Abstract
Graphene was the first material predicted to realize a topological insulator (TI), but unfortunately the gap is unobservably small due to carbon's weak spinorbital coupling (SOC). Based on firstprinciples calculations, we propose a stable spsp^{2} hybrid carbon network as a graphene analog whose electronic band structures in proximity of the Fermi level are characterized by Dirac cones. We demonstrate that this unique carbon framework has topologically nontrivial electronic structures with the Z_{2} topological invariant of v = 1 which is quite promising for hosting the quantum spin Hall effect (QSHE) in an experimentally accessible low temperature regime (<7 K). This provides a viable approach for searching for new TIs in 2D carbon allotropes.
Introduction
Topological insulators (TIs) comprising of a class of remarkably spinorbitalcoupled nonmagnetic materials are insulating in the bulk but on the boundary they have gapless edge or surface states that are topologically protected by time reversal symmetry^{1,2,3}. The exotic phenomena of these systems, including Majorana fermions^{4,5}, charge fractionalization^{6}, and novel magnetoelectric^{7,8,9}, are quite promising for the uses of quantum computing^{2} and spintronic devices^{10}. Twodimensional (2D) TIs are ideally suitable for such applications due to robustness of conducting edge states from backscattering, and graphene was the first material predicted to realize a TI^{1}, in which the spinorbital coupling (SOC) opens a band gap at the Dirac points (K and K^{/}). The quantum spin Hall effect (QSHE), a new quantum state of matter with a nontrivial topological property, was expected to be observed in graphene, where the time reversal invariant electronic state with a bulk electronic band gap will support the transport of charge and spin in gapless edge state. Subsequent works, however, indicated that the gap is unobservavbly small (~10^{−3} meV) and the QSHE in graphene can occur only at an unrealistically low temperature, because the SOC is rather weak, which is in fact a second order process for graphene^{11,12,13,14}. Further theoretical works suggest that the SOC gap of graphene can be dramatically enhanced to detectable values by dilute concentrations of 3d or 5d transition metal adatoms^{15,16,17,18,19}, which can host a quantum spin Hall state or a quantum anomalous Hall state. This was attributed to the hybridization of graphene with impurity bands arising from heavy adatoms possessing partially filled d shells, as well as the interplay between proximityinduced magnetization and Rashba SOC. The quantum spin Hall states in some metalorganic frameworks with remarkable SOC has also been proposed based on firstprinciples^{20,21,22,23,24}. However, if the QSHE in 2D carbon materials without dopants can occur at experimentally accessible low temperature remains doubtful.
Recent works show that carbon has abundant 2D allotropes beyond graphene. The most representative allotropes are the socalled graphynes, which are built from triple and doublebonded units of carbon atoms^{25,26,27,28}. Many graphynes can be assumed to be chemically stable, even though they are less favorable in formation energy than graphene due to the presence of sphybridized carbons. Indeed, finite building blocks and cutouts have already been synthesized and first steps towards the preparation of extended graphynes have been proposed and developed^{29,30,31,32,33,34,35}. The presence of acetylenic bonds associated with sphybridized carbon atoms in these 2D frameworks introduces a rich variation in electronic and optical properties. Considerable efforts have been devoted to the electronic structures of graphynes^{26,28,36,37,38,39,40,41,42,43}. Both γgraphyne and the alreadysynthesized graphdiyne film are semiconducting with a moderate band gap of about 1.1 eV^{42,43}. Interestingly, there exists a special class of graphynes associated with Diracconecharacterized electronic band structures in proximity of Fermi level. Up to now, this special class includes αgraphyne, βgraphyne, and 6,6,12graphyne belonging to different lattices^{28}. Unlike graphene, the Dirac cones are not located at the K or K^{/} point but on the lines from Γ to M for the hexagonal βgraphyne. Rectangular 6,6,12graphyne lattice, however, has two selfdoped nonequivalent distorted Dirac cones^{28}. In a sense, honeycomb αgraphyne is by far an ideal analog of graphene in both the shape and positions of Dirac cones, but it is energetically most unstable among the graphynes. The abundant 2D carbon allotropes with versatile electronic properties offer candidate materials for hosting QSHE superior to graphene, unfortunately, such possibility has never been considered, partly due to the weak SOC revealed in graphene.
