2.2.2 Distorted Cubic Structure - Ferroelectricity

Figure 2.5: Tetragonally distorted crystal

As already outlined in Section 2.1, ferroelectricity is caused by asymmetries in the lattice structure. If for example the $\mathrm{Sr}$ in $\mathrm{SrTiO_3}$ is replaced by a $\mathrm{Ba}$ ion which has a bigger diameter, the cubic structure gets tetragonally distorted at room temperature and the height of the crystal increases compared to its base (base length=3.98Å, height=4.03Å) [Die97]. This distortion is schematically outlined in Fig. 2.4, the resulting polarization in Fig. 2.5. Fig. 2.6 shows the potential distribution around the small center ion B in the z-direction. The minimum does not occur in the center of the structure. Instead, the outlined double well energy structure, will evolve [LG96] along the z-axis of the crystal. Now there are two different energetically favored states for the positive center ion, both of them resulting in a dipole moment. Consequently the center ion will be trapped in one of these two positions as long as the thermal energy is lower than the barrier height. The crystal cell will carry a permanent polarization, which is called the spontaneous polarization $P_S$.

Figure 2.6: Potential distribution in the direction of the spontaneous polarization

Depending on the temperature there are two other distorted crystal phases for $\mathrm{BaTiO_3}$, each of them with a different geometry of the unit cell. For temperatures beyond $-80^\circ$C the crystal becomes rhombohedral, between $-80^\circ$C and about $-5^\circ$C monoclinic. This will raise different orientations and absolute values of the spontaneous polarization [Kit86]. The respective unit cells are sketched in Fig. 2.7 and Fig. 2.8.

Figure 2.7: Rhombohedrally distorted crystal

Figure 2.8: Monoclinicly distorted crystal

Initially, all the spontaneous polarizations of individual cells will be randomly distributed throughout the material, so the resulting overall displacement will be zero.

If an electric field is applied, the ions will be pushed towards the energetically better position, and if the applied field is big enough, the ions will cross the potential barrier. In an undistorted, perfect crystal this transition field is the same for each lattice cell. Impurities and stress modify the energy barriers locally and smoothen the resulting $D/E$ characteristics.

When all the dipoles are organized into the same direction, the maximum contribution of the dipole moment to the displacement is reached. This component is called saturation polarization $P_\mathrm{Sat}$.

If the electric field is reduced to zero again, many of the dipoles will be trapped in the last state causing a resulting polarization of the material, called remanent polarization $P_\mathrm{Rem}$. If the electric field is turned into the other direction, the resulting polarization is decreased to zero. The field necessary to achieve this is called the coercive field $E_c$. This behavior leads to a hysteresis of the $D/E$ characteristics, outlined in Fig. 2.9. The effects related to hysteresis can be summarized as follows:

• Hysteresis consists of many different branches (locus curves).
• By default these locus curves are nonsymmetric functions.
• The locus curves depend on the history.

Similar to magnetism the $D/E$ characteristics can be separated into a linear and a nonlinear part,

$$D=\epsilon_\mathrm{r}\cdot \epsilon_0 \cdot E + P_\mathrm{nonlin}.$$ (2.1)

Even though wrong from a rigorous point of view, it has become quite common in the literature on ferroelectrics to denote the nonlinear part $P_\mathrm{nonlin}$ as polarization. As the nonlinear part stems from the switching dipoles, it will be denoted as $P_\mathrm{Ferro}$ throughout this work.

Figure 2.9: Hysteresis loop