Honeycomb Boron Allotropes with Dirac Cones: A True Analogue to

Letter pubs.acs.org/JPCL

Honeycomb Boron Allotropes with Dirac Cones: A True Analogue to Graphene Wen-cai Yi,†,‡ Wei Liu,§,∥ Jorge Botana,‡,§ Lei Zhao,‡ Zhen Liu,‡ Jing-yao Liu,*,† and Mao-sheng Miao*,‡,§ †

Laboratory of Theoretical and Computational Chemistry, Institute of Theoretical Chemistry, Jilin University, Changchun 130023, PR China ‡ Department of Chemistry & Biochemistry, California State University, Northridge, California 91330-8262, United States § Beijing Computational Science Research Center, Beijing 100193, PR China ∥ Department of Physics and Astronomy, University of California, Irvine, California 92697, United States S Supporting Information *

ABSTRACT: We propose a series of planar boron allotropes with honeycomb topology and demonstrate that their band structures exhibit Dirac cones at the K point, the same as graphene. In particular, the Dirac point of one honeycomb boron sheet locates precisely on the Fermi level, rendering it as a topologically equivalent material to graphene. Its Fermi velocity (vf) is 6.05 × 105 m/s, close to that of graphene. Although the freestanding honeycomb B allotropes are higher in energy than α-sheet, our calculations show that a metal substrate can greatly stabilize these new allotropes. They are actually more stable than α-sheet sheet on the Ag(111) surface. Furthermore, we find that the honeycomb borons form low-energy nanoribbons that may open gaps or exhibit strong ferromagnetism at the two edges in contrast to the antiferromagnetic coupling of the graphene nanoribbon edges.

G

complexity. Boron can exist in many forms, including zerodimensional (0D) clusters (such as B38, B40, B84, B80, etc.),13−16 one-dimensional (1D) nanoribbons,17−19 2D sheets (such as αseries, δ-series, χ-series.etc.),20−26 three-dimensional (3D) crystals (such as α-phase and β-phase, tetragonal T-phase, recently synthesized γ-phase, etc.),27,28 and even hydrogenated borophene.29 In one recent paper, the polymorphism of planar boron layers was demonstrated by Yakobson and co-workers using a cluster expansion method combined with first-principles density-functional theory (DFT) calculations, namely, there exist a number of stable 2D boron allotropes with similar stabilities.30,31 Interestingly, 2D boron allotropes show the most structural variations. Many have been investigated theoretically or experimentally,32−34 including α-, β-, δ-, and χ-type sheets20 and Pmmm, Pmmn,35 P6/mmm,36 and P21/c37 boron phases. 2D boron sheets are composed of triangular and hexagonal motifs and exhibit very different electronic properties. They can be semiconductors or metals, depending on the structure. Several 2D boron allotropes, such as Pmmn and P6/mmm, exhibit distorted Dirac cone and double distorted Dirac cones, respectively. However, both Pmmn and P6/mmm phases are not strictly planar, and their structures are distinctively different from the single-atom layer honeycomb lattice of graphene.

raphene, a single-layer two-dimensional (2D) material composed of carbon atoms with a strictly planar honeycomb lattice, has many significant properties and is a promising candidate for applications in future technologies.1,2 Besides the peculiar chemical and mechanical properties, graphene also possesses exceptional carrier mobility and ballistic transport properties, which originate from its unique electronic band structure, the Dirac cone structure.3,4 Thus, the search for the other 2D materials that present Dirac cones has attracted considerable interest. Many carbon allotropes with Dirac cones or distorted Dirac cones in their band structure have been investigated, including α-, β-, and 6,6,12-graphyne; S-, D-, E-, and r-graphene; and phagraphene.5−7 Many of them are not strictly planar and are composed of not only sp2 hybridized carbon atoms but also sp and sp3 hybridized carbon atoms. Moreover, some of these structures have rectangular lattices, which demonstrates that the hexagonal symmetry is not a prerequisite for the Dirac cone. On the other hand, 2D Dirac materials composed of elements other than carbon are also predicted.8 Using first-principles calculations, Cahangirov and co-workers reported that silicene and germanene possess stable low-buckled honeycomb structures and are semimetals with Dirac cones.9−11 2D Dirac materials with strictly planar structures are very rare compared with the other 2D materials.12 Boron, the neighbor of carbon in the periodic table, has also received considerable attention for its chemical and structural © XXXX American Chemical Society

