Reasoning of the self ordering in multilayer quantum dots: Part I: Elastic fields

Abstract The present work aims to explain the reasons for the elastic interactions between quantum dots (QDs) that give rise to their self ordering in different materials with cubic symmetry. This is done by a detailed consideration of the components of the elastic displacements, strains and stresses associated with a dilating QD on a substrate. The interaction between QDs is a result of the anisotropic response of the substrate that is due both to material and geometrical anisotropy. The elastic anisotropy is revealed by a distinction between the soft directions along which the strains are large and the displacements have a short range and the hard directions, along which the strains are small and displacements have a long range. It is shown that the elastic fields and the attractive interactions are dominated by the long range displacements. If a hard direction is normal to the free surface, a single hill-shaped protrusion is formed, which stretches the surface on top of a buried dot and gives rise to favored vertical stacking of QDs. If the normal to the free surface is not a hard direction, a square or a triangular protrusion is formed on the free surface where the hard directions intersect the surface. The formation of favored sites with a square or triangular symmetry gives rise to a staggered vertical stacking and, when the dilatation in these sites is intense enough, may induce lateral ordering of QDs into superlattices. As an outcome of the discussion, a criterion is suggested for the elastic properties of a substrate that can induce the formation of a superlattice.