732 | Phys. Chem. Chem. Phys., 2018, 20, 732--737 This journal is© the Owner Societies 2018

Cite this:Phys.Chem.Chem.Phys.,

2018, 20, 732

Square transition-metal carbides MC6 (M = Mo, W)as stable two-dimensional Dirac cone materials†

Peng-Fei Liu, ab Yang Wu,ab Tao Bo,ab Ling Hou,ab Juping Xu,ab Hui-jie Zhangab

and Bao-Tian Wang *ab

Searching for new two-dimensional (2D) Dirac cone materials has

been popular since the exfoliation of graphene. Herein, based on

density functional theory, we predict a novel family of 2D Dirac

cone materials in square transition-metal carbides MC6 (M = Mo, W)

which show inherent stability confirmed by phonon spectrum analy-

sis and ab initio molecular dynamics calculations. The Dirac point,

located exactly at the Fermi level, mainly arises from the hybridization

of M-dz2,x2�y2 and C-pz orbitals which gives rise to an ultrahigh Fermi

velocity comparable to that of graphene. Moreover, strong spin–orbit

coupling related to M-d electrons can generate large band gaps of

35 and 89 meV for MoC6 and WC6 monolayers, respectively, which

allows MC6 materials to be operable at room temperature (26 meV),

as candidates for nanoelectronics in the upcoming post-silicon era.

The conceived novel stable metal–carbon framework materials

provide a platform for designing 2D Dirac cone materials.

Introductions

Since the experimental observation of monolayer graphene,1

two-dimensional (2D) Dirac cone materials, characterized bylinear band dispersion at the Fermi level, have attracted intenseattention due to their unique physical properties and potentialapplications in nanoscale devices.2,3 In particular, the Diracfermions in graphene give rise to an ultra-high carrier mobility4,5

and half-integer/fractional/fractal quantum Hall effects6–9 whichoffer a broad perspective for the development of dissipationlessapplications.10 Meanwhile, previous reports have theoreticallyrealized the quantum anomalous Hall effect by proximity cou-pling graphene to an antiferromagnetic insulator (BiFeO3).11

These exotic behaviors in graphene have motivated the contin-uous search for novel 2D Dirac materials.12 This has yielded acrowd of 2D carbon allotropes discovered experimentally ortheoretically, including graphyne (e.g. a-, a2-, b-, d-, 6,6,12-,

4,12,2- and 4,12,4-graphyne),13–16 ph-graphene,17 x-graphene (x = S,D, and E)18 and phagraphene,19 which also show linearly dispersedband structures with intrinsic massless fermions. Meanwhile,some other 2D systems, such as silicene,20 germanene,20 staneneallotropes,21–23 boron allotropes,24 organometallic crystals,25,26

and transition metal compounds,27–30 have been verified to beDirac cone materials.

Among the abundant candidates, 2D transition metalmaterials stand out from the competition due to their strongspin–orbit coupling (SOC) effects which usually open up largeband gaps suitable for realistic applications in the next genera-tion of faster and smaller field-effect transistors.27–30 Consider-ing the importance of carbon in modern device applications,transition metal carbides are a long sought goal. MXenes,31 onekind of 2D transition-metal–carbon material, have already beensynthesized experimentally and possess potential applicationsin supercapacitor electrodes,32 energy storage,33 metal ionbatteries34 and superconducting crystals.35 In the meantime,theoretical computations have revealed several generations of 2Dtransition metal–carbon materials to be intrinsic topologicalinsulators (TIs) including triangular MXenes with d-band topo-logical order,36,37 hexagonal organometallic TIs with benzenemolecular building blocks25,26 and rectangular transition metalcarbides with quasiplanar tetracoordinate C atoms.29,30 Theseresults greatly enrich the physics of 2D transition metal carbidesand pave new ways for designing C-based materials for realisticapplications. However, rare Dirac materials have been found intransition metal–carbon compounds and only a few rectangular2D phases29 and organometallic lattices25,26,38 have been reportedwithout experimental synthesis. Therefore, exploring new 2DDirac cone materials in C-based transition metal compoundswith sizable band gaps is still a priority to guide efforts inexperiment and theory.

