3.2.2 One-Dimensional POISSON's Equation

As an example the discretized form of POISSON's equation (2.14) will be written using the finite difference method. Assuming constant permittivity POISSON's equation in the one-dimensional case is given by

$$\displaystyle \psi$

(3.11)

The discretized representation is obtained by replacing the second order derivative by the central difference quotient eqn. (3.10) at all inner points

$\displaystyle \frac{\frac{\psi_{i + 1} - \psi_i}{h_i} - \frac{\psi_i - \psi_{i ... ... + N_{A \, i}^- - N_{D \, i}^+\bigr) = 0 \ , \qquad i = 2, 3, \ldots, N - 1 \ .$

(3.12)

The equations for

and

are determined by the boundary conditions. By rewriting this equation in the form

$\displaystyle b_i \, \psi_{i - 1} + z_i \, \psi_i + a_i \, \psi_{i + 1} - r_i = 0 \ ,$

(3.13)

the coefficients are found to be

$\displaystyle a_i$	$\displaystyle = \frac{1}{h_i} \ ,$	$\displaystyle \textcolor{lightgrey}{.......}b_i$	$\displaystyle = \frac{1}{h_{i - 1}} = a_{i - 1} \ ,$	(3.14)
$\displaystyle z_i$	$\displaystyle = - a_i - b_i \ ,$	$\displaystyle \textcolor{lightgrey}{.......}r_i$	$\displaystyle = \frac{\mathrm{q}}{\varepsilon} \, \bigl(n_i - p_i + N_{A \, i}^- - N_{D \, i}^+\bigr) \, \frac{h_i + h_{i - 1}}{2} \ .$	(3.15)

The coefficient matrix resulting from the linear equation system eqn. (3.13) is symmetric and only the diagonal and the secondary diagonal are filled with elements different from zero.

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


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