4.2.3 Polarization in an Orthogonal Direction

The basic key for developing a rigorous approach to two-dimensional hysteresis is to find a useful formulation for the following problem: A piece of ferroelectric material has a remanent polarization ${\vec{P}_\mathrm{Rem}}$, as sketched in Fig. 4.4. Then an electric field is applied in the perpendicular direction.

Figure 4.4: Application of an electric field orthogonal to the remanent polarization

Figure 4.5: Construction of the polarization components

The newly applied electric field will raise a polarization component $P_{\parallel}$ in the same direction as the field. This is plotted in Fig. 4.5.

Regarding the fact that there was no prior polarization in this direction, an initial polarization curve (dashed line) is used. The finite number of dipoles introduces the saturation polarization as a hard limit. Regarding the domain structure of the material and neglecting the rotation of the dipoles, it can be assumed that the sum of magnitudes of the newly raised and the remanent component will not exceed the saturation polarization $P_\mathrm{Sat}$.

$$\Vert\vec{P}_\mathrm{Rem}\Vert+\Vert\vec{P}_{\parallel}\Vert \leq P_\mathrm{Sat}$$ (4.14)

For the model presented in this thesis the component $\Vert\vec{P}_\mathrm{Rem}\Vert$ will be reduced in order to fulfill (4.14), due to the fact that there is no field component in this direction. The resulting polarization is plotted in Fig. 4.6.

Even in the case that (4.14) does not fulfill the exact physical properties, there has to be a similar, more general expression the form of:

$$\Vert\vec{P}_\mathrm{Rem}\Vert+\Vert\vec{P}_{\parallel}\Vert \leq f(\Vert\vec{E}\Vert).$$ (4.15)

Figure 4.6: Construction of the lag angle