Received: April 12, 2017 Accepted: May 30, 2017 Published: May 31, 2017 2647

DOI: 10.1021/acs.jpclett.7b00891 J. Phys. Chem. Lett. 2017, 8, 2647−2653

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The Journal of Physical Chemistry Letters Recently, two other important boron planar structures that locally reserve the honeycomb hexagonal symmetry have been proposed and studied. They show Dirac cones in their band structure.38,39 However, the Dirac points in these structures do not locate at the Fermi level. Despite the numerous studies on 2D boron allotropes, several questions still remain: Can a boron sheet with planar structure possess a Dirac point at the Fermi level? Is there a 2D boron allotrope that is truly analogous to graphene in both atomic and electronic structures? In this Letter, we report, for the first time, that a set of planar boron allotropes can exhibit Dirac cones, and the Dirac point of one allotrope locates at the Fermi level. Remarkably, these structures have honeycomb topology similar to that of graphene. The stability and electronic structures of a set of similar structures are studied by first-principles density functional calculations. Their dynamical and thermal stabilities were confirmed by the phonon-spectrum and ab initio molecular dynamics simulation. Furthermore, inspired by their similarity to graphene, we also studied the structures and properties of the nanoribbons of these honeycomb borons. All the proposed allotropes are metallic, and one shows ferromagnetism with a considerable spin-moment. All the DFT calculations were carried out within the generalized gradient approximation (GGA) parametrized by Perdew, Burke, and Ernzerhof (PBE)40 implemented by the Vienna ab initio Simulation Package code (VASP).41,42 A planewave basis set with an energy cutoff of 400 eV was used, and the Monkhorst−Pack scheme with the grid of 9 × 9 × 1 and 15 × 15 × 1 k-mesh was sampled for the Brillouin zone integration in structure relaxation and densities of states (DOS) calculations. Periodic boundary conditions (PBC) were applied, and a vacuum space of about 15 Å was included in all the models to remove any interactions between layers. The geometry optimization convergence criterion was 0.01 eV/Å for all the interatomic forces, and the self-consistent convergence was set to 1 × 10−5 eV/atom. Spin polarization was considered in all the calculations. Because standard DFT may fail to accurately describe magnetism and band structure, the hybrid functional calculations based on the Heyd− Scuseria−Ernzerhof (HSE06)43 functional have been also carried out. As illustrated in Figure 1, a series of graphene-like boron allotropes can be constructed. They have the same topology as the honeycomb lattice of graphene. We use the coordination number (CN) of boron atoms to classify the type of these boron allotropes, following the same nomenclature defined in ref 20. In these boron allotropes, the coordination numbers of boron atoms are either 4 or 5; therefore, these structures could be classified as χ-type. We name them as χ-hn (χ-type with holes, n = 0, 1, 2, 3, 4, ...). All the proposed structures have the same space group of P6/m, and most of them except χ-h2 are planar with no corrugation along the z-direction after thorough geometry optimizations. These structures are closely related to the δ5 boron sheet (also named B1/7) in a previous study.20,44 χ-h0 is identical to δ5, in which all boron atoms have a coordination number of 5. The δ5 boron is the most stable structure in all four δ-type boron sheets.20 The currently proposed structures are composed of triangular and hexagonal motifs. The χ-hn structures can be viewed as inserting boron dimers at the side of hexagonal rings in the χ-h0 (δ5) structure. With increasing number n, the atomic density decreases, while the topology remains the same as that of the graphene

Figure 1. Structures of (a) χ-h0, (b) χ-h1, (c) χ-h2, (d) χ-h3, (e) χ-h4, and (f) χ-h5. The green and pink balls are boron atoms with the coordination number of 4 and 5, respectively. The structures of each honeycomb boron allotrope can be found in Table S1.

