Tunable Bandgap and Optical Properties of Black Phosphorene Nanotubes

materials

Article Tunable Bandgap and Optical Properties of Black Phosphorene Nanotubes

Chunmei Li *, Zhongjing Xie, Zhiqian Chen, Nanpu Cheng, Jinghui Wang and Guoan Zhu Faculty of Materials and Energy, Southwest University, Chongqing 400715, China; [email protected] (Z.X.); [email protected] (Z.C.); [email protected] (N.C.); [email protected] (J.W.); [email protected] (G.Z.) * Correspondence: [email protected]

Received: 12 January 2018; Accepted: 9 February 2018; Published: 19 February 2018

Abstract: Black phosphorus (BP), a new two-dimensional material, has been the focus of scientists’ attention. BP nanotubes have potential in the field of optoelectronics due to their low-dimensional effects. In this work, the bending strain energy, electronic structure, and optical properties of BP nanotubes were investigated by using the first-principles method based on density functional theory. The results show that these properties are closely related to the rolling direction and radius of the BP nanotube. All the calculated BP nanotube properties show direct bandgaps, and the BP nanotubes with the same rolling direction express a monotone increasing trend in the value of bandgap with a decrease in radius, which is a stacking effect of the compression strain on the inner atoms and the tension strain on the outer atoms. The bending strain energy of the zigzag phosphorene nanotubes (zPNTs) is higher than that of armchair phosphorene nanotubes (aPNT) with the same radius of curvature due to the anisotropy of the BP’s structure. The imaginary part of the dielectric function, the absorption range, reflectivity, and the imaginary part of the refractive index of aPNTs have a wider range than those of zPNTs, with higher values overall. As a result, tunable BP nanotubes are suitable for optoelectronic devices, such as lasers and diodes, which function in the infrared and ultra-violet regions, and for solar cells and photocatalysis.

Keywords: nanotube; tunable bandgap; optical property; electric ﬁeld; black phosphorus

1. Introduction Two-dimensional (2D) nanomaterials are one of the most attractive research topics due to their outstanding potential applications in many fields, such as flexible electronics [1], sensing [2], and optics [3], as a result of their desirable physical and structural properties [4–7]. Among the 2D materials, graphene is outstanding as it has the highest charge carrier mobility [8], but fails to act as a semiconductor due to a lack of bandgap in its electronic structure [9]. This deficiency hinders its use in many applications, especially for optoelectronic devices. Transition metal dichalcogenides (TMDCs), another main member in the 2D material family, have also attracted the interest of scientists. Molybdenum disulfide (MoS2), the most common TMDC [10], has a noticeable bandgap, enabling the conversion of electrons into photons of light and resulting in extraordinary on/off ratios (>108)[11]. However, the sandwich structure of MoS2 restricts its charge carrier mobility [12]. Phosphorene, a monolayer or few-layer material, appears to be adesirable alternative. Black phosphorus (BP), one of the three main allotropes of phosphorus, is thermodynamically the most stable compared to its red and white counterparts. One of the extraordinary properties of black phosphorus is its high mobility [13–17], which is responsible for a relatively large part of BP’s unique electronic properties. Previous studies showed few-layer BP possesses high mobility, ranging from 600 cm2/Vs to 1000 cm2/Vs at room temperature [14–16,18], allowing BP to be applied in electrode materials. Additionally, another outstanding property of BP is its direct and tunable bandgap in both

Materials 2018, 11, 304; doi:10.3390/ma11020304 www.mdpi.com/journal/materials Materials 2018, 11, 304 2 of 16 mono- and multi-layered forms [14,15] making BP an ideal semiconductor for potential applications in extraordinary light emission and efficient photo-electrical conversion. Furthermore, due to the small bandgap attributed to the large excitation binding energy, both p-type and n-type configurations can be tuned in BP to satisfy the wide range of demands in optoelectronic devices, including the tubular devices. BP thin films have been receiving the attention of scientists worldwide. The puckered structure provides BP with in-plane anisotropy, resulting in its thermal conductivity [19] and unique angle-dependent transport anisotropy [20], especially in carrier mobility [21,22]. Based on the special properties of black phosphorus, the characteristics of BP nanotubes have attracted more attention [23–25], as have carbon nanotubes. Comparatively, the research on BP tubes is limited. However, the urgent demand for tubular electronics stresses the importance of BP tube research, especially for its electronic and optical properties. Because of the anisotropy of black phosphorus, the rolling direction of BP nanotubes is a determining factor of their electrical properties. Additionally, the bandgap and optical properties of BP nanotubes can be modulated by their diameter. Therefore, we systematically studied the properties of BP tubes with two different rolling directions and various radii.

2. Computational Models and Method

2.1. Model Construction Similar to graphene, bulk black phosphorus is a layered material composed of individual stacking atomic layers by van der Waals interactions. Figure1 illustrates the structure of black phosphorus from different side views. Since there are ﬁve electrons on the 3p orbitals of the phosphorus (P) atom, within the phosphorene structure, each P atom covalently bonds with three adjacent P atoms with a 2.20 Å or 2.24 Å bond length in the monolayer model, leaving one lone pair [20], forming a puckered orthorhombic lattice [23,24], resulting in a layer-to-layer spacing of approximately 5 Å [20,25]. The corresponding parameters are shown in Table1. Given the asymmetrical crystal structures caused by the corrugated pattern of phosphorus atoms, BP has different properties compared to graphene and TMDCs. In calculations, a separation of 30 Å, 30 Å, 40 Å, 40 Å, and 50 Å for the adjacent nanotubes with radii of 5.2 Å, 7.3 Å, 10.06 Å, 11.5 Å, and 12.85 Å, respectively, was used to minimize the mirroring interaction.

Table 1. The corresponding parameters of black phosphorus (BP).

Lattice Parameter (Å) Bond Length (Å) Interlayer Spacing (Å) Crystal Space Items System Group Armchair Zigzag Nearest Layer-to-Layer P1–P2 P2–P3 Direction Direction Atomic Distance This work Orthorhombic CMCA 4.55 3.28 2.20 2.24 3.10 5.24 4.58 [14] 3.32 [14] Other work - - 4.54 [26] 3.28 [26] - - 3.20 [14]- 4.54 [27] 3.31 [27] Experimental value - - 4.37 [28] 3.31 [28]----

Figure2 presents a ﬂow diagram for constructing BP nanotubes. Based on the puckered structure of BP, BP nanotubes are separately rolled vertically to the zigzag or armchair direction to construct the nanotubes correspondingly named armchair phosphorene nanotubes (aPNT) (Figure2a) and zigzag phosphorene nanotubes (zPNT) (Figure2b). In addition to the rolling direction factor, the radius of curvature was also considered. As a result, the BP nanotubes were built with different radii of 5.2 Å, 7.3 Å, 10.06 Å, 11.5 Å, and 12.85 Å, to analyze the effect of the bending strain energy on the electronic and optical properties. The radius values are consistent with the middle circles of the models, and non-integral values were chosen to ensure the integrity of the atomic periodicity of the puckered structure. Materials 2018, 11, x FOR PEER REVIEW 3 of 17

Materials 2018, 11, 304 3 of 16 Materials 2018, 11, x FOR PEER REVIEW 3 of 17

Figure 1. The structures of black phosphorus (BP): (a) bulk BP; (b) top view of monolayer BP; (c) computing Figure 1. The structures of black phosphorus (BP): (a) bulk BP; (b) top view of monolayer BP; (c) computing Figure 1.unit The of structuresthe puckered of orthorhombic black phosphorus lattice; (BP):(d) side (a )view bulk from BP; the(b) armchairtop view direction; of monolayer and (e )BP; side (c view) computing unit of the puckered orthorhombic lattice; (d) side view from the armchair direction; and (e) side view unit of fromthe puckered zigzag direction. orthorhombic Atoms in the lattice; top and (d bo) sidettom viewlayers fromare in theblue armchair and pink, respectively. direction; and (e) side view from zigzag direction.direction. AtomsAtoms inin thethe toptop andand bottombottom layerslayers areare inin blueblue andand pink,pink, respectively.respectively.

(a) (b)

Figure 2. Schematic diagram of the construction of BP nanotubes.

(a) (b)

Figure 2. Schematic diagram of the construction of BP nanotubes. Figure 2. Schematic diagram of the construction of BP nanotubes.

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2.2. Energy Calculation Method All the calculations were performed using the pseudo-potential plane-wave method [29] and implemented in the Cambridge Serial Total Energy Package Program (CASTEP; Materials Studio 8.0, Accelrys, San Diego, CA, USA) [30]. The electronic exchange-correlation energy was determined using the generalized gradient approximation of Perdew-Burke-Ernzerh (GGA-PBE) [29]. All nanotubes were fully relaxed with respect to the volume and all cell-internal atomic coordinates. The convergence of the results with respect to energy cutoff and k-points was carefully considered. A plane-wave basis set was used with an energy cutoff of 600 eV. We used 1 × 1 × 32 and 1 × 1 × 24 Monkhorst-pack k-point meshes for aPNT and zPNT with a spacing of 0.01 nm−1 for all calculations. All atomic positions were optimized using the Broyden-Flecher-Goldfarb-Shanno (BFGS) scheme based on the cell optimization criterion. The convergence was conﬁrmed with the system total energy ﬂuctuation within 5 × 10–6 eV. The force on each atom in the unit cell was less than 0.01 eV/Å, the residual stress of the unit cell was lower than 0.02 GPa, and the tolerance offset was lower than 5 × 10−6 Å.

