TU CE GSC CE People at the Graduate School CE Alumni Daniel Franzbach Research Project of Daniel Franzbach

Electric Field and Mechanical Stress in Ferroelectric Microstructures

Todays modern life is unimaginable without applications based on ferroelectric materials. Due to their large dielectric constant in the range between 100 and 100 000 they find their application in high performance and thin film capacitors. Additionally, the nonlinear constitutive behavior of ferroelectric materials allows the realization of tunable electric components even at high frequencies (10 GHz). This is, for example, used in modern multi-band cellar phone antennas. The existence of a spontaneous dipole moment, which can be switched between several stable remanent states, makes these materials also suitable for electric data storage in a novel non-volatile version of random accesss memory (FeRAM - ferroelectric RAM). In addition to purely electrical applications, ferroelectric materials belong to a larger class of electroactive materials, called piezoelectrics. In contrast to most of the other piezoelectric materials ferroelectrics are not required to be mono-crystalline to show piezoelectric behavior. This characteristic allows a cheaper and more flexible production of polycrystalline piezoelectric components. Piezoelectric actuators based on polycrystalline ferroelectric ceramics are today dominating the application field of ultrasound sources/detectors, loudspeakers for active sound damping, pressure sensors, common rail fuel-injection systems and as positioners in scanning tunneling or atomic force microscopes.

Like most materials used for active applications ferroelectrics are affected by aging and fatigue effects, i.e., some of their material properties decrease as a function of time (aging) or after performing a certain number of working cycles (fatigue). Due to the complex microstructure (polycrystallinity, inhomogeneities, mobile charges) and the internal dependencies between the dielectric and the mechanic system of a ferroelectric material, it is difficult to find a model that explains aging and fatigue effects. Therefore a number of various explanations exist, but few numerical models have been presented that can help understand aging and fatigue behavior of ferroelectric materials.

Ferroelectric systems are very interesting for new modeling approaches. The interactions in the system of electric dipoles are similar to those between magnetic dipoles in ferromagnetic systems. But the fact that the polarization system is strongly coupled to the system mechanics prohibits a simple carry-over from ferromagnetic theories. Instead it is necessary to include the mechanics into the theoretical model. Until now one of the most promising approaches published in the literature to describe ferroelectric systems are ab initio calculations, phenomenological based micro-mechanical models, and Ginzburg-Landau (GL) based phase field models. Ab initio calculations can determine the behavior of single atoms in a solid and therefore the basic material constants and study the dynamic behavior of structural defects. But these models are limited to a very small structure size of a few elementary cells. In contrast micro-mechanical models are able to simulate the macroscopic behavior of a polycrystalline ferroelectric structure, but can only indirectly include microscopic effects such as domain walls and intergranular stresses. Thus the effects of defects and different sorts of grain boundaries can not be studied.

GL models work basically on a mesoscopic scale and are able to close the gap between the other two models. They are based on the thermodynamic approximation that all the essential physical degrees of freedom can be described by continuous fields; the physical processes on the atomic scale are threatedstatistically. The system dynamic on the atomic length scale therefore remains unobtainable, but the necessary material parameters can be used from ab initio calculations. Because of the complexity of the coupled partial differential equations and the fact that the resolution of the FEM mesh has to be in the vicinity of domain wall size, GLT is limited to structures below the size of a single grain. However the predicted switching criteria can be used to adjust the parameters of micro-mechanical models.

Recent studies at TU Darmstadt have shown that GL theory can capture field induced phase transition behavior, in addition to domain dynamics. Such electric field and stress induced phase transitions have been observed in single crystal ferroelectrics, such as Pb(Mg_{1/3}Nb_{2/3})O3-PbTiO_{3} and Pb(Zn_{1/3}Nb_{2/3})O_{3}-PbTiO_{3}, which also display exceptional coupling coefficients and large electrically induced strains with reduced hysteresis for compositions close to the morphotropic phase boundary between the ferroelectric tetragonal (T) and rhombohedral (R) phases. To explain these experimental observations a field induced RT phase transformation via polarization rotation has been proposed by Park and Shrout. Additional experimental and structural investigations support this explanation.

Using a time-dependent phase field model the macroscopic field-induced tetragonal-orthorhombic phase transformations were simulated for perovskite materials. Semi-discontinuous hysteretic phase transformations were observed, providing a possible explanation for some of the variability in single crystal experimental measurements. Through adjustment of the energy landscape (defined by Landau parameters) it is possible to determine the significance of crystallographic misalignment or changes in 90° domain switching energy on field induced phase transformations.

Multi-Physics – Special Coupled Systems: Continuum Mechanic, Elektrostatic and Polarization Phase Field Dynamic

Daniel Franzbach

Dipl.-Phys.

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