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 provides a rich system for exploring the behavior of strongly interacting nanocrystals. In the temperature regime above 500 degreesC, there are two different (metastable) shapes of defect-free nanocrystals, termed pyramids and domes, that dominate the system depending on the temperature of the substrate during growth and the amount of Ge deposited. In contrast to the usual case considered in nucleation theory, the relaxation of the strain energy at the surface of the nanocrystals makes those surfaces stabilizing, i.e. the surface contribution to the free energy of the Ge nanocrystals is negative. Given that the edges of the nanocrystals are destabilizing (positive free energy), 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 minimum with respect to the size (volume) of the nanocrystal. At finite temperatures, this free energy yields a size distribution with a characteristic centroid, width, and skewness for each nanocrystal shape. The smaller pyramids transform into domes when they grow to the point where they can surmount a kinetic energy barrier between the two structures. However, the Ge nanocrystals also effectively repel one another strongly via the strain fields that are produced in the Si substrate. This repulsive interaction makes the ensemble of Ge nanocrystals a highly nonideal thermodynamic system and, in turn, makes the free energies of the nanocrystals a function of their number density, or equivalently a function of the amount of Ge deposited. The interplay of the stabilizing effect of the nanocrystal surfaces and the destabilizing influence of their repulsive interactions yields a complex behavior for the nanocrystal-size distributions that can nonetheless be modeled using simple thermodynamic expressions.