In this work, from firstprinciples calculations within densityfunctional theory (DFT), we propose a novel 2D carbon allotrope, named as δgraphyne (Figure 1(a)), which has higher energetic favorability compared with most of the graphynes including α and βgraphynes, as well as the alreadysynthesized graphdiyne. Like graphene, the band structure of δgraphyne in proximity of the Fermi level can be characterized by Dirac cones at the K and K^{/} points of the Brillouin zone (BZ), except that the slope of the Dirac cones (Fermi velocity) is only 56% of that of graphene. Combined with a tightbinding method, we demonstrate that δgraphyne has topologically nontrivial electronic structures with a Z_{2} topological invariant of v = 1. The band gap opened up at the Dirac points of δgraphyne due to the SOC for the π orbital is about 0.59 meV, which can host the QSHE in an experimentally accessible low temperature regime (<7 K). This offers a viable approach for searching new TIs in 2D carbon allotropes.
Results
Generally, the frameworks of graphynes comprise of sphybridized carbon atoms accompanied by acetylenic bonds beside sp^{2}hybridized carbon atoms, but the ratios of the sphybridized carbon atoms and the arrangements of the acetylenic bonds differ significantly for different allotropes. The honeycomb framework of δgraphyne is plotted in Figure 1(a). We classify the carbons atoms of δgraphyne into three types: (1) sp^{2}hybridized carbon atoms at the hexagons (denoted as C1 hereafter); (2) sphybridized carbon atoms forming the acetylenic bonds (C2); and (3) sp^{2}hybridized carbon atoms joint to three sphybridized carbon atoms (C3). The length of the covalent bond between C1 atoms is 1.437 Å, slightly longer than that in graphene,1.420 Å. The length of the acetylenic bond (C2C2) is about 1.229 Å. The lengths of the bonds between C1 and C2 and that between C2 and C3 are 1.389 Å and 1.395 Å, respectively, both of which are shorter than the C1C1 covalent bond, due to the stabilization effect of the triple bonds. All the atoms are on the same plane without any buckling. The unit cell contains 20 carbon atoms, and the optimized lattice constant (the length of lattice vectors) is 9.440 Å. The ratio of sphybridized carbon atoms is 0.6, lower than that of αgraphyne (0.75), due the existence of carbon hexagons.
To compare the energetic stability of δgraphyne with other graphynes, we plotted their formation energies with respect to graphene in Fig. 2. The datum of graphdiyne monolayer whose multilayered counterpart (graphdiyne film) has been fabricated in recent experiments^{35} is presented as a reference point (represented by the dashed line). The graphynes with formation energy lower than that of graphdiyne monolayer are expected to be plausible. From this figure, we can see the following features: (1) These graphynes are energetically less favorable than graphene, due to the presence of sphybridized carbon atoms, however, δgraphyne, 6,6,12graphyne, and γgraphyne are more stable in energy than the reference point of graphdiyne monolayer. Especially, δgraphyne is energetically more favorable than the reference point by about 0.022 eV/atom. (2) The formation energy (E_{form}) of these graphynes varies linearly with respect to the sphybridized carbon atoms ratio x, and can be fitted using the formula: E_{form} = kx, with k = 1.25 eV/atom, as indicated by the solid line in Fig. 2. This linear relationship can be easily understood by supposing that the energy increase of graphynes with respect to graphene is due to the sphybridized atoms. In this case, the value of k represents the energy increase of a sphybridized atom relative to the sp^{2}hybridized carbon atom in graphene. This empirical formula is useful for evaluating the energetic priority of graphynes simply from the sphybridized carbon atom ratio, i.e., a smaller x corresponds to higher energetic favorability. Noting that δgraphyne is not energetically most stable among the graphynes under study (less stable than 6,6,12graphyne and γgraphyne), however, γgraphyne is semiconducting and 6,6,12graphyne has two selfdoped nonequivalent distorted Dirac cones both of which are unsuitable for hosting QSHE.
