THE JOURNAL OF CHEMICAL PHYSICS 139, 184701 (2013)
Identifying Dirac cones in carbon allotropes with square symmetry Jinying Wang,1 Huaqing Huang,2 Wenhui Duan,2 and Zhirong Liu1,3,a) 1
College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China Department of Physics, Tsinghua University, Beijing 100084, China 3 State Key Laboratory for Structural Chemistry of Unstable and Stable Species and Beijing National Laboratory for Molecular Sciences (BNLMS), Peking University, Beijing 100871, China 2
(Received 18 June 2013; accepted 22 October 2013; published online 8 November 2013) A theoretical study is conducted to search for Dirac cones in two-dimensional carbon allotropes with square symmetry. By enumerating the carbon atoms in a unit cell up to 12, an allotrope with octatomic rings is recognized to possess Dirac cones under a simple tight-binding approach. The obtained Dirac cones are accompanied by flat bands at the Fermi level, and the resulting massless Dirac-Weyl fermions are chiral particles with a pseudospin of S = 1, rather than the conventional S = 1/2 of graphene. The spin-1 Dirac cones are also predicted to exist in hexagonal graphene antidot lattices. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828861] Graphene has attracted tremendous interest in the past few years.1, 2 Dirac cones are an important aspect of graphene in terms of fundamental physics. Their unusual electronic structure has valence and conduction bands that meet at discrete points (the Dirac points) at the Fermi level and display linear dispersion in the vicinity.3 The charge carriers obey the massless Dirac equation, so it provides a convenient way to mimic the quantum electrodynamics phenomena. Many interesting phenomena have been reported regarding graphene, such as the quantum Hall effect and the minimum conductivity exhibited at the limit of vanishing charge carriers.4, 5 The existence of Dirac cones is a peculiarity of the hexagonal honeycomb lattice of graphene and it is rather robust to perturbations.6, 7 Recently, some studies have been conducted to search for Dirac cones in other two-dimensional (2D) carbon allotropes and to explore the relationship between the existence of Dirac cones and lattice symmetry.8–13 It was found that some graphynes possess Dirac cones but have a rectangular structural symmetry instead of hexagonal.9, 12 However, when one examines the connection topologies in the graphynes, they were shown to be equivalent to that of graphene.12 It was also claimed that Dirac cones exist in buckled T graphene C8 , a carbon allotrope with a square lattice.10 But, unfortunately, it comes out that buckled T graphene C8 is simply metallic.14 In this paper, we conducted a study to search for Dirac cones in 2D carbon allotropes with square symmetry. To achieve this goal, numerical calculations and analytic deductions were adopted by using a tight-binding (TB) model under the guidance of group theory. Although the TB model produces less accurate band structures than first-principles calculations, it is useful for revealing fundamental properties of graphene.15 The TB model has been successfully applied to analyze the existence of Dirac cones in graphynes11, 12 and graphene antidot lattices (GALs).16 Group theory is another
a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2013/139(18)/184701/6/$30.00
powerful tool for predicting the electronic structure of superlattices. For example, a group-theoretical analysis in 1993 predicted that bipartite honeycomb carbon super-structures with a formula unit of C6m + 2 or C6m + 5 (m is an integer) would be semimetals with Dirac cones, while those with a formula unit of C6m or C6m + 3 are semiconductors.17 This prediction was consistent with the existence of Dirac cones in graphene, and was later validated by first-principles calculations on hexagonal GALs.16 For systems with square symmetry, a criterion for Dirac cones has been proposed by Aoki and Shima.18 According to their criterion, Dirac cones are expected to exist widely in many square systems. This proposal will be tested in our study. In this study, we consider 2D carbon allotropes with square symmetry, which are termed as square graphene (SG) in the rest of the paper. According to the symmetry, the C atoms in a unit cell are classified by location: center (at most one atom); vertice (at most one atom, since there are four vertice where each vertex is shared by four unit cells); edge centers (where atoms appear in pairs with at most, one pair); horizontal/vertical reflection axes (where atoms appear in fours); diagonal lines (where atoms appear in fours); other nonsymmetric locations (where atoms appear in eights). To maintain the planar structure and reserve the π electrons, the bonding number of each C atom should be less than four. So the center and vertice locations are ruled out. The bonding-number restriction also requires that the atom at the edge centers only bonds to atoms at horizontal/vertical reflection axes. Based on these rules, SG with C number in a unit cell (N) up to 12 are enumerated as (Fig. 1): r N = 4. C atoms are located at horizontal/vertical reflection axes or diagonal lines. The former arrangement results in the system SG-4 in Fig. 1, which is actually equivalent to the planar T graphene studied by Liu et al.10 The latter arrangement violates the bondingnumber restriction and does not produce a legal structure. r N = 6. The solution is to arrange two atoms at edge centers and the other four atoms at horizontal/vertical
