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Two dimensional Dirac carbon allotropes from graphene† Li-Chun Xu,ag Ru-Zhi Wang,*ab Mao-Sheng Miao,*bc Xiao-Lin Wei,bf Yuan-Ping Chen,bf Hui Yan,a Woon-Ming Lau,be Li-Min Liu*b and Yan-Ming Ma*bd Using a structural search method in combination with first-principles calculations, we found lots of low energy 2D carbon allotropes and examined all possible Dirac points around their Fermi levels. Three amazing 2D Dirac carbon allotropes have been discovered, named as S-graphene, D-graphene and E-graphene. By analyzing the topology correlations among S-, T, net W graphene and graphene, we found that a general rule is valuable for constructing 2D carbon allotropes that are keen to possess Dirac cones in their electronic structures. Based on this rule, we have successfully designed many new 2D carbon allotropes possessing Dirac cones. Their energy order can be well described by an Ising-like model, and some allotropes are energetically more stable than those recently reported. The related electronic structures of these Dirac allotropes are anisotropy distinguished from those of graphene.
Received 22nd August 2013 Accepted 24th October 2013
Moreover, the fact that D- and E-graphene present Dirac cones suggests that sp hybridization or sp3 hybridization could not suppress the emerging of Dirac features. Our results demonstrate that the Dirac
DOI: 10.1039/c3nr04463g
cone and carrier linear dispersion is a very common feature in 2D carbon allotropes and can exist
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beyond the limitations of fundamental structure features of graphene.
1. Introduction Graphene, a single layer of carbon in a hexagonal lattice, shows many remarkable properties.1–5 Besides the unusual mechanical and chemical properties, the existence of both electron and hole carriers possessing exceedingly large mobility makes it a promising material for future electronic devices.6–8 The large and ballistic transport properties originate deeply from the unique electronic structure of graphene: the energy dispersion is linear close to the Fermi level.4 This Dirac cone structure makes graphene gapless and was found to be robust under
a
College of Materials Science and Engineering, Beijing University of Technology, Beijing 100124, China. E-mail: [email protected]; [email protected]; [email protected]; [email protected]
b
Beijing Computational Science Research Centre, Beijing, 100084, China
c
Materials Research Laboratory, University of California Santa Barbara, Santa Barbara, California 93110, USA d
State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China
e
Chengdu Green Energy and Green Manufacturing Technology R&D Centre, Chengdu, Sichuan, 610207, China
f
Department of Physics, Xiangtan University, Xiangtan 411105, China
g
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China † Electronic supplementary information (ESI) available: Phonon band structure of S-graphene. Snapshots of S-graphene, D-graphene and E-graphene aer a 3 ps MD simulation at 500 K. Relative energies of all predicted new carbon allotropes. Crystal structures and electronic structures of A-, B-, T4, S3, T3, uududu([[Y[Y [) graphenes. See DOI: 10.1039/c3nr04463g
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deformations and chemical modications.9–12 In order to implement graphene in complicated electronic devices, the capability of manipulating a large variety of electronic properties is needed. However, the chemical modication of graphene is limited by its hexagonal backbone, which is known as the necessary condition for the existence of the Dirac cone. Recently, two novel 2D carbon allotropes were proposed to possess Dirac points,9,11,13 which do not have hexagonal symmetry. The breaking of this traditional view raises more general questions. One of those is: to what extent does the Dirac feature remain robustly in a 2D carbon lattice? Based on the tight binding models, several structural features of graphene are considered as a necessity for Dirac allotropes, including the 2D dimensionality, the hexagonal symmetry, the bilattice honeycomb structure, the chemical equivalence, as well as the sp2 hybridization of the carbon atoms.4,13 Some of these commonly accepted guidelines are violated by the recent discoveries of alternative Dirac carbon allotropes. The T graphene as proposed by Liu et al., and although it maintains the bilattice feature, it does not have hexagonal symmetry.11 In another set of Dirac carbon allotropes, so called “graphynes”, the carbon atoms are not equivalent, and are in both sp and sp2 hybridization.9 Though the graphynes are strictly 2D, the T graphene is not, having a puckered plane. These recent progresses pointed out the possibility of nding new Dirac allotropes towards a wider scope of carbon lattices.14 There is an emergent request for a systematic design of the Dirac carbon allotropes in a large
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variety of 2D lattices, and a general guideline of constructing Dirac allotropes is greatly desirable. In this work, we systematically designed various Dirac allotropes in a wide range of 2D carbon lattices by employing a recently developed 2D structure searching technique15 based on the CALYPSO algorithm16,17 and rst-principles calculations. By examining the obtained low-energy carbon allotropes and picking up the ones with Dirac cones, we got three Dirac allotropes as typical examples. By investigating the structural features of one allotrope associated with Dirac points, we found unexpectedly a construction scheme that can systematically generate structures that are keen to have Dirac points. Their energy order can be well described in an Ising-like model, allowing us to design low-energy allotropes in large unit cells. To further illustrate the structural features of Dirac materials, two Dirac allotropes with special atomic conguration have been discussed. Finally, a general guideline on the association of Dirac cones with the atomic structure of 2D carbon allotropes has been expounded, which opens up the possibilities of controllable design of 2D carbon-based nano-electronic devices.
2.
Methods
The structure searching method has been implemented in the CALYPSO code,16,17 and its capability in predicting novel 2D structures has been demonstrated.15,18–20 Although the search is based on two dimensional space groups, the new module allows the relaxation of the atoms in the perpendicular direction, making it capable of exploring low dimension structures in real materials.15 The underlying rst-principles calculations were carried out using a Perdew–Burke–Ernzerhof (PBE)21 exchange– correlation density functional and the projector-augmentedwave (PAW)22 potentials as implemented in VASP.23 The PAW with 2s22p2 electrons as valence electrons was adopted. The kinetic-energy cutoff of 800 eV and a ne Monkhorst–Pack24 k-points meshes were chosen. The atomic relaxation was carried out until the change in the total energy per one unit cell was smaller than 0.01 meV. The dynamic stability of new structures has been examined by the phonon calculations and molecular dynamics. The phonon calculations were carried out by using a supercell approach as implemented in the PHONON code.25 The rst-principles molecular dynamics26 (FPMD) simulations adopt the constant temperature and volume (NVT) ensemble, and are carried out with time steps of 3 fs for a total simulation time of 3 ps at 500 K.
