Derivation of a continuum model for the long-range elastic interaction on stepped epitaxial surfaces in 2 + 1 dimensions

In heteroepitaxy, the mismatch of lattice constants in the crystal film and the substrate causes a misfit stress in the bulk of the film, driving the self-organization of the film surface into various nanostructures. Below roughening transition temperature, the epitaxial surface consists of terraces separated by atomic-height steps, and the misfit results in a long-range elastic interaction between surface steps. In this case, the surface morphology is determined by the motion of the steps, and the widely used continuum models for surface evolution above the roughening transition temperature do not apply directly. In this paper, we present a continuum model for this long-range elastic interaction on a stepped heteroepitaxial surface in 2+1 dimensions. The continuum model is derived rigorously by taking the continuum limit from the discrete model for the interaction between steps, thus incorporating the discrete features of the stepped surface.