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reflection axes (SG-6) or diagonal lines (violating the bonding-number restriction). r N = 8. Possible location combinations are: all eight atoms at horizontal/vertical reflection axes (SG-8a); four atoms at horizontal/vertical reflection axes and the other four atoms at diagonal lines (SG-8b); all eight atoms at diagonal lines (SG-8c), whose topology is identical to that of SG-4; all 8 atoms at nonsymmetric locations, whose topology is identical to that of SG-8c and thus is not given in Fig. 1. r N = 10. The solution is to arrange two atoms at edge centers and the other eight atoms at locations as enumerated in N = 8. As the atoms at the edge centers only bond to atoms at horizontal/vertical reflection axes, there are two available legal structures (SG-10a and SG-10b). r N = 12. A similar analysis results in eight unique structures from SG-12a to SG-12h. Aoki and Shima have analyzed the band structure of square super-lattices from group-theoretical considerations.18 In bipartite cases, systems with a formula unit of C4m in a unit cell were predicted to be semimetallic (for odd m) or semiconducting (for even m), while those with a formula of C4m + 2 were predicted to be semimetallic for any m. In nonbipartite cases, C4m were predicted to be semimetallic or semiconducting, while C4m + 2 were predicted to be metallic. Accordingly, it is estimated that almost half the SG are semimetals with Dirac cones. Is the possibility of Dirac cones really so high? We examine the systems one by one that were enumerated in Fig. 1 to search for possible Dirac cones. For the smallest SG as shown in Fig. 1, SG-4, extensive studies have been performed.8, 10, 14 It is nonbipartite and shows metallic behavior with bands crossing the Fermi level.14 It is noted that the group-theoretical study only considered the band degeneracy at special k points of high symmetry.18 For SG-4, the crossing of bands at the Fermi level does not occur at special k points. So the metallic properties come from accidental degeneracy, which does not conflict with the group-theoretical analysis. The metallicity is not protected by the square symmetry, which can be demonstrated by its dependence on the hopping parameters.19 For N = 6, it was predicted from group-theoretical studies that the systems act as semimetals.17 There is only one kind of SG with N = 6 (Fig. 1). We adopted a TB model20 to numerically calculate the band structures of SG. The properties of SG-6 are summarized in Fig. 2. SG-6 is bipartite (Fig. 2(a)), so the band structure is symmetric about the Fermi level under a TB approximation considering only the nearestneighboring hoppings (Fig. 2(b)). The valence and conductance bands contact at the Fermi level as predicted (Fig. 2(b)). However, the contact occurs at two lines, rather than at discrete points (Figs. 2(b) and 2(c)), i.e., at kx = 0 and ky = 0. It is easy to verify that an eigen solution of the contact at kx = 0 is {x1 = −x3 = √12 , x2 = · · · = 0} for any ky values. This property is protected by the square symmetry that is not affected by the TB parameters (t1 and t2 ). Therefore, there are no Dirac cones in SG-6, which is not caused by the absence of contacts between valence and conductance bands, but is
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SG−4
SG−8a
SG−6
SG−8b
SG−10b
SG−10a
SG−12a
SG−12e
SG−8c
SG−12b
SG−12f
SG−12c
SG−12g
SG−12d
SG−12h
FIG. 1. Schematic atomic structures for 2D carbon allotropes with square symmetry where the carbon number in a unit cell (N) is numbered up to 12. Unit cells are depicted as dashed squares. The systems are named in SGN[suffix], where an optional suffix is used to discriminate different allotropes under the same N.
caused by the existence of too much contacts. Band crossing with linear dispersion can be observed along some k-lines, but the Fermi velocity changes with the adopted k-line (Fig. 2(d)). As such, SG-6 acts as a metal instead of a semimetal. For N = 8, there are four kinds of SG (Fig. 1). SG-8a and SG-8b are nonpartite, and their TB band structures are given in Fig. 3. SG-8a has bands intersecting at the Fermi level (Fig. 3(a)), which allows it to show metallic behavior similar to that of SG-4.14 SG-8b has a flat band at the Fermi level and a band intersecting at the Fermi level (Fig. 3(b)). SG-8c is topologically equivalent to SG-4, so their band structures are essentially analogous to that of SG-4 under an enlarged supercell with eight atoms as previously demonstrated.14 Therefore, Dirac cones could not be achieved in SG with N = 8. Dirac cones were achieved in SG with N = 10. The band structure of SG-10a (see Fig. S1 of the supplementary material21 ) is similar to that of SG-6 where valence and
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(a)
(b)
(a)
(b) t2
2
t2 6 1
5
t1
1
4 5
2
6
t1 4
(c)
9
10 3
8
3
(d)
(c)
7
(d)
4 3
E (eV)
2 1 0
−1 −2 −3
−0.4 −0.2
0
kx
FIG. 2. Properties of SG-6. (a) Schematic atomic structure. Six carbon atoms in a unit cell are labeled with numbers from 1 to 6. t1 and t2 represent the intra- and inter-tetraring hopping parameters, respectively. (b) The band structures from TB calculations. (c) The 3D structure of the conductance band. (d) The energy (E) of the first band above the Fermi level as a function of kx when (from top to bottom) ky = 0.5, 0.4, 0.3, 0.2, 0.1. kx and ky are measured in units of 2π /a in this paper where a is the lattice constant.
conductance bands contact at two lines, so it does not possess Dirac cones. However, SG-10b shows typical Dirac cones in the band structure (Fig. 4). There are two flat bands located at the Fermi level, which is caused by an imbalance between the atom number of the two sublattices.22, 23 Aside from the flat bands, two dispersive bands come into contact at the Fermi level (Figs. 4(b) and 4(c)). The contact occurs at the M point. The dispersion in the vicinity of the contact point is linear and cone-like (Figs. 4(c) and 4(d)). The presence of flat bands at the Fermi energy does not force a bandgap opening in Dirac cone, being at variance with some previously calculated systems.24 The coexistence of flat bands and conelike bands in SG-10b is a result of strong neighboring hopping under the given topology symmetry as will be shown in the analytic solution below. It is intriguing that there is only one Dirac cone in SG-10b, which is different from graphene and graphynes where Dirac cones always appear in pairs. For N = 12, we cannot find any Dirac cones near the Fermi
FIG. 3. The TB band structures for (a) SG-8a and (b) SG-8b.
