0
Review Article

Advances in Developing Electromechanically Coupled Computational Methods for Piezoelectrics/Ferroelectrics at Multiscale

[+] Author and Article Information
Daining Fang

State Key Laboratory for
Turbulence and Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China;
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China
e-mail: fangdn@pku.edu.cn

Faxin Li

State Key Laboratory for
Turbulence and Complex Systems,
College of Engineering,
Peking University,
Beijing 100871, China
e-mail: lifaxin@pku.edu.cn

Jiawang Hong

Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China

Xianghua Guo

State Key Laboratory Explosion and
Safety Science,
Beijing Institute of Technology,
Beijing 100081, China

1Corresponding author.

Manuscript received November 19, 2012; final manuscript received September 23, 2013; published online October 28, 2013. Assoc. Editor: Bart Prorok.

Appl. Mech. Rev 65(6), 060802 (Oct 28, 2013) (52 pages) Paper No: AMR-12-1062; doi: 10.1115/1.4025633 History: Received November 19, 2012; Revised September 23, 2013

Piezoelectrics and ferroelectrics have been widely used in modern industries because of their peculiar electromechanical coupling properties, quick response, and compact size. In this work, we give a comprehensive review of our works and others' works in the past decade on the multiscale computational mechanics methods for electromechanical coupling behavior of piezoelectrics and ferroelectrics. The methods are classified into three types based on their applicable scale (i.e., macroscopic methods, mesoscopic methods, and atomic-level methods). In macroscopic methods, we first introduce the basic linear finite element method and employ it to analyze the crack problems in piezoelectrics. Then, the nonlinear finite element methods are presented for electromechanically coupled deformation and the domain switching processes were simulated. Based on our developed nonlinear electromechanically coupled finite element method, the domain switching instability problem was specially discussed and a constrained domain-switching model was proposed to overcome it. To specially address the crack problem in piezoelectrics, we further proposed a meshless electromechanical coupling method for piezoelectrics. In mesoscopic methods, the phase field methods (PFM) were firstly presented and the simulation results on the defects effect and size effect of deformation in ferroelectrics were given. Then, to solve the computational complexity problem of PFM in polycrystals, we proposed an optimization-based computational method taking the interactions between grains in an Eshelby inclusion manner. The domain texture evolution process can be calculated, and the Taylor's rule of plasticity has been reproduced well by this optimization-based model. Alternatively, the domain switching in polycrystalline ferroelectrics can be simulated by a proposed Monte Carlo method, which treated domain switching as a stochastic process. In atomic-level methods, we firstly introduce the first-principles method to calculate polarization and studied the topological polarization and strain gradient effect in ferroelectrics. Then, we present a modified electromechanically coupled molecular dynamic (MD) method for ferroelectrics based on the shell model and investigated the size effect of electromechanical deformation in ferroelectric thin films and nanowires. Finally, we introduced our recently proposed novel atomic finite element method (AFEM), which has higher computational efficiency than the MD. The deformation as well as domain evolution processes in ferroelectrics calculated by AFEM were also presented. The development of electromechanically coupled computational mechanics methods at multiscale is greatly beneficial, not only to the deformation and fracture of piezoelectrics/ferroelectrics, but also to structural design and reliability analysis of smart devices in engineering.

Copyright © 2013 by ASME
Your Session has timed out. Please sign back in to continue.

