Towards a high-supersaturation theory of crystal growth: Nonlinear one-step flow model in 1+1 dimensions

TL;DR

This paper derives a mesoscale model for crystal surface growth from a microscopic kinetic model, validating it near equilibrium and proposing nonlinear corrections for high supersaturation conditions.

Contribution

It provides a rigorous derivation of the classical BCF step-flow model from a microscopic kinetic framework and introduces empirical nonlinear corrections for high supersaturation regimes.

Findings

BCF model derived from microscopic kinetics near equilibrium

Linear adatom flux relation remains valid under low supersaturation

Nonlinear flux corrections improve model accuracy at higher supersaturation

Abstract

Starting with a many-atom master equation of a kinetic, restricted solid-on-solid (KRSOS) model with external material deposition, we investigate nonlinear aspects of the passage to a mesoscale description for a crystal surface in 1+1 dimensions. This latter description focuses on the motion of an atomic line defect (i.e. a step), which is defined via appropriate statistical average over KRSOS microstates. Near thermodynamic equilibrium and for low enough supersaturation, we show that this mesoscale picture is reasonably faithful to the Burton-Cabrera-Frank (BCF) step-flow model. More specifically, we invoke a maximum principle in conjunction with asymptotic error estimates to derive the elements of the BCF model: (i) a diffusion equation for the density of adsorbed adatoms; (ii) a step velocity law; and (iii) a linear relation for the mass flux of adatoms at the step. In this vein, we also provide a criterion by which the adatom flux remains linear in supersaturation, suggesting a range of non-equilibrium conditions under which the BCF model remains valid. Lastly, we make use of kinetic Monte Carlo simulations to numerically study effects that drive the system to higher supersaturations -- e.g. deposition of material onto the surface from above. Using these results, we describe empirical corrections to the BCF model that amount to a nonlinear relation for the adatom flux at the step edge.