Rigorous derivation of the rate equations for epitaxial growth

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.