3.
Results and discussion
Selected from the CALYPSO structural searches, a number of structures that possess Dirac points are identied. Three representative allotropes have been shown in Fig. 1, named as S-graphene [Fig. 1(b)], D-graphene [Fig. 1(c)] and E-graphene [Fig. 1(d)], respectively. Some of the structures such as S- and D-graphenes are strictly 2D, whereas E-graphene is not. They are all found to be dynamically stable as proven by one of two widely accepted criteria: (i) there are no imaginary phonons in the calculated phonon spectra (see ESI Fig. S1†) and (ii) the lattices
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Fig. 1 Structures of (a) (2 1) graphene cell and selected low energy Dirac carbon allotropes as found by a CALYPSO structure search. The allotropes are named as (b) S-graphene, (c) D-graphene and (d) E-graphene, respectively.
remain very stable aer 3 ps FPMD simulations at heating up to 500 K (see ESI Fig. S2†). All the three structures are metastable compared with graphene, but their energies are in the same range as T graphene11 and three graphynes9 (see ESI Fig. S3†). At rst sight, these structures seem not to have any relation. However, by comparing the structural features of S-graphene and a (2 1) graphene cell, as well as the previously proposed T graphene11 and net W graphene,27 we found that they are closely related and can be constructed systematically from graphene by a series of reconstruction of the neighboring C–C bonds. Taking S-graphene as an example, as shown in Fig. 2(a), the S-graphene can be constructed by breaking one C–C bond in a (2 1) cell and shiing oppositely the two carbon atoms towards the nonbonding directions shown by arrows. In a similar manner, T graphene can be created by cutting and shiing two C–C bonds, as shown in Fig. 2(b). The net W is more complicated and can only be constructed from a (4 1) unit cell. Therefore, one can construct new 2D carbon allotropes by choosing a (n 1) graphene unit cell and applying a series of aforementioned cutand-paste operations. All these operations act on the parallel C–C bonds, there are only two states before and aer the cut-and-paste operation on one bond, which is somewhat like the “spin” status of the chain structure. It should be noted that the denition of “spin” is an articial geometry index. As shown in Fig. 2, there are 2n lines along the cell-repetition direction for (n 1) cells, and these 2n lines of carbon in the (n 1) cell can be marked by 2n “spins”. Depending on whether they have undergone the cut-and-paste operation or not, each line can be marked by “spin up” (remained intact, blue arrow) or “spin down” (undergone the operation, red arrow). Therefore, we are able to represent the constructed carbon allotropes by a chain of “spin”. For example, graphene itself can be marked as a chain of “up spins”, [[[[. The S- and the T-graphenes can be notated as Y[[[ and YY[[,
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Fig. 2 Deriving new allotropes from graphene. (a) S-graphene, (b) T graphene, (c) net W.
while the net W corresponds to YYYY[[[[. For these spin-like chains, the total energy of the carbon allotropes can be expressed by the interactions of the neighboring spins based on the Ising model:28,29 E ¼ E0
1 X 1 X 1 X J1 si siþ1 J2 si siþ2 J3 si siþ3 N N N i i i
The parameters J represent the inter-array interactions between the rst, the second and the third neighboring C–C arrays, and E is a reference energy. A positive J favors the “ferromagnetic (FM)” interactions, meaning the two neighboring “spins” tend to align to the same orientation; whereas, a negative J favors the antiferromagnetic (AFM) alignment of the “spins”. By tting the density functional theory (DFT) total energies of original, S-, T, net W and other allotropes, we found the J values are: J1 ¼ 0.49249 eV, J2 ¼ 0.15017 eV and J3 ¼ 0.05577 eV. As shown in Table 1, all energies of related carbon allotropes derived from the Ising-like model are in excellent agreement with those obtained from DFT calculations. The fact that J1 is much larger than J2 and J3 reveals that the energies of the new carbon structures are mainly determined by the relative positions of the carbon atoms in the nearest neighboring arrays. They are energetically most stable while they form an hexagonal lattice (FM spin chain) as in the original graphene. Interestingly, the J2 is negative, showing that the next nearest carbon arrays tend to deviate from the hexagonal lattice, which is a factor governing the structure deviation from original graphene. Based on our structural design strategy and Ising-like model, a series of related structures has been constructed and optimized by DFT calculations. Among such constructed allotropes, two series, Sn and Tn, are particularly interesting. Both of them exhibit rectangular symmetry instead of hexagonal symmetry. A Sn graphene consists of one ipped spin in a (n 1) unit cell; whereas a Tn graphene is formed by ip half of the neighboring spins. Both constructions minimize the neighboring AFM interactions and then can maintain a lower energy. It is worth noting that the earlier proposed T graphene and net W graphene are special cases of our generalized Tn series with n ¼ 2 and n ¼ 4, respectively. The energy of Sn graphene will converge towards that of graphene with the increasing n, then Sn graphene with large n can be considered as graphene with defects. Although in general the designed carbon allotropes become more and more stable with increasing n due to the fact of an increasing FM domain region, we do nd that the Tn series tends to be energetically more favorable over the Sn series. Due to dynamic stability and comparable formation energy to the other graphene allotropes, these new allotropes might be experimentally synthesized through chemical vapor deposition
Table 1 Summary of the configuration, expression and energy of the Ising-like model for graphene and predicted allotropes, the energy based on DFT calculations is listed as reference
Allotrope
Conguration
Expression
EIsing
EDFT
EError
Graphene S2 S3 S4 S5 S6 S7 S8 S9 T2 T4 (net W)
[[[[ Y[[[ Y[[[[[ Y[[[[[[[ Y[[[[[[[[[ Y[[[[[[[[[[[ Y[[[[[[[[[[[[[ Y[[[[[[[[[[[[[[[ Y[[[[[[[[[[[[[[[[[ YY[[ YYYY[[[[
2J1 2J2 2J3 0 2/3J1 2/3J2 2/3J3 J1 J2 J3 1.2J1 1.2J2 1.2J3 4/3J1 4/3J2 4/3J3 10/7J1 10/7J2 10/7J3 1.5J1 1.5J2 1.5J3 14/9J1 14/9J2 14/92J3 +2J2 J1 + J3
0.01708 0.81326 0.54787 0.41517 0.33555 0.28247 0.24456 0.21612 0.19401 0.51292 0.37654
0 0.79987 0.57466 0.43128 0.34719 0.29050 0.24964 0.21882 0.19476 0.51293 0.37655
0.01708 0.01339 0.02679 0.01611 0.01164 0.00803 0.00508 0.0027 0.00075 1 105 1 105
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on the suitable substrate30 or through electron irradiation on the graphene.31 By examining the electronic structures, we nd that Sn may possess Dirac-like dispersions in their band structures. For example, S-graphene exhibits four Dirac points in its rst Brillouin zone [Fig. 3(a) and (b)]. However, due to the symmetry relation, only two of them are independent, located on the lines from X to M (point A) and from G to Y (point B). ˚ and Both Dirac points are asymmetric, with slopes of +14 eVA ˚ around point A and slopes of +15 eVA ˚ and 18 eVA ˚ 21 eVA around point B. It is important to note that Dirac point A is slightly below the Fermi level, indicating that S-graphene has the self-doping character. Therefore, it contains electron carriers even without n-type doping or electron transfer from the side groups. Meanwhile, the electronic structure of Tn series is substantially different. In a strict 2D structure, these allotropes are metallic around the Fermi level.32 Some of the T series allotropes, such as T2 and T4 graphene may relax in the perpendicular direction and lead to a puckered layer with multiple unit cells. The corresponding folded band structure can contain a circle of continuous crossover points (see ESI Fig. S6†). Although each point of this corral is the meeting point of linear valence and conduction bands, they can be never described by the Dirac–Weyl equation and thus are not Dirac cones.
