenomenological models of switching.