Facets and Roughening in Crystals and Quasicrystals.

The history and modern theory of crystal shapes is reviewed. The roughening phase transition, whereby a faceted interface changes from flat to macroscopically curved and microscopically rough, is studied in detail in the context of a sine-Gordon Hamiltonian for a one-component system and its generalization for a two-component system, such as SnTe along the (111) direction. For the two-component system, a new type of phase transition is described as a function of a decreasing subharmonic potential strength. The theory is generalized for aperiodic systems, such as for quasicrystals, predicting an infinite roughening temperature. Extensive Monte Carlo simulations are performed on a simple quasiperiodic three-dimensional solid-on-solid model of an ideal quasicrystal facet, both in equilibrium and during growth. A similar but periodic model is simulated for comparison. In equilibrium, interface widths, surface energies, and surface heat capacities are studied. While the periodic system exhibited a roughening transition, optimal (faceted at temperature T = 0) interfaces in the quasiperiodic system remain faceted at all temperatures. Although faceted, the interface width in the quasiperiodic system increases markedly with temperature through a series of pseudoroughening transitions. During growth, average surface heights and growth rates are monitored, as well as perspective views of the interfaces. Consistent with an infinite roughening temperature, at all temperatures the system grows via two -dimensional nucleation at low chemical potential driving forces. At sufficiently large driving forces the interfaces dynamically roughen.