Scaling Law and Reduced Models for Epitaxially Strained Crystalline Films

Abstract

A variational model for the epitaxial deposition of a film on a rigid substrate in the presence of a crystallographic misfit is studied. The scaling behavior of the minimal energy in terms of the volume of the film and the amplitude of the misfit is considered, and reduced models in the various regimes are derived by Γ-convergence methods. Depending on the relation between the thickness of the film and the amplitude of the misfit, the surface or the elastic energy contribution dominate, and in the critical case the two contributions balance. In particular, the formation of islands is proven if the amplitude of the misfit is large compared to the volume of the film.

Study of Island Formation in Epitaxially Strained Films on Unbounded Domains

“…We point out that, as noted before, in contrast to many previous works (see [10,14,18]), we do not assume a periodic pattern of islands and do not restrict to a single island on a compact domain. The main difference is that in (1.1) the support of the height profile function h may be unbounded, which can lead to a loss of compactness for low energy sequences.…”

“…We study analytically a two-dimensional variational model introduced in [31] (see also [3,10,14]), to describe the surface morphologies of the epitaxially strained film. The main difference to the previous analytical works (see [10,14,18]) is that the model explicitly allows for wetting, which corresponds to film profiles with unbounded support. We assume that the film occupies a domain Ω h which can be described as a subgraph of a height profile function h : R → [0, ∞), i.e., Ω h := {x := (x, y) ∈ R 2 : 0 < y < h(x)}.…”

Derivation of a heteroepitaxial thin-film model

“…In [14] the existence and the shape of island profiles, which enforces the presence of nonzero contact angles, has been analyzed in the constraint of faceted profiles. In [18] a mathematical justification of island nucleation was provided by deriving scaling laws for the minimal energy in terms of e 0 and the film volume, and then extended in [2] to the situation of unbounded domains, in the two regimes of small-and large-slope approximations for the profile function h. Finally, the evolutionary problem for thin-film profiles has been studied in dimension two in [11] for the evolution driven by surface diffusion, and in [22] for the growth in the evaporation-condensation case (see also [7,13] for a related model describing vicinal surfaces in epitaxial growth). Recently the analysis of [11] has been extended to three dimensions in [12].…”

The Surface Diffusion Flow with Elasticity in the Plane

“…They are very well studied in the physical and numerical literature, see for instance [26,29,40,41,42]. Concerning rigorous mathematical analysis, we refer to [6,8,10,17,21,25,28] for some existence, regularity and stability results 1 related to a variational model describing the equilibrium configurations of two-dimensional epitaxially strained elastic films, and to [9,16] for results in three-dimensions. A hierarchy of variational principles to describe equilibrium shapes in the aforementioned contexts has been introduced in [30].…”