Abstract
Based on the firstprinciples evolutionary materials design, we report a stable boron Kagome lattice composed of triangles in triangles on a twodimensional sheet. The Kagome lattice can be synthesized on a silver substrate, with selecting Mg atoms as guest atoms. While the isolated Kagome lattice is slightly twisted without strain, it turns into an ideal triangular Kagome lattice under tensile strain. In the triangular Kagome lattice, we find the exotic electronic properties, such as topologically nontrivial flat band near the Fermi energy and halfmetallic ferromagnetism, and predict the quantum anomalous Hall effect in the presence of spinorbit coupling.
Introduction
Recently, twodimensional (2D) materials have attracted much attention because of their unusual characteristics, such as Dirac fermions, topological states, and valley polarization^{1,2,3}. The discovery of graphene, a monolayer of carbon atoms in the honeycomb lattice^{1}, has resulted in the synthesis of different 2D materials and various graphene analogues, such as hexagonal BN, transition metal dichalcogenides, silicene, germanene, stanene, and phosphorene^{3,4,5}. Elemental boron exhibits a variety of allotropes with structural complexity^{6, 7}, but a 2D honeycomb lattice is inherently prohibited because boron lacks one valence electron compared to its adjacent carbon in the periodic table. A flat triangular boron lattice has been reported to be unstable to a buckled shape due to excessive electrons occupying the antibonding states^{8}. Since the surplus of electrons can be balanced by introducing hexagonal holes in the triangular lattice, more stable 2D forms composed of triangular and hexagonal motifs have been proposed, including the B αsheet and its analogues^{8, 9}.
A Kagome lattice also consists of triangular and hexagonal motifs in a network of cornersharing triangles. Many exotic phenomena have been predicted for the Kagome lattice, such as frustrated magnetic ordering^{10,11,12}, ferromagnetism^{13, 14}, and topologically nontrivial states^{15,16,17,18}. However, the experimental realization of the Kagome lattice is confined to the Kagome layers of pyrochlore oxides^{19}, the Cu ions sitting on a Kagome lattice in Herbertsmithite^{12}, the selfassembled metalorganic molecules on a substrate^{20}, and the cold atoms of an optical Kagome lattice^{21}. A B_{3} Kagome lattice, in which a triangular motif is made of three B atoms, is known to be dynamically unstable due to the lack of electrons^{22}. Theoretical calculations indicate that, in the case of 2D metalB systems such as MoB_{4} ^{23}, TiB_{2} ^{24}, FeB_{6} ^{25}, FeB_{2} ^{26}, and MnB_{6} ^{27}, triangular or honeycomb B networks can be stabilized by the electron transfer from metal ions to B networks and the interaction between metal and B layers. Similarly, a bilayer form of MgB_{6} sandwiching the Mg layer between two B_{3} Kagome layers was found to be stable^{22}. Despite a number of theoretical attempts to predict 2D boron allotropes, only a few 2D boron sheets have been synthesized on metal substrates, such as a quasi2D layer of γB_{28} ^{28}, a 2D triangular sheet (generally referred to as borophene)^{29}, and borophene with stripepatterned vacancies^{30}. Given the structural diversity of metalB systems, a proper choice of substrate and metal elements as guest atoms can open the way to realizing stable 2D boron Kagome sheets that have not been discovered yet.
In this work, we perform an evolutionary crystal structure search for 2D boron phases with the Mg atoms as guest atoms on a silver substrate. We find a new 2D boron sheet consisting of triangular B networks and the Mg atoms embedded in large hexagonal voids. The boron sheet separated from the substrate forms a twisted Kagome lattice and turns into an ideal triangular Kagome lattice under tensile strain, accompanied with a metaltohalfmetal transition. The ferromagnetism of the triangular Kagome lattice is characterized by a nearly flat band at the Fermi level, which is topologically nontrivial and thus induces the quantum anomalous Hall effect in the presence of spinorbit coupling.