−0.2 0.2 0.4 −0.4
0
0.2
0.4
ky
FIG. 4. Dirac cone in SG-10b. (a) Schematic atomic structure. The unit cell is indicated by the dashed square. Carbon atoms in a unit cell are labeled from 1 to 10. t1 and t2 are the hopping parameters. (b) The TB band structures. (c) 3D structure of the bands near the Fermi level. (d) Contour plot of the upper band in (c).
level in all eight allotropes (see Fig. S2 of the supplementary material21 for TB band structures). The Dirac cone in SG-10b can be analytically solved under a TB approach. x1 , x2 , ···, x10 are denoted as the wavefunction values of the corresponding carbon atoms, and t1 and t2 are the nearest-neighboring hopping parameters (Fig. 4(a)). Then there are four eigen solutions with E = 0 at M, x2 = −x4 = x6 = −x8 = 12 ψ0 : , (1) x1 = x3 = x5 = x7 = x9 = x10 = 0 x1 = −x3 = x5 = −x7 = 12 , (2) ψ1 : x2 = x4 = x6 = x8 = x9 = x10 = 0 ⎧ t1 ⎪ x9 = x10 = ⎪ ⎪ ⎪ ⎪ 2t12 + 2t22 ⎪ ⎪ ⎨ ψ2 : −x = x = t2 , (3) 2 6 ⎪ ⎪ 2 2 ⎪ ⎪ 2t + 2t ⎪ 1 2 ⎪ ⎪ ⎩ x1 = x3 = x4 = x5 = x7 = x8 = 0 ⎧ t1 ⎪ x9 = −x10 = ⎪ ⎪ ⎪ ⎪ 2t12 + 2t22 ⎪ ⎪ ⎨ , (4) ψ3 : x = −x = t2 4 8 ⎪ ⎪ 2 2 ⎪ ⎪ 2t1 + 2t2 ⎪ ⎪ ⎪ ⎩ x1 = x2 = x3 = x5 = x6 = x7 = 0 here ψ 0 is an easily recognized solution to E = 0 for any k, so it composes a trivial flat band at the Fermi level. ψ 1 , ψ 2 , and ψ 3 are not eigen solutions when k deviates from M. Using ψ 1 , ψ 2 , and ψ 3 as bases, the Hamiltonian for k in the vicinity
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of M is expanded as ⎡ 0 e−ikx a −eiky a ⎢ ikx a −iky a 0 H =C ⎣ e −e 0 2 + eikx a + e−iky a
2 + e−ikx a +eiky a 0
⎤ ⎥ ⎦,
0 (5)
where t1 t2 C= . 2 2t12 + 2t22
(6)
For k = M + q with |q| |M|, the eigen energy is determined to be t1 t 2 a E = 0, ± |q|. (7) 2 t12 + t22 So the obtained Dirac cone is isotropic with the Fermi velocity given by (set ¯ = 1), t1 t2 a . vF = 2 t12 + t22
(8)
Despite the cone-like structure, the present Dirac fermions do not satisfy the two-component Dirac-Weyl equation. In fact, the Hamiltonian in Eq. (5) can be simplified in the vicinity of the cone to ⎤ ⎡ 1 1 0 √ (iqx +iqy ) √ (iqx −iqy )⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ 0 0 √ (−iqx −iqy ) ⎥, H =vF ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ 1 0 0 √ (−iqx +iqy ) 2 (9) which can be further re-written as a Dirac-Weyl Hamiltonian, H = vF q · S,
(10)
where S is a pseudospin of the quantum number 1 with three eigenvalues. Therefore, the massless Dirac fermions obtained here are the chiral particles of pseudospin S = 1. Massless Dirac-Weyl fermions with a pseudospin S = 1 have been predicted to exist in ultracold atoms confined in artificial optical lattices.25–28 It has also been suggested that arbitrary spin could be realized in stacked triangular lattices.29 The Dirac cone with a pseudospin S = 1 has interesting characteristics that are different from S = 1/2. The Berry’s phase enclosing the Dirac point vanishes. The enlarged pseudospin leads to an enhanced Klein tunneling where the barrier is transparent for all incident angles.26, 30 The presence of the flat band under S = 1 is also of great interest.26, 31 It serves as a good starting point to study correlated electron systems. Previous predictions of Dirac fermions with S = 1 have mainly been restricted to ultracold atoms. The results obtained in SG-10b allow for the possibility that Dirac fermions with S = 1 may be realized in carbon allotropes where the characteristics could be measured at higher temperatures. Reexamining the published literature data, we recognized that
FIG. 5. Strain effect on the cone-like band structure of SG-10b. Only the first dispersive band above the Fermi level is shown under a strain of εx = 100% (red), ε x = −50% (green) and γ = 50% (blue), respectively. The unstrained result (black) is also provided for a comparison.
the single-walled hexagonal GALs, with odd numbers of carbon atoms in each edge, also possess massless Dirac fermions with S = 1 at the Fermi level. Their first-principles band structure and analytical solution have been provided by Ouyang et al.16 As such, carbon allotropes are promising systems for realizing Dirac fermions with S = 1. The Dirac cone in SG-10b is very robust under strain. We added a uniform strain on SG-10b with a tensor of εx γ , (11) 0 εy where εx and εy are the uniaxial strains along the x- and ydirections, respectively, and γ is the shear strain. The hopping parameters are assumed to vary according to the Harrison relation,32 1 , (12) l2 where l is the bond length. The band structures under various strains are given in Fig. 5, where the reduced reciprocal vector k∗ is used to restore the regular square shape of the Brillouin zone7 . It can be seen that the cone-like structure is preserved even under uniaxial strain as extreme as 100% or −50%, or under a shear strain of 50%, even though the isotropy is destroyed. The Dirac point is pinned to the M point in the k∗ -space. The robustness of the Dirac cone in the current system may be related to the fact that there is only one Dirac double-cone, so it will not collide with another double-cone and vanish as occurs for S = 1/2.13 To obtain more realistic results, we also conducted firstprinciples calculations. Results of SG-10b and two other systems possessing Dirac cones were given in Fig. 6. The band structure of SG-10b from first-principles calculation is shown in Fig. 6(a). It is noted that hydrogen atoms were used to saturate the dangling bonds in order to remove them from the Fermi level (see Fig. S3 of the supplementary material21 for the first-principles band structure without hydrogen saturation) as well as to stabilize the system. It can be seen from Fig. 6(a) that the band structure near the Fermi level is similar to the TB result except that a gap is opened in first-principles calculation. A flat band and a dispersive band near the Fermi level (the upper cone in the TB result) shift upwards and keep in contact, while the other flat band and t (l) ∝
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(c)
FIG. 6. First-principles calculation for (a) SG-10b, (b) a SG with N = 14 which was obtained by adding two carbon atoms to each link between the octatomic ring of SG-10b, and (c) a single-walled hexagonal GALs. Hydrogen atoms were used for bonding saturation. Geometry optimization and electronic structure calculations were performed using density functional theory (DFT) implemented in the Vienna ab initio simulation package (VASP)33 . The projector-augmented-wave (PAW) pseudopotential with the general gradient approximation (GGA) of the PW91 exchange correlation functional was adopted, and the cutoff energy was 520 eV. The Monkhorst-Pack grid was 21 × 21 × 1, 9 × 9 × 1, and 15 × 15 × 1 for (a)–(c), respectively. A vacuum separation of 20 Å was used to avoid any interactions between adjacent sheets.
dispersive band shift downwards. This leaves a gap of 0.79 eV at the M point. The gap observed here may be caused by the effective on-site energy,34 which was previously shown to play a major role in the self-doping of graphyne.12 In a study on kagome lattices, Green et al. found that flat bands can be classified into three types:31 (I) touching two linearly dispersing bands with S = 1 behavior, (II) separating from other bands, and (III) touching a single dispersive band either above or below. The on-site energy of SG-10b leads to a transition from type I into type III. Type III has been predicted for ultracold atoms with hexagonal and kagome lattices,31, 35, 36 and it has also been shown to act as a topological insulator under spin-orbit interactions.36 Therefore, SG-10b may be expected to possess other interesting properties. The gap can be reduced by modifying the dimension and/or geometry of the system. When the link between the octatomic ring increases from one atom to three atoms, the gap between the two dispersive bands is reduced to be 0.52 eV (Fig. 6(b)). The Dirac point moves from M to which can be easily understood under a TB approach. As indicated above, Dirac fermions with S = 1 can be also achieved in systems with hexagonal symmetry. For a single-walled hexagonal GALs as shown in Fig. 6(c), the gap is as small as 0.09 eV and the cone-like structure is much better than that of SG. By increasing the carbon number in each hexagonal edge, as shown in Ouyang et al.,16 the gap can be further reduced. To address the stability of the systems, we also calculated the formation energy of the proposed systems with respective to graphene and benzene. For SG-10b as shown in Fig. 6(a) with hydrogen saturation, the formation energy is determined to be 0.66 eV per C atom. This is comparable to that of other carbon allotropes,10 e.g., planar T graphene (the current SG-4) (0.53 eV/atom), bct C4 (0.34 eV/atom), bcc C8 (0.77 eV/atom), and graphdiyne (0.77 eV/atom). It
is noted that graphdiyne nanoscale films have been successfully synthesized.37 So SG-10b could be thermodynamically metastable. The formation energy of the single-walled hexagonal GALs as shown in Fig. 6(c) is 0.19 eV per C atom, which is much more stable due to its structural similarity with graphene. It is also noted that although aromaticity usually enhances the stability, no correlation between aromaticity and the existence/absence of Dirac cones (no matter whether S = 1/2 or S = 1) was revealed in the current study as well as those in the literature.9, 12, 16, 35 In summary, this paper explored the possibility of Dirac cones existing in 2D carbon allotropes with square symmetry. For atom numbers in a unit cell up to 12, there is only one system (SG-10b) that possesses Dirac cones under a simple tight-binding approach. This was a much smaller probability than expected from group analysis, and is caused by the valence and conduction bands meeting at lines rather than at discrete points. The cone-like band structure in SG-10b is accompanied by flat bands at the Fermi level, and the obtained massless Dirac fermions are chiral particles with a pseudospin of S = 1, rather than S = 1/2 as those in graphene. There is only one Dirac point in SG-10b, and it is robust under strains. The existence of Dirac cones with S = 1 has also been discovered in hexagonal GALs. We thank Professor Jianlong Li and Professor Zhongfan Liu for valuable discussions. This work was supported by the Ministry of Science and Technology of China (Grant No. 2011CB921900) and the National Natural Science Foundation of China (Grant Nos. 21373015 and 11074139). 1 A.
