A microscopic time scale approximation to the behavior of the local slope on the faceted surface under a nonuniformity in supersaturation

The morphological stability of a growing faceted crystal is discussed. We argue that the interplay between nonuniformity in supersaturation on a growing facet and anisotropy of surface kinetics derived from the lateral motion of steps leads to a faceted instability. Qualitatively speaking, as long as the nonuniformity in supersaturation on the facet is not too large, it can be compensated for by a variation of step density along the facet, and the faceted crystal can grow in a stable manner. The problem can be modeled as a Hamilton-Jacobi equation for height of the crystal surface. The notion of a maximal stable region of a growing facet is introduced for microscopic time scale approximation of the original Hamilton-Jacobi equation. It is shown that the maximal stable region keeps its shape, determined by profile of the surface supersaturation, with constant growth rate by studying large time behavior of solution of macroscopic time scale approximation. A quantitative criterion for the facet stability is given.