Rigorous derivation of the rate equations for the epitaxial growth

TL;DR

This paper rigorously derives rate equations for epitaxial growth, defining capture numbers precisely and revealing significant differences from traditional models, especially at low surface coverage, supported by kinetic Monte Carlo simulations.

Contribution

It provides a rigorous derivation of rate equations for epitaxial growth, clarifies the definition of capture numbers, and challenges existing models with new insights supported by simulations.

Findings

Exact equations for island density evolution obtained

Discrepancies found between rigorous capture numbers and literature values

New rate equation including monomer diffusion term proposed

Abstract

In the framework of the second-quantization representation of the master equation governing the irreversible epitaxial growth, exact equations describing the evolution of the island densities has been obtained. Their decoupling within a mean field-type approximation with the unknown correlation functions replaced by capture numbers (CNs) has been used to derive a closed set of rate equations. The latter has been compared with the exact equations to obtain rigorous definitions of the CNs. The CN that describes the nucleation of dimer islands from two mobile monomers has been measured in the exact kinetic Monte Carlo simulations with the use of the rigorous definition. Strong disagreement with the literature values calculated within alternative techniques has been found, especially at low surface coverage. Plausible causes for the discrepancies are suggested. Another important result of the rigorous approach is the rate equation with the term describing the monomer diffusion. The equation significantly differs from the widely used equations of this kind known from literature. Arguments in favour of our approach are given.