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Structural, mechanical, and electronic properties of nanotubes based on buckled arsenene: A first-principles study
Materials Today Communications 22 (2020) 100791
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Structural, mechanical, and electronic properties of nanotubes based on buckled arsenene: A first-principles study
T
Bo Chen*, Lin Xue, Yan Han, Xiang-Qian Li, Zhi Yang College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanotubes Stability Structural property Mechanical property Electronic property Density functional theory
At present work, the structural, mechanical, and electronic properties of zigzag and armchair buckled arsenene nanotubes are investigated using density functional theory. All nanotubes are found to be stable by means of vibrational spectra and room temperature molecular dynamics simulations. With the enlargement of tube diameter, armchair nanotubes are energetically more favorable than zigzag nanotubes at first and then the strain energies of both zigzag and armchair nanotubes are about the same; Young’s modulus increases in general and zigzag nanotube is stiffer than the armchair one with a comparable diameter; Poisson ratio is relatively insensitive to tube diameter at the beginning and decreases about one order of magnitude for zigzag nanotubes, while the Poisson ratio of armchair nanotubes has a opposite behavior. Zigzag nanotubes and some armchair nanotubes with small diameter exhibit an indirect band gap, whereas the armchair nanotubes with larger diameter exhibit a direct band gap. Transition between indirect to direct band gap or semiconductor to metal is tunable by uniaxial strain, and the effective mass of electron is smaller than that of hole. In particular, faceted nanotubes could be constructed by introducing defect lines or joining different structural phases of arsenene.
1. Introduction Arsenene, a single layer of arsenic atoms arranged in buckled or puckered honeycomb structure, was recently proposed [1,2]. Buckled arsenene is the minimum-energy configuration [2], which possesses an indirect band gap of about 1.60 eV and could be transformed to a direct band gap semiconductor [1] or a unique two-dimensional topological insulator [3] by applying 4 % or 11.7 % biaxial tensile strain, respectively. Besides, buckled arsenene has high carrier mobilities (635/ 1700 cm2/V·s for electron/hole) [4] and is suggested to be competitive channel material for the sub-5 nm transistors [5,6]. In experiments, few-layer arsenene nanosheets have been synthesized [7–11] and exhibited great potential to be applied to switching and light-emitting devices [7], as well as vapor sensors [9]. Like most two-dimensional materials, an arsenene sheet could be rolled up into a nanotube, and the nanotube may own versatile properties due to the quantum confinement effect. Unfortunately, rare work has been done about the nanotubes based on arsenene. Bhuvaneswari et al. studied adsorption properties of explosive vapors on buckled arsenene nanotubes and suggested that buckled arsenene nanotubes can be employed in the detection of explosive vapors [12]; Nagarajan et al. investigated the electronic properties of functionalized buckled arsenene nanotubes to demonstrate their ability as base material for ⁎
designing spintronic devices and chemical sensor [13]. Further research should be done to explore the properties and potential applications of nanotubes based on arsenene. In this work, we report a comprehensive study of the structural, mechanical, and electronic properties of zigzag and armchair buckled arsenene nanotubes, as well as the strain effect on these one-dimensional materials. It should be noted that rolling up buckled arsenene generates only round nanotubes, so faceted nanotubes are constructed by introducing defect lines or joining different structural phases of arsenene and their stability and electronic properties are also investigated. 2. Computational methods All calculations are performed using the Vienna ab initio simulation package (VASP) [14]. Interactions between electrons and nuclei are described by the projector augmented wave (PAW) potentials [15,16], and exchange-correlation between electrons is treated using generalized gradient approximation (GGA) given by Perdew, Burke, and Ernzerhof (PBE) [17]. Three-dimensional periodic boundary conditions are applied, which give rise to a nanotube in the z-direction. A vacuum region exceeding 20 Å is incorporated to avoid interactions between adjacent nanotubes. Cutoff energy for the plane-wave basis set is 450 eV
Corresponding author. E-mail address: [email protected] (B. Chen).
https://doi.org/10.1016/j.mtcomm.2019.100791 Received 19 September 2019; Received in revised form 21 November 2019; Accepted 21 November 2019 Available online 22 November 2019 2352-4928/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 2. Phonon spectra of (a) zigzag and (b) armchair nanotubes.
Fig. 1(a), with buckling of the honeycomb lattice illustrated using different colors for atoms in different sublattices. In relaxed unit cell, the calculated lattice constant, As-As bond length, As-As-As bond angle and buckling height are 3.608 Å, 2.508 Å, 91.98° and 1.397 Å, respectively, which are in good agreement with previous works [1,2]. Nanotubes are built by rolling up monolayer buckled arsenene along either the zigzag or armchair direction, which can be denoted as zigzag (n,0) or armchair (n,n) according to the definition used in carbon nanotubes [19], as seen in Fig. 1(b) and (c). Because of the buckling, there are two classes of atoms with small and large distances to the tube axis. For zigzag and armchair nanotubes, the calculated magnetic moments are zero and the ground state energies between spin-unpolarized and spin-polarized calculations are almost the same, indicating that both kinds of nanotubes are nonmagnetic. Phonon dispersion calculations are carried out to study the stability of zigzag and armchair nanotubes. If there is an instability related phonon mode in Brillouin zone, then the frequencies at some k points are imaginary. So this particular mode cannot generate restoring force to execute lattice vibration and hence the system is vulnerable to go away from its original configuration. As shown in Fig. 2, no imaginary phonon modes are observed in the computed phonon dispersion spectra, which verifies the dynamical stability of zigzag and armchair nanotubes. Besides, we can see that the acoustic and optical modes are well separated in both the zigzag and armchair cases. The larger the nanotube is, the more obvious the separation will be. And the separation for armchair (14,14) nanotube is comparable to that of monolayer buckled arsenene, see Fig. S1 in the supplementary material. For the long-wavelength acoustic modes in one-dimensional system, the longitudinal acoustic (LA) mode and the torsional acoustic (TA) mode are linear with wave number, while the flexural acoustic (ZA) mode has a parabolic dispersion relation with wave number. In our phonon dispersion calculations, we find that all acoustic modes are linear in the long-wavelength limit. This may be due to the fact that the approach to obtain frequency spectrum through the atomistic calculation of force-constant matrix often fails for long-wavelength acoustic modes. So we use the continuum elasticity theory to correct the long-
Fig. 1. Geometric structures of (a) monolayer buckled arsenene, (b) zigzag (10,0) and (c) armchair (10,10) nanotubes. The two atomic layers are distinguished by color.
