Tuning and breakdown of faceting under externally applied stress

The theory of thermodynamic faceting is developed for an epitaxial film grown coherently on a lattice-mismatched substrate. The situation is considered where the planar top surface of the epitaxial film in the absence of the lattice mismatch (\ensuremath{\Delta}a=0) is unstable against faceting, and the stable state of the surface is a periodic array of facets. It is shown that, for a finite lattice mismatch (\ensuremath{\Delta}a\ensuremath{\ne}0), the continuous epitaxial film with a periodically faceted top surface is a metastable state of the heterophase system. The global energy minimum corresponds then to a periodic system of coherent strained islands. If attaining the global energy minimum is kinetically forbidden, the metastable continuous epitaxial film with a periodically faceted top surface is formed. In the case where the period of the faceted structure without external stress ${\mathit{L}}_{0}$ exceeds the order of \ensuremath{\approxeq}50 \AA{}, the dependence of the period L on the lattice mismatch is determined by the linear theory of elasticity. The period L of the metastable faceted structure increases with \ensuremath{\Vert}\ensuremath{\Delta}a\ensuremath{\Vert} for both tensile and compressive mismatch-induced strain. The dependence of L on \ensuremath{\Delta}a gives a possibility of controlling the period of faceting by varying \ensuremath{\Delta}a. If the lattice mismatch exceeds a certain critical value [\ensuremath{\Vert}\ensuremath{\Delta}a\ensuremath{\Vert}>(\ensuremath{\Delta}a${)}_{\mathit{c}}$], the breakdown of formation of metastable faceted structures occurs; the metastable state disappears, and the surface shape is governed by kinetic mechanism.In the case where the period of the faceted structure without external stress is ${\mathit{L}}_{0}$\ensuremath{\lesssim}50 \AA{}, the dependence of L on \ensuremath{\Delta}a is determined by nonlinear elastic effects. The period L increases for one sign of \ensuremath{\Delta}a up to the breakdown of formation of metastable faceted structures and decreases for the other sign of \ensuremath{\Delta}a, where the macroscopic faceting transforms gradually into a microscopic surface reconstruction and the surface becomes apparently flat. The typical critical value of the lattice mismatch for nanometer-scale faceting varies from (\ensuremath{\Delta}a/a${)}_{\mathit{c}}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}4}$ for L\ensuremath{\sim}${10}^{3}$ \AA{} to (\ensuremath{\Delta}a/a${)}_{\mathit{c}}$\ensuremath{\sim}${10}^{\mathrm{\ensuremath{-}}2}$ \AA{} for L\ensuremath{\sim}10 \AA{}. A similar dependence of faceting on externally applied stress occurs for a loaded sample.