3.1 The Diffuse Interface Concept

The diffuse interface concept avoids a moving interface problem, because there is not a sharp interface between silicon and SiO$_2$ in contrast to the standard models [64,65]. Because of the missing sharp interface there different segments for silicon and SiO$_2$ do not exist. In order to determine where is silicon and where is SiO$_2$, a parameter named normalized silicon is defined [66]

$$\eta(\vec{x},t) = \frac{C_{Si}(\vec{x},t)}{C_{0,Si}}.$$ (3.1)

Here $C_{Si}(\vec{x},t)$ is the silicon concentration at time t and point $\vec{x} \mathrm{(x, y, z)}$ and $C_{0,Si}$ is the concentration in pure silicon. $\eta$ is 1 in pure silicon and 0 in pure silicon dioxide.

Instead of a sharp interface there is a so-called reaction layer where the diffusion of oxidants, the chemical reaction, and the volume increase occur simultaneously. This reaction layer has a spatial finite width (see Fig. 3.1), where the values of $\eta$ lie between 0 and 1 [66]. The $\eta$ curve always starts with 0 near silicon and ends at 1 near oxide, as shown in Fig. 3.2. The shape of this curve is given by the calculated $\eta$ distribution in the reaction layer, which depends on the parameters in the model.