References

Xu, Y., 1991, Ferroelectric Materials and Their Applications, North-Holland, Amsterdam.
Jaffe, B., Cook, W. R., and Jaffe, H., 1971, Piezoelectric Ceramics, Academic, London.
Lines, M. E., and Glass, A. M., 1977, Principles and Applications of Ferroelectrics and Related Materials, Clarendon, Oxford.
Otsuka, K., and Ren, B., 2005, “Physical Metallurgy of Ti-Ni-Based Shape Memory Alloys,” Prog. Mater. Sci., 50, pp. 511–678. [CrossRef]
Chen, P. J., and Peerey, P. S., 1979, “One-Dimensional Dynamic Electromechanical Constitutive Relations of Ferroelectric Materials,” Acta Mech., 31, pp. 231–241. [CrossRef]
Chen, P. J., 1980, “Three Dimensional Dynamic Electromechanical Constitutive Relations for Ferroelectric Materials,” Int. J. Solids Struct., 16, pp. 1059–1067. [CrossRef]
Bassiouny, E., Ghaleb, A. F., and Maugin, G. A., 1988, “Thermodynamical Formulation for Coupled Electromechanical Hysteresis Effects-I. Basic Equations,” Int. J. Eng. Sci., 26(12), pp. 1279–1295. [CrossRef]
Cocks, C. F., and McMeeking, R. M., 1999, “A Phenomenological Constitutive Law for the Behaviour of Ferroelectric Ceramics,” Ferroelectrics, 228, pp. 219–228. [CrossRef]
Kamlah, M., and Tsakmakis, C., 1999, “Phenomenological Modeling of the Nonlinear Electromechanical Coupling in Ferroelectrics,” Int. J. Solids Struct., 36, pp. 669–695. [CrossRef]
Huber, J. E., and Fleck, N. A., 2001, “Multi-axial Electrical Switching of a Ferroelectric: Theory Versus Experiment,” J. Mech. Phys. Solids, 49, pp. 785–811. [CrossRef]
Landis, C. M., 2002, “Fully Coupled, Multi-axial, Symmetric Constitutive Laws for Polycrystalline Ferroelectric Ceramics,” J. Mech. Phys. Solids, 50, pp. 127–152. [CrossRef]
Hwang, S. C., Lynch, C. S., and McMeeking, R. M., 1995, “Ferroelectric/Ferroelastic Interactions and a Polarization Switching Model,” Acta Metall. Mater., 43(5), pp. 2073–2084. [CrossRef]
Hwang, S. C., Huber, J. E., McMeeking, R. M., and Fleck, N. A., 1998, “The Simulation of Switching in Polycrystalline Ferroelectric Ceramics,” J. Appl. Phys., 83(3), pp. 1530–1540. [CrossRef]
Chen, X., Fang, D. N., and Hwang, K. C., 1997, “Micromechanics Simulation of Ferroelectric Polarization Switching,” Acta Mater., 45(8), pp. 3181–3189. [CrossRef]
Lu, W., Fang, D. N., Li, C. Q., and Hwang, K. C., 1999, “Nonlinear Electric-Mechanical Behavior and Micromechanics Modeling of Ferroelectric Domain Evolution,” Acta Mater., 47, pp. 2913–2926. [CrossRef]
Li, F. X., and Rajapakse, R. K. N. D., 2007, “A Constrained Domain Switching Model for Polycrystalline Ferroelectric Ceramics: Part I—Model Formulation and Application to Tetragonal Materials,” Acta Mater., 55, pp. 6472–6480. [CrossRef]
Li, F. X., and Rajapakse, R. K. N. D., 2007, “A Constrained Domain Switching Model for Polycrystalline Ferroelectric Ceramics: Part II—Combined Switching and Application to Rhombohedral Materials,” Acta Mater., 55, pp. 6481–6488. [CrossRef]
Park, S. B., and Sun, C. T., 1995, “Fracture Criteria for Piezoelectric Ceramics,” J. Am. Ceram. Soc., 78, pp. 1475–1480. [CrossRef]
Zhang, T. Y., and Tong, P., 1996, “Fracture Mechanics for a Mode III Crack in a Piezoelectric Material,” Int. J. Solids Struct., 33, pp. 343–359. [CrossRef]
Zhang, T. Y., and Qian, C. F., 1998, “Linear Electro-elastic Analysis of a Cavity or a Crack in a Piezoelectric Material,” Int. J. Solids Struct., 35, pp. 2121–2149. [CrossRef]
Zhang, T. Y., Zhao, M. H., and Tong, P., 2002, “Fracture of Piezoelectric Ceramics,” Adv. Appl. Mech., 38, pp. 147–289. [CrossRef]
Chen, Y. H., and Lu, T. R., 2003, “Cracks and Fracture in Piezoelectrics,” Adv. Appl. Mech., 39, pp. 121–215. [CrossRef]
Zhang, T. Y., and Gao, C. F., 2004, “Fracture Behavior of Piezoelectric Materials,” Theor. Appl. Fract. Mech., 41, pp. 339–379. [CrossRef]
Zhang, T. Y., Zhao, M. H., and Gao, C. F., 2005, “The Strip Dielectric Breakdown Model,” Int. J. Fract., 132, pp. 311–327. [CrossRef]
Yang, W., and Zhu, T., 1998, “Switch-Toughening of Ferroelectrics Subjected to Electric Fields,” J. Mech. Phys. Solids, 46(2), pp. 291–311. [CrossRef]
Zhu, T., and Yang, W., 1997, “Toughness Variation of Ferroelectrics by Polarization Switch Under Non-uniform Electric Field,” Acta Mater., 45(11), pp. 4695–4702. [CrossRef]
Mao, G. Z., and Fang, D. N., 2004, “Fatigue Crack Growth Induced by Domain Switching Under Electromechanical Load in Ferroelectrics,” Theor. Appl. Fract. Mech., 41, pp. 115–123. [CrossRef]
Fang, D. N., Zhang, Y. H., and Mao, G. Z., 2011, “A COD Fracture Model of Ferroelectric Ceramics With Applications in Electric Field Induced Fatigue Crack Growth,” Int. J. Fract., 167, pp. 211–220. [CrossRef]
Gao, H., Zhang, T. Y., and Tong, P., 1997, “Local and Global Energy Release Rate for an Electrically Yield Crack in a Piezoelectric Ceramic,” J. Mech. Phys. Solids, 45, pp. 491–510. [CrossRef]
Gong, X., and Suo, Z., 1996, “Reliability of Ceramic Multiplayer Actuators: A Nonlinear Finite Element Simulation,” J. Mech. Phys. Solids, 44(5), pp. 751–769. [CrossRef]
Hom, C. L., and Shankar, N., 1996, “A Finite Element Method for Electrostrictive Ceramics,” Int. J. Solids Struct., 33(12), pp. 1757–1779. [CrossRef]
Qi, H., Fang, D. N., and Yao, Z. H., 1997, “FEM Analysis of Electro-mechanical Coupling Effect of Piezoelectric Materials,” Comp. Mater. Sci., 8, pp. 283–290. [CrossRef]
Hwang, S. C., and McMeeking, R. M., 1999, “A Finite Element Model of Ferroelastic Polycrystals,” Int. J. Solids Struct., 36(10), pp. 1541–1556. [CrossRef]
Li, F. X., and Fang, D. N., 2004, “Simulations of Domain Switching in Ferroelectrics by a Three-Dimensional Finite Element Model,” Mech. Mater., 36, pp. 959–973. [CrossRef]
Chen, L. Q., 2002, “Phase-Field Models for Microstructure Evolution,” Annu. Rev. Mater. Res., 32, pp. 113–140. [CrossRef]
Wang, J., Shi, S. Q., Chen, L. Q., Li, Y. L., and Zhang, T. Y., 2004, “Phase Field Simulations of Ferroelectric/Ferroelastic Polarization Switching,” Acta Mater., 52, pp. 749–764. [CrossRef]
Wang, J., Li, Y. L., Chen, L. Q., and Zhang, T. Y., 2005, “The Effect of Mechanical Strains on the Ferroelectric and Dielectric Properties of a Model Single Crystal – Phase Field Simulation,” Acta Mater., 53, pp. 2495–2507. [CrossRef]
Wang, J., and Zhang, T. Y., 2006, “Size Effects in Epitaxial Ferroelectric Islands and Thin Films,” Phys. Rev. B, 73, p. 144107. [CrossRef]
Wang, J., and Zhang, T.-Y., 2007, “Phase Field Simulations of Polarization Switching-Induced Toughening in Ferroelectric Ceramics,” Acta Mater., 55, pp. 2465–2477. [CrossRef]
Zhang, W., and Bhattacharya, K., 2005, “A Computational Model of Ferroelectric Domains. Part I: Model Formulation and Domain Switching,” Acta Mater., 53, pp. 185–198. [CrossRef]
Zhang, W., and Bhattacharya, K., 2005, “A Computational Model of Ferroelectric Domains. Part II: Grain Boundaries and Defect Pinning,” Acta Mater., 53, pp. 199–209. [CrossRef]
Song, Y. C., Soh, A. K., and Ni, Y., 2007, “Phase Field Simulation of Crack Tip Domain Switching in Ferroelectrics,” J. Phys. D: Appl. Phys., 40, pp. 1175–1182. [CrossRef]
Su, Y., and Landis, C. M., 2007, “Continuum Thermodynamics of Ferroelectric Domain Evolution: Theory, Finite Element Implementation, and Application to Domain Wall Pinning,” J. Mech. Phys. Solids, 55, pp. 280–305. [CrossRef]
Dayal, K., and Bhattacharya, K., 2007, “A Real-Space Non-local Phase-Field Model of Ferroelectric Domain Patterns in Complex Geometries,” Acta Mater., 55, pp. 1907–1917. [CrossRef]
Zhang, Y. H., Li, J. Y., and Fang, D. N., 2010, “Oxygen-Vacancy-Induced Memory Effect and Large Recoverable Strain in a Barium Titanate Single Crystal,” Phys. Rev. B, 82, p. 064103. [CrossRef]
Zhang, Y. H., Li, J. Y., and Fang, D. N., 2010, “Size Dependent Domain Configuration and Electric Field Driven Evolution in Ultrathin Ferroelectric Films: A Phase Field Investigation,” J. Appl. Phys., 107, p. 034107. [CrossRef]
Abdollahi, A., and Arias, I., 2011, “Phase-Field Modeling of the Coupled Microstructure and Fracture Evolution in Ferroelectric Single Crystals,” Acta Mater., 59, pp. 4733–4746. [CrossRef]
Abdollahi, A., and Arias, I., 2012, “Numerical Simulation of Intergranular and Transgranular Crack Propagation in Ferroelectric Polycrystals,” Int. J. Fract., 174, pp. 3–15. [CrossRef]
Abdollahi, A., and Arias, I., 2012, “Phase-Field Modeling of Crack Propagation in Piezoelectric and Ferroelectric Materials With Different Electromechanical Crack Conditions,” J. Mech. Phys. Solids, 60, pp. 2100–2126. [CrossRef]
Li, W., and Landis, C. M., 2011, “Nucleation and Growth of Domains Near Crack Tips in Single Crystal Ferroelectrics,” Eng. Fract. Mech., 78, pp. 1505–1513. [CrossRef]
Tang, W., Fang, D. N., and Li, J. Y., 2009, “Two-Scale Micromechanics-Based Probabilistic Modeling of Domain Switching in Ferroelectric Ceramics,” J. Mech. Phys. Solids, 57, pp. 1683–1701. [CrossRef]
