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

TL;DR

This paper generalizes the Wulf-Kaishew theorem to describe how epitaxial stress influences the equilibrium shape of 3D crystals on mismatched substrates, revealing size-dependent shape changes and the impact of dislocations.

Contribution

It introduces a generalized theorem accounting for elastic relaxation and strain effects on crystal shape, extending prior models to include size dependence and dislocation effects.

Findings

Epitaxial strain causes thickening of the crystal's equilibrium shape.

The equilibrium shape varies with size, altering facet extensions.

Dislocation entry causes abrupt shape changes, disrupting size self-similarity.

Abstract

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: (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 in case of box shaped and truncated pyramidal crystals. Some experimental evidences are discussed.