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Because of the close relation of their structures, the Dirac points of Sn graphenes are also related to those in graphene. In graphene, the origin of the Dirac point can be traced back to the degeneracy of the orbitals. The triply degenerated valence band maximum (VBM) states at the G point (vertical, le tilted and right tilted) reduce to two-fold degeneracy at the Dirac points (K points). The two corresponding wavefunctions are dimmerlike and they can be transformed to each other by a rotation of 60 degrees. The dimmer feature is kept for the wavefunctions at the Dirac points in S-graphene. One is very much like the vertical one in graphene except the difference in bond topology. The second one can be considered as the linear combination of the le and right tilted wavefunctions in graphene, which is a result of the change of symmetry. For Sn graphenes, we examined their electronic structures up to n ¼ 10, and found that only for those n ¼ 3m 1 and n ¼ 3m 2 (see ESI Fig. S7†), the corresponding allotropes possess Dirac points at the Fermi level. Dirac points are absent for n ¼ 3m. This magic number is quite similar to the Dirac point trend in graphene nanoribbons and is due to the preservation of the degeneracy of graphene states under deformation. In addition, by applying more cutand-paste operations in a (4 1) graphene cell, two new allotropes (A- and B-graphene in ESI Fig. S4 and S5†) with Dirac points at the Fermi level can also be obtained. Hence, with the increase of the original cell unit of graphene, more allotropes
Electronic structures of S-graphene. (a) Band structure and density of states, (b) Dirac points in 3D band structure and those projections in first Brillouin zone, (c) degenerate wavefunctions at the Dirac points of S-graphene.
Fig. 3
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with Dirac points can be reasonably obtained due to the topology correlations of atomic structure and wavefunctions. Besides the many Dirac allotropes that can be constructed from the above proposed method, our CALYPSO structure search method revealed that the Dirac points can exist beyond those structures, revealing that Dirac physics is a very rich phenomena in 2D carbon allotropes. In the two representative structures, D-graphene [Fig. 1(c)] consists of both sp2 and sp carbons, which can be constructed by inserting one pair of C/C (marked as red) into a graphene layer containing arrays of 5–7 ring defects. Despite the complex structure type, the energy of this allotrope is only 0.53 eV per atom above graphene, very close to the energy of T graphene. Different to all the other allotropes, E-graphene can no longer be viewed as a puckered layer since the shi of the atoms in the perpendicular direction is so large [Fig. 1(d)]. The structure of E-graphene can also be related to graphene. If the extra carbon is removed (marked red) and the other atoms shi to the center of the close-by hexagon, E-graphene can be transformed back to graphene. Because of the existence of the extra carbon atom in the graphene lattice, the atomic density is larger than that of graphene and all the above 2D allotropes. That is why the structure of E graphene is not 2D or puckered 2D. In order to accommodate the extra carbon atoms, some of the carbon atoms have to form four C–C s bonds, and therefore they must be in sp3 hybridization. This
Fig. 5 Electronic structures of E-graphene. (a) Band structure. (b) Density of states. (c) Dirac points in 3D band structure and those projections. (d) First Brillouin zone with red points designating Dirac points. (e) Linear dispersion relation in the vicinity of the Dirac points.
allotrope can also be viewed as an array of N ¼ 2 zigzag nanoribbons being connected by chains of sp3 carbon atoms. The energy of E-graphene is 1.28 eV per atom higher than that of graphene, showing that the mixture of the sp2 carbons and sp3 carbons in a quasi-2D structure is not energetically favored. Despite the complexity of their structures, both D- and E-graphenes possess Dirac points. As shown in Fig. 4, D-graphene exhibits one independent Dirac point at the boundary of the rst Brillouin zone with a distorted cone. Around the Dirac point, the conduction and valence bands show linear dispersion with the ˚ in direction from M to Y. The Dirac point slopes of 11 eVA locates slightly above the Fermi level, indicating the p-type selfdoping character of the allotrope. As shown in Fig. 5, the Dirac point of E-graphene is on the G–Y line and located exactly at the Fermi level. Along this direction, the band presents linear ˚ which is close to the value of graphene dispersion of 29 eVA, ˚ The Dirac cone also shows anisotropy perpendicular (34 eVA). to the line from G to Y. Therefore both D- and E-graphene should show strongly anisotropic transport properties.
Fig. 4 Electronic structures of D-graphene. (a) Band structure. (b) Density of states. (c) Dirac points in 3D band structure and those projections. (d) First Brillouin zone with red points designating Dirac points. (e) Linear dispersion relation in the vicinity of the Dirac points.