3. Results and Discussion

3.1. Bending Strain Energy Bending strain energy is deﬁned as the energy difference per atom between the nanotube models and the planar monolayer BP, which can be calculated as follows:

E − NE E = nt s (1) b N where Eb is the bending strain energy, Ent is the total energy of nanotubes, Es is the energy of the planar monolayer BP per atom, and N is the number of BP atoms in the calculated unit cell. The formation energy per atom of the PNTs can be obtained from the following equation:

NE − E E = a nt (2) f N where Ef is the formation energy, and Ea is the energy per free P atom. The calculated results of the bending strain energy and formation energy are shown in Table2 and Figure3. As shown in Figure3a, the bending strain energy of the BP nanotubes monotonically increases with the decrease in the models’ radii with the same bending direction. The reason for this variation is that when the BP planar monolayer is rolled into a nanotube, the puckered structure of BP forms a nanotube with a speciﬁed thickness, including inner atoms located in the inner wall and outer atoms on the ektexine of the nanotube. The different radii of the inner wall and ektexine result in different stress modes, with compressive strain on the inner atoms and tension strain on the outer atoms. The compressive and tensile strain on the inner and outer atoms can be expressed as the ratio of thickness to diameter, as shown in Figure1 and Equation (3).

h/2 S = = h/d (3) r where h is the thickness of the planar monolayer BP, and r and d represent the radius and diameter of the nanotube, respectively. With the decrease in the nanotube radius, the rate of the compressive strain on the inner atoms and tension strain on the outer atoms both increase, leading to amonotone increase in the bending energy shown in Figure3a. Materials 2018, 11, 304 5 of 16

Table 2. Data of BP nanotubes with different bending directions and curvature radii. The energy per atom of planar monolayer BP (Ea) is −179.66 eV with a band gap of 0.763 eV.

r Atom E/Atom E E E E Band Gap Modes Strain b b f f (Å) Number (eV) (eV/atom) (KJ/mol) (eV/atom) (KJ/mol) (eV) aPNT 5.2 20.2% 36 −179.60 0.060 5.73 5.22 502.13 0.331 aPNT 7.3 14.4% 48 −179.62 0.034 3.30 5.24 504.05 0.492 aPNT 10.06 10.5% 64 −179.64 0.020 1.94 5.26 505.97 0.641 aPNT 11.5 9.1% 72 −179.65 0.016 1.55 5.27 506.93 0.678 aPNT 12.85 8.2% 80 −179.66 0.014 1.29 5.28 507.89 0.720 zPNT 7.3 14.4% 64 −179.49 0.163 15.66 5.11 491.55 0.291 zPNTMaterials 2018 11.5, 11, x FOR 9.1% PEER REVIEW 96 −179.56 0.101 9.69 5.18 498.28 0.5675 of 17

20 (a) Strain 15

10 Strain (%)

0 5.2 7.3 10.06 11.5 12.85 Radii of aPNTs and zPNTs

16 E of aPNTs (b) b E of zPNTs 14 b 12 10

8 (J/mol) b

E 6 4 2 0 5.2 7.3 10.06 11.5 12.85 Radii of aPNTs and zPNTs

FigureFigure 3. (a) 3. Bending (a) Bending strain strain and and (b ()b bending) bending strain strain energy of of aPNTs aPNTs wi withth different different bending bending directions directions and radii.and radii.

Table 2. Data of BP nanotubes with different bending directions and curvature radii. The energy per

Furthermore,atom of planar the monolayer bending BP strain (Ea) is energy −179.66 ofeV thewith zPNTsa band gap is higherof 0.763 thaneV. that of the aPNTs with the same radii of curvature, due to the anisotropy of the BP structure. As shown in Figure1 and Table3, r Atom E/Atom Eb Eb Ef Ef Band Gap Modes Strain in one structural(Å) unit, three bondsNumber have a bond(eV) length(eV/atom) of 2.242 (KJ/mol) Å and four(eV/atom) bonds (KJ/mol) have a bond(eV) length of 2.203 Å.aPNT The lattice5.2 constant 20.2% along 36 the armchair−179.60 is much0.060 larger 5.73 than that 5.22 along the 502.13 zigzag direction 0.331 in aPNT 7.3 14.4% 48 −179.62 0.034 3.30 5.24 504.05 0.492 the BPaPNT periodic 10.06 unit shown10.5% in Figure 64 1.− Along179.64 the 0.020armchair 1.94direction, 5.26 bonds with 505.97 a bond0.641 length of 2.242 ÅaPNT dominate 11.5 the 9.1%strain of the 72 structure−179.65 and release0.016 the compressive 1.55 5.27 or tension 506.93 strain by 0.678 changing the angleaPNT between12.85 the 8.2% adjacent 80 bonds − which179.66 partially0.014 releases 1.29 the compressive 5.28 507.89 and tensile 0.720 stress zPNT 7.3 14.4% 64 −179.49 0.163 15.66 5.11 491.55 0.291 betweenzPNT the inner11.5 and 9.1% outer atoms 96 and−179.56 results in0.101 a smaller 9.69 distortion 5.18 of the bond 498.28 length 0.567 between the adjacent atoms, as shown in Table3. However, for zPNTs, the bending strain during rolling is mainly imposedFurthermore, on the the shorter bending bonds strain with energy a bond of the length zPNTsof is 2.203higher Å than as shownthat of the in FigureaPNTs1 with. Generally, the a shortersame covalent radii of curvature, bond length due meansto the anisotropy a greater cohesiveof the BP structure. strength, As which shown can in be Figure theoretically 1 and Table deduced 3, fromin the one bond structural population unit, three shown bonds in have Table a bond3. So, length for zPNTs, of 2.242 despiteÅ and four experiencing bonds have a the bond same length strain, of 2.203 Å. The lattice constant along the armchair is much larger than that along the zigzag direction both the compressive strain energy under gone by the inner wall atoms and the tension strain energy in the BP periodic unit shown in Figure 1. Along the armchair direction, bonds with a bond length of experienced2.242 Å dominate by the ektexine the strain atoms of the structure are higher and than release those the compressive of the atoms or tension in aPNTs. strain Therefore, by changing much the angle between the adjacent bonds which partially releases the compressive and tensile stress between the inner and outer atoms and results in a smaller distortion of the bond length between the adjacent atoms, as shown in Table 3. However, for zPNTs, the bending strain during rolling is mainly imposed on the shorter bonds with a bond length of 2.203 Å as shown in Figure 1. Generally, a shorter covalent bond length means a greater cohesive strength, which can be theoretically deduced from the bond population shown in Table 3. So, for zPNTs, despite experiencing the same strain, both the compressive strain energy under gone by the inner wall atoms and the tension strain energy

Materials 2018, 11, 304 6 of 16

largerMaterials deviations 2018, 11 in, x bondFOR PEER lengths REVIEW are imposed on the zPNTs atoms than those in aPNTs, as shown6 of 17 in Table3. As a result, the bending strain energy of the zPNTs is much higher than that of the aPNTs with theexperienced same curvature by the ektexine radius. Theatoms higher are higher the bending than those strain of the energy, atoms the in aPNTs. greater Therefore, the instability. muchThis is thelarger reason deviations why the in zPNTs bond lengths with a are radius imposed under on the 7.3 zPNTs Å cannot atoms achieve than those energy in aPNTs, convergence as shown during in the optimizingTable 3. As process. a result, Thethe bending formation strain energy energy indicates of the zPNTs the strengthis much higher of atomic than bondingthat of the and aPNTs reﬂects the stabilitywith the of same the phase.curvature The radius. higher The the higher value the of bendin the formationg strain energy, energy, the thegreater more the stable instability. the structure. This is the reason why the zPNTs with a radius under 7.3 Å cannot achieve energy convergence during In Table2, the formation energy of zPNT with radius of 11.5 Å is even lower than that of aPNT with the optimizing process. The formation energy indicates the strength of atomic bonding and reflects radiusthe of5.2 stability Å. As of a the result, phase. the The aPNTs higher are the more valuestable of the formation than the zPNTsenergy, with the more the samestable radius.the structure. In Table 2, the formation energy of zPNT with radius of 11.5 Å is even lower than that of aPNT with Table 3. The lattice parameters of armchair phosphorene nanotubes (aPNTs) and zigzag phosphorene radius of5.2 Å. As a result, the aPNTs are more stable than the zPNTs with the same radius. nanotubes (zPNTs) with radii of 7.3 Å compared to planar monolayer BP. Table 3. The lattice parameters of armchair phosphorene nanotubes (aPNTs) and zigzag phosphorene Bond Deviation Bond Deviation Bond Bondsnanotubes Planar (zPNTs) with radii of 7.3aPNT Å compared to planar monolayer BP.zPNT Population (%) Population (%) Population Bond Deviation Bond Deviation Bond P1–P2Bonds 2.20Planar 0.84 2.23aPNT 1.3 0.87zPNT 2.41 9.2 0.32 Population (%) Population (%) Population P2–P3 2.24 0.40 2.25 0.4 0.34 2.20 −2.1 0.43 P1–P2 2.20 0.84 2.23 1.3 0.87 2.41 9.2 0.32 P3–P4 2.20 0.84 2.19 −0.7 1.05 2.15 −2.2 0.36 P2–P3 2.24 0.40 2.25 0.4 0.34 2.20 −2.1 0.43 P1–P2–P3 103.86 - 100.09 −3.6 - 96.93 −6.7 - P3–P4 2.20 0.84 2.19 −0.7 1.05 2.15 −2.2 0.36 P3–P4–P5 103.86 - 109.13 5.1 - 107.57 3.6 - P1–P2–P3 103.86 - 100.09 −3.6 - 96.93 −6.7 - P3–P4–P5 103.86 - 109.13 5.1 - 107.57 3.6 - 3.2. Electronic Properties 3.2. Electronic Properties 3.2.1. Bands Structure 3.2.1. Bands Structure All the calculated BP nanotubes showed direct bandgaps, with the top of valence band located All the calculated BP nanotubes showed direct bandgaps, with the top of valence band located at the same site with the bottom of the conduction band in reciprocal space, which is consistent with at the same site with the bottom of the conduction band in reciprocal space, which is consistent with the planarthe planar monolayer monolayer BP characteristics.BP characteristics. This This means means thethe BP nanotube nanotube can can directly directly couple couple with with light, light, makingmaking BP an BP ideal an ideal semiconductor semiconductor for for potential potential applicationsapplications in in extraordinary extraordinary light light emission emission and and efﬁcientefficient photo-electrical photo-electrical conversion. conversion The. The band band structure structure ofof aPNTaPNT and and zPNT zPNT with with a radius a radius of 7.3 of Å 7.3 are Å are shownshown in Figure in Figure4. 4.

