Abstract
This paper reviews the previous work on phase field simulations of low-dimensional ferroelectrics of two-dimensional epitaxial ferroelectric islands, thin films, and nanoparticles. The simulations are conducted in real space with exact boundary conditions of a low-dimensional ferroelectric, but consuming a much longer simulation time in comparison with that conducted in Fourier space. For ferroelectric islands and thin films, the simulations exhibit spatial polarization distributions with different types of domain walls and find two critical thicknesses, at which the simulated material changes from a multidomain state to a single-domain state and from ferroelectric phase to paraelectric phase, respectively. The remanent polarization and the coercive field of the simulated ferroelectric films both decrease with decreasing film thickness. The simulations exhibit vortex patterns of polarizations, which have purely toroidal moments of polarizations and macroscopically negligible averaged polarizations, in stress-free nanoparticles when long-range electrostatic interactions are fully taken into account. However, a single-domain structure without any toroidal moment of polarizations is formed in small nanoparticles with strong long-range elastic interactions.
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Jiang B., Peng J.L., Bursill L.A., Zhong W.L.: Size effects on ferroelectricity of ultrafine particles of PbTiO3. J. Appl. Phys. 87, 3462–3467 (2000)
Zhong W.L., Wang Y.G., Zhang P.L., Qu B.D.: Phenomenological study of the size effect on phase-transitions in ferroelectric particles. Phys. Rev. B 50, 698–703 (1994)
Naumov I.I., Bellaiche L., Fu H.: Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737–740 (2004)
Wang Y.G., Zhong W.L., Zhang P.L.: Surface and size effects on ferroelectric-films with domain-structures. Phys. Rev. B 51, 5311–5314 (1995)
Cheng J., Wang B., Du S.: Effective electroelastic properties of polycrystalline ferroelectric ceramics predicted by a statistical model. Acta Mech. 138, 163–175 (1999)
Wang J., Shi S.Q., Chen L.Q., Li Y.L., Zhang T.Y.: Phase field simulations of ferroelectric/ferroelastic polarization switching. Acta. Mater. 52, 749–764 (2004)
Wang J., Zhang T.Y.: Effect of long-range elastic interactions on the toroidal moment of polarization in a ferroelectric nanoparticle. Appl. Phys. Lett. 88, 182904 (2006)
Li Y.L., Hu S.Y., Liu Z.K., Chen L.Q.: Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films. Acta Mater. 50, 395–411 (2002)
Nambu S., Sagala D.A.: Domain formation and elastic long-range interaction in ferroelectric perovskites. Phys. Rev. B 50, 5838–5847 (1994)
Hu H.L., Chen L.Q.: Three-dimensional computer simulation of ferroelectric domain formation. J. Am. Ceram. Soc. 81, 492–500 (1998)
Cao W., Tavener S., Xie S.: Simulation of boundary condition influence in a second-order ferroelectric phase transition. J. Appl. Phys. 86, 5739–5746 (1999)
Zhang W., Bhattacharya K.: A computational model of ferroelectric domains. Part I: model formulation and domain switching. Acta Mater. 53, 185–198 (2005)
Wang J., Zhang T.Y.: Size effects in epitaxial ferroelectric islands and thin films. Phys. Rev. B 73, 144107 (2006)
Wang J., Kamlah M., Zhang T.Y., Li Y., Chen L.Q.: Size-dependent polarization distribution in ferroelectric nanostructures: phase field simulations. Appl. Phys. Lett. 92, 162905 (2008)
Wang J., Kamlah M., Zhang T.Y.: Phase field simulations of ferroelectric nanoparticles with different long-range-electrostatic and -elastic interactions. J. Appl. Phys. 105, 014104 (2009)
Lines M.E., Glass A.M.: Principles and Applications of Ferroelectrics and Related Materials. Clarendon press, Oxford (1977)
Mueller R., Gross D., Schrade D., Xu B.X.: Phase field simulation of domain structures in ferroelectric materials within the context of inhomogeneity evolution. Int. J. Fract. 147, 173–180 (2007)
Zhang T.Y.: Strained ferroelectric thin films. Int. J. Appl. Mech. 1, 21–40 (2009)
Roelofs A., Schneller T., Szot K., Waser R.: Piezoresponse force microscopy of lead-titanate nanograins possibly reaching the limit of ferroelectricity. Appl. Phys. Lett. 81, 5231–5233 (2002)
Ahn C.H., Rabe K.M., Triscone J.-M.: Ferroelectricity at the nanoscale: local polarization in oxide thin films and heterostructures. Science 303, 488–491 (2004)
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Wang, J., Kamlah, M. & Zhang, TY. Phase field simulations of low-dimensional ferroelectrics. Acta Mech 214, 49–59 (2010). https://doi.org/10.1007/s00707-010-0322-9
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DOI: https://doi.org/10.1007/s00707-010-0322-9