Determining absolute orientation-dependent step energies: a general theory for the Wulff-construction and for anisotropic two-dimensional island shape fluctuations

We describe an analytical form of the Wulff plot construction procedure and derive a general expression for the surface energy from the three-dimensional equilibrium crystal shape in generalized orthogonal curvilinear coordinates. Particular expressions in Cartesian, spherical polar, and circular cylindrical coordinates are also presented. Corresponding results for a two-dimensional (2D) island on a flat terrace provide relative orientation-dependent step energies within a scale factor λ, the equilibrium chemical potential of the island per unit area. In order to determine λ and, hence obtain absolute step energies, we have developed an exact theoretical approach, applicable to both isotropic and anisotropic 2D island shapes, relating the temporal change in island free energy to thermal fluctuations about the equilibrium shape.