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.
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
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(3.2) |
is the transition field in the 'up' direction,
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
might, e.g., be
positive as well as negative.
is the polarization of
a single dipole. The next assumption is that these transition fields
are statistically distributed.
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
A possible distribution function is outlined in Fig. 3.4. It
shows a peak at a certain point
and is furthermore
expected to be symmetrical around the line
.
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
for the transition
field
, and from 0 to
, for the other transition
field
. 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].