A Non-Linear Model for Elastic Dielectric Crystals with Mobile Vacancies

Abstract A framework is developed for electromechanical behavior of dielectric crystalline solids subjected to finite deformations. The theory is formulated in the context of electrostatics; however, vacancies in the lattice may carry an electric charge, and their concentrations may be large. The material is treated as a continuous body with a continuous distribution of point vacancies, but volumes and charges of individual defects enter the description. The deformation gradient is decomposed multiplicatively into terms accounting for recoverable thermoelasticity and irreversible volume changes associated with vacancies. Thermodynamic arguments lead to constitutive relations among electromechanical quantities framed in the elastically unloaded intermediate configuration, with the Cauchy stress tensor consistently non-symmetric as a result of electrostatic effects. The requirement of non-negative dissipation imposes constraints on vacancy migration. Following postulation of a quadratic form for the free energy potential, a kinetic equation for vacancy flux is derived in the intermediate configuration, with diffusion driven by gradients of vacancy concentration, electrostatic potential, hydrostatic pressure, and crystal structure. Effects of geometric non-linearity (i.e. finite elastic strains and large vacancy concentrations) are found to affect vacancy diffusion in a body subjected to biaxial lattice strain, for example a film device with lattice mismatch at its interfaces.

[1]  Michael J. Aziz,et al.  THERMODYNAMICS OF DIFFUSION UNDER PRESSURE AND STRESS : RELATION TO POINT DEFECT MECHANISMS , 1997 .

[2]  R. McMeeking,et al.  A principle of virtual work for combined electrostatic and mechanical loading of materials , 2007 .

[3]  Yunjun Xiao.  The influence of oxygen vacancies on domain patterns in ferroelectric perovskites , 2005 .

[4]  A. W. Warner,et al.  Dislocations, Relaxations, and Anelasticity of Crystal Quartz , 1956 .

[5]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. I. The Simple Fluid , 1940 .

[6]  S. TOLANSKY,et al.  Dislocations , 1966, Nature.

[7]  G. Dienes.  A Theoretical Estimate of the Effect of Radiation on the Elastic Constants of Simple Metals , 1952 .

[8]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[9]  S. B. Krupanidhi,et al.  Microstructure related influence on the electrical properties of pulsed laser ablated (Ba, Sr)TiO3 thin films , 2000 .

[10]  Mark F. Horstemeyer,et al.  Modeling stress state dependent damage evolution in a cast Al–Si–Mg aluminum alloy , 2000 .

[11]  Byung Chan Eu,et al.  Thermodynamics of Irreversible Processes , 1995 .

[12]  K. Johnson.  Variation of Dielectric Constant with Voltage in Ferroelectrics and Its Application to Parametric Devices , 1962 .

[13]  Dragan Damjanovic,et al.  FERROELECTRIC, DIELECTRIC AND PIEZOELECTRIC PROPERTIES OF FERROELECTRIC THIN FILMS AND CERAMICS , 1998 .

[14]  J. Koehler,et al.  Space Charge in Ionic Crystals. I. General Approach with Application to NaCl , 1965 .

[15]  A. F. Devonshire.  Theory of ferroelectrics , 1954 .

[16]  M. Muir.  Physical Chemistry , 1888, Nature.

[17]  Carl Eckart,et al.  The Thermodynamics of Irreversible Processes. IV. The Theory of Elasticity and Anelasticity , 1948 .

[18]  J. D. Eshelby.  The Continuum Theory of Lattice Defects , 1956 .

[19]  Elias C. Aifantis,et al.  A damage model for ductile metals , 1989 .

[20]  By E. Pan.  Mindlin ’ s problem for an anisotropic piezoelectric half-space with general boundary conditions , 2001 .

[21]  M. Cole,et al.  GRAIN GROWTH AND RESIDUAL STRESS IN BST THIN FILMS , 2005 .

[22]  J. Weertman,et al.  Theory of Steady‐State Creep Based on Dislocation Climb , 1955 .

[23]  James S. Speck,et al.  DOMAIN CONFIGURATIONS DUE TO MULTIPLE MISFIT RELAXATION MECHANISMS IN EPITAXIAL FERROELECTRIC THIN FILMS. I: THEORY , 1994 .

[24]  Paul Steinmann,et al.  Nonlinear electro-and magneto-elastostatics : Material and spatial settings , 2007 .

