3.3.2.3 Growth Function

As already mentioned, some works explicitly assume a particular interpolation of the generalized concentration between two grid points, for instance an exponential variation. No such assumption was necessary to calculate the fluxes in the presented discretization. Nonetheless, the resulting interpolation is needed in some cases and will be investigated in this section.

First, a variable upper boundary $x$ is used in the definite integral (3.51)

$$- \frac{\Phi}{\mathrm{C}_\Phi} \, \frac{T_\Phi^{\alpha + 1}}{\alp... ...Delta x}{\Delta T_\Phi} = \xi \, T_\Phi^{\alpha + 1} \biggr\vert _{x_i}^{x} \ ,$$ (3.69)

which evaluates to

$$- \underbrace{\frac{\Phi} {\mathrm{C}_\Phi} \, \frac{\Delta x}{(\... ...r) = \xi(x) \, T_\Phi^{\alpha + 1}(x) - \xi_i \, T_{\Phi \, i}^{\alpha + 1} \ .$$ (3.70)

By using eqn. (3.52) the coefficient $a$ is found to be

$$a = - \frac{\xi_j \, T_{\Phi \, j}^{\alpha + 1} - \xi_i \, T_{\Ph... ... i}^{\alpha + 1}} {T_{\Phi \, j}^{\alpha + 1} - T_{\Phi \, i}^{\alpha + 1}} \ .$$ (3.71)

A normalization of $\xi(x)$ to the range $[0, 1]$ can be achieved by using the substitution

$$g(x) = \frac{\xi(x) - \xi_i}{\xi_j - \xi_i} \ , \textcolor{lightg... ...lor{lightgrey}{.......}\xi(x) = g(x) \, \bigl( \xi_j - \xi_i \bigr) + \xi_i \ ,$$ (3.72)

where $g(x)$ is called growth function [8, p.156]. Inserting eqn. (3.71) and eqn. (3.72) into eqn. (3.70) yields the expression for the growth function

$$g(x) = \frac{T_\Phi^{\alpha + 1}(x) - T_{\Phi \, i}^{\alpha + 1}}... ...\alpha + 1}} {1 - \bigl( T_{\Phi \, i} / T_{\Phi \, j} \bigr)^{\alpha + 1}} \ .$$ (3.73)

A graph of this function with $\alpha$ as parameter is depicted in Fig. 3.5. The curves in the upper-left region are obtained for $T_{\Phi \, i} < T_{\Phi \, j}$ and vice versa.

Figure 3.5: Functional shape of the growth function $g(x)$ displayed in a normalized interval with $\alpha$ as parameter.

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