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C-57 carbon: A two-dimensional metallic carbon allotrope with pentagonal and heptagonal rings
Computational Materials Science 160 (2019) 115–119
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C-57 carbon: A two-dimensional metallic carbon allotrope with pentagonal and heptagonal rings
T
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Chun-Xiang Zhaoa, Yi-Qi Yanga, Chun-Yao Niua, , Jia-Qi Wanga, Yu Jiaa,b a
International Joint Research Laboratory for Quantum Functional Materials of Henan, and School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China b Key Laboratory for Special Functional Materials of Ministry of Education, School of Physics and Electronics, Henan University, Kaifeng 475004, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Two-dimensional Carbon allotrope Metallicity Carbon nanotube
By means of the first-principles calculations, we have theoretically investigated the structural stability and electronic properties of a two-dimensional planar metallic carbon allotrope named C-57 carbon which possesses the P 6¯2m (D33h ) symmetry. This carbon allotrope is an all-sp2 hybridized bonding network consisting of 5–7 rings of carbon atoms. The stability of C-57 carbon is confirmed through phonon-mode analysis, total energy and elastic constants calculations, as well as first-principles molecular dynamics simulations. We conceived that the metallicity of C-57 carbon is attributed to the large states across Fermi-level contributed by py orbital due to the bond distortion, which is much different from that of graphite. This new carbon sheet can also serve as a precursor for stable one-dimensional nanotubes with metallic character. These results broaden our understanding of two-dimensional carbon allotropes and will attract more researchers to focus the research on the field of two-dimensional carbon materials. Besides, the C-57 carbon may be useful for designing of nano-electronic devices.
1. Introduction In the periodic table of the elements, the sixth element carbon is known to form a number of allotropes due to its ability to exist in different hybridizations. A series of carbon materials such as zero-dimensional (0D) fullerenes [1], one-dimensional (1D) carbon nanotubes [2], two-dimensional (2D) graphene [3] and three dimensional (3D) superhard and metallic carbon structures have been experimentally fabricated or theoretically predicted [4–10], all of them exhibiting excellent properties. Among these carbon allotropes, the prototype twodimensional (2D) inorganic honeycomb crystal, graphene, a monolayer of sp2 bonding networks, is the most extensively studied carbon material. Graphene possesses many fascinating physical properties [3,11,12], for example, the anomalous quantum Hall effect, ambipolar effect, ballistic electronic conductivity and 2D electron-gas behavior, so it is considered as a revolutionary material for future generation of high-speed nanoelectronics [13], transparent electrodes [14], and so on. Inspired by the success of graphene, an intense research enthusiasm has now been focused on the search for newer promising 2D carbon allotropes. Wang et al. have theoretically predicted a new 2D metallic carbon named net W with Cmmm symmetry [15], which contains
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squares C4 , hexagons C6 , and octagons C8 similar to while energetically more favorable than net C carbon [16]. In 2015, Zhang et al. predicted a 2D carbon sheet (penta-graphene) composed entirely of pentagons, which can withstand temperatures up to 1000 K [17]. The penta-graphene possesses novel properties, for example, it exhibits semiconductor characters, negative Poisson’s ratio, and ultrahigh ideal strength. In the study of Wang et al., they proposed a new low-energy 2D carbon networks named phagraphene, composed of 5-6-7 carbon rings [18]. The direction-dependent Dirac cones of phagraphene are proved to be robust against the external strain with tunable Fermi velocities. Recently, a new 2D planar carbon sheet, twin graphene, has also been predicted by Jiang et al. [19]. The excellent mechanical properties of twin graphene make it suitable to be used in nanoelectromechanical systems. Besides, many other 2D carbon allotropes such as graphyne [20–23], graphenylene [24], biphenylene [25], radialenes [26], haeckelites [27], etc. [28–32], have also been theoretically predicted or experimentally synthesized. These research results continue to stimulate more studies in exploring novel 2D carbon-based nanomaterials owning to their outstanding chemical and physical properties. In this work, based on first-principles calculations, we predict a sTable 2D planar metallic carbon allotrope in P 6¯2m (D33h ) symmetry. The carbon allotrope composed of penta and hepta carbon rings,
Corresponding author. E-mail address: [email protected] (C.-Y. Niu).
https://doi.org/10.1016/j.commatsci.2018.12.035 Received 13 September 2018; Received in revised form 6 December 2018; Accepted 17 December 2018 0927-0256/ © 2018 Published by Elsevier B.V.
Computational Materials Science 160 (2019) 115–119
C.-X. Zhao et al.
whence we named it as C-57 carbon. It should point out that the structure of C-57 carbon can be obtained by modifying graphite sheet [33,34]. The dynamical, mechanical and thermal stability of C-57 carbon have been thoroughly checked by phonon mode, elastic constants and molecular dynamic simulations, respectively. The results of the calculated electronic band structures show that C-57 carbon is a metallic carbon allotrope. Meanwhile, its electronic density of states at Fermi level is larger than that of all plausible metallic carbon nanotubes, and is also larger than that of recently reported metallic planar Tgraphene and net W carbon sheet. The C-57 carbon sheet can also be rolled into stable metallic nanotubes. Assuredly, our results will attract more attention in the research field of 2D carbon allotropes, and the C57 carbon has great potential to be applied in nanoelectronics. 2. Computational method The calculations are carried out using the density functional theory which within the all-electron projector augmented wave (PAW) [35] method as implemented in the Vienna ab initio simulation package (VASP) [36–38]. The exchange-correlation potential is adopted with the generalized gradient approximation (GGA) developed by Perdew, Burke, and Ernzerhof (PBE) [39]. A plane-wave basis set with an energy cutoff of 800 eV was used and gave well converged total energies (∼1 meV/atom). A vacuum slab of 20 Å is built in the perpendicular direction to avoid the interactions between neighboring layers. The Brillouin zone (BZ) is sampled with a 9 × 9 × 1 Monkhorst-Pack (MP) special k-point grid including Γ -point. The geometries are optimized with no symmetry constraints until the total energy and force components on each atom were less than 10−6 eV and 10−3 eV/Å, respectively. Phonon dispersion curves were calculated by the PHONON code [40,41] with the forces calculated from VASP. The first-principles molecular dynamics simulations are performed in the canonical (NVT) ensemble with the Nosé thermostat. All the calculations in this work are performed at zero pressure.
Fig. 1. Schematic depiction for the structure of C-57 carbon. (a) The top view of the atomic configuration of C-57 carbon sheet constructed from the primitive unit cell (the yellow area) of 12 atoms, with the lattice vectors a1 and a2 . Two vertical directions in the plane of C-57 carbon sheet are represented by x and y , respectively. (b) The primitive unit cell of C-57 carbon, the carbon atoms occupy the 1b(0.0000, 0.0000, 0.5000), 3g(0.0000, 0.7789, 0.5000), 6k (0.1439, 0.4070, 0.5000), and 2d(0.3333, 0.6667, 0.5000) Wyckoff positions, which are denoted by C1 (red), C2 (yellow), C3 (blue), and C4 (green), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Results and discussion The optimized crystal structure of C-57 carbon is shown in Fig. 1. This structure is flat and one-atom thick, composed of pentagonal and heptagonal rings and possesses P 6¯2m (D33h ) symmetry. What should be point out is that the arrangement of carbon rings of C-57 carbon is distinct from that of the previously proposed planar carbon pentaheptite [42] which is also composed of pentagonal and heptagonal carbon rings. In pentaheptite, each pentagon is adjacent to one pentagon and four heptagons, each heptagon is adjacent to four pentagons and three heptagons. While in the C-57 carbon, each pentagon is adjacent to two pentagons and three heptagons, each heptagon is adjacent to three pentagons and four heptagons. The equilibrium primitive cell length a of C-57 carbon is estimated to be 6.155 Å at zero pressure. C-57 carbon has four nonequivalent carbon atoms which are labeled as C1-C4 (see Fig. 1) and their Wyckoff positions are 1b(0.0000, 0.0000, 0.5000), 3 g(0.0000, 0.7789, 0.5000), 6 k(0.1439, 0.4070, 0.5000), and 2d (0.3333, 0.6667, 0.5000), respectively. The bond length between C1-C2 , C2 -C3 , C3 -C4 and C3 -C3 are 1.361 Å, 1.442 Å, 1.432 Å and 1.534 Å, respectively. The average bond length of C-57 carbon is very close to that of graphene (1.420 Å). The C-57 carbon has six distinct bond angles, they are 103.42°, 106.58° , and 120° in pentagon, 118.56°, 120°, 134.86° , and 153.17° in heptagon, respectively. While for pentaheptite, the bond angles in pentagon and heptagon are 105.32°, 106.97°, 110.38° and 122.57°, 127.35°, 130.47° , 139.24°, respectively. Compared with pentaheptite, the bond angles of C-57 deviate much larger from 120° for perfectly sp2 hybridization, implying a higher strain exists in C-57 carbon, which would lead to an enhancement in total energy. The total energy per atom as a function of the 2D crystal area per atom of C-57 carbon in comparison with α -graphyne, graphene, penta-
Fig. 2. The total energy per atom as a function of the 2D crystal area per atom for C-57 carbon in comparison with some other 2D carbon allotropes.
graphene, planar T-graphene and pentaheptite have been calculated and the results are shown in Fig. 2. What we can see from Fig. 2 is that graphene is energetically more stable than the other five 2D carbon allotropes and C-57 carbon is energetically preferable than α -graphyne, penta-graphene, and planar T-graphene. The equilibrium total energy of pentaheptite is lower than that of C-57 carbon due to the greater distortion of bond angles in C-57 carbon compared with pentaheptite, as mentioned above. The calculated lattice parameters, equilibrium atom density, and total energy per atom for graphene, α -graphyne,
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Table 1 The calculated lattice parameters a and b (in Å), equilibrium atom density ( ρ in g/cm3), and total energy (Et in eV) for C-57 carbon in comparison with graphene, α -graphyne, penta-graphene, planar T-graphene, and pentaheptite as well as the available experimental and other theoretical values. Structure
Space group
References
a (Å)
b (Å)
ρ (atom/ Å2)
Et (eV/atom)
Graphene
P 63/ mmc
α -graphyne
P 6/ mmm
pentagraphene
P − 421m
Exp. [43] This work GGA [44] This work GGA [17]
2.460 2.467 6.997 6.962 3.640
2.460 2.467 6.997 6.962 3.640
0.330 0.329 0.163 0.165 0.453
– −9.227 – −8.305 −8.320
planar Tgraphene
P 4/ mmm
This work GGA [45]
3.641 3.447
3.641 3.447
0.453 0.337
−8.324 −8.712
pentaheptite
Cmmm
C-57 carbon
P 6¯2m
This work –[42] This work This work
3.446 7.525 7.473 6.155
3.446 5.899 5.858 6.155
0.337 0.361 0.365 0.317
−8.714 – −8.984 −8.768
Fig. 4. Strain energy of C-57 carbon under different kinds of in-plane strain in the harmonic region.
