Orientational competition in quantum dot growth in Si-Ge heteroepitaxy on pit-patterned Si(001) substrates

Towards precisely controlled nanostructure growth, patterned substrates are used as templates to direct heteroepitaxial self--assembly. This affects the size, shape and ordering of nanostructures which are formed as a consequence of the mismatch strain. In the well-studied case of Si-Ge heteroepiaxy on Si, the lattice mismatch leads to spontaneous formation of quantum dots. On patterned substrates, the competition between the length scale of the pattern and the intrinsic quantum dot size leads to rich behavior where the localization of dots can be modified with respect to the features of the patterns. We show by continuum modeling that, in cubic elastic materials such as silicon and germanium, there is also a competition between the pattern orientation and the elastically soft directions of the film which affects the precise location of quantum dots on the surface. When the pattern is between the elastically soft directions, the quantum dots can form purely in the narrow region directly between two neighboring pits, referred to as the saddle region. On the other hand, when the pattern is along the elastically soft directions, the quantum dots prefer to form in the region at the centre of four pits, referred to as the crown region. This resolves a discrepancy between theory and experiments and gives another dimension to control quantum dot formation in strained nanocrystalline systems.