Equilibrium nano-shape changes induced by epitaxial stress (generalised Wulf–Kaishew theorem)

A generalised Wulf–Kaishew theorem is given describing the equilibrium shape (ES) of an isolated 3D crystal A deposited coherently onto a lattice mismatched planar substrate. For this purpose a free polyhedral crystal is formed then homogeneously strained to be accommodated onto the lattice mismatched substrate. During its elastic inhomogeneous relaxation the epitaxial contact remains coherent so that the 3D crystal drags the atoms of the contact area and produces a strain field in the substrate. The ES of the deposit is obtained by minimising at constant volume the total energy (bulk and surface energies) taking into account the bulk elastic relaxation. Our main results are as follows. (1) Epitaxial strain acts against wetting (adhesion) so that globally it leads to a thickening of the ES. (2) Owing to strain the ES changes with size. More precisely the various facets extension changes, some facets decreasing, some others increasing. (3) Each dislocation entrance, necessary for relaxing plastically too large crystals abruptly modifies the ES and thus the different facets extension in a jerky way. (4) In all cases the usual self-similarity with size is lost when misfit is considered. We illustrate these points for box-shaped and truncated pyramidal crystals. Some experimental evidence is discussed.