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4. Conclusion In summary, by a large-scale structure search for 2D carbon allotropes using the CALYPSO algorithm, the identied several
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families of structures possessing the Dirac cones in their electronic structures have been found. Originating from inherent relation of wavefunctions and atomic structures between graphene and these new allotropes, we then established a general route to construct a series of novel 2D carbon allotropes that are keen to have Dirac cones. The energy of these allotropes can be well described by an Ising-like model. Many such constructed allotropes possess linear energy dispersion and Dirac cones. Furthermore, we found the construction of allotropes by adding either sp or sp3 hybridized carbon atoms into graphene may also lead to Dirac structures. Our results show that neither of the graphene structural features, including the hexagonal symmetry and the bilattice backbone, nor the sp2 hybridization, is necessary for a Dirac dispersion of their carriers. Enough examples indicate the phenomenon existed the Dirac points in 2D materials is universal. This may be due to the nature of chemical bonding in thin-enough 2D materials. The existence of pz electrons could give these electrons the possibility of noscattering migration in 2D surface of real space, and appropriate potential eld controls the conductivity of 2D materials.
Acknowledgements The work was nancially supported by the National Natural Science Foundation of China (NSFC) (Grant no. 11274029, 11074017, 11025418, 91022029, 11274028, 51222212, and 51032002), the Beijing Nova Program (Grant no. 2008B10), the Ministry of Science & Technology of China (973 Project, no. 2011CB922200, 863 Project, no. 2012AA050704), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (no. CIT&TCD201204037), the Key Programs of Beijing Plan of Science and Technology (D121100001812002), the Qualied Personnel Foundation of Taiyuan University of Technology (QPFT) (no. tyut-rc201333a), and the Basic Research Foundation of Beijing University of Technology.
References 1 A. S. Mayorov, R. V. Gorbachev, S. V. Morozov, L. Britnell, R. Jalil, L. A. Ponomarenko, P. Blake, K. S. Novoselov, K. Watanabe, T. Taniguchi and A. K. Geim, Nano Lett., 2011, 11, 2396–2399. 2 R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres and A. K. Geim, Science, 2008, 320, 1308. 3 A. K. Geim and K. S. Novoselov, Nat. Mater., 2007, 6, 183–191. 4 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200. 5 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666–669. 6 L. Britnell, R. V. Gorbachev, R. Jalil, B. D. Belle, F. Schedin, A. Mishchenko, T. Georgiou, M. I. Katsnelson, L. Eaves, S. V. Morozov, N. M. R. Peres, J. Leist, A. K. Geim,
1118 | Nanoscale, 2014, 6, 1113–1118
Paper
7
8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26
27 28 29 30 31 32
K. S. Novoselov and L. A. Ponomarenko, Science, 2012, 335, 947–950. Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill and P. Avouris, Science, 2010, 327, 662. F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia and P. Avouris, Nat. Nanotechnol., 2009, 4, 839–843. D. Malko, C. Neiss, F. Vi˜ nes and A. G¨ orling, Phys. Rev. Lett., 2012, 108, 086804. S.-M. Choi, S.-H. Jhi and Y.-W. Son, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 081407. Y. Liu, G. Wang, Q. Huang, L. Guo and X. Chen, Phys. Rev. Lett., 2012, 108, 225505. S. Cahangirov, M. Topsakal, E. Akt¨ urk, H. ¸ Sahin and S. Ciraci, Phys. Rev. Lett., 2009, 102, 236804. A. N. Enyashin and A. L. Ivanovskii, Phys. Status Solidi B, 2011, 248, 1879–1883. H. Huang, W. Duan and Z. Liu, New J. Phys., 2013, 15, 023004. Y. Wang, M. Miao, J. Lv, L. Zhu, K. Yin, H. Liu and Y. Ma, J. Chem. Phys., 2012, 137, 224108. Y. Wang, J. Lv, L. Zhu and Y. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116. Y. Wang, J. Lv, L. Zhu and Y. Ma, Comput. Phys. Commun., 2012, 183, 2063–2070. H. Xiang, B. Huang, Z. Li, S. H. Wei, J. Yang and X. Gong, Phys. Rev. X, 2012, 2, 011003. X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang and X. C. Zeng, ACS Nano, 2012, 6, 7443–7453. X. Luo, J. Yang, H. Liu, X. Wu, Y. Wang, Y. Ma, S. H. Wei, X. Gong and H. Xiang, J. Am. Chem. Soc., 2011, 133, 16285– 16290. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758. G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci., 1996, 6, 15–50. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192. A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106. P. Pavone, K. Karch, O. Schutt, W. Windl, D. Strauch, P. Giannozzi and S. Baroni, Phys. Rev. B: Condens. Matter, 1993, 48, 3156. X.-Q. Wang, H.-D. Li and J.-T. Wang, Phys. Chem. Chem. Phys., 2013, 15, 2024–2030. S. Limpijumnong and W. R. Lambrecht, Phys. Rev. B: Condens. Matter, 1998, 57, 12017–12022. M. Miao, S. Limpijumnong and W. R. Lambrecht, Appl. Phys. Lett., 2001, 79, 4360–4362. J. Lahiri, Y. Lin, P. Bozkurt, I. I. Oleynik and M. Batzill, Nat. Nanotechnol., 2010, 5, 326–329. J. Kotakoski, A. V. Krasheninnikov, U. Kaiser and J. C. Meyer, Phys. Rev. Lett., 2011, 106, 105505. H. Huang, Y. Li, Z. Liu, J. Wu and W. Duan, Phys. Rev. Lett., 2013, 110, 029603.