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Identifying Dirac cones in carbon allotropes with square symmetry Jinying Wang,1 Huaqing Huang,2 Wenhui Duan,2 and Zhirong Liu1,3,a) 1
College of Chemistry and Molecular Engineering, Peking University, Beijing 100871, China Department of Physics, Tsinghua University, Beijing 100084, China 3 State Key Laboratory for Structural Chemistry of Unstable and Stable Species and Beijing National Laboratory for Molecular Sciences (BNLMS), Peking University, Beijing 100871, China 2
(Received 18 June 2013; accepted 22 October 2013; published online 8 November 2013) A theoretical study is conducted to search for Dirac cones in two-dimensional carbon allotropes with square symmetry. By enumerating the carbon atoms in a unit cell up to 12, an allotrope with octatomic rings is recognized to possess Dirac cones under a simple tight-binding approach. The obtained Dirac cones are accompanied by flat bands at the Fermi level, and the resulting massless Dirac-Weyl fermions are chiral particles with a pseudospin of S = 1, rather than the conventional S = 1/2 of graphene. The spin-1 Dirac cones are also predicted to exist in hexagonal graphene antidot lattices. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4828861] Graphene has attracted tremendous interest in the past few years.1, 2 Dirac cones are an important aspect of graphene in terms of fundamental physics. Their unusual electronic structure has valence and conduction bands that meet at discrete points (the Dirac points) at the Fermi level and display linear dispersion in the vicinity.3 The charge carriers obey the massless Dirac equation, so it provides a convenient way to mimic the quantum electrodynamics phenomena. Many interesting phenomena have been reported regarding graphene, such as the quantum Hall effect and the minimum conductivity exhibited at the limit of vanishing charge carriers.4, 5 The existence of Dirac cones is a peculiarity of the hexagonal honeycomb lattice of graphene and it is rather robust to perturbations.6, 7 Recently, some studies have been conducted to search for Dirac cones in other two-dimensional (2D) carbon allotropes and to explore the relationship between the existence of Dirac cones and lattice symmetry.8–13 It was found that some graphynes possess Dirac cones but have a rectangular structural symmetry instead of hexagonal.9, 12 However, when one examines the connection topologies in the graphynes, they were shown to be equivalent to that of graphene.12 It was also claimed that Dirac cones exist in buckled T graphene C8 , a carbon allotrope with a square lattice.10 But, unfortunately, it comes out that buckled T graphene C8 is simply metallic.14 In this paper, we conducted a study to search for Dirac cones in 2D carbon allotropes with square symmetry. To achieve this goal, numerical calculations and analytic deductions were adopted by using a tight-binding (TB) model under the guidance of group theory. Although the TB model produces less accurate band structures than first-principles calculations, it is useful for revealing fundamental properties of graphene.15 The TB model has been successfully applied to analyze the existence of Dirac cones in graphynes11, 12 and graphene antidot lattices (GALs).16 Group theory is another
a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2013/139(18)/184701/6/$30.00
powerful tool for predicting the electronic structure of superlattices. For example, a group-theoretical analysis in 1993 predicted that bipartite honeycomb carbon super-structures with a formula unit of C6m + 2 or C6m + 5 (m is an integer) would be semimetals with Dirac cones, while those with a formula unit of C6m or C6m + 3 are semiconductors.17 This prediction was consistent with the existence of Dirac cones in graphene, and was later validated by first-principles calculations on hexagonal GALs.16 For systems with square symmetry, a criterion for Dirac cones has been proposed by Aoki and Shima.18 According to their criterion, Dirac cones are expected to exist widely in many square systems. This proposal will be tested in our study. In this study, we consider 2D carbon allotropes with square symmetry, which are termed as square graphene (SG) in the rest of the paper. According to the symmetry, the C atoms in a unit cell are classified by location: center (at most one atom); vertice (at most one atom, since there are four vertice where each vertex is shared by four unit cells); edge centers (where atoms appear in pairs with at most, one pair); horizontal/vertical reflection axes (where atoms appear in fours); diagonal lines (where atoms appear in fours); other nonsymmetric locations (where atoms appear in eights). To maintain the planar structure and reserve the π electrons, the bonding number of each C atom should be less than four. So the center and vertice locations are ruled out. The bonding-number restriction also requires that the atom at the edge centers only bonds to atoms at horizontal/vertical reflection axes. Based on these rules, SG with C number in a unit cell (N) up to 12 are enumerated as (Fig. 1): r N = 4. C atoms are located at horizontal/vertical reflection axes or diagonal lines. The former arrangement results in the system SG-4 in Fig. 1, which is actually equivalent to the planar T graphene studied by Liu et al.10 The latter arrangement violates the bondingnumber restriction and does not produce a legal structure. r N = 6. The solution is to arrange two atoms at edge centers and the other four atoms at horizontal/vertical