(a) (b) Z2 2 2 G2 G2 Z3 0 0.492eV 0 0.291eV G1 Z1 G1

-2 -2

Energy (eV) Energy Z2 Energy (eV) Energy -4 -4

Z1 -6 -6

G Z G Z Figure 4. Band structures of BP nanotubes with a radius of 7.3 Å: (a) aPNT and (b) zPNT. The red Figure 4. Band structures of BP nanotubes with a radius of 7.3 Å: (a) aPNT and (b) zPNT. The red arrows represent the inter-band transitions between the valence and conduction bands, and the black arrows represent the inter-band transitions between the valence and conduction bands, and the black arrows point out the band-gaps of band structures of aPNT and zPNT. arrows point out the band-gaps of band structures of aPNT and zPNT. The bandgaps of the BP nanotubes with different rolling direction and radii are shown in Figure 5. TheBP nanotubes bandgaps in of the the same BP nanotubesrolling direction with demonstr differentate rolling a monotone direction increasing and radii trend are in shown the bandgaps in Figure 5. BP nanotubes in the same rolling direction demonstrate a monotone increasing trend in the bandgaps Materials 2018, 11, x FOR PEER REVIEW 7 of 17

with a decrease in radius, which is consistent with a previous finding [24]. When the radius of the nanotube is sufficiently large, its bandgap value should approach to that of the planar BP monolayer. The trend is consistent with that of the bending strain energy. With higher bending strain energy, the BP nanotubes in the same rolling direction exhibit a narrower bandgap. This is the stacking effect of the compression strain on the inner atoms and the tension strain on the outer atoms. To further explain the influence of stress on the BP bandgap, uniaxial tension and compression were placed on the planar monolayer BP. Figure 6 shows the effects of the uniaxial compressive and tensile stress Materialsalong the2018 armchair, 11, 304 and zigzag directions on the bandgaps. With compressive strain or tension strain7 of 16 along zigzag direction, the bandgaps of the modes all monotonically decreased with a large slope. Therefore, when the BP monolayers were rolled into nanotubes along the armchair direction, the Materials 2018, 11, x FOR PEER REVIEW 7 of 17 withcompressive a decrease strain in radius,of the inner which atoms is consistent and the tensio with an previousstrain of the finding outer [ 24atoms]. When induced the radiusa decrease of the in nanotubetheir withbandgaps. a is decrease sufficiently A smaller in radius, large, radius which its results bandgap is consistent in a valuelarger with shouldbending a previous approach strain finding energy, to that [24]. leadin of When theg to planar thea narrower radius BP monolayer.of bandgap. the The trendnanotube is consistent is sufficiently with large, that its ofbandgap the bending value should strain approach energy. to With that higherof the planar bending BP monolayer. strain energy, the BPThe nanotubes trend is consistent in the same with0.8 rollingthat of the direction bending exhibit strain energy. a narrower With bandgap.higher bending This strain is the energy, stacking the effect of theBP compression nanotubes in strainthe same on rolling the inner direction Bandgap atoms exhibit of and aPNTs a the narrower tension bandgap. strain on This the is outerthe stacking atoms. effect To furtherof 0.7 Bandgap of zPNTs explainthe thecompression influence strain of stress on the on theinner BP atoms bandgap, and the uniaxial tension tension strain on and the compression outer atoms. wereTo further placed on the planarexplain monolayer the influence BP. of Figure stress0.6 6 on shows the BP the bandgap, effects of uniaxial the uniaxial tension compressive and compression and tensilewere placed stress on along the planar monolayer BP. Figure 6 shows the effects of the uniaxial compressive and tensile stress the armchair and zigzag directions0.5 on the bandgaps. With compressive strain or tension strain along along the armchair and zigzag directions on the bandgaps. With compressive strain or tension strain zigzag direction, the bandgaps of the modes all monotonically decreased with a large slope. Therefore, along zigzag direction, the0.4 bandgaps of the modes all monotonically decreased with a large slope. whenTherefore, the BP monolayers when the BP were monolayers rolled into were nanotubes rolled into along nanotubes the armchair along the direction,armchair direction, the compressive the 0.3 straincompressive of the inner strain atoms of andthe inner the tension atoms and strain the of tensio the outern strain atoms of the induced outer atoms a decrease induced in a decrease their bandgaps. in Bandgap (eV) Bandgap A smallertheir bandgaps. radius results A smaller in a 0.2radius larger results bending in a larger strain be energy,nding strain leading energy, to aleadin narrowerg to a narrower bandgap. bandgap.

0.10.8 Bandgap of aPNTs 0.00.7 Bandgap of zPNTs 5.2 7.3 10.06 11.5 12.85 0.6 Radii of aPNTs and zPNTs 0.5 Figure 5. The variation in bandgaps for aPNTs and zPNTs with different radii. 0.4

When the nanotubes were0.3 rolled in the zigzag direction, the strain energy imposed on the atoms Bandgap (eV) Bandgap was more severe, as discussed0.2 in the previous section. With the same radius, the zPNTs have narrower bandgaps than the aPNTs. Under tensile strain, the variation in the bandgap trend of the planar 0.1 monolayer BP was different, quickly declining followed by a gradual recovery. With tension at 14%, 0.0 the bandgap reached a valley with5.2 a value of7.3 about10.06 0.15 eV.11.5 With 12.85compressive strain, the bandgap of the zPNT decreased slowly compared toRadii the aPNT,of aPNTs as shownand zPNTs in Figure 5. Because of this, the rate of decrease of the bandgap did not keep pace with the increase in the bending strain energy shown in Figure 5. The variation in bandgaps for aPNTs and zPNTs with different radii. Table 2, thoughFigure the bandgap 5. The variation of the zPNT in bandgaps was narr for aPNTsower than and zPNTsthat of with aPNT different with the radii. same radius. When the nanotubes were rolled in the zigzag direction, the strain energy imposed on the atoms 1.1 1.0 was more severe, as discussed in the previous sect(a)ion. With1.0 the same radius, the zPNTs have(b) narrower 0.9 bandgaps than the aPNTs. Under tensile strain, the 0.9variation in the bandgap trend of the planar monolayer0.8 BP was different, quickly declining followed0.8 by a gradual recovery. With tension at 14%, the bandgap0.7 reached a valley with a value of about 0.150.7 eV. With compressive strain, the bandgap of the zPNT0.6 decreased slowly compared to the aPNT, as0.6 shown in Figure 5. Because of this, the rate of 0.5 decrease 0.5of the bandgap did not keep pace with the increase in the bending strain energy shown in 0.4 Table 2, though0.4 the bandgap of the zPNT was narrower0.3 than that of aPNT with the same radius. Band-gap (eV) Band-gap Band-gap (eV) 0.3 0.2 1.1 1.0 0.1 0.2 (a) 1.0 (b) 0.9 0.0 0.1 0.9 0.000.8 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.80.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.7Strain along zigzag direction (%) 0.7 Strain along armchair direction (%) 0.6 0.6 0.5 0.5 FigureFigure 6.6.Band-gaps Band-gaps of of monolayer monolayer phosphorus phosphorus under under different0.4 different uniaxial uniaxial compressive compressive and tensile andstresses tensile alongstresses (a )along zigzag0.4 (a and) zigzag (b) armchair and (b) armchair directions. directions. 0.3 Band-gap (eV) Band-gap Band-gap (eV) 0.3 0.2 0.2 0.1 0.0 When the nanotubes0.1 were rolled in the zigzag direction, the strain energy imposed on the atoms 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 was more severe, as discussed in the previous section. With the same radius, the zPNTs have narrower Strain along zigzag direction (%) Strain along armchair direction (%) bandgaps than the aPNTs. Under tensile strain, the variation in the bandgap trend of the planar monolayerFigure BP was6. Band-gaps different, of quicklymonolayer declining phosphorus followed under different by a gradual uniaxial recovery.compressive With and tensiontensile at 14%, the bandgapstresses reached along ( a) valleyzigzag and with (b) a armchair value of directions. about 0.15 eV. With compressive strain, the bandgap of the zPNT decreased slowly compared to the aPNT, as shown in Figure5. Because of this, the rate of decrease of the bandgap did not keep pace with the increase in the bending strain energy shown in Table2, though the bandgap of the zPNT was narrower than that of aPNT with the same radius. Materials 2018, 11, 304 8 of 16