Results
Crystal structure search
First, we explored twodimensional MgB allotropes with low energies on the substrate by using an ab initio evolutionary crystal structure search method, as implemented in the AMADEUS code^{31}. Distinct configurations were generated under the constraint of layer group symmetry, with the number of configurations setting to 20 in the population size of global optimization. For each configuration, the energy minimization was performed by using the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE)^{32} for the exchangecorrelation potential and the projector augmented wave pseudopotentials^{33}, as implemented in the VASP code^{34}. The details of calculations are given in Method.
We considered monolayer and bilayer coverages of the B atoms with the Mg atoms as guest atoms for two lateral supercells on the Ag(111) surface, rhombic 2 × 2 and rectangular \(2\times 2\sqrt{3}\) (Fig. 1a). If a 2D triangular sheet, which has been synthesized on a Ag substrate^{29}, is taken as the most stable structure for the monolayer coverage, the B atoms can be deposited up to 12 in the 2 × 2 supercell (Fig. 2a). Since the Mg atoms prefer to occupy the hollow sites on the Ag(111) surface, they can be deposited up to 4 for the monolayer coverage, and three B atoms per Mg atom are depleted. In the bilayer coverage, the Mg atoms pop into the empty space between the two B layers, keeping the same maximum coverage of 4, and each Mg atom depletes six B atoms. Considering such combinations of Mg and B, 2D MgB phases can be represented as Mg_{ m }B_{ n } with 0 ≤ m ≤ 4 and 0 ≤ n ≤ 12–3 m (0 ≤ n ≤ 24–6 m) for the B monolayer (bilayer) coverage in the 2 × 2 supercell.
We searched for the lowestenergy configurations for Mg_{ m }B_{ n } systems and compared their formation energies defined as,
where E _{ tot }(A) is the total energy of the system A and x = m/4, y = n/24, and z = 1 − x − y. Here, Mg_{4}B_{0} and Mg_{0}B_{24} correspond to the Mg monolayer and the B bilayer on the Ag surface (Fig. 2b and c), respectively, whereas Mg_{0}B_{0} denotes the bare Ag substrate. The bilayer form of Mg_{0}B_{24} is energetically more favorable than the Mg_{0}B_{12} monolayer on the Ag substrate. Thus, the chemical potentials of Mg and B on the substrate are defined as, μ _{Mg} = [E _{ tot }(Mg_{4}B_{0}) − E _{ tot }(Mg_{0}B_{0})]/4 and μ _{B} = [E _{ tot }(Mg_{0}B_{24}) − E _{ tot }(Mg_{0}B_{0})]/24. The phase diagram of Mg_{ m }B_{ n } is shown in Fig. 1b, and the configurations on the convex hull are energetically stable against the decomposition into other phases. Among Mg_{ m }B_{ n } systems, we find that Mg_{1}B_{9} lies on the convex hull for the monolayer coverage (Fig. 1c). The stability of Mg_{1}B_{9} was confirmed through the tests for both the 2 × 2 and \(2\times 2\sqrt{3}\) supercells. The Mg_{1}B_{9} allotrope consists of nine B atoms in slightly buckled triangular networks and one Mg atom occupying a large hole in the unit cell (Fig. 1d). As the boron coverage increases, we obtained a bilayer form of Mg_{1}B_{18} on the convex hull, in which the Mg atoms are sandwiched between the two B layers, lying in between the empty holes. In the Mg_{1}B_{18} allotrope, each B layer has the same triangular network as that of Mg_{1}B_{9}, however, its buckling is enhanced due to interlayer interactions (Fig. 2d).
