Interaction of point defects in ferroelectrics -parameter identification and simulation

Ferroelectric materials are used in a wide field of applications, where they are exposed to a high number of mechanical and electrical load cycles. This involves degradation of the material and a decrease of the electromechanical coupling capability, which is usually called electric fatigue. The causes are assumed to be ionic and electronic charge carriers, which interact with each other, with microstructural elements in the bulk and with interfaces. Accumulation of defects can lead to degradation, mechanical damage and dissociation reactions, for more details see e.g. [3]. In order to get a better understanding of the defect accumulation processes, a model based on material forces is used in [6] to simulate the interaction of defects in periodic and in infinite cells. Applying thermodynamically reasonable kinetic laws, defect migration is simulated in a deterministic way in order to understand the general tendency of defect formations. The transversally isotropic material is modelled with linear electromechanical coupling. Here, the defect parameters used in the continuum model are obtained by fitting the results of molecular dynamics (MD) simulations to the continuous spatial fields. Transferring data from the atomic to the continuum level is a field of active research and no unique solution can be presented. On the atomic level, Coulomb–interaction causes a displacement field incompatible to an elastic solution. To address this difficulty, the volume change of a domain around the defect is used to determine defect parameters. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

3D Simulation of Point Defect Migration in Ferroelectrics

Electric fatigue in functional materials involves a set of phenomena which lead to the degradation of materials with an increasing number of electrical cycles. Ionic and electronic charge carriers, later 0 as point defects, interact with each other and with microstructural elements in the bulk and with interfaces, which can lead to degradation, or finally to mechanical damage and dissociation reactions, see e.g. [1]. With this in mind, efforts are made to calculate the fields caused by point defects to simulate their interaction as well as to verify the used material parameters. Here, a material with linear electro mechanical coupling is used. The applied methods are integral transforms (Radon Transform) and a combination of Difference Methods and a Fast Fourier Transform to obtain solutions in an infinite domain and under periodic boundary conditions, respectively. The point defect interaction is studied within the framework of material or configurational forces. These forces are used in combination with reasonable kinetic laws to simulate defect migration, cf. [2]. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Laminate-based modelling of microstructure and switching in ferroelectrics

The purpose of the work is the thermodynamics-based modelling of the polarisation and the deformation microstructure in the ferroelectric single crystal with the help of a laminate-based approach. The incremental variational-based rate-dependent macroscopic model for dissipative ferroelectric material [1] and the laminate-based microscopic model [2] established in the literature are taken as basis and shall be further extended to a single crystal laminate structure dependent on the loading frequency based on the coupled electromechanical framework taking the effect of polarisation into account. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Phase field simulations on the interaction of ferroelectric domains with point defects

One of the suspected micro-mechanical mechanisms causing electric fatigue in ferroelectric materials is the hindering and blocking of domain wall movement. These blocking or pinning phenomena are thought to be due to point defects which interact with domain walls and applied external loads. A phase field model employing the spontaneous polarization as an order parameter is used to simulate the inhomogeneous material behavior. The coupled field equations are solved using the Finite Element Method. The influence of a stationary point defect on a domain wall is shown in a numerical simulation. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A laminate-based modelling approach for rate-dependent switching in ferroelectric materials

This contribution focuses on the sequential laminate-based modelling approach for the numerical simulation of the complex electromechanical material behaviour of ferroelectric single crystals. The construction of engineered domain configurations by using the method of sequential lamination in order to study the domain evolution and polarisation switching in ferroelectric single crystals has recently been carried out in the works of [1–4]. By fulfilling the kinematic and polarisation compatibility conditions between the domain structures in a crystal, the proposed laminate-based formulation is governed by an energy-enthalpy function and by a dissipation potential. The mixed energy-enthalpy, written in terms of the total strains, electric field and a set of internal variables, here the multi-rank laminate volume fractions, governs the dissipative electromechanical response of the ferroelectric crystal, whereas the rate-dependent dissipation potential formulated in terms of the flux of the internal variables describes the time-dependent evolution of the multi-rank laminate volume fractions, subjected to inequality constraints. The model reproduces experimentally observed hysteresis and butterfly curves, characteristic for single crystal ferroelectric materials, when subjected to homogeneous electromechanical loading conditions. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A constitutive and damage model for high cycle fatigue of tetragonal ferroelectrics

Reliability and life time of smart materials are crucial features for the development and design of actuator and sensor devices. Being widely used and exhibiting brittle failure characteristics, ceramic ferroelectrics are of particular interest in this field. Due to manifold interactions of the complex nonlinear constitutive behavior on the one hand and the damage evolution in terms of microcrack growth on the other, modeling and simulation are inevitable to investigate influence parameters on strength, reliability and life time. A condensed approach is used for the simulations considering just one characteristic point in the material, nonetheless accounting for polycrystalline grain interactions. On this basis, a model to predict the life time in terms of high cycle fatigue under electromechanical loading conditions is introduced. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A condensed approach to model the constitutive behavior of multiphase ferroelectric and multiferroic materials

