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Introduction
Ginzburg-Landau (GL) Theory is a conceptual bridge between microscopic models & observed macroscopic phenomena. At its most basic level, it models the domain structure of a material, based on an order parameter. This parameter may refer to the: (a) magnetic domains (containing magnetic dipoles) in the case of a magnetic/ferromagnetic material, (b) ferroelectric domains (containing electric dipoles) in the case of a ferroelectric material, (c) the domains in superconducting materials, or (d) the strains in a martensitic transformation.

Domain Evolution in 2D of Ferroelectric Material without Depolarization Field
The TDGL equations drive a ferroelectric system from an initial state (with polarization equal to zero plus a small random fluctuation) to the free energy minima. The image on the left shows the domain state of a 2D system with 2 degrees of freedom (Polarizations: Px & Py) after 0, 5000, 10000, 20000, 50000 and 100000 time steps (the display updates at approx. 1 second intervals... click to load gif file), using periodic boundary conditions.
Domain Evolution in 2D of Ferroelectric Material with Depolarization Field
The image on the left shows the effect of adding the depolarization field, obtained by adding the restriction of the Maxwell Equation, div(D)=0, at 4000 to 30000 time steps (the display updates at approx. 0.5 second intervals... click to load gif file). The blue regions represent large positive values of the electric potential (of the depolarization field), while the red regions represent large negative values of the electric potential. The black regions represent zero electric potential of the depolarization field, which can be seen primarily at the regions of the grain boundaries.
Domain Evolution in 2D of Ferroelectric Material with Depolarization Field and Elastic Compatibility
The next step involves the addition of the elastic energy term into the free energy expression, with the displacements subject to the elastic compatibility constraint, Sxx,x + Sxy,y = 0 & Sxy,x + Syy,y = 0. The images on the left show the domain evolution for a 128x64 grid under stress free conditions (top image), and clamped conditions (bottom image). The frames for the gif animation are extracted from the 200th to 10000th time step in intervals of 200 time steps (click to load gif file).
Hysteresis Loops
The hysteresis loops of ferroelectric material are also be modeled using TDGL theory. A typical result is shown on the left.

Publications
based on this work
1. Ng, Nathaniel, Rajeev Ahluwalia, Haibin Su, and Freddy Boey. “Lateral size and thickness dependence in ferroelectric nanostructures formed by localized domain switching.” Acta Materialia 57, no. 7 (April 2009): 2047-2054. [http://dx.doi.org/10.1016/j.actamat.2008.10.022]
2. Nguyen, C. A., P. S. Lee, N. Ng, H. Su, S. G. Mhaisalkar, J. Ma, and F. Y. C. Boey. “Anomalous polarization switching in organic ferroelectric field effect transistors.” Applied Physics Letters 91, no. 4 (July 23, 2007): 042909-3. [http://dx.doi.org/10.1063/1.2757092]


Videos from Publication 1
These are simulations showing that fringing electric fields at the edge of an electrode can influence the switching mechanism towards nucleation via 90 degree domain wedges.

Phase field simulation of the switching process in a ferroelectric - PFM-type geometry


Phase field simulation of the switching process in a ferroelectric - capacitor geometry



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