Atomic structure and stability
A 2D boron Kagome lattice can be obtained from Mg_{1}B_{9} by exfoliating the MgB sheet from the substrate and removing the guest atoms. The isolation of a 2D boron sheet can be made by using various exfoliation techniques^{35, 36}. In a freestanding Mg_{1}B_{9} sheet, the Mg ions bind weakly to the boron networks with a smaller binding energy of about 0.7 eV/Mg (Supplementary Table S1), compared with other magnesium borides which were suggested as the potential cathode materials for Mgion batteries^{37}. This result indicates that the Mg ions in the Mg_{1}B_{9} sheet can be dissolved in conventional electrolytes used for Mg batteries^{38, 39}. In the optimized Mgfree B_{9} sheet, called a twisted Kagome lattice (denoted as B_{9}tKL), three B atoms are depleted per empty hole, and each void is surrounded by six large triangles (Fig. 3a). As biaxial tensile strain (ε) is applied, the voids are enlarged, and the bonds connecting large triangular units are subsequently broken, resulting in an ideal triangular Kagome lattice (denoted as B_{9}KL)^{40, 41}, as shown in Fig. 3b. We find that the B_{9}tKL and B_{9}KL sheets are perfectly flat, while the remaining Mg ions cause buckling or twisting of the B networks due to the charge transfer. Both B_{9}tKL and B_{9}KL belong to the general Kagome system with the subnet 2, in which each triangle of the Kagome arrangement contains a stack of four triangles^{42}. The lattice parameters, plane groups, and Wyckoff positions of B_{9}tKL and B_{9}KL are given in Table 1.
To estimate the critical strain (ε _{ c }) for the transition from B_{9}tKL to B_{9}KL, we calculated the 2D biaxial stress (σ ^{2D}) as a function of strain. In the stressstrain curve, we find two stress drops at ε = 9.5% and 13% (Fig. 3d). The prominent drop at ε = 9.5% is accompanied with bondbreaking relaxations between the large triangular units, whereas the weak drop at ε = 13% is related to a transition to the ferromagnetic state (which will be discussed shortly). When the bonds between the edge B atoms of large triangles are broken, the coordination number of the edge B atoms is reduced from 5 to 4. Then, a charge transfer of 2.5 electrons occurs from six edge atoms to three corner atoms within the unit cell (Table 1). As strain increases above 9.5%, the angle between the large triangular units (θ) increases rapidly and reaches 120° at the critical strain of 16.5%, where B_{9}KL is formed (Fig. 3d). For strain above 16.5%, B_{9}KL experiences only elastic deformation and maintains the angle of θ = 120°. For strain up to 24%, overall the calculated stress is below 16N/m, lying in the stress range accessible by using an atomic force microscope, as demonstrated for various 2D materials, such as graphene and transitionmetal dichalcogenides^{43, 44}.
The stability of B_{9}tKL and B_{9}KL was verified by calculating the full phonon spectra and performing firstprinciples molecular dynamics simulations at high temperatures (Fig. 4 and Supplementary Figure S1). Among 2D boron sheets composed of triangular and hexagonal motifs, a B_{3} Kagome lattice (denoted as B_{3}KL) also has a network of cornersharing triangles, similar to B_{9}tKL and B_{9}KL. In B_{3}KL, however, each triangle of the Kagome arrangement is made of three B atoms, and a single B atom is depleted in each hexagonal hole (Fig. 3c). It is known that B_{3}KL is dynamically unstable due to oneelectron deficiency to fully occupy the bonding states. On the Ag substrate, we find a Mg_{3}B_{9} structure consisting of the B_{3} Kagome lattice and the Mg atoms located underneath the hexagonal holes (Fig. 2e), similar to MgB_{6} ^{22}. However, this allotrope is energetically less stable by 1.64 eV per 2 × 2 cell than the combined structure of Mg_{1}B_{9} with two additional Mg atoms sandwiched between the Mg_{1}B_{9} layer and substrate (Fig. 2f).
The valence band of B_{3}KL consists of five bonding states: two threecenter σbonding states, two σbonding states captured in hexagonal holes, and one delocalized πbonding state. In B_{9}KL, six edge atoms in two adjacent large triangular motifs form one sixcenter σbonding state, replacing for two threecenter σbonding states of B_{3}KL. While the number of the σbonding states is reduced by one, two πbonding states are fully occupied, in contrast to B_{3}KL, and the πnonbonding state is halffilled due to an exchange splitting (Supplementary Figure S2). Meanwhile, one σbonding state is further reduced due to the multicenter bonds between the edge atoms of large triangles in B_{9}tKL. Although one σantibonding state is halffilled, two πbonding and one πnonbonding states are fully filled. Therefore, it is inferred that the occupation of the delocalized πorbital states plays a crucial role in stabilizing the flat forms of both B_{9}tKL and B_{9}KL, similar to the cases of B αsheet^{45} and B clusters^{46}.
