2.4.1 Analysis of Domain Configuration

As a consequence of the above considerations, a domain configuration evolves, which minimizes the total free energy $G$. The free energy includes the two entries outlined above, the energy of the depolarization field $W_E$, stemming from the surface, and the energy of the domain walls $W_W$. With the energy term related to the shift of the ions $W_\mathrm{dip}$, it finally reads as

$$G = W_\mathrm{dip} + W_E +W_W.$$ (2.6)

The exact calculation of the energy term related to the dipoles, $W_\mathrm{dip}$, is far from trivial. A simplified model that allows at least a rough analysis will be introduced later in Section 3.3.

The depolarization energy $W_E$ depends on the crystal and domain geometry at the crystal surfaces. If the geometry is rather simple, analytic calculations are possible, thus giving a good insight into the mechanisms involved in the formation of domain structures. For this consideration it is assumed that the currents inside the ferroelectric material are neglectable and accordingly the surface charge is not compensated.

In the simplest case of a thin crystal with uniform polarization perpendicular to the surface, it is not surprising that the electric depolarization field is proportional to the polarization [LG96]

$$E=-\frac{1}{\epsilon_0} P.$$ (2.7)

Figure 2.13: Periodic domain structure

Things get more complicated for multi domain structures. Still for a periodic domain structure as outlined in Fig. 2.13 an analytic description is possible. One obtains

$$W_E= \frac{\epsilon^* \cdot d \cdot P_0 \cdot V}{t_\mathrm{cryst}}$$ (2.8)

for the depolarization energy, where $d$ is the domain width, $t_\mathrm{cryst}$ is the crystal thickness, $P_0$ is the polarization in the center of a domain, $V$ is the crystal volume, and $\epsilon^*$ is a coefficient depending on the dielectric permittivity [MF53]. Using the domain wall energy per unit area $\sigma_\mathrm{Wall}$, the overall domain energy $W_W$ in Fig. 2.13 is

$$W_W=(\sigma_\mathrm{Wall}/d)\cdot V.$$ (2.9)

The wall energy itself consists of several independent contributions. These are the depolarization energy stemming from $\mathrm{div }\vec{P}$ at the domain boundaries, the dipolar energy caused by the misalignment of the ferroelectric dipoles on both sides of the domain wall and the elastic energy.

The minimum of these two entries is

$$W_E +W_W = \frac{\epsilon^* \cdot d \cdot P_0 \cdot V}{t} + (\sigma_\mathrm{Wall}/d)\cdot V,$$ (2.10)

and with respect to the domain width one reads

$$d=\sqrt{\frac{\sigma_\mathrm{Wall} \cdot t_\mathrm{cryst}}{\epsilon^* \cdot{P_0}^2}}.$$ (2.11)

Even this simple analysis delivers remarkable results. First, the resulting domain width is finite if the polarization in the material is not zero. This effect prevents any finite structure from showing a uniform polarization. The next remarkable result is that neither the energy of a domain wall nor the domain width can be zero. Finally, this states that a pure random distribution of the orientations of the dipoles is not possible, as the area between two of them already has to be regarded as a domain wall. This leads to the strong result that there have to be domains of finite size.