A continuum description of the energetics and evolution of stepped surfaces in strained nanostructures

As a departure from existing continuum approaches for describing the stability and evolution of surfaces of crystalline materials, this article provides a description of surface evolution based on the physics of the main feature imposed by the discrete nature of the material, namely, crystallographic surface steps. It is shown that the formation energy of surface steps depends on the sign of extensional strain of the crystal surface, and this behavior plays a crucial role in surface evolution. The nature of this dependence implies that there is no energetic barrier to nucleation of islands on the growth surface during deposition, and that island faces tend toward natural orientations which have no counterpart in unstrained materials. This behavior is expressed in terms of a small number of parameters that can be estimated through atomistic analysis of stepped surfaces. The continuum framework developed is then applied to study the time evolution of surface shape of an epitaxial film being deposited onto a substrate. The kinetic equation for mass transport is enforced in a weak form by means of a variational formulation. It is found that islands form without nucleation barriers and they evolve to shapes with natural surface orientations. The implications of the calculations are shown to be consistent with the behavior observed during deposition of semiconductor materials in recently reported experiments. Finally, it is verified that the predictions of the continuum model are essentially the same as those of the discrete step model for an isolated strained island. The development in this article is limited to two-dimensional plane strain deformation to keep the arguments transparent, but this is not a fundamental limitation of the approach.