for mechanical stability, further confirming the mechanical stability of C-57 carbon. The Young’s modulus of C-57 carbon can be derived from 2 2 )/C11, and the result is 243 N/m, its elastic constants using E = (C11 -C12 which is more than two-thirds of that of graphene (345 N/m) [49], but smaller than that of pentaheptite (298 N/m) and penta-graphene (264 N/m) [17]. To examine the thermal stability of C-57 carbon, we have also performed the first-principles molecular dynamics simulations with the canonical (NVT) ensemble. A 4 × 4 supercell containing 192 carbon atoms was built to reduce the constraint of periodic boundary condition and explore possible structure reconstruction. The fluctuations of the temperature and total energy as a function of simulation time is depicted in Fig. 5. After heating at room temperature (300 K) for 6 ps with a time step of 1 fs, no structure reconstruction is found, except for slight fluctuations of the total energy (Fig. 5(a)), which strongly suggests that
penta-graphene, planar T-graphene, pentaheptite and C-57 carbon are summarized in Table 1 for comparison. Our calculated results are well agree with the previous experimental and theoretical ones, indicating that our calculation is valid. To further check the dynamical stability, we perform a thorough analysis of its phonon modes. The calculated phonon dispersions of C57 carbon are shown in Fig. 3. It can be seen that there is no imaginary mode, demonstrating that this structure is dynamically stable in the ground state. What’s more, it is instructive to note that the highest phonon frequency of C-57 carbon is estimated to be ∼1663 cm−1, locating at M(0.5, 0.0, 0.0) of Brillioun zone, slightly larger than ∼1600 cm−1 for graphene [46], and there exist a narrow indirect band gap (∼170 cm−1) in the phonon band structures. These features could be useful to identify C-57 carbon in experiments. The elastic constants of theoretical materials are not only important for their potential applications, but also can be used to test their mechanical stabilities. The strain energy of C-57 carbon under different kinds of in-plane strain has been calculated, as shown in Fig. 4. For 2D 1 2 + 2 C22 sheets, the strain energy can be expressed as [47]: Uε = 1 C11εxx 2
2 2 ε yy + C12 εxx εyy + 2C66εxy , where Uε is the strain energy, x and y represent two vertical directions in the plane of C-57 carbon sheet as shown in Fig. 1 (a), εij are small strains in the harmonic region, C11, C12 , C22 , and C66 are components of the elastic modulus tensor. By fitting the strain energy to the parabolic function of the axial and biaxial strains, we get C11 = C22 = 275 N/m, C12 = 94 N/m, and C66 = 90 N/m. These elastic constants satisfy the Born-Huang criteria [48] of C11 > |C12 | and C66 > 0
Fig. 5. The fluctuations of total potential energy of the C-57 carbon supercell as a function of the molecular dynamic simulation step at (a) 300 K and (b) 800 K, respectively. The inset shows the snapshot of the simulated system.
Fig. 3. Phonon dispersions of C-57 carbon. The predicted highest vibrational frequency for C-57 carbon is ∼1663 cm−1. 117
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Fig. 7. (a) Illustration of chiral vectors of C-57-tubes. The lattice basis vectors are denoted by a1 and a2 . Optimized structure of (b) (6, 0) and (c) (6, 6) C-57tubes from side view.
Fig. 6. (a) The electronic band structures and the projected density of states (PDOS) for C1, C2 , C3 , and C4 atom obtained by HSE06 potential. The Fermi level is set at 0 eV with dash line. (b) Band decomposed charge density distribution (the partially occupied band).
This new carbon sheet can also be used as a precursor for materials of different dimensions, for instance, quasi-1D nanotubes. We have constructed a series of carbon nanotubes by rolling the C-57 carbon sheet. The chiral vector Ch and its corresponding translation vector Th are shown in Fig. 7(a). The chirality of a nanotube is defined as: Ch = ma1 + na2 , where, m and n are the chiral indices. Here, we restrict our study to (m,0) and (m,m) nanotubes with diameters from 3.918 Å to 20.361 Å. After fully optimized, the C-57 carbon based nanotubes (C57-tubes) are stable, as shown in Fig. 7(b) (6, 0) and (c) (6, 6) tubes. The diameters of (6, 0) and (6, 6) C-57-tubes are 11.755 Å and 20.361 Å, respectively. The dynamic stability of these nanotubes is confirmed by carrying out phonon calculations and the calculated results are shown in Fig. 8(a) and (b). The calculated band structures (see Fig. 8(c) and (d)) revealed that both of (6, 0) and (6, 6) tubes are metallic. The metallic behavior of C-57-tubes can be attributed to the strong metallicity of C-57 carbon. The chirality-independent metallic C-57 based nanotubes as well as the C-57 sheet may have potential application in nanoelectronics.
