3.3.1 POISSON's Equation

To find a discrete approximation of MAXWELL's fourth equation, eqn. (2.4) is integrated over a control volume $\ensuremath{\mathcal{ V }}_i$

$$\ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} \ensuremath{\ensu... ...h{\int_{\ensuremath{\mathcal{ V }}_i} \varrho \,\, \ensuremath{\mathrm{d}}V}\ .$$ (3.18)

Applying the theorem of GAUSS to the left hand side turns eqn. (3.18) into

$$\ensuremath{\oint_{\partial \ensuremath{\mathcal{ V }}_i} \ensure... ...h{\int_{\ensuremath{\mathcal{ V }}_i} \varrho \,\, \ensuremath{\mathrm{d}}V}\ .$$ (3.19)

The integrals are approximated as follows:

$$\sum_j \, D_{ij} \, A_{ij} = \varrho_i V_i \ ,$$ (3.20)

where $D_{ij}$ is the projection of the flux $\ensuremath{\boldsymbol{\mathrm{D}}}$ onto the grid edge $d_{ij}$, evaluated at the midpoint of the edge, $A_{ij}$ is the boundary line which belongs to both subdomains $\ensuremath{\mathcal{ V }}_i$ and $\ensuremath{\mathcal{ V }}_j$, and $\varrho_i$ is the space charge density at the grid point $P_i$ (Fig. 3.4).

Figure 3.4: Control volume of grid point $P_i$ used for the box integration method.

The remaining task is to find an approximation for the projection of the dielectric flux density $D_{ij}$. This is done by the finite difference approximation

$$D_{ij} = - \varepsilon \, \frac{\psi_j - \psi_i}{d_{ij}} \ ,$$ (3.21)

$$\varepsilon = \frac{\varepsilon_i + \varepsilon_j}{2} \ .$$ (3.22)

With eqn. (3.20) and eqn. (3.21), the discretization of POISSON's equation can be concluded.

M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF