Novel Carbon Nanotubes Rolled from 6,6,12-Graphyne: Double Dirac

Article pubs.acs.org/JPCC

Novel Carbon Nanotubes Rolled from 6,6,12-Graphyne: Double Dirac Points in 1D Material Dong-Chun Yang,† Ran Jia,*,† Yu Wang,† Chui-Peng Kong,† Jian Wang,† Yuchen Ma,‡ Roberts I. Eglitis,§ and Hong-Xing Zhang*,† †

Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, 130023 Changchun, PR China ‡ School of Chemistry and Chemical Engineering, Shandong University, Jinan 250100, China § Institute of Solid State Physics, University of Latvia, 8 Kengaraga Str., Riga LV1067, Latvia S Supporting Information *

ABSTRACT: Two kinds of novel carbon nanotubes, namely, (N, 0) and (0, N) 6,6,12-graphyne nanotubes (6,6,12-GNTs), are constructed by rolling up the rectangular 6,6,12-graphyne sheets along two different sides into cylinders. The mechanical and electronic properties of 6,6,12GNTs with varied N from 3 to 20 are investigated by using density functional theory. Unlike the single-wall carbon nanotubes, the Young’s moduli of 6,6,12-GNTs do not remain constant in the case of (N, 0), but the (0, N) tubes possess almost the same one around 0.32 TPa. The band structures and density of states are also exhibited in this work. When the tube sizes N are bigger than four, Dirac points appear at Fermi level in the band maps of (N, 0) type 6,6,12-GNTs following an even−odd law, while the (0, N) tubes are narrow-gap semiconductors with tiny band gaps between 5.5 and 247.3 meV.

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INTRODUCTION The versatile flexibility of carbon hybridization states (sp, sp2, and sp3) between carbon atoms results in a wide variety of allotropic structures in nature,1 which greatly enriches the carbon family. In recent years, many new carbon allotropes such as graphene (2D),2 carbon nanotubes (1D),3 and fullerene (0D)4 have been successfully synthesized and characterized. In particular, the monolayer graphene with honeycomb structure containing sp2 carbon atoms has attracted tremendous attention in scientific communities since it was isolated by Novoselov and coworkers5 due to its extraordinary electronic properties.6−12 The amazing electronic structure is characterized by its peculiar band structure featuring the so-called Dirac point, where the electron and hole spectra meet linearly at a single point in the momentum space, thereby making graphene a semiconductor with a zero band gap. Compared with graphene, another interesting 2D carbon allotrope-graphyne, comprising one atom-thick sheets with both sp- in addition to sp2-hybridized carbon atoms, was first proposed by Baughman et al. in 1987.13 Several different structures of graphyne (e.g., α-, β-, γ-, and 6,6,12-graphynes) can be obtained by inserting acetylenic linkages into the honeycomb structure of the pristine graphene.14 The single and triple bonds between the corner atoms in the honeycomb structure endow graphyne systems with many excellent properties, including remarkable chemical stabilities, extremely large surface, and uniformly located pores as well as superior electronic properties.15−17 Recently, the electronic properties of © 2017 American Chemical Society

different graphyne allotropes have been carefully studied with the aid of different theoretical methods.18−21 Results show that the possible alterations of the atomic bonds can lead to some impressive changes in their electronic structures. For example, all of the α-, β-, and 6,6,12-graphynes exhibit the Dirac cones in their band maps. In particular, the 6,6,12-graphyne contains four cones that show the self-doping, nonequivalent, and distorting features in its first Brillouin zone. It is unlikely that the γ-graphyne is a semiconductor with an intrinsic nonzero band gap due to the Peierls instability that corresponds to the Kekule distortion effect.22,23 These results also demonstrate that Dirac cones are not unique to graphene and that neither all carbon atoms being chemically equivalent nor the hexagonal symmetry is the prerequisite for the existence of Dirac cones. These special characters of graphynes make them a good candidate in lithium batteries,24 energy storage,25 catalysts,26 fuel cells,27 and so on. Meanwhile, much more experimental effort has been paid to synthesize graphyne allotropic.28,29 It is known that graphyne films and nanoribbons have already been demonstrated by coupling reactions of precursor molecules on a copper surface,30,31 and they will provide credible evidence for us to further understand the carbon materials in the near future. As we all know, the carbon nanotubes (CNTs) have been widely researched in the last two decades. However, the Received: February 21, 2017 Revised: June 14, 2017 Published: June 22, 2017 14835