Electronic structure
We find that B_{9}KL exhibits a halfmetallic band structure, whereas B_{9}tKL is metallic but nonmagnetic (Fig. 5a). In B_{9}tKL, a less dispersive band appears at about −1.0 eV below the Fermi level, and three Diraclike bands are formed at the K point: one characterized by the p _{ x } and p _{ y } orbitals in the valence band and two by the p _{ z } orbital in both the valence and conduction bands. While strain changes the positions of three Diraclike bands, a noticeable effect is that the less dispersive band moves upward and becomes flattened. Thus, the density of states at the Fermi level singificantly increases, causing the Stoner instability^{47}. For strain above 13%, the exchange splitting occurs for the flat band, resulting in a halffilled band structure and thereby a ferromagnetic transition, similar to the flatband ferromagnetism^{13, 14}. With the hybrid functional of Heyd, Scuseria, and Ernzerhof (HSE06) for the exchangecorrelation potential^{48}, for B_{9}KL, we find that the band width of the flat band is 175 meV and the band gap of spindown electrons is 1.42 eV (Fig. 5b).
To understand the origin of the flat band in B_{9}KL, we analyzed the wave function associated with the flat band, which is mainly derived from the localized p _{ z } orbitals at the corner B atoms of triangles (Fig. 5c). By using maximally localized Wannier functions (MLWFs)^{49}, we derived an effective tightbinding Hamiltonian, with the hopping parameters, t _{1} = −2.31 eV, t _{2} = −2.10 eV, t _{3} = −0.08 eV, t _{4} = 0.29 eV, t _{5} = 0.52 eV, and t _{6} = 0.39 eV (Fig. 3b and Supplementary Method). Nine MLWFs are sufficient enough to represent the bonding characteristics between the p _{ z } orbitals and well reproduces the HSE06 band structure near the Fermi level, as illustrated in Fig. 6a. If only the nearestneighbor hopping terms are considered, a completely flat band is formed at the Fermi level because of the destructive interference of the hopping terms to adjacent hexagonal holes (Supplementary Figure S3).
In the effective tightbinding Hamiltonian, we introduced the spinorbit coupling (SOC), with the parameters λ _{SO,3} = 0.004 eV and λ _{SO,4} = 0.015 eV, which correspond to 5% of the second nearestneighbor hopping terms, t _{3} and t _{4}, respectively. Although the SOC is small for B systems, it could be enhanced by various methods, such as hydrogenation^{50}, introduction of transition metal adatoms^{51}, and substrate proximity effects^{52}, which have been used for graphene. With including the SOC, we find that the band gap of 57 meV opens at the Γ point and the flat band has the nontrivial Chern number of C = 1 (Fig. 6a). This result implies that the quantum anomalous Hall effect could be realized in B_{9}KL, provided that the SOC opens the band gap. In fact, the quantization of Hall conductance is found in B_{9}KL with the SOC (Fig. 6a and b), which is the hallmark of the quantum anomalous Hall effect. In addition, the appearance of chiral edge states inside the bulk band gap is confirmed in a onedimensional ribbon of B_{9}KL (Fig. 6c). Similaly, Guterding and coworkers predicted the quantum anomalous Hall and quantum spin Hall effects in doped herbertsmithite^{18}. They achieved the nontrivial band topology by the precise control of doping so that the Fermi level lies at the band touching points, while doping is not required in B_{9}KL.
