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Dirac Cones in Two-Dimensional Lattices: Janugraphene and Chlorographene Yandong Ma, Ying Dai, and Baibiao Huang J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/jz401099j • Publication Date (Web): 14 Jul 2013 Downloaded from http://pubs.acs.org on July 15, 2013

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The Journal of Physical Chemistry Letters

Dirac Cones in Two-Dimensional Lattices: Janugraphene and Chlorographene Yandong Ma, Ying Dai* , and Baibiao Huang School of Physics, State Key Laboratory of Crystal Materials, Shandong University, Jinan 250100, People’s Republic of China

Abstract Using density functional theory, we predict that two supposedly ordinary two-dimensional materials, janugraphene and chlorographene, feature Dirac points in their band structure even though they possess nonequivalent bonds and rectangular lattices. Our results provide a new perspective on the formation of Dirac points since the honeycomb structure composed of two equivalent sublattices and hexagonal symmetry are commonly considered to be necessary preconditions for the presence of Dirac points in two-dimensional materials. The Dirac fermions of these materials are rather robust in response to external electric field. Furthermore, the possible underlying physical mechanisms are discussed in detail. We envision that janugraphene and chlorographene will greatly broaden the scientific and technological impact of two-dimensional Dirac semimetals.

*

E-mail: [email protected] 1

ACS Paragon Plus Environment


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TOC Graphic

Keywords: density functional theory; janugraphene; chlorographene; Dirac points; rectangular lattices.

Two-dimensional (2D) materials have attracted a lot of research interest because of their intriguing physics as well their application potential.1-6 In particular, the extremely high mobility has made graphene a promising material for the next-generation electronics with properties that may exceed those of conventional semiconductors.7-12 Graphene is an atomically thin layer of carbon atoms arranged in a honeycomb lattice with two carbon atoms per unit cell. Of the four valence states, three sp2 orbitals form a σ state with three neighboring carbon atoms, and one p orbital develops into delocalized π and π* states that form the highest occupied valence band and the lowest unoccupied conduction band. The π and π* states of freestanding graphene are conical in the proximity of the Fermi energy, with the vertices touching exactly at the Fermi level, resulting in a point-like metallic Fermi surface due to graphene’s crystal symmetry. Under this crystal symmetry, in applying an effective mass description for the highest occupied valence band and the lowest

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unoccupied conduction band, one can arrive at a formal equivalence between the resulting differential equation and the Dirac equation.10,13 Hence, the charge carries in the vicinity of Fermi level can be described by a Dirac-like Hamiltonian operator containing a two dimensional pseudospin operator and are termed “Dirac fermions”.7,14 Moreover, the linearly dispersed band structure allows a mapping of carriers’ behavior to a model of massless fermions at the Dirac point. Such peculiar band structure leading to a plethora of new, observed, or predicted phenomena, such as anomalous half-integer quantum Hall effect or the Klein paradox.15 Its honeycomb structure composed of two equivalent triangular carbon sublattices and hexagonal lattice symmetry are commonly considered to be necessary preconditions for the presence of Dirac points and the large mobility of charge carries. If the hexagonal layer structure is composed of nonequivalent elements, such as in boron nitride, the lateral, in-plane symmetry is broken, resulting in the formation of a large gap between π and π* states.16 On the other hand if the regular hexagonal symmetry is stretched out of equilibrium, such as in graphene ribbons, a semimetal-to-insulator transition occurs.17 Then a question of fundamental interest is whether it is possible for the existence of Dirac points in the band structure of two-dimensional (2D) lattices does neither depend on the honeycomb structure with two equivalent sublattices nor depend on hexagonal symmetry. Following this idea, then we try to construct 2D lattices without the existence of equivalent sublattices or hexagonal symmetry while considering their feasibility to obtain them in experiment. We notice that a nonsymmetrically modified graphene