Ferroelectric as well as ferromagnetic materials are widely used in smart structures and devices as actuators, sensors etc. Most of the developed models, describing the nonlinear behavior, are implemented within the framework of the Finite Element Method. Most investigations, however, are restricted to simple boundary value problems under uniaxial or biaxial loading and their goal is the calculation of hysteresis loops or to determine e.g. electromechanical coupling coefficients. Regarding these circumstances, the so-called condensed method (CM) is introduced to investigate the macroscopic polycrystalline ferroelectric material behavior at a macroscopic material point without any kind of discretization scheme. In the presented paper, the CM is extended towards multiphase ferroelectric material behavior. Moreover, first numerical results of a multiphase ferroelectric material at the morphotropic phase boundary are presented. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Model for Predicting the Production Yield of Integrated Circuits

In the production of integrated circuits (I.C.s), the ability to estimate the yield of working devices, or die, is economically important. This yield depends on the distribution of material and processing defects across the surface of the wafer of devices.A number of models have been proposed to explain the distribution of point defects on I.C.s. Most of these models are based on the Poisson distribution for describing the number of defects expected on a die. These models do not inherently allow for dependence between the number of defects expected on two adjacent die.This paper develops a model for generating an I.C. defect distribution that allows for dependence in the number of defects on die that are near one another. The correlation matrix and yields of observed data are found to compare favourably with theoretical results of the proposed dependent model. Simulation results are used to compare the new dependent model to a previously proposed independent model. These dependent-model simulations show defective die clustered near each other on the wafer, a property that has been observed on real wafers. Methods of parameter and yield estimation in the dependent model are discussed.     

Synchronized states in a ring of four mutually coupled self-sustained electromechanical devices

In this paper, we study the dynamics of a ring of four mutually coupled identical self-sustained electromechanical devices both in their autonomous and nonautonomous chaotic states. The transition boundaries that can occur between instability and complete synchronization states when the coupling strength varies are derived. Numerical simulations are then performed to support the accuracy of the analytical approach.     

Constitutive models of ferroelectric composites with a viscoelastic and dielectric relaxation matrix (II)

Experimental analysis of ferroelectric composites with a viscoelastic and dielectric relaxation matrix is carried out, and the electromechanical coupling behavior of the ferroelectric composites is calculated by means of the constitutive model proposed in this paper. Comparisons between the experimental results and the calculations show that the constitutive model can reflect the electromechanical coupling behavior of the ferroelectric composites. The analysis indicates that the effect of viscoelasticity and dielectric relaxation of the matrix on the electromechanical coupling behavior of ferroelectric composites cannot be neglected.     

A condensed microelectromechanical constitutive and damage model for tetragonal ferroelectrics

A condensed model for ferroelectric solids with tetragonal unit cells is presented. The approach is microelectromechanically and physically motivated, considering discrete switching processes on the level of unit cells and quasi-continuous evolution of inelastic fields on the domain wall level. To calculate multiple grain interactions an interaction tensor is introduced. Hysteresis loops are simulated for pure electric and electromechanical loading, demonstrating e.g. the influence of a compressive preload on the poling process and interaction between statistically arranged crystallits. The residual stresses and the corresponding principle stresses are used to simulate fatigue damage in ferroelectric materials. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Small Defects and Inhomogeneities in Fatigue Strength: Experiments, Models and Statistical Implications

The method explained in this paper for quantitative evaluation of fatigue limit for materials containing defects is based on the experimental evidences that inhomogeneities and micro-notches can be treated like cracks. First, the basic concept of the area parameter model is explained introducing the various data obtained by the first author's group for over last 15 years. Evidences are shown that small cracks, defects and nonmetallic inclusions having the same value of the square root of projection area, area, have the identical influence on the fatigue limit regardless of different stress concentration factors. Various applications of these concepts to various defect types and microstructural inhomogeneities are shown. Since the estimation of fatigue strength is related to the estimation of the size of maximum defects occurring in a piece, the methods for searching the defects and the quality control of materials with respect to inclusion or defect rating as well as their statistical implications are discussed.     

Random Features of the Fatigue Limit

The classical fatigue limit is often an important characteristic in fatigue design regarding metallic material. The limit is usually obtained from a staircase test in combination with some assumption about the statistical distribution of the limit. This distribution can be of a normal, log-normal or of extreme value type and no particular physical argument gives favor to any specific distribution. This leads to a certain ambiguity in the evaluation of test results which forces the designer to introduce large safety factors. In order to find a physically based statistical distribution for use in staircase tests to determine the fatigue limit we present here a random model for the fatigue limit based on the following assumptions; (i) The square root area model according to Murakami and co-workers is valid, (ii) the randomness in the fatigue limit is induced by the randomness of the maximum defect size, (iii) the random maximum defect size has an extreme value distribution of Gumbel type. This leads to the fatigue limit distribution based on Gumbel (FLG), which is recommended to replace the normal distribution in the evaluation of staircase fatigue tests in case of hard materials. It turns out that the skewness of the resulting distribution depends on the coefficient of variation; with a normal-like non-skewed distribution at the coefficient of variation of five percent.     