and five electrons (4s24p3) of arsenic atom are treated as valence electrons. The convergence threshold is set to 10−5 eV for energy and 0.02 eV/Å for force, respectively. A 1 × 1×7/1 × 1×14 Gamma centered Monkhorst-Pack grid is used to sample the Brillouin zone for zigzag/armchair nanotubes. All structures are fully relaxed using the conjugated gradient method [18], and both spin-unpolarized and spinpolarized calculations are performed to determine the ground state of nanotubes. 3. Results and discussion 3.1. Geometric structures and stability The structure of monolayer buckled arsenene is depicted in 2
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wavelength acoustic modes of our nanotubes [20]. For monolayer buckled arsenene, assuming its coefficients of elastic stiffness matrix, areal mass density, and flexural rigidity are cij, ρ2D, and T, respectively, then the vibration frequencies of LA mode, TA mode, and ZA mode of buckled arsenene nanotube of radius R in the long-wavelength limit can be expressed as
ωLA =
2 2 − c12 c11 k c11 ρ2D
ωTA =
c66 k ρ2D
ωZA =
R2c11 T (1 + ) k2 2ρ2D c11 R2
The flexural rigidity T can be obtained by [21]
T=
1 ∈b D 2 2
Fig. 4. Strain energy versus diameter of zigzag and armchair nanotubes.
in which ∈b is the bending strain energy divided by the surface area of the tube, and D is the tube diameter. The calculated elements of the elastic stiffness matrix are c11 = c22 = 51.97 N/m, c12 = 9.26 N/m, and c66 = 21.35 N/m; the areal mass density ρ2D and flexural rigidity T are calculated to be 2.21 × 10−6 kg/m2 and 0.53 eV, respectively. Our results are presented in Fig. 3, superposed on the acoustic modes of armchair (10,10) nanotube obtained by atomistic calculation. The corresponding group velocities at k → 0 give the longitudinal speed of sound of 4.78 km/s and the speed of sound with torsional polarization of 3.11 km/s. To find out which kind of nanotube is energetically more stable at the same diameter, we calculate the strain energy for zigzag and armchair nanotubes as a function of their diameters, as illustrated in Fig. 4. Strain energy is defined as the energy difference per As atom between a nanotube and monolayer buckled arsenene. As expected, the strain energy decreases with increasing the tube diameter, and the data can be fitted by a power law of 1/Dx, where x is 2.63 and 2.34 for zigzag and armchair nanotubes. Armchair nanotubes are energetically more favorable than zigzag nanotubes at small diameter. When the diameter is sufficiently large, the strain energy difference between zigzag and armchair nanotubes is negligible. Interestingly, the calculated strain energies of buckled arsenene nanotubes are smaller than those of blue phosphorene nanotubes when D ≥ 16 Å [22], suggesting that it may be easier to fabricate big buckled arsenene nanotubes than big blue phosphorene nanotubes (Fig. S2).
Since a structure with stable phonon spectrum still may become unstable by thermal fluctuations at elevated temperature, ab initio molecular dynamics simulation using Nosé algorithm is carried out at 300 K to investigate the thermal stability of buckled arsenene nanotubes at room temperature. To avoid artifacts associated with constraints imposed on nanotubes by finite-size unit cells, we use supercells containing 200 As atoms for zigzag (10,0) nanotube. The time period is of at least 3 ps with 2 fs time step. From Fig. 5 we can see that the equilibrium structure of zigzag (10,0) nanotube is stable at room temperature. Because the stabilities of armchair nanotubes are better or equal to those of zigzag nanotubes based on strain energy analysis, we believe that both zigzag and armchair nanotubes are stable at room temperature. 3.2. Mechanical properties Young’s modulus and Poisson ratio are calculated to investigate the mechanical properties of buckled arsenene nanotubes, with the applied largest uniaxial strain of ± 1 %. Young’s modulus, which indicates how much resistance the nanotube presents to its deformation under uniaxial strain, is defined as the second derivative of strain energy with respect to the strain at equilibrium configuration [23]
Y=
1 ∂ 2E ⋅ V0 ∂ε 2
where Y is the Young’s modulus, V0 is the relaxed equilibrium volume, E is the strain energy per unit cell, and ε is the uniaxial strain. For a unit cell of nanotube, V0 can be defined as V0 = πD0LC0, where D0 and C0 are the diameter and unit cell length of the unstrained nanotube and L is the tube thickness. However, it should be noted that the definition of tube thickness is not completely consistent in previous publications. Taking carbon nanotubes as an example, Lu chose the distance between successive planes in graphite (3.40 Å) as tube thickness [24] while
Fig. 3. Frequency of the long-wavelength acoustic modes of armchair (10,10) nanotube obtained by continuum elasticity theory (dash lines) and atomistic calculation (black solid lines).