In this work, we have designed a novel family of 2D materialsof square transition-metal carbides MC6 (M = Mo, W) which arezero-gap semimetals with Dirac cones at off-symmetry reciprocalsites in the absence of SOC. We demonstrated that the distinctDirac cones at the Fermi level are mainly derived from

a Institute of High Energy Physics, Chinese Academy of Sciences (CAS),

Beijing 100049, China. E-mail: wangbt@ihep.ac.cnb Dongguan Neutron Science Center, Dongguan 523803, China

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp07466b

Received 5th November 2017,Accepted 4th December 2017

DOI: 10.1039/c7cp07466b

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M-dz2,x2�y2 and C-pz orbitals. The SOC effects open large bandgaps of 35 and 89 meV for MoC6 and WC6, respectively. Ourpredicted square transition metal–carbon monolayers withhigh Fermi velocity could provide a new choice for realizingDirac materials beyond hexagonal lattice symmetry.

Methods

First-principles calculations were performed within the frameworkof the Perdew–Burke–Ernzerhof (PBE)39 generalized gradientapproximation (GGA),40,41 as implemented in the Vienna ab initiosimulation package (VASP).42 The property calculations were per-formed with plane-wave basis sets of 500 eV on the 11 � 11 � 1Monkhorst–Pack k-point mesh. A slab model with a vacuumlayer larger than 15 Å was employed and all geometry structureswere fully relaxed until the residual forces on each atom wereless than 0.01 eV Å�1. The dynamic stabilities of the systemswere checked by phonon calculations via the force-constantmethod43 as implemented in the PHONOPY code.44 Thermalstability was also examined with ab initio molecular dynamics(AIMD) using the Nose algorithm45 in the NVT ensemble at hightemperature. Based on the projected densities of states (PDOS)calculated within density functional theory (DFT), we haveconstructed the tight-binding Hamiltonian on the basis of themaximally localized Wannier functions (MLWFs)46,47 and thencomputed the topological Z2 invariant by tracing the Wanniercharge centers using the non-Abelian Berry connection,48,49 asprovided by the software package WannierTools.50

Results and discussion

The novel 2D transition-metal carbides MC6 (M = Mo, W)crystallize in the tetragonal space group P4/mbm (no. 127) witha = b = 8.546–8.540 Å from Mo to W (Table 1). As shown inFig. 1, the MC6 monolayers are triple atomic layer structures withMo/W atomic layers sandwiched between two C atomic layers.All of the transition-metal atoms are coordinated in a slightlydistorted [MC6] octahedron with the six nearest neighboring Catoms for MC6 monolayers. There are one independent M atom(M, site symmetry 4h) and two crystallographically unique Catoms (C1, site symmetry 16l; C2, site symmetry 8k) with a

chemical formula of MC6 in one unit. In order to distinguishour structures from the reported hexagonal MC6 (M = Mo, W),51

we denote square MC6 as s-MC6 and hexagonal MC6 as h-MC6

(Fig. S1, ESI†). The length of the C–C bonds in s-MC6 rangefrom 1.410 to 1.420 Å. These results, analogous to those ofh-MC6 (1.410 Å),51 are quite in accordance with the length ofC–C bonds (1.420 Å) in graphene. The similar values of the C–Cbond length indicate sp2 hybridization between three adjacentC atoms that favors a strong extended p bond. The length of theM–C bonds is calculated to be 2.068–2.129 Å and 2.084–2.113 Åfor s-MoC6 and s-WC6 monolayers, respectively. These resultsare close to those in h-MC6

51 and TM2CO2 (M = Mo, W),36,37,52

in which the transition-metal atoms bonding with C atomshave been proven to be stable.

To evaluate the stability of the hypothetical structures, wefirst computed the formation energy with respect to the bulkmetals and graphene, defined as below:

Ef ¼E2D � nMEM � nCEC

nM þ nC;

where E2D is the total energy of the monolayers, and EM and EC

are the energies of bulk M metals and graphene per atom,respectively. Our results show that s-MoC6 and s-WC6 mono-layers have formation energies of 1.38 and 1.33 eV per atom,respectively, which are slightly larger than those of h-MoC6

(1.30 eV per atom) and h-WC6 (1.25 eV per atom) monolayers.The very similar formation energies suggest that s-MC6 andh-MC6 monolayers51 are equally stable. The positive values meanthat the experimental synthesis of MC6 monolayers is anendothermic process and can be achieved by epitaxial growthon an appropriate semiconductor surface.53