Li, F. X., and Soh, A. K., 2010, “An Optimization-Based Computational Model for Domain Evolution in Polycrystalline Ferroelastics,” Acta Mater., 58, pp. 2207–2215. [CrossRef]
Li, F. X., Zhou, X. L., and Soh, A. K., 2010, “An Optimization-Based “Phase Field” Model for Polycrystalline Ferroelectrics,” Appl. Phys. Lett., 96, p. 152905. [CrossRef]
Zhou, X. L., and Li, F. X., 2011, “Simulations of Domain Evolution in Morphotropic Ferroelectric Ceramics Under Electromechanical Loading Using an Optimization-Based Model,” J. Appl. Phys., 109, p. 084107. [CrossRef]
Migoni, R., Bilz, H., and Bauerle, D., 1976, “Origin of Raman-Scattering and Ferroelectricity in Oxidic Perovskites,” Phys. Rev. Lett., 37, pp. 1155–1158. [CrossRef]
Khatib, D., Migoni, R., Kugel, G. E., and Godefroy, L., 1989, “Lattice-Dynamics of BaTiO3 in the Cubic Phase,” J. Phys.: Condens. Matter, 1, pp. 9811–9822. [CrossRef]
Tinte, S., Stachiotti, M. G., Sepliarsky, M., Migoni, R. L., and Rodriguez, C. O., 1999, “Atomistic Modelling of BaTiO3 Based on First-Principles Calculations,” J. Phys.: Condens. Matter, 11, pp. 9679–9690. [CrossRef]
Sepliarsky, M., Phillpot, S. R., Wolf, D., Stachiotti, M. G., and Migoni, R. L., 2001, “Long-Ranged Ferroelectric Interactions in Perovskite Superlattices,” Phys. Rev. B, 64, p. 060101. [CrossRef]
Sepliarsky, M., Asthagiri, A., Phillpot, S. R., Stachiotti, M. G., and Migoni, R. L., 2005, “Atomic-Level Simulation of Ferroelectricity in Oxide Materials,” Curr. Opin. Solid State Mater. Sci., 9, pp. 107–113. [CrossRef]
Shimada, T., Wakahara, K., Umeno, Y., and Kitamura, T., 2008, “Shell Model Potential for PbTiO3 and Its Applicability to Surfaces and Domain Walls,” J. Phys.: Condens. Matter, 20, p. 325225. [CrossRef]
Zhang, Y. H., Hong, J. W., Liu, B., and Fang, D. N., 2009, “Molecular Dynamics Investigations on the Size-Dependent Ferroelectric Behavior of BaTiO3 Nanowires,” Nanotechnology, 20, p. 405703. [CrossRef] [PubMed]
Zhang, Y. H., Hong, J. W., Liu, B., and Fang, D. N., 2010, “Strain Effect on Ferroelectric Behaviors of BaTiO3 Nanowires: A Molecular Dynamics Study,” Nanotechnology, 21, p. 015701. [CrossRef] [PubMed]
Zhang, Y. H., Liu, B., and Fang, D. N., 2011, “Stress-Induced Phase Transition and Deformation Behavior of BaTiO3 Nanowires,” J. Appl. Phys., 110, p. 054109. [CrossRef]
Zhang, Y. H., Sang, Y. L., Liu, B., and Fang, D. N., 2011, “Critical Thickness and the Size-Dependent Curie Temperature of BaTiO3 Nanofilms,” J. Comput. Theor. Nanosci., 8, pp. 867–872. [CrossRef]
Stachiotti, M. G., and Sepliarsky, M., 2011, “Toroidal Ferroelectricity in PbTiO3 Nanoparticles,” Phys. Rev. Lett., 106, p. 137601. [CrossRef]
Cohen, R. E., 1992, “Origin of Ferroelectricity in Perovskite Oxides,” Nature, 358, pp. 136–138. [CrossRef]
Junquera, J., and Ghosez, P., 2003, “Critical Thickness for Ferroelectricity in Perovskite Ultrathin Films,” Nature, 422, pp. 506–509. [CrossRef] [PubMed]
Naumov, I. I., Bellaiche, L., and Fu, H., 2004, “Unusual Phase Transitions in Ferroelectric Nanodisks and Nanorods,” Nature, 432, pp. 737–740. [CrossRef] [PubMed]
Geneste, G., Bousquet, E., Junquera, J., and Ghosez, P., 2006, “Finite-Size Effects in BaTiO3 Nanowires,” Appl. Phys. Lett., 88, p. 112906. [CrossRef]
Pilania, G., Alpay, S. P., and Ramprasad, R., 2009, “Ab Initio Study of Ferroelectricity in BaTiO3 Nanowires,” Phys. Rev. B, 80, p. 014113. [CrossRef]
Allik, H., and Hughes, T. J. R., 1970, “Finite Element Method for Piezoelectric Vibration,” Int. J. Numer. Meth. Eng., 2, pp. 151–157. [CrossRef]
Hom, C. L., and Shankar, N., 1995, “A Numerical Analysis of Relaxor Ferroelectric Multilayered Actuators and 2-2 Composite Arrays,” Smart Mater. Struct., 4, pp. 266–273. [CrossRef]
Kamlah, M., and Bohle, U., 2001, “Finite Element Analysis of Piezoceramic Components Taking Into Account Ferroelectric Hysteresis Behavior,” Int. J. Solids Struct., 38, pp. 605–633. [CrossRef]
Chen, W., and Lynch, C. S., 1999, “Finite Element Analysis of Cracks in Ferroelectric Ceramic Materials,” Eng. Fract. Mech., 64, pp. 539–562. [CrossRef]
Li, F. X., and Rajapakse, R. K. N. D., 2008, “Nonlinear Finite Element Modeling of Polycrystalline Ferroelectrics Based on Constrained Domain Switching,” Comp. Mater. Sci., 44(2), pp. 322–329. [CrossRef]
Guo, X. H., 2004, “Study on Mechanical Behavior of Thin Films”, Ph.D. thesis, Tsinghua University, Beijing, China.
Guo, X. H., and Fang, D. N., 2004, “Simulation of Interface Cracking in Piezoelectric Layers,” Int. J. Nonlin. Sci. Numer. Simul., 5(3), pp. 235–242. [CrossRef]
Guo, X. H., Fang, D. N., Soh, A. K., Kim, H. C., and Lee, J. J., 2006, “Analysis of Piezoelectric Ceramic Multilayer Actuators Based on an Electro-mechanical Coupled Meshless Method,” Acta Mech. Sinica, 22, pp. 34–39. [CrossRef]
Li, Y. L., Hu, S. Y., Liu, Z. K., and Chen, L. Q., 2001, “Phase-Field Model of Domain Structures in Ferroelectric Thin Films,” Appl. Phys. Lett., 78, pp. 3878–3880. [CrossRef]
Li, Y. L., Hu, S. Y., Liu, Z. K., and Chen, L. Q., 2002, “Effect of Electrical Boundary Conditions on Ferroelectric Domain Structures in Thin Films,” Appl. Phys. Lett., 81, pp. 427–429. [CrossRef]
Li, Y. L., Hu, S. Y., Liu, Z. K., and Chen, L. Q., 2002, “Effect of Substrate Constraint on the Stability and Evolution of Ferroelectric Domain Structures in Thin Films,” Acta Mater., 50, pp. 395–411. [CrossRef]
Li, Y. L., Cross, L. E., and Chen, L. Q., 2005, “A Phenomenological Thermodynamic Potential for BaTiO3 Single Crystals,” J. Appl. Phys., 98, p. 064101. [CrossRef]
Zhao, X. F., Soh, A. K., and Li, L., 2010, “Influence of Dipole Defects on Polarization Switching in the Vicinity of a Crack in Relaxor Ferroelectrics,” Philos. Mag. Lett., 90, pp. 251–260. [CrossRef]
Wang, Y. U., Jin, Y. M., and Khachaturyan, A. G., 2002, “Phase Field Microelasticity Theory and Modeling of Elastically and Structurally Inhomogeneous Solid,” J. Appl. Phys., 92, pp. 1351–1360. [CrossRef]
Choudhury, S., Li, Y. L., Krill, C. E., and Chen, L. Q., 2005, “Phase-Field Simulation of Polarization Switching and Domain Evolution in Ferroelectric Polycrystals,” Acta Mater., 53, pp. 5313–5321. [CrossRef]
Schrade, D., Mueller, R., Xu, B. X., and Gross, D., 2007, “Domain Evolution in Ferroelectric Materials: A Continuum Phase Field Model and Finite Element Implementation,” Comput. Mech. Appl. Mech. Eng., 196, pp. 4365–4374. [CrossRef]
Wang, J., and Zhang, T.-Y., 2008, “Phase Field Simulations of a Permeable Crack Parallel to the Original Polarization Direction in a Ferroelectric Mono-domain,” Eng. Fract. Mech., 75, pp. 4886–4897. [CrossRef]
Zhang, Y. H., Xu, R., Liu, B., and Fang, D. N., 2012, “An Electromechanical Atomic-Scale Finite Element Method for Simulating Evolutions of Ferroelectric Nanodomains,” J. Mech. Phys. Solids, 60, pp. 1383–1399. [CrossRef]
Wang, X., and Shen, Y., 1995, “Some Basic Theory for Thermal Magnetic Electric Elastic Media,” Chinese J. Appl. Mech., I2(2), pp. 28–39 (in Chinese).
Sosa, H., 1991, “Plane Problem in Piezoelectric Media With Defects,” Int. J. Solids Struct., 28(4), pp. 491–505. [CrossRef]
Ghandi, K., and Hagood, N. W., 1996, “Nonlinear Finite Element Modeling of Phase Transitions in Electro-mechanically Coupled Material,” Proc. SPIE, 2715, pp. 121–140. [CrossRef]
Ghandi, K., and Hagood, N. W., 1997, “A Hybrid Finite-Element Model for Phase Transition in Nonlinear Electro-mechanically Coupled Material,” Proc. SPIE, 3039, pp. 97–112. [CrossRef]
Chen, W., and Lynch, C. S., 2001, “Multiaxial Constitutive Behavior of Ferroelastic Materials,” ASME J. Eng. Mater. Technol., 123, pp. 169–175. [CrossRef]
Fett, T., and Munz, D., 2003, “Deformation of PZT Under Tension, Compression, Bending, and Torsion Loading,” Adv. Eng. Mater., 5, pp. 718–722. [CrossRef]
Li, F. X., and Fang, D. N., 2005, “Effects of Lateral Stress on the Electromechanical Response of Ferroelectric Ceramics: Experiments Versus Model,” J. Intell. Mater. Syst. Struct., 16(7–8), pp. 583–588. [CrossRef]
Steinkopff, T., 1999, “Finite-Element Modeling of Ferroic Domain Switching in Piezoelectric Ceramics,” J. Eur. Ceram. Soc., 19, pp. 1247–1249. [CrossRef]
Zeng, W., Manzari, M. T., Lee, J. D., and Shen, Y. L., 2003, “Fully Coupled Non-linear Analysis of Piezoelectric Solids Involving Domain Switching,” Int. J. Numer. Meth. Eng., 56, pp. 13–34. [CrossRef]
Kamlah, M., Liskowsky, A. C., McMeeking, R. M., and Balke, H., 2005, “Finite Element Simulations of a Polycrystalline Ferroelectric Based on a Multidomain Single Crystal Switching Model,” Int. J. Solids Struct., 42, pp. 2949–2964. [CrossRef]
Fang, D. N., and Li, C. Q., 1999, “Nonlinear Electric-Mechanical Behavior of a Soft PZT-51 Ferroelectric Ceramic,” J. Mater. Sci., 34, pp. 4001–4010. [CrossRef]
Axelsson, O., 1994, Iterative Solution Methods, Cambridge University, Cambridge, UK.
Crisfield, M. A., 1983, “An Arc-Length Method Including Line Searches and Accelerations,” Int. J. Numer. Meth. Eng., 19, pp. 1269–1289. [CrossRef]
