Energy Scaling and Asymptotic Properties of One-Dimensional Discrete System with Generalized Lennard--Jones $(m,n)$ Interaction

Abstract

It is well known that elastic effects can cause surface instability. In this paper, we analyze a one-dimensional discrete system which can reveal pattern formation mechanism resembling the "step-bunching" phenomena for epitaxial growth on vicinal surfaces. The surface steps are subject to long-range pairwise interactions taking the form of a general Lennard--Jones (LJ) type potential. It is characterized by two exponents $m$ and $n$ describing the singular and decaying behaviors of the interacting potential at small and large distances, and henceforth are called generalized LJ $(m,n)$ potential. We provide a systematic analysis of the asymptotic properties of the step configurations and the value of the minimum energy, in particular, their dependence on $m$ and $n$ and an additional parameter $\alpha$ indicating the interaction range. Our results show that there is a phase transition between the bunching and non-bunching regimes. Moreover, some of our statements are applicable for any critical points of the energy, not necessarily minimizers. This work extends the technique and results of [Luo et al, SIAM MMS, 2016] which concentrates on the case of LJ (0,2) potential (originated from the elastic force monopole and dipole interactions between the steps). As a by-product, our result also leads to the well-known fact that the classical LJ (6,12) potential does not demonstrate step-bunching type phenomena.