The phonon spectrum and projected phonon density ofstates (PhDOS), shown in Fig. 1b and c, are calculated to focuson the examination of the dynamic stability for the s-MC6

monolayers. All of the branches over the entire Brillouin zoneare appreciably positive phonon vibration modes, establishingthat the s-MC6 monolayers are at local minimums of energy.From the PhDOS, it can be seen that the phonon mode mainlycontains the M atom vibrations in the low-energy region

Table 1 The calculated lattice parameters of lattice constant, bond lengthof M–C and C–C bonds, formation energy (Ef) and band gap with andwithout SOC based on PBE for MC6 sheets

Comp. la (Å) lM–C (Å) lC–C (Å)Ef (eV peratom) PBE

PBE + SOC(meV)

h-MoC6 4.382/4.385a

2.103/2.103a

1.408/1.408a

1.30 Semimetal 69a

s-MoC6 8.546 2.068,2.129

1.414,1.410

1.38 Semimetal 35

h-WC6 4.380/4.382a

2.100/2.101a

1.411/1.411a

1.25 Semimetal 226a

s-WC6 8.540 2.084,2.113

1.421,1.412

1.33 Semimetal 89

a Data taken from ref. 51.

Fig. 1 (a) The side and top view of s-MC6. The phonon band dispersionsand projected phonon density of states of (b) s-MoC6 and (c) s-WC6

monolayers.

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(below 5 THz). In the intermediate-frequency region from 5 to17 THz, the interaction between M and C atom vibrations becomesdominant. This feature is about the same as other sandwichstructures of transition metal compounds.28 The high-energyregions of 28–32 THz and 43–47 THz mainly consist of the Catom vibrations without Mo vibrations, suggesting that thecontributions mainly stem from strong p-conjugated bonds.Notably, the highest frequencies in the phonon spectra of thes-MoC6 and s-WC6 sheets can reach up to 46.42 and 46.62 THz,respectively, which are slightly less than those in h-MC6 (Mo,47.08 THz; W, 47.33 THz),51 indicating almost a similar stabilityfor the square and hexagonal structures. Compared to theconventional sandwich structures of transition metal compounds,the values of the highest phonon vibration in the s-MC6 mono-layers are obviously greater than those in h-MoN2 (20–25 THz)54

and MoS2 (13.25–13.85)55 monolayers.Additionally, we checked the mechanical stability of s-MC6

monolayers by calculating the linear elastic constants. The 2Dlinear elastic constants in units of N m�1 are calculated tobe C11 = 84.38, C12 = 69.57 and C66 = 34.44 for the s-MoC6

monolayer, and C11 = 91.12, C12 = 73.88 and C66 = 38.18 forthe s-WC6 monolayer. For the square lattice, the Born criteriafor structural stability is defined as follows:56,57

C11 4 |C12|, C66 4 0

It is obvious that the elastic constants of s-MC6 monolayersfulfil the condition of the Born criteria. So, the s-MC6 sheetsare mechanically stable square lattices. Correspondingly, thein-plane axial Young’s modulus, defined as E = (C11

2 � C122)/

C11, is calculated to be 27.02 and 31.22 N m�1 for s-MoC6 ands-WC6, respectively, which are much lower than that of graphene(340 N m�1),58 penta-graphene (263.8 N m�1)57 and the MoS2

monolayer (123 N m�1).59 Such values indicate the relativelygood elastic properties and demonstrate the feasibility of s-MC6

monolayers being rolled into nanotubes.Finally, we examine the thermal stability of the s-MC6

monolayers by carrying out AIMD simulations (Fig. S2, ESI†).After heating for 5 ps at a temperature of 500 K, the s-MC6

structures do not suffer drastic structural distortion or trans-formation and can be optimized back to the initial structuresthrough a relaxation calculation. However, the snapshots of thes-MC6 monolayers at 1000 K show a serious distortion of thegeometry. The total potential energies of the s-MC6 monolayersonly fluctuate around a certain constant magnitude for AIMDrunning at 500 K (Fig. S3, ESI†). Therefore, it can be concludedthat both of the estimated melting temperatures of the twomonolayers are between 500 and 1000 K.