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Two dimensional Dirac carbon allotropes from graphene† Li-Chun Xu,ag Ru-Zhi Wang,*ab Mao-Sheng Miao,*bc Xiao-Lin Wei,bf Yuan-Ping Chen,bf Hui Yan,a Woon-Ming Lau,be Li-Min Liu*b and Yan-Ming Ma*bd Using a structural search method in combination with first-principles calculations, we found lots of low energy 2D carbon allotropes and examined all possible Dirac points around their Fermi levels. Three amazing 2D Dirac carbon allotropes have been discovered, named as S-graphene, D-graphene and E-graphene. By analyzing the topology correlations among S-, T, net W graphene and graphene, we found that a general rule is valuable for constructing 2D carbon allotropes that are keen to possess Dirac cones in their electronic structures. Based on this rule, we have successfully designed many new 2D carbon allotropes possessing Dirac cones. Their energy order can be well described by an Ising-like model, and some allotropes are energetically more stable than those recently reported. The related electronic structures of these Dirac allotropes are anisotropy distinguished from those of graphene.
Received 22nd August 2013 Accepted 24th October 2013
Moreover, the fact that D- and E-graphene present Dirac cones suggests that sp hybridization or sp3 hybridization could not suppress the emerging of Dirac features. Our results demonstrate that the Dirac
DOI: 10.1039/c3nr04463g
cone and carrier linear dispersion is a very common feature in 2D carbon allotropes and can exist
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beyond the limitations of fundamental structure features of graphene.
1. Introduction Graphene, a single layer of carbon in a hexagonal lattice, shows many remarkable properties.1–5 Besides the unusual mechanical and chemical properties, the existence of both electron and hole carriers possessing exceedingly large mobility makes it a promising material for future electronic devices.6–8 The large and ballistic transport properties originate deeply from the unique electronic structure of graphene: the energy dispersion is linear close to the Fermi level.4 This Dirac cone structure makes graphene gapless and was found to be robust under
a
College of Materials Science and Engineering, Beijing University of Technology, Beijing 100124, China. E-mail: [email protected]; [email protected]; [email protected]; [email protected]
b
Beijing Computational Science Research Centre, Beijing, 100084, China
c
Materials Research Laboratory, University of California Santa Barbara, Santa Barbara, California 93110, USA d
State Key Laboratory of Superhard Materials, Jilin University, Changchun 130012, China
e
Chengdu Green Energy and Green Manufacturing Technology R&D Centre, Chengdu, Sichuan, 610207, China
f
Department of Physics, Xiangtan University, Xiangtan 411105, China
g
College of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, China † Electronic supplementary information (ESI) available: Phonon band structure of S-graphene. Snapshots of S-graphene, D-graphene and E-graphene aer a 3 ps MD simulation at 500 K. Relative energies of all predicted new carbon allotropes. Crystal structures and electronic structures of A-, B-, T4, S3, T3, uududu([[Y[Y [) graphenes. See DOI: 10.1039/c3nr04463g
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deformations and chemical modications.9–12 In order to implement graphene in complicated electronic devices, the capability of manipulating a large variety of electronic properties is needed. However, the chemical modication of graphene is limited by its hexagonal backbone, which is known as the necessary condition for the existence of the Dirac cone. Recently, two novel 2D carbon allotropes were proposed to possess Dirac points,9,11,13 which do not have hexagonal symmetry. The breaking of this traditional view raises more general questions. One of those is: to what extent does the Dirac feature remain robustly in a 2D carbon lattice? Based on the tight binding models, several structural features of graphene are considered as a necessity for Dirac allotropes, including the 2D dimensionality, the hexagonal symmetry, the bilattice honeycomb structure, the chemical equivalence, as well as the sp2 hybridization of the carbon atoms.4,13 Some of these commonly accepted guidelines are violated by the recent discoveries of alternative Dirac carbon allotropes. The T graphene as proposed by Liu et al., and although it maintains the bilattice feature, it does not have hexagonal symmetry.11 In another set of Dirac carbon allotropes, so called “graphynes”, the carbon atoms are not equivalent, and are in both sp and sp2 hybridization.9 Though the graphynes are strictly 2D, the T graphene is not, having a puckered plane. These recent progresses pointed out the possibility of nding new Dirac allotropes towards a wider scope of carbon lattices.14 There is an emergent request for a systematic design of the Dirac carbon allotropes in a large
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variety of 2D lattices, and a general guideline of constructing Dirac allotropes is greatly desirable. In this work, we systematically designed various Dirac allotropes in a wide range of 2D carbon lattices by employing a recently developed 2D structure searching technique15 based on the CALYPSO algorithm16,17 and rst-principles calculations. By examining the obtained low-energy carbon allotropes and picking up the ones with Dirac cones, we got three Dirac allotropes as typical examples. By investigating the structural features of one allotrope associated with Dirac points, we found unexpectedly a construction scheme that can systematically generate structures that are keen to have Dirac points. Their energy order can be well described in an Ising-like model, allowing us to design low-energy allotropes in large unit cells. To further illustrate the structural features of Dirac materials, two Dirac allotropes with special atomic conguration have been discussed. Finally, a general guideline on the association of Dirac cones with the atomic structure of 2D carbon allotropes has been expounded, which opens up the possibilities of controllable design of 2D carbon-based nano-electronic devices.
2.
Methods
The structure searching method has been implemented in the CALYPSO code,16,17 and its capability in predicting novel 2D structures has been demonstrated.15,18–20 Although the search is based on two dimensional space groups, the new module allows the relaxation of the atoms in the perpendicular direction, making it capable of exploring low dimension structures in real materials.15 The underlying rst-principles calculations were carried out using a Perdew–Burke–Ernzerhof (PBE)21 exchange– correlation density functional and the projector-augmentedwave (PAW)22 potentials as implemented in VASP.23 The PAW with 2s22p2 electrons as valence electrons was adopted. The kinetic-energy cutoff of 800 eV and a ne Monkhorst–Pack24 k-points meshes were chosen. The atomic relaxation was carried out until the change in the total energy per one unit cell was smaller than 0.01 meV. The dynamic stability of new structures has been examined by the phonon calculations and molecular dynamics. The phonon calculations were carried out by using a supercell approach as implemented in the PHONON code.25 The rst-principles molecular dynamics26 (FPMD) simulations adopt the constant temperature and volume (NVT) ensemble, and are carried out with time steps of 3 fs for a total simulation time of 3 ps at 500 K.
3.