3.2.2.Materials Electron 2018 Density, 11, x FOR PEER and DifferenceREVIEW in Electron Density 8 of 17 Materials 2018, 11, x FOR PEER REVIEW 8 of 17 To further analyze the inﬂuence of bending direction and curvature radius on the bending strain 3.2.2. Electron Density and Difference in Electron Density energy3.2.2. and Electron bandgap, Density the electronand Difference density in Electron and electron Density density difference were investigated. From the electron densityTo further shown analyze in Figurethe influence7, the isosurfacesof bending direct of theion electron and curvature density radius demonstrated on the bending strong strain covalent energyTo andfurther bandgap, analyze the the electron influence density of bending and electr directonion density and curvature difference radius were on investigated. the bending strainFrom bonding between the P atoms. Only a small quantity of electrons transferred between neighboring theenergy electron and bandgap,density shown the electron in Figure density 7, the and isosur electrfaceson of density the electron difference density were demonstrated investigated. strong From P atoms.covalentthe electron As bonding shown density inbetween shown Figure in8the, Figure the P atoms. maximum 7, the Only isosur transferreda facessmall of quantity the charge electron of forelectrons density aPNT demonstratedtransferred with a radius between strong of 7.3 Å was onlyneighboringcovalent 0.212, bonding andP atoms. for between zPNTAs shown withthe inP radius Figureatoms. 8,ofOnly the 7.3 ma Åa smximum wasall onlyquantity transferred 0.230. of Furthermore,electrons charge for transferred aPNT all with electrons betweena radius shifted fromofneighboring the 7.3 periphery Å was onlyP atoms. of 0.212, the As atomand shown for to zPNTin the Figure bond with 8, radius direction,the ma ofximum 7.3 demonstrating Å wastransferred only 0.230. charge the Furthermore, strong for aPNT covalent with all electrons a radius interaction betweenshiftedof 7.3 the Å fromwas P atoms onlythe periphery0.212, once again.and offor the AszPNT shownatom with to in radiusthe Table bond of3 7.3, thedirection, Å zPNTswas only demonstrating distorted 0.230. Furthermore, more the than strong all the electronscovalent aPNTs with the sameinteractionshifted radius. from between the So, periphery with the theP atoms increaseof the once atom inagain. theto theAs distance showbondn betweendirection,in Table 3, thedemonstrating the twozPNTs neighboring distorted the strong more atoms covalentthan located the at interaction between the P atoms once again. As shown in Table 3, the zPNTs distorted more than the the ektexineaPNTs with in zPNTs the same (Figure radius.7), So, the with necking the increase phenomenon in the distance was observedbetween the on two the neighboring isosurface ofatoms electron locatedaPNTs withat the the ektexine same radius. in zPNTs So, with(Figure the 7), increase the ne ckingin the phenomenondistance between was theobserved two neighboring on the isosurface atoms density,located meaning at the ektexine a weakened in zPNTs interaction (Figure 7), occurs the ne undercking phenomenon tensile strain. was Furthermore, observed on the we isosurface deduced that the aPNTsof electron would density, be more meaning stable a weakened than the zPNTsinteraction because occurs both under the tensile inner strain. atoms Furthermore, with a bond we length deducedof electron that density, the aPNTs meaning would a weakenedbe more stable interact thanion the occurs zPNTs under because tensile both strain. the inner Furthermore, atoms with we a 2.19 Å and the outer atoms with a bond length of 2.23 Å have much higher bond populations, at 1.05 bonddeduced length that 2.19the aPNTsÅ and wouldthe outer be more atoms stable with than a bond the zPNTslength becauseof 2.23 bothÅ have the muchinner atomshigher with bond a and 0.87,populations,bond respectively,length at2.19 1.05 Å than and and 0.87, the the corresponding respectively,outer atoms thanwith values the a bondcorresponding of 0.36 length and of 0.32, values2.23 in Å zPNTs,of have 0.36 andmuch shown 0.32, higher inin TablezPNTs, bond3. This reﬂectsshownpopulations, the in higher Table at stability1.053. This and reflects 0.87, of aPNTs respectively, the higher than zPNTsstabilit than they based of corresponding aPNTs on thethan bond zPNTs values strength. based of 0.36 on and the 0.32,bond in strength. zPNTs, shown in Table 3. This reflects the higher stability of aPNTs than zPNTs based on the bond strength. (a) (b) (a) (b)

0.868 0.868 0.6510.868 0.6510.868 0.4340.651 0.4340.651 0.4340.217 0.4340.217 0.0000.217 0.0000.217 necking effect 0.000 0.000 necking effect (c) (d) (c) (d)

0.905 0.905 0.6790.905 0.6790.905 0.4530.679 0.4530.679 0.4530.226 0.4530.226 0.226 0.226 0.000 0.000 0.000 0.000 FigureFigure 7. Electron 7. Electron density density diagrams diagrams for for (a, b(a), aPNTb) aPNT and and (c, d(c),d zPNT) zPNT with with a radiusa radius of of 7.3 7.3 Å. Å. (a )(a The) The rotated aPNTrotatedFigure with electron7.aPNT Electron with density electrondensity isosurface diagramsdensity isosurface tofor show (a,b) toaPNT the show weak and the difference(weakc,d) zPNT difference inwith the ina electron theradius electron of distribution 7.3 distribution Å. (a) The around rotated aPNT with electron density isosurface to show the weak difference in the electron distribution the Paround atoms the located P atoms at thelocated inner at andthe inner outer and rings outer of Apnt;rings of (b Apnt;) The electron(b) The electron density density proﬁle profile vertical to around the P atoms located at the inner and outer rings of Apnt; (b) The electron density profile the centralvertical axeto the of central aPNT; axe (c) of the aPNT; rotated (c) the zPNT rotated with zPNT electron with electron density density isosurface isosurface to show to show the the necking vertical to the central axe of aPNT; (c) the rotated zPNT with electron density isosurface to show the effectnecking shown effect around shown the around P atoms the located P atoms at lo thecated outer at the ring outer of ring zPNT of zPNT and (andd) the (d) the electron electron density density proﬁle profilenecking vertical effect shown to central around axe ofthe zPNT. P atoms located at the outer ring of zPNT and (d) the electron density verticalprofile to centralvertical axeto central of zPNT. axe of zPNT. (a) (b) (a) (b)

0.212 0.230 0.1270.212 0.1380.230 0.0410.127 0.0460.138 -0.0450.041 -0.0460.046 -0.045 -0.046 -0.130 -0.138 -0.130 -0.138 Figure 8. Electron density difference for aPNT and zPNT with a radius of 7.3 Å. The profiles of the FigureelectronFigure 8. Electron 8. density Electron density difference density difference differencevertical tofor forthe aPNTaPNT central and axe zPNT zPNTof (a) withaPNT; with a radius aand radius (b )of zPNT. 7.3 of 7.3Å. The The Å. attached Theprofiles proﬁles rulersof the of the electronshowelectron density the density transferred difference difference charge vertical vertical corresponding to tothe the centralcentral to the axe axecolors. of of ( a ()a )aPNT; aPNT; and and (b) ( bzPNT.) zPNT. The The attached attached rulers rulers show the transferred charge corresponding to the colors. show the transferred charge corresponding to the colors.

Materials 2018, 11, 304 9 of 16

3.3. Optical Properties Notably, due to the in-plane anisotropic structure of the BP thin ﬁlm, the angle-resolved polarized, or the polarization-resolved, optical property is considered an effective technique to investigate the crystalline orientation of the sample [26–28]. The dependence of the anisotropic optical spectrum on incident waves helps to determine crystallographic orientation. As a result, sufﬁcient caution is required when the optical spectrum is used for BP nanotubes with different rolling directions and radii. Alternatively, the anisotropic optical absorption spectroscopy along the armchair and zigzag directions of the BP crystals should result in diverse optical properties of aPNTs and zPNTs.

