Weakly non-linear bifurcation analysis of pattern formation in strained alloy film growth

We consider a model for the growth of alloy films that includes surface diffusion, the effect of stresses due to misfit and stresses due to composition gradients when the alloy components have different sizes. Linear stability theory predicts a bifurcation from the planar homogeneous film to a non-planar compositionally modulated film at a critical deposition rate. In this paper, we perform a weakly non-linear bifurcation analysis of hexagonal and band patterns using an asymptotic analysis of the system close to its critical state. A novel feature of the analysis is that the formulation of the adjoint problem involves the solution of the composition-driven elasticity problem in the presence of surface diffusion and requires multiple scales in the growth direction. Our results characterize the transcritical bifurcation to hexagons and the pitchfork bifurcation to bands near threshold. Finally, we apply our results to the growth of Si_1-XGe_X films on Si_0.5Ge_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.