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derivative called Janus graphene with distinct and different functionalities on its sides was fabricated.18 This material draw attention due to the fact that it provides an ideal 2D bifacial Janus system to study asymmetric chemistry as the decorations on each side offer useful platforms for further chemical functionalisation.18 We also notice that nondestructive and patternable conversion of graphene is possible by using various photochemical chlorination techniques,19-21 while the research on various chlorinated graphene structures is rapidly growing.22-24 Here, we present a comprehensive analysis of the structural and electronic properties of a janus graphene named janugraphene and a chlorinated graphene named chlorographene based on first-principles calculations. We predict the existence of Dirac fermions in such supposedly ordinary systems though they have nonequivalent bonds and non-hexagonal symmetry. Further theoretical analysis shows that the created Dirac fermions consist of predominantly layer plane carbon states, whereas both the honeycomb structure and hexagonal symmetry are actually not necessary. This goes beyond existing views, where honeycomb structure composed of two equivalent sublattices and hexagonal lattice symmetry are commonly believed to be necessary prerequisites for the presence of Dirac points and the large mobility of charge carries in 2D materials. Furthermore, the Dirac fermions in both materials can be maintained as applying external electric field. Our findings provide a new perspective on the formation of Dirac fermions in the band structure of 2D lattices. The optimized geometry structures of chlorographene and janugraphene are depicted in Figure 1 (a) and 1(b). Chlorographene is a graphene derivative containing

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chlorine atoms on both sides. And janugraphene is also derived from graphene, in this case, by removing all chlorine atoms on one side contained in chlorographene and by saturating the resulting dangling bonds by phenyl. Both chlorographene and janugraphene exhibit a rectangular unit cell instead of a hexagonal unit cell as in graphene. In detail, chlorographene after geometry optimization can be described by the symmetry group pma2 with lattice constants a=4.948 Å, b=8.425 Å containing twenty atoms; while janugraphene belongs to the same symmetry group as chlorographene, namely pma2, with lattice constants a=4.970 Å, b=8.391 Å containing forty atoms. Geometry optimizations indicate that both materials are strongly buckled with the sp3-bonded carbon atoms shifting respectively upward and downward with respective to the layer plane, as addressed in Figure 1. This is because the sp2-bonded carbon networks are strongly modified by grafting sp3-type defects. The Brillouin zone of graphene is a honeycomb lattice, and the high-symmetry points Γ, K, and M are sampled. Since both chlorographene and janugraphene exhibit a rectangular unit cell instead of a hexagonal unit cell as in graphene, the high-symmetry points Γ, X, M and X' are sampled for both chlorographene and janugraphene. The electronic band structures of chlorographene are addressed in Figure 2. For chlorographene, remarkably, one point in the Brillouin zone where the valence and conduction bands meet in a single point located at asymmetric position (-0.227, 0, 0) on the lines from Γ to X' is predicated, see Figure 2a. The meeting point of the valence and conduction bands has a Dirac fermions character similar to those of

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graphene but is located at different positions in the Brillouin zone, as shown in Figure 2b. Examining the states near the Fermi level reveals that when approaching the cross points, the slopes, i.e., first derivatives of the band energies essentially remain unchanged. To get a deeper insight into the distribution of the linear dispersion relation in reciprocal space, the corresponding bands around the Fermi level in three dimensions are presented in Figure 2c. Comparison with the band structure of graphene shows that the cross points of graphene and chlorographene display different symmetries. As observed in well-known grahene, the Dirac points exhibit a threefold symmetry, arising from their location at the K and K' points. Namely, the curvatures of the valence band and the conduction band approached the Dirac points are zero not only along the line perpendicular to Γ-K and Γ-K' directions, but also along the lines obtained by rotating such lines by 120° and 240°.25 The two Dirac points of graphene are rotated by an angle of 60° against each other. In contrast to graphene, however, in chlorographene, two cross points are present which are related by a reflection at the line through the Γ and X points in the complete Brillouin zone. In this case, on the other hand, the two cross points of chlorographene are rotated by an angle of 180° against each other. In both materials, their corresponding symmetry prevents a significant directional dependence of the electronic properties in the plane of the material. The electronic band structures of janugraphene are depicted in Figure 3. From Figure 3a, we can see linear dispersion relation near the Fermi level, suggesting that a Dirac feature is found in janugraphene alone a line from Γ to X' points in the

