A New Method for Simulating Strongly Anisotropic Cahn-Hilliard Equations

We present a new approach for modeling strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn-Hilliard-type equations. Such problems arise during the growth and coarsening of thin films. When the surface anisotropy is sufficiently strong, sharp corners form and unregularized anisotropic Cahn-Hilliard equations become ill-posed. Our models contain a high order Willmore regularization to remove the ill-posedness. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. We present 2D numerical results using an adaptive, nonlinear multigrid finite-difference method. In particular, we find excellent agreement between the computed equilibrium shapes using the Cahn-Hilliard approach, with a finite but small Willmore regularization, and an analytical sharp-interface theory recently developed by Spencer [1]. Introduction In order to overcome the fundamental limits that will prevent continued shrinking of microelectronics, revolutionary new quantum computing schemes will be needed within the next two decades. However, if we are to realize the extraordinary information processing potential offered by quantum logic, control at near-atomic length scales is required to build devices based on ordered assemblies of nanoscale quantum dots. While such structures are well beyond the capabilities of standard lithographic techniques, directed self-assembly approaches are already demonstrating how complex 3D nanostructures can be constructed via manipulation of the natural processes associated with their growth [4]. Self-organized semiconductor nanostructures are a promising inexpensive and effective approach to manufacture novel electronic and magnetic devices. Quantum dots (pyramids/domes) and other nanostructures such as quantum dot molecules (pits surrounded by four islands) (see Figure 1), for example, have a sharper density of states than higherdimensional structures and may be used in diode lasers, amplifiers, biological sensors and data storage devices. (a) (b) Fig. 1 (a) Quantum dot pyramids (light) and domes (dark) [3]; (b) Quantum dot molecules [5] The production of quantum-dot-based devices is still challenging. A fundamental understanding of the self-organization process (nucleation, growth and coarsening) during epitaxial growth is necessary to achieve controlled quantum scale structures. The influence of strain, surface energies and kinetics on the surface evolution have to be considered and may play a significant role in the evolution and equilibria. These effects can be analyzed by modeling and numerical studies that complement experimental investigations. The long-term goal is the production of large numbers of controlled self-assembly and spatially ordered nanostructures with narrow size distribution. In many technologically important materials (e.g. Si-based materials for electronic devices and Fe-based materials for magnetic devices), the surface energy is strongly anisotropic and there are missing orientations. This poses a problem for standard sharp-interface or CahnHilliard type models as the equations become ill-posed (e.g. see [12,13] and the references therein). In this work, we present a new approach for modeling strongly anisotropic crystal and epitaxial growth using regularized, anisotropic Cahn-Hilliard-type equations. Our models contain a high order Willmore regularization [2] to remove the ill-posedness. A key feature of our approach is the development of a new formulation in which the interface thickness is independent of crystallographic orientation. We present 2D numerical results using an adaptive, nonlinear multigrid finite-difference method. In particular, we find excellent agreement between the computed equilibrium shapes using the Cahn-Hilliard approach, with a finite but small Willmore regularization, and an analytical sharp-interface theory recently developed by Spencer [1]. Background The interfacial energy γ can be a function of the crystallographic orientation of the interface, Figure 2. In 2D, the parametric form of the surface or interfacial energy can be γ(θ) =1+ εm cos(mθ) (1) where γ is the interfacial energy density, εm is the anisotropy parameter, m is number of fold symmetry, θ is the crystallographic orientation (e.g. the angle the normal vector makes with a coordinate axis). In 2D, total energy for the sharp interface model is