Fig. 5. Snapshot of the final frame of molecular dynamics simulation from zigzag (10,0) nanotube at 300 K. 3
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Fig. 6. (a) Young’s modulus and (b) Poisson ratio as a function of tube diameter.
Yakobson et al. used the wall thickness (0.66 Å) as tube thickness [25]. For the ease of comparing with previous results, we adopt the tube thickness as interlayer distance of gray arsenic (2.04 Å). The calculated Young’s modulus as a function of tube diameter is plotted in Fig. 6(a). In general, Young’s modulus increases with tube diameter and converges to a constant value of approximate 120 GPa for both kinds of nanotubes; with a comparable diameter, zigzag nanotubes are stiffer than the armchair counterparts. Compared with other nanotubes, our buckled arsenene nanotubes are stiffer than MoTe2 and SiGe nanotubes [23,26] and comparable to silicon and black phosphorene nanotubes [27–29], but softer than MoS2 and TiS2 nanotubes [30,31]. As a consequence of uniaxial elongation or compression, the nanotube undergoes a deformation perpendicular to the tube axis. The ratio of transverse contraction and axial elongation is described by the Poisson ratio
[32,33], the band gap of our zigzag buckled arsenene nanotubes also approaches exponentially the value of corresponding monolayer buckled arsenene for growing diameter. The band gap decrease for zigzag nanotubes with small diameter may be due to the enhanced interatomic interactions between non-nearest neighbors (see Fig. S3 in the supplementary material) [34]. For armchair nanotubes, they transform from indirect band gap to direct band gap and the band gap decreases slightly first and then approaches the value of corresponding monolayer buckled arsenene with the increase of diameter. The valence band maximum lies at the Γ point, and the conduction band minimum appears at about 62.02 % and 64.56 % of Γ-Z for armchair (4,4) and (6,6) nanotubes while at the Γ point for armchair (8,8), (10,10), (12,12), and (14,14) nanotubes. Moreover, the differences between indirect band gap and direct gap at Γ point are 27.6 meV and 66.9 meV for armchair (4,4) and (6,6) nanotubes, which are of the order of thermal energy at room temperature. Like buckled arsenene [2], from the total and partial densities of states we can see that for both kinds of nanotubes, the states near Fermi level have contributions from both s and p orbitals, and the contributions from p orbitals to the total densities of states are much higher than that from s orbitals. Applying mechanical strain is a powerful method to modulate the electronic properties of materials, where the strain is defined as (C-C0)/ C0. So the positive or negative value indicates the tensile or compressive strain. Take zigzag (10,0) and armchair (10,10) nanotubes as examples, the valence band maxima locate at the Γ point for the unstrained and strained nanotubes other than armchair (10,10) nanotube under tensile strains of 1 % and 2 %, where the valence band maxima lie at 2.54 % and 1.26 % of Γ-Z, respectively. As seen from Fig. 9, the maximal band gap is obtained at tensile strains of 1 % for zigzag (10,0) nanotube and in the pristine case for armchair (10,10) nanotube. Zigzag (10,0) or armchair (10,10) nanotube transforms from an indirect band gap semiconductor to metal at 13 % or 11 % compressive strain, respectively. Armchair (10,10) nanotube mostly remains a direct band gap semiconductor under tensile strain, while zigzag (10,0) nanotube transforms from indirect to direct band gap at 6 % tensile strain (band structures of the strained nanotubes are shown in Figs. S4 and S5 in the supplementary material). Besides, the changes of diameter does not exceed 3.1 % for zigzag (10,0) nanotube and 1.98 % for armchair (10,10) nanotube within the strain of −10 % to 10 %, and the structures of both nanotubes remain intact without any bond breaking, which demonstrates the large elastic range of buckled arsenene nanotubes. However, it should be noted that the larger the strain is, the more difficulty of applying strain on nanotubes (see Fig. S6 of the strain energy as a function of strain in the supplementary material). The effective masses of electron at conduction band minimum and hole at valence band maximum are also computed, as shown in Fig. 10. For zigzag nanotubes, the effective masses of electron and hole decrease
ΔD ΔC = −v D C where ν is the Poisson ratio, D is the tube diameter and C is the corresponding unit cell length of nanotube. The dependence of Poisson ratio on tube diameter is shown in Fig. 6(b), which shows an opposite behavior for the two kinds of nanotubes. With the increase of tube diameter, Poisson ratios for zigzag (6,0), (7,0), (10,0), and (12,0) nanotubes are among 0.21-0.27 and then decrease about one order of magnitude for zigzag (16,0) and (18,0) nanotubes; while Poisson ratio for armchair (4,4) nanotube is about 0.228 and decreases about one order of magnitude for armchair (8,8), (10,10), (12,12) and (14,14) nanotubes. It seems that for D < 10 Å and D > 20 Å, the Poisson ratios for zigzag and armchair nanotubes are comparable, and large nanotubes are more resistant to tube elongation or compression than small nanotubes. The reason for the sudden decrease of Poisson ratio with increase of tube diameter is unclear now, which needs further research. 3.3. Electronic properties The band structure, total and partial densities of states, and the dependence of band gap on tube diameter for zigzag and armchair nanotubes are shown in Figs. 7 and 8 , respectively. For zigzag nanotubes, all of them are indirect semiconductors, with valence band maxima appear at the Γ point and conduction band minima occur along the Γ-Z direction. With the increase of tube diameter, the percentage of location of conduction band minimum along Γ-Z decreases and approaches the percentage of location of conduction band minimum along Γ-M of buckled arsenene. Note that the Γ-M direction of buckled arsenene corresponds to the Γ-Z direction of zigzag nanotubes [32], so the conduction band edge of buckled arsenene qualitatively resembles that of zigzag nanotubes. Like zigzag blue and black phosphorene nanotubes 4
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Fig. 7. Band structure, total and partial densities of states, and the dependences of band gap as well as the percentage of location of conduction band minimum along Γ-Z on tube diameter for zigzag nanotubes, the Fermi energy level is set at zero. The black and red dashed lines in the right bottom panel refer to the band gap and the percentage of location of conduction band minimum along Γ-M of buckled arsenene, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
first with increase of tube diameter and stabilized at about 0.2m0 and 0.51m0, respectively, where m0 is the mass of free electron; while for armchair nanotubes the hole effective mass increases with tube diameter but the electron effective mass decreases and then stabilized at 0.16m0. Note that the effective masses of electron and hole are 0.53m0 and 0.83 m0 for MoS2 (6,6) nanotube and 0.51m0 and 1.55m0 for MoS2 (10,0) nanotube [31], thus the charge mobilities of our buckled arsenene nanotubes may be comparable to or higher than those of MoS2 nanotubes.