DOI: 10.1021/acs.jpcc.7b01687 J. Phys. Chem. C 2017, 121, 14835−14844

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The Journal of Physical Chemistry C graphyne tubes are just coming into our field of vision. Because single-wall nanotubes (SWCNTs) can be considered as a graphene sheet rolling up into a seamless cylinder,32 one can also get several types of graphyne nanotubes (GNTs) by the same approach. As early as 2003, Coluci and coworkers investigated the electronic properties of the graphyne-based nanotubes by using tight-binding and ab initio density functional theory (DFT) methods for the first time,33 which opened up new ground in the research of carbon materials. Since then, significant progress has been made through unremitting efforts. For instance, a successful synthesis of graphyne nanotubes with high-performance field emission properties has been reported by Li,34 and Wang claims that γ-GNTs may be a promising media for hydrogen storage.35 Very recently, several theoretical studies have been carried out to investigate the electronic structures of the α-, β-, and γGNTs.36−38 Their results show that the band structures of GNTs are diverse, owing to their complex and volatile geometric configurations. The armchair α-GNTs are concluded to be semiconductors with very small band gaps when tube size is small. The band gaps of the zigzag α-GNTs abide by the following rank order: 3m − 1 > 3m + 1 > 3m, where m is a positive integer used to define the tube index as N = 3m − 1, 3m, or 3m + 1. Nevertheless, all β-GNTs have quite small band gaps without any correlation with tube size for both armchair and zigzag configurations. Additionally, the band gaps and thermoelectric figures of merit, ZT, of γ-GNTs depend on their diameters.39 The distinctive feature of the Dirac cones in 6,6,12-graphyne that deviate from the perfectly conical Dirac cones may lead to a conductance with current direction dependence. This property just makes the 6,6,12-graphyne easier than graphene to be exploited in nanoscale circuits. However, only a few works talk about its derivative-6,6,12-GNTs, which motivate us to undertake the present study. In this work, to further investigate the structures of 6,6,12-GNTs, we discuss the structural stabilities by calculating the cohesive energies, strain energies, and Young’s moduli. Furthermore, the electronic properties are investigated through the analysis of band structures and density of states (DOS). We hope that our fundamental study on the 6,6,12-GNTs can make a clear understanding of this kind of materials, and the outstanding functionalities of 6,6,12-GNTs could be efficiently applied in science and technology in the future.

Figure 1. (a) Sketches for the 6,6,12-graphyne and its first Brillouin zone. The lattice constants are a = 6.898 Å and b = 9.426 Å according to our simulation. (b) Band structure and DOS maps for pristine 6,6,12-graphyne. (c) Sketches for (N, 0) and (0, N) 6,6,12-GNTs; here, for example, N = 9.

two Dirac points I and II appearing in the band map. The point I is located between the Γ and X′ points at Fermi level; meanwhile, the point II could be found between the M and X points, as introduced by Malko.18 The band structure and DOS map of the 6,6,12-graphyne from our simulation are shown in Figure 1b. Our simulation provides exactly the same electronic band structure and the same DOS arrangement, as shown in the pioneering study using PBE method and plane-wave basis set.18 Therefore, the basis set and other parameters employed in this study are reasonable and reliable. For insurance purposes, the 1D MP nets for 6,6,12-GNTs are doubled in the coming simulations as 64 × 1 × 1. Note that the full geometry optimizations for 6,6,12-GNTs are also performed by sampling the 64 × 1 × 1 nets. The truncation tolerances for the Coulomb overlap, Coulomb penetration, exchange overlap, and the first- and second-exchange pseudo-overlaps are set to 8, 8, 8, 8, and 16. Because of the anisotropic polygon geometry structure of the 6,6,12-graphyne, two different types of 6,6,12-GNTs, namely, (N, 0) and (0, N), could be obtained by rolling up 6,6,12graphyne sheet along the different sides of the rectangular unit cell into cylinders. Examples can be seen in Figure 1c. Note that the character N in uppercase is referred to as the tube index of a 6,6,12-GNT in this work to distinguish itself from the lowercase (n, m) commonly for SWCNT. The reason for this convention is that the unit cell of 6,6,12-graphyne is bigger than graphene’s. For all calculations, the 500 Å vacuum space is automatically imposed by the code CRYSTAL14 in each nonperiodic direction to avoid interactions between tubes in adjacent cells. The overlap populations between the nearest neighbors are obtained by using the standard Mulliken analysis. The chemical bonding between different carbon atoms is studied with the help of the periodic natural bond orbital (NBO) analysis.46 The second-order (self-consistent charge) density-functionalbased tight binding (scc-DFTB) method is also employed to test the thermal stability of the 6,6,12-GNTs. The molecular

