Nonlinear constitutive behavior of soft and hard PZT: experiments and modeling

This work characterizes and models the nonlinear inelastic behavior of lead zirconate–titanate (PZT) ceramics. A detailed investigation of the hysteresis loop of the stress–strain curve shows that the process of ferroelastic non-180° polarization switching and of switching saturation are, respectively, a smooth softening process and a gradual hardening process. The inflection point in the curve is found to be a boundary point that distinguishes the transition from a softening process to a hardening one. Based on underlying physical mechanisms, this nonlinear behavior is simulated by a mechanical model, consisting of several parallel-arranged Maxwell chains with each chain taking a main role in a different, successive stage of the switching process. Each chain includes a nonlinear spring and a frictional slider. The generalized displacement of the slider, indicating an internal variable, is introduced to describe the local process of the cooperative switching of the corresponding grain group. The hardening and softening processes are controlled by variations of stiffness of the nonlinear springs. Furthermore, the observed similarities between the hysteresis loop of the stress–strain curve and that of stress–electric displacement curve are used to develop an equivalent stress concept and to formulate constitutive laws for PZT ceramics. Comparisons between the experimental hysteresis loops and those calculated by the proposed model show satisfactory agreement for both hard PZT and C5800 and soft C5500.     

Nonlinear electric–mechanical behavior and micromechanics modelling of ferroelectric domain evolution

Domains exist in ferroelectric ceramics. External loads, such as electric field and stress, can cause domain switching. Domain switching always results in nonlinear ferroelectricity and ferroelasticity of ferroelectric ceramics. In this investigation, nonlinear electric–mechanical behavior related to ferroelectric and ferroelastic domain switching is experimentally and theoretically studied. In the experimental work, the electric–mechanical response of a soft PZT ferroelectric ceramic subjected to combined electric–mechanical loads was observed. The effect of different compressive stress levels on the electromechanical response was examined. In the theoretical modelling, the orientation of each domain is defined by its local coordinate relative to a fixed global coordinate. Orientation distribution function (ODF) is used to describe the domain pattern. For mathematical simplicity, the Reuss average is used in the modelling. According to the proposed theory, a domain has different Gibbs' energy at different orientation states and the energy difference forms the domain switching driving force. The domain pattern and its evolution are determined by the joint action of the domain switching driving force and the dissipation during domain switching. In ferroelectricity and ferroelasticity, 90° and 180° domain switchings play different roles and have different switching dissipations associated with them. A criterion considering the difference between the 90° switching and the 180° switching is established by the thermodynamic approach. There is an agreement between theoretical and experimental results. It should be pointed out that the micromechanical model proposed in this paper is restricted to ferroelectric materials exhibiting transformation from cubic to tetragonal only.     

Role of grain orientation distribution in the ferroelectric and ferroelastic domain switching of ferroelectric polycrystals

The non-linear electromechanical behavior of ferroelectric polycrystals stems from polarization/domain switching, which are affected by the grain boundaries and grain orientations. The effects of grain orientation distribution on the domain switching and non-linear behavior of a two-dimensional ferroelectric polycrystal subjected to an electric or/and mechanical load are investigated by computer simulations with a real-space phase-field model. Phase-field simulations indicate that the macroscopic coercive field, remanent polarization and remanent strain in the polycrystal with a random distribution of grain orientation are correspondingly smaller than those in the polycrystal with a uniform distribution of grain orientation. However, the polycrystal with randomly distributed grain has a larger strain variation with the electric field than the polycrystal with uniformly distributed grains, which suggests that the random polycrystal has a better piezoelectric property than the uniform one. The different macroscopic non-linear behaviors of the ferroelectric polycrystals are attributed to different microscopic domain switching processes. For the polycrystal with randomly distributed grains, the domain switching takes place from the regions near the large angle grain boundaries, while new domains nucleate from the cross sections between the grain boundaries and the material surface in the polycrystal with uniform grain orientation.     

Estimation of strain from piezoelectric effect and domain switching in morphotropic PZT by combined analysis of macroscopic strain measurements and synchrotron X-ray data

Morphotropic PZT ceramics are the state of the art materials for ferroelectric actuators. Essential performance parameters for these materials are strain and hysteresis. On a microscopic scale the strain provided by an electric field is due to two different mechanisms. The piezoelectric effect causes an elongation of the unit cells, whereas domain switching changes their crystallographic orientation by aligning the polarization axis towards the field direction. A method is outlined to estimate the contribution of the two mechanisms to total strain by combining macroscopic strain measurements and X-ray diffraction (XRD) data. Results from macroscopic measurements of remanent and unipolar strain with the corresponding data on texture, derived from in situ synchrotron radiation XRD patterns, are analyzed and evaluated by a semi-empirical approach. The method was applied to six morphotropic, LaSr doped PZT materials of different Zr/Ti ratios. Results are discussed with respect to the differences between the materials.     