defect lines that invert the buckling orientation on one side with respect to the other. Thus these two defect lines should appear in pairs to construct a faceted nanotube. As pictured in Fig. 11(b), a quadrilateral nanotube can be made by introducing these two defect lines, whose stability is confirmed by phonon spectrum shown in Fig. 11(c). This quadrilateral nanotube is an indirect semiconductor with a band gap of about 1.50 eV (Fig. 11(d)). For zigzag nanotubes, we use the defect line along armchair direction shown in Fig. 12(a) to make a quadrilateral nanotube as depicted in Fig. 12(b). This quadrilateral nanotube is dynamically stable and has an indirect gap of 1.27 eV, as seen form Fig. 12(c) and (d). Based on these two examples, we are sure that faceted nanotubes based on buckled arsenene can be constructed by introducing defect lines. In addition to introducing defect lines, joining different structural phases of arsenene may be also a feasible way to create faceted nanotubes [34]. As mentioned in the introduction, buckled and puckered arsenenes are the two stable allotropes, so we try to construct faceted nanotubes by joining these two phases. We prove the feasibility of this method, as seen part seven in the supplementary material.
3.4. Faceted nanotubes Aierken et al. demonstrated that for blue phosphorene nanotubes, faceted nanotubes are energetically more favorable than small round nanotubes, and faceted nanotubes can be generated by creating defect lines [22]. Since phosphorus and arsenic are in the same group of periodic table, we wonder if it is also possible to create faceted nanotubes based on buckled arsenene by introducing defect lines. For armchair nanotubes, the defect lines should along the zigzag direction. As described in Ref. 22 and depicted in Fig. 11(a), we use two kinds of 5
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Fig. 8. Band structure, total and partial densities of states, and the dependence of band gap on tube diameter for armchair nanotubes, the Fermi energy level is set at zero. The black and blue dots as well as the black dashed line in the right bottom panel refer to the indirect and direct band gap and the band gap of buckled arsenene, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
4. Conclusions
nanotubes are stable at room temperature and as stiff as silicon and black phosphorene nanotubes. Zigzag nanotubes and some small armchair nanotubes are indirect semiconductors while large armchair nanotubes are direct semiconductors. The band gap value, positions of
In conclusion, first we investigated the structural, mechanical, and electronic properties of buckled arsenene nanotubes. Buckled arsenene
Fig. 9. Dependences of band gap and the percentage of location of conduction band minimum along Γ-Z on tube diameter for (a) zigzag (10,0) and (b) armchair (10,10) nanotubes. The black and blue dots refer to the indirect and direct band gap, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
6
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Fig. 10. Hole and electron effective masses with respect to tube diameter for (a) zigzag and (b) armchair nanotubes, m0 is the mass of free electron.
created by introducing defect lines in buckled arsenene or joining buckled and puckered arsenenes. These results show that nanotubes based on buckled arsenene are promising materials for applications in nanoelectronics and optoelectronics.
conduction band minimum and valence band maximum can be tunable by uniaxial strain. The electron effective mass is smaller than that of MoS2 nanotube while the hole effective mass is comparable to that of MoS2 nanotube. Finally, we proved that faceted nanotubes can be
Fig. 11. (a) Top and side view of two kinds of defect lines along zigzag direction. The defect lines are framed by red rectangles, and the two atomic layers are distinguished by color. (b) Cross-section view of the quadrilateral nanotube made by introducing defect lines shown in (a). (c) Phonon spectrum and (d) band structure of the quadrilateral nanotube in (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). 7
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Fig. 12. (a) Top and side view of defect line along armchair direction. The defect line is framed by red rectangle, and the two atomic layers are distinguished by color. (b) Cross-section view of the quadrilateral nanotube made by introducing defect line shown in (a). (c) Phonon spectrum and (d) band structure of the quadrilateral nanotube in (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
Data availability
Physics Research Program of China (Grant No. 11847065, 11547134, and 11547213), and the Shanxi Scholarship Council of China (Grant No. 2017-050).
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.
Appendix A. Supplementary data CRediT authorship contribution statement Supplementary material related to this article can be found, in the online version, at doi:10.1016/j.mtcomm.2019.100791.
Bo Chen: Conceptualization, Writing - original draft, Supervision, Funding acquisition. Lin Xue: Funding acquisition, Resources. Yan Han: Funding acquisition, Project administration. Xiang-Qian Li: Formal analysis, Visualization. Zhi Yang: Resources.
