Growth of quantum dots (QDs) in multilayered semiconductors yields ordered arrays both in lateral directions and in the vertical direction. The QDs are observed to rearrange in different types of superlattices. The ordering phenomenon is attributed to substrate mediated elastic interactions among the QDs and between the dots and the substrate. The elastic stress fields induced by the mismatch between the lattice parameters of the QDs and the matrix contain favored sites for new dots to nucleate and grow due to the elastic anisotropy of the materials. These sites depend on the elastic anisotropy factor and the ratio between the lateral dot distances and the spacer  thickness in the vertical direction.

The present work aims at explaining the direct cause for self ordering into different superlattice symmetries in materials with cubic crystallographic symmetry. This is done by exploring the elastic displacements, strains and stress fields generated by epitaxial dots with pure dilatational misfit.  Three locations relative to the substrate are : dots embedded in infinite matrix, dots buried near a free surface and dots on a free surface. The study of the three cases qualitatively explains the formation of different superlattices as function of the elastic anisotropy and the geometrical parameters of the system. We also develop three methods for the quantitative computation of the interaction energy between the QDs that are important to calculate the rate of ordering.