Thermodynamics of the size and shape of nanocrystals: Epitaxial Ge on Si(001)

The growth and evolution of strained epitaxial Ge on a Si(001) surface prov ides a rich system for exploring the behavior of strongly interacting nanoc rystals. In the temperature regime above 500 degreesC, there are two differ ent (metastable) shapes of defect-free nanocrystals, termed pyramids and do mes, that dominate the system depending on the temperature of the substrate during growth and the amount of Ge deposited. In contrast to the usual cas e considered in nucleation theory, the relaxation of the strain energy at t he surface of the nanocrystals makes those surfaces stabilizing, i.e. the s urface contribution to the free energy of the Ge nanocrystals is negative. Given that the edges of the nanocrystals are destabilizing (positive free e nergy), the interaction of the surfaces and edges of the nanocrystals in an ensemble renders an internal free energy for the system that has a local m inimum with respect to the size (volume) of the nanocrystal. At finite temp eratures, this free energy yields a size distribution with a characteristic centroid, width, and skewness for each nanocrystal shape. The smaller pyra mids transform into domes when they grow to the point where they can surmou nt a kinetic energy barrier between the two structures. However, the Ge nan ocrystals also effectively repel one another strongly via the strain fields that are produced in the Si substrate. This repulsive interaction makes th e ensemble of Ge nanocrystals a highly nonideal thermodynamic system and, i n turn, makes the free energies of the nanocrystals a function of their num ber density, or equivalently a function of the amount of Ge deposited. The interplay of the stabilizing effect of the nanocrystal surfaces and the des tabilizing influence of their repulsive interactions yields a complex behav ior for the nanocrystal-size distributions that can nonetheless be modeled using simple thermodynamic expressions.