Low Energy Periodic Microstructure in Ferroelectric Single Crystals

Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 24)

Abstract

Two distinct modelling approaches are used to find minimum energy (equilibrium) microstructural states in tetragonal ferroelectric single crystals. The first approach treats domain walls as sharp interfaces and uses analytical solutions of the compatibility conditions at domain walls to identify multi-rank laminate microstructures that are free of residual stress and electric field. The second method treats domain walls as diffuse interfaces, using a phase-field model in 3-dimensions. This is computationally intensive, but takes the full field equations into account and allows a more general class of periodic microstructure to be explored. By searching for minimum energy configurations of a cube of tetragonal material, candidate unit cells of a periodic microstructure are identified. Adding periodic boundary conditions allows the assembly of the unit cells into a macro-structure of low energy. A noteworthy structure identified in this way is a “hexadomain” vortex consisting of six tetragonal domains meeting along the major diagonal of a cube. Several of the structures identified by the phase-field model are found to be special cases of multi-rank laminate structure. Thus the analytical approach offers a fast method for finding equilibrium microstructures, while the phase-field model provides a validation of these solutions.

Keywords

Domain Wall Barium Titanate Polarization Vortex Diffuse Interface Periodic Assembly 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgements

J.E.H. Acknowledges support of EPSRC project No. EP/E026095/1. I.M. gratefully acknowledges the Karlsruhe House of Young Scientists for awarding a research fellowship to support his stay at the University of Oxford.

References

  1. 1.
    Ahluwalia R, Cao W (2001) Size dependence of domain patterns in constrained ferroelectric system. J Appl Phys 89:8105CrossRefGoogle Scholar
  2. 2.
    Arlt G, Sasko P (1980) Domain configuration and equilibrium size of domains in BaTiO3ceramics. J Appl Phys 51:4956CrossRefGoogle Scholar
  3. 3.
    Ball JM, James RD (1987) Fine phase mixtures as minimizers of energy. Arch Rat Mech Anal 100:13MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bhattacharya K (1993) Comparison of the geometrically nonlinear and linear theories of martensitic transformation. Cont Mech Thermo 5:205MATHCrossRefGoogle Scholar
  5. 5.
    Chen L-Q (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32:113CrossRefGoogle Scholar
  6. 6.
    Gruverman A, Wu D, Fan HJ, Vrejoiu I, Alexe M (2008) Vortex ferroelectric domains. J Phys Condens Matter 20:342201CrossRefGoogle Scholar
  7. 7.
    Hu H-L, Chen L-Q (1998) Three-dimensional computer simulation of ferroelectric domain formation. J Am Ceram Soc 81:492CrossRefGoogle Scholar
  8. 8.
    Jona F, Shirane G (1962) Ferroelectric crystals. Pergamon, OxfordGoogle Scholar
  9. 9.
    Kay HF (1948) Preparation and properties of crystals of barium titanate BaTiO3. Acta Cryst 1:229CrossRefGoogle Scholar
  10. 10.
    Kontsos A, Landis CM (2009) Computational modeling of domain wall interactions with dislocations in ferroelectric crystals. Int J Solids Struct 46:1491MATHCrossRefGoogle Scholar
  11. 11.
    Li JY, Liu D (2004) On ferroelectric crystals with engineered domain configurations. J Mech Phys Solids 8:1719CrossRefGoogle Scholar
  12. 12.
    Münch I, Huber JE (2009) A hexadomain vortex in tetragonal ferroelectrics. Appl Phys Lett 95:022913CrossRefGoogle Scholar
  13. 13.
    Naumov II (2004) Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432:737CrossRefGoogle Scholar
  14. 14.
    Prosandeev S, Ponomareva I, Naumov I, Kornev I, Bellaiche L (2008) Original properties of dipole vortices in zero-dimensional ferroelectrics. J Phys Condens Matter 20:193201CrossRefGoogle Scholar
  15. 15.
    Shu YC, Bhattacharya K Domain patterns and macroscopic behaviour of ferroelectric materials. Philos Mag B 81:2021Google Scholar
  16. 16.
    Shu YC, Yen JH (2007) Pattern formation in martensitic thin films. Appl Phys Lett 91:021908CrossRefGoogle Scholar
  17. 17.
    Su Y, Landis CM (2007) Continuum thermodynamics of ferroelectric domain evolution: Theory, finite element implementation, and application to domain wall pinning. J Mech Phys Solids 55:280MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Tsou NT, Huber JE (2009) Construction of compatible microstructures for tetragonal ferroelectric single crystals. Proc SPIE 7289:72890BCrossRefGoogle Scholar
  19. 19.
    Wang J, Shi S-Q, Chen L-Q, Li Y, Zhang T-Y (2004) Phase field simulations of ferroelectric/ferroelastic polarization switching. Acta Mat 52:749CrossRefGoogle Scholar
  20. 20.
    Wang J, Kamlah M, Zhang T, Li Y, Chen L (2008) Size-dependent polarization distribution in ferroelectric nanostructures: Phase field simulations. Appl Phys Lett 92:162905CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Department of Engineering ScienceUniversity of OxfordOxfordUK
  2. 2.Karlsruhe Institute of TechnologyUniversität KarlsruheKarlsruheGermany

Personalised recommendations