Asymptotic analysis of patterns and islands in strained alloy films

We consider a recent model for the strained alloy film growth, which describes the moving boundary with underlying composition and strain, and the coupling between strain, composition, and the morphology of the free boundary. We use asymptotic analysis to study two problems of nontrivial film morphologies. The first problem considers the formation of surface and composition patterns during film growth. The second problem considers the formation of equilibrium drops or "islands" in the limit of slow growth.; In the problem of pattern formation during alloy film growth, linear stability theory predicts a bifurcation from the planar homogeneous film to a nonplanar compositionally modulated film at a critical deposition rate. We perform a weakly nonlinear analysis of this bifurcation for the case of hexagonal and band patterns using an asymptotic analysis of the system close to its critical state together with the method of multiple scales. Formulation of adjoint problem involves the solution of the composition driven elasticity problem and requires multiple scales in the growth direction. We find that hexagons have a transcritical behavior near threshold and both branches of the hexagon solutions are unstable. We find that bands, if restricted to the two dimensional case only, have a classic pitchfork behavior and stability. We apply our theoretical results to the growth of Si1- XGeX films on Si0.5Ge 0.5 substrates and describe how the amplitude of surface undulations and the amplitude of compositional modulations corresponding to hexagons and bands depend on Ge composition X.; In the problem of island formation in alloy films, we analyze the equilibrium morphology of a strained single component solid film for the case where it wets the substrate (Stranski-Krastanow growth). We determine the shape and the composition profile of a small axisymmetric three-dimensional equilibrium island, by considering an asymptotic solution based on the island height being much smaller than the island width. To solve asymptotically the elasticity problem in the film, we employ a thin domain scaling. We find that the strained alloy island has the same nondimensional shape as the single component film island (in the limit of small island size) and the composition profile of the alloy island is directly determined by the island shape. We also interpret our theoretical results to the growth of Si1- XGeX films on Si1- YGeY substrates, and make qualitative comparisons to the composition profile in large "dome" islands. Although our theory is valid only for small islands, we do obtain good qualitative agreement of our theory with these experiments.