Continuum model for the long-range elastic interaction on stepped epitaxial surfaces in2+1dimensions

Abstract

In heteroepitaxy, the mismatch of lattice constants between the crystal film and the substrate causes misfit strain and stress in the bulk of the film, driving the surface of the film to self-organize into various nanostructures. Below the roughening transition temperature, an epitaxial surface consists of facets and steps and changes its morphology by lateral motion of steps. In this paper, we present a 2 + 1-dimensional continuum model for the long-range elastic interaction on stepped surface of a strained film. The continuum model is derived rigorously from the discrete model for the interaction between steps; thus it incorporates the discrete features of the stepped surfaces. Examples show that our continuum model is much more accurate as an approximation to the discrete model than the traditional continuum approximation. Moreover, in the linear instability of a planar surface, our continuum model gives the transition from step bunching instability to step undulation instability as the distance between adjacent steps increases, which agrees with the experimental observations and the results of discrete models and is missing using the traditional continuum approximation. Numerical simulations of the surface evolution using our model in the nonlinear regime show several different surface morphologies, including the morphology of step bunching which cannot be obtained using the traditional continuum approximation.

Energy Scaling and Asymptotic Properties of Step Bunching in Epitaxial Growth with Elasticity Effects

“…The continuum model is able to predict linear instability of a uniform step train and evolution of a stepped surface in the nonlinear regime, which are in agreement with the results of the discrete model. Zhu, Xu, and Xiang [22] performed linear instability analysis and numerical simulations using the 2+1 dimensional continuum model. Dal Maso, Fonseca, and Leoni [5] and Fonseca, Leoni, and Lu [9] proved the existence and regularity of weak solutions of Xiang and E's 1+1 dimensional continuum model [19,20].…”

Existence, uniqueness, and energy scaling of 2+1 dimensional continuum model for stepped epitaxial surfaces with elastic effects

“…The first term in E m [h] is the traditional expression of the misfit elastic energy above the roughening transition temperature on surfaces [3,10,21,23]. The second term in E m [h] is the contribution to the step line energy from the force monopole interaction, and this additional term incorporates the atomic feature of the stepped surfaces [26][27][28][29].…”

“…Instability analysis and the numerical simulations based on this continuum model performed by Xiang and E [27] showed that this continuum model is able to correctly describe the step bunching instabilities compared with the results of discrete models and experimental observations. Xu and Xiang [28], Zhu, Xu and Xiang [29] further developed a 2+1 dimensional continuum model for the stepped surfaces with elastic effects, which is able to account for both the step bunching and step meandering instabilities as well as their competition. Kukta and Bhattacharya [12] proposed a three-dimensional model for step flow mediated crustal growth under stress and terrace diffusion.…”

A continuum model for the dynamics of dislocation arrays

“…We will derive a continuum model from the discrete model given in equation (2.8). The derivation method is a generalization of those used to obtain continuum models for the Peach-Koehler force on dislocations in a single slip plane [46,53] and the elastic interaction on epitaxial surfaces [45,48,49,55].…”