the C-57 carbon can exist at room temperature. Furthermore, we find that the c-57 sheet can withstand temperatures as high as 800 K (Fig. 5(b)), implying that this carbon allotrope is separated by highenergy barriers from other local minima on the potential energy surface of elemental carbon. Finally, let us discuss the electronic properties of C-57 carbon. We show in Fig. 6 the electronic band structures for C-57 carbon and the projected density of states (PDOS) of C1, C2 , C3 , and C4 atoms, which are calculated by the hybrid density functional method based on the HeydScuseria-Ernzerhof scheme (HSE06) [50,51]. As we can see from Fig. 6(a), C-57 carbon is metallic and some flat bands cross the Fermi level along the M-K and K-Γ direction. There also exist some steep bands near the Fermi level along M-K and K-Γ paths. It is believed that the coexistence of steep and flat bands at the Fermi level is the necessary condition for the superconductivity of materials [52,53]. The C-57 carbon possesses a very high density of states of 0.23 states/eV per atom at Fermi level owing to the flat bands, which is not only larger than that of planar T-graphene (0.11 states/eV per atom) [45] and net W carbon sheet (0.12 states/eV per atom) [15] but also almost three times that of the 5.3 Å diameter (4, 4) carbon nanotube [54]. In order to explore the origin of the metallic feature displayed in C57 carbon, we have also calculated the charge density of the partially occupied bands in the energy windows of Ef − 0.5 to Ef + 0.5 eV. We can see from Fig. 6(b) that the unequivalent carbon atoms make different contribution to the charge density: the electron accumulation around C4 atom is relatively low, while the electron accumulation around C1, C2 , and C3 atoms is relatively high, which is in accordance with the discrepant contributions to states at Fermi level in PDOS. The calculated PDOS and band decomposed charge density reveal that the conductive electrons of C-57 carbon are predominately contributed by the py orbital of C1, C2 , and C3 atoms, which is quite different from graphite, where the metallicity originates from the delocalization of electrons in pz orbital. The bond angle around C1, C2 , and C3 atoms deviate a lot from the perfect sp2 bond, which raise the energy of py orbital, resulting in the metallic property of C-57 carbon; while the C4 atom form almost perfect sp2 bond with the C3 atom, contributing low electron density around Fermi level.
4. Conclusion In summary, we have systemically studied the structural stability and electronic properties of a 2D metallic carbon allotrope in P 6¯2m (D33h ) symmetry, termed C-57 carbon here, which is composed of penta and hepta carbon rings similar to the previously proposed pentaheptite while their arrangement of carbon rings is different. The energetic calculations show that C-57 carbon is metastable compared to graphene, but more stable than α -graphyne and penta-graphene. The dynamic, mechanical and thermal stability of C-57 carbon have been verified by the analyses of the phonon mode, elastic constants and the first-principles molecular dynamics simulations, respectively. The electronic structure calculations reveal that C-57 carbon is metallic with much higher DOS of ∼0.23 eV/states per atom at the Fermi level, and there are several flat bands across the Fermi level. The metallic nanotubes based on C-57 carbon can also be produced for application in nanoeletronics. These findings further expand our understanding of the structural and electronic properties of 2D carbon allotropes, and will stimulate more efforts focus on the research field of 2D carbon 118
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Fig. 8. Calculated phonon spectrums of (a) (6, 0) and (b) (6, 6) C-57 carbon based nanotubes. Band structures of (c) (6, 0) and (d) (6,6) C-57-tubes.
materials. The C-57 carbon and the chirality-independent metallic C-57 based nanotubes have potential to be used in nanoelectronics.
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CRediT authorship contribution statement Chun-Xiang Zhao: Methodology, Formal analysis, Writing original draft. Yi-Qi Yang: Formal analysis, Data curation. ChunYao Niu: Conceptualization, Writing - review & editing, Supervision. Jia-Qi Wang: Data curation, Validation. Yu Jia: Conceptualization, Methodology. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 11504332) and Outstanding Young Talent Research Fund of Zhengzhou University (1521317006). References [1] H.W. Kroto, J.R. Heath, S.C. Obrien, R.F. Curl, R.E. Smaliey, Nature 318 (1985) 162–163. [2] S. Iijima, Nature 354 (1991) 56–58. [3] K. Novoselov, A. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I. V Grigorieva, A.A. Firsov, Science 306 (2004) 666–669. [4] Q. Li, Y.M. Ma, A.R. Oganov, H.B. Wang, H. Wang, Y. Xu, T. Cui, H.K. Mao, G.T. Zou, Phys. Rev. Lett. 102 (2009) 175506. [5] J.T. Wang, C.F. Chen, Y. Kawazoe, Phys. Rev. Lett. 106 (2011) 075501. [6] J.Q. Wang, C.X. Zhao, C.Y. Niu, Q. Sun, Y. Jia, J. Phys.: Condens. Matter 28 (2016) 475402. [7] S.H. Zhang, Q. Wang, X.S. Chen, P. Jena, Proc. Natl. Acad. Sci. USA 110 (2013) 18809–18813. [8] C.Y. Niu, X.Q. Wang, J.T. Wang, J. Chem. Phys. 140 (2014) 054514. [9] Y. Cheng, R. Melnik, Y. Kawazoe, B. Wen, Cryst. Growth Des. 16 (2016) 1360–1365. [10] C.X. Zhao, C.Y. Niu, Z.J. Qin, X.Y. Ren, J.T. Wang, J.H. Cho, Y. Jia, Sci. Rep. 6 (2016) 21879. [11] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [12] A.K. Geim, Science 324 (2009) 1530. [13] F. Schwierz, Nat. Nanotechnol. 5 (2010) 487. [14] X. Wang, L.J. Zhi, K. Müllen, Nano Lett. 8 (2008) 323. [15] X.Q. Wang, H.D. Lib, J.T. Wang, Phys. Chem. Chem. Phys. 15 (2013) 2024. [16] N. Tyutyulkov, F. Dietz, K. Müllen, M. Baumgarten, Chem. Phys. Lett. 272 (1997) 111–114.