Temperature-dependent ferroelastic switching of soft lead zirconate titanate

The longitudinal strain and polarization of a soft polycrystalline ferroelectric ceramic (lead zirconate titanate, PZT) were measured under uniaxial compressive stress at elevated temperatures utilizing a novel testing fixture. Ferroelectric ceramics have not previously been characterized under these conditions due to experimental complexity. In addition to nonlinear macroscopic constitutive behavior, the linear elastic moduli have been measured throughout the loading cycle, allowing for the determination of the relative contributions from the linear and nonlinear ferroelastic behavior as a function of stress and temperature. Experimental results show a strong temperature dependence of ferroelastic switching. The ferroelastic properties of unpoled and poled materials for temperatures up to the Curie point are contrasted with the spontaneous strain, elucidating the role of tetragonality in ferroelastic switching. When thermal changes are considered, marked changes in the maximum strain are observed.     

THREE DIMENSIONS PHASE FIELD SIMULATION OF MECHATRONIC COUPLE FOR FERROELECTRIC MATERIALS

利用三维相场理论模拟了铁电材料的自发极化、电滞回线以及力电耦合对畴变的影响. 结果表明, 外场下的畴变是通过新畴形核及畴壁移动所引起的长大实现的. 逐渐改变外场时, 在矫顽场附近各类电畴均发生90°畴变, 从而导致极化强度突变. 沿垂直电场方向加 拉、压应变能阻止或促进沿电场方向的畴变.     

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

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.     

Critical properties of symmetric nanoscale metal–ferroelectric–metal capacitors

The size, surface and interface effects on the magnitude and stability of spontaneous polarization in a symmetric nanoscale ferroelectric capacitor were studied by analyzing its evolutionary trajectory based on a thermodynamic model. Analytic expressions of the Curie temperature, spontaneous polarization, critical thickness and the Curie–Weiss relation were derived, taking into account the effects of the depolarization field, built-in electric field, interfaces and surfaces. Our results show that the critical properties are not only functions of the ambient temperature, misfit strain and electromechanical boundary conditions, but also depend on the characteristics of electrodes, surfaces and interfaces, through the incomplete charge compensation, near-surface variation of polarization and work function steps of ferroelectric–electrode interfaces, which are adjustable.     

Collective dynamics in nanostructured polycrystalline ferroelectric thin films using local time-resolved measurements and switching spectroscopy

Grain-to-grain long-range interactions and the ensuing collective dynamics in the domain behavior of nanostructured polycrystalline Pb(Zr,Ti)O₃ ferroelectric thin films have been investigated. To identify the key factors and interactions controlling local polarization dynamics we utilize a synergistic approach based on focused ion beam (FIB) milled damage-free nanostructures to isolate single grains and grain clusters, time-resolved piezoresponse force microscopy and switching spectroscopy PFM (SSPFM) (PFM) to address polarization dynamics within individual grains, and finite-element simulations to quantify the local ferroelectric interactions and hence assess the weight of several possible switching mechanisms. The experiments find that of the three possible switching mechanisms, namely direct electromechanical coupling, local built-in electric field and strain, and grain boundary electrostatic charges, the last one is the dominant mechanism. Although finite-element simulations find that direct electromechanical coupling and local built-in field-induced switching are possible, calculations confirm that for the utilized material properties, the aforementioned mechanisms are energetically unfavored.     

Phase-field simulation of polarization switching and domain evolution in ferroelectric polycrystals

A phase-field model is developed for predicting the polarization switching and domain structure evolution under an applied electric field in ferroelectric polycrystals. The model takes into account realistic grain structures as well as various energetic contributions, including elastic energy, electrostatic energy, and domain wall energy. A hysteresis loop – average polarization as a function of applied electric field – is computed, and the detailed domain evolution process during switching is analyzed. In particular, the role of grain boundaries in the nucleation and growth of new domains is studied. It is shown that switching takes place through the nucleation of 90° domains at grain boundaries and subsequent growth into the grain interiors instead of direct 180° domain switching. A correlation between the domain structures in neighboring grains was observed, and polarization switching in one grain was found to affect the switching in neighboring grains.     

