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The effect of long-range elastic interactions on the toroidal moment of polarization in a ferroelectric nanoparticle
The effect of long-range (LR) elastic interactions on the toroidal moment of polarization in a two-dimensional ferroelectric particle is investigated using a phase field model. The phase field simulations exhibit vortex patterns with purely toroidal moments of polarization and negligible macroscopic polarization when the spontaneous strains are low and the simulated ferroelectric size is small. However, a monodomain structure with a zero toroidal moment of polarization is formed when the spontaneous strains are high in small simulated ferroelectrics, indicating that, because of the LR elastic interactions, high values of spontaneous strains hinder the formation of polarization vortices in ferroelectric particles. Applied Physics Letters 88, 182904 (2006)
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This kind of domain patterns
This kind of domain patterns have already been observed in an early work Prof. Zhigang Suo and I dis. See doi:10.1115/1.1469000. We found out that this phenomenon is rather a consequence of the parameters chosen in the Landau free energy. In other words, whether strained or strain-freee do not qualitatively change this type of "toroidal moment". Perhaps I'm missing some subtly here.
Dear Yanfei, Thanks for your
Dear Yanfei,
Thanks for your comments. I have read your paper. The simulation model in our paper is different from that of the paper you and Prof. Zhigang Suo. First, we study free-standing particles. There is no external mechanical constraint. The stresses in the particle are generated by inhomogeneous spontaneous strains, which are similar to the thermal stresses induced by an inhomogeneous temperature field. The boundaries of the simulated model are traction free, not strained at all. Secondly, the formation of domain patterns in a nonperiodic ferroelectric system is more complex than that of a periodic one. There exists much more depolarization energy in the nonperiodic system under open-circuit condition. The competition between depolarization energy and elastic energy is the main driving force to determine which kind of domain patterns will be formed in the nonperiodic ferroelectric system. The vortex pattern has minimum depolarization energy and maximum elastric energy. By contraries, the single domain pattern has minimum elastic energy and maximum depolarization energy.
Jie