Landis, C. M., 2002, “A New Finite-Element Formulation for Electromechanical Boundary Value Problems,” Int. J. Numer. Meth. Eng., 55, pp. 613–628. [CrossRef]
Matthies, H., and Strang, G., 1979, “The Solution of Non-linear Finite Element Equations,” Int. J. Numer. Meth. Eng., 14, pp. 1613–1626. [CrossRef]
Wempner, G. A., 1971, “Discrete Approximations Related to Nonlinear Theories of Solids,” Int. J. Solids Struct., 7, pp. 1581–1599. [CrossRef]
Riks, E., 1972, “The Application of Newtons Method to the Problem of Elastic Instability,” J. Appl. Mech., 39, pp. 1060–1066. [CrossRef]
Kessler, H., and Balke, H., 2001, “On the Local and Average Energy Release in Polarization Switching Phenomena,” J. Mech. Phys. Solids, 49, pp. 953–978. [CrossRef]
Lucy, L. B., 1977, “A Numerical Approach to the Fission Hypothesis,” J. Astron., 8(12), pp. 1013–1024. [CrossRef]
Monaghan, J. J., 1992, “Smoothed Particle Hydrodynamics,” Ann. Rev. Astron. Astrophys, 30, pp. 543–574. [CrossRef]
Nayroles, B., Touzot, G., and Villion, P., 1992, “Generalizing the Finite Element Method: Diffuse Approximation and Diffuse Element,” Comput. Mech., 10, pp. 307–318. [CrossRef]
Belytschko, T., Gu, L., and Lu, Y. Y., 1994, “Fracture and Crack Growth by Element Free Galerkin Methods,” Model. Simul. Mater. Sci. Eng., 2(3A), pp. 519–534. [CrossRef]
Belytschko, T., Lu, Y. Y., and Gu, L., 1994, “Element-Free Galerkin Methods,” Int. J. Numer. Meth. Eng., 37, pp. 229–256. [CrossRef]
Belytschko, T., Organ, D., and Gerlach, C., 2000, “Element-Free Galerkin Methods for Dynamic Fracture in Concrete,” Comp. Meth. Appl. Mech. Eng., 187, pp. 385–399. [CrossRef]
Onate, E., Idelsohn, S., Zienkiewicaz, O. C., and Taylor, R. L., 1996, “A Finite Point Method in Computational Mechanics: Applications to Convective Transport and Fluid Flow,” Int. J. Numer. Meth. Eng., 39, pp. 3839–3866. [CrossRef]
Zhang, X., Liu, X. H., Song, K. Z., and Lu, M. W., 2011, “Least-Square Collocation Meshless Method,” Int. J. Numer. Meth. Eng., 51(9), pp. 1089–1100. [CrossRef]
Xu, T., Zou, P., Xu, T. S., and Jiye, C. M., 2010, “Study on Weight Function of Meshless Method Based on B-Spline Wavelet Function,” The 3rd Int. Joint Conference on Computational Science and Optimization, pp. 36–40.
Razmjoo, H., Movahhedi, M., and Hakimi, A., 2010, “Improved Meshless Method Using Direct Shape Function for Computational Electromagnetics,” Proceedings of the Asia-Pacific Microwave Conference, Yokohama, Japan, pp. 2157–2160.
Sladek, J., Sladek, V., Solek, P., and Pan, E., 2008, “Fracture Analysis of Cracks in Magneto-electro-elastic Solids by the MLPG,” Comput. Mech., 42, pp. 697–714. [CrossRef]
Feng, W. J., Han, X., and Li, Y. S., 2009, “Fracture Analysis for Two-Dimensional Plane Problems of Nonhomogeneous Magneto-Electro-Thermo-Elastic Plates Subjected to Thermal Shock by Using the Messless Local Petrov Galerkin Method,” Computer Model. Eng. Sci., 48(1), pp. 1–26.
Sosa, H. A., and Pak, Y. E., 1990, “Three Dimensional Eigenfunction Analysis of a Crack in a Piezoelectric Material,” Int. J. Solids Struct., 26, pp. 1–15. [CrossRef]
Li, Y. L., and Chen, L. Q., 2006, “Temperature-Strain Phase Diagram for BaTiO3 Thin Films,” Appl. Phys. Lett., 88, p. 072905. [CrossRef]
Li, Y. L., Choudhury, S., Haeni, J. H., Biegalski, M. D., Vasudevarao, A., Sharan, A., Ma, H. Z., Levy, J., Gopalan, V., Trolier-McKinstry, S., Schlom, D. G., Jia, Q. X., and Chen, L. Q., 2006, “Phase Transitions and Domain Structures in Strained Pseudocubic (100) SrTiO3 Thin Films,” Phys. Rev. B, 73, p. 184112. [CrossRef]
Liu, P.-L., Wang, J., Zhang, T.-Y., Li, Y., Chen, L.-Q., Ma, X.-Q., Chu, W.-Y., and Qiao, L.-J., 2008, “Effects of Unequally Biaxial Misfit Strains on Polarization Phase Diagrams in Embedded Ferroelectric Thin Layers: Phase Field Simulations,” Appl. Phys. Lett., 93, p. 132908. [CrossRef]
Sheng, G., Zhang, J. X., Li, Y. L., Choudhury, S., Jia, Q. X., Liu, Z. K., and Chen, L. Q., 2008, “Misfit Strain-Misfit Strain Diagram of Epitaxial BaTiO(3) Thin Films: Thermodynamic Calculations and Phase-Field Simulations,” Appl. Phys. Lett., 93, p. 232904. [CrossRef]
Wang, J. J., Wu, P. P., Ma, X. Q., and Chen, L. Q., 2010, “Temperature-Pressure Phase Diagram and Ferroelectric Properties of BaTiO(3) Single Crystal Based on a Modified Landau Potential,” J. Appl. Phys., 108, p. 114105. [CrossRef]
Wang, J., and Kamlah, M., 2009, “Three-Dimensional Finite Element Modeling of Polarization Switching in a Ferroelectric Single Domain With an Impermeable Notch,” Smart Mater. Struct. 18, p. 104008. [CrossRef]
Xu, B.-X., Schrade, D., Gross, D., and Mueller, R., 2010, “Phase Field Simulation of Domain Structures in Cracked Ferroelectrics,” Int. J. Fract., 165, pp. 163–173. [CrossRef]
Miehe, C., Welschinger, F., and Hofacker, M., 2010, “A Phase Field Model of Electromechanical Fracture,” J. Mech. Phys. Solids, 58, pp. 1716–1740. [CrossRef]
Hong, L., Soh, A. K., Song, Y. C., and Lim, L. C., 2008, “Interface and Surface Effects on Ferroelectric Nano-thin Films,” Acta Mater., 56, pp. 2966–2974. [CrossRef]
Wang, J., Kamlah, M., Zhang, T. Y., Li, Y., and Chen, L. Q., 2008, “Size-Dependent Polarization Distribution in Ferroelectric Nanostructures: Phase Field Simulations,” Appl. Phys. Lett., 92, p. 162905. [CrossRef]
Wang, J., Kamlah, M., and Zhang, T.-Y., 2009, “Phase Field Simulations of Ferroelectric Nanoparticles With Different Long-Range-Electrostatic and -Elastic Interactions,” J. Appl. Phys., 105, p. 014104. [CrossRef]
Li, Y. L., Hu, S. Y., Tenne, D., Soukiassian, A., Schlom, D. G., Chen, L. Q., Xi, X. X., Choi, K. J., Eom, C. B., Saxena, A., Lookman, T., and Jia, Q. X., 2007, “Interfacial Coherency and Ferroelectricity of BaTiO(3)/SrTiO(3) Superlattice Films,” Appl. Phys. Lett., 91, p. 252904. [CrossRef]
Xiao, Y., Shenoy, V. B., and Bhattacharya, K., 2005, “Depletion Layers and Domain Walls in Semiconducting Ferroelectric Thin Films,” Phys. Rev. Lett., 95, p. 247603. [CrossRef] [PubMed]
Xiao, Y., and Bhattacharya, K., 2008, “A Continuum Theory of Deformable, Semiconducting Ferroelectrics,” Arch. Ration. Mech. Anal., 189, pp. 59–95. [CrossRef]
Hong, L., Soh, A. K., Du, Q. G., and Li, J. Y., 2008, “Interaction of O Vacancies and Domain Structures in Single Crystal BaTiO3: Two-Dimensional Ferroelectric Model,” Phys. Rev. B, 77, p. 094104. [CrossRef]
Ren, X. B., 2004, “Large Electric-Field-Induced Strain in Ferroelectric Crystals by Point-Defect-Mediated Reversible Domain Switching,” Nature Mater., 3, pp. 91–94. [CrossRef]
Zhang, L. X., and Ren, X., 2005, “In Situ Observation of Reversible Domain Switching in Aged Mn-Doped BaTiO3 Single Crystals,” Phys. Rev. B, 71, p. 174108. [CrossRef]
Shu, Y. C., and Bhattacharya, K., 2001, “Domain Patterns and Macroscopic Behaviour of Ferroelectric Materials,” Philos. Mag. B, 81, pp. 2021–2054. [CrossRef]
Devonshire, A. F., 1954, “Theory of Ferroelectrics,” Adv. Phys., 3, pp. 85–130. [CrossRef]
Pertsev, N. A., Zembilgotov, A. G., and Tagantsev, A. K., 1998, “Effect of Mechanical Boundary Conditions on Phase Diagrams of Epitaxial Ferroelectric Thin Films,” Phys. Rev. Lett., 80, pp. 1988–1991. [CrossRef]
Koukhar, V. G., Pertsev, N. A., and Waser, R., 2001, “Thermodynamic Theory of Epitaxial Ferroelectric Thin Films With Dense Domain Structures,” Phys. Rev. B, 64, p. 214103. [CrossRef]
Shu, Y. C., Yen, J. H., Chen, H. Z., Li, J. Y., and Li, L. J., 2008, “Constrained Modeling of Domain Patterns in Rhombohedral Ferroelectrics,” Appl. Phys. Lett., 92, p. 052909. [CrossRef]
Li, J. Y., and Liu, D., 2004, “On Ferroelectric Crystals With Engineered Domain Configurations,” J. Mech. Phys. Solids, 52, pp. 1719–1742. [CrossRef]
Ma, Y. F., and Li, J. Y., 2007, “Magnetization Rotation and Rearrangement of Martensite Variants in Ferromagnetic Shape Memory Alloys,” Appl. Phys. Lett., 90, p. 172504. [CrossRef]
Li, L. J., Yang, Y., Shu, Y. C., and Li, J. Y., 2010, “Continuum Theory and Phase-Field Simulation of Magnetoelectric Effects in Multiferroic Bismuth Ferrite,” J. Mech. Phys. Solids, 58, pp. 1613–1627. [CrossRef]
Hu, H. L., and Chen, L. Q., 1997, “Computer Simulation of 90 deg Ferroelectric Domain Formation in Two-Dimensions,” Mater. Sci. Eng. A, 238, pp. 182–191. [CrossRef]
Rabe, K. M., Ahn, C. H., and Triscone, J. M., 2007, Physics of Ferroelectrics: A Modern Perspective, Springer, Heidelberg.
Zhang, L. X., and Ren, X. B., 2006, “Aging Behavior in Single-Domain Mn-Doped BaTiO3 Crystals: Implication for a Unified Microscopic Explanation of Ferroelectric Aging,” Phys. Rev. B, 73, p. 094121. [CrossRef]
Zhang, L. X., Erdem, E., Ren, X. B., and Eichel, R. A., 2008, “Reorientation of (Mn-Ti(’)-V-O(Center Dot Center Dot))(x) Defect Dipoles in Acceptor-Modified BaTiO3 Single Crystals: An Electron Paramagnetic Resonance Study,” Appl. Phys. Lett., 93, p. 202901. [CrossRef]
