Derivation of a heteroepitaxial thin-film model

TL;DR

This paper derives a variational model for heteroepitaxial thin films on rigid substrates, capturing the balance between elastic stresses from lattice mismatch and surface energy effects, with initial regularity results for optimal profiles.

Contribution

It introduces a new variational model derived via 3-convergence and relaxation methods, unifying different descriptions of epitaxially strained thin films.

Findings

Model accounts for elastic and surface energy competition.

First regularity results for optimal film profiles.

Provides a rigorous mathematical derivation of the thin-film model.

Abstract

A variational model for epitaxially-strained thin films on rigid substrates is derived both by {\Gamma}-convergence from a transition-layer setting, and by relaxation from a sharp-interface description available in the literature for regular configurations. The model is characterized by a configurational energy that accounts for both the competing mechanisms responsible for the film shape. On the one hand, the lattice mismatch between the film and the substrate generate large stresses, and corrugations may be present because film atoms move to release the elastic energy. On the other hand, flatter profiles may be preferable to minimize the surface energy. Some first regularity results are presented for energetically-optimal film profiles.