4.2.5 Algorithm for Materials without any Favored Direction

As already mentioned, grain sizes of ferroelectric ceramics can get as small as 10nm. This means that in many cases we do not deal with ideal monocrystals but with structures formed by several grains. The orientations of these grains are not easy to find. As outlined above, the material as a whole shows uniform and, due to the randomly spread orientations, isotropic properties.

Following the ideas outlined in the previous section, the basic principle of this model is to split the polarization $\vec{P}_\mathrm{old}$ and the electric field $\vec{E}_\mathrm{old}$ of the previous operating point into components in the direction of the next applied electric field $\vec{E_1}$, resulting in

$$\vec{P}_{\mathrm{old},\parallel} = \vec{P}_\mathrm{old} \cdot \vec{E}_1 \cdot \frac{\vec{E}_1}{\Vert\vec{E}_1\Vert}$$ (4.18)

$$\vec{E}_{\mathrm{old},\parallel} =\vec{E}_\mathrm{old}\cdot \vec{E}_1 \cdot \frac{\vec{E}_1}{\Vert\vec{E}_1}$$ (4.19)

as well as

$$\vec{P}_{\mathrm{old},\perp} = \vec{P}_{\mathrm{old}} \cdot \vec{E}_{1,\perp} \cdot \frac{\vec{E}_{1,\perp}}{\Vert\vec{E}_{1,\perp}\Vert}$$ (4.20)

$$\vec{E}_{\mathrm{old},\perp} =\vec{E}_{\mathrm{old}}\cdot \vec{E}_{1,\perp} \cdot \frac{\vec{E}_{1,\perp}}{\Vert\vec{E}_{1,\perp}\Vert}$$ (4.21)

in the orthogonal direction

$$\vec{E}_{1,\perp}=\left(\vec{E}_{1,y} \atop{-\vec{E}_{1,x}}\right),$$ (4.22)

which is graphically outlined in Fig. 4.8.

Figure 4.8: Splitting of the field vectors

With these geometric operations, the two-dimensional problem gets reduced to two one-dimensional scalar problems, with the additional advantage that this geometric approach also works in three dimensions. Now the concept of Preisach hysteresis, outlined in the previous chapter, can be applied to each of these scalar problems.

For a general approach to two-dimensional hysteresis effects an inhomogeneous field distribution has to be assumed. This prevents the usage of a simple one-dimensional hysteresis model which would use an identical locus curve for the complete ferroelectric region. According to the algorithm presented above, two different locus curves $f_{\mathrm{loc},\parallel}$ and $f_{\mathrm{loc},\perp}$ have to be calculated for each grid point. These are outlined in Fig. 4.9.

Figure 4.9: Calculation of the initial guess

According to the Preisach model, the parameters $w$ and $k$ of the locus curves are calculated using the projections of the old directions of the old electric field $\vec{E}_{\mathrm{old},\parallel}$, $\vec{E}_{\mathrm{old},\perp}$, the old polarization field $\vec{P}_{\mathrm{old},\parallel}$, $\vec{P}_{\mathrm{old},\perp}$, and the turning points $\vec{P}_{\mathrm{turn},\parallel}$, $\vec{P}_{\mathrm{turn},\perp}$.

The component $P_{\parallel,1}$ in the direction of the electric field is calculated by entering the signed length of the electric field vector into the equation of the local locus curve $f_\mathrm{local}$

$$P_{\parallel,1} = f_ {\mathrm{loc},\parallel}(\pm\Vert\vec{E}_1\Vert).$$ (4.23)

The actual algorithm required to achieve the signed length quantity will be discussed in Section 6.2.2. The input for the locus curve is, according to geometrical properties, zero in the perpendicular direction. Thus with the component

$$P_{1,\perp,\mathrm{init}} = f_{\mathrm{loc},\perp}(0)$$ (4.24)

the polarization in the perpendicular direction is obtained. The two polarization components, which were given in (4.23) and (4.24), constitute an initial guess

$$\vec{P}_\mathrm{init} = P_{\parallel,1} \cdot \frac{\vec{E}_{1,\p... ...P_{\perp,1,\mathrm{init}} \frac{\vec{E}_{1,\perp}}{\Vert\vec{E}_{1,\perp}\Vert}$$ (4.25)

for the next polarization which is plotted in Fig. 4.10.

Figure 4.10: Calculation of the polarization

Following the model for the polarization in perpependicular direction, outlined in Section 4.2.3, the scalar values of the two components are added and compared to the saturation polarization $P_\mathrm{Sat}$.

The vanishing electric field in the perpendicular direction makes it easier to switch the dipoles which are oriented in this direction than to switch the dipoles which are oriented in the direction of the electric field and, consequently, hold by it. The perpendicular component $\vec{P}_\perp$ is reduced appropriately with respect to the limit

$$\Vert P_\perp\Vert = P_\mathrm{Sat} - \Vert{P}_{\parallel,1}\Vert.$$ (4.26)

This is shown schematically in Fig. 4.10 and Fig. 4.11 and leads to the actual polarization vector

$$\vec{P}_1 = \vec{P}_{\parallel,1} + \Vert P_\perp\Vert \cdot \frac{\vec{P}_{\perp,1}}{\Vert\vec{P}_{\perp,1}\Vert}.$$ (4.27)

Figure 4.11: Reduction of the orthogonal component