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C-57 carbon: A two-dimensional metallic carbon allotrope with pentagonal and heptagonal rings
T
⁎
Chun-Xiang Zhaoa, Yi-Qi Yanga, Chun-Yao Niua, , Jia-Qi Wanga, Yu Jiaa,b a
International Joint Research Laboratory for Quantum Functional Materials of Henan, and School of Physics and Engineering, Zhengzhou University, Zhengzhou 450001, China b Key Laboratory for Special Functional Materials of Ministry of Education, School of Physics and Electronics, Henan University, Kaifeng 475004, China
A R T I C LE I N FO
A B S T R A C T
Keywords: Two-dimensional Carbon allotrope Metallicity Carbon nanotube
By means of the first-principles calculations, we have theoretically investigated the structural stability and electronic properties of a two-dimensional planar metallic carbon allotrope named C-57 carbon which possesses the P 6¯2m (D33h ) symmetry. This carbon allotrope is an all-sp2 hybridized bonding network consisting of 5–7 rings of carbon atoms. The stability of C-57 carbon is confirmed through phonon-mode analysis, total energy and elastic constants calculations, as well as first-principles molecular dynamics simulations. We conceived that the metallicity of C-57 carbon is attributed to the large states across Fermi-level contributed by py orbital due to the bond distortion, which is much different from that of graphite. This new carbon sheet can also serve as a precursor for stable one-dimensional nanotubes with metallic character. These results broaden our understanding of two-dimensional carbon allotropes and will attract more researchers to focus the research on the field of two-dimensional carbon materials. Besides, the C-57 carbon may be useful for designing of nano-electronic devices.
1. Introduction In the periodic table of the elements, the sixth element carbon is known to form a number of allotropes due to its ability to exist in different hybridizations. A series of carbon materials such as zero-dimensional (0D) fullerenes [1], one-dimensional (1D) carbon nanotubes [2], two-dimensional (2D) graphene [3] and three dimensional (3D) superhard and metallic carbon structures have been experimentally fabricated or theoretically predicted [4–10], all of them exhibiting excellent properties. Among these carbon allotropes, the prototype twodimensional (2D) inorganic honeycomb crystal, graphene, a monolayer of sp2 bonding networks, is the most extensively studied carbon material. Graphene possesses many fascinating physical properties [3,11,12], for example, the anomalous quantum Hall effect, ambipolar effect, ballistic electronic conductivity and 2D electron-gas behavior, so it is considered as a revolutionary material for future generation of high-speed nanoelectronics [13], transparent electrodes [14], and so on. Inspired by the success of graphene, an intense research enthusiasm has now been focused on the search for newer promising 2D carbon allotropes. Wang et al. have theoretically predicted a new 2D metallic carbon named net W with Cmmm symmetry [15], which contains
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squares C4 , hexagons C6 , and octagons C8 similar to while energetically more favorable than net C carbon [16]. In 2015, Zhang et al. predicted a 2D carbon sheet (penta-graphene) composed entirely of pentagons, which can withstand temperatures up to 1000 K [17]. The penta-graphene possesses novel properties, for example, it exhibits semiconductor characters, negative Poisson’s ratio, and ultrahigh ideal strength. In the study of Wang et al., they proposed a new low-energy 2D carbon networks named phagraphene, composed of 5-6-7 carbon rings [18]. The direction-dependent Dirac cones of phagraphene are proved to be robust against the external strain with tunable Fermi velocities. Recently, a new 2D planar carbon sheet, twin graphene, has also been predicted by Jiang et al. [19]. The excellent mechanical properties of twin graphene make it suitable to be used in nanoelectromechanical systems. Besides, many other 2D carbon allotropes such as graphyne [20–23], graphenylene [24], biphenylene [25], radialenes [26], haeckelites [27], etc. [28–32], have also been theoretically predicted or experimentally synthesized. These research results continue to stimulate more studies in exploring novel 2D carbon-based nanomaterials owning to their outstanding chemical and physical properties. In this work, based on first-principles calculations, we predict a sTable 2D planar metallic carbon allotrope in P 6¯2m (D33h ) symmetry. The carbon allotrope composed of penta and hepta carbon rings,
Corresponding author. E-mail address: [email protected] (C.-Y. Niu).
https://doi.org/10.1016/j.commatsci.2018.12.035 Received 13 September 2018; Received in revised form 6 December 2018; Accepted 17 December 2018 0927-0256/ © 2018 Published by Elsevier B.V.
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whence we named it as C-57 carbon. It should point out that the structure of C-57 carbon can be obtained by modifying graphite sheet [33,34]. The dynamical, mechanical and thermal stability of C-57 carbon have been thoroughly checked by phonon mode, elastic constants and molecular dynamic simulations, respectively. The results of the calculated electronic band structures show that C-57 carbon is a metallic carbon allotrope. Meanwhile, its electronic density of states at Fermi level is larger than that of all plausible metallic carbon nanotubes, and is also larger than that of recently reported metallic planar Tgraphene and net W carbon sheet. The C-57 carbon sheet can also be rolled into stable metallic nanotubes. Assuredly, our results will attract more attention in the research field of 2D carbon allotropes, and the C57 carbon has great potential to be applied in nanoelectronics. 2. Computational method The calculations are carried out using the density functional theory which within the all-electron projector augmented wave (PAW) [35] method as implemented in the Vienna ab initio simulation package (VASP) [36–38]. The exchange-correlation potential is adopted with the generalized gradient approximation (GGA) developed by Perdew, Burke, and Ernzerhof (PBE) [39]. A plane-wave basis set with an energy cutoff of 800 eV was used and gave well converged total energies (∼1 meV/atom). A vacuum slab of 20 Å is built in the perpendicular direction to avoid the interactions between neighboring layers. The Brillouin zone (BZ) is sampled with a 9 × 9 × 1 Monkhorst-Pack (MP) special k-point grid including Γ -point. The geometries are optimized with no symmetry constraints until the total energy and force components on each atom were less than 10−6 eV and 10−3 eV/Å, respectively. Phonon dispersion curves were calculated by the PHONON code [40,41] with the forces calculated from VASP. The first-principles molecular dynamics simulations are performed in the canonical (NVT) ensemble with the Nosé thermostat. All the calculations in this work are performed at zero pressure.