To study the electronic properties of the s-MC6 monolayers,we calculated the orbital-resolved band structures and depictthem in Fig. 2. The monolayers are zero-band-gap semimetalswith the valence band (VB) and conduction band (CB) touchingeach other at off-symmetry points of [0.21, 0.21] and [0.20, 0.20]from the R to the G point for s-MoC6 and s-WC6, respectively, in theabsence of SOC (Fig. 2a and b). It can be seen that the linear banddispersion appears in the range of �0.1 to 0.2 eV, which causesDirac cones around the Fermi level. The three-dimensional (3D)

plot for the Dirac cones is illustrated in Fig. 3, which presentsa notable circular-cone character around the Fermi level. Asshown in Fig. S4 (ESI†), the Dirac band structures of the s-MC6

monolayers were checked at the GGA+U level with U = 2.81 eV,J = 0.59 eV (U = 2.67 eV, J = 0.54 eV), and this was carried out inorder to consider the Coulomb interaction effect for the Mo (W)atom, for which the Hubbard U values have been tested andused in previous theoretical results.60,61 The Fermi velocity (nf)values can be evaluated via the slope of the bands by theexpression nf = qE/(h� qk).28 The nf values of the monolayers arecalculated to be 2.84 � 105 m s�1 (�2.61 � 105 m s�1) and3.23 � 105 m s�1 (�2.43 � 105 m s�1) in the kR direction fors-MoC6 and s-WC6, respectively, which are comparable to those ofgraphene (8.22 � 105 m s�1)13 and silicene (5.31 � 105 m s�1),62

and less than those of B-based 2D materials including borophane(3.0 � 106 m s�1, �1.7 � 106 m s�1)63 and Pmmn borophene(0.46 � 106 m s�1, �1.16 � 106 m s�1).24 To benchmark ourcomputational results, the Fermi velocity, calculated for grapheneat the same theoretical level, is 8.43 � 105 m s�1 and �8.42 �105 m s�1 along the line perpendicular to the G–K direction,which is in good agreement with the previous results.13

Fig. 2 The calculated band structures of (a and c) the s-MoC6 and (b and d)the s-WC6 monolayers, without and with SOC.

Fig. 3 The 3D plot of the low-energy band structures for (a) s-MoC6 and(b) s-WC6 sheets without SOC.

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When taking into account the SOC effect, the Dirac bandstructure is well retained except for an enlarged band gap of 35 meVat the Dirac point for the s-MoC6 monolayer (Fig. 2c). In the case ofthe s-WC6 monolayer, the degenerate states at the touching point arelifted out with an energy gap of 89 meV, which provides the bandstructures around the Fermi level from the linear crossed bands tothe quadratic bands. The band gaps, recalculated at the GGA+U levelwith SOC, are 37 meV and 87 meV for the s-MoC6 and s-WC6

monolayers, respectively (Fig. S4, ESI†). These values are much largerthan those of graphene, silicene and germanene which have smallgaps in the order of meV.3,64 For both of the s-MoC6 and s-WC6

monolayers in the vicinity of the Fermi level, the valence bandmaximum (VBM) and conduction band minimum (CBM) are mainlyoriginated from the Mo-dz2,x2�y2 orbitals as well as having indispen-sable contributions from the C-pz orbital by the projection analysis.

In order to gain more insight into the Dirac behavior of thes-MC6 sheets, the electron localization function (ELF)65 and thepartial charge densities at the Dirac points were calculated(Fig. 4). The ELF for s-MoC6 shows that there are high electronlocalization regions between two adjacent carbon atoms, indi-cating obvious p electrons in carbon rings in analogy to theh-MC6 monolayers.51 In the real space, the partial chargedensities of the CBM (Fig. 4c) and VBM (Fig. 4d) clearly confirmthat the cooperation of Mo-dz2,x2�y2 and C-pz orbitals is respon-sible for the formation of the Dirac cones. The same character-istics also appear in the square so-MoS2 monolayer,27 in whichthe four pear-shaped lobes of the in-plane Mo-dx2�y2 orbitals arecompatible with the square lattice symmetry which leads to theformation of the Dirac cones. This is different from conven-tional graphene which does not easily achieve Dirac fermionsin a square lattice with full symmetry.66,67