Results and discussion
Selected from the CALYPSO structural searches, a number of structures that possess Dirac points are identied. Three representative allotropes have been shown in Fig. 1, named as S-graphene [Fig. 1(b)], D-graphene [Fig. 1(c)] and E-graphene [Fig. 1(d)], respectively. Some of the structures such as S- and D-graphenes are strictly 2D, whereas E-graphene is not. They are all found to be dynamically stable as proven by one of two widely accepted criteria: (i) there are no imaginary phonons in the calculated phonon spectra (see ESI Fig. S1†) and (ii) the lattices
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Fig. 1 Structures of (a) (2 1) graphene cell and selected low energy Dirac carbon allotropes as found by a CALYPSO structure search. The allotropes are named as (b) S-graphene, (c) D-graphene and (d) E-graphene, respectively.
remain very stable aer 3 ps FPMD simulations at heating up to 500 K (see ESI Fig. S2†). All the three structures are metastable compared with graphene, but their energies are in the same range as T graphene11 and three graphynes9 (see ESI Fig. S3†). At rst sight, these structures seem not to have any relation. However, by comparing the structural features of S-graphene and a (2 1) graphene cell, as well as the previously proposed T graphene11 and net W graphene,27 we found that they are closely related and can be constructed systematically from graphene by a series of reconstruction of the neighboring C–C bonds. Taking S-graphene as an example, as shown in Fig. 2(a), the S-graphene can be constructed by breaking one C–C bond in a (2 1) cell and shiing oppositely the two carbon atoms towards the nonbonding directions shown by arrows. In a similar manner, T graphene can be created by cutting and shiing two C–C bonds, as shown in Fig. 2(b). The net W is more complicated and can only be constructed from a (4 1) unit cell. Therefore, one can construct new 2D carbon allotropes by choosing a (n 1) graphene unit cell and applying a series of aforementioned cutand-paste operations. All these operations act on the parallel C–C bonds, there are only two states before and aer the cut-and-paste operation on one bond, which is somewhat like the “spin” status of the chain structure. It should be noted that the denition of “spin” is an articial geometry index. As shown in Fig. 2, there are 2n lines along the cell-repetition direction for (n 1) cells, and these 2n lines of carbon in the (n 1) cell can be marked by 2n “spins”. Depending on whether they have undergone the cut-and-paste operation or not, each line can be marked by “spin up” (remained intact, blue arrow) or “spin down” (undergone the operation, red arrow). Therefore, we are able to represent the constructed carbon allotropes by a chain of “spin”. For example, graphene itself can be marked as a chain of “up spins”, [[[[. The S- and the T-graphenes can be notated as Y[[[ and YY[[,
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Fig. 2 Deriving new allotropes from graphene. (a) S-graphene, (b) T graphene, (c) net W.
while the net W corresponds to YYYY[[[[. For these spin-like chains, the total energy of the carbon allotropes can be expressed by the interactions of the neighboring spins based on the Ising model:28,29 E ¼ E0
1 X 1 X 1 X J1 si siþ1 J2 si siþ2 J3 si siþ3 N N N i i i
The parameters J represent the inter-array interactions between the rst, the second and the third neighboring C–C arrays, and E is a reference energy. A positive J favors the “ferromagnetic (FM)” interactions, meaning the two neighboring “spins” tend to align to the same orientation; whereas, a negative J favors the antiferromagnetic (AFM) alignment of the “spins”. By tting the density functional theory (DFT) total energies of original, S-, T, net W and other allotropes, we found the J values are: J1 ¼ 0.49249 eV, J2 ¼ 0.15017 eV and J3 ¼ 0.05577 eV. As shown in Table 1, all energies of related carbon allotropes derived from the Ising-like model are in excellent agreement with those obtained from DFT calculations. The fact that J1 is much larger than J2 and J3 reveals that the energies of the new carbon structures are mainly determined by the relative positions of the carbon atoms in the nearest neighboring arrays. They are energetically most stable while they form an hexagonal lattice (FM spin chain) as in the original graphene. Interestingly, the J2 is negative, showing that the next nearest carbon arrays tend to deviate from the hexagonal lattice, which is a factor governing the structure deviation from original graphene. Based on our structural design strategy and Ising-like model, a series of related structures has been constructed and optimized by DFT calculations. Among such constructed allotropes, two series, Sn and Tn, are particularly interesting. Both of them exhibit rectangular symmetry instead of hexagonal symmetry. A Sn graphene consists of one ipped spin in a (n 1) unit cell; whereas a Tn graphene is formed by ip half of the neighboring spins. Both constructions minimize the neighboring AFM interactions and then can maintain a lower energy. It is worth noting that the earlier proposed T graphene and net W graphene are special cases of our generalized Tn series with n ¼ 2 and n ¼ 4, respectively. The energy of Sn graphene will converge towards that of graphene with the increasing n, then Sn graphene with large n can be considered as graphene with defects. Although in general the designed carbon allotropes become more and more stable with increasing n due to the fact of an increasing FM domain region, we do nd that the Tn series tends to be energetically more favorable over the Sn series. Due to dynamic stability and comparable formation energy to the other graphene allotropes, these new allotropes might be experimentally synthesized through chemical vapor deposition
Table 1 Summary of the configuration, expression and energy of the Ising-like model for graphene and predicted allotropes, the energy based on DFT calculations is listed as reference
Allotrope
Conguration
Expression
EIsing
EDFT
EError
Graphene S2 S3 S4 S5 S6 S7 S8 S9 T2 T4 (net W)
[[[[ Y[[[ Y[[[[[ Y[[[[[[[ Y[[[[[[[[[ Y[[[[[[[[[[[ Y[[[[[[[[[[[[[ Y[[[[[[[[[[[[[[[ Y[[[[[[[[[[[[[[[[[ YY[[ YYYY[[[[
2J1 2J2 2J3 0 2/3J1 2/3J2 2/3J3 J1 J2 J3 1.2J1 1.2J2 1.2J3 4/3J1 4/3J2 4/3J3 10/7J1 10/7J2 10/7J3 1.5J1 1.5J2 1.5J3 14/9J1 14/9J2 14/92J3 +2J2 J1 + J3
0.01708 0.81326 0.54787 0.41517 0.33555 0.28247 0.24456 0.21612 0.19401 0.51292 0.37654
0 0.79987 0.57466 0.43128 0.34719 0.29050 0.24964 0.21882 0.19476 0.51293 0.37655
0.01708 0.01339 0.02679 0.01611 0.01164 0.00803 0.00508 0.0027 0.00075 1 105 1 105
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on the suitable substrate30 or through electron irradiation on the graphene.31 By examining the electronic structures, we nd that Sn may possess Dirac-like dispersions in their band structures. For example, S-graphene exhibits four Dirac points in its rst Brillouin zone [Fig. 3(a) and (b)]. However, due to the symmetry relation, only two of them are independent, located on the lines from X to M (point A) and from G to Y (point B). ˚ and Both Dirac points are asymmetric, with slopes of +14 eVA ˚ around point A and slopes of +15 eVA ˚ and 18 eVA ˚ 21 eVA around point B. It is important to note that Dirac point A is slightly below the Fermi level, indicating that S-graphene has the self-doping character. Therefore, it contains electron carriers even without n-type doping or electron transfer from the side groups. Meanwhile, the electronic structure of Tn series is substantially different. In a strict 2D structure, these allotropes are metallic around the Fermi level.32 Some of the T series allotropes, such as T2 and T4 graphene may relax in the perpendicular direction and lead to a puckered layer with multiple unit cells. The corresponding folded band structure can contain a circle of continuous crossover points (see ESI Fig. S6†). Although each point of this corral is the meeting point of linear valence and conduction bands, they can be never described by the Dirac–Weyl equation and thus are not Dirac cones.