Conclusion
In conclusion, we have predicted the ideal planar shape of a triangular B_{9} Kagome lattice, which contains triangles in triangles. Based on the ab initio evolutionary crystal structure search, we propose that the B_{9} Kagome lattice can be synthesized on the Ag(111) surface, with selecting the Mg atoms as guest atoms. The triangular B_{9} Kagome lattice offers the exotic electronic characteristics, such as flat band at the Fermi level and halfmetallic ferromagnetism, providing opportunities for spintronics applications. Because of the nontrivial band topology of the flat band, the quantum anomalous Hall effect can be realized in the presence of spinorbit coupling.
Method
Density functional calculations
We employed the generalized gradient approximation of Perdew, Burke, and Ernzerhof (PBE)^{32} for the exchangecorrelation potential and the projector augmented wave pseudopotentials^{33}, as implemented in the VASP code^{34}. In addition, van der Waals forces were taken into account to describe more accurately interlayer interactions^{53}. In a slab geometry, a vacuum region larger than 15 Å was inserted, with a dipole correction^{54}, ensuring prohibiting interactions between adjacent supercells, and only the topmost layer of the silver substrate was relaxed. The wave functions were expanded in plane waves up to an energy cutoff of 300 eV and the MonkhorstPack mesh^{55} with a grid spacing of 2π × 0.03 Å^{−1} was used for Brillouin zone integration. The atomic coordinates were optimized until the residual forces were less than 0.02 eV/Å. For the freestanding B_{9}tKL, the inplane lattice vectors were relaxed until stress was below 0.5 kbar. At the final stage of optimization, we used a higher energy cutoff of 400 eV.
References
Novoselov, K. S. et al. Twodimensional gas of massless Dirac fermions in graphene. Nature (London) 438, 197–200 (2005).
Qian, X., Liu, J., Fu, L. & Li, J. Quantum spin Hall effect in twodimensional transition metal dichalcogenides. Science 346, 1344–1347 (2014).
Geim, A. K. & Grigorieva, I. V. Van der Waals heterostructures. Nature (London) 499, 419–425 (2013).
Novoselov, K. S., Mishchenko, A., Carvalho, A. & Castro Neto, A. H. 2D materials and van der Waals heterostructures. Science 353, aac9439–aac9439 (2016).
Balendhran, S., Walia, S., Nili, H., Sriram, S. & Bhaskaran, M. Elemental Analogues of Graphene: Silicene, Germanene, Stanene, and Phosphorene. Small 11, 640–652 (2015).
Albert, B. & Hillebrecht, H. Boron: Elementary Challenge for Experimenters and Theoreticians. Angew. Chem. Int. Ed. 48, 8640–8668 (2009).
Oganov, A. R. et al. Ionic highpressure form of elemental boron. Nature (London) 457, 863–7 (2009).
Tang, H. & IsmailBeigi, S. Novel Precursors for Boron Nanotubes: The Competition of TwoCenter and ThreeCenter Bonding in Boron Sheets. Phys. Rev. Lett. 99, 115501 (2007).
Penev, E. S., Bhowmick, S., Sadrzadeh, A. & Yakobson, B. I. Polymorphism of TwoDimensional Boron. Nano Lett. 12, 2441–2445 (2012).
Balents, L. Spin liquids in frustrated magnets. Nature (London) 464, 199–208 (2010).
Nisoli, C., Moessner, R. & Schiffer, P. Colloquium: Artificial spin ice: Designing and imaging magnetic frustration. Rev. Mod. Phys. 85, 1473–1490 (2013).
Norman, M. R. Colloquium: Herbertsmithite and the search for the quantum spin liquid. Reviews of Modern Physics 88, 041002 (2016).
Mielke, A. Exact ground states for the Hubbard model on the Kagome lattice. J. Phys. A 25, 4335–4345 (1992).
Tanaka, A. & Ueda, H. Stability of Ferromagnetism in the Hubbard Model on the Kagome Lattice. Phys. Rev. Lett. 90, 067204 (2003).
Ohgushi, K., Murakami, S. & Nagaosa, N. Spin anisotropy and quantum Hall effect in the kagomé lattice: Chiral spin state based on a ferromagnet. Phys. Rev. B 62, R6065–R6068 (2000).