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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by the Special Foundation for Theoretical 8
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Structural, mechanical, and electronic properties of nanotubes based on buckled arsenene: A first-principles study
T
Bo Chen*, Lin Xue, Yan Han, Xiang-Qian Li, Zhi Yang College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Nanotubes Stability Structural property Mechanical property Electronic property Density functional theory
At present work, the structural, mechanical, and electronic properties of zigzag and armchair buckled arsenene nanotubes are investigated using density functional theory. All nanotubes are found to be stable by means of vibrational spectra and room temperature molecular dynamics simulations. With the enlargement of tube diameter, armchair nanotubes are energetically more favorable than zigzag nanotubes at first and then the strain energies of both zigzag and armchair nanotubes are about the same; Young’s modulus increases in general and zigzag nanotube is stiffer than the armchair one with a comparable diameter; Poisson ratio is relatively insensitive to tube diameter at the beginning and decreases about one order of magnitude for zigzag nanotubes, while the Poisson ratio of armchair nanotubes has a opposite behavior. Zigzag nanotubes and some armchair nanotubes with small diameter exhibit an indirect band gap, whereas the armchair nanotubes with larger diameter exhibit a direct band gap. Transition between indirect to direct band gap or semiconductor to metal is tunable by uniaxial strain, and the effective mass of electron is smaller than that of hole. In particular, faceted nanotubes could be constructed by introducing defect lines or joining different structural phases of arsenene.
1. Introduction Arsenene, a single layer of arsenic atoms arranged in buckled or puckered honeycomb structure, was recently proposed [1,2]. Buckled arsenene is the minimum-energy configuration [2], which possesses an indirect band gap of about 1.60 eV and could be transformed to a direct band gap semiconductor [1] or a unique two-dimensional topological insulator [3] by applying 4 % or 11.7 % biaxial tensile strain, respectively. Besides, buckled arsenene has high carrier mobilities (635/ 1700 cm2/V·s for electron/hole) [4] and is suggested to be competitive channel material for the sub-5 nm transistors [5,6]. In experiments, few-layer arsenene nanosheets have been synthesized [7–11] and exhibited great potential to be applied to switching and light-emitting devices [7], as well as vapor sensors [9]. Like most two-dimensional materials, an arsenene sheet could be rolled up into a nanotube, and the nanotube may own versatile properties due to the quantum confinement effect. Unfortunately, rare work has been done about the nanotubes based on arsenene. Bhuvaneswari et al. studied adsorption properties of explosive vapors on buckled arsenene nanotubes and suggested that buckled arsenene nanotubes can be employed in the detection of explosive vapors [12]; Nagarajan et al. investigated the electronic properties of functionalized buckled arsenene nanotubes to demonstrate their ability as base material for ⁎
designing spintronic devices and chemical sensor [13]. Further research should be done to explore the properties and potential applications of nanotubes based on arsenene. In this work, we report a comprehensive study of the structural, mechanical, and electronic properties of zigzag and armchair buckled arsenene nanotubes, as well as the strain effect on these one-dimensional materials. It should be noted that rolling up buckled arsenene generates only round nanotubes, so faceted nanotubes are constructed by introducing defect lines or joining different structural phases of arsenene and their stability and electronic properties are also investigated. 2. Computational methods All calculations are performed using the Vienna ab initio simulation package (VASP) [14]. Interactions between electrons and nuclei are described by the projector augmented wave (PAW) potentials [15,16], and exchange-correlation between electrons is treated using generalized gradient approximation (GGA) given by Perdew, Burke, and Ernzerhof (PBE) [17]. Three-dimensional periodic boundary conditions are applied, which give rise to a nanotube in the z-direction. A vacuum region exceeding 20 Å is incorporated to avoid interactions between adjacent nanotubes. Cutoff energy for the plane-wave basis set is 450 eV
Corresponding author. E-mail address: [email protected] (B. Chen).
https://doi.org/10.1016/j.mtcomm.2019.100791 Received 19 September 2019; Received in revised form 21 November 2019; Accepted 21 November 2019 Available online 22 November 2019 2352-4928/ © 2019 Elsevier Ltd. All rights reserved.
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Fig. 2. Phonon spectra of (a) zigzag and (b) armchair nanotubes.
Fig. 1(a), with buckling of the honeycomb lattice illustrated using different colors for atoms in different sublattices. In relaxed unit cell, the calculated lattice constant, As-As bond length, As-As-As bond angle and buckling height are 3.608 Å, 2.508 Å, 91.98° and 1.397 Å, respectively, which are in good agreement with previous works [1,2]. Nanotubes are built by rolling up monolayer buckled arsenene along either the zigzag or armchair direction, which can be denoted as zigzag (n,0) or armchair (n,n) according to the definition used in carbon nanotubes [19], as seen in Fig. 1(b) and (c). Because of the buckling, there are two classes of atoms with small and large distances to the tube axis. For zigzag and armchair nanotubes, the calculated magnetic moments are zero and the ground state energies between spin-unpolarized and spin-polarized calculations are almost the same, indicating that both kinds of nanotubes are nonmagnetic. Phonon dispersion calculations are carried out to study the stability of zigzag and armchair nanotubes. If there is an instability related phonon mode in Brillouin zone, then the frequencies at some k points are imaginary. So this particular mode cannot generate restoring force to execute lattice vibration and hence the system is vulnerable to go away from its original configuration. As shown in Fig. 2, no imaginary phonon modes are observed in the computed phonon dispersion spectra, which verifies the dynamical stability of zigzag and armchair nanotubes. Besides, we can see that the acoustic and optical modes are well separated in both the zigzag and armchair cases. The larger the nanotube is, the more obvious the separation will be. And the separation for armchair (14,14) nanotube is comparable to that of monolayer buckled arsenene, see Fig. S1 in the supplementary material. For the long-wavelength acoustic modes in one-dimensional system, the longitudinal acoustic (LA) mode and the torsional acoustic (TA) mode are linear with wave number, while the flexural acoustic (ZA) mode has a parabolic dispersion relation with wave number. In our phonon dispersion calculations, we find that all acoustic modes are linear in the long-wavelength limit. This may be due to the fact that the approach to obtain frequency spectrum through the atomistic calculation of force-constant matrix often fails for long-wavelength acoustic modes. So we use the continuum elasticity theory to correct the long-
Fig. 1. Geometric structures of (a) monolayer buckled arsenene, (b) zigzag (10,0) and (c) armchair (10,10) nanotubes. The two atomic layers are distinguished by color.