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irreducible part of Brillouin zone. Closer inspection reveals that, on the other hand, the cross point is not located at the same position like in chlorographene but at the different asymmetric position between Γ and X', (-0.215, 0, 0). For the symmetry of the Dirac points, we obtain hardly any change when chlorographene is transformed into janugraphene. Namely, in the complete Brillouin zone of janugraphene, as in chlorographene, a second cross point feature is present that is related by symmetry more precisely a reflection at the line through G and X. We wish to point out that the primary Dirac cone of graphene has been deformed upon chemisorption. The Dirac cones of janugraphene and chlorographene are newly formed in association with the characteristics of the derived lattice structure as well as the chemical environment. These new Dirac cones of janugraphene and chlorographene should be governed by many factors, including the charge transfer between carbon and Cl. From above, we can see that the band structure of chlorographene along the line from

Γ to X' points are qualitatively identical to that of janugraphene. In other

words, the Dirac character of chlorographene remains intact when the chlorographene is transformed into janugraphene, although the modification of the chemical composition is giant and the lattice structure is changed. This fact that the substitution of all the chlorine atoms on one side in chlorographene by phenyl, resulting in janugraphene, leads to almost no change to the Dirac points, can be understood by Figure 4 which plots the partial charge density distribution of the valence and conduction bands near the Fermi level of chlorographene and janugraphene. As Figure 4 illustrates, the conduction bands of chlorographene near the Fermi level is

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predominately by the layer plane carbon p states, and the valence bands near the Fermi level is also mainly contributed by the layer plane carbon p orbitals, indicating that the orbitals of the cross points of chlorographene are not localized at the chlorine atoms. In janugraphene, like in chlorographene, both the conduction and valence bands are dominated by the layer plane carbon p states. And the orbitals of the cross points of janugraphene are almost not related to the chlorine atoms or the phenyl ligands. Therefore, the shift from chlorographene to janugraphene hardly affects the cross points. Provided that the orbitals near the cross points, on the other hand, are localized at the chlorine atoms or delocalized over the complete unit cell, then the cross points can be significantly swayed by the modification turning chlorographene into janugraphene. Indeed, the result is also confirmed by the partial density of states of chlorographene and janugraphene. Plots of the partial density of states for chlorographene and janugraphene in Figure 5 clearly shows that most chlorine and phenyl states are located in the lower energy range of the valence band and in the higher energy range of the conduction band, while only a little weight of the chlorine and phenyl states contributes to the states near the cross points. The energy levels of the cross points thus mainly consist of carbon orbitals of the layer plane. It is important to note that, similar as graphene,26 silicene,27 and germanene,28 the Dirac cone can be greatly affected by the substrate. However, like graphene,26 silicene,27 and germanene,28 this does not mean that the Dirac cone could not exist. The above results establish that both chlorographene and janugraphene are demonstrated to be with amazing properties of Dirac fermions. For these sp2/sp3

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hybrid bonded frameworks, the σ state is supposed to share the responsibility for the formation of these structures. While, on the other hand, the pz state, which is roughly vertical to the layer plane, binds covalently with each other and develops into delocalized π and π* states. Such bands are responsible for the linear band dispersion near the Fermi level illustrated in Figure 2 and Figure 3. With these results in hand, it thus can be concluded that the presence of the Dirac fermions in both chlorographene and janugraphene is attributed to the delocalized π and π* states. This mechanism is similar to that of graphene. Commonly, it is assumed that the existence of π and π* states is not sufficient for the presence of the linear band dispersion, and the honeycomb structure composed of two equivalent sublattices and hexagonal lattice symmetry, which are the general characteristics of graphene,26 silicene,27 and germanene,28 are considered to be necessary preconditions for the emergence of linear band dispersion. In contrast, both chlorographene and janugraphene does neither have the honeycomb structure with two equivalent sublattices nor possess hexagonal symmetry. Besides, the results for both chlorographene and janugraphene demonstrate that the cross points of the linear band dispersion occur at the low-symmetry points instead of the high-symmetry K and K' points in the Brillouin zone of 2D materials. They are completely different from that of pure graphene in a 2D square lattice which also possess Dirac cone. Because graphene in a square lattice and in a hexagonal lattice essentially have the same primary unit cells. Therefore, for graphene in a square lattice, honeycomb structure composed of two equivalent triangular carbon sublattices and hexagonal lattice symmetry are still necessary preconditions for the