Guo, H.M. & Franz, M. Topological insulator on the kagome lattice. Phys. Rev. B 80, 113102 (2009).
Tang, E., Mei, J.W. & Wen, X.G. HighTemperature Fractional Quantum Hall States. Phys. Rev. Lett. 106, 236802 (2011).
Guterding, D., Jeschke, H. O. & Valentí, R. Prospect of quantum anomalous Hall and quantum spin Hall effect in doped kagome lattice Mott insulators. Scientific Reports 6, 25988 (2016).
Gardner, J. S., Gingras, M. J. P. & Greedan, J. E. Magnetic pyrochlore oxides. Rev. Mod. Phys. 82, 53–107 (2010).
Mao, J. et al. Tunability of supramolecular kagome lattices of magnetic phthalocyanines using graphenebased moir? patterns as templates. J. Am. Chem. Soc. 131, 14136–14137 (2009).
Jo, G.B. et al. Ultracold Atoms in a Tunable Optical Kagome Lattice. Phys. Rev. Lett. 108, 045305 (2012).
Xie, S.Y. et al. A novel twodimensional MgB_{6} crystal: metallayer stabilized boron kagome lattice. Phys. Chem. Chem. Phys. 17, 1093–1098 (2015).
Xie, S.Y. et al. Firstprinciples calculations of a robust twodimensional boron honeycomb sandwiching a triangular molybdenum layer. Phys. Rev. B 90, 035447 (2014).
Zhang, L. Z., Wang, Z. F., Du, S. X., Gao, H.J. & Liu, F. Prediction of a Dirac state in monolayer TiB_{2}. Phys. Rev. B 90, 161402 (2014).
Zhang, H., Li, Y., Hou, J., Tu, K. & Chen, Z. FeB_{6} Monolayers: The Graphenelike Material with Hypercoordinate Transition Metal. J. Am. Chem. Soc. 138, 5644–5651 (2016).
Zhang, H., Li, Y., Hou, J., Du, A. & Chen, Z. Dirac State in the FeB_{2} Monolayer with GrapheneLike Boron Sheet. Nano Lett. 16, 6124–6129 (2016).
Li, J. et al. Voltagegated spinfiltering properties and global minimum of planar MnB_{6}, and halfmetallicity and roomtemperature ferromagnetism of its oxide sheet. J. Mater. Chem. C 4, 10866–10875 (2016).
Tai, G. et al. Synthesis of Atomically Thin Boron Films on Copper Foils. Angew. Chem. Int. Ed. 54, 15473–15477 (2015).
Mannix, A. J. et al. Synthesis of borophenes: Anisotropic, twodimensional boron polymorphs. Science 350, 1513–1516 (2015).
Feng, B. et al. Experimental realization of twodimensional boron sheets. Nat. Chem. 8, 563–568 (2016).
Lee, I.H., Oh, Y. J., Kim, S., Lee, J. & Chang, K. J. Ab initio materials design using conformational space annealing and its application to searching for direct band gap silicon crystals. Comput. Phys. Comm. 203, 110–121 (2016).
Perdew, J. P., Burke, K. & Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 77, 3865–3868 (1996).
Blöchl, P. E. Projector augmentedwave method. Phys. Rev. B 50, 17953–17979 (1994).
Kresse, G. & Joubert, D. From ultrasoft pseudopotentials to the projector augmentedwave method. Phys. Rev. B 59, 1758–1775 (1999).
Nicolosi, V., Chhowalla, M., Kanatzidis, M. G., Strano, M. S. & Coleman, J. N. Liquid Exfoliation of Layered. Materials. Science 340, 1226419–1226419 (2013).
Tao, L. et al. Silicene fieldeffect transistors operating at room temperature. Nat. Nanotech. 10, 1–5 (2015).
Zhao, Y., Ban, C., Xu, Q., Wei, S.H. & Dillon, A. C. Chargedriven structural transformation and valence versatility of boron sheets in magnesium borides. Phys. Rev. B 83, 035406 (2011).
