Ginzburg-Landau (GL) Theory is a conceptual bridge between
& 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
(b) ferroelectric domains (containing electric dipoles) in the case of a
(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).
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.