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reflection axes (SG-6) or diagonal lines (violating the bonding-number restriction). r N = 8. Possible location combinations are: all eight atoms at horizontal/vertical reflection axes (SG-8a); four atoms at horizontal/vertical reflection axes and the other four atoms at diagonal lines (SG-8b); all eight atoms at diagonal lines (SG-8c), whose topology is identical to that of SG-4; all 8 atoms at nonsymmetric locations, whose topology is identical to that of SG-8c and thus is not given in Fig. 1. r N = 10. The solution is to arrange two atoms at edge centers and the other eight atoms at locations as enumerated in N = 8. As the atoms at the edge centers only bond to atoms at horizontal/vertical reflection axes, there are two available legal structures (SG-10a and SG-10b). r N = 12. A similar analysis results in eight unique structures from SG-12a to SG-12h. Aoki and Shima have analyzed the band structure of square super-lattices from group-theoretical considerations.18 In bipartite cases, systems with a formula unit of C4m in a unit cell were predicted to be semimetallic (for odd m) or semiconducting (for even m), while those with a formula of C4m + 2 were predicted to be semimetallic for any m. In nonbipartite cases, C4m were predicted to be semimetallic or semiconducting, while C4m + 2 were predicted to be metallic. Accordingly, it is estimated that almost half the SG are semimetals with Dirac cones. Is the possibility of Dirac cones really so high? We examine the systems one by one that were enumerated in Fig. 1 to search for possible Dirac cones. For the smallest SG as shown in Fig. 1, SG-4, extensive studies have been performed.8, 10, 14 It is nonbipartite and shows metallic behavior with bands crossing the Fermi level.14 It is noted that the group-theoretical study only considered the band degeneracy at special k points of high symmetry.18 For SG-4, the crossing of bands at the Fermi level does not occur at special k points. So the metallic properties come from accidental degeneracy, which does not conflict with the group-theoretical analysis. The metallicity is not protected by the square symmetry, which can be demonstrated by its dependence on the hopping parameters.19 For N = 6, it was predicted from group-theoretical studies that the systems act as semimetals.17 There is only one kind of SG with N = 6 (Fig. 1). We adopted a TB model20 to numerically calculate the band structures of SG. The properties of SG-6 are summarized in Fig. 2. SG-6 is bipartite (Fig. 2(a)), so the band structure is symmetric about the Fermi level under a TB approximation considering only the nearestneighboring hoppings (Fig. 2(b)). The valence and conductance bands contact at the Fermi level as predicted (Fig. 2(b)). However, the contact occurs at two lines, rather than at discrete points (Figs. 2(b) and 2(c)), i.e., at kx = 0 and ky = 0. It is easy to verify that an eigen solution of the contact at kx = 0 is {x1 = −x3 = √12 , x2 = · · · = 0} for any ky values. This property is protected by the square symmetry that is not affected by the TB parameters (t1 and t2 ). Therefore, there are no Dirac cones in SG-6, which is not caused by the absence of contacts between valence and conductance bands, but is
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SG−4
SG−8a
SG−6
SG−8b
SG−10b
SG−10a
SG−12a
SG−12e
SG−8c
SG−12b
SG−12f
SG−12c
SG−12g
SG−12d
SG−12h
FIG. 1. Schematic atomic structures for 2D carbon allotropes with square symmetry where the carbon number in a unit cell (N) is numbered up to 12. Unit cells are depicted as dashed squares. The systems are named in SGN[suffix], where an optional suffix is used to discriminate different allotropes under the same N.
caused by the existence of too much contacts. Band crossing with linear dispersion can be observed along some k-lines, but the Fermi velocity changes with the adopted k-line (Fig. 2(d)). As such, SG-6 acts as a metal instead of a semimetal. For N = 8, there are four kinds of SG (Fig. 1). SG-8a and SG-8b are nonpartite, and their TB band structures are given in Fig. 3. SG-8a has bands intersecting at the Fermi level (Fig. 3(a)), which allows it to show metallic behavior similar to that of SG-4.14 SG-8b has a flat band at the Fermi level and a band intersecting at the Fermi level (Fig. 3(b)). SG-8c is topologically equivalent to SG-4, so their band structures are essentially analogous to that of SG-4 under an enlarged supercell with eight atoms as previously demonstrated.14 Therefore, Dirac cones could not be achieved in SG with N = 8. Dirac cones were achieved in SG with N = 10. The band structure of SG-10a (see Fig. S1 of the supplementary material21 ) is similar to that of SG-6 where valence and
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(a)
(b)
(a)
(b) t2
2
t2 6 1
5
t1
1
4 5
2
6
t1 4
(c)
9
10 3
8
3
(d)
(c)
7
(d)
4 3
E (eV)
2 1 0
−1 −2 −3
−0.4 −0.2
0
kx
FIG. 2. Properties of SG-6. (a) Schematic atomic structure. Six carbon atoms in a unit cell are labeled with numbers from 1 to 6. t1 and t2 represent the intra- and inter-tetraring hopping parameters, respectively. (b) The band structures from TB calculations. (c) The 3D structure of the conductance band. (d) The energy (E) of the first band above the Fermi level as a function of kx when (from top to bottom) ky = 0.5, 0.4, 0.3, 0.2, 0.1. kx and ky are measured in units of 2π /a in this paper where a is the lattice constant.
conductance bands contact at two lines, so it does not possess Dirac cones. However, SG-10b shows typical Dirac cones in the band structure (Fig. 4). There are two flat bands located at the Fermi level, which is caused by an imbalance between the atom number of the two sublattices.22, 23 Aside from the flat bands, two dispersive bands come into contact at the Fermi level (Figs. 4(b) and 4(c)). The contact occurs at the M point. The dispersion in the vicinity of the contact point is linear and cone-like (Figs. 4(c) and 4(d)). The presence of flat bands at the Fermi energy does not force a bandgap opening in Dirac cone, being at variance with some previously calculated systems.24 The coexistence of flat bands and conelike bands in SG-10b is a result of strong neighboring hopping under the given topology symmetry as will be shown in the analytic solution below. It is intriguing that there is only one Dirac cone in SG-10b, which is different from graphene and graphynes where Dirac cones always appear in pairs. For N = 12, we cannot find any Dirac cones near the Fermi
FIG. 3. The TB band structures for (a) SG-8a and (b) SG-8b.
