Tiling phosphorene.

We present a scheme to categorize the structure of different layered phosphorene allotropes by mapping their nonplanar atomic structure onto a two-color 2D triangular tiling pattern. In the buckled structure of a phosphorene monolayer, we assign atoms in "top" positions to dark tiles and atoms in "bottom" positions to light tiles. Optimum sp3 bonding is maintained throughout the structure when each triangular tile is surrounded by the same number N of like-colored tiles, with 0≤N≤2. Our ab initio density functional calculations indicate that both the relative stability and electronic properties depend primarily on the structural index N. The proposed mapping approach may also be applied to phosphorene structures with nonhexagonal rings and 2D quasicrystals with no translational symmetry, which we predict to be nearly as stable as the hexagonal network.