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COMPUTATIONAL DETAILS Throughout this study, the Perdew−Burke−Ernzerhof (PBE) functional, which is dependably implemented in the computer code CRYSTAL14,40−42 is employed to describe the exchangecorrelation potential for our simulations. The used all-electron Gaussian basis set 6−21G* for carbon atoms is suggested by Catti.43 A sketch for 6,6,12-graphyne is illustrated in Figure 1a. The unit cell of the 6,6,12-graphyne contains 18 carbon (C) atoms. The first Brillouin zone is sampled with the aid of the Monkhorst−Pack (MP) Scheme. As a starting point, the 2D periodic 6,6,12-graphyne system is carefully tested with respect to three MP nets 16 × 16 × 1, 32 × 32 × 1, and 64 × 64 × 1. It turns out that a 32 × 32 × 1 MP net is already large enough to obtain acceptable geometric and electronic structures for the rectangle unit cell, even as the setting in the pioneering article.18 After a full relaxation, the calculated lattice constants are a = 6.898 Å and b = 9.426 Å, which are in good agreement with the literature reports44,45 with deviations 30 Å.59 The collapsed structure of the (0, 20) 6,6,12-GNT

Figure 9. Top and side views of the collapsed (0, 20) 6,6,12-GNT. The layer spacing in the middle segment is ∼3.34 Å. 14840


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The Journal of Physical Chemistry C

As a potential functional nanomaterial for electronic devices, the carrier transport properties of the 6,6,12-GNTs are important. Around the Dirac points, the electronic energy exhibits a linear relationship with the reciprocal vectors. Therefore, the Fermi velocity at Dirac point can be expressed as follows

form with smaller radius. The collapse mode is more energy efficient than the circular mode. However, the energy difference is very small, ∼9 meV/atom. This suggests that both circular and collapsed forms could be coexist in practice. Therefore, larger (0, N) 6,6,12-GNTs than (0, 20) are not considered in this work. The radius of a (N, 0) 6,6,12-GNT is smaller than that of the (0, N) one with the same index N. Thus the circular form of the (20, 0) 6,6,12-GNT might be more stable than the collapsed form. However, to pairwise compare the electronic properties of the 6,6,12-GNTs with the same tube index, we stop our research at N = 20. A separate article is deserved to discuss the collapse of the tubes rolling up from all kinds of the graphynes in detail. As mentioned above, the band structures of the 6,6,12-GNTs exhibit clear distinctions from monolayer 6,6,12-graphyne. The origin of this phenomenon can be explained from the point of the atomic orbital. When graphyne is rolled up from a 2D sheet into an 1D tube, the pz orbitals, which form a delocalized π bond, are no longer parallel to each other. They have to alter their shapes and geometric configurations to stabilize the tube and accordingly influence the band structures. From the simulation for the monolayer 6,6,12-graphyne, the bond length in hexagon (e.g., C3−C4 in Figure 1a) is ∼1.43 Å, and the Mulliken’s overlap population on the bond is ∼450 me. The bond between C4 and C5 is ∼1.40 Å long, and the overlap population is ∼440 me. It indicates that there exists not only a single σ-bond but also π-conjugation between the sp and sp2 carbons to participate in the formation of the delocalized πbond on the surface. The triple bond C1−C2 or C5−C6 is 1.23 Å long, which is significantly shorter than the others. Its overlap population is ∼770 me. Contrary to the expectation, the bond lengths and the overlap populations between the carbons in the tube forms have only very small changes according to our simulations. The NBO analysis confirms that the molecular orbital bond orders between different carbons are not significantly changed after rolling up to 1D form. The detailed bond-order information is listed in Table 1 for pristine 6,6,12-

vF = ±

a

bond

6,6,12-graphyne

(3, 0) 6,6,12-GNT

(0, 3) 6,6,12-GNT

2.5521 1.1635 1.2438 1.2102 2.4813 1.2098 1.2149

2.5985 1.1727 1.2521 1.1898 2.5178 1.2078 1.3292

2.5704 1.1817 1.2482 1.1844 2.4847 1.1745 1.2777

(6)

where E(k)⃗ is the energy of a charge carrier (i.e., electron or electron hole) at wavevector k⃗ in the band map and ℏ is the reduced Planck constant. To obtain the first derivative in the formal, 100 points located at ±0.01 Å−1 are employed in our fitting procedure. The Fermi velocities estimated from the 2D band maps of the (N, 0) 6,6,12-GNTs are plotted in Figure 10.