3.3.1. Complex Dielectric Constants The optical properties of solids can be obtained through electronic transitions. Within the range of the linear optical response, the complex dielectric function, ε(ω) = ε1(ω) + iε2(ω) always represents the macroscopic optical response function in a solid, where ε1(ω) is the real component of the dielectric function, and ε2(ω) isits imaginary part. The complex dielectric function can be calculated from the momentum matrix elements of occupied and unoccupied wave functions according to Equation (4) and can be thought of as detailing the real transition between occupied and unoccupied electronic states.

4π2 Z 2 ( ) = · 3 | · ( )|2 × [ ( ) − ( ) − ] ε2 ω 2 2 ∑ d k e MCV K EC K EV K }ω (4) m ω V,C BZ 2π where e is the electric quantity of an electron, ω is the frequency of the incident photon, C and V are the conduction and valence bands, respectively, EC(K) and EV(K) represent the intrinsic energy of the conduction band and the valence band, respectively, BZ is the ﬁrst Brillouin zone, K is the reciprocal lattice vector, and |e · MCV (K)| is the momentum matrix element [18]. According to the Kramer-Krönig dispersion relationship, the real part of the dielectric function can be obtained by Equation (5):

2 Z 2 3 8πe 3 2 |e · MCV (K)| } ε1(ω) = 1 + · d K × (5) 2 ∑ [ ( ) − ( )] 2 2 2 m V,C BZ 2π EC K EV K [EC(K) − EV (K)] − } ω

The real and imaginary parts of the dielectric function, which are tightly connected with band structure, for aPNTs and zPNTs with different radii are shown in Figure9. The differences in the dielectric function of aPNTs and zPNTs originate from the anisotropy of the 2D nature of the atomic conﬁguration. The static dielectric constant (the value of dielectric constant at zero frequency ε(0)) changes with the rolling direction and the radius of the BP nanotubes. Along with the decrease in the aPNTs radius, the static dielectric constant ﬂuctuates, but the aPNTs static dielectric constant is higher than that of zPNTs. The dielectric constant under high frequency ε(∞) tends to become a constant at 0.8. Three peaks exist in the imaginary part of aPNTs; however, only two peaks appear in that of zPNTs. The imaginary part of the dielectric function, in the energy range of 0 to 20 eV for BP nanotubes, are clearly related to their band structure, indicating the absorption behavior, so that the electronic transitions from valance to conduction bands contribute to the main part of the optical spectra. The threshold energies of the dielectric function correspond to the system bandgaps. The threshold energy for the transition between the highest valence band and the lowest conduction band is the fundamental absorption edge. The peaks are related to different electronic transitions from occupied states (valence bands) to the unoccupied states (conduction bands). The highest peak in the imaginary part of aPNTs at a radius of 7.3 Å is located at 1.1 eV, which shows a slight red shift with an increase in radius. The other two peaks are situated at 4.4 eV and 6.5 eV. These three peaks are related to the inter-band transitions between the valence and conduction bands, from G1 to G2 at the high-symmetry point G in the reciprocal space, from Z1 to Z3 at the high-symmetry point Z, and from Z2 to Z3 at the high-symmetry point Z (Figure4a). Comparatively, the two peaks in the imaginary part of zPNTs are Materials 2018, 11, x FOR PEER REVIEW 9 of 17

3.3. Optical Properties Notably, due to the in-plane anisotropic structure of the BP thin film, the angle-resolved polarized, or the polarization-resolved, optical property is considered an effective technique to investigate the crystalline orientation of the sample [26–28]. The dependence of the anisotropic optical spectrum on incident waves helps to determine crystallographic orientation. As a result, sufficient caution is required when the optical spectrum is used for BP nanotubes with different rolling directions and radii. Alternatively, the anisotropic optical absorption spectroscopy along the armchair and zigzag directions of the BP crystals should result in diverse optical properties of aPNTs and zPNTs.

3.3.1. Complex Dielectric Constants The optical properties of solids can be obtained through electronic transitions. Within the range

of the linear optical response, the complex dielectric function, () =() +() always represents the macroscopic optical response function in a solid, where () is the real component of the dielectric function, and () isits imaginary part. The complex dielectric function can be calculated from the momentum matrix elements of occupied and unoccupied wave functions according to Equation (4) and can be thought of as detailing the real transition between occupied and unoccupied electronic states. 2 42π 2 εω()=⋅ dk3 eM⋅×−− () K E () K E () K  ω 2 BZ π CV C V  (4) m22ω V,C 2

where e is the electric quantity of an electron, ω is the frequency of the incident photon, C and V are the conduction and valence bands, respectively, EC(K) and EV(K) represent the intrinsic energy of the conduction band and the valence band, respectively, BZ is the first Brillouin zone, K is the reciprocal Materials 2018 11 , , 304 ⋅ 10 of 16 lattice vector, and eMCV () K is the momentum matrix element [18]. According to the Kramer-Krönig dispersion relationship, the real part of the dielectric function can be obtained by locatedEquation at 2.6 eV (5): and 4.2 eV, attributed to the inter-band transitions from G1 to G2 at the high-symmetry point G, and Z1 to Z2 at the high-symmetry point Z (Figure2 4b). The lower energy values are caused π 23eM⋅ () K by the near-bandεω electronic=+82e direct ⋅ transitions.3 ComparingCV the× imaginary parts of aPNTs with different 1() 1  BZ dK (5) radii, the peak values gradually22 decline with2π  anEK increase()− EK () in the radius. The peak shape22 broadens as a m V,C CVEK()−− EK ()  ω result of the changes in the band structure and the density of theCV different states.

5 2.5 (a) (b) 4.56 aPNT r = 5.2 aPNT r = 5.2 aPNT r = 7.3 1.1eV aPNT r = 7.3 4 4.00 aPNT r = 10.05 2.0 aPNT r = 10.08 3.83 aPNT r = 11.5 aPNT r = 11.5 3.29 aPNT r = 12.85 4.4eV aPNT r = 12.85 3 1.5 2.54

2 1.0 6.5eV

1 0.5

0 0.0 MaterialsReal Part of Dielectric Function 20180, 11, x FOR 5 PEER REVIEW 10 15 20 0 5 10 15 2010 of 17

Frequency (eV) Function Dielectric of Part Imaginary Frequency (eV)

2.5 (c) aPNT r = 7.3 1.2 (d) aPNT r = 11.5 1.1eV zPNT r = 7.3 4.5eV zPNT r = 11.5 2.0 1.1eV 4.4eV 2.6eV 1.5 0.8 4.5eV

2.6eV

1.0 4.2eV6.3eV 0.4 6.4eV 0.5

0.0 0.0 0 5 10 15 20 0 5 10 15 20

Frequency (eV) maginary Part of Dielectric Function Frequency (eV) Imaginary Part of Dielectric Function Dielectric of Imaginary Part I Figure 9. Dielectric functions: (a) Real (Re) part and (b) imaginary (Im) part for aPNTs with different Figure 9. Dielectric functions: (a) Real (Re) part and (b) imaginary (Im) part for aPNTs with different radii; (c) Real (Re) and (d) imaginary (Im) parts of the dielectric function for aPNTs and zPNTs with radii; (c) Real (Re) and (d) imaginary (Im) parts of the dielectric function for aPNTs and zPNTs with the the same radius. same radius. The real and imaginary parts of the dielectric function, which are tightly connected with band 3.3.2.structure, Absorption for CoefficientaPNTs and zPNTs with different radii are shown in Figure 9. The differences in the dielectric function of aPNTs and zPNTs originate from the anisotropy of the 2D nature of the atomic Theconfiguration. optical absorption The static spectrum dielectric isconstant important (the forvalue estimating of dielectric the constant optical propertiesat zero frequency of a material ε(0)) to be usedchanges as an with optoelectronic the rolling direction device. Theand the absorption radius of coefficientthe BP nanotubes. indicates Along the with fraction the decrease of energy in the lost by the waveaPNTs when radius, it passes the static through dielectric a material. constant The fluctu absorptionates, but coefficientthe aPNTs staticI(ω) candielectric be derived constant from is the real parthigherε1( ωthan) and that imaginary of zPNTs. partThe εdiel2(ωectric) of theconstant dielectric under function high frequency as follows: ε(∞) tends to become a