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Because of the close relation of their structures, the Dirac points of Sn graphenes are also related to those in graphene. In graphene, the origin of the Dirac point can be traced back to the degeneracy of the orbitals. The triply degenerated valence band maximum (VBM) states at the G point (vertical, le tilted and right tilted) reduce to two-fold degeneracy at the Dirac points (K points). The two corresponding wavefunctions are dimmerlike and they can be transformed to each other by a rotation of 60 degrees. The dimmer feature is kept for the wavefunctions at the Dirac points in S-graphene. One is very much like the vertical one in graphene except the difference in bond topology. The second one can be considered as the linear combination of the le and right tilted wavefunctions in graphene, which is a result of the change of symmetry. For Sn graphenes, we examined their electronic structures up to n ¼ 10, and found that only for those n ¼ 3m 1 and n ¼ 3m 2 (see ESI Fig. S7†), the corresponding allotropes possess Dirac points at the Fermi level. Dirac points are absent for n ¼ 3m. This magic number is quite similar to the Dirac point trend in graphene nanoribbons and is due to the preservation of the degeneracy of graphene states under deformation. In addition, by applying more cutand-paste operations in a (4 1) graphene cell, two new allotropes (A- and B-graphene in ESI Fig. S4 and S5†) with Dirac points at the Fermi level can also be obtained. Hence, with the increase of the original cell unit of graphene, more allotropes
Electronic structures of S-graphene. (a) Band structure and density of states, (b) Dirac points in 3D band structure and those projections in first Brillouin zone, (c) degenerate wavefunctions at the Dirac points of S-graphene.
Fig. 3
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with Dirac points can be reasonably obtained due to the topology correlations of atomic structure and wavefunctions. Besides the many Dirac allotropes that can be constructed from the above proposed method, our CALYPSO structure search method revealed that the Dirac points can exist beyond those structures, revealing that Dirac physics is a very rich phenomena in 2D carbon allotropes. In the two representative structures, D-graphene [Fig. 1(c)] consists of both sp2 and sp carbons, which can be constructed by inserting one pair of C/C (marked as red) into a graphene layer containing arrays of 5–7 ring defects. Despite the complex structure type, the energy of this allotrope is only 0.53 eV per atom above graphene, very close to the energy of T graphene. Different to all the other allotropes, E-graphene can no longer be viewed as a puckered layer since the shi of the atoms in the perpendicular direction is so large [Fig. 1(d)]. The structure of E-graphene can also be related to graphene. If the extra carbon is removed (marked red) and the other atoms shi to the center of the close-by hexagon, E-graphene can be transformed back to graphene. Because of the existence of the extra carbon atom in the graphene lattice, the atomic density is larger than that of graphene and all the above 2D allotropes. That is why the structure of E graphene is not 2D or puckered 2D. In order to accommodate the extra carbon atoms, some of the carbon atoms have to form four C–C s bonds, and therefore they must be in sp3 hybridization. This
Fig. 5 Electronic structures of E-graphene. (a) Band structure. (b) Density of states. (c) Dirac points in 3D band structure and those projections. (d) First Brillouin zone with red points designating Dirac points. (e) Linear dispersion relation in the vicinity of the Dirac points.
allotrope can also be viewed as an array of N ¼ 2 zigzag nanoribbons being connected by chains of sp3 carbon atoms. The energy of E-graphene is 1.28 eV per atom higher than that of graphene, showing that the mixture of the sp2 carbons and sp3 carbons in a quasi-2D structure is not energetically favored. Despite the complexity of their structures, both D- and E-graphenes possess Dirac points. As shown in Fig. 4, D-graphene exhibits one independent Dirac point at the boundary of the rst Brillouin zone with a distorted cone. Around the Dirac point, the conduction and valence bands show linear dispersion with the ˚ in direction from M to Y. The Dirac point slopes of 11 eVA locates slightly above the Fermi level, indicating the p-type selfdoping character of the allotrope. As shown in Fig. 5, the Dirac point of E-graphene is on the G–Y line and located exactly at the Fermi level. Along this direction, the band presents linear ˚ which is close to the value of graphene dispersion of 29 eVA, ˚ The Dirac cone also shows anisotropy perpendicular (34 eVA). to the line from G to Y. Therefore both D- and E-graphene should show strongly anisotropic transport properties.
Fig. 4 Electronic structures of D-graphene. (a) Band structure. (b) Density of states. (c) Dirac points in 3D band structure and those projections. (d) First Brillouin zone with red points designating Dirac points. (e) Linear dispersion relation in the vicinity of the Dirac points.
