Strain-induced formation and tuning of ordered nanostructures on crystal surfaces

Abstract

The energy of an array of 3D coherent strained islands on a lattice-mismatched substrate equals: E = ΔE V EL + ΔE RENORM FACETS + ΔE EDGES EL + E EDGES + E INTER, where ΔE V EL is the volume elastic relaxation energy, ΔE RENORM FACETS is the change of the surface energy of the system due to the formation of islands, which includes the strain-induced renormalization of the surface energy of the island facets and of the planar surface, ΔE EDGES EL is the contribution of the island edges to the elastic relaxation energy, E EDGES is the short-range energy of the edges, and E INTER is the energy of the elastic interaction between islands via the substrate. The energy ΔE EDGES EL ≈ - L −2 · ln L always has a minimum as a function of the size of L, and the total energy E = E( L) may have a minimum at an optimum size L opt. E INTER is the driving force for the latera ordering of 3D islands. Among different arrays of islands on the (001) - surface of a cubic crystal, the total energy is minimum for the periodic square lattice with primitive lattice vectors along the “soft” directions [100] and [010]. Thus, a periodic square lattice of equal-shaped and equal-sized 3D islands is, under certain conditions, the stable array of islands which do not undergo ripening. The theory explains the spontaneous formation of ordered arrays of 3D islands in the InAs GaAs(001) system.