Theoretical Study and Computer Simulation of Generalized Solid-on-solid Models

The subject of this thesis is investigation of the morphology of a crystal surface by means of statistical mechanics and Monte Carlo simulations. We employ solid-on-solid models, modified to include the effects of corner and edge energies of faceted surfaces. We also account for surface configurational entropy associated with various surface configurations (colonies of facets). This is an extension of the work of Herring who ignored corner and edge energies and effectively treated periodic hill-and-valley structures, which have no configurational entropy. The excess energies from the corners and edges of a surface also affect the equilibrium shape of very small crystals. These and other related effects are studied on solid-on-solid models for nearest-neighbor forces with central symmetry and additive bond energies. We obtain theoretical formulae for configurational entropy and theoretical distributions of the heights and lengths of facets on one-dimensional crystal surfaces (two-dimensional crystals). These results are tested by comparison with simulation data and good agreement results. A modified solid-on-solid model with nearest neighbor energy proportional to the nearest neighbor height difference raised to a power p is used to account for effects of corner and edge energies for two-dimensional surfaces (three-dimensional crystals). On an initially flat (100) surface, a slight change of p-value has a significant effect on surface morphology. Especially for p = 0:9, which corresponds to positive corner energies, a “macroscopic smoothing” transition from a faceted surface at low temperatures to a non-faceted surface at high temperatures is observed. This transition is only evident for surfaces that are initially tilted with respect to a close-packed surface. We also develop a symmetric solid-on-solid model that preserves crystal symmetry. For this symmetric model, the “macroscopic smoothing” transition for p = 0:9 is still observed on (111) and (112) surfaces, but now the surface structure is consistent with crystal symmetry. We find a hysteresis effect in these transitions, which is less pronounced for large systems. We calculate the correlation time of the surface by several different measures to study the relaxation of the system. A discrete Fourier analysis of the surface is implemented and we verify that there exists a long-wave fluctuation in the surface. We also study the distribution of facet areas and facet heights, which turns out to be exponential. A histogram method is employed to extend results at a given temperature to nearby temperatures.