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presence of Dirac points. On the other hand, chlorographene and janugraphene does neither have honeycomb structure with two equivalent sublattices nor have hexagonal lattice symmetry; yet the Dirac features could still be found in both materials. At this point, it seems that both the honeycomb structure with two equivalent sublattices and the hexagonal symmetry is not at all relevant for the formation of linear band dispersion in both chlorographene and janugraphene. More remarkably, according to the example of chlorographene and janugraphene, it is worth highlighting that that, even 2D materials that are made out of other atoms than carbon, could have linear band dispersion and, as a result, could possess highly interesting electronic properties. Note that applying external electric field (E-field) has proven an efficient method towards tuning the electronic properties of graphene-related materials.29-32 For instance, recently, Li et al. computationally revealed that the applied E-field can effectively modulate the energy gap of graphane/fluorographene bilayer, and correspondingly causes a semiconductor−metal transition.32 Inspired by these studies, we further investigated the effect of E-field on the electronic behaviors of chlorographene and janugraphene. Two directions of E-field (with the value of 0.003 a.u.) perpendicular to the basal planes of chlorographene or janugraphene were considered. Here, we define the positive direction of the E-field as pointing along the +z direction, as illustrated in Figure 6a. Details of the electronic band structures of chlorographene and janugraphene around the Fermi level are depicted in Figure 6b and Figure 6c, respectively, in which the case with E-field=0.003, -0.003 and 0 a.u. is indicated with orange, olive and red lines, respectively. Our computational results

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reveal that, in sharp contrast to the case of graphane/fluorographene bilayer,32 the electronic properties of chlorographene or janugraphene are rather robust in response to E-field, since negligible modulation in the band structure is induced by the E-field, as addressed in Figure 6b and Figure 6c. This discrepancy probably arises from the difference between the graphane/fluorographene bilayer with and chlorographene (or janugraphene)

without significant spontaneous

interlayer

polarization.

This

observation is in line with the above findings for the partial charge density distribution which are mainly localized at the layer carbon atoms. At last, we wish to stress that some configurations of janugraphene and chlorographene have been already fabricated experimentally.18-21 We also notice that nondestructive and patternable conversion of graphene is possible by using various photochemical chlorination techniques,19-21 while the research on various chlorinated graphene structures is rapidly growing. 22-24 These results indicate that practical applications of these materials would be realizable in the near future. In conclusion, we have employed first-principles calculations to show that the Dirac features are found in two novel 2D materials, chlorographene and janugraphene, alone a line from Γ to X' points in the Brillouin zone. The orbitals of near the Dirac points is predominately by the carbon orbitals of the layer plane and are almost not related to the chlorine atoms or the phenyl ligands. Our calculation shows that both materials contain carbon atoms in different chemical environments and exhibit rectangular structures. Thus the commonly considered necessary preconditions for the emergence of Dirac points in two-dimensional materials, honeycomb structure