The kinetic stability of the δgraphyne is also confirmed from a vibrational analysis of the phonon spectrum calculated from firstprinciples. We didn't find the modes with imaginary frequencies in the phonon spectrum along the highlysymmetric points of BZ as shown in Fig. 1(b). We also heated a large supercell (4 × 4) of δgraphyne at 500 K for 1.0 ps by using the Nose thermostat to check the thermal stability. No obvious collapse tendency of the framework was found at this time scale. These results imply the stability and plausibility of the δgraphyne from other points.
The electronic structures of the δgraphyne are then calculated. The band lines along the highlysymmetric point of BZ are plotted in Fig. 3 (a). There are six points (three equivalent K and three equivalent K^{/}) in the BZ where conduction and valence bands meet in a single point at the Fermi level without SOC (Fig. 3(a) and 3(b)). These meeting points of valence and conduction bands are Dirac points equivalent to those of graphene. Around the meeting point, both the valence and conduction bands (referred to as Dirac bands) are linear and can be characterized by Dirac cones. The charge carriers (electrons and holes) in these bands will behave as massless Dirac fermions, similar to the cases of the wellknown Diracfermion materials such as graphene^{44}, silicene^{45}, germanene^{45}, and silicon germanide^{46}. We also evaluated the Fermi velocity (ν_{f}) of the δgraphyne by fitting these two bands at k = K + q to the expression of . The Fermi velocities of electrons and holes are identical in our calculations, which are 6.96 × 10^{5} m/s. This value is about 56% of value of graphene, 8.22 × 10^{5} m/s from the present calculations. Noting that that the Dirac cones of the δgraphyne differ significantly from those of the 6,6,12graphyne proposed in the Ref. 28 in both positions and shapes. Due to the rectangular symmetry, the 6,6,12graphyne has four Dirac cones which are distorted and directiondependent in its BZ^{28}. Therefore, δgraphyne can be regarded as the most stable counterpart of graphene in electronic structures among the graphynes.
Charge density distribution of the two Dirac bands in proximity of the Fermi level and the electron density of states (PDOS) projected onto the s and p (p_{x}, p_{y}, and p_{z}) orbitals of the atoms labeled as C1, C2 and C3 are then employed to reveal the origins of the Dirac cones. From Fig. 4 (a) and 4(b), it is obvious that electronic states in proximity of the Fermi level arise mainly from the p_{z} orbitals of carbon atoms which are perpendicular to the basal plane of the δgraphyne. The electronic states contributed by the p_{x}, p_{y}, and s states locate in the region ~2 eV away from the Fermi level. The interactions between the p_{z} orbitals lead to the formation of πconjugated framework. These features allow us to model the electronic structures in proximity of the Fermi level using a simple tightbinding (TB) Hamiltonian of the πelectrons: where and c_{i} represent the creation and annihilation operators of an electron at the ith atom, respectively. The parameter t_{ij} is the hopping energy of an electron between the ith and jth atoms. For simplification, we only considered the hopping between the nearestneighboring (NN) atoms. Corresponding to the three types of carbon atoms (C1, C2, and C3) in the δgraphyne framework (Fig. 1(a)), there are four types of NN hopping energies which are t_{C1C1}, t_{C1C2}, t_{C2C2}, and t_{C2C3}, respectively. Noting that the lengths of the bonds (C1C1, C1C2, and C2C3) are very close with the difference less than 0.04 Å, whereas that of the C2C2 bond is much shorter, we therefore set t_{C1C1}, t_{C1C2}, and t_{C2C3} to be identical to the value in graphene, 2.7 eV, and t_{C2C2} to 3.0 eV. Through diagonalizing a 20 × 20 matrix in the reciprocal space, the electronic band structures can be obtained, as indicated by the red dash lines in Fig. 3(a). Obviously, this simple TB Hamiltonian can reproduce well the band lines of the δgraphyne in proximity of the Fermi level given by DFT calculations, especially the appearance of the Dirac cones as well as the profiles of the two Dirac bands. Further studies show that the appearance of the Dirac cones is independent of the NN hopping energies, suggesting that it is the intrinsic feature of such πconjugated δgraphyne framework.
