3.2 Compact Modeling - Preisach Hysteresis

The basis for this approach was laid for magnetism by F. Preisach as early as 1938 [Pre35]. Later it was intensively tested for adsorption hysteresis and finally verified for magnetic MnAlGe by Barker et al. [BSHE83] in 1983. In 1997 Jiang et al [JZJ+97] proved that this model is applicable for ferroelectric materials as well and an excellent fit can be obtained.

Figure 3.3: Hysteresis loop for an individual particle

The basic idea of this approach is modeling the material through a cluster of independent dipoles. Each of these dipoles can switch between two opposing states, thus showing a rectangular hysteresis as in Fig. 3.3. The according mathematical formulation is

$$\begin{array}{lclcl} \frac{dE}{dt} > 0 & \wedge & E < E_u & ... ... E > E_d & \Rightarrow & P = + P_\mathrm{Part}. \\ \end{array}$$ (3.2)

Figure 3.4: Polarization density function $F(E_d,E_u)$

$E_u$ is the transition field in the 'up' direction, $E_d$ the transition field in the 'down' direction. As outlined in the figure, the transition fields are not symmetrical and furthermore not even restricted to a specific sign. This means that $E_d$ might, e.g., be positive as well as negative. $P_\mathrm{Part}$ is the polarization of a single dipole. The next assumption is that these transition fields are statistically distributed. $F(E_u,E_d)$ is the distribution function, its integral has to be 1. For the Preisach model this distribution function contains all the relevant hysteresis information.

According to this description the overall polarization can be easily calculated as

$$P= P_\mathrm{Sat} \cdot \int_{A_\mathrm{up}}F(E_d,E_u)\cdot ... ...P_\mathrm{Sat})\cdot \int_{A_\mathrm{down}}F(E_d,E_u)\cdot dA,$$ (3.3)

where $A_\mathrm{up}$ is the area in the $E_u-E_d$ plane where the dipoles are oriented upwards and $A_\mathrm{down}$ the area where the dipoles are oriented downwards.

A possible distribution function is outlined in Fig. 3.4. It shows a peak at a certain point $E_u = -E_d = E_0$ and is furthermore expected to be symmetrical around the line $E_u = - E_d$.

This simple model leads to remarkable results for the analysis of ferroelectric materials, and, as mentioned, also shows an excellent correspondence to measured data. In order to allow a graphical analysis of the hysteresis, the range of the values is restricted from 0 to a maximum transition field $E_\mathrm{max}$ for the transition field $E_u$, and from 0 to $-E_\mathrm{max}$, for the other transition field $E_d$ . As shown below, this does not reduce the universality of the obtained conclusions. In order to illustrate the state of the dipoles, a top view of the distribution function, the Preisach-Everett diagram is plotted [BSHE83].