6.2.2 Nonsymmetry of the Locus Curves - Sign Handling

In contrast to most of the functions used in device simulation, the locus curves of hysteresis are nonsymmetrical functions of the applied electric field $\vec{E}$ if $E_c \ne 0$. As will be shown in this Subsection, this leads to an increase of numerical complexity.

The straightforward scheme used to calculate the flux quantity of the box integration, the displacement $\vec{D}$ would be

$$\vec{D}=f(\vec{E}) \rightarrow \vec{D}=f_\mathrm{scalar}(\Vert\vec{E}\Vert)\cdot \frac{\vec{E}}{\Vert\vec{E}\Vert}.$$ (6.5)

$f$ is the vector function describing the dependence between $\vec{D}$ and $\vec{E}$, $f_\mathrm{scalar}$ is its scalar equivalent. Obviously this scheme has to fail, as it only allows positive arguments of the function $f_\mathrm{scalar}$. A criterion has to be established which decides whether the argument of the function is treated as positive or negative, thus modifying $f_\mathrm{scalar}$:

$$f_\mathrm{scalar}(\Vert\vec{E}\Vert) \rightarrow f_\mathrm{scalar}(\pm\Vert\vec{E}\Vert).$$ (6.6)

In order to achieve this, the direction of the field is compared of the with the box boundary. In local coordinates using the box boundary as y axis the criterion reads as

$$E_x \geq 0 \rightarrow +\Vert\vec{E}\Vert$$ (6.7)

$$E_x < 0 \rightarrow -\Vert\vec{E}\Vert .$$ (6.8)

In order to obtain the correct sign of the function argument also the term ${\vec{E}}/{\Vert\vec{E}\Vert}$ of (6.5) has to be modified, assuring that it always points to the same half-room. The complete discretization reads as

$$\vec{D}=f_\mathrm{scalar}(\pm\Vert\vec{E}\Vert)\cdot \frac{\pm\vec{E}}{\Vert\vec{E}\Vert},$$ (6.9)

where the sign in the last term is chosen accordingly to the rule defined in (6.7) and (6.8).