Macroscopic description of the diffusion of interstitial impurity atoms considering the influence of elastic stress on the drift of interstitial species

TL;DR

This paper derives a comprehensive macroscopic diffusion equation for interstitial impurity atoms that incorporates elastic stress effects, charge states, and electric fields, providing a more efficient tool for modeling impurity diffusion in materials.

Contribution

It introduces a generalized diffusion equation considering charge states, electric fields, and elastic stress, simplifying numerical solutions for impurity diffusion modeling.

Findings

Derived a generalized diffusion equation for interstitial impurity atoms.

Showed the equation accounts for charge states, electric fields, and elastic stress.

Validated the equation's consistency with existing models of impurity diffusion.

Abstract

The diffusion equation for nonequilibrium interstitial impurity atoms taking into account their charge states and drift of all mobile interstitial species in the built-in electric field and in the field of elastic stress was obtained. The obtained generalized equation is equivalent to the set of diffusion equations written for the interstitial impurity atoms in each individual charge state. Due to a number of the characteristic features the generalized equation is more convenient for numerical solution than the original system of separate diffusion equations. On this basis, the macroscopic description of stress-mediated impurity diffusion due to a kick-out mechanism was obtained. It is supposed that the interstitial impurity atom makes a number of jumps before conversion to the substitutional position. At the same time, a local equilibrium prevails between substitutionally dissolved impurity atoms, nonequilibrium self-interstitials, and interstitial impurity atoms. Also, the derived equation for impurity diffusion due to the kick-out mechanism takes into account all charge states of interstitial impurity atoms as well as drift of interstitial species in the electric field and in the field of elastic stress. Moreover, this equation exactly matches the equation of stress-mediated impurity diffusion due to generation, migration, and dissociation of the equilibrium pairs "impurity atom - self-interstitial".