Fig. 1. Schematic depiction for the structure of C-57 carbon. (a) The top view of the atomic configuration of C-57 carbon sheet constructed from the primitive unit cell (the yellow area) of 12 atoms, with the lattice vectors a1 and a2 . Two vertical directions in the plane of C-57 carbon sheet are represented by x and y , respectively. (b) The primitive unit cell of C-57 carbon, the carbon atoms occupy the 1b(0.0000, 0.0000, 0.5000), 3g(0.0000, 0.7789, 0.5000), 6k (0.1439, 0.4070, 0.5000), and 2d(0.3333, 0.6667, 0.5000) Wyckoff positions, which are denoted by C1 (red), C2 (yellow), C3 (blue), and C4 (green), respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
3. Results and discussion The optimized crystal structure of C-57 carbon is shown in Fig. 1. This structure is flat and one-atom thick, composed of pentagonal and heptagonal rings and possesses P 6¯2m (D33h ) symmetry. What should be point out is that the arrangement of carbon rings of C-57 carbon is distinct from that of the previously proposed planar carbon pentaheptite [42] which is also composed of pentagonal and heptagonal carbon rings. In pentaheptite, each pentagon is adjacent to one pentagon and four heptagons, each heptagon is adjacent to four pentagons and three heptagons. While in the C-57 carbon, each pentagon is adjacent to two pentagons and three heptagons, each heptagon is adjacent to three pentagons and four heptagons. The equilibrium primitive cell length a of C-57 carbon is estimated to be 6.155 Å at zero pressure. C-57 carbon has four nonequivalent carbon atoms which are labeled as C1-C4 (see Fig. 1) and their Wyckoff positions are 1b(0.0000, 0.0000, 0.5000), 3 g(0.0000, 0.7789, 0.5000), 6 k(0.1439, 0.4070, 0.5000), and 2d (0.3333, 0.6667, 0.5000), respectively. The bond length between C1-C2 , C2 -C3 , C3 -C4 and C3 -C3 are 1.361 Å, 1.442 Å, 1.432 Å and 1.534 Å, respectively. The average bond length of C-57 carbon is very close to that of graphene (1.420 Å). The C-57 carbon has six distinct bond angles, they are 103.42°, 106.58° , and 120° in pentagon, 118.56°, 120°, 134.86° , and 153.17° in heptagon, respectively. While for pentaheptite, the bond angles in pentagon and heptagon are 105.32°, 106.97°, 110.38° and 122.57°, 127.35°, 130.47° , 139.24°, respectively. Compared with pentaheptite, the bond angles of C-57 deviate much larger from 120° for perfectly sp2 hybridization, implying a higher strain exists in C-57 carbon, which would lead to an enhancement in total energy. The total energy per atom as a function of the 2D crystal area per atom of C-57 carbon in comparison with α -graphyne, graphene, penta-
Fig. 2. The total energy per atom as a function of the 2D crystal area per atom for C-57 carbon in comparison with some other 2D carbon allotropes.
graphene, planar T-graphene and pentaheptite have been calculated and the results are shown in Fig. 2. What we can see from Fig. 2 is that graphene is energetically more stable than the other five 2D carbon allotropes and C-57 carbon is energetically preferable than α -graphyne, penta-graphene, and planar T-graphene. The equilibrium total energy of pentaheptite is lower than that of C-57 carbon due to the greater distortion of bond angles in C-57 carbon compared with pentaheptite, as mentioned above. The calculated lattice parameters, equilibrium atom density, and total energy per atom for graphene, α -graphyne,
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Table 1 The calculated lattice parameters a and b (in Å), equilibrium atom density ( ρ in g/cm3), and total energy (Et in eV) for C-57 carbon in comparison with graphene, α -graphyne, penta-graphene, planar T-graphene, and pentaheptite as well as the available experimental and other theoretical values. Structure
Space group
References
a (Å)
b (Å)
ρ (atom/ Å2)
Et (eV/atom)
Graphene
P 63/ mmc
α -graphyne
P 6/ mmm
pentagraphene
P − 421m
Exp. [43] This work GGA [44] This work GGA [17]
2.460 2.467 6.997 6.962 3.640
2.460 2.467 6.997 6.962 3.640
0.330 0.329 0.163 0.165 0.453
– −9.227 – −8.305 −8.320
planar Tgraphene
P 4/ mmm
This work GGA [45]
3.641 3.447
3.641 3.447
0.453 0.337
−8.324 −8.712
pentaheptite
Cmmm
C-57 carbon
P 6¯2m
This work –[42] This work This work
3.446 7.525 7.473 6.155
3.446 5.899 5.858 6.155
0.337 0.361 0.365 0.317
−8.714 – −8.984 −8.768
Fig. 4. Strain energy of C-57 carbon under different kinds of in-plane strain in the harmonic region.
