3. Advanced Oxidation Model

THE MODEL described in this chapter is designed for a realistic physical and three-dimensional simulation of thermal oxidation. Advantageously, this model takes into account that the diffusion of oxidants, the chemical reaction, and the volume increase occur simultaneously. Furthermore, this model does not use moving Si/SiO$_2$-interfaces for the SiO$_2$-growth like the standard models [59,60], which are all based on the Deal-Grove model. The handling of moving interfaces problems in complex three-dimensional structures becomes very complicated and causes an enormous data update which are the most restricting factors for such applications [61,62].

In case of oxidation there exist two segments, one for silicon and one for SiO$_2$, with an interface. It is not a problem to make a mesh for such structures, but the SiO$_2$-growth results in a moving boundary problem, which means that the interface should move after each simulation step. In order to reach the new position of the interface, new grid points are inserted and a remeshing step has to be performed [61,63]. These mesh operations demand complicated algorithms.

The basic idea of this model is to define the regions of and SiO$_2$ on a single and static mesh with a separating parameter $\eta$. In this model $\eta$ plays a key role, because the main interest of oxidation simulation is to predict the shape of the SiO$_2$-domain. Since the newly formed SiO$_2$ leads to a significant volume increase and so to large displacements or stresses, the modeling of the mechanics also plays an important role. Besides the oxidant diffusion and the change of $\eta$, the mechanics is an important part of the mathematical formulation.