Stability and electronic structure of phosphorus nanotubes

First-principles density-functional theory (DFT) calculations of single-walled phosphorus nanotubes constructed from the black-phosphorus (b-P) layered allotrope show that their strain energies per atom for radii above 0.6$\un{nm}$ are comparable to the strain energies predicted for experimentally observed single-walled carbon nanotubes with radii of 0.5$\un{nm}$. Our DFT calculations further predict that the nanotube structures are energetically more stable than the corresponding strips for radii larger than 0.55$\un{nm}$, suggesting that the synthesis of phosphorus nanotubes (PNTs) could be possible. We find that polarized basis sets including d functions are necessary for accurate treatment of the strain energy, and these basis sets lead to strain energies per atom substantially larger than DFT strain energies of single-walled carbon nanotubes at similar diameters. We have also found that all the PNTs studied are semiconducting regardless of their helicity, in contrast with the band structures of carbon nanotubes. The band gaps increase and converge to the value of the band gap of a black-phosphorus single puckered layer, 1.8$\un{eV}$, as the radius is increased. For a fixed radius, the band gaps increase when increasing the chiral angle. Our DFT calculations are in very good qualitative and quantitative agreement with earlier density-functional tight-binding calculations of black-phosphorus single puckered layers and nanotubes