for mechanical stability, further confirming the mechanical stability of C-57 carbon. The Young’s modulus of C-57 carbon can be derived from 2 2 )/C11, and the result is 243 N/m, its elastic constants using E = (C11 -C12 which is more than two-thirds of that of graphene (345 N/m) [49], but smaller than that of pentaheptite (298 N/m) and penta-graphene (264 N/m) [17]. To examine the thermal stability of C-57 carbon, we have also performed the first-principles molecular dynamics simulations with the canonical (NVT) ensemble. A 4 × 4 supercell containing 192 carbon atoms was built to reduce the constraint of periodic boundary condition and explore possible structure reconstruction. The fluctuations of the temperature and total energy as a function of simulation time is depicted in Fig. 5. After heating at room temperature (300 K) for 6 ps with a time step of 1 fs, no structure reconstruction is found, except for slight fluctuations of the total energy (Fig. 5(a)), which strongly suggests that
penta-graphene, planar T-graphene, pentaheptite and C-57 carbon are summarized in Table 1 for comparison. Our calculated results are well agree with the previous experimental and theoretical ones, indicating that our calculation is valid. To further check the dynamical stability, we perform a thorough analysis of its phonon modes. The calculated phonon dispersions of C57 carbon are shown in Fig. 3. It can be seen that there is no imaginary mode, demonstrating that this structure is dynamically stable in the ground state. What’s more, it is instructive to note that the highest phonon frequency of C-57 carbon is estimated to be ∼1663 cm−1, locating at M(0.5, 0.0, 0.0) of Brillioun zone, slightly larger than ∼1600 cm−1 for graphene [46], and there exist a narrow indirect band gap (∼170 cm−1) in the phonon band structures. These features could be useful to identify C-57 carbon in experiments. The elastic constants of theoretical materials are not only important for their potential applications, but also can be used to test their mechanical stabilities. The strain energy of C-57 carbon under different kinds of in-plane strain has been calculated, as shown in Fig. 4. For 2D 1 2 + 2 C22 sheets, the strain energy can be expressed as [47]: Uε = 1 C11εxx 2
2 2 ε yy + C12 εxx εyy + 2C66εxy , where Uε is the strain energy, x and y represent two vertical directions in the plane of C-57 carbon sheet as shown in Fig. 1 (a), εij are small strains in the harmonic region, C11, C12 , C22 , and C66 are components of the elastic modulus tensor. By fitting the strain energy to the parabolic function of the axial and biaxial strains, we get C11 = C22 = 275 N/m, C12 = 94 N/m, and C66 = 90 N/m. These elastic constants satisfy the Born-Huang criteria [48] of C11 > |C12 | and C66 > 0
Fig. 5. The fluctuations of total potential energy of the C-57 carbon supercell as a function of the molecular dynamic simulation step at (a) 300 K and (b) 800 K, respectively. The inset shows the snapshot of the simulated system.
Fig. 3. Phonon dispersions of C-57 carbon. The predicted highest vibrational frequency for C-57 carbon is ∼1663 cm−1. 117
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Fig. 7. (a) Illustration of chiral vectors of C-57-tubes. The lattice basis vectors are denoted by a1 and a2 . Optimized structure of (b) (6, 0) and (c) (6, 6) C-57tubes from side view.
Fig. 6. (a) The electronic band structures and the projected density of states (PDOS) for C1, C2 , C3 , and C4 atom obtained by HSE06 potential. The Fermi level is set at 0 eV with dash line. (b) Band decomposed charge density distribution (the partially occupied band).
This new carbon sheet can also be used as a precursor for materials of different dimensions, for instance, quasi-1D nanotubes. We have constructed a series of carbon nanotubes by rolling the C-57 carbon sheet. The chiral vector Ch and its corresponding translation vector Th are shown in Fig. 7(a). The chirality of a nanotube is defined as: Ch = ma1 + na2 , where, m and n are the chiral indices. Here, we restrict our study to (m,0) and (m,m) nanotubes with diameters from 3.918 Å to 20.361 Å. After fully optimized, the C-57 carbon based nanotubes (C57-tubes) are stable, as shown in Fig. 7(b) (6, 0) and (c) (6, 6) tubes. The diameters of (6, 0) and (6, 6) C-57-tubes are 11.755 Å and 20.361 Å, respectively. The dynamic stability of these nanotubes is confirmed by carrying out phonon calculations and the calculated results are shown in Fig. 8(a) and (b). The calculated band structures (see Fig. 8(c) and (d)) revealed that both of (6, 0) and (6, 6) tubes are metallic. The metallic behavior of C-57-tubes can be attributed to the strong metallicity of C-57 carbon. The chirality-independent metallic C-57 based nanotubes as well as the C-57 sheet may have potential application in nanoelectronics.
