Simulating interfacial anisotropy in thin-film growth using an extended Cahn-Hilliard model

We present an extended Cahn-Hilliard model for simulating interfacial anisotropy in thin-film dynamics by incorporating high-order terms in the energy from an expansion of the energy about an equilibrium state, following earlier work by Abinandanan and Haider [Philos. Mag. Sect. A81, 2457 (2001)]. For example, to simulate SiGe/Si thin films, where diamond cubic symmetry is needed, fourth order derivatives are included in the energy. This results in a sixth order evolution equation for the order parameter. For less symmetric crystals, one needs to add terms of higher order than fourth order. One advantage of this approach is its intrinsic regularized behavior. In particular, even for strongly anisotropic surface energy, sharp corners will not form and the extended anisotropic Cahn-Hilliard equations are well-posed. For this system we develop an energy-stable numerical scheme in which the energy decreases for any time step. We present two-dimensional (2D) and three-dimensional (3D) numerical results using an adaptive, nonlinear multigrid finite-difference method. We find excellent agreement between the computed equilibrium shapes using the new model and results from an analysis associated with a Wulff construction for energy minimization. The model predictions also compare well with experimental results for silicon voids. In the context of thin films, we observe the formation of interconnected ridges, wires, and fortresses similar to those observed in SiGe/Si thin films.