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$

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

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

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

The integrals are approximated as follows:

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

$\displaystyle D_{ij}$ $\displaystyle = - \varepsilon \, \frac{\psi_j - \psi_i}{d_{ij}} \ ,$ (3.21)
$\displaystyle \varepsilon$ $\displaystyle = \frac{\varepsilon_i + \varepsilon_j}{2} \ .$ (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