honeycomb lattice. It is interesting to note that the inserted boron atoms are 4-fold coordinated, while the original boron atoms as in the δ5 structure are 5-fold coordinated. The calculated equilibrium lattice constants are listed in Table 1. The lattice vectors a (b) increase from 7.31 Å in χ-h1 Table 1. Calculated Lattice Constants, Atom Densities, and Energy of the Boron Structures from GGA (PBE) Results allotrope

a (Å)

density (atom/Å2)

energy (eV/atom)

on Ag(111) (eV/atom)

α-sheet χ-h0 χ-h1 χ-h2 χ-h3 χ-h4 χ-h5

5.06 4.47 7.31 10.05 12.85 15.63 18.41

0.36 0.35 0.26 0.21 0.17 0.14 0.12

−6.31 −6.29 −6.17 −6.08 −6.10 −6.07 −6.07

−6.32 −6.31 −6.46 −6.47 − − −

to 18.41 Å in χ-h5. The B−B bond lengths strongly depend on the coordination numbers. The B−B length is the longest (∼1.79 Å) while both B atoms are 5-fold coordinated. While both B atoms are 4-fold coordinated, the B−B lengths are around 1.58−1.68 Å, depending on the allotrope and the location of the B−B pair. The B−B lengths between 4-fold and 5-fold B atoms are around 1.65 Å. Therefore, the longest B−B bonds locate at the apex of the hexagonal rings, while the B−B lengths at the sides of the hexagonal rings are shorter. With increasing n, the sides of the hexagonal rings become more substantial. They can be viewed as a double-line nanoribbon; a B38 cluster consisting of such nanoribbon and forming a double-ring tubular (DRT) has been found in literature to have a quite low formation energy,13 and that one-dimensional boron ribbon was calculated and analyzed by Yakobson and coworkers recently.19 Among all the allotropes, only χ-h2 significantly deviates from a planar structure. 2648

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Figure 2. (a) DOS and (b) band structures of χ-h0, χ-h1, χ-h2, χ-h3, χ-h4, and χ-h5. The dashed line shows the Fermi level.

hn structures, we found the highest temperature that they maintain the honeycomb lattice are 2000, 1300, 1200, 1000, and 1000 K for χ-h0, χ-h2, χ-h3, χ-h4, and χ-h5, respectively (Figure S2). Furthermore, we also examined the dynamic stability of honeycomb B allotrope by calculating the phonon spectrum. This study is conducted for χ-h0, χ-h1, χ-h2, and χ-h3, the most significant structures. As shown in Figure S3a−d, there are no imaginary frequencies, indicating the structure is dynamically stable. The highest frequency of the χ-h1 sheet reaches 40.69 THz (1357 cm−1), higher than the highest frequency of 33.97 THz (1167 cm−1) in α1-sheet and 34.40 THz (1148 cm−1) in α′-sheet.20 The highest frequencies happen to the B−B stretching modes of the 4-fold B−B bonds. Because of the lower coordination number, the B−B bonds are shorter and stronger, corresponding to higher stretching frequencies. We now turn our attention to the band structure and density of states of the honeycomb boron sheets. As illustrated in Figure 2a,b, all the χ-hn boron sheets are metallic. Strikingly, the band structure of χ-h1 shows two linear bands crossing at the high symmetric point K, indicating a possible Dirac cone. This is consistent with the DOS of χ-h1 that shows zero state at the Fermi level. Similar linear band crossing can also be seen at the K point in the band structures of other honeycomb boron allotropes. However, the cross points are not at the Fermi level. For example, they are slightly lower than the Fermi level for χh3 and χ-h5, which is the result of the electron transfer from the adjacent valence bands to the unoccupied states above the cross points. Inspired by the quantum spin Hall effect in graphene, we also examined the influence of spin−orbit interaction (SOI) on the band structure of honeycomb boron allotropes. The calculations show that the effect is very small, and there is no gap opening in contrast to graphene, which can be understood by the fact that B is even lighter than C and has smaller SOI.