constant at 0.8. Three peaks exist in the imaginary part of aPNTs; however,1 only two peaks appear in √ q 2 that of zPNTs. The imaginary part of the dielectric2 function,2 in the energy range of 0 to 20 eV for BP I(ω) = 2ω ε1(ω) − ε2(ω) − ε1(ω) (6) nanotubes, are clearly related to their band structure, indicating the absorption behavior, so that the electronic transitions from valance to conduction bands contribute to the main part of the optical For BP nanotubes, the absorption coefficient is related to both the extinction index and the spectra. imaginary part of the dielectric function (Figure 10). The absorption spectra of aPNT and zPNT at a The threshold energies of the dielectric function correspond to the system bandgaps. The radiusthreshold of 7.3 Å energy in Figure for the 10 transitionb are obviously between different. the highest The valence absorption band and range the lowest of aPNT conduction is much band larger than thatis the of fundamental zPNT, and absorption aPNT has edge. a higher The peaks absorption are related coefficient. to different As aelectronic result, the transitions aPNTs arefrom more efficientoccupied for the states absorption (valence of bands) low frequency to the unoccupied incident states light. (conduction For aPNTs, bands). the first The absorption highest peak peak in the occurs at an energyimaginary of 1.5part eV, of aPNTs and the at highesta radius absorptionof 7.3 Å is located peak isat located1.1 eV, which at an shows energy a ofslight 4.7 red eV. However,shift with for zPNTs,anonly increase two in peaks radius. appear The other in the two absorption peaks are spectrum,situated at with4.4 eV the and first 6.5 peakeV. These occurring three peaks at 3.4 are eV and the highestrelated peak to the appearing inter-band at transitions 4.7 eV. The between visible the light vale regionnce and ranges conduction from bands, 1.62 eV from to 3.11G1 to eV. G2 According at the to Figurehigh-symmetry 10b, aPNTs point have G a in better the reciprocal absorption space, from fr theom infraredZ1 to Z3 regionat the high-symmetry to the visible region,point Z, and and even from Z2 to Z3 at the high-symmetry point Z (Figure 4a). Comparatively, the two peaks in the to the ultra-violet region, which is reconfirmed by comparing the absorption index of aPNT and zPNT imaginary part of zPNTs are located at 2.6 eV and 4.2 eV, attributed to the inter-band transitions from G1 to G2 at the high-symmetry point G, and Z1 to Z2 at the high-symmetry point Z (Figure 4b). The lower energy values are caused by the near-band electronic direct transitions. Comparing the imaginary parts of aPNTs with different radii, the peak values gradually decline with an increase in the radius. The peak shape broadens as a result of the changes in the band structure and the density of the different states.

3.3.2. Absorption Coefficient The optical absorption spectrum is important for estimating the optical properties of a material to be used as an optoelectronic device. The absorption coefficient indicates the fraction of energy lost by the wave when it passes through a material. The absorption coefficient I()ω can be derived from εω εω the real part 1() and imaginary part 2 () of the dielectric function as follows:

1 ω = ω( ε ω 2 − ε ω 2 − ε ω ) 2 (6) I( ) 2 1 ( ) 2 ( ) 1 ( )

Materials 2018, 11, x FOR PEER REVIEW 11 of 17

For BP nanotubes, the absorption coefficient is related to both the extinction index and the imaginary part of the dielectric function (Figure 10). The absorption spectra of aPNT and zPNT at a radius of 7.3 Å in Figure 10b are obviously different. The absorption range of aPNT is much larger than that of zPNT, and aPNT has a higher absorption coefficient. As a result, the aPNTs are more efficient for the absorption of low frequency incident light. For aPNTs, the first absorption peak occurs at an energy of 1.5 eV, and the highest absorption peak is located at an energy of 4.7 eV. However, for zPNTs, only two peaks appear in the absorption spectrum, with the first peak occurring at 3.4 eV and the highest peak appearing at 4.7 eV. The visible light region ranges from 1.62 eV to 3.11 eV. According to Figure 10b, aPNTs have a better absorption from the infrared region to the visible region, and even to the ultra-violet region, which is reconfirmed by comparing the absorption index of aPNT and zPNT at a radius of 11.5 Å as shown in Figure 10c. So, BP nanotubes are suitable for optoelectronic devices such as lasers and diodes that work in the infrared and ultra-violet regions, and are also appropriate for solar cells and photocatalysis. These peaks could originate from the transition of covalence bands to conduction bands, as discussed in the imaginary part of the dielectric function in Section 3.3.1. This can also be explained by the partial density of states (PDOS). As shown in Figure 11, the transition of 2s-P (occupied) and 2p-P (occupied) states into 3p-P (unoccupied) states in the PDOS of aPNTs results in the absorption peak at 1.5 eV. Similarly, the peaks at 4.7 eV and 6.7 eV originate from the transition of 2s-P (occupied) and 2p-P (occupied) states into 3p-P (unoccupied) states, as shown in Figure 11.

The absorption edge can be calculated using Equation (7): ℎ = ℎ − (7) where hυ is the photo energy, C is a constant, and α is the absorption coefficient. For materials with a direct bandgap, such as the BP nanotubes, n is 0.5; for materials with an indirect bandgap, n is set to 2. The absorption edge can be determined using the linear extrapolation method, and the results were 0.5 eV for aPNT and 0.3 eV for zPNT with a radius of 7.3 Å, which corresponds to the values of Materials the2018 bandgap, 11, 304 between the highest valance band and the lowest conduction band, at 0.492 eV and 0.291 11 of 16 eV, as shown in Table 2. Additionally, the absorption edge shows a slight red shift with a decrease in the radius of the nanotubes, which is also consistent with trend in the bandgap value. at a radius ofComparing 11.5 Å as the shown absorption in Figure spectra 10 ofc. aPNTs So, BP with nanotubes different areradii suitable (Figure 10), for all optoelectronic the absorption devices such aspeaks lasers of and aPNTs diodes monotonously that work decline in the with infrared the increa andse ultra-violet in the radii of regions, the BP nanotubes and are alsoas long appropriate as for solarthe cells radius and is photocatalysis.bigger than 7.3 Å.

(a) 50,000 aPNT r = 5.2 aPNT r = 7.3 aPNT r = 10.06 )

-1 40,000 aPNT r = 11.5 aPNT r = 12.85

15,000 30,000

10,000 20,000

5,000 Absorption (cm 10,000

0 0.0 0.5 1.0 1.5 2.0 0 Materials 2018, 11, x FOR PEER REVIEW 0 5 10 15 20 12 of 17 Frequency (eV)

(b) 4.7eV 6.7eV aPNT r = 7.3 50,000 zPNT r = 7.3 ) -1 40,000 4.7eV 30,000

3.0eV 20,000 1.5eV Absorption (cm 10,000

0 0 4 8 121620 Frequency (eV)

20,000 6.6eV 3.0eV

Absorption (cm Absorption 1.6eV 10,000

0 0 4 8 121620 Frequency (eV)

Figure 10.FigureAbsorption 10. Absorption coefﬁcient coefficient of BPof BP nanotubes nanotubes with electric electric field ﬁeld direction direction parallel parallel to the toradius. the radius. (a) aPNTs(a) with aPNTs different with different radii; radii; (b) aPNT (b) aPNT and and zPNT zPNT with with a radiusa radius of of 7.3 7.3 Å, Å, andand ((c) aPNT and and zPNT zPNT with a radiuswith of 11.5 a radius Å. of 11.5 Å.

40 s s (a) 50 (b) These peaks could originate from the p transition of covalence bands to conduction p bands, as discussed in the imaginary part of the dielectric total function in Section 3.3.1. This can also total be explained 30 40 by the partial density of states (PDOS). As shown in Figure 11, the transition of 2s-P (occupied) and 2p-P (occupied) states into 3p-P (unoccupied) states in30 the PDOS of aPNTs results in the absorption 20

4.7eV peak at 1.5 eV. Similarly, the peaks at 4.7 eV and 6.7 eV originate from the transition of3.0eV 2s-P (occupied) 6.7eV 20 4.7eV and 2p-P (occupied) states into 3p-P (unoccupied)1.5eV states, as shown in Figure 11. 10 The absorption edge Eop can be calculated using Equation10 (7): Partial Density of States Density Partial Partial Density of States Density Partial n 0 αhv = C hv − E0op (7) -15 -10Energy -5 (eV) 0 5 -15 -10 -5 0 5 Energy (eV) where hυ is the photo energy, C is a constant, and α is the absorption coefﬁcient. For materials with a direct bandgap,Figure such 11. Partial as the Density BP nanotubes,of States (PDOS)n ofis BP 0.5; nanotubes for materials with radius with of 7.3 anÅ: (a indirect) aPNTs and bandgap, (b) zPNTs. n is set to 2.

3.3.3. Reflection Coefficient

The reflectivity at normal incidence is given by:

2 − 2 − 2 ε ω + ε ω − (n 1) k 1 ( ) i 2 ( ) 1 R(ω) = = (8) + 2 + 2 ε ω + ε ω + (n 1) k 1 ( ) i 2 ( ) 1

where n and k are the real and imaginary parts of the complex refractive index, which are known as the refractive index and the extinction index, respectively, and are given by the following relationship:

Materials 2018, 11, x FOR PEER REVIEW 12 of 17

(b) 4.7eV 6.7eV aPNT r = 7.3 50,000 zPNT r = 7.3 ) -1 40,000 4.7eV 30,000

3.0eV 20,000 1.5eV Absorption (cm 10,000

0 0 4 8 121620 Frequency (eV)

Materials 2018, 11, 304 30,000 12 of 16 4.6eV

20,000 6.6eV 3.0eV The absorption edge can be determined using the linear extrapolation method, and the results

Absorption (cm Absorption 1.6eV were 0.5 eV for aPNT and 0.3 eV10,000 for zPNT with a radius of 7.3 Å, which corresponds to the values of the bandgap between the highest valance band and the lowest conduction band, at 0.492 eV and 0 0.291 eV, as shown in Table2. Additionally,0 the 4 absorption 8 121620 edge shows a slight red shift with a decrease Frequency (eV) in the radius of the nanotubes, which is also consistent with trend in the bandgap value. Comparing the absorption spectra of aPNTs with different radii (Figure 10), all the absorption peaks of aPNTsFigure monotonously 10. Absorption coefficient decline of with BP nanotubes the increase with electric in the field radii direction of the parallel BP nanotubes to the radius. as long as the (a) aPNTs with different radii; (b) aPNT and zPNT with a radius of 7.3 Å, and (c) aPNT and zPNT radius is biggerwith a than radius 7.3 of 11.5 Å. Å.

