The evolution of (001) and (111) facets for the epitaxial growth inside submicron trenches is systematically studied in this report. The analysis with the method of “Lagrange multiplier” indicates the equilibrium crystal shape. In the case of non-equilibrium without external fluxes, we employed
the “weighted mean curvature” method to mathematically model the inter-facet migration rate for two extreme kinetic cases: “surface diffusion limited” and “surface attachment/detachment limited.” Coupled with external supply of atoms, the self-limited behavior of facet size is theoretically predicted. Moreover, we find that the self-limited stable facet size in trenches of different widths
has a specific relationship determined by the surface energy ratio, kinetic rate ratio, and isolated growth rate difference. The two limited cases could be discriminated according to the mathematical fitting of one exponent in this relationship based on the stable facet size in trenches of different widths.