It should be mentioned that previous studies have revealedan intrinsic quantum spin Hall (QSH) insulating feature ingraphene,10 silicene,68 germanene,68 and stanene allotropes,21–23

with Dirac states. With respect to the s-MC6 monolayers withinversion symmetry in the structures, we also checked whethersuch a non-trivial topological state exists in the sheets via calcula-tion of the well-defined Z2 topological invariant established usingthe scheme proposed by Fu and Kane.69,70 Herein, the topologicalinvariant Z2 = 1 characterizes a non-trivial topological phase, whileZ2 = 0 indicates a trivial topological phase. The detailed paritiesof the occupied bands for the s-MC6 monolayers are listed inFig. S5 (ESI†). Our computational results yield a trivial topo-logical invariant Z2 = 0, indicating that the monolayers are justtrivial nanostructures without the presence of the QSH effect.The results have also been verified by the analysis of helicalgapless edge states using the WannierTools code (Fig. S6,ESI†),48–50 which is different from other QSH insulators.71,72

To examine the stability and robustness of the Dirac conesfor device applications, external biaxial strain was applied tothe s-MC6 monolayers by changing the lattices as e = (a � a0)/a0,where a(a0) is the strained (equilibrium) lattice constant.21 It isfound that the Dirac cones are robust to strain and can be wellpreserved within the strain range of �10% to 7% and �10% to9% for the s-MoC6 and s-WC6 monolayers, respectively (Fig. 5).For the s-MoC6 monolayer, when the stretching strain isincreased up to 6%, the Dirac cones are slightly tilted at theboundary between electron and hole pockets. With the stretch-ing strain increasing over 8%, the Dirac cones disappear withthe parabolic VB and CB touching at the G point in the absenceof SOC. Although applying SOC can open up the gaps at the Gpoint, the Fermi level still passes through the conduction band,leading s-MoC6 into a metal phase (Fig. S7, ESI†). For the s-WC6

sheet, there is always a band gap in a large strain range from�10% to 10% (Fig. S7, ESI†). When the stretching strainreaches up to 8%, strongly tilted Dirac cones emerge with anew type of Dirac fermion named type-II Dirac fermions, incontrast to the type-I Dirac fermions in graphene.2,3 Whenapplying a stretching strain up to 10%, the VB and CB touch

Fig. 4 (a) The electron localization function (ELF) of the s-MoC6 mono-layer with an isovalue of 0.003 a.u. (b) ELF maps sliced perpendicular to theout-of-plane direction for the s-MoC6 monolayer. (c) VBM and (d) CBMcharge density contours at the Dirac point of the s-MoC6 monolayer.

Fig. 5 The selected low-energy band structures for (a) s-MoC6 and(b) s-WC6 monolayers in the strain range of �10% to 10% without SOC.

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at the G point and the Dirac cones vanish at off-symmetryreciprocal sites (Fig. 5b). Such robust topology against externalbiaxial strain makes s-MC6 easier for experimental realizationand characterization.

Experimentally, high-quality graphene58 and MoS273 have been

successfully grown on substrates in their intrinsic strain limit(B15% for graphene and B11% for MoS2) without substantiallydamaging their crystal structures. Thus, it is expected that the s-MC6

monolayers can also be fabricated or transferred on the substrate.SiO2, as a common large-gap insulator, has been successfully used asthe substrate to grow high-quality polycrystalline MoS2 monolayersor organic molecules.74,75 Therefore, we selected T-silica (3 � 3:8.387 Å)76,77 as the substrate to support the s-MoC6 monolayer as anexample. To correctly describe the van der Waals interaction, areliably dispersion-corrected DFT method (DFT-D2) has beenused.78 The optimized distance (d) between adjacent layers wasfound to be 3.037 Å. Fig. S8 (ESI†) displays the optimized crystalstructure and calculated band structure for the MoC6@SiO2

film with intrinsic Dirac cones, which demonstrates the robust-ness of the Dirac structures.

Conclusions

In summary, we predict s-MC6 (M = Mo, W) as a new family of2D materials with intrinsic Dirac states based on first-principlescalculations. The hybridization of M-dz2,x2�y2 and C-pz bandsendows sheets with an ultrahigh Fermi velocity comparable tothat of graphene. Owing to the strong SOC of transition metalatoms (Mo and W), sizable band gaps of the s-MC6 monolayers arecalculated to be 35 and 89 meV for s-MoC6 and s-WC6, respectively.Our results demonstrate that the Dirac states are robust againstapplied external biaxial strain. These interesting results would notonly stimulate further efforts on transition metal carbides but alsoprovide a new pathway to engineer 2D materials.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The support of the Supercomputer Centre of the China Spalla-tion Neutron Source (CSNS) is acknowledged. We acknowledgefruitful discussion with Liujiang Zhou. We would like to thankDr Quansheng Wu at ETH Zurich for help with the softwarepackage WannierTools.

