Stochastic Models of Epitaxial Growth

Abstract

We study theoretical aspects of step fluctuations on vicinal surfaces by adding conservative white noise to the Burton-Cabrera-Frank model in one spatial dimension. We consider material deposition from above, as well as entropic and elastic-dipole step repulsions. Two approaches are discussed: (i) the linearization of stochastic equations when fluctuations are small, which captures correlations; and (ii) a mean field approach, which leaves out correlations but captures nonlinearities. Comparisons to kinetic Monte-Carlo simulations are presented.

Connection of Kinetic Monte Carlo Model for Surfaces to One-Step Flow Theory in 1+1 Dimensions

“…An adatom density, which obeys a diffusion equation, represents adatoms on the terraces. Solving the BCF model yields the adatom density and step positions as functions of time; see Figure 1(a), as well as [11,25,27,31,32,33,34] for examples of BCF-type models.…”

“…In the many years since it was first proposed, the Burton-Cabrera-Frank (BCF) model [5] of step flow has been recognized as a useful tool for describing near-equilibrium crystal surface evolution. The BCF theory has been invoked in studies that span a range of topics, including electromigration of atoms and step bunching [28,43,44,45], surface relaxation [16,17], stochastic fluctuations of surface defects [11,25,27,28,31,32,33,34], and faceting [17,45], to name a few. Moreover, the theory has served as a starting point from which fully continuum models, such as partial differential equations for the surface height, are derived; see, e.g., [8,21,28,46].…”

Small fluctuations in epitaxial growth via conservative noise

“…In this thesis, we studied small stochastic fluctuations of line defects on a crystal surface when material is deposited from above in 1+1 dimensions. This work has been inspired by [9,10,16], and aims to complement recent kinetic studies in the steady-state distribution of terrace widths on vicinal crystal surfaces [34][35][36].…”

“…From a kinetic theory view, our work forms an extension of the treatment in [34][35][36] where the mean field approach of [16] is placed on the grounds of Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchies for joint probability densities of terrace widths. In [34], the steps are energetically non-interacting, and the white noise is non-conservative.…”

“…In [35], the steps are interacting, albeit in the absence of deposition (with F = 0). Both step interactions and nonzero flux F are discussed briefly in [36], which only reports (without derivations) the TWD from an ad hoc second-order conservative noise. Here, we consider different, physically inspired and more transparent models of noise and, in addition, describe terrace width pair correlations.…”