  Contrary to recent reports, we show that the electronic properties of
phosphorene nanotubes are surprisingly rich and much more complex than
previously assumed. We find that all phosphorene nanotubes exhibit an intricate
direct-to-indirect bandgap transition as the nanotube diameter decreases - a
unique property not identified in any prior studies (which claimed either
direct or indirect band gaps only) that we uncover with large-scale DFT
calculations. We address these previous inconsistencies by detailed analyses of
orbital interactions, which reveal that the strain associated with decreasing
the nanotube diameter causes a transition from a direct to an indirect band gap
for all of the phosphorene nanotubes. We show that our findings are completely
general, and extensive calculations across several exchange-correlation
functionals verify our conclusions. Most importantly, our results and analyses
resolve a long-standing question on the electronic properties of phosphorus
nanotubes and brings closure to previously conflicting findings in these unique
nanostructures.