Erhart, P., Eichel, R. A., Traskelin, P., and Albe, K., 2007, “Association of Oxygen Vacancies With Impurity Metal Ions in Lead Titanate,” Phys. Rev. B, 76, p. 174116. [CrossRef]
Hooton, J. A., and Merz, W. J., 1955, “Etch Patterns and Ferroelectric Domains in Batio3 Single Crystals,” Phys. Rev., 98, pp. 409–413. [CrossRef]
Zhong, W. L., Wang, Y. G., Zhang, P. L., and Qu, B. D., 1994, “Phenomenological Study of the Size Effect on Phase-Transitions in Ferroelectric Particles,” Phys. Rev. B, 50, pp. 698–703. [CrossRef]
Sang, Y. L., Liu, B., and Fang, D. N., 2008, “The Size and Strain Effects on the Electric-Field-Induced Domain Evolution and Hysteresis Loop in Ferroelectric BaTiO3 Nanofilms,” Comp. Mater. Sci., 44, pp. 404–410. [CrossRef]
Asada, T., and Koyama, Y., 2007, “Ferroelectric Domain Structures Around the Morphotropic Phase Boundary of the Piezoelectric Material PbZr1-xTixO3,” Phys. Rev. B, 75, p. 214111. [CrossRef]
Lupascu, D. C., 2004, Fatigue in Ferroelectric Ceramics and Related Issues, Springer-Verlag, Berlin.
Eshelby, J. D., 1957, “The Determination of the Elastic Field of an Ellipsoidal Inclusion and Related Problems,” Proc. R. Soc. A, 241, pp. 376–396. [CrossRef]
Uchida, N., and Ikeda, T., 1967, “Electrostriction in Perovskite-Type Ferroelectric Ceramics,” Jpn. J. Appl. Phys., 6, pp. 1079–1088. [CrossRef]
Huber, J. E., Fleck, N. A., and McMeeking, R. M., 1999, “A Crystal Plasticity Model for Ferroelectrics,” Ferroelectrics, 228, pp. 39–52. [CrossRef]
Li, F. X., Fang, D. N., and Soh, A. K., 2004, “An Analytical Axisymmetric Model for the Poling-History Dependent Behavior of Ferroelectric Ceramics,” Smart Mater. Struct., 13, pp. 668–675. [CrossRef]
Han, S. P., 1977, “A Global Convergent Method for Nonlinear Programming,” J. Optim. Theory Appl., 22, pp. 297–309. [CrossRef]
Bunge, H. J., 1982, Texture Analysis in Materials Science, Butterworth, Berlin.
Li, F. X., and Rajapakse, R. K. N. D., 2007, “Analytically Saturated Domain Orientation Textures and Electromechanical Properties of Ferroelectric Ceramics Due to Electric/Mechanical Loading,” J. Appl. Phys., 101, p. 054110. [CrossRef]
Hoffmann, M. J., Hammer, M., Endriss, A., and Lupascu, D. C., 2001, “Correlation Between Microstructure, Strain Behavior, and Acoustic Emission of Soft PZT Ceramics,” Acta Mater., 49, pp. 1301–1310. [CrossRef]
Li, F. X., and Zhou, X. L., 2011, “Simulations of Gradual Domain-Switching in Polycrystalline Ferroelectrics Using an Optimization-Based, Multidomain-Grain Model,” Comput. Struct., 89, pp. 1142–1147. [CrossRef]
Li, J. Y., Rogan, R. C., Üstündag, E., and Bhattacharya, K., 2005, “Domain Switching in Polycrystalline Ferroelectric Ceramics,” Nature Mater., 4, pp. 776–781. [CrossRef]
Webber, K. G., Aulbach, E., Key, T., Marsilius, M., Granzow, T., and Rödel, J., 2009, “Temperature-Dependent Ferroelastic Switching of Soft Lead Zirconate Titanate,” Acta Mater., 57, pp. 4614–4623. [CrossRef]
Li, Y. W., Zhou, X. L., and Li, F. X., 2010, “Temperature Dependent Mechanical Depolarization of Ferroelectric Ceramics,” J. Phys. D: Appl. Phys., 43, p. 175501. [CrossRef]
Taylor, G. I., 1938, “Plastic Strain in Metals,” J. Inst. Met., 62, pp. 307–324.
Noheda, B., Cox, D. E., Shirane, G., Guo, R., Jones, B., and Cross, L. E., 2001, “Stability of the Monoclinic Phase in the Ferroelectric Perovskite PbZr1-xTixO3,” Phys. Rev. B, 63, p. 014103. [CrossRef]
Zhou, D. Y., Kamlah, M., and Munz, D., 2005, “Effects of Uniaxial Prestress on the Ferroelectric Hysteretic Response of Soft PZT,” J. Eur. Ceram. Soc., 25, pp. 425–432. [CrossRef]
Li, Y. W., and Li, F. X., 2010, “Large Anisotropy of Fracture Toughness in Mechanically Poled/Depoled Ferroelectric Ceramics,” Script. Mater., 62, pp. 313–316. [CrossRef]
Berlincourt, D. A., Cmolik, C., and Jaffe, H., 1960, “Piezoelectric Properties of Polycrystalline Lead Titanate Zirconate Compositions,” Proc. Inst. Radio Eng., 48, pp. 220–229. [CrossRef]
Berlincourt, D. A., and Krueger, H. H. A., 1959, “Domain Processes in Lead Titanate Zirconate and Barium Titanate Ceramics,” J. Appl. Phys., 30, pp. 1804–1810. [CrossRef]
King-Smith, R. D., and Vanderbilt, D., 1993, “Theory of Polarization of Crystalline Solids,” Phys. Rev. B, 47, pp. 1651–1654. [CrossRef]
Resta, R., 1993, “Macroscopic Electric Polarization as a Geometric Quantum Phase,” EPL, 22, pp. 133–138. [CrossRef]
King-Smith, R. D., and Vanderbilt, D., 1994, “First-Principles Investigation of Ferroelectricity in Perovskite Compounds,” Phys. Rev. B, 49, pp. 5828–5844. [CrossRef]
Meyer, B., and Vanderbilt, D., 2001, “Ab Initio Study of BaTiO3 and PbTiO3 Surfaces in External Electric Fields,” Phys. Rev. B, 63, p. 205426. [CrossRef]
Hong, J. W., Catalan, G., Fang, D. N., Artacho, E., and Scott, J. F., 2010, “Topology of the Polarization Field in Ferroelectric Nanowires From First Principles,” Phys. Rev. B, 81, p. 172101. [CrossRef]
Neaton, J. B., and Rabe, K. M., 2003, “Theory of Polarization Enhancement in Epitaxial BaTiO3/SrTiO3 Superlattices,” Appl. Phys. Lett., 82, pp. 1586–1588. [CrossRef]
Bousquet, E., Dawber, M., Stucki, N., Lichtensteiger, C., Hermet, P., Gariglio, S., Triscone, J.-M., and Ghosez, P., 2008, “Improper Ferroelectricity in Perovskite Oxide Artificial Superlattices,” Nature, 452, pp. 732–736. [CrossRef] [PubMed]
Hao, F., Hong, J., and Fang, D., 2011, “Size Effect of Elastic and Electromechanical Properties of BaTiO3 Films From First-Principles Method,” Integr. Ferroelectrics, 124, pp. 79–86. [CrossRef]
Spanier, J. E., Kolpak, A. M., Urban, J. J., Grinberg, I., Lian, O. Y., Yun, W. S., Rappe, A. M., and Park, H., 2006, “Ferroelectric Phase Transition in Individual Single-Crystalline BaTiO3 Nanowires,” Nano Lett., 6, pp. 735–739. [CrossRef] [PubMed]
Fu, H., and Bellaiche, L., 2003, “Ferroelectricity in Barium Titanate Quantum Dots and Wires,” Phys. Rev. Lett., 91, p. 257601. [CrossRef] [PubMed]
Benedek, N. A., and Fennie, C. J., 2011, “Hybrid Improper Ferroelectricity: A Mechanism for Controllable Polarization-Magnetization Coupling,” Phys. Rev. Lett., 106, p. 107204. [CrossRef] [PubMed]
Martin, R. M., 1974, “Comment on Calculations of Electric Polarization in Crystals,” Phys. Rev. B, 9, pp. 1998–1999. [CrossRef]
Vanderbilt, D., and King-Smith, R. D., 1993, “Electric Polarization as a Bulk Quantity and Its Relation to Surface Charge,” Phys. Rev. B, 48, pp. 4442–4455. [CrossRef]
Resta, R., 1994, “Macroscopic Polarization in Crystalline Dielectrics: The Geometric Phase Approach,” Rev. Mod. Phys., 66, pp. 899–915. [CrossRef]
Resta, R., and Vanderbilt, D., 2007, “Theory of Polarization: A Modern Approach,” Physics of Ferroelectrics, Topics in Applied Physics, Springer, Berlin/Heidelberg, pp. 31–68.
Spaldin, N. A., “A Beginner's Guide to the Modern Theory of Polarization,” J. Solid State Chem. (in press).
Zhong, W., King-Smith, R. D., and Vanderbilt, D., 1994, “Giant LO-TO Splittings in Perovskite Ferroelectrics,” Phys. Rev. Lett., 72, pp. 3618–3621. [CrossRef] [PubMed]
Hong, J., and Fang, D., 2008, “Size-Dependent Ferroelectric Behaviors of BaTiO3 Nanowires,” Appl. Phys. Lett., 92, p. 012906. [CrossRef]
Hong, J., and Fang, D., 2008, “Systematic Study of the Ferroelectric Properties of Pb(Zr0.5Ti0.5)O3 Nanowires,” J. Appl. Phys., 104, p. 064118. [CrossRef]
Wang, Z. Y., Hu, J., and Yu, M. F., 2006, “One-Dimensional Ferroelectric Monodomain Formation in Single Crystalline BaTiO3 Nanowire,” Appl. Phys. Lett., 89, p. 263119. [CrossRef]
Mermin, N. D., 1979, “The Topological Theory of Defects in Ordered Media,” Rev. Mod. Phys., 51, pp. 591–648. [CrossRef]
Kogan, S. M., 1964, “Piezoelectric Effect During Inhomogeneous Deformation and Acoustic Scattering of Carriers in Crystals,” Sov. Phys. Solid State, 5, pp. 2069–2070.
Tagantsev, A. K., 1986, “Piezoelectricity and Flexoelectricity in Crystalline Dielectrics,” Phys. Rev. B, 34, pp. 5883–5889. [CrossRef]
Cross, L. E., 2006, “Flexoelectric Effects: Charge Separation in Insulating Solids Subjected to Elastic Strain Gradients,” J. Mater. Sci., 41, pp. 53–63. [CrossRef]
Catalan, G., Noheda, B., McAneney, J., Sinnamon, L., and Gregg, J., 2005, “Strain Gradients in Epitaxial Ferroelectrics,” Phys. Rev. B, 72, p. 020102. [CrossRef]
Zubko, P., Catalan, G., Buckley, A., Welche, P. R. L., and Scott, J. F., 2007, “Strain-Gradient-Induced Polarization in SrTiO3 Single Crystals,” Phys. Rev. Lett., 99, p. 167601. [CrossRef] [PubMed]
Ma, W., 2008, “A Study of Flexoelectric Coupling Associated Internal Electric Field and Stress in Thin Film Ferroelectrics,” Physica Status Solidi B, 245, pp. 761–768. [CrossRef]
Hong, J., Catalan, G., Scott, J. F., and Artacho, E., 2010, “The Flexoelectricity of Barium and Strontium Titanates From First Principles,” J. Phys.: Condens. Matter, 22, p. 112201. [CrossRef] [PubMed]
Resta, R., 2010, “Towards a Bulk Theory of Flexoelectricity,” Phys. Rev. Lett., 105, p. 127601. [CrossRef] [PubMed]