Evolution of compatible laminate domain structures in ferroelectric single crystals

The electromechanical properties and behaviour of ferroelectric single crystals are dominated by their domain structures. The domain structure can evolve according to the applied electrical and mechanical boundary conditions, but typically maintains a low energy state by adopting compatible configurations of microstructure in the form of multi-rank laminates. In this work, a model of domain structure evolution is developed, using a variational method to capture the dissipative nature of domain wall motion. The model describes the evolution of domain patterns with the constraint that they remain in low energy, compatible configurations. The electromechanical behaviour, such as microstructure evolution and hysteresis response, of periodic compatible laminate domain patterns in the tetragonal crystal system is studied. Estimates of the material response based on uniform field approximations are developed, and compared with a numerical model in which finite element analysis is used for accurate computation of the free energy. Microstructural transitions from one type of laminate domain pattern to another are included in the model by considering “pivot states”, which are the limiting states shared by more than one laminate pattern. The transition between distinct microstructural patterns at a pivot state is modelled as a bifurcation in which a material element notionally evolves along multiple paths simultaneously, representing sub-regions of the element evolving in different ways. The model is applied to study the hysteresis responses, such as dielectric hysteresis loops and butterfly strain loops, and switching mechanism of barium titanate (BaTiO₃) single crystals subjected to a variety of loads. The relationship between domain patterns and the behaviour of ferroelectric switching is discussed. The results show good general agreement with experimental data in the literature, reproducing several features such as the effect of stress on electrical hysteresis.     

Influence of radial stress on the poling behaviour of lead zirconate titanate ceramics

The development of new piezoelectric materials is often hampered by high electrical conductivity or coercive fields, which make it virtually impossible to produce sufficient levels of electric polarization by application of an external electric field. As a remedy, we utilize a new poling method: stress-assisted poling, which is based on the simultaneous application of electrical field and mechanical stress. The method is demonstrated on a commercial ferroelectric (PZT PIC 151). The poling field E¹ where most domain switching occurs is reduced by about 40% if a radial compressive stress of about 90 MPa is applied to the sample. The piezoelectric coefficient d₃₃ is determined for different load magnitudes. Variations of both remnant and maximum polarization are determined with respect to the magnitude of compressive radial stress.     

R-curve behavior of BaTiO3- and PZT ceramics under the influence of an electric field applied parallel to the crack front

The fracture resistance curves (R-curves) of BaTiO₃ and commercial PZT–PIC 151 were measured with compact tension specimens under the influence of an electric field applied parallel to the crack front. A strong influence of the electric field on the starting and plateau value was found as well as on the length of the R-curve. Generally a toughness increase was detected with increasing electric field. The toughening effect is estimated from the change in crack tip stress intensity induced by ferroelastic domain switching near the crack tip using the weight function formalism developed for stress-induced transformation toughening of zirconia ceramics. In order to obtain a quantitative prediction of toughening, ferroelastic and ferroelectric properties were measured.     

A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part I: Model formulation and application to tetragonal materials

A micromechanical model is proposed to study the constrained domain-switching process in polycrystalline ferroelectric ceramics. It is assumed that the depolarization field induced by domain switching is completely compensated by free charges, while the stress caused by non-180° switching is considered in an Eshelby inclusion manner. The model assumes that each grain contains multi-domains and the domain-switching criterion is based on potential energy density. Two switching options, which are based on Hwang et al. [Hwang SC, Lynch CS, McMeeking RM. Acta Metall Mater 1995;43:2073] and Berlincourt and Krueger [Berlincourt D, Krueger HHA. J Appl Phys 1959;30:1804], are used in the model development. Details of the switching process are analyzed for tetragonal ferroelectric/ferroelastic ceramics under electric loading or uniaxial compression (tension) by using an inverse-pole-figure method. Numerical results show that during electric poling, only a few per cent 90° switching can occur in BaTiO₃ ceramics, which agrees well with experimental observations.     

