Vertical correlations in superlattices using the Grinfeld method

In this paper, we analyse the energetics of a multilayered structures like, for instance, B/A/B/A substrate. It is well-known that a coherent pre-strained B layer on an A substrate will generally results in a corrugation of the free-surface of the B layer. This behavior is the result of stress relaxation in the B-layer and the phenomenon is known as the Asaro-Tiller-Grinfeld instability. We extend the methods used for a two-layer structure to a multilayered structure and the main application is the vertical correlation in superlattices. We analyse the energetics of a corrugated B layer which is grown on a A/B/A substrate, where the A layers are flat but the intermediate B layer is already corrugated. We show that the self-organization of the second B layer, due to elastic interactions in the bulk, depends on the corrugation of the first B layer and the generic best situation is that of a top-on-top (also called correlated layers) vertical alignment. We also prove that the interaction energy between two successive B layers attains a maximum at a critical thickness of the intermediate A layer. This interaction energy has the same order of magnitude as the elastic energy release due to free-surface corrugation at each upper surface of a B layer.