Now that such δgraphyne can be regarded as an ideal counterpart of graphene with superior stability among the graphynes, the SOC in δgraphyne becomes interesting. We therefore switch on SOC in our DFT calculations. As a benchmark, we first calculated the SOC gap in graphene and fond that it is only 0.008 meV, close to the value of previous works^{11}. Using the same method, we calculated the band structures of δgraphyne in the vicinity of the Fermi level. The band gap opened by SOC for the π orbital at the Dirac point is about 0.59 meV, as shown in Fig. 5(a), which corresponds to 7 K. This value is much larger than that of graphene, suggesting that the effective SOC in δgraphyne is stronger than that in graphene. Consequently, QSHE can be observed in an experimentally accessible low temperature regime (<7 K) in δgraphyne. We also evaluated the SOC gap of αgraphyne at the Dirac points using the same method and found that it is only 0.014 meV which is much smaller than that in δgraphyne. This implies that the presence of acetylenic bonds is not always associated with drastic increase of SOC strength. For 6,6,12graphyne, although the SOC gap opened in each Dirac cone is in the order of ~0.5 meV, there is no gap throughout the BZ due to the selfdoping effect.
In the presence of SOC, the TB Hamiltonian for the δgraphyne is written as^{5}:
In addition to the nearestneighbor hopping term (the first term), the spinorbital interaction with spindependent secondnearestneighbor hopping is involved (the second term). λ_{SO} is spinorbital coupling strength and S is the spin. The unit vector is defined as: where and are bond vectors along the two bonds that the electron traverses when going from atom j to i. According to this definition, . Based on the above Hamiltonian with λ_{SO} = 0.4 meV, the two Dirac bands are well reproduced as shown in Fig. 5(a).
The low energy effective Harmiltonian with SOC in graphene in the vicinity of Dirac point K can be described by where ν_{F} is the Fermi velocity of π electrons near the Dirac points, and σ_{z} is Pauli matrix. The effective SOC ξ for graphene has the explicit from , with Δ_{ε} being the energy difference between the 2s and 2p orbitals and half of the intrinsic spinorbital strength, respectively^{13}. The parameter V_{spσ} corresponds to the σ bond formed by the 2s and 2p orbitals. This effective Harmiltonian leads to a spectrum . The energy gap at the Dirac point is 2ξ. For graphene, ξ was estimated to be of order of 10^{−3} meV^{13}. Supposing that this effective Harmiltonian still holds for δgraphyne, because they have similar electronic bands structures in the vicinity of Dirac point K, ξ is estimated to be about 0.3 meV from our firstprinciples calculations, almost three orders of graphene. The physical reasons behind the vastly enhanced SOC gap size compared to graphene are related to the unique framework of δgraphyne. Basically, the SOC is a relativistic effect described by a Hamiltonian with the form , is the Pauli matrix. The SOC strength is proportional to the grad of electric potential . The presence of acetylenic bonds in the δgraphyne framework enhances the local inhomogenity of electron distribution and thus the grad of electronic potential compared to the case of graphene. This is also the reason why the SOC gap sizes of other 2D carbon allotropes (α and 6,6,12graphynes) are larger than that of graphene. Obviously, the enhancement of electric potential grad depends on the distribution pattern of the acetylenic bonds, because αgraphyne possesses the smallest SOC gap size among the three graphynes.
The topologically nontrivial electronic structures of δgraphyne can be confirmed by calculating the Z_{2} topological invariant. The work of Fu and Kane showed that the inversion symmetry can simplify the calculations of the Z_{2} invariants^{5}. In their strategy, the Z_{2} invariant v is defined by with for 2N occupied bands. ζ_{2m}(Γ_{i}) = ±1 is the parity eigenvalue of the 2mth occupied energy band at the timereversal invariant momentum Γ_{i}. The two states of a Kramers doublet have the same parity, ζ_{2m} = ζ_{2m−1}. It means that with inversion symmetry, the Z_{2} topological invariants can be deduced from the knowledge of the parity of the four timereversal and parity invariant points at BZ. This provides a simple method for determining the topological phase of any inversion symmetric insulator, without having to know about the global properties of the energy bands. For the honeycomb δgraphyne lattice, the four timereversal invariant momenta occur at Γ_{i = (n1 n2)} = (n_{1}b_{1} + n_{2}b_{2})/2 with n_{1}, n_{2} = 0, 1 and b_{1}, b_{2} are primitive reciprocal lattice vectors, which correspond to the Γ and three M points in BZ, as shown in Fig. 5(b). Using this strategy, we calculated the Z_{2} topological invariant based on the abovementioned TB Hamiltonian and found that v = 1. This confirms that δgraphyne is a topological insulator in the quantum spinHall phase. Noting that the Fermi level is right within the SOC band gap, doping is therefore unnecessary in realizing QSHE in δgraphyne.