and five electrons (4s24p3) of arsenic atom are treated as valence electrons. The convergence threshold is set to 10−5 eV for energy and 0.02 eV/Å for force, respectively. A 1 × 1×7/1 × 1×14 Gamma centered Monkhorst-Pack grid is used to sample the Brillouin zone for zigzag/armchair nanotubes. All structures are fully relaxed using the conjugated gradient method [18], and both spin-unpolarized and spinpolarized calculations are performed to determine the ground state of nanotubes. 3. Results and discussion 3.1. Geometric structures and stability The structure of monolayer buckled arsenene is depicted in 2
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wavelength acoustic modes of our nanotubes [20]. For monolayer buckled arsenene, assuming its coefficients of elastic stiffness matrix, areal mass density, and flexural rigidity are cij, ρ2D, and T, respectively, then the vibration frequencies of LA mode, TA mode, and ZA mode of buckled arsenene nanotube of radius R in the long-wavelength limit can be expressed as
ωLA =
2 2 − c12 c11 k c11 ρ2D
ωTA =
c66 k ρ2D
ωZA =
R2c11 T (1 + ) k2 2ρ2D c11 R2
The flexural rigidity T can be obtained by [21]
T=
1 ∈b D 2 2
Fig. 4. Strain energy versus diameter of zigzag and armchair nanotubes.
in which ∈b is the bending strain energy divided by the surface area of the tube, and D is the tube diameter. The calculated elements of the elastic stiffness matrix are c11 = c22 = 51.97 N/m, c12 = 9.26 N/m, and c66 = 21.35 N/m; the areal mass density ρ2D and flexural rigidity T are calculated to be 2.21 × 10−6 kg/m2 and 0.53 eV, respectively. Our results are presented in Fig. 3, superposed on the acoustic modes of armchair (10,10) nanotube obtained by atomistic calculation. The corresponding group velocities at k → 0 give the longitudinal speed of sound of 4.78 km/s and the speed of sound with torsional polarization of 3.11 km/s. To find out which kind of nanotube is energetically more stable at the same diameter, we calculate the strain energy for zigzag and armchair nanotubes as a function of their diameters, as illustrated in Fig. 4. Strain energy is defined as the energy difference per As atom between a nanotube and monolayer buckled arsenene. As expected, the strain energy decreases with increasing the tube diameter, and the data can be fitted by a power law of 1/Dx, where x is 2.63 and 2.34 for zigzag and armchair nanotubes. Armchair nanotubes are energetically more favorable than zigzag nanotubes at small diameter. When the diameter is sufficiently large, the strain energy difference between zigzag and armchair nanotubes is negligible. Interestingly, the calculated strain energies of buckled arsenene nanotubes are smaller than those of blue phosphorene nanotubes when D ≥ 16 Å [22], suggesting that it may be easier to fabricate big buckled arsenene nanotubes than big blue phosphorene nanotubes (Fig. S2).
Since a structure with stable phonon spectrum still may become unstable by thermal fluctuations at elevated temperature, ab initio molecular dynamics simulation using Nosé algorithm is carried out at 300 K to investigate the thermal stability of buckled arsenene nanotubes at room temperature. To avoid artifacts associated with constraints imposed on nanotubes by finite-size unit cells, we use supercells containing 200 As atoms for zigzag (10,0) nanotube. The time period is of at least 3 ps with 2 fs time step. From Fig. 5 we can see that the equilibrium structure of zigzag (10,0) nanotube is stable at room temperature. Because the stabilities of armchair nanotubes are better or equal to those of zigzag nanotubes based on strain energy analysis, we believe that both zigzag and armchair nanotubes are stable at room temperature. 3.2. Mechanical properties Young’s modulus and Poisson ratio are calculated to investigate the mechanical properties of buckled arsenene nanotubes, with the applied largest uniaxial strain of ± 1 %. Young’s modulus, which indicates how much resistance the nanotube presents to its deformation under uniaxial strain, is defined as the second derivative of strain energy with respect to the strain at equilibrium configuration [23]
Y=
1 ∂ 2E ⋅ V0 ∂ε 2
where Y is the Young’s modulus, V0 is the relaxed equilibrium volume, E is the strain energy per unit cell, and ε is the uniaxial strain. For a unit cell of nanotube, V0 can be defined as V0 = πD0LC0, where D0 and C0 are the diameter and unit cell length of the unstrained nanotube and L is the tube thickness. However, it should be noted that the definition of tube thickness is not completely consistent in previous publications. Taking carbon nanotubes as an example, Lu chose the distance between successive planes in graphite (3.40 Å) as tube thickness [24] while
Fig. 3. Frequency of the long-wavelength acoustic modes of armchair (10,10) nanotube obtained by continuum elasticity theory (dash lines) and atomistic calculation (black solid lines).
