Equilibrium theory of strained epitaxial layers

We develop a simple theory of the equilibrium stability of strained epitaxial layers on a rigid substrate based on a generalization of the continuum theory. Each layer is treated as a continuum elastic medium while the coupling between layers is treated in a discrete manner. Using a periodic parabolic interaction between adjacent layers, we obtain exact numerical results for the stability boundary of the epitaxial phase, δ(N), expressed as a function of the misfit δ and the number of layers N. In addition, we develop a variational approach which agrees very well with the exact results. Our method interpolates between a few layers and the thick-film limit of the continuum theory, and in this limit we recover the standard result for the relation between misfit and critical thickness. Considerable deviations from continuum theory can occur in the thin-film limit. For very weak coupling to the substrate, we find δ(N)∝$N^{\mathrm{-}1/2}$. The present approach has the advantage of allowing different misfits and elastic constants for each layer and arbitrary variations in the interlayer couplings.