## Dissertation

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.