Saha, P. et al. Rechargeable magnesium battery: Current status and key challenges for the future. Prog. Mater. Sci. 66, 1–86 (2014).
Zhang, R. & Ling, C. Status and challenge of Mg battery cathode. MRS Energy Sustain. 3, E1 (2016).
Norman, R. E. & Stenkamp, R. E. Structure of a copper(II) complex of 2Ccarboxypentonic acid (H_{3}cpa); [Cu_{9}Br_{2}(cpa)_{6}]_{2−n }. xH_{2}O. Acta Cryst. C 46, 6–8 (1990).
Mekata, M. et al. Magnetic ordering in triangulated kagomé lattice compound, Cu_{9}Cl_{2}(cpa)_{6}. nH_{2}O. J. Magn. Magn. Mater. 177–181, 731–732 (1998).
Ziff, R. M. & Gu, H. Universal condition for critical percolation thresholds of kagomélike lattices. Phys. Rev. E 79, 020102 (2009).
Lee, C., Wei, X., Kysar, J. W. & Hone, J. Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene. Science 321, 385–388 (2008).
Bertolazzi, S., Brivio, J. & Kis, A. Stretching and Breaking of Ultrathin MoS_{2}. ACS Nano 5, 9703–9709 (2011).
Galeev, T. R. et al. Deciphering the mystery of hexagon holes in an allboron graphene αsheet. Phys. Chem. Chem. Phys. 13, 11575 (2011).
Zhai, H.J., Kiran, B., Li, J. & Wang, L.S. Hydrocarbon analogues of boron clusters — planarity, aromaticity and antiaromaticity. Nat. Mater. 2, 827–833 (2003).
Stoner, E. C. Ferromagnetism. Rep. Prog. Phys. 11, 43–112 (1947).
Heyd, J., Scuseria, G. E. & Ernzerhof, M. Hybrid functionals based on a screened Coulomb potential. J. Chem. Phys 118, 8207–8215 (2003).
Mostofi, A. A. et al. wannier90: A tool for obtaining maximallylocalised Wannier functions. Comput. Phys. Comm. 178, 685–699 (2008).
Balakrishnan, J., Kok Wai Koon, G., Jaiswal, M., Castro Neto, A. H. & Özyilmaz, B. Colossal enhancement of spinorbit coupling in weakly hydrogenated graphene. Nat. Phys. 9, 284–287 (2013).
Weeks, C., Hu, J., Alicea, J., Franz, M. & Wu, R. Engineering a Robust Quantum Spin Hall State in Graphene via Adatom Deposition. Phys. Rev. X 1, 021001 (2011).
Calleja, F. et al. Spatial variation of a giant spin–orbit effect induces electron confinement in graphene on Pb islands. Nat. Phys. 11, 43–47 (2014).
Grimme, S. Semiempirical GGAtype density functional constructed with a longrange dispersion correction. J. Comput. Chem. 27, 1787–1799 (2006).
Neugebauer, J. & Scheffler, M. Adsorbatesubstrate and adsorbateadsorbate interactions of Na and K adlayers on Al(111). Phys. Rev. B 46, 16067–16080 (1992).
Monkhorst, H. J. & Pack, J. D. Special points for Brillouinzone integrations. Phys. Rev. B 13, 5188–5192 (1976).
Acknowledgements
This work was supported by Samsung Science and Technology Foundation under Grant No. SSTFBA140108.
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S.K., W.H.H., I.H.L., and K.J.C. contributed equally to this manuscript. K.J.C. conceived the work and designed the research strategy. S.K. performed theoretical calculations and S.K. and W.H.H. did data analysis. All authors discussed the results and cowrote the manuscript.
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Kim, S., Han, W.H., Lee, IH. et al. Boron Triangular Kagome Lattice with HalfMetallic Ferromagnetism. Sci Rep 7, 7279 (2017). https://doi.org/10.1038/s41598017075189
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DOI: https://doi.org/10.1038/s41598017075189
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