A micromechanically motivated model for ferroelectrics combined with polygonal finite elements

Piezoelectric materials are one of the most prominent smart materials due to their strong electromechanical coupling behaviour. Ferroelectric ceramics behave like piezoelectric materials under low electrical and mechanical loads, but exhibit pronounced nonlinear response at higher loads due to microscopic domain switching. Modern smart devices consist of complex geometries that may force the ferroelectrics employed within them to experience higher fields than they were originally designed for, so that the material responds within its nonlinear region. Hence, models predicting the nonlinear effects of ferroelectrics under complex loading cases are important from the design point of view. Within standard finite element models dealing with electromechanical problems, each grain may be subdiscretized by several finite elements. This problem can be approximated or rather overcome by a polygonal finite element method, where each grain is modelled by solely one single finite element. In this contribution, a micromechanically motivated switching model for ferroelectric ceramics, as based on volume fraction concepts, is combined with polygonal finite element approach. Related representative numerical examples allow to further study and understand the nonlinear response of this material under complex loading cases. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling and simulations of interface properties with first‐principles electronic structure computations

We report extensive first‐principles electronic structure modeling and calculations for the SiC–SiO₂ interface, a solid–solid interface formed during oxidation of silicon carbide (SiC). The interface modeling provides atomic‐scale understanding about the nature of the interface defects as well as passivation effects due to the modification of the interface bonding. In particular, simulation results show that incorporation of hydrogen and fluorine decreases the defect density, thus enhancing the performance of SiC‐based electronic devices. Copyright © 2013 John Wiley & Sons, Ltd.     

Simulation of the diffusion of point defects in structures with local elastic stresses

The stress-mediated diffusion of nonequilibrium point defects into the bulk of a semiconductor is investigated by computer simulation. It is assumed that the point defects are generated on the surface of a semiconductor and that in the course of diffusion they pass through the local region of elastic stresses because the average length of defect migration is greater than the thickness and depth of the strained layer. Within the strained layer, point defect segregation or heavy defect depletion occurs if defect drift under stresses is directed in or out of the layer, respectively. The calculations also show that, in contrast to the case of local defect sink, the local region of elastic stresses practically does not change the distribution of defects beyond this region if there is no generation/absorption of point defects within the strained layer.     

The interaction of a system of weakened zones at the interface of elastic media in a tensile stress field

The interaction of a system of crack-like defects with distributed cohesive forces over the whole surface of the edges, located at the interface of two elastic half-planes and which open under the action of forces at infinity, is considered. A dislocation approach is used to describe the model of each defect: the discontinuity in the asymmetric shifts is specified in the form of a basis function with free parameters that satisfies a number of physical constraints. The free parameters of the model are determined when finding an analytical solution of the problem. The key questions are: what is the minimum load at which just one of these weakened zones is converted into the nucleus of a crack or when one of the connecting bridges separating these zones is fractured and, also, under what conditions can the interaction of the defects be neglected ? The model is extended with a relation which enables an explicit opening - bonding force dependence to be obtained.     

The vibrational behaviour of bladed disks with multiple coupling devices

In turbomachinery applications turbine blades are subjected to high static and dynamic loads. Static loads are due to centrifugal stresses and thermal strains. Especially the dynamic excitation caused by fluctuating gas forces results in high vibration amplitudes which can lead to high cycle fatigue failures (HCF). Therefore, in practical applications, coupling devices like underplatform dampers, lacing wires and tip shrouds are installed to the structure. In case of blade vibrations the relative displacements between these coupling devices and the blades generate friction forces. The resulting energy dissipation provides additional damping to the structure. Furthermore, coupling devices, in particular tip shrouds, snubbers and lacing wires, increase the stiffness of the structure. Hence, they lead to a shift of the resonance frequencies. So far, only effects of single coupling devices and the influencing properties have been examined. Within this paper the effect of multiple couplings is determined and compared with single couplings. The forced response of turbine bladings with multiple couplings is calculated under consideration of geometrical and mechanical parameters of the blading and contacts, respectively. The results are compared with the single coupled blading. Furthermore, a multiple coupled device with under-platform damper and connecting pin is compared with respect to his effectiveness. Especially the influence on the resonance frequency and the achievable damping is analysed. The results of the simulation are verified by measurements at a two-blade non-rotating test rig with an underplatform damper and connecting pin. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Dynamics of a self-sustained electromechanical system with flexible arm and cubic coupling

This paper studies the dynamics of a self-sustained electromechanical system with nonlinear coupling. The mechanical part is a flexible beam. The Krylov–Bogoliubov averaging method is used to derive oscillatory solutions. Focus is made on the effects of the detuning parameter and nonlinear coupling. The largest Lyapunov exponent is used to determine chaotic domains of the system.