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4. Conclusion In summary, by a large-scale structure search for 2D carbon allotropes using the CALYPSO algorithm, the identied several
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families of structures possessing the Dirac cones in their electronic structures have been found. Originating from inherent relation of wavefunctions and atomic structures between graphene and these new allotropes, we then established a general route to construct a series of novel 2D carbon allotropes that are keen to have Dirac cones. The energy of these allotropes can be well described by an Ising-like model. Many such constructed allotropes possess linear energy dispersion and Dirac cones. Furthermore, we found the construction of allotropes by adding either sp or sp3 hybridized carbon atoms into graphene may also lead to Dirac structures. Our results show that neither of the graphene structural features, including the hexagonal symmetry and the bilattice backbone, nor the sp2 hybridization, is necessary for a Dirac dispersion of their carriers. Enough examples indicate the phenomenon existed the Dirac points in 2D materials is universal. This may be due to the nature of chemical bonding in thin-enough 2D materials. The existence of pz electrons could give these electrons the possibility of noscattering migration in 2D surface of real space, and appropriate potential eld controls the conductivity of 2D materials.
Acknowledgements The work was nancially supported by the National Natural Science Foundation of China (NSFC) (Grant no. 11274029, 11074017, 11025418, 91022029, 11274028, 51222212, and 51032002), the Beijing Nova Program (Grant no. 2008B10), the Ministry of Science & Technology of China (973 Project, no. 2011CB922200, 863 Project, no. 2012AA050704), the Importation and Development of High-Caliber Talents Project of Beijing Municipal Institutions (no. CIT&TCD201204037), the Key Programs of Beijing Plan of Science and Technology (D121100001812002), the Qualied Personnel Foundation of Taiyuan University of Technology (QPFT) (no. tyut-rc201333a), and the Basic Research Foundation of Beijing University of Technology.
References 1 A. S. Mayorov, R. V. Gorbachev, S. V. Morozov, L. Britnell, R. Jalil, L. A. Ponomarenko, P. Blake, K. S. Novoselov, K. Watanabe, T. Taniguchi and A. K. Geim, Nano Lett., 2011, 11, 2396–2399. 2 R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres and A. K. Geim, Science, 2008, 320, 1308. 3 A. K. Geim and K. S. Novoselov, Nat. Mater., 2007, 6, 183–191. 4 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos and A. A. Firsov, Nature, 2005, 438, 197–200. 5 K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science, 2004, 306, 666–669. 6 L. Britnell, R. V. Gorbachev, R. Jalil, B. D. Belle, F. Schedin, A. Mishchenko, T. Georgiou, M. I. Katsnelson, L. Eaves, S. V. Morozov, N. M. R. Peres, J. Leist, A. K. Geim,
1118 | Nanoscale, 2014, 6, 1113–1118
Paper
7
8 9 10 11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26
27 28 29 30 31 32
K. S. Novoselov and L. A. Ponomarenko, Science, 2012, 335, 947–950. Y.-M. Lin, C. Dimitrakopoulos, K. A. Jenkins, D. B. Farmer, H.-Y. Chiu, A. Grill and P. Avouris, Science, 2010, 327, 662. F. Xia, T. Mueller, Y.-m. Lin, A. Valdes-Garcia and P. Avouris, Nat. Nanotechnol., 2009, 4, 839–843. D. Malko, C. Neiss, F. Vi˜ nes and A. G¨ orling, Phys. Rev. Lett., 2012, 108, 086804. S.-M. Choi, S.-H. Jhi and Y.-W. Son, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 81, 081407. Y. Liu, G. Wang, Q. Huang, L. Guo and X. Chen, Phys. Rev. Lett., 2012, 108, 225505. S. Cahangirov, M. Topsakal, E. Akt¨ urk, H. ¸ Sahin and S. Ciraci, Phys. Rev. Lett., 2009, 102, 236804. A. N. Enyashin and A. L. Ivanovskii, Phys. Status Solidi B, 2011, 248, 1879–1883. H. Huang, W. Duan and Z. Liu, New J. Phys., 2013, 15, 023004. Y. Wang, M. Miao, J. Lv, L. Zhu, K. Yin, H. Liu and Y. Ma, J. Chem. Phys., 2012, 137, 224108. Y. Wang, J. Lv, L. Zhu and Y. Ma, Phys. Rev. B: Condens. Matter Mater. Phys., 2010, 82, 094116. Y. Wang, J. Lv, L. Zhu and Y. Ma, Comput. Phys. Commun., 2012, 183, 2063–2070. H. Xiang, B. Huang, Z. Li, S. H. Wei, J. Yang and X. Gong, Phys. Rev. X, 2012, 2, 011003. X. Wu, J. Dai, Y. Zhao, Z. Zhuo, J. Yang and X. C. Zeng, ACS Nano, 2012, 6, 7443–7453. X. Luo, J. Yang, H. Liu, X. Wu, Y. Wang, Y. Ma, S. H. Wei, X. Gong and H. Xiang, J. Am. Chem. Soc., 2011, 133, 16285– 16290. J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865. G. Kresse and D. Joubert, Phys. Rev. B: Condens. Matter Mater. Phys., 1999, 59, 1758. G. Kresse and J. Furthm¨ uller, Comput. Mater. Sci., 1996, 6, 15–50. H. J. Monkhorst and J. D. Pack, Phys. Rev. B: Solid State, 1976, 13, 5188–5192. A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. Matter Mater. Phys., 2008, 78, 134106. P. Pavone, K. Karch, O. Schutt, W. Windl, D. Strauch, P. Giannozzi and S. Baroni, Phys. Rev. B: Condens. Matter, 1993, 48, 3156. X.-Q. Wang, H.-D. Li and J.-T. Wang, Phys. Chem. Chem. Phys., 2013, 15, 2024–2030. S. Limpijumnong and W. R. Lambrecht, Phys. Rev. B: Condens. Matter, 1998, 57, 12017–12022. M. Miao, S. Limpijumnong and W. R. Lambrecht, Appl. Phys. Lett., 2001, 79, 4360–4362. J. Lahiri, Y. Lin, P. Bozkurt, I. I. Oleynik and M. Batzill, Nat. Nanotechnol., 2010, 5, 326–329. J. Kotakoski, A. V. Krasheninnikov, U. Kaiser and J. C. Meyer, Phys. Rev. Lett., 2011, 106, 105505. H. Huang, Y. Li, Z. Liu, J. Wu and W. Duan, Phys. Rev. Lett., 2013, 110, 029603.
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