The polarization process of ferroelectric materials in the framework of variational inequalities

An energy-based model of the ferroelectric polarization process is presented in the current contribution. In an energy-based setting, dielectric displacement and strain (or displacement) are the primary independent unknowns. As an internal variable, the remanent polarization vector is chosen. The model is then governed by two constitutive functions: the free energy function and the dissipation function. Choices for both functions are given. As the dissipation function for rate-independent response is non-differentiable, it is proposed to regularize the problem. Then, a variational equation can be posed, which is subsequently discretized using conforming finite elements for each quantity. We point out which kind of continuity is needed for each field (displacement, dielectric displacement and remanent polarization) is necessary to obtain a conforming method, and provide corresponding finite elements. The elements are chosen such that Gauss’ law of zero charges is satisfied exactly. The discretized variational equations are solved for all unknowns at once in a single Newton iteration. We present numerical examples gained in the open source software package Netgen/NGSolve.

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Variational inequalities for ferroelectric constitutive modeling

Journal of Intelligent Material Systems and Structures ◽

10.1177/1045389x20951252 ◽

2020 ◽

pp. 1045389X2095125

Author(s):

Martin Meindlhumer ◽

Astrid Pechstein ◽

Alexander Humer

Keyword(s):

Free Energy ◽

Variational Inequality ◽

Variational Inequalities ◽

Lagrange Multiplier ◽

Constitutive Modeling ◽

Domain Switching ◽

Polarization Process ◽

Consistent Model ◽

Nonlinear Iteration ◽

Variational Formulations

This paper is concerned with modeling the polarization process in ferroelectric media. We develop a thermodynamically consistent model, based on phenomenological descriptions of free energy as well as switching and saturation conditions in form of inequalities. Thermodynamically consistent models naturally lead to variational formulations. We propose to use the concept of variational inequalities. We aim at combining the different phenomenological conditions into one variational inequality. In our formulation we use one Lagrange multiplier for each condition (the onset of domain switching and saturation), each satisfying Karush-Kuhn-Tucker conditions. An update for reversible and remanent quantities is then computed within one, in general nonlinear, iteration.

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Electron Microscope study of commensurate-incommensurate phase transition in BaZnGeO4

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100173996 ◽

1990 ◽

Vol 48 (4) ◽

pp. 172-173

Author(s):

Naoki Yamamoto ◽

Makoto Kikuchi ◽

Tooru Atake ◽

Akihiro Hamano ◽

Yasutoshi Saito

Keyword(s):

Phase Transition ◽

Electron Microscope Study ◽

Room Temperature ◽

Hexagonal Lattice ◽

Ferroelectric Materials ◽

Temperature Phase ◽

X Ray Diffraction ◽

X Ray ◽

Periodic Arrays ◽

Modulation Wave Vector

BaZnGeO4 undergoes many phase transitions from I to V phase. The highest temperature phase I has a BaAl2O4 type structure with a hexagonal lattice. Recent X-ray diffraction study showed that the incommensurate (IC) lattice modulation appears along the c axis in the III and IV phases with a period of about 4c, and a commensurate (C) phase with a modulated period of 4c exists between the III and IV phases in the narrow temperature region (—58°C to —47°C on cooling), called the III' phase. The modulations in the IC phases are considered displacive type, but the detailed structures have not been studied. It is also not clear whether the modulation changes into periodic arrays of discommensurations (DC’s) near the III-III' and IV-V phase transition temperature as found in the ferroelectric materials such as Rb2ZnCl4.At room temperature (III phase) satellite reflections were seen around the fundamental reflections in a diffraction pattern (Fig.1) and they aligned along a certain direction deviated from the c* direction, which indicates that the modulation wave vector q tilts from the c* axis. The tilt angle is about 2 degree at room temperature and depends on temperature.

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Investigation of ferroelectrics using conventional and in situ electron microscopy

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100170669 ◽

1994 ◽

Vol 52 ◽

pp. 586-587

Author(s):

V. Saikumar ◽

H. M. Chan ◽

M. P. Harmer

Keyword(s):

Nonvolatile Memory ◽

Ferroelectric Materials ◽

Gate Insulator ◽

Sensors And Actuators ◽

In Situ Electron Microscopy ◽

Ferroelectric Domains ◽

Domain Boundaries ◽

Domain Interactions ◽

Memory Applications

In recent years, there has been a growing interest in the application of ferroelectric thin films for nonvolatile memory applications and as a gate insulator in DRAM structures. In addition, bulk ferroelectric materials are also widely used as components in electronic circuits and find numerous applications in sensors and actuators. To a large extent, the performance of ferroelectric materials are governed by the ferroelectric domains (with dimensions in the micron to sub-micron range) and the switching of domains in the presence of an applied field. Conventional TEM studies of ferroelectric domains structures, in conjunction with in-situ studies of the domain interactions can aid in explaining the behavior of ferroelectric materials, while providing some answers to the mechanisms and processes that influence the performance of ferroelectric materials. A few examples from bulk and thin film ferroelectric materials studied using the TEM are discussed below.Figure 1 shows micrographs of ferroelectric domains obtained from undoped and Fe-doped BaTiO3 single crystals. The domain boundaries have been identified as 90° domains with the boundaries parallel to <011>.

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The need of electron microscopy in the study of domains and domain walls in ferroelectrics

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100170487 ◽

1994 ◽

Vol 52 ◽

pp. 550-551

Author(s):

Wenwu Cao

Keyword(s):

Domain Wall ◽

Domain Walls ◽

High Mobility ◽

Continuum Theory ◽

Polarization Vector ◽

Ferroelectric Materials ◽

Dielectric Constants ◽

Nonlocal Coupling ◽

Cubic Symmetry ◽

Gradient Energy

Domain structures play a key role in determining the physical properties of ferroelectric materials. The formation of these ferroelectric domains and domain walls are determined by the intrinsic nonlinearity and the nonlocal coupling of the polarization. Analogous to soliton excitations, domain walls can have high mobility when the domain wall energy is high. The domain wall can be describes by a continuum theory owning to the long range nature of the dipole-dipole interactions in ferroelectrics. The simplest form for the Landau energy is the so called ϕ model which can be used to describe a second order phase transition from a cubic prototype,where Pi (i =1, 2, 3) are the components of polarization vector, α's are the linear and nonlinear dielectric constants. In order to take into account the nonlocal coupling, a gradient energy should be included, for cubic symmetry the gradient energy is given by,

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