Fig. 5. Snapshot of the final frame of molecular dynamics simulation from zigzag (10,0) nanotube at 300 K. 3
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Fig. 6. (a) Young’s modulus and (b) Poisson ratio as a function of tube diameter.
Yakobson et al. used the wall thickness (0.66 Å) as tube thickness [25]. For the ease of comparing with previous results, we adopt the tube thickness as interlayer distance of gray arsenic (2.04 Å). The calculated Young’s modulus as a function of tube diameter is plotted in Fig. 6(a). In general, Young’s modulus increases with tube diameter and converges to a constant value of approximate 120 GPa for both kinds of nanotubes; with a comparable diameter, zigzag nanotubes are stiffer than the armchair counterparts. Compared with other nanotubes, our buckled arsenene nanotubes are stiffer than MoTe2 and SiGe nanotubes [23,26] and comparable to silicon and black phosphorene nanotubes [27–29], but softer than MoS2 and TiS2 nanotubes [30,31]. As a consequence of uniaxial elongation or compression, the nanotube undergoes a deformation perpendicular to the tube axis. The ratio of transverse contraction and axial elongation is described by the Poisson ratio
[32,33], the band gap of our zigzag buckled arsenene nanotubes also approaches exponentially the value of corresponding monolayer buckled arsenene for growing diameter. The band gap decrease for zigzag nanotubes with small diameter may be due to the enhanced interatomic interactions between non-nearest neighbors (see Fig. S3 in the supplementary material) [34]. For armchair nanotubes, they transform from indirect band gap to direct band gap and the band gap decreases slightly first and then approaches the value of corresponding monolayer buckled arsenene with the increase of diameter. The valence band maximum lies at the Γ point, and the conduction band minimum appears at about 62.02 % and 64.56 % of Γ-Z for armchair (4,4) and (6,6) nanotubes while at the Γ point for armchair (8,8), (10,10), (12,12), and (14,14) nanotubes. Moreover, the differences between indirect band gap and direct gap at Γ point are 27.6 meV and 66.9 meV for armchair (4,4) and (6,6) nanotubes, which are of the order of thermal energy at room temperature. Like buckled arsenene [2], from the total and partial densities of states we can see that for both kinds of nanotubes, the states near Fermi level have contributions from both s and p orbitals, and the contributions from p orbitals to the total densities of states are much higher than that from s orbitals. Applying mechanical strain is a powerful method to modulate the electronic properties of materials, where the strain is defined as (C-C0)/ C0. So the positive or negative value indicates the tensile or compressive strain. Take zigzag (10,0) and armchair (10,10) nanotubes as examples, the valence band maxima locate at the Γ point for the unstrained and strained nanotubes other than armchair (10,10) nanotube under tensile strains of 1 % and 2 %, where the valence band maxima lie at 2.54 % and 1.26 % of Γ-Z, respectively. As seen from Fig. 9, the maximal band gap is obtained at tensile strains of 1 % for zigzag (10,0) nanotube and in the pristine case for armchair (10,10) nanotube. Zigzag (10,0) or armchair (10,10) nanotube transforms from an indirect band gap semiconductor to metal at 13 % or 11 % compressive strain, respectively. Armchair (10,10) nanotube mostly remains a direct band gap semiconductor under tensile strain, while zigzag (10,0) nanotube transforms from indirect to direct band gap at 6 % tensile strain (band structures of the strained nanotubes are shown in Figs. S4 and S5 in the supplementary material). Besides, the changes of diameter does not exceed 3.1 % for zigzag (10,0) nanotube and 1.98 % for armchair (10,10) nanotube within the strain of −10 % to 10 %, and the structures of both nanotubes remain intact without any bond breaking, which demonstrates the large elastic range of buckled arsenene nanotubes. However, it should be noted that the larger the strain is, the more difficulty of applying strain on nanotubes (see Fig. S6 of the strain energy as a function of strain in the supplementary material). The effective masses of electron at conduction band minimum and hole at valence band maximum are also computed, as shown in Fig. 10. For zigzag nanotubes, the effective masses of electron and hole decrease
ΔD ΔC = −v D C where ν is the Poisson ratio, D is the tube diameter and C is the corresponding unit cell length of nanotube. The dependence of Poisson ratio on tube diameter is shown in Fig. 6(b), which shows an opposite behavior for the two kinds of nanotubes. With the increase of tube diameter, Poisson ratios for zigzag (6,0), (7,0), (10,0), and (12,0) nanotubes are among 0.21-0.27 and then decrease about one order of magnitude for zigzag (16,0) and (18,0) nanotubes; while Poisson ratio for armchair (4,4) nanotube is about 0.228 and decreases about one order of magnitude for armchair (8,8), (10,10), (12,12) and (14,14) nanotubes. It seems that for D < 10 Å and D > 20 Å, the Poisson ratios for zigzag and armchair nanotubes are comparable, and large nanotubes are more resistant to tube elongation or compression than small nanotubes. The reason for the sudden decrease of Poisson ratio with increase of tube diameter is unclear now, which needs further research. 3.3. Electronic properties The band structure, total and partial densities of states, and the dependence of band gap on tube diameter for zigzag and armchair nanotubes are shown in Figs. 7 and 8 , respectively. For zigzag nanotubes, all of them are indirect semiconductors, with valence band maxima appear at the Γ point and conduction band minima occur along the Γ-Z direction. With the increase of tube diameter, the percentage of location of conduction band minimum along Γ-Z decreases and approaches the percentage of location of conduction band minimum along Γ-M of buckled arsenene. Note that the Γ-M direction of buckled arsenene corresponds to the Γ-Z direction of zigzag nanotubes [32], so the conduction band edge of buckled arsenene qualitatively resembles that of zigzag nanotubes. Like zigzag blue and black phosphorene nanotubes 4
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Fig. 7. Band structure, total and partial densities of states, and the dependences of band gap as well as the percentage of location of conduction band minimum along Γ-Z on tube diameter for zigzag nanotubes, the Fermi energy level is set at zero. The black and red dashed lines in the right bottom panel refer to the band gap and the percentage of location of conduction band minimum along Γ-M of buckled arsenene, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
first with increase of tube diameter and stabilized at about 0.2m0 and 0.51m0, respectively, where m0 is the mass of free electron; while for armchair nanotubes the hole effective mass increases with tube diameter but the electron effective mass decreases and then stabilized at 0.16m0. Note that the effective masses of electron and hole are 0.53m0 and 0.83 m0 for MoS2 (6,6) nanotube and 0.51m0 and 1.55m0 for MoS2 (10,0) nanotube [31], thus the charge mobilities of our buckled arsenene nanotubes may be comparable to or higher than those of MoS2 nanotubes.