The calculated energies per atom and the atomic densities (number of B atoms per unit area), are listed in Table 1. The results show that χ-h0 (δ5) has the lowest energy among the χhn series. Its energy is only 20 meV/atom higher than the αsheet, which is close to that in previous reports.20 χ-h1 is about 0.14 eV/atom higher in energy than the α-sheet. Although the energy is higher, it is still quite low when compared with the situation in 2D carbon allotropes. For example, the recently proposed phagraphene is 0.20 eV/atom higher in energy than graphene,7 whereas the graphyne is 0.71 eV/atom higher than graphene.45 The energies of larger honeycomb B sheets are typically 0.21−0.25 eV higher than that of the α-sheet. Similar to many other 2D materials, honeycomb borons might be stabilized by the substrates. We performed calculations of χ-h0, χ-h1, and χ-h2 on Ag(111) surface (Figure S1). For comparison, similar calculations are performed for αsheet on the same surface. The lattice mismatch of the boron allotropes are controlled within 1% by matching different size of in-plane unit cells. The energies of boron are obtained by removing the Ag slab energies from the total energies of B/ Ag(111). The results are also listed in Table 1. Our calculations reveal that while placing on the Ag(111) surface, the honeycomb boron allotropes become lower in energy than αsheet, indicating the strong possibility of growing these materials on metal surfaces. Now, we examine the stability of these honeycomb structures by molecular dynamics (MD) simulations. The simulations were run for 3 ps containing 2000 MD steps. The χ-h1 remains the same after 3 ps of MD simulation at 1600 K (Figure S2b), although the atoms shifted along a perpendicular direction, developing ripples. Similar ripples also happened in MD simulation for a 5 × 5 unit cell of graphene. At 1800 K, the lattice of χ-h1 starts to severely deform around the apex of the hexagonal ring. Performing similar MD simulations for other χ2649

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Figure 3. (a) Snap of planar and nonplanar structure of χ-h2; (b) densities of states of planar χ-h2 without considering spin; (c) densities of states of planar χ-h2 considering spin; (d) densities of states of nonplanar χ-h2.

Figure 4. (a) Comparison of band structures from DFT (black line) and HSE (orange line) model and projected density of states of χ-h1 (calculated at PBE level). The corresponding DOS is zero at the Fermi level (dashed line). Inserted first BZ with high-symmetry k points: Γ (0,0,0), K (−0.33,0.67,0), M (0.50,0,0). (b) Dirac cone formed by the valence and conduction bands in the vicinity of the Dirac point of χ-h1. (c) Dirac cone formed by the valence and conduction bands in the vicinity of the Dirac point of χ-h3. (d) Partial charge density distributions near the Dirac cone of χ-h1. (e) Partial charge density distributions near the Dirac cone of χ-h3.

Among all the honeycomb borons, χ-h2 is the only one that exhibits a buckled structure (Figure 3a). The overall thickness of the layer is about 0.5 Å. The origin of this instability can be revealed by the DOS of planar structure that shows a peak at the Fermi level (Figure 3b). This high DOS at the Fermi level can cause instability. There are two ways to alleviate this issue, and they both include the split of the DOS. In one way, the materials can become magnetic and split into majority and minority spin channels (Figure 3c), causing the shift of the Fermi level to a low DOS. With spin polarized calculations, we found this method can lower the energy by about 18.3 meV. In another way, χ-h2 allotrope can lower the structure symmetry by relaxing in z directions, which will also split the DOS (Figure 3d). The calculations show that the energy can be lowered by 139.0 meV, which is more significant. The cross of two linear bands at the K point in χ-h1 indicates a Dirac cone. The same band structure calculation using the HSE hybrid functional shows the same band crossing, ruling

out the possibility of accidental degeneracy by PBE (Figure 4a). In order to prove band cross is a Dirac point, we calculated the states in the two-dimensional reciprocal space around the K point. As shown in Figure 4b, the results clearly elaborate a Dirac cone that is highly isotropic. The Fermi velocities of the particles at the Dirac point can be obtained by the expression vf = E(q)/ℏ|q|, which can be evaluated via the slope of the bands. While fitting the bands to the above equation with ki = Ki + q, we found a value of ±25 eV Å in the kx direction, which corresponds to a velocity of vfx = 6.05 × 105 m/s. This velocity is very close to the velocity of 8.22 × 105 m/s in graphene.6 Therefore, the honeycomb boron sheet may have exceptional transport properties, similar to graphene. The exhibition of the Dirac cone at K point may arise from the topology of the structure. As a matter of fact, all the honeycomb boron sheets possess Dirac cones at K points, although for most of them, the Dirac cones do not locate at the Fermi level, and often it is degenerate with other bands. For 2650