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composed of two equivalent sublattices and hexagonal lattice symmetry, are not at all relevant for the formation of linear band dispersion in chlorographene and janugraphene. As a consequence, we propose to view the formation of Dirac fermion states from a new perspective: the existence of Dirac points in the band structure is not restricted to materials with the mentioned prerequisites. Further calculation indicates that negligible modulation in the band structure of chlorographene and janugraphene can be induced by the E-field. Computational Methods All the calculations on janugraphene and chlorographene are performed by using the plane-wave basis Vienna ab initio simulation package (VASP) code33,34 implementing the local density approximation (LDA)35 for the exchange-correlation potential. The projector augmented wave method is used to describe the electron-ion interaction.36 The cutoff energy for plane waves is chosen to be 450 eV. The periodic boundary condition is applied along xy plane, and we applied periodic boundary conditions with a vacuum space of more than 18 Å along the z directions (i.e., the direction perpendicular to the layer of the atoms) to avoid the interactions between periodical images. A (9×5×1) Monkhorst-Pack grid37 is used for the sampling of the Brillouin zone during geometry optimization, a (13×7×1) Monkhorst-Pack grid is used for the sampling of the Brillouin zone during the static total energy calculations, and more than 1400 uniform k-points along one-dimensional Brillouin zone are used to obtain the accurate band structure. For partial occupancies the Gaussian smearing method is used. The geometries, including the lattices and the positions of all the

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atoms, of all the structures are fully relaxed, and the convergence of force was set to 0.02 eV/Å. Therefore the ground-state electronic structures are calculated by applying a dipole correction38 to eliminate the artificial electrostatic field between the periodic supercells. Acknowledgement This work is supported by the National Basic Research Program of China (973 program, 2013CB632401), National Science foundation of China under Grant 11174180 and the Fund for Doctoral Program of National Education 20120131110066, Natural Science

Foundation of Shandong Province

under Grant

number

ZR2011AM009, 111 Project B13029, and the Ministry of Education Academic Award for Postgraduates. References (1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666-669. (2) Ma, Y. D.; Dai, Y.; Guo, M.; Niu, C. W.; Zhu, Y. T.; Huang, B. B. Evidence of the Existence of Magnetism in Pristine VX2 Monolayers (X = S, Se) and Their Strain-Induced Tunable Magnetic Properties. ACS Nano 2012, 6, 1695-1701. (3) Li, Y. F.; Li, F. Y.; Zhou, Z.; Chen, Z. F. SiC2 Silagraphene and Its One-Dimensional Derivatives: Where Planar Tetracoordinate Silicon Happens. J. Am. Chem. Soc. 2011, 133, 900-908. (4) Malko, D.; Neiss, C.; Gorling, A. Two-Dimensional Materials with Dirac Cones:

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Graphene Nanoribbons by Using a Ferroelectric Polymer. ACS Nano 2010, 4, 1345-1350. (31) Hod, O.; Scuseria, G. E. Half-Metallic Zigzag Carbon Nanotube Dots. ACS Nano 2008, 2, 2243-2249. (32) Li, Y. F.; Li, F. Y.; Chen, Z. F. Graphane/Fluorographene Bilayer: Considerable C–H···F–C Hydrogen Bonding and Effective Band Structure Engineering. J. Am. Chem. Soc. 2012, 134, 11269-11275. (33) Kresse, G.; Furthmuller, J. Efficiency of ab-Initio Total Energy Calculations for Metals and Semiconductors Using a Plane-Wave Basis Set. Comput. Mater. Sci. 1996, 6, 15-50. (34) Kresse, G.; Furthmuller, J. Efficient Iterative Schemes for ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set. Phys. Rev. B 1996, 54, 11169-11186. (35) Ceperley, D. M.; Alder, B. J. Ground State of the Electron Gas by a Stochastic Method. Phys. Rev. Lett. 1980, 45, 566-569. (36) Blöchl, P. E. Projector Augmented-Wave Method. Phys. Rev. B 1994, 50, 17953-17979. (37) Monkhorst, H. J.; Pack, J. D. Special Points for Brillouin-Zone Integrations. Phys. Rev. B 1976, 13, 5188-5192. (38) Makov, G.; Payne, M. C. Periodic Boundary Conditions in ab Initio Calculations. Phys. Rev. B 1995, 51, 4014-4022.

