next up previous contents
Next: 3.3 The Basic Semiconductor Up: 3. The Box Integration Previous: 3.1 The Poisson Equation


3.2 The Diffusion Equation

In this section, the discretization of parabolic time-variant problems is described. In its simplest representation, the right-hand side of the diffusion equation is time-invariant. Solving a diffusion problem the diffusion equation becomes time-variant. It describes the out-diffusion of matter, driven by its own concentration gradient. The diffusion flux $ {\mathrm{\bf J}}$ can be written as

$\displaystyle {\mathrm{\bf J}}=-D \;\operatorname{grad}c$ (3.45)

where $ D$ denotes the diffusion coefficient and $ c$ is the concentration of the diffusing material.
Additionally, the conservation of material must be fulfilled

$\displaystyle \operatorname{div}{\mathrm{\bf J}} = -\frac{\partial c}{\partial t}.$ (3.46)

After insertion of (3.46) in (3.45), the diffusion equation can be reformulated as

$\displaystyle \operatorname{div}\operatorname{grad}c = \frac{1}{D} \;\frac{\partial c}{\partial t}.$ (3.47)

As described in the previous section, this equation can be discretized as

$\displaystyle \sum_{\forall \;j\;:\;\exists\; \mathrm{edge} \;< p_ip_j >} \frac...
..._{j} - c_{i}) = \frac{1}{D} \;\left(\frac{\partial}{\partial t}c_i\right) \;V_i$ (3.48)

The time derivative in this formula can be discretized by several methods. By the backward Euler method, the time derivative is approximated by [66]

$\displaystyle \frac{\partial}{\partial t}c(t+\Delta t)\approx \frac{c(t+\Delta t)-c(t)}{\Delta t}=\frac{c^{k+1}-c^{k}}{\Delta t},$ (3.49)

with $ \Delta t$ the sampling interval. The discrete notation by backward Euler time discretization follows by

$\displaystyle \sum_{\forall \;j\;:\;\exists\; \mathrm{edge} \;< p_ip_j >} \frac...
...+1} - c_i^{k+1}\right) = \frac{1}{D} \;\frac{c_i^{k+1}-c_i^{k}}{\Delta t}\;V_i.$ (3.50)

By separating the unknowns to the left-hand side, this expression becomes

$\displaystyle \frac{1}{\Delta t} \;c_i^{k+1}\;V_i + D \left( - \sum_{\forall \;...
...\frac{A_{ij}}{d_{ij}} \;c_i^{k+1} \right) = \frac{1}{\Delta t} \;c_i^{k} \;V_i.$ (3.51)

In matrix notation, (3.51) can be written as

$\displaystyle {\mathrm{\bf K}} \cdot {\mathrm{\bf x}} = {\mathrm{\bf B}},$ (3.52)


$\displaystyle {\mathrm{\bf K}}=\frac{1}{\Delta t} \;{\mathrm{\bf M}} + D \;{\mathrm{\bf S}},\qquad$ (3.53)

$\displaystyle \qquad\qquad\qquad\qquad x_{i}$ $\displaystyle =c_i^{k+1}$   $\displaystyle \qquad\forall \; i,$ (3.54)
$\displaystyle m_{ij}$ $\displaystyle = 0$   $\displaystyle \qquad\forall \;i,j \;:\; i\neq j,$ (3.55)
$\displaystyle m_{ii}$ $\displaystyle =V_i$   $\displaystyle \qquad\forall \; i,$ (3.56)
$\displaystyle s_{ij}$ $\displaystyle = -\frac{A_{ij}}{d_{ij}}$   $\displaystyle \qquad\forall \;i,j \; :\;\exists\;\mathrm{edge}\;< p_i p_j >,\qquad\qquad$ (3.57)
$\displaystyle s_{ii}$ $\displaystyle =\sum_{\forall\;j\;:\;\exists\;\mathrm{edge} \; < p_ip_j >} \frac...
...ij}}{d_{ij}}=-\sum_{\forall\;j\;:\;\exists\;\mathrm{edge} \; < p_ip_j >} s_{ij}$   $\displaystyle \qquad\forall \; i,$ (3.58)
$\displaystyle b_{i}$ $\displaystyle = \frac{1}{\Delta t} \;V_i \;c_i^{k}$   $\displaystyle \qquad\forall \;i.$ (3.59)

The necessary boundary conditions for this parabolic equation can be handled as shown in the previous section. Also the requirements for an M-matrix (see Section 2.3) are satisfied. The conditions for $ {\mathrm{\bf S}}$ can be handled as in the previous section and $ {\mathrm{\bf M}}$ consists of positive diagonal entries only. Additionally, an initial condition is required.

3.2.1 Initial Conditions

The initial concentration distribution at initial time $ \tau$ is defined as

$\displaystyle c({\mathrm{\bf x}},\tau)=C({\mathrm{\bf x}}).$ (3.60)

The discrete system is satisfied by the discrete formulation

$\displaystyle c_i(\tau)=c_i^0=C_{i}$   for all grid points $ p_i$$\displaystyle .$ (3.61)

The concentration distributions $ c_i^1, c_i^2, \ldots, c_i^{k+1}$ can be computed by an sequential evaluation of .

next up previous contents
Next: 3.3 The Basic Semiconductor Up: 3. The Box Integration Previous: 3.1 The Poisson Equation

J. Cervenka: Three-Dimensional Mesh Generation for Device and Process Simulation