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Figure 5. (a) Structure of zigzag nanoribbon; (b) structure of armchair-1 nanoribbon; (c) structure of armchair-2 nanoribbon; (d) densities of states of zigzag borophene nanoribbon; (e) densities of states of armchair-1 borophene nanoribbon; (f) densities of states of armchair-2 borophene nanoribbon. All the nanoribbons are constructed from χ-h1 structure.

lengths are about 1.6−1.8 Å, which are significantly longer than the C−C bond length in graphene. The longer B−B bond lengths indicate weaker p−p bonding, leading to smaller energy scale of the states compared with that of graphene. Inspired by the similarity of honeycomb borons and graphene, we proceed to study the structure and properties of the analogue of graphene nanoribbons. As shown in Figure 5, three ribbons are created from the χ-h1 structure. The first one is analogous to the zigzag graphene nanoribbon (Figure 5a), and the last two are analogous to the armchair nanoribbons (Figure 5b,c). The two armchair structures (armchair-1 and armchair-2) are different on the boron edges. The edge of the former structure consists of double boron strip, whereas the edge of the later consists of a single B chain. The energies of these structures are −5.99, −6.07, and −5.94 eV/atom, respectively, which are not significantly higher than that of χh1 (−6.17 eV/atom, as shown in Table 1). For comparison, 8zigzag graphene nanoribbon is 0.40 eV/atom higher than graphene. Therefore, it is relatively easier to form honeycomb boron nanoribbons. The electronic structures of honeycomb boron nanoribbons are shown in Figure 5d−f. Similar to graphene, the zigzag nanoribbon and the armchair-1 nanoribbon open gaps of 0.18 and 0.05 eV for χ-h1, respectively, as calculated by the PBE functional. It is worth noting that the 2D boron allotropes reported so far are all metallic. More interestingly, armchair-2 nanoribbon yields a ferromagnetic state on its edge; the spin-polarized electron density is plotted in Figure S4. This ferromagnetic state is 304.11 meV lower in energy than the nonspin-polarized state. The total spin moment in the ferromagnetic state of armchair-2 borophene nanoribbon is 2.92 μB per unit. We also calculated the antiferromagnetic state with opposite spins locating at two different edges, and we found it is 48.58 meV higher in energy than the ferromagnetic state. This is quite different from zigzag graphene nanoribbon,

example, there is a Dirac cone point located at 0.65 eV above the Fermi level in χ-h0. The corresponding Fermi velocity is 11.21 × 105 m/s. Similarly, Dirac cones exist in the bands of χh2 and χ-h3 boron sheets. The Dirac cone in χ-h3 is only 0.18 eV below the Fermi level (Figure 4c). It is caused by the electron transfer from the valence band maximum at the point to the Dirac cone states. Therefore, the system is a self-doped Dirac material, in which the Dirac electrons and Fermi holes coexist. The partial charge densities (mode square of the wave functions) for the two degenerated states at Dirac points (Figure 4d,e) show that the two states have different symmetry but have the same number of nodes, similar to the degenerated states at the Dirac point in graphene. With the increasing size of honeycomb boron sheet, the number of nodes also increases. Dirac cones in band structures have received great attention in many recent works.46 In previous studies, Pmmn boron sheet,35 P6/mmm boron sheet,36 and P21/c boron sheet37 were reported to possess Dirac cones in their electronic structures. However, these allotropes are not strictly planar, and their structures are distinctively different from the single-atom layer honeycomb lattice of graphene. Recently, two boron planar structures that locally reserve the honeycomb hexagonal lattice have been proposed and studied. They also show Dirac cones in their band structure.38,39 However, the Dirac points in these structures do not locate at the Fermi level. In contrast, χ-h1 is a strict planar boron sheet that possesses a Dirac cone with its crossing point exactly at the Fermi level. Therefore, it is the best exemplar boron allotrope that resembles the honeycomb lattice and the electronic structure of graphene. The projected DOS (Figure 4a) not only shows that the states around the Dirac point mainly consist of B 2pz orbitals, the same as in graphene, but also resembles that of graphene. However, the two signature peaks shown in the figure are much closer to the Dirac point. This is consistent with the fact that the B−B bond 2651