Hong, J., and Vanderbilt, D., 2011, “First-Principles Theory of Frozen-Ion Flexoelectricity,” Phys. Rev. B, 84, p. 180101. [CrossRef]
Lee, D., Yoon, A., Jang, S. Y., Yoon, J.-G., Chung, J.-S., Kim, M., Scott, J. F., and Noh, T. W., 2011, “Giant Flexoelectric Effect in Ferroelectric Epitaxial Thin Films,” Phys. Rev. Lett., 107, p. 057602. [CrossRef] [PubMed]
Catalan, G., Lubk, A., Vlooswijk, A. H. G., Snoeck, E., Magen, C., Janssens, A., Rispens, G., Rijnders, G., Blank, D. H. A., and Noheda, B., 2011, “Flexoelectric Rotation of Polarization in Ferroelectric Thin Films,” Nature Mater., 10, pp. 963–967. [CrossRef]
Lu, H., Bark, C.-W., Esque De Los Ojos, D., Alcala, J., Eom, C. B., Catalan, G., and Gruverman, A., 2012, “Mechanical Writing of Ferroelectric Polarization,” Science, 336, pp. 59–61. [CrossRef] [PubMed]
Zhou, H., Hong, J., Zhang, Y., Li, F., Pei, Y., and Fang, D., 2012, “Flexoelectricity Induced Increase of Critical Thickness in Epitaxial Ferroelectric Thin Films,” Physica B: Condens. Matter, 407, pp. 3377–3381. [CrossRef]
Zhou, H., Hong, J., Zhang, Y., Li, F., Pei, Y., and Fang, D., 2012, “External Uniform Electric Field Removing the Flexoelectric Effect in Epitaxial Ferroelectric Thin Films,” EPL, 99, p. 47003. [CrossRef]
Maranganti, R., and Sharma, P., 2009, “Atomistic Determination of Flexoelectric Properties of Crystalline Dielectrics,” Phys. Rev. B, 80, p. 054109. [CrossRef]
Ma, W., and Cross, L. E., 2006, “Flexoelectricity of Barium Titanate,” Appl. Phys. Lett., 88, p. 232902. [CrossRef]
Parker, C. B., Maria, J.-P., and Kingon, A. I., 2002, “Temperature and Thickness Dependent Permittivity of (Ba,Sr)TiO3 Thin Films,” Appl. Phys. Lett., 81, pp. 340–342. [CrossRef]
Sinnamon, L. J., Bowman, R. M., and Gregg, J. M., 2002, “Thickness-Induced Stabilization of Ferroelectricity in SrRuO3/Ba0.5Sr0.5TiO3/Au Thin Film Capacitors,” Appl. Phys. Lett., 81, pp. 889–891. [CrossRef]
Ramirez, F., Heyliger, P. R., and Pan, E., 2006, “Discrete Layer Solution to Free Vibrations of Functionally Graded Magneto-Electro-Elastic Plates,” Mech. Adv. Mater. Struct., 13, pp. 249–266. [CrossRef]
Ahn, C. H., Rabe, K. M., and Triscone, J.-M., 2004, “Ferroelectricity at the Nanoscale: Local Polarization in Oxide Thin Films and Heterostructures,” Science, 303, pp. 488–491. [CrossRef] [PubMed]
Fong, D. D., Stephenson, G. B., Streiffer, S. K., Eastman, J. A., Auciello, O., Fuoss, P. H., and Thompson, C., 2004, “Ferroelectricity in Ultrathin Perovskite Films,” Science, 304, pp. 1650–1653. [CrossRef] [PubMed]
Lee, H. N., Christen, H. M., Chisholm, M. F., Rouleau, C. M., and Lowndes, D. H., 2005, “Strong Polarization Enhancement in Asymmetric Three-Component Ferroelectric Superlattices,” Nature, 433, pp. 395–399. [CrossRef] [PubMed]
Zhu, X. H., Evans, P. R., Byrne, D., Schilling, A., Douglas, C., Pollard, R. J., Bowman, R. M., Gregg, J. M., Morrison, F. D., and Scott, J. F., 2006, “Perovskite Lead Zirconium Titanate Nanorings: Towards Nanoscale Ferroelectric ‘Solenoids'?” Appl. Phys. Lett., 89, p. 129913. [CrossRef]
Sepliarsky, M., Stachiotti, M. G., and Migoni, R. L., 1995, “Structural Instabilities in KTaO3 and KNbO3 Described by the Nonlinear Oxygen Polarizability Model,” Phys. Rev. B, 52, pp. 4044–4049. [CrossRef]
Sepliarsky, M., Stachiotti, M. G., and Migoni, R. L., 1997, “Ferroelectric Soft Mode and Relaxation Behavior in a Molecular-Dynamics Simulation of KNbO3 and KTaO3,” Phys. Rev. B, 56, pp. 566–571. [CrossRef]
Sepliarsky, M., Stachiotti, M. G., and Migoni, R. L., 2005, “Surface Reconstruction and Ferroelectricity in PbTiO3 Thin Films,” Phys. Rev. B, 72, p. 014110. [CrossRef]
Tinte, S., and Stachiotti, M. G., 2001, “Surface Effects and Ferroelectric Phase Transitions in BaTiO3 Ultrathin Films,” Phys. Rev. B, 64, p. 235403. [CrossRef]
Sepliarsky, M., Phillpot, S. R., Stachiotti, M. G., and Migoni, R. L., 2002, “Ferroelectric Phase Transitions and Dynamical Behavior in KNbO3/KTaO3 Superlattices by Molecular-Dynamics Simulation,” J. Appl. Phys., 91, pp. 3165–3171. [CrossRef]
Sepliarsky, M., and Tinte, S., 2009, “Dynamical Behavior of the Phase Transition of Strained BaTiO(3) From Atomistic Simulations,” Physica B, 404, pp. 2730–2732. [CrossRef]
Sang, Y.-L., Liu, B., and Fang, D.-N., 2008, “Strain and Size Effects on Ferroelectric Properties of BaTiO3 Nanofilms,” Chin. Phys. Lett., 25, pp. 1113–1116. [CrossRef]
Tinte, S., Stachiotti, M. G., Phillpot, S. R., Sepliarsky, M., Wolf, D., and Migoni, R. L., 2004, “Ferroelectric Properties of BaxSr1-xTiO3 Solid Solutions Obtained by Molecular Dynamics Simulation,” J. Phys.: Condens. Matter, 16, pp. 3495–3506. [CrossRef]
Wolf, D., Keblinski, P., Phillpot, S. R., and Eggebrecht, J., 1999, “Exact Method for the Simulation of Coulombic Systems by Spherically Truncated, Pairwise r(-1) Summation,” J. Chem. Phys., 110, pp. 8254–8282. [CrossRef]
Angoshtari, A., and Yavari, A., 2011, “Convergence Analysis of the Wolf Method for Coulombic Interactions,” Phys. Lett. A, 375, pp. 1281–1285. [CrossRef]
Ewald, P. P., 1921, “Die Berechnung Optischer und Elektrostatischer Gitterpotentiale,” Ann. Phys., 369, pp. 253–287. [CrossRef]
Deleeuw, S. W., Perram, J. W., and Smith, E. R., 1980, “Simulation of Electrostatic Systems in Periodic Boundary-Conditions 1. Lattice Sums and Dielectric-Constants,” Proc. R. Soc. A, 373, pp. 27–56. [CrossRef]
Padilla, J., and Vanderbilt, D., 1997, “Ab Initio Study of BaTiO3 Surfaces,” Phys. Rev. B, 56, pp. 1625–1631. [CrossRef]
Strukov, B. A., Davitadze, S. T., Taraskin, S. A., Goltzman, B. M., Shulman, S. G., and Lemanov, V. V., 2003, “Thermodynamical Properties of the Thin Polycrystalline BaTiO3 Films on Substrates,” Ferroelectrics, 286, pp. 967–972. [CrossRef]
Drezner, Y., and Berger, S., 2005, “Thermodynamic Stability of BaTiO3 Nano-domains,” Mater. Lett., 59, pp. 1598–1602. [CrossRef]
Zhang, Y. H., Hong, J. W., Liu, B., and Fang, D. N., 2010, “A Surface-Layer Model of Ferroelectric Nanowire,” J. Appl. Phys., 108, p. 124109. [CrossRef]
Liu, B., Huang, Y., Jiang, H., Qu, S., and Hwang, K. C., 2004, “The Atomic-Scale Finite Element Method,” Comput. Mech. Appl. Mech. Eng., 193, pp. 1849–1864. [CrossRef]
Liu, B., Jiang, H., Huang, Y., Qu, S., Yu, M. F., and Hwang, K. C., 2005, “Atomic-Scale Finite Element Method in Multiscale Computation With Applications to Carbon Nanotubes,” Phys. Rev. B, 72, p. 035435. [CrossRef]
Zhang, X., Hashimoto, T., and Joy, D. C., 1992, “Electron Holographic Study of Ferroelectric Domain Walls,” Appl. Phys. Lett., 60, pp. 784–786. [CrossRef]
Merz, W. J., 1954, “Domain Formation and Domain Wall Motions in Ferroelectric BaTiO3 Single Crystals,” Phys. Rev., 95, pp. 690–698. [CrossRef]
Floquet, N., and Valot, C., 1999, “Ferroelectric Domain Walls in BaTiO3: Structural Wall Model Interpreting Fingerprints in XRPD Diagrams,” Ferroelectrics, 234, pp. 107–122. [CrossRef]
Floquet, N., Valot, C. M., Mesnier, M. T., Niepce, J. C., Normand, L., Thorel, A., and Kilaas, R., 1997, “Ferroelectric Domain Walls in BaTiO3: Fingerprints in XRPD Diagrams and Quantitative HRTEM Image Analysis,” J. Phys. III, 7, pp. 1105–1128.
Hlinka, J., and Marton, P., 2006, “Phenomenological Model of a 90 deg Domain Wall in BaTiO3-Type Ferroelectrics,” Phys. Rev. B, 74, p. 104104. [CrossRef]
Tsou, N. T., Potnis, P. R., and Huber, J. E., 2011, “Classification of Laminate Domain Patterns in Ferroelectrics,” Phys. Rev. B, 83, p. 184120. [CrossRef]
Pilania, G., and Ramprasad, R., 2010, “Complex Polarization Ordering in PbTiO3 Nanowires: A First-Principles Computational Study,” Phys. Rev. B, 82, p. 155442. [CrossRef]
Jiang, B., Bai, Y., Chu, W. Y., Su, Y. J., and Qiao, L. J., 2008, “Direct Observation of Two 90 Degrees Steps of 180 Degrees Domain Switching in BaTiO3 Single Crystal Under an Antiparallel Electric Field,” Appl. Phys. Lett., 93, p. 152905. [CrossRef]
Dieguez, O., and Vanderbilt, D., 2006, “First-Principles Calculations for Insulators at Constant Polarization”, Phys. Rev. Lett., 96, p. 056401. [CrossRef] [PubMed]
Stengel, M., Spaldin, N. A., and Vanderbilt, D., 2009, “Electric Displacement as the Fundamental Variable in Electronic-Structure Calculations,” Nature Phys., 5, pp. 304–308. [CrossRef]
Hong, J., and Vanderbilt, D., 2011, “Mapping the Energy Surface of PbTiO3 in Multidimensional Electric-Displacement Space,” Phys. Rev. B, 84, p. 115107. [CrossRef]
Hong, J., and Vanderbilt, D., 2013, “Electrically Driven Octahedral Rotations in SrTiO3 and PbTiO3,” Phys. Rev. B, 87, p. 064104. [CrossRef]
McCash, K., Srikanth, A., and Ponomareva, I., 2012, “Competing Polarization Reversal Mechanisms in Ferroelectric Nanowires,” Phys. Rev. B, 86, p. 214108. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Schematic depictions of typical domain structures observed in ferroelectric (a) single crystal and (b) polycrystal ceramic

