Dynamics and stability of nanostructures on crystal surfaces

Morphological stability and evolution of nanostructures on crystalline surfaces is studied analytically and numerically. Crystalline surfaces present modeling challenges owing to the infinite stiffness of facets. This work uses variational formulations that naturally handle both rough and singular orientations. The physics of crystallographic steps is used to provide a fundamental basis for the energetics and kinetics of morphological evolution. Investigations are pursued to understand recent experiments where surface energy anisotropy is seen to play an important role. Specifically, the relaxation behavior of surface undulations, the evolution of grain-boundary grooves and Stranski-Krastanov growth in heteroepitaxial systems are studied. The relaxation of surface gratings is studied in conserved and non-conserved kinetic regimes. The results of these analyses provide resolution to several open questions in the literature. The relaxation of more complicated surface structures such as ion-sputtered ripples are also studied and shown to be in excellent agreement with experimental observations. The anisotropic theory of grain-boundary grooving is less well understood as compared to the isotropic theory---the latter has thus been conventionally used to determine grain-boundary energies and diffusion constants even for faceted grooves for which it is strictly not applicable. Faceted groove-roots lead to ambiguities in the equilibrium conditions at the triple-junction and in the relationship between the grain-boundary and facet energies. Analytical studies are used to show that this intricate relationship requires careful consideration of "chemical-torque" at the triple-junction. A simple graphical method for determining groove-profiles is developed. Numerical studies show that Mullins' one-fourth power-law holds even for faceted grooves at high junction-mobilities. Low-mobility junctions not only cause deviations from this scaling but also lead to kinetic groove-shapes that cannot be predicted from the usual evolution equations. Stranski-Krastanov growth in heteroepitaxial thin films provides a simple and cost-effective way of producing nanostructures such as quantum dots and quantum wires by self-assembly. Numerical studies of are used to understand the initial growth instability, spatial ordering and coarsening behavior of quantum dots in SiGe/Si systems.