−0.2 0.2 0.4 −0.4
0
0.2
0.4
ky
FIG. 4. Dirac cone in SG-10b. (a) Schematic atomic structure. The unit cell is indicated by the dashed square. Carbon atoms in a unit cell are labeled from 1 to 10. t1 and t2 are the hopping parameters. (b) The TB band structures. (c) 3D structure of the bands near the Fermi level. (d) Contour plot of the upper band in (c).
level in all eight allotropes (see Fig. S2 of the supplementary material21 for TB band structures). The Dirac cone in SG-10b can be analytically solved under a TB approach. x1 , x2 , ···, x10 are denoted as the wavefunction values of the corresponding carbon atoms, and t1 and t2 are the nearest-neighboring hopping parameters (Fig. 4(a)). Then there are four eigen solutions with E = 0 at M, x2 = −x4 = x6 = −x8 = 12 ψ0 : , (1) x1 = x3 = x5 = x7 = x9 = x10 = 0 x1 = −x3 = x5 = −x7 = 12 , (2) ψ1 : x2 = x4 = x6 = x8 = x9 = x10 = 0 ⎧ t1 ⎪ x9 = x10 = ⎪ ⎪ ⎪ ⎪ 2t12 + 2t22 ⎪ ⎪ ⎨ ψ2 : −x = x = t2 , (3) 2 6 ⎪ ⎪ 2 2 ⎪ ⎪ 2t + 2t ⎪ 1 2 ⎪ ⎪ ⎩ x1 = x3 = x4 = x5 = x7 = x8 = 0 ⎧ t1 ⎪ x9 = −x10 = ⎪ ⎪ ⎪ ⎪ 2t12 + 2t22 ⎪ ⎪ ⎨ , (4) ψ3 : x = −x = t2 4 8 ⎪ ⎪ 2 2 ⎪ ⎪ 2t1 + 2t2 ⎪ ⎪ ⎪ ⎩ x1 = x2 = x3 = x5 = x6 = x7 = 0 here ψ 0 is an easily recognized solution to E = 0 for any k, so it composes a trivial flat band at the Fermi level. ψ 1 , ψ 2 , and ψ 3 are not eigen solutions when k deviates from M. Using ψ 1 , ψ 2 , and ψ 3 as bases, the Hamiltonian for k in the vicinity
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of M is expanded as ⎡ 0 e−ikx a −eiky a ⎢ ikx a −iky a 0 H =C ⎣ e −e 0 2 + eikx a + e−iky a
2 + e−ikx a +eiky a 0
⎤ ⎥ ⎦,
0 (5)
where t1 t2 C= . 2 2t12 + 2t22
(6)
For k = M + q with |q| |M|, the eigen energy is determined to be t1 t 2 a E = 0, ± |q|. (7) 2 t12 + t22 So the obtained Dirac cone is isotropic with the Fermi velocity given by (set ¯ = 1), t1 t2 a . vF = 2 t12 + t22
(8)
Despite the cone-like structure, the present Dirac fermions do not satisfy the two-component Dirac-Weyl equation. In fact, the Hamiltonian in Eq. (5) can be simplified in the vicinity of the cone to ⎤ ⎡ 1 1 0 √ (iqx +iqy ) √ (iqx −iqy )⎥ ⎢ 2 2 ⎥ ⎢ ⎥ ⎢ 1 ⎥ ⎢ 0 0 √ (−iqx −iqy ) ⎥, H =vF ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ ⎦ ⎣ 1 0 0 √ (−iqx +iqy ) 2 (9) which can be further re-written as a Dirac-Weyl Hamiltonian, H = vF q · S,
(10)
where S is a pseudospin of the quantum number 1 with three eigenvalues. Therefore, the massless Dirac fermions obtained here are the chiral particles of pseudospin S = 1. Massless Dirac-Weyl fermions with a pseudospin S = 1 have been predicted to exist in ultracold atoms confined in artificial optical lattices.25–28 It has also been suggested that arbitrary spin could be realized in stacked triangular lattices.29 The Dirac cone with a pseudospin S = 1 has interesting characteristics that are different from S = 1/2. The Berry’s phase enclosing the Dirac point vanishes. The enlarged pseudospin leads to an enhanced Klein tunneling where the barrier is transparent for all incident angles.26, 30 The presence of the flat band under S = 1 is also of great interest.26, 31 It serves as a good starting point to study correlated electron systems. Previous predictions of Dirac fermions with S = 1 have mainly been restricted to ultracold atoms. The results obtained in SG-10b allow for the possibility that Dirac fermions with S = 1 may be realized in carbon allotropes where the characteristics could be measured at higher temperatures. Reexamining the published literature data, we recognized that
FIG. 5. Strain effect on the cone-like band structure of SG-10b. Only the first dispersive band above the Fermi level is shown under a strain of εx = 100% (red), ε x = −50% (green) and γ = 50% (blue), respectively. The unstrained result (black) is also provided for a comparison.