40 s s (a) 50 (b) p p total total 30 40

30 20

4.7eV 3.0eV 6.7eV 20 4.7eV 1.5eV 10 10 Partial Density of States Density Partial Partial Density of States Density Partial

0 0 -15 -10Energy -5 (eV) 0 5 -15 -10 -5 0 5 Energy (eV)

Figure 11. FigurePartial 11. Density Partial Density of States of States (PDOS) (PDOS) of of BP BP nanotubes nanotubes with with radius radius of 7.3 of Å: 7.3 (a) aPNTs Å: (a) and aPNTs (b) zPNTs. and ( b) zPNTs.

3.3.3. Reflection Coefficient 3.3.3. Reﬂection Coefﬁcient The reflectivity at normal incidence is given by:

The reflectivity at normal incidence is given by: 2 − 2 − 2 ε ω + ε ω − (n 1) k 1 ( ) i 2 ( ) 1 ω = = 2 R( ) 2 2 2 2 p (8) (n(−n +1)1)−+kk εε1(ωω)+ +εiε2(ωω)+− 1 R(ω) = = 1 ( ) i 2 ( ) 1 (8) 2 2 p (n + 1) + k ε1(ω) + iε2(ω) + 1 where n and k are the real and imaginary parts of the complex refractive index, which are known as where ntheand refractivek are the index real and and the imaginaryextinction index, parts respectively, of the complex and are refractivegiven by the index, following which relationship: are known as the refractive index and the extinction index, respectively, and are given by the following relationship: v u q r u 2 ( ) + 2 ( ) + ( ) |ε(ω) + ε (ω)| t ε1 ω ε2 ω ε1 ω n(ω) = 1 = 2 2 v (9) u q r u 2 ( ) + 2 ( ) − ( ) |ε(ω) − ε (ω)| t ε1 ω ε2 ω ε1 ω k(ω) = 1 = 2 2 The reflectivities of aPNTs and zPNTs with perpendicular polarization are shown in Figure 12. Reflectivity is sensitive to the rolling direction and the radius of PNTs, including the static reflectivity. Comparing Figure 12 to Figure 10, the reflectivity peaks are located at the same sites as those of the absorption coefficients if the frequency of the incident light is higher than the value of the first peak of the absorption coefficient. This is because a solid can effectively reflect the light in the same scope if it shows a strong absorption fora specific light range. For the aPNTs shown in Figure 12a, when the frequency is lower than the value of the first peak of the absorption coefficient, the reflectivity sharply increases, resulting in the intense reflection of low frequency incident light because of the smaller transition from the valence band to the conduction band. Conversely, zPNTs do not exhibit any sign of increase at the beginning of the increase in frequency, as shown in Figure 12b,c. For aPNTs, after a drop in the static reflectivity (0–0.1 eV) and at the beginning of the reflectivity curves, a platform gently increases (ranging from 0.1 to 0.8 eV) followed by a decrease from 0.8 to 2.5 eV. Instead, for zPNTs, an evident increase occurs from 0 to 1 eV, and then the value is maintained within a narrow range. In the 2D black phosphorene layer, the optical reflectivity spectra display a strong directional dependency (Figure 12d). These characteristics correspond to the different Materials 2018, 11, x FOR PEER REVIEW 13 of 17

ε (ω) + ε (ω) ε 2 (ω) + ε 2 (ω) + ε (ω) n(ω) = 1 = 1 2 1 2 2 (9) ε (ω) − ε (ω) ε 2 (ω) + ε 2 (ω) − ε (ω) k(ω) = 1 = 1 2 1 2 2

The reflectivities of aPNTs and zPNTs with perpendicular polarization are shown in Figure 12. Reflectivity is sensitive to the rolling direction and the radius of PNTs, including the static reflectivity. Comparing Figure 12 to Figure 10, the reflectivity peaks are located at the same sites as those of the absorption coefficients if the frequency of the incident light is higher than the value of the first peak of the absorption coefficient. This is because a solid can effectively reflect the light in the same scope if it shows a strong absorption fora specific light range. For the aPNTs shown in Figure 12a, when the frequency is lower than the value of the first peak of the absorption coefficient, the reflectivity sharply increases, resulting in the intense reflection of low frequency incident light because of the smaller transition from the valence band to the conduction band. Conversely, zPNTs do not exhibit any sign of increase at the beginning of the increase in Materials 2018, 11, 304 13 of 16 frequency, as shown in Figure 12b,c. For aPNTs, after a drop in the static reflectivity (0–0.1 eV) and at the beginning of the reflectivity curves, a platform gently increases (ranging from 0.1 to 0.8 eV) followed by a decrease from 0.8 to 2.5 eV. Instead, for zPNTs, an evident increase occurs from 0 to variations1in eV, the and reflectivity then the value of is maintained 2D black within phosphorus a narrow withrange. itsIn the polarization 2D black phosphorene direction layer, along the armchair or zigzag directions.optical reflectivity The spectra peaks display in the a aPNTsstrong direct reflectivityional dependency curves (Figure show 12d). an overall These characteristics red shift,due to the strain sustainedcorrespond by to the the nanotubes. different variations Additionally, in the reflecti aPNTsvity of 2D exhibit black phosphorus a significantly with its higher polarization reflectivity to direction along armchair or zigzag directions. The peaks in the aPNTs reflectivity curves show an low frequency incident light. Overall, the reflectivity of aPNTs is higher than that of zPNTs, except overall red shift, due to the strain sustained by the nanotubes. Additionally, aPNTs exhibit a at low frequenciessignificantly in higher therange reflectivity of 2.0 to low to 3.2frequency eV for inci PNTsdent light. with Overall, radius the of reflectivity 7.3 Å, and of aPNTs from 0is to 1.75 eV for PNTs withhigher radius than that of of 11.5 zPNTs, Å except (Figure at low 12b,c). frequencies For aPNTs, in the range three of 2.0 peaks to 3.2 aroundeV for PNTs 0.8, with 4.8, radius and 7.3 eV in the reflectivityof 7.3 Å, curves and from show 0 to 1.75 a red eV for shift PNTs relative with radius to the of 11.5 2D Å reflectivity (Figure 12b,c). polarized For aPNTs, alongthree peaks the armchair around 0.8, 4.8, and 7.3 eV in the reflectivity curves show a red shift relative to the 2D reflectivity direction.polarized This is due along to the the armchair narrowed direction. bandgap This is due shown to the innarrowed Figure bandgap5, which shown is induced in Figure by5, which the stains the nanotubesis sustainedinduced by the during stains theirthe nanotubes formation sustained process. during Correspondingly, their formation process. a similarCorrespondingly, conclusion a can be drawn forsimilar zPNT. conclusion can be drawn for zPNT.

0.16 (a) aPNT r = 5.2 (b) aPNT r = 7.3 0.14 aPNT r = 7.3 0.15 zPNT r = 7.3 0.132 aPNT r = 10.08 0.12 0.8eV 0.132 0.112 aPNT r = 11.5 4.8eV 7.3eV aPNT r = 12.85 0.7eV 0.10 0.107 4.7eV 7.2eV 0.10 0.08 0.085

0.06

Reflectivity 0.053 0.039

Reflectivity 0.05 0.04

0.02

0.00 0.00 0 5 10 15 20 048121620 Materials 2018, 11, x FOR PEER REVIEW Frequency (eV) 14 of 17 Frequency (eV)

0.8 (c) 4.9eV aPNT r = 11.5 (d) parrallel to zigzag direction of 2D BP 0.06 zPNT r = 11.5 0.7 parrallel to armchair direction of 2D BP perpendicular to the plane of 2D BP 0.053 0.6 1.0eV 8.6eV 10.5eV 0.5 0.04 5.8eV 9.7eV 2.5eV

0.4

5.5eV 4.5eV 0.3

Reflectivity 0.023 0.84eV 0.02 Reflectivity 0.225 6.8eV 0.2 0.143 0.1 0.064 0.00 0.0 048121620 Frequency (eV) 0 4 8 12 16 20 Frequency (eV)

Figure 12.FigureReﬂectivity 12. Reflectivity of nanotubes of nanotubes and and two-dimensional two-dimensional (2D) (2D) BP: ( BP:a) aPNTs (a) aPNTs with different with differentradii; radii; (b) aPNT and(b) aPNT zPNT and with zPNT the with same the same radius radius of of 7.3 7.3 Å;Å; ( (cc)) aPNT aPNT and and zPNT zPNT with the with same the radius same of radius11.5 Å, of 11.5 Å, and (d) reﬂectivityand (d) reflectivity of 2D BPof 2D along BP along different different polarizations. polarizations. 3.3.4. Refractive Index 3.3.4. Refractive Index The refractive index can be expressed as: The refractive index can be expressedN as:() =() +() ()N(−ω)(=)n(=ω)+(ik) +(ω)() (10)