Grahic Jump Location
Fig. 2

Variations of (a) σθ and (b) Dθ on the rim of the hole subjected to tension at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 3

Variations of (a) Er and (b) Eθ on the rim of the hole subjected to tension at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 4

Variations of (a) Dθ and (b) Eθ on the rim of the hole subjected to electric displacement loading at infinity. Reprinted from Ref. [32] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 5

Stable electric field versus electric displacement hysteresis loops (the thin line is the simulation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 6

Stable longitudinal strain versus electric field curves (the thin line is the calculation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 7

The simulated D–E hysteresis loop (a) and butterfly loop (b) for an unpoled specimen with marked loading points. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 8

The whole process of domain switching under uniaxial electric loading. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 9

Domain-switching process when the applied electric field is close to the coercive field. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 10

Longitudinal strain versus uniaxial stress curves (the thin line is the calculation result and the bold line the experimental result). Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 11

Simulated depolarization versus uniaxial stress curve. Reprinted from Ref. [34] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 12

Illustrations of Newton–Raphson iterative instability in nonlinear FE models: (a) taking the electric potential as nodal variables and (b) taking the electric displacement potential as nodal variables. Reprinted from Ref. [75] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 13

High depolarization electric field and mechanical field induced by (a) 180 deg switching or ferroelectric switching and (b) 90 deg switching or ferroelastic switching. Reprinted from Ref. [75] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 14

An infinite PZT-5 H plate with circular hole

Grahic Jump Location
Fig. 15

Comparison of EFG results with analytical results: the variation of hoop (a) stress σθ and (b) dielectric displacement Dθ with angle θ. Reprinted from Ref. [78] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 16

Schematic diagram of interface crack

Grahic Jump Location
Fig. 17

The distribution of stress σyy near the crack tip [76]

Grahic Jump Location
Fig. 18

The distributions of electric field Ey around the crack tip [76]

Grahic Jump Location
Fig. 19

The electric field vector distribution near the crack tip [76]

Grahic Jump Location
Fig. 20

The variation of hoop stress around the crack tip when different electromechanical loadings are applied [76]

Grahic Jump Location
Fig. 21

The engineering domain structures with both 90 deg and 180 deg domain walls: (a) calculated by the phase field simulation and (b) observed by experiments in BaTiO3 single crystal. The arrow represents the polarization direction. Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 22

The distributions of electrostatic potential (a) and oxygen vacancy density (b) in the rank-2 domain structure. The initial oxygen vacancy density is uniformly distributed with a magnitude of Nd = 1 × 1024 m–3. Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 23

