A generalized mathematical framework for thermal oxidation kinetics

Abstract

We present a generalized mathematical model for thermal oxidation and the growth kinetics of oxide films. The model expands long-standing classical models by taking into account the reaction occurring at the interface as well as transport processes in greater detail. The standard Deal-Grove model (the linear-parabolic rate law) relies on the assumption of quasi-static diffusion that results in a linear concentration profile of, for example, oxidant species in the oxide layer. By relaxing this assumption and resolving the entire problem, three regimes can be clearly identified corresponding to different stages of oxidation. Namely, the oxidation starts with the reaction-controlled regime (described by a linear rate law), is followed by a transitional regime (described by a logarithmic or power law depending on the stoichiometry coefficient m), and ends with the well-known diffusion-controlled regime (described by a parabolic rate law). The theory of Deal-Grove is shown to be the lower order approximation of the proposed model. Various oxidation rate laws are unified into a single model to describe the entire oxidation process.