Figure 10. Fermi velocities around the Dirac points (DPs) as functions of tube radius. The positive and negative signs point out the different approach directions. We define that the Γ → DP direction is positive and Z → DP is negative. The hollow squares represent the Fermi velocities at the first Dirac points near the Z point in the band maps of the (N, 0) 6,6,12-GNTs. The solid lines on the data sets are fitting cures as vF = −3.66831 exp(−r/3.66831) + 5.38061 and vF = 6.6592 exp(−x/4.1447) − 6.633998. The solid squares are for the second Dirac points near the Γ point, if index N are even numbers. The fitting cures are set as vF = −3.1463 exp(−x/7.13385) + 2.76589 and vF = 7.85122exp(−x/4.33) − 2.09. As references, the solid circles and triangles are for the armchair and zigzag SWCNTs, respectively. The data can also be found in Table 2.

Table 1. Molecular Orbital Bond Orders between Different Carbon Atoms in Pristine 6,6,12-Graphyne and (3, 0), (0, 3) 6,6,12-GNTs from Our Periodic NBO Calculationsa C1−C2 C2−C3 C3−C4 C4−C5 C5−C6 C6−C7 C7−C8

1 ∂E(k ⃗) ℏ ∂k ⃗

The detailed values and some reference data can also be found in Table 2. As one can see in Figure 10, the Fermi velocity of a (N, 0) 6,6,12-GNT is direction-dependent. Because of the lattice distortion owing to the curvature, the energy near the second Dirac point also exhibits the linear relationship with respect to the vector displacements in the reciprocal space instead of parabolic dispersion as in monolayer 6,6,12graphyne. Moreover, the direction dependence of the Fermi velocities at the second Dirac point is contrary to that at the first one. According to our fitting results, the Fermi velocities at the first Dirac point (DP) will approach 5.38 × 105 and 6.63 × 105 m/s in directions Γ → DP and Z → DP, respectively. The Fermi velocity at the first Dirac point in direction Z → DP is even higher than the one in monolayer 6,6,12-graphyne. The ones at the second Dirac point are much lower, which tend to 2.77 × 105 and 2.09 × 105 m/s in directions Γ → DP and Z → DP, respectively.

Labels for the carbons are shown in Figure 1a.

graphyne and the tubes with N = 3 with the largest curvatures. The delicate deformation can profoundly affect the electronic structure owing to the Peierls dimerization.58,60,61 More importantly, the different bond lengths and different bond types between the differently hybridized carbon atoms lead to diverse curvatures (i.e., the bending degrees). After the full relaxations for the 6,6,12-GNTs, the radius of a 6,6,12-GNT is not a constant precisely due to the different curvatures. This situation is different from the SWCNTs. The structure distortion and the broken symmetry bring the new phenomena into the energy band maps of the 6,6,12-GNTs. 14841


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The Journal of Physical Chemistry C Table 2. Fermi Velocities of the (N, 0) 6,6,12-GNTs Determined by Using Equation 6a vI (105 m/s) 6,6,12-GNTs

(5,0) (6,0) (7,0) (8,0) (9,0) (10,0) (11,0) (12,0) (13,0) (14,0) (15,0) (16,0) (17,0) (18,0) (19,0) (20,0)

Γ → DP

Z → DP

3.27 4.23 4.63 4.88 5.00 5.11 5.18 5.22 5.26 5.31 5.32 5.33 5.34 5.36 5.38 5.39

3.51 4.69 5.23 5.54 5.78 5.93 6.01 6.04 6.09 6.13 6.17 6.19 6.21 6.23 6.26 6.27

vII (105 m/s) Γ → DP

vref. (105 m/s)

Z→ DP SWCNTs

1.82

0.91

2.08

1.67

2.27

1.71

2.45

1.87

graphene

2.49

1.91

α β

(12,0) (15,0) (30,0) (45,0) (10,10) (15,15) (20,20) (25,25) (30,30)

7.96 7.95 8.10 8.02 8.21 8.37 8.40 8.43 8.46 8.60 (8.3b)

graphynes

2.61

2.00

2.62

2.02

6,6,12

Γ → DP M→ DP Γ → DP X′ → DP

7.07 5.36 4.45 5.42 6.38

(7.1b, 6.67c) (5.07c, 5.24d) (3.80c, 4.45d) (4.9e) (5.8e)

Reference data for graphene and α-, β-, and 6,6,12-graphynes are selected from refs 23 and 62−64 and listed in brackets. bRef 62. cRef 23. dRef 63. Ref 64.

a e

However, the effective masses of (3, 0) and (4, 0) tubes at Γ point are not identical for electrons and holes. In (3, 0) 6,6,12GNT, the effective mass for electrons is 0.74me at Γ point, which is ∼0.30me heavier than that for holes. The values in (4, 0) 6,6,12-GNT are 0.24me and 0.17me for electrons and holes at Γ point.