the single-walled hexagonal GALs, with odd numbers of carbon atoms in each edge, also possess massless Dirac fermions with S = 1 at the Fermi level. Their first-principles band structure and analytical solution have been provided by Ouyang et al.16 As such, carbon allotropes are promising systems for realizing Dirac fermions with S = 1. The Dirac cone in SG-10b is very robust under strain. We added a uniform strain on SG-10b with a tensor of εx γ , (11) 0 εy where εx and εy are the uniaxial strains along the x- and ydirections, respectively, and γ is the shear strain. The hopping parameters are assumed to vary according to the Harrison relation,32 1 , (12) l2 where l is the bond length. The band structures under various strains are given in Fig. 5, where the reduced reciprocal vector k∗ is used to restore the regular square shape of the Brillouin zone7 . It can be seen that the cone-like structure is preserved even under uniaxial strain as extreme as 100% or −50%, or under a shear strain of 50%, even though the isotropy is destroyed. The Dirac point is pinned to the M point in the k∗ -space. The robustness of the Dirac cone in the current system may be related to the fact that there is only one Dirac double-cone, so it will not collide with another double-cone and vanish as occurs for S = 1/2.13 To obtain more realistic results, we also conducted firstprinciples calculations. Results of SG-10b and two other systems possessing Dirac cones were given in Fig. 6. The band structure of SG-10b from first-principles calculation is shown in Fig. 6(a). It is noted that hydrogen atoms were used to saturate the dangling bonds in order to remove them from the Fermi level (see Fig. S3 of the supplementary material21 for the first-principles band structure without hydrogen saturation) as well as to stabilize the system. It can be seen from Fig. 6(a) that the band structure near the Fermi level is similar to the TB result except that a gap is opened in first-principles calculation. A flat band and a dispersive band near the Fermi level (the upper cone in the TB result) shift upwards and keep in contact, while the other flat band and t (l) ∝
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(c)
FIG. 6. First-principles calculation for (a) SG-10b, (b) a SG with N = 14 which was obtained by adding two carbon atoms to each link between the octatomic ring of SG-10b, and (c) a single-walled hexagonal GALs. Hydrogen atoms were used for bonding saturation. Geometry optimization and electronic structure calculations were performed using density functional theory (DFT) implemented in the Vienna ab initio simulation package (VASP)33 . The projector-augmented-wave (PAW) pseudopotential with the general gradient approximation (GGA) of the PW91 exchange correlation functional was adopted, and the cutoff energy was 520 eV. The Monkhorst-Pack grid was 21 × 21 × 1, 9 × 9 × 1, and 15 × 15 × 1 for (a)–(c), respectively. A vacuum separation of 20 Å was used to avoid any interactions between adjacent sheets.
dispersive band shift downwards. This leaves a gap of 0.79 eV at the M point. The gap observed here may be caused by the effective on-site energy,34 which was previously shown to play a major role in the self-doping of graphyne.12 In a study on kagome lattices, Green et al. found that flat bands can be classified into three types:31 (I) touching two linearly dispersing bands with S = 1 behavior, (II) separating from other bands, and (III) touching a single dispersive band either above or below. The on-site energy of SG-10b leads to a transition from type I into type III. Type III has been predicted for ultracold atoms with hexagonal and kagome lattices,31, 35, 36 and it has also been shown to act as a topological insulator under spin-orbit interactions.36 Therefore, SG-10b may be expected to possess other interesting properties. The gap can be reduced by modifying the dimension and/or geometry of the system. When the link between the octatomic ring increases from one atom to three atoms, the gap between the two dispersive bands is reduced to be 0.52 eV (Fig. 6(b)). The Dirac point moves from M to which can be easily understood under a TB approach. As indicated above, Dirac fermions with S = 1 can be also achieved in systems with hexagonal symmetry. For a single-walled hexagonal GALs as shown in Fig. 6(c), the gap is as small as 0.09 eV and the cone-like structure is much better than that of SG. By increasing the carbon number in each hexagonal edge, as shown in Ouyang et al.,16 the gap can be further reduced. To address the stability of the systems, we also calculated the formation energy of the proposed systems with respective to graphene and benzene. For SG-10b as shown in Fig. 6(a) with hydrogen saturation, the formation energy is determined to be 0.66 eV per C atom. This is comparable to that of other carbon allotropes,10 e.g., planar T graphene (the current SG-4) (0.53 eV/atom), bct C4 (0.34 eV/atom), bcc C8 (0.77 eV/atom), and graphdiyne (0.77 eV/atom). It
is noted that graphdiyne nanoscale films have been successfully synthesized.37 So SG-10b could be thermodynamically metastable. The formation energy of the single-walled hexagonal GALs as shown in Fig. 6(c) is 0.19 eV per C atom, which is much more stable due to its structural similarity with graphene. It is also noted that although aromaticity usually enhances the stability, no correlation between aromaticity and the existence/absence of Dirac cones (no matter whether S = 1/2 or S = 1) was revealed in the current study as well as those in the literature.9, 12, 16, 35 In summary, this paper explored the possibility of Dirac cones existing in 2D carbon allotropes with square symmetry. For atom numbers in a unit cell up to 12, there is only one system (SG-10b) that possesses Dirac cones under a simple tight-binding approach. This was a much smaller probability than expected from group analysis, and is caused by the valence and conduction bands meeting at lines rather than at discrete points. The cone-like band structure in SG-10b is accompanied by flat bands at the Fermi level, and the obtained massless Dirac fermions are chiral particles with a pseudospin of S = 1, rather than S = 1/2 as those in graphene. There is only one Dirac point in SG-10b, and it is robust under strains. The existence of Dirac cones with S = 1 has also been discovered in hexagonal GALs. We thank Professor Jianlong Li and Professor Zhongfan Liu for valuable discussions. This work was supported by the Ministry of Science and Technology of China (Grant No. 2011CB921900) and the National Natural Science Foundation of China (Grant Nos. 21373015 and 11074139). 1 A.
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