Faceting of Equilibrium and Metastable Nanostructures: A Phase-Field Model of Surface Diffusion Tackling Realistic Shapes

Several crystalline structures are metastable or kinetically frozen out-of-equilibrium states in the phase space: When the corresponding lifetime is sufficiently long, typical equilibrium features such as regular and extended faceting can be observed. However, interpreting the extension of the facets and the overall shape in terms of a standard Wulff analysis is not justified. Here, we introduce a convenient general formulation of the anisotropic surface energy density, combined with a suitable phase-field model of surface diffusion. This allows for the investigation of the evolution toward equilibrium of realistically shaped nanostructures, describing an actual kinetic path and including the proper faceting. Numerical solution by the finite element method allows for efficient simulations even for the so-called strong anisotropy condition. After illustrating applications yielding equilibrium crystal shapes (corresponding to the Wulff construction), we focus our attention on faceting of structures in long-lived metastable states. The generality and numerical robustness of the approach is proven by a few applications to crystalline systems of great importance (quantum dots, quantum wires, patterned substrates) in present materials science.