One-dimensional model of interacting-step fluctuations on vicinal surfaces: Analytical formulas and kinetic Monte Carlo simulations

Abstract

We study analytically and numerically a one-dimensional model of interacting line defects (steps) fluctuating on a vicinal crystal. Our goal is to formulate and validate analytical techniques for approximately solving systems of coupled nonlinear stochastic differential equations (SDEs) governing fluctuations in surface motion. In our analytical approach, the starting point is the Burton-Cabrera-Frank (BCF) model by which step motion is driven by diffusion of adsorbed atoms on terraces and atom attachment-detachment at steps. The step energy accounts for entropic and nearest-neighbor elastic-dipole interactions. By including Gaussian white noise to the equations of motion for terrace widths, we formulate large systems of SDEs under different choices of diffusion coefficients for the noise. We simplify this description via (i) perturbation theory and linearization of the step interactions and, alternatively, (ii) a mean-field (MF) approximation whereby widths of adjacent terraces are replaced by a self-consistent field but nonlinearities in step interactions are retained. We derive simplified formulas for the time-dependent terrace-width distribution (TWD) and its steady-state limit. Our MF analytical predictions for the TWD compare favorably with kinetic Monte Carlo simulations under the addition of a suitably conservative white noise in the BCF equations.

Stochastic Models of Epitaxial Growth

“…In this vein, a systematic kinetic description of terrace-width fluctuations in terms of a mean field was proposed recently [5] on the basis of hierarchies for terrace correlation functions in one space dimension (1D). This formalism was recently applied to a large system of interacting steps in the absence of material deposition under the assumption that the noise obeys a second-order conservative scheme [6]. Here, we extend the formulation of [6] to the case with material deposition of flux F from above.…”

“…This formalism was recently applied to a large system of interacting steps in the absence of material deposition under the assumption that the noise obeys a second-order conservative scheme [6]. Here, we extend the formulation of [6] to the case with material deposition of flux F from above. In addition, we discuss possible physical implications that stem from the analysis, and illustrate limitations of approximations for the (intrinsically nonlinear) stochastic equations of motion.…”

“…However, lower-order (non-conservative and first-order) schemes for white noise fail to yield a finite variance for long times [6]. Details of the formulation are given in [2,5,6,8]. By change of variables from w j to w j /w, denoted s j , and use of non-dimensional time via scaling of t by some scale t * , e.g., w …”

“…This choice yields a finite variance of the TWD but is not unique. However, lower-order (non-conservative and first-order) schemes for white noise fail to yield a finite variance for long times [6]. Details of the formulation are given in [2,5,6,8].…”

“…The j-th terrace is the region x j <x<x j+1 which has width w j , where j=0,…, N-1, and N is large. We apply screw periodic boundary conditions, mapping steps to point particles on a ring [5,6]. For a vicinal crystal, the average terrace width is fixed at a constant value, w, and w j equals w initially (at t=0).…”