The bending strain energy E_b can be considered as the energy of wrapping the BP layer into a nanotube and defined as

where N and n are the numbers of atoms and sheet unit cells in a nanotube unit cell, respectively; E_s and E_nt denote the free energies of a unit cell in the BP layer and the corresponding nanotube, respectively. Figure  2 shows snapshots of BP wrapped from its initial planar layer into tubes along the armchair direction (armchair bending) and zigzag direction (zigzag bending), respectively.

Fig.  2.

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	Fig.  2. Structures of BP in bent states. Bending moment M is perpendicular to (a) armchair direction (armchair bending) and (b) zigzag direction (zigzag bending). ArmPNT means the armchair single-walled phosphorus nanotube, and zigPNT the zigzag one.

The radius of curvature is first chosen as an independent reference variable to facilitate the discussion, which is defined as beginequation

and r > 2.1  Å is taken for the two typical bending modes to guarantee that the initial geometric structures are in circular shape. Changes of bending strain energy with radius of curvature are shown in Fig.  3.

Fig.  3.

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Fig.  3. Bending strain energies changing with radius of curvature for BP bent into nanotubes. (a) The curve of bending strain energy (in pink) in zigzag bending induced phase transition zone (PT zone) becomes significantly lower than the curve without phase transitions (in blue). For tubes with radius of curvature larger than the critical value rcr, no bending-induced phase transition (NPT zone) appears. (b) Comparison of bending strain energies of armchair and zigzag tubes between before and after the phase transition. Blue diamonds and circles indicate armPNT and zigPNT before phase transition, and pink pentagrams indicate phase transition phosphorus nanotube (zptPNT) under zigzag bending.

Figure  3(a) shows that for zigPNTs smaller than zigPNT(0, 10), the bending strain energy curve presents a large bifurcation and a new branch with much lower energies. Figure  3(b) shows a robust hyperbolic law for armchair bending, or armPNTs, even down to armPNT(4, 0) at a radius of 3  Å . This indicates that bending-induced buckling occurs only due to large curvature along the zigzag direction, which has been in detail analyzed in the light of continuum elasticity theory in our previous work.^[26] It is surprising that the zigPNTs with small radius of curvature transfer into zptPNTs of interesting structural phases (see Fig.  5) with significantly lower energy. Recently, Guan et al.^[27] presented a paradigm in constructing very stable faceted nanotube and fullerene structures by laterally joining nanoribbons or patches of different planar phosphorene phases, so it is necessary to further discuss the zptPNTs.

In order to examine thermal stabilities of zptPNTs, especially the smallest zptPNT(0, 5), the BOMD simulations are also carried out. As shown in Fig.  4, the structure of zptPNT(0, 5) is still intact at 500  K after 10  ps of simulation but shows some deformations. At low temperature of 50  K, the overall structure of zptPNT(0, 5) is well kept after 10-ps simulation. Our BOMD simulation on zptPNT(0, 7) also shows a similar stability and the detailed results are presented in Appendix A, although the zptPNT (0, 5) is stabler. As shown in the appendixes, the stabilities of other zptPNTs are further investigated by using different methods. The phonon dispersions of zptPNT(0, 4) and zptPNT(0, 8) are calculated and presented in Appendix B, which shows no negative phonon spectrum. What is more, we check the stability of the zptPNT(0, 9) by post-hydrogenation passivation (see Appendix C for the details).

Fig.  4.

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	Fig.  4. DFT-based BOMD simulations at 10  ps for the zptPNT (0, 5) [top view on the left and side view on the right] controlled at (a) low temperature of 50  K and (b) 500  K.

In Fig.  3(a), the bending-induced phase transition zone (PT zone) under zigzag bending is separated from the non-phase transition zone (NPT zone) by a critical radius of curvature r_cr = 5.25  Å of zigPNT(0, 10). Although zigPNTs still exist in the PT zone as shown by the blue curve in the figure, they are metastable by comparison with the corresponding zptPNTs, which have significantly lower energies. However, no metastable zigPNT smaller than zigPNT(0, 6) (with radius of 3.15  Å ) exists.

From the isosurface electronic density diagram of zptPNTs in Fig.  5, three obvious features can be observed which are helpful for understanding the bending-induced phase transition.

Fig.  5.

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	Fig.  5. Isosurface electronic density diagrams with sectional drawings and relaxed structures of the phase transition nanotubes zptPNTs (top and side views). The detailed structures of the nanotubes are decomposed into I, II, III, IV, and Arm building-blocks. The isosurface level is set to be 0.40  e/Å 3.

(i) The detailed structures of the zptPNTs can be decomposed into I, II, III, IV and armchair-like building-blocks (indicated as Arm) based on their chemical bonding states^{[6, 28]} except for zptPNT(0, 5) which will be further discussed later. The zptPNT(0, 7) is a combination of IV and Arm building-blocks, and zptPNT(0, 9) is composed of I and Arm, and zptPNT(0, 8) consists of II and III. The zptPNT(0, 4) is made up solely of II building-blocks. The tubes with odd numbered BP unit cells are apt to transfer toward armchair-like phases together with I or IV building-blocks, but the tubes with even numbered BP unit cells transfer into phases II or a combination of II and III.

