Tiling Phosphorene

We introduce a scheme to categorize the structure of different layered phosphorene allotropes by mapping their non-planar atomic structure onto a two-color 2D triangular tiling pattern. In the puckered structure of a phosphorene monolayer, we assign atoms in ``top'' positions to dark tiles and atoms in ``bottom'' positions to light tiles. Optimum $sp^3$ bonding is maintained throughout the structure when each triangular tile is surrounded by the same number $N$ of like-colored tiles, with $0{\le}N{\le}2$. Our {\em 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 non-hexagonal rings and 2D quasicrystals with no translational symmetry, which we predict to be nearly as stable as the hexagonal network.