Discussion
We should stress that although δgraphyne is kinetically stable and energetically more favorable than the alreadysynthesized graphdiyne, its synthesis remains challenging for chemists. Neither γgraphyne nor 6,6,12graphyne has been realized in experiments, even though both of them are lower in energy than δgraphyne. However, the recent progresses on the growth of graphene on metal surfaces by chemical vapor deposition or organics precursor molecules may also be helpful for reaching this goal. Anyway, our work suggests that the observation of TIs may be within rich in 2D carbon allotropes beyond graphene. The acetylenic bonds associated with sphybridized carbon atoms in these allotropes introduce a rich variation in electronic properties superior to graphene. Interestingly, the unique framework with Diraccone characterized electronic structures is also useful for the design of other 2D materials, because the appearance of the Dirac cones is independent of the NN hopping energies as revealed by TB model. For example, one can replace the C3 atom of δgraphyne with silicon or other atoms to get novel 2D Dirac materials.
In conclusion, using firstprinciples calculations combined with a tightbinding model, we propose a twodimensional carbon topological insulator (δgraphyne) beyond graphene with superior energetic stability and remarkable spinorbital interactions. The electronic band structures in proximity of the Fermi level of the πconjugated framework resembles well those of graphene characterized by the appearance of Dirac cones at the K and K^{/} points, but the spinorbital interaction is more remarkable. The band gap opened up at the Dirac points due to SOC is about 0.59 meV and can host the QSHE in an experimentally accessible low temperature regime (<7 K). The topologically nontrivial electronic structures of δgraphyne in the quantum spinHall phase is identified by a Z_{2} topological invariant of v = 1. This provides a viable approach for searching new TIs in 2D carbon allotropes.
Methods
We performed firstprinciples calculations within DFT using the Vienna ab initio simulation package (VASP)^{47,48}. The electronelectron interactions are treated within a generalized gradient approximation (GGA) in the form of PerdewBurkeErnzerhof (PBE) for the exchangecorrelation functional^{49}. Projectoraugmentedwave (PAW) potentials are adopted to describe the electronion interaction^{50}. The electron wavefunctions are expanded using the planewaves with the energy cutoff of 800 eV. Selfconsistent solution of the KohnSham equation is carried out with the convergence of 10^{−8}. The supercells are repeated periodically on the xy plane while a vacuum region of about 15 Å is applied along the zdirection to avoid mirror interaction between neighboring images. A symmetry reduced (9 × 9 × 1) MonkhorstPack sampling is used for the Brillouin zone (BZ) integration. The convergence of gap size with respect to the number of k points and cutoff energy is better than 1 × 10^{−3} meV. Structural optimizations are carried out using a conjugated gradient (CG) method until the remaining force on each atom is lower than 0.001 eV/Å.
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Acknowledgements
This work is supported by the National Basic Research Program of China (No. 2012CB932302), the National Natural Science Foundation of China (No.91221101), the 111 project (no. B13209), and National Super Computing Centre in Jinan.
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School of Physics & State Key Laboratory of Crystal Materials, Shandong University, Jinan, Shandong 250100, China
 Mingwen Zhao
 , Wenzheng Dong
 & Aizhu Wang
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Contributions
M.Z. conceived the study and performed most of the calculations, analyzed the data, drafted and revised the manuscript. W.D. calculated the phonon spectrum. A.W. contributed to the calculations and analysis of PDOS. All authors read and approved the final manuscript.
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The authors declare no competing financial interests.
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Correspondence to Mingwen Zhao.
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