References

1 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.

2 A. K. Geim and K. S. Novoselov, Nat. Mater., 2007, 6,183–191.

3 A. C. Neto, F. Guinea, N. M. Peres, K. S. Novoselov andA. K. Geim, Rev. Mod. Phys., 2009, 81, 109.

4 K. I. Bolotin, K. Sikes, Z. Jiang, M. Klima, G. Fudenberg,J. Hone, P. Kim and H. Stormer, Solid State Commun., 2008,146, 351–355.

5 E. Hwang and S. D. Sarma, Phys. Rev. B: Condens. MatterMater. Phys., 2008, 77, 115449.

6 K. I. Bolotin, F. Ghahari, M. D. Shulman, H. L. Stormer andP. Kim, Nature, 2009, 462, 196–199.

7 Y. Zhang, Y. W. Tan, H. L. Stormer and P. Kim, Nature, 2005,438, 201.

8 X. Du, I. Skachko, F. Duerr, A. Luican and E. Y. Andrei,Nature, 2009, 462, 192–195.

9 C. Dean, L. Wang, P. Maher, C. Forsythe, F. Ghahari, Y. Gao,J. Katoch, M. Ishigami, P. Moon and M. Koshino, et al.,Nature, 2013, 497, 598.

10 C. L. Kane and E. J. Mele, Phys. Rev. Lett., 2005, 95, 226801.11 Z. Qiao, W. Ren, H. Chen, L. Bellaiche, Z. Zhang, A. H.

MacDonald and Q. Niu, Phys. Rev. Lett., 2014, 112, 116404.12 J. Wang, S. Deng, Z. Liu and Z. Liu, Natl. Sci. Rev., 2015, 2,

22–39.13 D. Malko, C. Neiss, F. Vines and A. Gorling, Phys. Rev. Lett.,

2012, 108, 086804.14 N. V. R. Nulakani and V. Subramanian, J. Phys. Chem. C,

2016, 120, 15153–15161.15 M. Zhao, W. Dong and A. Wang, Sci. Rep., 2013, 3, 3532.16 L. Zhang, Z. Wang, Z. M. Wang, S. Du, H.-J. Gao and F. Liu,

J. Phys. Chem. Lett., 2015, 6, 2959–2962.17 X. Zhang, L. Wei, J. Tan and M. Zhao, Carbon, 2016, 105,

323–329.18 L.-C. Xu, R.-Z. Wang, M.-S. Miao, X.-L. Wei, Y.-P. Chen,

H. Yan, W.-M. Lau, L.-M. Liu and Y.-M. Ma, Nanoscale, 2014,6, 1113–1118.

19 Z. Wang, X.-F. Zhou, X. Zhang, Q. Zhu, H. Dong, M. Zhaoand A. R. Oganov, Nano Lett., 2015, 15, 6182–6186.

20 S. Cahangirov, M. Topsakal, E. Akturk, H. Sahin andS. Ciraci, Phys. Rev. Lett., 2009, 102, 236804.

21 H. Zhao, C. Zhang, W. Ji, R. Zhang, S. Li, S. Yan, B. Zhang,P. Li and P. Wang, Sci. Rep., 2016, 6, 20152.

22 Y.-P. Wang, W.-X. Ji, C.-W. Zhang, P. Li, F. Li, P.-J. Wang,S.-S. Li and S.-S. Yan, Appl. Phys. Lett., 2016, 108, 073104.

23 R.-W. Zhang, C.-W. Zhang, W.-X. Ji, S.-S. Li, S.-J. Hu,S.-S. Yan, P. Li, P.-J. Wang and F. Li, New J. Phys., 2015,17, 083036.

