High-order surface relaxation versus the Ehrlich–Schwoebel effect

We consider a class of continuum models of epitaxial growth of thin films with two competing mechanisms: (1) the surface relaxation described by high-order gradients of the surface profile and (2) the Ehrlich–Schwoebel (ES) effect which is the asymmetry in the adatom attachment and detachment to and from atomic steps. Mathematically, these models are gradient-flows of some effective free-energy functionals for which large slopes are preferred for surfaces with low energy.We characterize the large-system asymptotics of the minimum energy and the magnitude of gradients of energy-minimizing surfaces. We also show that, in the large-system limit, the renormalized energy with an infinite ES barrier is the Γ-limit of those with a finite one, indicating the enhancement of the ES effect in a large system. Introducing λ-minimizers as energy minimizers among all candidates that are spatially λ-periodical, we show the existence of a sequence of such λ-minimizers that are in fact equilibriums. For the case of a finite ES effect, we prove the well-posedness of the initial-boundary-value problem of the continuum model and obtain bounds for the scaling laws of interface width, surface slope and energy, all of which characterize the surface coarsening during the film growth. We conclude with a discussion on the implications of our rigorous analysis.