The electric hysteresis loops (a) and butterfly loops (b) of rank-2 domain structures with different oxygen vacancy densities, and the domain evolutions for Nd = 1.2 × 1024 m–3 under a positive alternating field (c). Domains indexed by (I) and (II) in (c) correspond to the two states (I) and (II) in (a) and (b). Reprinted from Ref. [45] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 24

Two extreme cases simulated by the phase field method: (a) The domain pattern of a ultrathin film calculated by the phase field method with 2.4 nm < h < 4.8 nm [46]; (b) the domain pattern of the ultrathin film calculated by molecular dynamics with h = 4.4 nm [152]; (c) the domain pattern of a thin film calculated by the phase field method with h > 20 nm without considering the out-of-plane depolarization [46]; (d) the domain pattern of the thin film calculated by the 2D phase field simulation without concerning the size effect [141]. The symbol ⊙ (⊗) denotes the direction of polarization flowing out of (into) the x1x2 surface. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 25

The size-dependent distribution of the out-of-plane polarization at different thicknesses. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 26

The size-dependent out-of-plane polarization and domain pattern. The domain exhibits a zigzag pattern with eight variants coexisting when h > 7.6 nm, a zigzag pattern with four variants coexisting when 6.8 nm < h < 8.0 nm, a vortex pattern with four variants coexisting when 4.4 nm < h < 7.2 nm, and a stripe pattern with two variants coexisting when 2.4 nm < h < 4.8 nm; when the thickness is smaller than 2.8 nm, the domain disappears, indicating that the ferroelectricity is suppressed. Reprinted from Ref. [46] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 27

2D Illustration of material model for morphotropic ferroelectric ceramics

Grahic Jump Location
Fig. 28

(a) Six types of tetragonal domains and (b) eight types of rhombohedral domains in ferroelectric crystals

Grahic Jump Location
Fig. 29

Simulated D-E hysteresis loops (a) and butterfly curves (b) of tetragonal PZT ceramics during the first three cycles of electric loading. The overlapped curves after 1.5 cycles loading indicates the good convergence of the proposed model. Reprinted from Ref. [163] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 30

Simulated (a) polarization and (b) strain curves of tetragonal and rhombohedral PZT ceramics. Reprinted from Ref. [53] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 31

(a) (001) pole figure of tetragonal PZT and (b) (111) pole figures of rhombohedral PZT ceramics during cyclic electric loading. Reprinted from Ref. [53] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 32

Switching strain curves of tetragonal and rhombohedral PZT ceramics under uniaxial tension/compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 33

(a) (001) pole figures of tetragonal PZT and (b) (111) pole figures of rhombohedral PZT ceramics during uniaxial tension/compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 34

(a) Switching strain-stress curves and (b) the cumulative pole figures of (001) and (111) axes in morphotropic PZT ceramics under uniaxial tension and compression. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 35

Comparison of switching strain versus stress curves of tetragonal, rhombohedral, and MPB PZT ceramics (the nominal coercive stress σC = 70 MPa). Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 36

Simulated switching strain versus stress curves of tetragonal, rhombohedral, and MPB PZT ceramics using a small nominal coercive stress of σC = 5 MPa. Reprinted from Ref. [52] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 37

(a) D-E hysteresis loops, (b) butterfly curves, and (c) reversed butterfly curves of a morphotropic PZT ceramic under electric loading with different levels of precompression. Reprinted from Ref. [54] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 38

The cumulative pole figures of [001] and [111] axes of a morphotropic PZT ceramic with different levels of precompression: (a) before electric loading; (b) under maximum electric field; (c) after removing the electric field. Reprinted from Ref. [54] with the kind permission of The American Institute of Physics.

Grahic Jump Location
Fig. 39

The simulated hysteresis (left) and butterfly (right) loops of PZT ceramics. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 40

Evolution of March coefficient of PZT ceramics in tetragonal (left) and rhombohedral (right) phases during electric poling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 41

Comparison between simulation and experiment to hysteresis (left) and butterfly (right) loops of soft PLZT during electric poling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 42

Depolarization (left) and stress–strain (right) curves of PZT ceramics during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 43

Evolution of March coefficient of PZT ceramics in tetragonal (left) and rhombohedral (right) phases during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 44

Comparison between simulation and experiment of depolarization (left) and stress-strain (right) curves of soft PLZT during mechanical depoling. Reprinted from Ref. [51] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 45

Calculation of the polarization for different unit cells in one-dimensional chain of alternating anions and cations. In the top panel, anions and cations are spaced a distance a/2 apart (a is the lattice constant). In the bottom panel, the cations are displaced by a distance Δ. The dashed rectangles indicate unit cells that are used for calculation of the polarization (see text).

Grahic Jump Location
Fig. 46

The largest balls are for Ba ions, the intermediate ones represent Ti ions, and the smallest ones are for O. (a) is TiO2 terminated; (b) is BaO terminated, except the edges, where the corner Ba atoms have been removed; (c) is stoichiometric, with two BaO surfaces and two TiO2 ones. (d) is BaO terminated. Reprinted from Ref. [177] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 47

Left column (a)–(d): Ti off-center displacements for the four respective nanowires in Fig. 46. (e) is similar to (d) but for a thinner wire. The right column sketches the corresponding 2D field lines around topological point defects of winding numbers n = 1, 1, −1, and −3, respectively, (j) showing the n = −3 decomposition into a central n = +1 and four n = −1 defects. The respective largest arrows correspond to (a) 21 pm, (b) 35 pm, (c) 23 pm, (d) 6.8 pm, and (e) 6.6 pm off-centering displacement. Reprinted from Ref. [177] with the kind permission of American Physical Society.

Grahic Jump Location
Fig. 48

Supercell and strain profile used in the calculations. For BaTiO3, the unit cell contains only one formula unit (a), while for SrTiO3, due to its antiferrodistortive transition at low temperature, the in-plane size of the unit cell is doubled (b). (c) and (d) supercell in xz plan, (e) cosine gradient strain in the supercell, (f) atom displacement in the supercell. Reprinted from Ref. [200] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 49

Longitudinal FEC μ3333 of rhombohedral BaTiO3 for different supercell sizes (N = 6, 10, 14) under various strain gradients (εmax = 0.5%, 1.0%, 1.5%, 2.0%). In each panel, the line with square symbols (up) is for the applied strain, the line with dot symbols (down) is for the “relaxed strain” (see text). The two values converge as the supercell size is increased. Reprinted from Ref. [200] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 50

Size effect of elastic stiffness C11 for BaTiO3 films. Reprinted from Ref. [180] with the kind permission of Taylor & Francis.

Grahic Jump Location
Fig. 51

The piezoelectric properties of BaTiO3 bulk and thin films, e33 for bulk and e11 for films of n = 2 and 4. Reprinted from Ref. [180] with the kind permission of Taylor & Francis.

Grahic Jump Location
Fig. 52

Two different representative unit cells of BaTiO3

Grahic Jump Location
Fig. 53

(a) The cell-by-cell out-of-plane polarization profile of a z-direction chain in film with different thicknesses. (b) Schematic side views of the out-of-plane polarization pattern for film in which h ≥ 3.2 nm. (c) Schematic side views of the out-of-plane polarization pattern for film in which h ≤ 2.8 nm. Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 54

The evolution of domains in the 4.4-nm-thick film under the external electric field. (a) and (b): The average polarization of the film as a function of the external electric field. (c) ∼ (k): The sequent top views of the out-of-plane polarization patterns under different electric fields marked in (a) and (b). The value in each grid represents the average z-direction polarization over a chain along z-direction (unit: μC/cm2). Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 55

The hysteresis loops (a) of BaTiO3 films with thicknesses of 4.4 nm, 3.6 nm, and 2.8 nm and the variations of remnant polarization (b) and coercive field (c) with the reciprocal of film thickness. Reprinted from Ref. [152] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 56

(a) Out-of-plane average polarization as a function of temperature for stress-free BaTiO3 nanofilms with different thicknesses, and (b) the Curie temperature as a function of the reciprocal of film thickness. Reprinted from Ref. [64] with the kind permission of American Scientific Publishers.

Grahic Jump Location
Fig. 57

The size-dependent polarization components of BaTiO3 nanowire. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 58

The size-dependent ferroelectric hysteresis loops of BaTiO3 nanowires. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 59

Hysteresis loop of nanowire with diameter d = 2.0 nm and the domain structures at different electrical loading levels. Reprinted from Ref. [61] with the kind permission of Institute of Physics (IOP) Publishing.

Grahic Jump Location
Fig. 60

A comparison of computational efficiency between the developed AFEM and MD method. The inset table demonstrates the ratio of time cost in the MD to that in the AFEM. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 61

Three types of stable nanoscale domain structure: (a) single domain; (b) 90° domain; (c) vortex domain. The three figures above illustrate the polarization distributions with each arrow representing the polarization of a unit cell, while the ones below demonstrate the schematic domain patterns. Notice that only 1/4 area of each figure are calculated by the present AFEM, and based on the periodical boundary conditions, four identical patterns are packed together to obtain a better image of the polarization topology. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Grahic Jump Location
Fig. 62

Evolution of single domain structure under electric fields with different loading directions. Reprinted from Ref. [88] with the kind permission of Elsevier Science Ltd.

Tables

Errata

Discussions

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In