THE goal of scientific work has been expressed by the father of quantum electron dynamics, Richard P. Feynman [Fey63]: ``Now in the further development of science, we want more than just a formula. First we have an observation, then we have numbers that we measure, then we have a law which summarizes all numbers. But the real glory of science is that we find a way of thinking such that the law is evident.'' Unfortunately, in case of diffusion, we are not really near this goal.
Diffusion process modeling started out on the physical basis of local concentration dependence of diffusivity [Hu68], [Fai73]. The dopant diffusivity was taken to be proportional to the local concentration of point defects, which were assumed to be vacancies. The concentration of point defects was taken to be at thermal equilibrium and dependent only on the local Fermi level, and hence on the local dopant concentrations.
In 1974, two concepts were introduced [Hu74] which have since then become the foundation of diffusion process modeling in silicon: (1) Substitutional atoms diffuse in silicon via a dual mechanism involving both the lattice vacancy and the silicon self-interstitial, with a fractional interstitialcy component that is characteristic for each atomic species. (2) Thermal oxidation of silicon generates excess self-interstitials, tending to enhance the diffusivities of those atoms which diffuse with a significant interstitialcy component. Oxidation retarded diffusion was reported [Ant82], which is compatible with the predominantly vacancy mechanism of antimony diffusion in silicon. Two other surface processes, thermal nitridation [Miz82] and silicidation of silicon [Mae89], were found to produce excess vacancies and to enhance antimony diffusion.
In recent years the pair diffusion model [Mor86] advanced in treatment of point defect mediated dopant diffusion by hypothesizing that dopant diffusion proceeds by diffusion of dopant point-defect pairs. A derivation of point defect assisted diffusion from the master equation [Orl90], considering the same mechanisms and the same functional dependence for the basic mechanism as the aforementioned dual diffusion model, however at a different level of theoretical description, indicates that the dual model is inconsistent with the basic mechanisms invoked, and that pair diffusion models happen to be consistent.
Already from this rudimentary recapitulation of major steps in diffusion modeling using macroscopic equations, i.e. diffusion equations, we can see that the basic mechanisms of diffusion in silicon's diamond lattice are not quite well understood, not to mention the processes of activation and deactivation of dopants at high concentrations. Repeatedly intuitive models have been proposed which afterwards proved indefensible. Other models have so many parameters that everything can be fitted without any evidence that the underlying physics is correct. Modeling from first principles, like model building techniques based on master equations or molecular dynamic studies [Chi91] can safeguard against pitfalls of intuitive approaches.
In process modeling the simulation serves as a tool for model development or generally for insight. Here, the simulation software must accommodate the input of a new model for its speedy evaluation. On the other hand, within the simulation framework for process development simulation is used for analysis and for optimization. Here, it is necessary to have verified, robust default models. In addition, the need for swift adaption of simulation capabilities to changing technological requirements asks for a simulation software which allows easy implementation of new or updated models which can be handed out again to a device or process engineer.
The PROMIS Diffusion Module addresses these demands. It allows easy implementation of new physical models by a well defined model interface for speedy model evaluation, e.g. in one space dimension. When verified, the new model is made available to the device designers for use in two-dimensional nonplanar structures by putting it into a library, without additional coding effort.
In this chapter we describe the Diffusion Module of PROMIS. In Section 3.1 we show the set of partial differential equations (PDE) and boundary conditions which can be treated by the diffusion module. In Section 3.2 we describe those diffusion models which are currently supplied with the Model Library of PROMIS. The major part of this chapter is dedicated the `numerical modeling engine' in Sections 3.3 - 3.6. There, we describe the transformation method which is applied to solve equations in nonplanar domains, and give a survey of the applied numerical methods. We close this chapter with some applications (Section 3.7).