## Doctoral Thesis

Point defects in piezoelectric materials – continuum mechanical modelling and numerical simulation
(2010)

The topic of this work is the continuum mechanic modelling of point defects in piezoelectric materials. Devices containing piezoelectric material and especially ferroelectrics require a high precision and are exposed to a high number of electrical and mechanical load cycles. As a result, the relevant material properties may decrease with increasing load cycles. This phenomenon is called electric fatigue. The transported ionic and electric charge carriers can interact with each other, as well as with structural elements (grain boundaries, inhomogeneities) or with material interfaces (domain walls). A reduced domain wall mobility also reduces the electromechanical coupling effect, which leads to the electric fatigue effect. The materials considered here are barium titanate and lead zirconate titanate (PZT), in which oxygen vacancies is the most mobile and most frequently appearing defect species. Intentionally introduced foreign atoms (dopants) can adjust the material properties according to their field of application by generating electric dipoles with the vacancies. Agglomerations of point defects can strongly influence the domain wall motion. The domain wall can be slowed down or even be stopped by the locally varying fields in the vicinity of the clusters. Accumulations of point defects can be detected at electrodes, pores or in the bulk of fatigued samples. The present thesis concentrates focuses on the self interaction behaviour of point defects in the bulk. A micro mechanical continuum model is used to show the qualitative and the quantitative interaction behaviour of defects in a static setup and during drift processes. The modelling neglects the ferroelectric switching mechanisms, but is applicable to every piezoelectric material. The underlying differential equations are solved by means of analytical (Green's functions) and numerical (Finite Differences with discrete Fourier Transform) methods, depending on the boundary conditions. The defects are introduced as localised Eigenstrains, as electric charges and as electric dipoles. The required defect parameters are obtained by comparisons with atomistic methods (lattice statics). There are no standardised procedures available for the parameter identification. In this thesis, the mechanical parameter is obtained by a comparison of relaxation volumes of the atomic lattice and the continuum solution. Parameters for isotropic and anisotropic defect descriptions are identified. The strength of the electric defect is obtained by a comparison of the electric internal energies of atomistics and continuum. The appearing singularities are eliminated by taking only the energy difference of a infinite crystal and a periodic cell into account. Both identification processes are carried out for the cubic structure of barium titanate, which decouples the mechanical and the electrical problem. The defect interaction is analysed by means of configurational forces. The mechanical defect parameter generates a directional short-range attraction between defects. An electrical defect parameter produces the long-range Coulomb interaction, which predicts a repulsion of two similar charges. Additionally, an interaction with defect dipoles is taken into account. It is shown that a defect agglomeration is possible for any static defect configuration. Finally, defect drift is simulated using a thermodynamically motivated migration law based on configurational forces. In this context, the migration of point defects due to self interaction, and the influence of external fields is investigated.