(ii) It is known clearly that phosphorus atoms can hold a maximum of 10 valence electrons in the building-blocks except the Arm one.^[29] Nitrogen atoms can hold a maximum of eight valence electrons. Phosphorus, however, has empty 3d atomic orbitals that can be used to expand the valence shell to hold 10 or more electrons. Thus, phosphorus can react with chlorine to form both PCl₃ and PCl₅. Phosphorus can even form the ion, in which there are 12 valence electrons on the central atom. As an example, partial-wave density of states (PDOS) of zptPNT(0, 8) for the typical atoms in the building-blocks is analyzed in detail in Fig.  6, where the s-, p-, and d-projected local densities of states are shown clearly at atom-a and atom-c (corresponding panels  (a) and (c)). But the d-projected local densities of states cannot be observed in atom-b (corresponding panel  (b)). Appendix  D also presents the PDOS of zptPNT(0, 5) in Fig.  S4 and no d-projected local densities of states can be observed.

(iii) Looking back at Figs.  1 and 2(a), it can be seen that the III and IV blocks both have the distorted unit cells in BP along the zigzag direction and the armPNTs are made up solely of Arm blocks. So the structures of zptPNT(0, 7) and zptPNT(0, 9) are very close to those of armPNT(5, 0) and armPNT(6, 0), but the blocks IV and I make the former slightly higher in energy than the latter as shown in Fig.  3(b). The block II has a double bond which makes the bending strain energies of zptPNT(0, 4) and zptPNT(0, 8) higher than those of the other zptPNTs but still significantly lower than those of the corresponding zigPNTs. For block I, the dangling bond can be passivated by hydrogen as shown in Appendix C and the zptPNT(0, 9) is stretched to an ellipse by two hydrogen atoms. This implies that zptPNT(0, 9) might be unstable in hydrogen environment. This is similar to the large distortion of lattice structure observed when water molecules are placed on the surface of black phosphorus.^[30]

Owing to their one-dimensional characteristic, metallic tubes are expected to undergo Peierls distortions at T = 0  K^[31] and the debate about whether carbon nanotubes should exhibit Peierls distortions arose shortly after their discovery as soon as their metallic behavior was predicted.^{[32, 33]} In carbon nanotubes Peierls distortions can lead to occurrence of bond alternation.^{[34, 35]} To probe the possibility of Peierls distortion in the present cases, the band structures of zigPNTs, zptPNTs, and armPNTs are calculated and shown in Fig.  7.

Fig.  6.

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	Fig.  6. Partial-wave density of states (PDOS) of zptPNT(0, 8) for the typical atoms in the building-blocks. The s-, p-, and d-projected local densities of states are shown in each typical atom.

Fig.  7.

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	Fig.  7. Band structures of (a) zigPNTs, (b) zptPNT(0, 5) and armPNTs. The dark green lines indicate that the characteristic energy bands consist of local d orbital levels in phosphorus atoms and the band gap in zptPNT(0, 5) is 0.75  eV marked by dark green.

Figure  7(a) shows that the band structures of smaller zigPNTs are significantly different from those of zigPNT(0, 10) and show metallic features. There are two energy bands close to each other as marked by the dark green lines, rather flat for zigPNT(0, 8) and zigPNT(0, 9), growing out of the highly localized d orbital levels in phosphorus atoms, similar to those observed by Seifert and Herná ndez in BP allotrope tubes where the d-basis functions also play an important role in describing the conduction band states, ^[36] but more localized here. When the local d orbital levels are close to the Fermi level, one of the lone-pair electrons in the sp³ orbitals can be easily turns into one of the d orbitals so that the I, II, III, and IV building-blocks including five covalent bonds can appear as shown in Fig.  5. Metallic characteristic in band structure is necessary for the phase transition, because it makes the electrons below Fermi level easy to jump, ^{[6, 31]} but the formation of new structures requires different bonding orbitals. As shown by Fig.  7(b), no narrow band consisting of local d orbital levels can be observed in the armPNTs, so it is difficult to supply new chemical bonds for bending-induced phase transition in armPNTs.

It can be seen from Fig.  7(b) that the transition from zigPNT(0, 5) to zptPNT(0, 5) is more likely to be a Peierls transition as the lattice has been distorted seriously with a metal-to-semiconductor transition which opens a band gap of 0.75  eV, very similar to armchair carbon nanotubes which become semiconducting at low temperature.^[37] However, the metallic characteristics of other zptPNTs still remain in band structures when the zigPNTs transfer into zptPNTs which cannot be easily explained by Peierls transition (see Appendix  E). As a quasi-one-dimensional material, the zigPNTs remain the properties and structures of phosphorus monolayer so that phosphorus atoms are much more tightly linked together after wrapping the BP monolayer into nanotubes. In some cases a stabilizing deformation leads to the formation of a real band gap. In other cases, a deformation is effective in producing bonds, thereby pulling some states down from the Fermi level region. Because of the preserved two-dimensional linkage it may not be possible to remove all the states from the Fermi level region. Some density of states remains there and the material may still be a conductor. So the zigPNT(0, 7), zigPNT(0, 8), and zigPNT(0, 9) transfer into their new stabilizing structural phases, and the metal properties are still preserved.