4.2.1 Volume Models

Equation (4.1) shows the structure of the partial differential equation system describing the transport of $N$ quantities $W_j$ within a segment of the device geometry.

$$\sum_{j=1}^{N}\alpha_{ij}\cdot \frac{\partial W_j}{\partial... ...p{\rm div}} \vec{J}_i + {\gamma}_i = 0, {\quad} i=1 {\dots} N$$ (4.1)

$$\vec{J}_i = \sum_{j=1}^{N} \left( a_{ij} \cdot \mathop{\rm g... ... + \right. \left. \vec{c}_{ij} \cdot W_j \right) + \vec{d}_i$$ (4.2)

$W_j$ denotes the dependent variables which are the values of the affected quantities and $N$ is the number of equations. $\psi$ is either the electrostatic potential resulting from (4.5) or can be chosen to be one of the dependent variables $W_j$ by a corresponding statement in the input deck.

For a system of $N$ impurities the total net concentration $C_{net}$ of all impurities is given by equation (4.3). $z_i$ denotes the charge state of the impurity $C_i$ ($0$ for neutral impurities, $+1$ for singly ionized acceptors, and $-1$ for singly ionized donors).

$$C_{net} = \sum_{i=1}^{N} z_{i} {\cdot} C_{i}$$ (4.3)

The electrostatic potential $\psi$ is determined by the Poisson equation (4.4)

$$\mathop{\rm div}( \mathop{\rm grad}\psi ) = \frac{q}{\epsilon} \cdot ( n - p - C_{net} )$$ (4.4)

By assuming vanishing space charge and the applicability of Boltzmann statistics the Poisson equation can be solved explicitly (4.5).

$$\psi = \frac{k{\cdot}T}{q} {\cdot} \mathop{\rm asinh}\left(\frac{C_{net}}{2{\cdot}n_i}\right)$$ (4.5)