From Discrete Hopping to Continuum Modeling on Vicinal Surfaces with Applications to Si(001) Electromigration

TL;DR

This paper develops a continuum model for vicinal surface dynamics based on a discrete hopping model, explaining electromigration-induced step bunching on Si(001) surfaces with novel boundary conditions.

Contribution

It introduces a discrete hopping model that links microscopic parameters to continuum boundary conditions, including negative kinetic coefficients and permeability rates.

Findings

Negative kinetic coefficients and permeability rates can occur near steps.

The model explains step bunching under electromigration on Si(001).

Continuum parameters are derived from microscopic hopping dynamics.

Abstract

Coarse-grained modeling of dynamics on vicinal surfaces concentrates on the diffusion of adatoms on terraces with boundary conditions at sharp steps, as first studied by Burton, Cabrera and Frank (BCF). Recent electromigration experiments on vicinal Si surfaces suggest the need for more general boundary conditions in a BCF approach. We study a discrete 1D hopping model that takes into account asymmetry in the hopping rates in the region around a step and the finite probability of incorporation into the solid at the step site. By expanding the continuous concentration field in a Taylor series evaluated at discrete sites near the step, we relate the kinetic coefficients and permeability rate in general sharp step models to the physically suggestive parameters of the hopping models. In particular we find that both the kinetic coefficients and permeability rate can be negative when diffusion is faster near the step than on terraces. These ideas are used to provide an understanding of recent electromigration experiment on Si(001) surfaces where step bunching is induced by an electric field directed at various angles to the steps.