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(3) Zhang, Y.; Tan, Y. W.; Stormer, H. L.; Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 2005, 438, 201−204. (4) Bolotin, K. I.; Ghahari, F.; Shulman, M. D.; Stormer, H. L.; Kim, P. Observation of the fractional quantum Hall effect in graphene. Nature 2009, 462, 196−199. (5) Malko, D.; Neiss, C.; Viñes, F.; Görling, A. Competition for Graphene: Graphynes with Direction-Dependent Dirac Cones. Phys. Rev. Lett. 2012, 108, 086804. (6) Xu, L. C.; Wang, R. Z.; Miao, M. S.; Wei, X. L.; Chen, Y. P.; Yan, H.; Lau, W. M.; Liu, L. M.; Ma, Y. M. Two dimensional Dirac carbon allotropes from graphene. Nanoscale 2014, 6, 1113−1118. (7) Wang, Z.; Zhou, X. F.; Zhang, X.; Zhu, Q.; Dong, H.; Zhao, M.; Oganov, A. R. hagraphene: A Low-Energy Graphene Allotrope Composed of 5-6-7 Carbon Rings with Distorted Dirac Cones. Nano Lett. 2015, 15, 6182−6186. (8) Pu, C.; Zhou, D.; Li, Y.; Liu, H.; Chen, Z.; Wang, Y.; Ma, Y. TwoDimensional C4N Global Minima: Unique Structural Topologies and Nano-electronic Properties. J. Phys. Chem. C 2017, 121, 2669−2674. (9) Vogt, P.; De Padova, P.; Quaresima, C.; Avila, J.; Frantzeskakis, E.; Asensio, M. C.; Resta, A.; Ealet, B.; Le Lay, G. Silicene: compelling experimental evidence for graphenelike two-dimensional silicon. Phys. Rev. Lett. 2012, 108, 155501. (10) Cahangirov, S.; Audiffred, M.; Tang, P.; Iacomino, A.; Duan, W.; Merino, G.; Rubio, A. Electronic structure of silicene on Ag(111): Strong hybridization effects. Phys. Rev. B: Condens. Matter Mater. Phys. 2013, 88, 035432. (11) Dávila, M. E.; Xian, L.; Cahangirov, S.; Rubio, A.; Le Lay, G. Germanene: a novel two-dimensional germanium allotrope akin to graphene and silicene. New J. Phys. 2014, 16, 095002. (12) Wang, J.; Deng, S.; Liu, Z.; Liu, Z. The rare two-dimensional materials with Dirac cones. Natl. Sci. Rev. 2015, 2, 22−39. (13) Lv, J.; Wang, Y.; Zhu, L.; Ma, Y. B38: an all-boron fullerene analogue. Nanoscale 2014, 6, 11692−11696. (14) Zhai, H. J.; Zhao, Y. F.; Li, W. L.; Chen, Q.; Bai, H.; Hu, H. S.; Piazza, Z. A.; Tian, W. J.; Lu, H. G.; Wu, Y. B.; et al. Observation of an all-boron fullerene. Nat. Chem. 2014, 6, 727−731. (15) Gonzalez Szwacki, N.; Sadrzadeh, A.; Yakobson, B. I. B80 Fullerene: AnAb InitioPrediction of Geometry, Stability, and Electronic Structure. Phys. Rev. Lett. 2007, 98, 166804. (16) Rahane, A. B.; Kumar, V. B84: a quasi-planar boron cluster stabilized with hexagonal holes. Nanoscale 2015, 7, 4055−4062. (17) Xu, T. T.; Zheng, J.-G.; Wu, N.; Nicholls, A. W.; Roth, J. R.; Dikin, D. A.; Ruoff, R. S. Crystalline Boron Nanoribbons: Synthesis and Characterization. Nano Lett. 2004, 4, 963−968. (18) Tang, H.; Ismail-Beigi, S. Novel precursors for boron nanotubes: the competition of two-center and three-center bonding in boron sheets. Phys. Rev. Lett. 2007, 99, 115501. (19) Liu, M.; Artyukhov, V. I.; Yakobson, B. I. Mechanochemistry of One-Dimensional Boron: Structural and Electronic Transitions. J. Am. Chem. Soc. 2017, 139, 2111−2117. (20) Wu, X.; Dai, J.; Zhao, Y.; Zhuo, Z.; Yang, J.; Zeng, X. C. Twodimensional boron monolayer sheets. ACS Nano 2012, 6, 7443−7453. (21) Yu, X.; Li, L.; Xu, X.; Tang, C. Prediction of Two-Dimensional Boron Sheets by Particle Swarm Optimization Algorithm. J. Phys. Chem. C 2012, 116, 20075−20079. (22) Li, X. B.; Xie, S. Y.; Zheng, H.; Tian, W. Q.; Sun, H. B. Boron based two-dimensional crystals: theoretical design, realization proposal and applications. Nanoscale 2015, 7, 18863−18871. (23) Zhang, Z.; Yang, Y.; Gao, G.; Yakobson, B. I. Two-Dimensional Boron Monolayers Mediated by Metal Substrates. Angew. Chem. 2015, 127, 13214−13218. (24) Kou, L.; Ma, Y.; Tang, C.; Sun, Z.; Du, A.; Chen, C. Auxetic and Ferroelastic Borophane: A Novel 2D Material with Negative Possion’s Ratio and Switchable Dirac Transport Channels. Nano Lett. 2016, 16, 7910−7914. (25) Penev, E. S.; Kutana, A.; Yakobson, B. I. Can Two-Dimensional Boron Superconduct? Nano Lett. 2016, 16, 2522−2526.