It could be very important that the locations of the double Dirac points in 6,6,12-GNTs with respect to the Fermi level retain the same natures as them in 6,6,12-graphyne. The first Dirac point lies slightly below the Fermi level, while the other one is above the Fermi level. The same as 6,6,12-graphyne, 6,6,12-GNTs are also self-doped. At the first Dirac point, electrons are present as charge carriers. On the contrary, holes are present as charge carriers at the second Dirac point. Together with aforementioned different direction dependence of the Fermi velocities at two Dirac points in single-walled 6,6,12-GNT, the electrons and electron holes could be spontaneously separated. Therefore, this material in some aspects is more versatile due to its nonequivalent Dirac points. The effective masses of the charge carriers are closely related to the carrier transport of an electronic device. The inverse of the effective mass 1/m* of a charge carrier in semiconducting 6,6,12-GNTs can be characterized via the second-order derivative of energy as follows 1 1 ⎛ ∂ 2E(k ⃗) ⎞ ⎟⎟ = 2 ⎜⎜ m* ℏ ⎝ ∂k ⃗ 2 ⎠

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CONCLUSIONS Using the DFT simulation technique we have predicted two novel carbon nanotubes, namely, (N, 0) and (0, N) 6,6,12GNTs rolled up from the monolayer 6,6,12-graphyne, with some splendid properties. To verify the possibility of their existence, the cohesive energies as well as the strain energies of the tubes have been studied. It turns out that the binding energy between the dangling bonds overcomes the strain energy from the curvature, which leads to the negative cohesive energies of the 6,6,12-GNTs. The (0, N) 6,6,12-GNTs have a constant Young’s modulus by ∼0.32 TPa, while the Young’s moduli of the (N, 0) tubes fall on an exponential curve and can be extrapolated to 0.42 TPa. The calculated energy band maps show that the (0, N) 6,6,12-GNTs are semiconductors with narrow band gaps. On the contrary, the (N, 0) tubes are all metallic because they present an extra Dirac point in the Fermi level. The Dirac points appear in their band maps and obey an even−odd relation. The Fermi velocities and effective masses of the charge carriers have been estimated and discussed in the text. The nonequivalent Dirac points and the difference of the direction dependence of the Fermi velocities at these Dirac points could lead the 6,6,12-GNTs to be versatile.

(7)

Because the Dirac point appears in the (N, 0) 6,6,12-GNTs with N > 4, the effective masses of charge carriers are estimated only in (0, N) as well as (3, 0) and (4, 0) systems. The (0, N) 6,6,12-GNTs possess direct band gaps at point Z, as shown in Figure 6. The closest hundred points around the top of the valence band (TOV) and the bottom of conduction band (BOC) within ±0.01 Å−1 are adjusted to a quadratic function to numerically calculate the second-order derivative in eq 7. The effective masses at Z point are isotropic and remain almost unchanged if the tube index N is bigger than seven. The values are 0.12me and 0.11me, respectively, for electrons (in valence band) and electron holes (in conduction band), where me denotes the mass of a stationary electron. There are second gaps located at Γ, which are wider than the gaps at Z. However, the effective masses at Γ point are 0.05me for both electrons and holes, which are much smaller than that at Z point. Similarly, the effective masses of (3, 0) and (4, 0) tubes are 0.28me and 0.25me for both electrons and holes at Z point, respectively.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.7b01687. Details of the band structures and DOS maps for the (N, 0) and (0, N) 6,6,12-GNTs with N in the range of 3 to 20. (PDF) DFTB-MD simulations for (3, 0) 6,6,12-GNTs at 900 K to reveal the thermal stability of the tubes. (AVI) 14842


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DFTB-MD simulations for (0, 3) 6,6,12-GNTs at 900 K to reveal the thermal stability of the tubes. (AVI)

AUTHOR INFORMATION

Corresponding Authors

*R.J.: E-mail: [email protected]. Phone: +86 (0)431 88498962. *H.-X.Z.: E-mail: [email protected]. ORCID

Yuchen Ma: 0000-0002-9990-5023 Hong-Xing Zhang: 0000-0001-5334-733X Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work is supported by National Natural Science Foundation of China (Grant No. 21173096).

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Novel Carbon Nanotubes Rolled from 6,6,12-Graphyne: Double Dirac

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