Elastic fields and energies of coherent surface islands

We present simple models based on isotropic elasticity for determining the stress fields in the vicinity of, and the elastic energy associated with, coherent surface islands (i.e. a quantum dot or a nanorod). The first method treats the surface island as an areal point of dilatation and does not account for details of the island shape. Next, the finite element method is used to study simple island shapes such as spherical caps and cylinders, with a particular focus on the island aspect ratio. The latter is used in conjunction with the analytic results to develop empirical expressions for stress field and energy as functions of aspect ratio, which are somewhat insensitive to other features of the geometry. The analyses are used to assess the effect of lattice mismatch, dot volume and dot/surface contact area on the induced stresses and elastic energies. Furthermore, the interaction energy between surface islands is determined by finite element analyses. Outputs from these analyses are then used in an optimization of several cases of ordering of islands. The results show that for a range of idealized geometries, the shape of the surface island has little impact on the strain energy, and thereby the interaction energy.