Nowadays piezoelectric and ferroelectric materials are becoming more and more an interesting part of smart materials in scientific and engineering applications. Precision machining in manufacturing, micropositioning in metrology, common rail systems with piezo fuel injection control in automobile industry, and ferroelectric random access memories (FRAM) in microelectromechanical systems (MEMS) besides commercial piezo actuators and sensors can be very good examples for the application of piezoceramic and ferroelectric materials. In spite of having good characteristics, piezoelectric and ferroelectric materials have significant nonlinearities, which limit the applications in high performance usage. Domain switching (ferroelastic or ferroelectric) is the main reason for the nonlinearity of ferroelectric materials. External excessive electromechanical loads (mechanical stress and electric field) are driving forces for domain switching. In literature, various important experiments related to the non-linear properties of piezoelectric and ferroelectric materials are reported. Simulations of nonlinear properties of piezoelectric and ferroelectric materials based on physical insights of the material have been performed during the last two decades by using micromechanical and phenomenological models. The most significant experiments and models are deeply discussed in the literature survey. In this thesis the nonlinear behaviour of tetragonal perovskite type piezoceramic materials is simulated theoretically using two and three dimensional micromechanical models which are based on physical insights of the material. In the simulations a bulk piezoceramic material which has numerous grains is considered. Each grain has random orientation in properties of polarization and strain. Randomness of orientations is given by Euler angles equally distributed between \(0\) and \(2\pi\). Each element in the micromechanical model has been assumed to have the same properties of the real piezoelectric grain. In the first part of the simulations, quasi-static characteristics of piezoelectric materials are investigated by applying cyclic, rate independent, bipolar, uni-axial and external electrical loading with an amplitude of 2 kV/mm gradually starting from zero value in virgin state. Moreover, the simulations are undertaken for these materials which are subjected to quasi-static, uni-polar, uni-axial mechanical stress, namely compressive stress. The calculations are performed at each element based on linear constitutive equations, nonlinear domain switching and a probability theory for domain switching. In order to fit the simulations to the experimental data, some parameters such as spontaneous polarization, spontaneous strain, piezoelectric and dielectric constants are chosen from literature. The domain switching of each grain is determined by an electromechanical energy criterion. Depending on the actual energy related to a critical energy a certain probability is introduced for domain switching of the polarization direction. Same energy levels are assumed in the electromechanical energy relation for different types of domain switching like 90º and 180º for perovskite type tetragonal or 70.5º and 109.5º for rhombohedral microstructures. It is assumed that intergranular effects between grains can be modelled by such probability functions phenomenologically. The macroscopic response of the material to the applied electromechanical loading is calculated by using Euler transformations and averaging the individual grains. Properties of piezoelectric materials under fixed mechanical stresses are also investigated by applying constant compressive stress in addition to cyclic electrical loading in the simulations. Compressive stress is applied and kept constant before cyclic bipolar electrical loading is implemented. In the following chapters, a three-dimensional micromechanical model is extended for the simulation of the rate dependent properties of certain perovskite type tetragonal piezoelectric materials. The frequency dependent micromechanical model is now not only based on linear constitutive and nonlinear domain switching but also linear kinetics theories. The material is loaded both electrically and mechanically in separate manner with an alternating electrical voltage and mechanical stress values of various moderate frequencies, which are in the order of 0.01 Hz to 1 Hz. Electromechanical energy equation in combination with a probability function is again used to determine the onset of the domain switching inside the grains. The propagation of the domain wall during the domain switching process in grains is modelled by means of linear kinetics relations after a new domain nucleates. Electric displacement versus electric field hysteresis loops, mechanical strain versus mechanical stress and electric displacement versus mechanical stress for different frequencies and amplitudes of the alternating electric fields and compressive stresses are simulated and presented. A simple micromechanical model without using probabilistic approach is compared with the one that takes it into account. Both models give important insights into the rate dependency of piezoelectric materials, which was observed in some experiments reported in the literature. Intergranular effects are other significant factors for nonlinearities of polycrystalline ferroelectric materials. Even piezoelectric actuators and sensors show nonlinearities when they are operated with electrical loading, which is much lower than the coercive electric field level. Intergranular effects are the main cause of such small hysteresis loops. In the corresponding chapter, two basic field effects which are electrical and mechanical are taken into account for the consideration of intergranular effects micromechanically in the simulations of the two dimensional model. Therefore, a new electromechanical energy equation for the threshold of domain switching is introduced to explain nonlinearities stemming from both domain switching and intergranular effects. The material parameters like coercive electric field and critical spontaneous polarization or strain quantities are not implemented in the electromechanical energy relation. But, this relation contains new parameters which consider both mechanical and electrical field characteristics of neighbouring elements. By using this new model, mechanical strain versus electric field butterfly curves under small electrical loading conditions are also simulated. Hence, a rate dependent concept is applied in butterfly curves by means of linear kinetics model. As a result, the simulations have better matching with corresponding experiments in literature. In the next step, the model can be extended in three dimensional case and the parameters of electromechanical energy relation can be improved in order to get better simulations of nonlinear properties of polycrystalline piezoelectric materials.