3.3.2.1 Continuity and Balance Equations

The balance equations (2.184) to (2.186) are discretized in the same manner as POISSON's equation. Integrating over the control volume $\ensuremath{\mathcal{ V }}_i$ and applying the theorem of GAUSS yields

$$\ensuremath{\oint_{\partial \ensuremath{\mathcal{ V }}_i} \ensure... ...ath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n \,\, \ensuremath{\mathrm{d}}A} = \ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} - s_n \, \mathrm{q}\, (\ensuremath{\partial_{t} \, n} + R) \,\, \ensuremath{\mathrm{d}}V}\ ,$$ (3.23)

$$\ensuremath{\oint_{\partial \ensuremath{\mathcal{ V }}_i} \ensure... ...ath{\cdot}\ensuremath{\boldsymbol{\mathrm{S}}}_n \,\, \ensuremath{\mathrm{d}}A} = \ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} - \mathrm{C}_2 ... ...m{L}}{\tau_\mathcal{E}} + G_{\mathcal{E}\, n} \,\, \ensuremath{\mathrm{d}}V}\ ,$$ (3.24)

$$\ensuremath{\oint_{\partial \ensuremath{\mathcal{ V }}_i} \ensure... ...ath{\cdot}\ensuremath{\boldsymbol{\mathrm{K}}}_n \,\, \ensuremath{\mathrm{d}}A} = \ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} - \mathrm{C}_4 ... ...T_\mathrm{L}^2} {\tau_\beta} + G_{\beta \, n} \,\, \ensuremath{\mathrm{d}}V}\ .$$ (3.25)

The terms $\ensuremath{\boldsymbol{\mathrm{E}}} \ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n$ and $\ensuremath{\boldsymbol{\mathrm{E}}} \ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{S}}}_n$ are again discretized by the box integration method. Writing the electric field as the negative gradient of the electric potential and using the product rule the first term becomes

$$\ensuremath{\boldsymbol{\mathrm{E}}}\ensuremath{\cdot}\ensuremath... ...{\mathrm{\nabla}}}}\ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n}\ .$$ (3.26)

Integration over the control volume $\ensuremath{\mathcal{ V }}_i$, applying the theorem of GAUSS, and approximating the integrals by sums yields

$$\ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} \ensuremath{\bold... ...ath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n \,\, \ensuremath{\mathrm{d}}V} \approx - \ensuremath{\oint_{\partial \ensuremath{\mathcal{ V }}_... ...ath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n \,\, \ensuremath{\mathrm{d}}A}$$ (3.27)

$$\approx - \sum_j \, \frac{\psi_i + \psi_j}{2} \, J_{n \, ij} \, A_{ij} + \psi_i \, \sum_j \, J_{n \, ij} \, A_{ij} \ ,$$ (3.28)

where a linear variation of the potential between two grid points has been assumed. By combining the sums the discrete representation of $\ensuremath{\boldsymbol{\mathrm{E}}} \ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{J}}}_n$ is obtained

$$\ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} \ensuremath{\bold... ...}\approx - \sum_j \, \frac{\psi_i - \psi_j} {2} \, J_{n \, ij} \, \, A_{ij} \ .$$ (3.29)

By analogy, the discretization of $\ensuremath{\boldsymbol{\mathrm{E}}} \ensuremath{\cdot}\ensuremath{\boldsymbol{\mathrm{S}}}_n$ looks

$$\ensuremath{\int_{\ensuremath{\mathcal{ V }}_i} \ensuremath{\bold... ...}\approx - \sum_j \, \frac{\psi_i - \psi_j} {2} \, S_{n \, ij} \, \, A_{ij} \ .$$ (3.30)

Using eqns. (3.29) and (3.30) the discretization of the continuity equations can be concluded

$$\sum_j \, J_{n \, ij} \, A_{ij} = - s_n \, \mathrm{q}\, \bigl(\ensuremath{\partial_{t} \, n_i}+ R_i \bigr) \, V_i \ ,$$ (3.31)

$$\sum_j \, S_{n \, ij} \, A_{ij} = - \Bigl(\mathrm{C}_2 \, \ensure... ...igr) \, V_i - \sum_j \, \frac{\psi_i - \psi_j} {2} \, J_{n \, ij} \, A_{ij} \ ,$$ (3.32)

$$\sum_j \, K_{n \, ij} \, A_{ij} = - \Bigl(\mathrm{C}_4 \, \ensure... ... \mathrm{q}\, \sum_j \, \frac{\psi_i - \psi_j} {2} \, S_{n \, ij} \, A_{ij} \ ,$$ (3.33)

where $J_{n \, ij}$, $S_{n \, ij}$, $K_{n \, ij}$ are the projections of the fluxes $\ensuremath{\boldsymbol{\mathrm{J_n}}}$, $\ensuremath{\boldsymbol{\mathrm{S_n}}}$, $\ensuremath{\boldsymbol{\mathrm{K_n}}}$ onto the grid edge $\ensuremath{\boldsymbol{\mathrm{e}}}_{ij}$.

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