3.3 Lattice Models - Minimized Free Energy

The minimized free energy model concentrates on the double well structure of distorted perovskite structures. A one-dimensional lattice model was developed by Omura et al [OAI91], based on previous work on isomorphous ferroelectric phase transition by Ishibashi, which was published in 1992 [Ish92]. It relies on the minimization of the free energy which is modeled by a fourth order approach for the energy of the center ion stemming from the lattice cell and a coupling term describing the interaction between the ions of neighboring cells:

$$f= \sum_{n=1}^{N} {\frac{\alpha}{2}\cdot {p_n}^2 + \frac{\be... ... {p_n}^4 + \frac{\kappa}{2}\cdot(p_n-p_{n-1})^2- p_n \cdot E}.$$ (3.5)

$p_n$ is the dipole moment of the $n$th dipole, $\kappa$ is the coupling coefficient. $\alpha$ is a function of the temperature, namely

$$\alpha=a\cdot(T-T_0),\hspace{2cm} a>0.$$ (3.6)

$T_0$ is the Curie Temperature. Consequently

$$\alpha < 0$$ (3.7)

is true for the ferroelectric phase of the material, and finally it is assumed that

$$\beta > 0.$$ (3.8)

The overall polarization is given by

$$P=\sum_{n=1}^N p_n.$$ (3.9)

The model assumes the existence of permanent dipoles with random distribution (Fig. 3.14). These dipoles serve as centers for the nucleation of domains.

The two allowed dipole moments are restricted by the model to $p_n \geq 1$ for the positive nuclei and $p_n \leq -1$ for the negative ones.

Figure 3.14: One-dimensional lattice model, atoms in the double minimum potential interact with their neighbors, cyclic boundary condition $p_1=p_{n+1}$

The simulation starts at an equilibrium state where all the 'free' dipoles are negatively polarized and the time-dependent expansion of the domains that occurs around the nucleation centers is examined. Therefore a viscosity coefficient $\gamma$ is introduced which takes into account the switching delay of the individual dipoles. The resulting evolution of polarization is given by the Landau Khalatnikov kinetic equation [Bau99]

$$\gamma \cdot \frac{\mathrm{d}p_n}{\mathrm{d}t}=-\frac{\partial f}{\partial p_n}$$ (3.10)

and finally with (3.5) as

$$\gamma \cdot \frac{\mathrm{d}p_n}{\mathrm{d}t}= - \{\alpha \... ...ot {p_n}^3 - \kappa\cdot(p_{n+1}- 2 \cdot p_n + p_{n-1})-E\}.$$ (3.11)

Simulation is carried out by application of a time dependent signal

$$E=f(t)$$ (3.12)

and numerical solution of the resulting set of (3.11). The typical shape of the resulting hysteresis is plotted in Fig. 3.15.

Figure 3.15: Hysteresis of the free energy model

This approach was extended to two-dimensional lattice structures by Omura et al. [OAI92] in 1992. The functional for the free energy had to be modified in order to include the increased number of possible coupling partners as follows

$$f= \sum_{m,n}^{M,N} {\{\frac{\alpha}{2}\cdot {p_{m,n}}^2 + \frac{\beta}{4} \cdot {p_{m,n}}^4 + \frac{\kappa_1}{2}\cdot(p_{m,n}-p_{m-1,n})^2}$$ (3.13)

$$+ \frac{\kappa_2}{2}\cdot(p_{m,n}-p_{m,n-1})^2 + \frac{\kappa_1}{2}\cdot(p_{m,n}-p_{m-1,n-1})^2 - p_{m,n} \cdot E\}.$$ (3.14)

The Landau Khalatnikov equation reads then

$$\gamma \cdot \frac{\mathrm{d}p_{m,n}}{\mathrm{d}t}=-\frac{\partial f}{\partial p_{m,n}}.$$ (3.15)

This first approach did not consider the influence of the depolarization field. Instead the electric field was assumed to be constant in the entire simulation area. The influence of depolarization was finally added by Baudry in 1999 [Bau99], who implemented Poisson's equation into the system.

The basic intent of the free energy method is to gain insight into the material properties and the main focus is the correct qualitative reproduction of physical effects. Typical simulation setups analyze areas of a size of about $50\times50$ dipoles.