in which the spins at two edges are coupled antiferromagnetically. Also, the energy difference for the honeycomb boron is much larger, indicating superior magnetic properties of boron nanoribbons. In summary, we have found a set of planar boron allotropes adopting honeycomb lattices to possess Dirac cones at the K high-symmetry point in their band structure, in good analogue to graphene. These allotropes can be greatly stabilized by metal substrate and become lower in energy than boron when placed on a Ag(111) surface. All the allotropes are metallic. Depending on the size of the hexagonal cell, the Dirac point may locate at the Fermi level or closely above or below. The Fermi velocity of the Dirac particles in these materials is similar to that of graphene, indicating superior transport properties and a potential to be used as a Dirac material in future electronic devices. Furthermore, we found that the honeycomb boron can form nanoribbons with low energies. Depending on the direction and the edge structures, they may open energy gaps or exhibit strong ferromagnetism among their spins located at two edges.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpclett.7b00891. Structures of χ-h0, χ-h1, and χ-h2 grown on Ag(111) surface; snapshots of the final configurations for molecular dynamics simulation of χ-hn structures; phonon dispersion of χ-h0, χ-h1, χ-h2, and χ-h3; structure information for χ-hn structures (PDF)

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected]. *E-mail: [email protected]. ORCID

Wen-cai Yi: 0000-0003-3815-3435 Wei Liu: 0000-0001-9256-9066 Jorge Botana: 0000-0002-5406-3096 Jing-yao Liu: 0000-0003-4060-8830 Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (Grant 21373098). The authors are grateful to the referees for their valuable comments, which significantly improved the manuscript. Part of the calculations are performed on NSF-funded XSEDE resources (TGDMR130005) especially on the Stampede cluster run by Texas Advanced Computing Center.

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REFERENCES

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DOI: 10.1021/acs.jpclett.7b00891 J. Phys. Chem. Lett. 2017, 8, 2647−2653

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DOI: 10.1021/acs.jpclett.7b00891 J. Phys. Chem. Lett. 2017, 8, 2647−2653