Periodic overlayers and moire patterns: theoretical studies of geometric properties

Single crystal surfaces with periodic overlayers, such as graphene on hexagonal metal substrates, are found to exhibit, apart from their intrinsic periodicity, additional long-range order expressed by approximate surface lattices with large lattice constants. This phenomenon can be described as geometrically analogous to lateral interference effects resulting in periodic moire patterns which are characterized by two-dimensional moire lattices. Here we discuss in detail the mathematical formalism determining such moire patterns based on concepts of two-dimensional Fourier transformation including coincidence lattices. The formalism provides simple relations that allow one to calculate possible moire lattice vectors in their dependence on rotation angles alpha and scaling factors p1; p2 for periodic (p1 x p2)R alpha overlayers on substrate surfaces described by general Bravais lattices. Specific emphasis will be given to hexagonal lattices where experimental data are available.