[25]  Conyers Herring,et al.  Diffusional Viscosity of a Polycrystalline Solid , 1950 .

[26]  P. Chung,et al.  Kinematics, electromechanics, and kinetics of dielectric and piezoelectric crystals with lattice defects , 2008 .

[27]  D. McDowell,et al.  A geometric framework for the kinematics of crystals with defects , 2005 .

[28]  R. Chang.  Creep of Al2O3 Single Crystals , 1960 .

[29]  E. Kröner.  The differential geometry of elementary point and line defects in Bravais crystals , 1990 .

[30]  C. Hwang,et al.  Depletion layer thickness and Schottky type carrier injection at the interface between Pt electrodes and (Ba, Sr)TiO3 thin films , 1999 .

[31]  G. Maugin.  On the covariant equations of the relativistic electrodynamics of continua. III. Elastic solids , 1978 .

[32]  M. Mehl,et al.  Tight-binding calculations of the band structure and total energies of the various polytypes of silicon carbide , 2003 .

[33]  M. J. Grinfel'd,et al.  Thermodynamic methods in the theory of heterogeneous systems , 1991 .

[34]  R. Ogden,et al.  Nonlinear electroelasticity , 2005 .

[35]  M. Ervin,et al.  Low dielectric loss and enhanced tunability of Ba0.6Sr0.4TiO3 based thin films via material compositional design and optimized film processing methods , 2003 .

[36]  Kaushik Bhattacharya,et al.  A computational model of ferroelectric domains. Part I: model formulation and domain switching , 2005 .

[37]  D. McDowell,et al.  Army Research Laboratory Aberdeen Proving Ground , MD 21005-5066 ARL-RP-118 February 2006 Anholonomic Configuration Spaces and Metric Tensors in Finite Elastoplasticity , 2003 .

[38]  D. Lazarus,et al.  Calculating Activation Volumes and Activation Energies from Diffusion Measurements , 1970 .

[39]  P. Chung,et al.  Continuum modeling of charged vacancy migration in elastic dielectric solids, with application to perovskite thin films , 2008 .

[40]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[41]  R. Birss,et al.  The elastic dielectric. , 1967, Proceedings of the National Academy of Sciences of the United States of America.

[42]  J. Brice.  Crystals for quartz resonators , 1985 .

[43]  R. Raj,et al.  Space-charge-controlled diffusional creep: Volume diffusion case , 1996 .

[44]  M. L Kronberg,et al.  Plastic deformation of single crystals of sapphire: Basal slip and twinning , 1957 .

[45]  J. Bogdanoff,et al.  On the Theory of Dislocations , 1950 .

[46]  P. Chung,et al.  An Atomistic-to-Continuum Framework for Nonlinear Crystal Mechanics Based on Asymptotic Homogenization , 2006 .

[47]  M. Born,et al.  Dynamical Theory of Crystal Lattices , 1954 .

[48]  R. M. Bowen,et al.  Toward a thermodynamics and mechanics of mixtures , 1967 .

[49]  H. F. Tiersten,et al.  On the nonlinear equations of thermo-electroelasticity , 1971 .

[50]  K. Bhattacharya,et al.  Depletion layers and domain walls in semiconducting ferroelectric thin films. , 2005, Physical review letters.

[51]  A. Many,et al.  Theory of Transient Space-Charge-Limited Currents in Solids in the Presence of Trapping , 1962 .

[52]  J. Swinburne.  Electromagnetic Theory , 1894, Nature.

[53]  R. D. Mindlin.  Elasticity, piezoelectricity and crystal lattice dynamics , 1972 .

[54]  K. L. Chowdhury,et al.  On the thermodynamics of non-linear elastic dielectrics , 1978 .

[55]  Ilya Prigogine,et al.  Introduction to Thermodynamics of Irreversible Processes , 1967 .

[56]  H. Conrad.  Space charge and the dependence of the flow stress of ceramics on an applied electric field , 2001 .

[57]  Gérard A. Maugin,et al.  Continuum Mechanics of Electromagnetic Solids , 1989 .

[58]  Smirnov,et al.  Molecular approach to the modeling of elasticity and piezoelectricity of SiC polytypes. , 1995, Physical review. B, Condensed matter.

[59]  J. Ericksen.  The Cauchy and Born Hypotheses for Crystals. , 1983 .

[60]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[61]  A. Cemal Eringen,et al.  On the foundations of electroelastostatics , 1963 .