Thermodynamics and statistical mechanics of the faceting of stepped Si(111)

The thermodynamic formalism needed to analyze the faceting of surfaces is presented in a format readily applicable to experimental observations. Statistical mechanical predictions for the relationship of step and kink energies to the variation of the surface tension with orientation are reviewed and combined with the thermodynamic formalism to illustrate the physical mechanisms which can cause equilibrium faceting. The results show that faceting may be initiated by any process (such as reconstruction or chemical adsorption) which changes surface free energies by amounts comparable to the small (meV/Å) energies of step-step interactions. The application of these methods is illustrated by analysis of our prior observations of faceting of stepped Si(111) surfaces, which is driven by the formation of the surface reconstruction. Using a nearest-neighbor square-lattice model with elastic step-step interactions to describe the surface, the only adjustable parameters are the kink energies and the step-step interaction energies. Given previous measurements of the magnitudes of the step-step interactions, we are able to reproduce quantitatively the three different phase diagrams which occur on different azimuths of vicinal Si(111). We extract from the fit to the data the variation in the difference of surface tension of the 7 × 7 and “1 × 1” −(111) surface with temperature, and values for the anisotropy in the step energy in the presence of the 7 × 7 reconstruction. We find that steps in the [2̄11] direction are at least 10 meV/Å more costly than steps on the unreconstructed surface, and steps in the [21̄1̄] direction are approximately 5 meV/Å less costly than steps on the unreconstructed surface. We discuss the requirements for determining absolute energies for steps and kinks given thermodynamic observations.