High-Order Surface Relaxation vs. 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 Γlimit 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 λminimizers 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 implications of our rigorous analysis. MSC: 34K26, 49J45, 74G65, 74K30. PACS: 68.35.Ct; 68.43.Jk; 81.15.Aa.