Unusually stable helical coil allotrope of phosphorus

We have identified an unusually stable helical coil allotrope of phosphorus. Our {\em ab initio} density functional theory calculations indicate that the uncoiled, isolated straight 1D chain is equally stable as a monolayer of black phosphorus dubbed phosphorene. The coiling tendency and the attraction between adjacent coil segments add an extra stabilization energy of ${\approx}12$~meV/atom to the coil allotrope, similar in value to the ${\approx}16$~meV/atom inter-layer attraction in bulk black phosphorus. Thus, the helical coil structure is essentially as stable as black phosphorus, the most stable phosphorus allotrope known to date, and has a direct fundamental band gap similar to that of phosphorene monolayer structures. With an optimum radius of 2.4~nm, the helical coil of phosphorus may fit well and even form inside wide carbon nanotubes.