4.1.1 The arctan Shape Function

The $\textsf {arctan}$ function covers the physical properties of SBT( $\mathrm{SrBi_2Ta_2O_9}$) in a very accurate way [DBL98]:

$$P = \frac{2}{\pi} \cdot k \cdot P_\mathrm{Sat} \cdot\textsf{... ...gl( 2 \cdot(E \pm E_c)\cdot \frac{k}{w}\Bigr) + P_\mathrm{off}$$ (4.3)

$w$ is the shape parameter necessary to match the remanent polarization $P_\mathrm{Rem}$. $k$ and $P_\mathrm{off}$ are used for the simulation of subcycles. For the calculation of these parameters the equation system

$$P_\mathrm{old}= \frac{2}{\pi} \cdot k \cdot P_\mathrm{Sat}\cdot \textsf{arctan}\Bigl(2 \cdot(E_\mathrm{old} \pm E_c) \cdot \frac{k}{w}\Bigr) + P_\mathrm{off}$$

$$P_\mathrm{turn}= \frac{2}{\pi} \cdot k \cdot P_\mathrm{Sat}\cdot \textsf{arctan}\Bigl(2 \cdot(E_\mathrm{turn} \pm E_c) \cdot \frac{k}{w}\Bigr) + P_\mathrm{off}$$ (4.4)

has to be solved. As this equation system cannot be solved analytically, a Newton procedure is applied [Dir86]. This leads, if compared to the $\textsf {tanh}$ shape function (4.1.2), to a slight increase of computation time and more sophisticated numerical problems.