Continuum Limit of a Mesoscopic Model with Elasticity of Step Motion on Vicinal Surfaces

Abstract

This work considers the rigorous derivation of continuum models of step motion starting from a mesoscopic Burton-Cabrera-Frank (BCF) type model following the work [Xiang, SIAM J.Appl. Math. 2002]. We prove that as the lattice parameter goes to zero, for a finite time interval, a modified discrete model converges to the strong solution of the limiting PDE with first order convergence rate.

Weak Solution of a Continuum Model For Vicinal Surface in The Attachment-Detachment-Limited Regime

“…They involve fewer variables than discrete models so they can reveal the leading physics structure and are easier for numerical simulation. Many interesting continuum models can be found in the literature on surface morphological evolution; see [22,25,7,29,30,24,20,4,10] for one dimensional models and [19,31] for two dimensional models. The study of relation between the discrete ODE models and the corresponding continuum PDE has raised lots of interest.…”

Global strong solution with BV derivatives to singular solid-on-solid model with exponential nonlinearity

“…Logarithmic correction and explanation from mesoscopic view. From the mesoscopic view we can regard the surface evolution equation as continuum limit of discrete Burton-Cabrera-Frank (BCF) model [3,7,9], which tracks the dynamics of positions of each step x i with height h i = h 0 + i N . In ADL regime, it can be expressed by…”

Maximal monotone operator theory and its applications to thin film equation in epitaxial growth on vicinal surface

“…Although discrete models do have the advantage of reflecting physical principle directly, when we study the evolution of crystal growth from macroscopic view, continuum approximation for the discrete models involves fewer variables than discrete models and can briefly show the evolution of step flow. Many interesting continuum models can be found in the literature on surface morphological evolution; see [2][3][4][5][6][7][8][9][10] for one dimensional models and [11,12] for two dimensional models. Kohn clarified the evolution of surface height from the thermodynamic viewpoint in the book [13].…”

“…where u, considered as a [0, 1)-periodic function of the step height h, is the step slope of the surface. [10] provided a method to rigorously obtain the convergence rate of discrete model to its corresponding continuum limit. Two questions then arise.…”