2.4 Domains

Domains represent volumes of the same orientation of the dipoles. Similar to magnetism, domain structures also arise in ferroelectric materials, but of course the proper equations are different, and furthermore, the existence of free electric charge has consequences for the long time behavior of the material.

To allow an analysis of this subject, the material equation (2.1) and Poisson's equation

$$\mathrm{div }{\vec{D}} = \rho$$ (2.2)

are transformed with respect to the divergence of the electric field and the ferroelectric polarization $P_\mathrm{Ferro}$

$$\mathrm{div }{\vec{E}} = \frac{1}{\epsilon_r \cdot \epsilon_0}\cdot(\rho - \mathrm{div }\vec{P}_\mathrm{Ferro}).$$ (2.3)

In an infinite crystal the ferroelectric polarization $\vec{P}_\mathrm{Ferro}$ is uniform and $\mathrm{div }\vec{E} = \rho/\epsilon_r \epsilon_0$, as in non-ferroelectric dielectrics. In finite ferroelectrics, properties are more complicated. At the surface $P_\mathrm{Ferro}$ is reduced to zero, while in the neighborhood of defects $\mathrm{div }\vec{P}_\mathrm{Ferro}$ does not vanish and acts, according to (2.3), as the source of an electric field, the so-called depolarization field.

The depolarization energy $W_E$ plays an important role in the formation of the domains. When a crystal cools down from the paraelectric phase in the absence of fields, there is, as outlined previously, only a limited number of possible directions of the spontaneous ferroelectric polarization. In order to minimize the free energy, different regions polarize in one of those directions, thus forming the domain structure. If no electric field is applied, this structure usually shows no net polarization in a virgin crystal.

The other important contribution to the domain layout is the energy of the domain walls $W_W$. The final configuration will minimize the sum of both of these entries to the total energy.

Basically two different types of domain walls are common in ferroelectrics. The actual formation depends on the relative orientation of the distortion direction of two neighboring domains with different directions of the spontaneous polarization, the related angles being $90^\circ$ and $180^\circ$, respectively. These two wall types are outlined in Fig. 2.10. As the unit cell is not symmetric, the $90^\circ$ wall shows a distorted lattice structure. Furthermore, the center ions can be located at two equivalent positions, thus the respective figure shows two ions occupying the domain wall cells.

In principle also domain walls with a head to head scheme of ferroelectric polarization are possible (Fig. 2.11), but as they raise a large depolarization field, they are not favored in terms of the energy. Head to head walls have been observed in $\mathrm{BaTiO_3}$, but electro-microscopic examinations revealed a zigzagged domain wall layout (Fig. 2.12), which increased the overall wall length by a factor of 5.

In contrast to the magnetic equivalent, this field can be compensated by the flow of free charge inside and outside the medium

$$\rho = \int_0^t \sigma \cdot E \cdot dt,$$ (2.4)

where $\sigma$ is the electric conductivity of the material. As the conductivities of ferroelectrics and the surrounding air are usually low, the equilibrium where the energy of the depolarization field

$$W_E=\frac{1}{2}\cdot \int_V D \cdot E \cdot dV$$ (2.5)

is zero is reached very slowly.

Consequently, after the depolarization field is compensated by the free charges of a conductive ferroelectric material, theoretically a single domain structure should evolve. In reality, this is very unlikely since material properties are not ideal.

Figure 2.10: Structural model of a $180^\circ$ and a $90^\circ$ domain wall

Figure 2.11: Head to head domain wall

Figure 2.12: Scheme of a zigzagged domain wall