24 X.-F. Zhou, X. Dong, A. R. Oganov, Q. Zhu, Y. Tian andH.-T. Wang, Phys. Rev. Lett., 2014, 112, 085502.

25 Z. Wang, Z. Liu and F. Liu, Nat. Commun., 2013, 4, 1471.26 Z. Wang, Z. Liu and F. Liu, Phys. Rev. Lett., 2013, 110, 196801.27 W. Li, M. Guo, G. Zhang and Y.-W. Zhang, Phys. Rev. B:

Condens. Matter Mater. Phys., 2014, 89, 205402.28 P.-F. Liu, L. Zhou, T. Frauenheim and L.-M. Wu, Nanoscale,

2017, 9, 1007–1013.29 L. Zhou, B. Shao, W. Shi, Y. Sun, C. Felser, B. Yan and

T. Frauenheim, 2D Mater., 2016, 3, 035022.30 L. Zhou, W. Shi, Y. Sun, B. Shao, C. Felser, B. Yan and

T. Frauenheim, 2D Mater., 2016, 3, 035018.31 M. Naguib, V. N. Mochalin, M. W. Barsoum and Y. Gogotsi,

Adv. Mater., 2014, 26, 992–1005.

PCCP Communication

Publ

ishe

d on

04

Dec

embe

r 20

17. D

ownl

oade

d by

Ins

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e of

Hig

h E

nerg

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s, C

AS

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This journal is© the Owner Societies 2018 Phys. Chem. Chem. Phys., 2018, 20, 732--737 | 737

32 M. R. Lukatskaya, O. Mashtalir, C. E. Ren, Y. Dall’Agnese,P. Rozier, P. L. Taberna, M. Naguib, P. Simon, M. W.Barsoum and Y. Gogotsi, Science, 2013, 341, 1502–1505.