2 2()2() =() n(ω) − k(ω) = ε1(ω) + iε2(ω) (10) The refraction index n and extinction index k for BP nanotubes, calculated according to Equation 2n(ω)k(ω) = ε2(ω) (9), are shown in Figure 13. The static refractive index shows its dependence on the rolling direction The refractionand the radius index of PNTs,n and which extinction fluctuates index withk thefor change BP nanotubes, in the radius calculated of nanotubes, according similar to to the Equation (9), are showndependence in Figure on 13 the. Thechange static in static refractive dielectric index function shows shown its in dependenceFigure 9a. In addition, on the the rolling static direction refractive indexes of aPNTs, ranging from 1.6 to 2.2 eV, are higher than those of zPNTs, ranging from and the radiusabout 1.3 of to PNTs, 1.5 eV as which a whole, ﬂuctuates attributed to with the larger the change static refractive in the index radius of 2D of BP nanotubes, with polarization similar to the dependenceparallel on theto the change armchair in direction static dielectric shown in Figure function 13b, showncompared in to Figure the values9a. for In addition,polarization the static refractiveparallel indexes to ofthe aPNTs, zigzag direction. ranging The from refractive 1.6 to 2.2indexes eV, of are aPNTs higher and than zPNTs those in the of low zPNTs, frequency ranging from about 1.3 torange 1.5 show eV as completely a whole, different attributed trends. to theSimilar larger to the static reflectivity, refractive with index increasing of 2D frequency, BP with the polarization refractive index n of aPNTs sharply decrease when the frequency starts to increase, and then parallel tomaintains the armchair a gentle direction decrease, shown followed in by Figure a sharp 13 b,decrease compared (Figure to 13a). the valuesThis corresponds for polarization to the parallel to the zigzagvariation direction. in the refractive The refractive index for indexes2D BP polarized of aPNTs along and the armchair zPNTs direction in the low as shown frequency in Figure range show completely13b. different The reverse trends. change Similar in zPNTs to thecompared reﬂectivity, to aPNTs with for low increasing frequencies frequency, is also attributed the refractive to the index n of aPNTs sharplyvariation decreasein the refractive when index the frequencyfor 2D BP polarized starts to along increase, the zigzag and direction. then maintains Additionally, a gentle for decrease, aPNTs, the real parts of the refraction index n have the lowest values at 1.1, 4.5, and 6.6 eV, which correspond to the peaks of the imaginary parts of refractive index k. This is consistent with the peaks of the absorption coefficient shown in Figure 10, because the extinction index k represents the decrease in the incident light as it propagates through the material. The extinctive index is different because three peaks are distributed around 1.1, 4.5, and 6.6 eV for aPNTs; however, only two peaks appeared in the extinction curves of zPNTs. This is because the extinction indexes of aPNTs have a combined effect for the extinction peaks appearing in the extinction index of 2D BP with polarization along the armchair direction (2.2 and 6.1 eV) and with polarization perpendicular to the 2D BP plane (7.9 eV). Similar to the reflectivity of aPNTs, the red shift of the peaks in the extinction indexes for aPNTs and zPNTs are a result of the narrowed bandgaps under strain.

Materials 2018, 11, 304 14 of 16 followed by a sharp decrease (Figure 13a). This corresponds to the variation in the refractive index for 2D BP polarized along the armchair direction as shown in Figure 13b. The reverse change in zPNTs compared to aPNTs for low frequencies is also attributed to the variation in the refractive index for 2D BP polarized along the zigzag direction. Additionally, for aPNTs, the real parts of the refraction index n have the lowest values at 1.1, 4.5, and 6.6 eV, which correspond to the peaks of the imaginary parts of refractive index k. This is consistent with the peaks of the absorption coefﬁcient shown in Figure 10, because the extinction index k represents the decrease in the incident light as it propagates through the material. The extinctive index is different because three peaks are distributed around 1.1, 4.5, and 6.6 eV for aPNTs; however, only two peaks appeared in the extinction curves of zPNTs. This is because the extinction indexes of aPNTs have a combined effect for the extinction peaks appearing in the extinction index of 2D BP with polarization along the armchair direction (2.2 and 6.1 eV) and with polarization perpendicular to the 2D BP plane (7.9 eV). Similar to the reflectivity of aPNTs, the red shift of the peaks in the extinction indexes for aPNTs and zPNTs are a result of the narrowed bandgaps under strain. Materials 2018, 11, x FOR PEER REVIEW 15 of 17

2.5 (a) 3.5 aPNT r = 5.2 (b) parrallel to zigzag direction of 2D BP 2.14 aPNT r = 7.3 parrallel to armchair direction of 2D BP 2.0 2.00 3.0 1.96 aPNT r = 10.08 2.81 perpendicular to the plane of 2D BP 1.81 aPNT r = 11.5 aPNT r = 12.85 2.5 1.60 1.5 1.43 zPNT r = 7.3 2.21 zPNT r = 11.5 2.0

1.0 1.5 1.68 1.35 1.0 Refractive Index Refractive

0.5 Refractive Index 0.5

0.0 0.0 0 4 8 121620 048121620 Frequency (eV) Frequency (eV)

3.0 (f) parrallel to zigzag direction of 2D BP parrallel to armchair direction of 2D BP 2.5 perpendicular to the plane of 2D BP 5.1eV 2.0 6.9eV

1.5

1.0

Extinction Index 1.2eV 0.5

0.0 0 4 8 121620 Frequency (eV)

Figure 13. RefractiveFigure 13. Refractive index of index nanotubes of nanotubes with with different different radii radii and and 2D 2D BP: BP: ((aa)) refractiverefractive index index n ofn of aPNTs with differentaPNTs radii; with( differentb) refractive radii; (b index) refractiven of index 2D BP n of along 2D BP differentalong different polarizations; polarizations; ( c(c)) extinctiveextinctive index k of aPNTs withindex different k of aPNTs radii; with different (d) extinctive radii; (d) indexextinctivek of index aPNT k of andaPNT zPNT and zPNT with with the the same same radiusradius of 7.3 Å; of 7.3 Å; (e) extinctive index k of aPNT and zPNT with the same radius of 11.5 Å, and (f) extinctive (e) extinctiveindex index k of 2Dk ofBP aPNTalong different and zPNT polarizations. with the same radius of 11.5 Å, and (f) extinctive index k of 2D BP along different polarizations. 4. Conclusions Based on density functional theory, the bending strain energy, electronic structure, and optical properties of BP nanotubes were investigated. The results showed that these properties are closely related to the rolling direction and radius of the BP nanotube. The bending strain energy of the zPNTs is considerably higher than that of the aPNT with the same curvature radius, due to the anisotropy of the BP's structure, showing that the aPNTs are more stable than the zPNTs with the same radius. Additionally, the bending strain energy of BP nanotubes monotonically increases with the decrease in the models' radius with the same bending direction. All the calculated BP nanotubes showed direct bandgaps, and the BP nanotubes with the same rolling direction expressed a monotone increasing

Materials 2018, 11, 304 15 of 16

4. Conclusions Based on density functional theory, the bending strain energy, electronic structure, and optical properties of BP nanotubes were investigated. The results showed that these properties are closely related to the rolling direction and radius of the BP nanotube. The bending strain energy of the zPNTs is considerably higher than that of the aPNT with the same curvature radius, due to the anisotropy of the BP's structure, showing that the aPNTs are more stable than the zPNTs with the same radius. Additionally, the bending strain energy of BP nanotubes monotonically increases with the decrease in the models' radius with the same bending direction. All the calculated BP nanotubes showed direct bandgaps, and the BP nanotubes with the same rolling direction expressed a monotone increasing trend in the value of the bandgap with the decrease in radius, which is a stacking effect of the compression strain on the inner atoms and the tension strain on the outer atoms. Additionally, for the zPNTs, the nanotubes rolled along the zigzag direction, and experienced more severe strain on the lattice than aPNTs. The zPNTs showed narrower bandgaps than the aPNTs with the same radius. The imaginary part of the dielectric function, the absorption range, reﬂectivity, and the imaginary part of the refractive index of aPNTs had a wider range than those of zPNTs, having higher values overall. The absorption edge of aPNTs had a higher value than that of zPNTs with the same radius, which corresponds to the bandgap values. As a result, the aPNTs are more efﬁcient for the absorption of low frequency incident light. Additionally, all of the absorption peaks of aPNTs monotonously decreased with the increase in the radius of BP nanotubes as long as the radius was larger than 7.3 Å.

Acknowledgments: This work was financially supported by the National Natural Science Foundation of China (No. 51601153), Chongqing research program of basic research and frontier technology (No. cstc2017jcyjAX0195), and the Fundamental Research Funds for the Central Universities (No. SWU115068 and No. XDJK2017D020). Author Contributions: Chunmei Li and Zhiqian Chen conceived and designed the research programme; Zhongjing Xie performed the calculations; Chunmei Li and Nanpu Cheng analyzed the data; Jinghui Wang and Guoan Zhu built computational models; Chunmei Li wrote the paper. Conflicts of Interest: The authors have no conflicts of interest to declare.

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