Surface strain due to embedded epitaxial islands

We present a simple methodology for the evaluation of strain on the free surface above an embedded island. This work is motivated by the technological importance of self-organization of nanostructures. We first obtain the exact solution of the problem of a particle embedded in a half-space, using continuum theory. Then, we obtain a simple mathematical expression for the strain on the surface. In order to account for the influence of surface energy at the nanoscale, surface and interface stress are considered through the continuum Gurtin formulation. The results are compared with the classical approach, which typically ignores surface stress. This comparison illustrates a very dramatic (even qualitative) difference from the classical elasticity based prediction. It proves that surface stress must be taken into account when islands are small and softer than the substrate material. The procedure illustrated here should be useful to those who need usable expressions for surface strain and have no interest in solving the corresponding boundary value problem.