Evolution equation for a thin epitaxial film on a deformable substrate

We consider a continuum model for the evolution of an epitaxially-strained dislocation-free thin solid film on a deformable substrate in the absence of vapor deposition. By using a thin film approximation we derived a nonlinear evolution equation. We examined the nonlinear evolution equation and found that there is a critical film thickness below which every film thickness is stable and a critical wave number above which every film thickness is stable. We developed a numerical method for the evolution of strained solid films under the thin film approximation. The numerical method was used to characterize the family of equilibrium shapes in terms of the film thickness and the spatial periodicity for both two-dimensional (island ridge) and three-dimensional (quantum dot) morphologies.