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Figure captions：： Figure 1. Top and side view of the chlorographene (a and c) and janugraphene (b and d). Dashed lines show the unit cell. Gray, orange and blue balls denote H, C and Cl atoms, respectively. Figure 2. Electronic band structures of chlorographene. All data corresponds to one unit cell, with the (a) panel corresponding to the entire band structure. Details around the Fermi level are shown in the (b) panel. The (c) panel corresponds to the band structure around the Fermi level in three dimensions. The insert in (b) panel address the first Brillouin zone with the letters designating special points. The horizontal dashed red lines indicate the Fermi level. Figure 3. Electronic band structures of janugraphene. All data corresponds to one unit cell, with the (a) panel corresponding to the entire band structure. Details around the Fermi level are shown in the (b) panel. The (c) panel corresponds to the band structure around the Fermi level in three dimensions. The insert in (b) panel address the first Brillouin zone with the letters designating special points. The horizontal dashed red lines indicate the Fermi level. Figure 4. The charge density distribution of the conduction band and the valence band near the Fermi level of (a and b) chlorographene and (c and d) janugraphene. The charge densities are plotted for a isosurface value of 0.0002 Å-3. Gray, orange and blue balls denote H, C and Cl atoms, respectively. Figure 5. Total density of states (DOS) and corresponding partial density of states (PDOS) of (a) chlorographene and (b) janugraphene. The vertical dashed line

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represents the Fermi level. Figure 6. (a) Diagram of chlorographene with the E-field. The positive direction of E-field is denoted by two arrows and the infinitely extended direction of chlorographene is perpendicular to that of E-field. Electronic band structures of (b) chlorographene and (c) janugraphene. The orange, olive and red lines in (b) and (c) corresponds to the E-field=0.003, -0.003 and 0 a.u., respectively. The horizontal dashed lines indicate the Fermi level.

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Figure 1. Top and side view of the chlorographene (a and c) and janugraphene (b and d). Dashed lines show the unit cell. Gray, orange and blue balls denote H, C and Cl atoms, respectively.

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Figure 2. Electronic band structures of chlorographene. All data corresponds to one unit cell, with the (a) panel corresponding to the entire band structure. Details around the Fermi level are shown in the (b) panel. The (c) panel corresponds to the band structure around the Fermi level in three dimensions. The insert in (b) panel address the first Brillouin zone with the letters designating special points. The horizontal dashed red lines indicate the Fermi level.

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Figure 3. Electronic band structures of janugraphene. All data corresponds to one unit cell, with the (a) panel corresponding to the entire band structure. Details around the Fermi level are shown in the (b) panel. The (c) panel corresponds to the band structure around the Fermi level in three dimensions. The insert in (b) panel address the first Brillouin zone with the letters designating special points. The horizontal dashed red lines indicate the Fermi level.

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Figure 4. The charge density distribution of the conduction band and the valence band near the Fermi level of (a and b) chlorographene and (c and d) janugraphene. The charge densities are plotted for a isosurface value of 0.0002 e/Å3. Gray, orange and blue balls denote H, C and Cl atoms, respectively.

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Figure 5. Total density of states (DOS) and corresponding partial density of states (PDOS) of (a) chlorographene and (b) janugraphene. The vertical dashed line represents the Fermi level.

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Figure 6. (a) Diagram of chlorographene with the E-field. The positive direction of E-field is denoted by two arrows and the infinitely extended direction of chlorographene is perpendicular to that of E-field. Electronic band structures of (b) chlorographene and (c) janugraphene. The orange, olive and red lines in (b) and (c) corresponds to the E-field=0.003, -0.003 and 0 a.u., respectively. The horizontal dashed lines indicate the Fermi level.

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Two supposedly ordinary two-dimensional materials, janugraphene and chlorographene, feature Dirac pointseven though they possess nonequivalent bonds and rectangular lattices. 43x36mm (300 x 300 DPI)


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Dirac Cones in Two-Dimensional Lattices ... - ACS Publications

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