Reasons for self ordering in multilayer quantum dots Part II: Interaction energy

Two ways to calculate the elastic interaction between quantum dots (QDs) in the framework of linear elasticity are introduced and shown to vary in a similar way as the hydrostatic pressure. It is shown that the hydrostatic stress is the potential for the elastic interaction energy. The approach was used to estimate quantitatively the interaction energy between QDs in material systems that may form vertical order, anticorrelated order and FCC superlattice. The vertical interaction energy is very small compared with the thermal energy, nevertheless it is just enough to induce vertical ordering of QDs between layers. The attractive lateral interaction is an order of magnitude smaller than the vertical interaction, therefore ordering of multilatered QDs is made possible only by the vertical interactions. Several experimental observations are explained based on this understanding. The lateral interactions are small due to the small stresses set in the substrate after relief of the misfit strains in the free space. The vertical interactions are larger due to the large tensile stresses set up in the thin layer of matrix that separates an embedded QD from the free surface.