33 O. Mashtalir, M. R. Lukatskaya, M.-Q. Zhao, M. W. Barsoumand Y. Gogotsi, Adv. Mater., 2015, 27, 3501–3506.

34 Q. Tang, Z. Zhou and P. Shen, J. Am. Chem. Soc., 2012, 134,16909–16916.

35 C. Xu, L. Wang, Z. Liu, L. Chen, J. Guo, N. Kang, X.-L. Ma,H.-M. Cheng and W. Ren, Nat. Mater., 2015, 14, 1135.

36 C. Si, K.-H. Jin, J. Zhou, Z. Sun and F. Liu, Nano Lett., 2016,16, 6584–6591.

37 M. Khazaei, A. Ranjbar, M. Arai and S. Yunoki, Phys. Rev. B,2016, 94, 125152.

38 L. Zhang, Z. Wang, B. Huang, B. Cui, Z. Wang, S. Du,H.-J. Gao and F. Liu, Nano Lett., 2016, 16, 2072–2075.

39 G. Kresse and D. Joubert, Phys. Rev. B: Condens. MatterMater. Phys., 1999, 59, 1758.

40 P. E. Blochl, Phys. Rev. B: Condens. Matter Mater. Phys., 1994,50, 17953.

41 P. E. Blochl, O. Jepsen and O. K. Andersen, Phys. Rev. B:Condens. Matter Mater. Phys., 1994, 49, 16223.

42 G. Kresse and J. Furthmuller, Phys. Rev. B: Condens. MatterMater. Phys., 1996, 54, 11169.

43 S. Baroni, S. De Gironcoli, A. Dal Corso and P. Giannozzi,Rev. Mod. Phys., 2001, 73, 515.

44 A. Togo, F. Oba and I. Tanaka, Phys. Rev. B: Condens. MatterMater. Phys., 2008, 78, 134106.

45 S. Nose, J. Chem. Phys., 1984, 81, 511–519.46 A. A. Mostofi, J. R. Yates, Y.-S. Lee, I. Souza, D. Vanderbilt and

N. Marzari, Comput. Phys. Commun., 2008, 178, 685–699.47 A. A. Mostofi, J. R. Yates, G. Pizzi, Y.-S. Lee, I. Souza,

D. Vanderbilt and N. Marzari, Comput. Phys. Commun.,2014, 185, 2309–2310.

48 R. Yu, X. L. Qi, A. Bernevig, Z. Fang and X. Dai, Phys. Rev. B:Condens. Matter Mater. Phys., 2011, 84, 075119.

49 A. A. Soluyanov and D. Vanderbilt, Phys. Rev. B: Condens.Matter Mater. Phys., 2011, 83, 035108.

50 Q. Wu, S. Zhang, H.-F. Song, M. Troyer and A. A. Soluyanov,arXiv preprint arXiv:1703.07789, 2017.

51 X. Li, Y. Dai, Y. Ma, Q. Sun, W. Wei and B. Huang, Carbon,2016, 109, 788–794.

52 C. Si, J. You, W. Shi, J. Zhou and Z. Sun, J. Mater. Chem. C,2016, 4, 11524–11529.

53 M. Zhou, W. Ming, Z. Liu, Z. Wang, P. Li and F. Liu, Proc.Natl. Acad. Sci. U. S. A., 2014, 111, 14378–14381.

54 Y. Wang and Y. Ding, J. Mater. Chem. C, 2016, 4, 7485–7493.55 P.-F. Liu, L. Zhou, T. Frauenheim and L.-M. Wu, Nanoscale,

2016, 8, 4915–4921.

56 F. Mouhat and F.-X. Coudert, Phys. Rev. B: Condens. MatterMater. Phys., 2014, 90, 224104.

57 S. Zhang, J. Zhou, Q. Wang, X. Chen, Y. Kawazoe andP. Jena, Proc. Natl. Acad. Sci. U. S. A., 2015, 112, 2372–2377.

58 C. Lee, X. Wei, J. W. Kysar and J. Hone, Science, 2008, 321,385–388.

59 Q. Yue, J. Kang, Z. Shao, X. Zhang, S. Chang, G. Wang,S. Qin and J. Li, Phys. Lett. A, 2012, 376, 1166–1170.

60 M. V. Peskov, X.-H. Miao, D. Heryadi, J. Eppinger andU. Schwingenschlæzsgl, J. Mater. Chem. C, 2013, 117,5462–5469.

61 I. V. Solovyev, P. H. Dederichs and V. I. Anisimov, Phys. Rev.B: Condens. Matter Mater. Phys., 1994, 50, 16861–16871.

62 N. Drummond, V. Zolyomi and V. Fal’Ko, Phys. Rev. B:Condens. Matter Mater. Phys., 2012, 85, 075423.

63 L.-C. Xu, A. Du and L. Kou, Phys. Chem. Chem. Phys., 2016,18, 27284–27289.

64 C.-C. Liu, W. Feng and Y. Yao, Phys. Rev. Lett., 2011,107, 076802.

65 A. Savin, R. Nesper, S. Wengert and T. F. Fassler, Angew.Chem., Int. Ed., 1997, 36, 1808–1832.

66 Z. Liu, J. Wang and J. Li, Phys. Chem. Chem. Phys., 2013, 15,18855–18862.

67 J. Wang, H. Huang, W. Duan and Z. Liu, J. Chem. Phys.,2013, 139, 184701.

68 M. Ezawa, J. Phys. Soc. Jpn., 2015, 84, 121003.69 L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater.

Phys., 2006, 74, 195312.70 L. Fu and C. L. Kane, Phys. Rev. B: Condens. Matter Mater.

Phys., 2007, 76, 045302.71 R.-W. Zhang, W.-X. Ji, C.-W. Zhang, S.-S. Li, P. Li and

P.-J. Wang, J. Mater. Chem. C, 2016, 4, 2088–2094.72 Y. Guo, Y.-F. Zhang, X.-Y. Bao, T.-Z. Han, Z. Tang,

L.-X. Zhang, W.-G. Zhu, E. G. Wang, Q. Niu, Z. Q. Qiu,J.-F. Jia, Z.-X. Zhao and Q.-K. Xue, Science, 2004, 306,1915–1917.

73 S. Bertolazzi, J. Brivio and A. Kis, ACS Nano, 2011, 5,9703–9709.

74 S. E. Fritz, S. M. Martin, C. D. Frisbie, M. D. Ward andM. F. Toney, J. Am. Chem. Soc., 2004, 126, 4084–4085.

75 J. Zhang, H. Yu, W. Chen, X. Tian, D. Liu, M. Cheng, G. Xie,W. Yang, R. Yang and X. Bai, et al., ACS Nano, 2014, 8,6024–6030.

76 Y. Guo, S. Zhang, T. Zhao and Q. Wang, Nanoscale, 2016, 8,10598–10606.

77 G. Wang, G. Loh, R. Pandey and S. P. Karna, J. Phys.Chem. C, 2015, 119, 15654–15660.

78 S. Grimme, J. Comput. Chem., 2006, 27, 1787–1799.

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