Growth and ordering of Si-Ge quantum dots on strain patterned substrates

Manipulating the strain distribution along the surface of a substrate has been shown experimentally to promote spatial ordering of self-assembled nanostructures in heteroepitaxial film growth without having to resort to expensive nanolithographic techniques. We present here numerical studies of three-dimensional modeling of self-assembly in Si-Ge systems with the aim of understanding the effect of spatially varying mismatch strain-fields on the growth and ordering of quantum dots. We use a continuum model based on the underlying physics of crystallographic surface steps in our calculations. Using appropriate parameters from atomistic studies, the (100) orientation is found to be unstable under compressive strain; the surface energy now develops a new minimum at an orientation that may be interpreted as the (105)facet observed in SiGe/Si systems. This form of surface energy allows for the nucleationless growth of quantum dots which start off via a surface instability as shallow stepped mounds whose sidewalls evolve continuously toward their low-energy orientations. The interaction of the surface instability with one- and two-dimensional strain modulations is considered in detail as a function of the growth rate. One-dimensional strain modulations lead to the formation of rows of dots in regions of low mismatch-there is some ordering within these rows owing to elastic interactions between dots but this is found to depend strongly upon the kinetics of the growth process. Two-dimensional strain modulations are found to provide excellent ordering within the island array, the growth kinetics being less influential in this case. For purposes of comparison, we also consider self-assembly of dots for an isotropic surface energy. While the results do not differ significantly from those for the anisotropic surface energy with the two-dimensional strain variation, the one-dimensional strain variation produces profoundly different behavior. The surface instability is seen to start off initially as stripes in regions of low mismatch. However since stripes are less effective at relaxing the mismatch strain they eventually break up into islands. The spacing of these islands is determined by the wavelength of the fastest growing mode of the Asaro-Tiller-Grinfeld instability. However the fact that such a growth mode is not observed experimentally indicates the importance of accounting for surface energy anisotropy in growth models.