Effect of grain orientation and grain size on ferroelectric domain switching and evolution: Phase field simulations

Phase field simulations were conducted in order to understand the effect of grain orientation, grain boundary and grain size on ferroelectric domain switching, stress distribution and evolution behavior under an applied electric field. Tetragonal ferroelectric domains were considered. Hysteresis loops were obtained for a single crystal, a bi-crystal and a polycrystal and the differences in their coercive fields were examined. It was found that the magnitude of the coercive field was closely related to the domain structures at the maximum electric field. Nucleation of new domains at a grain boundary led to local high stress. The effect of a reduced ferroelectric transition temperature at the grain boundary on the polarization distribution, domain structure and switching was studied.     

Direct observation of asymmetric domain wall motion in a ferroelectric capacitor

We report in situ transmission electron microscopy observations of the 180° polarization switching process of a PbZr_0.2Ti_0.8O₃ (PZT) capacitor. The preferential, but asymmetric, nucleation and forward growth of switched c-domains were observed at the PZT/electrode interfaces, arising due to the built-in electric field induced at each interface. The subsequent sideways growth of the switched domains was inhibited by the depolarization field due to the imperfect charge compensation at the counter-electrode and also at the boundaries with preexisting a-domains, which contributed further to the asymmetric switching behavior. It was found that the preexisting a-domains split into fine a- and c-domains constituting a 90° stripe domain pattern during the 180° polarization switching process, revealing that these domains also actively participated in the out-of-plane polarization switching. The real-time observations uncovered the origin of the switching asymmetry and further clarified the importance of charged domain walls and the interfaces with electrodes in the ferroelectric switching processes.     

垂直电场下BaTiO_3单晶的畴变与压痕裂纹的演化

利用微分干涉相衬显微镜,在垂直于极化方向的电场下,对BaTiO_3单晶的畴变过程和单晶上压痕裂纹的变化过程进行了原位观察.通过对压痕周围和远离压痕区畴变过程的对比,研究了压痕对其周围畴变的影响.结果显示,面内极化试样在垂直于极化方向的面内电场作用下,原有压痕和压痕裂纹随整个试样经历90°畴变后也要发生变化,变化后其形貌与在新的极化状态下重新压制的压痕相似;对离面极化试样加面内电场时,畴变速度先增大后减小,畴变完成一半时,速度最大;而且畴变初始阶段速度呈现大小相间的振荡性.     

Polarization switching in PbTiO3: an ab initio finite element simulation

A continuum constitutive model for PbTiO₃ has been obtained from an effective Hamiltonian which was constructed from ab initio calculations. This model contains the nonlinearities necessary for switching from the ground-state tetragonal phase to the metastable rhombohedral and orthorhombic phases. The constitutive model was incorporated into a finite element formulation in order to study the large length-scale electro-mechanical response of this piezoelectric material. We use this approach to study the hysteresis of single-crystal PbTiO₃ as a function of applied electric field and temperature and we analyze the microscopic mechanisms responsible for polarization switching. The model successfully reproduces the qualitative features of a high-strain actuator recently proposed and tested experimentally.     

Effect of space charge on the polarization hysteresis characteristics of monolithic and compositionally graded ferroelectrics

A non-linear thermodynamic analysis of ferroelectric systems with localized space charges for monolithic and compositionally graded materials is described wherein the electrostatic interlayer interactions are specifically accounted for. The electrostatic coupling is established through the built-in polarization due to the space charges and the intrinsic polarization variations between the ferroelectric layers. The findings show that the polarization hysteresis response of monolithic stress-free barium strontium titanate (BST) ferroelectrics with asymmetrically distributed space charges result in a displacement of the hysteresis loop along the applied electric field axis. In compositionally graded BST multilayers, the hysteresis response is characterized by off-sets along both the polarization and the electric field axes, yet with magnitudes of displacement that are markedly larger than those for monolithic ferroelectrics.     

A constrained domain-switching model for polycrystalline ferroelectric ceramics. Part II: Combined switching and application to rhombohedral materials

A combined switching assumption (CSA) is incorporated into the constrained domain-switching model presented in part I to address the nonsymmetric deformations of ferroelectric ceramics under tension and compression. Using the CSA, the domain-switching process in rhombohedral ferroelectric/ferroelastic ceramics is analyzed in detail. The results show that in rhombohedral lead titanate zirconate (PZT) ceramics, most 109° switching can be accomplished with a minor fraction of 71° switching during electric poling, which is quite different from BaTiO₃ and tetragonal PZT ceramics in which only a few per cent of 90° switching can occur. Domain switching under combined uniaxial electric and mechanical tension/compression is also studied. Uniaxial tension is found to enhance electric poling while uniaxial compression inhibits it.