defect lines that invert the buckling orientation on one side with respect to the other. Thus these two defect lines should appear in pairs to construct a faceted nanotube. As pictured in Fig. 11(b), a quadrilateral nanotube can be made by introducing these two defect lines, whose stability is confirmed by phonon spectrum shown in Fig. 11(c). This quadrilateral nanotube is an indirect semiconductor with a band gap of about 1.50 eV (Fig. 11(d)). For zigzag nanotubes, we use the defect line along armchair direction shown in Fig. 12(a) to make a quadrilateral nanotube as depicted in Fig. 12(b). This quadrilateral nanotube is dynamically stable and has an indirect gap of 1.27 eV, as seen form Fig. 12(c) and (d). Based on these two examples, we are sure that faceted nanotubes based on buckled arsenene can be constructed by introducing defect lines. In addition to introducing defect lines, joining different structural phases of arsenene may be also a feasible way to create faceted nanotubes [34]. As mentioned in the introduction, buckled and puckered arsenenes are the two stable allotropes, so we try to construct faceted nanotubes by joining these two phases. We prove the feasibility of this method, as seen part seven in the supplementary material.
3.4. Faceted nanotubes Aierken et al. demonstrated that for blue phosphorene nanotubes, faceted nanotubes are energetically more favorable than small round nanotubes, and faceted nanotubes can be generated by creating defect lines [22]. Since phosphorus and arsenic are in the same group of periodic table, we wonder if it is also possible to create faceted nanotubes based on buckled arsenene by introducing defect lines. For armchair nanotubes, the defect lines should along the zigzag direction. As described in Ref. 22 and depicted in Fig. 11(a), we use two kinds of 5
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Fig. 8. Band structure, total and partial densities of states, and the dependence of band gap on tube diameter for armchair nanotubes, the Fermi energy level is set at zero. The black and blue dots as well as the black dashed line in the right bottom panel refer to the indirect and direct band gap and the band gap of buckled arsenene, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
4. Conclusions
nanotubes are stable at room temperature and as stiff as silicon and black phosphorene nanotubes. Zigzag nanotubes and some small armchair nanotubes are indirect semiconductors while large armchair nanotubes are direct semiconductors. The band gap value, positions of
In conclusion, first we investigated the structural, mechanical, and electronic properties of buckled arsenene nanotubes. Buckled arsenene
Fig. 9. Dependences of band gap and the percentage of location of conduction band minimum along Γ-Z on tube diameter for (a) zigzag (10,0) and (b) armchair (10,10) nanotubes. The black and blue dots refer to the indirect and direct band gap, respectively. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
6
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Fig. 10. Hole and electron effective masses with respect to tube diameter for (a) zigzag and (b) armchair nanotubes, m0 is the mass of free electron.
created by introducing defect lines in buckled arsenene or joining buckled and puckered arsenenes. These results show that nanotubes based on buckled arsenene are promising materials for applications in nanoelectronics and optoelectronics.
conduction band minimum and valence band maximum can be tunable by uniaxial strain. The electron effective mass is smaller than that of MoS2 nanotube while the hole effective mass is comparable to that of MoS2 nanotube. Finally, we proved that faceted nanotubes can be
Fig. 11. (a) Top and side view of two kinds of defect lines along zigzag direction. The defect lines are framed by red rectangles, and the two atomic layers are distinguished by color. (b) Cross-section view of the quadrilateral nanotube made by introducing defect lines shown in (a). (c) Phonon spectrum and (d) band structure of the quadrilateral nanotube in (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). 7
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Fig. 12. (a) Top and side view of defect line along armchair direction. The defect line is framed by red rectangle, and the two atomic layers are distinguished by color. (b) Cross-section view of the quadrilateral nanotube made by introducing defect line shown in (a). (c) Phonon spectrum and (d) band structure of the quadrilateral nanotube in (b). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
Data availability
Physics Research Program of China (Grant No. 11847065, 11547134, and 11547213), and the Shanxi Scholarship Council of China (Grant No. 2017-050).
The raw/processed data required to reproduce these findings cannot be shared at this time due to technical or time limitations.
Appendix A. Supplementary data CRediT authorship contribution statement Supplementary material related to this article can be found, in the online version, at doi:10.1016/j.mtcomm.2019.100791.
Bo Chen: Conceptualization, Writing - original draft, Supervision, Funding acquisition. Lin Xue: Funding acquisition, Resources. Yan Han: Funding acquisition, Project administration. Xiang-Qian Li: Formal analysis, Visualization. Zhi Yang: Resources.
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Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work is supported by the Special Foundation for Theoretical 8
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