Bunching instability and asymptotic properties in epitaxial growth with elasticity effects: continuum model

TL;DR

This paper analyzes a continuum model for epitaxial growth with elasticity effects, proving existence and properties of energy minimizers, and deriving energy scaling laws consistent with discrete models.

Contribution

It generalizes the epitaxial growth model to Lennard-Jones interactions and establishes mathematical properties of energy minimizers.

Findings

Existence and regularity of energy minimizers.

Symmetry and unimodality imply bunching profiles.

Derived energy scaling laws match discrete model results.

Abstract

We study the continuum epitaxial model for elastic interacting atomic steps on vicinal surfaces proposed by Xiang and E (Xiang, SIAM J. Appl. Math. 63:241-258, 2002; Xiang and E, Phys. Rev. B 69:035409, 2004). The non-local term and the singularity complicate the analysis of its PDE. In this paper, we first generalize this model to the Lennard-Jones (m,n) interaction between steps. Based on several important formulations of the non-local energy, we prove the existence, symmetry, unimodality, and regularity of the energy minimizer in the periodic setting. In particular, the symmetry and unimodality of the minimizer implies that it has a bunching profile. Furthermore, we derive the minimum energy scaling law for the original continnum model. All results are consistent with the corresponding results proved for discrete models by Luo et al. (Luo et al., Multiscale Model. Simul. 14:737 - 771, 2016).