4. Mesh Refinement for Diffusion

Diffusion in general is the process by which matter is transported from one part of a system to another as a result of random molecular motions [69]. The transfer of heat by conduction is due to random molecular motions, and there is an obvious analogy between these two processes. This was recognized by Adolf Eugen Fick who first put diffusion on a quantitative basis by adopting the mathematical equation of heat conduction derived some years earlier by Jean Baptiste Joseph Fourier. The mathematical theory of diffusion in isotropic substances is therefore based on the hypothesis that the rate of transfer of diffusing substance (dopant atoms) through unit area of a section is proportional to the concentration gradient measured perpendicular to the section. Fick's First Law gives

$$\vec{J}=-D \; \vec{\nabla}C$$ (4.1)

where $\vec{J}$ denotes the diffusion flux, $D$ is termed diffusion coefficient or diffusivity, and $C$ is the concentration of the diffusing species. It must be emphasized that (4.1) is in general consistent only for for an isotropic media, whose structure and diffusions properties in the neighborhood of any point are the same relative to all directions [70,69,71].

For the numerical solution of diffusion based problems as well as the issue of simulating the physical phenomenon of oxidation at the Institute for Microlectronics, an in-house simulator, called FEDOS (Finite Elements Diffusion and Oxidation Simulator) was developed. Since FEDOS uses the mathematical concept of finite elements (FE) [72], a heuristic a posteriori error estimator was developed. This estimator is used to control an anisotropic Hessian refinement method which is part of the integrated mesh adaptation module of FEDOS. Finally, the basic ideas behind FE error estimation and mesh refinement are illustrated by two three-dimensional diffusion examples.