the C-57 carbon can exist at room temperature. Furthermore, we find that the c-57 sheet can withstand temperatures as high as 800 K (Fig. 5(b)), implying that this carbon allotrope is separated by highenergy barriers from other local minima on the potential energy surface of elemental carbon. Finally, let us discuss the electronic properties of C-57 carbon. We show in Fig. 6 the electronic band structures for C-57 carbon and the projected density of states (PDOS) of C1, C2 , C3 , and C4 atoms, which are calculated by the hybrid density functional method based on the HeydScuseria-Ernzerhof scheme (HSE06) [50,51]. As we can see from Fig. 6(a), C-57 carbon is metallic and some flat bands cross the Fermi level along the M-K and K-Γ direction. There also exist some steep bands near the Fermi level along M-K and K-Γ paths. It is believed that the coexistence of steep and flat bands at the Fermi level is the necessary condition for the superconductivity of materials [52,53]. The C-57 carbon possesses a very high density of states of 0.23 states/eV per atom at Fermi level owing to the flat bands, which is not only larger than that of planar T-graphene (0.11 states/eV per atom) [45] and net W carbon sheet (0.12 states/eV per atom) [15] but also almost three times that of the 5.3 Å diameter (4, 4) carbon nanotube [54]. In order to explore the origin of the metallic feature displayed in C57 carbon, we have also calculated the charge density of the partially occupied bands in the energy windows of Ef − 0.5 to Ef + 0.5 eV. We can see from Fig. 6(b) that the unequivalent carbon atoms make different contribution to the charge density: the electron accumulation around C4 atom is relatively low, while the electron accumulation around C1, C2 , and C3 atoms is relatively high, which is in accordance with the discrepant contributions to states at Fermi level in PDOS. The calculated PDOS and band decomposed charge density reveal that the conductive electrons of C-57 carbon are predominately contributed by the py orbital of C1, C2 , and C3 atoms, which is quite different from graphite, where the metallicity originates from the delocalization of electrons in pz orbital. The bond angle around C1, C2 , and C3 atoms deviate a lot from the perfect sp2 bond, which raise the energy of py orbital, resulting in the metallic property of C-57 carbon; while the C4 atom form almost perfect sp2 bond with the C3 atom, contributing low electron density around Fermi level.
4. Conclusion In summary, we have systemically studied the structural stability and electronic properties of a 2D metallic carbon allotrope in P 6¯2m (D33h ) symmetry, termed C-57 carbon here, which is composed of penta and hepta carbon rings similar to the previously proposed pentaheptite while their arrangement of carbon rings is different. The energetic calculations show that C-57 carbon is metastable compared to graphene, but more stable than α -graphyne and penta-graphene. The dynamic, mechanical and thermal stability of C-57 carbon have been verified by the analyses of the phonon mode, elastic constants and the first-principles molecular dynamics simulations, respectively. The electronic structure calculations reveal that C-57 carbon is metallic with much higher DOS of ∼0.23 eV/states per atom at the Fermi level, and there are several flat bands across the Fermi level. The metallic nanotubes based on C-57 carbon can also be produced for application in nanoeletronics. These findings further expand our understanding of the structural and electronic properties of 2D carbon allotropes, and will stimulate more efforts focus on the research field of 2D carbon 118
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Fig. 8. Calculated phonon spectrums of (a) (6, 0) and (b) (6, 6) C-57 carbon based nanotubes. Band structures of (c) (6, 0) and (d) (6,6) C-57-tubes.
materials. The C-57 carbon and the chirality-independent metallic C-57 based nanotubes have potential to be used in nanoelectronics.
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CRediT authorship contribution statement Chun-Xiang Zhao: Methodology, Formal analysis, Writing original draft. Yi-Qi Yang: Formal analysis, Data curation. ChunYao Niu: Conceptualization, Writing - review & editing, Supervision. Jia-Qi Wang: Data curation, Validation. Yu Jia: Conceptualization, Methodology. Acknowledgement This work was supported by the National Natural Science Foundation of China (Grant Nos. 11504332) and Outstanding Young Talent Research Fund of Zhengzhou University (1521317006). References [1] H.W. Kroto, J.R. Heath, S.C. Obrien, R.F. Curl, R.E. Smaliey, Nature 318 (1985) 162–163. [2] S. Iijima, Nature 354 (1991) 56–58. [3] K. Novoselov, A. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I. V Grigorieva, A.A. Firsov, Science 306 (2004) 666–669. [4] Q. Li, Y.M. Ma, A.R. Oganov, H.B. Wang, H. Wang, Y. Xu, T. Cui, H.K. Mao, G.T. Zou, Phys. Rev. Lett. 102 (2009) 175506. [5] J.T. Wang, C.F. Chen, Y. Kawazoe, Phys. Rev. Lett. 106 (2011) 075501. [6] J.Q. Wang, C.X. Zhao, C.Y. Niu, Q. Sun, Y. Jia, J. Phys.: Condens. Matter 28 (2016) 475402. [7] S.H. Zhang, Q. Wang, X.S. Chen, P. Jena, Proc. Natl. Acad. Sci. USA 110 (2013) 18809–18813. [8] C.Y. Niu, X.Q. Wang, J.T. Wang, J. Chem. Phys. 140 (2014) 054514. [9] Y. Cheng, R. Melnik, Y. Kawazoe, B. Wen, Cryst. Growth Des. 16 (2016) 1360–1365. [10] C.X. Zhao, C.Y. Niu, Z.J. Qin, X.Y. Ren, J.T. Wang, J.H. Cho, Y. Jia, Sci. Rep. 6 (2016) 21879. [11] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod. Phys. 81 (2009) 109. [12] A.K. Geim, Science 324 (2009) 1530. [13] F. Schwierz, Nat. Nanotechnol. 5 (2010) 487. [14] X. Wang, L.J. Zhi, K. Müllen, Nano Lett. 8 (2008) 323. [15] X.Q. Wang, H.D. Lib, J.T. Wang, Phys. Chem. Chem. Phys. 15 (2013) 2024. [16] N. Tyutyulkov, F. Dietz, K. Müllen, M. Baumgarten, Chem. Phys. Lett. 272 (1997) 111–114.
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