Interplay between the Kagome flat band and the Dirac cone in porous graphitic networks

Accepted Manuscript Interplay between the Kagome flat band and the Dirac cone in porous graphitic networks Mina Maruyama, Susumu Okada PII:

S0008-6223(17)30826-6

DOI:

10.1016/j.carbon.2017.08.040

Reference:

CARBON 12302

To appear in:

Carbon

Received Date: 30 June 2017 Revised Date:

17 August 2017

Accepted Date: 20 August 2017

Please cite this article as: M. Maruyama, S. Okada, Interplay between the Kagome flat band and the Dirac cone in porous graphitic networks, Carbon (2017), doi: 10.1016/j.carbon.2017.08.040. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Interplay between the Kagome flat band and the Dirac cone in porous graphitic networks Mina Maruyamaa,*, Susumu Okadaa a

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Graduate School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan *

Corresponding author Email address: [email protected]

Abstract

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We have investigated the geometric and electronic structures of porous graphene consisting of triangular hydrocarbon molecules polymerized by oligoacene interconnects by density functional theory. The networks possess both Dirac cones and Kagome flat bands near the Fermi level of which electron filling is controllable by selecting appropriate oligoacene interconnects because the triangular units and interconnects are arranged in honeycomb and Kagome lattices, respectively. The partially filled Kagome band in the network with the shortest interconnects induces ferromagnetic spin ordering throughout the porous graphitic network with a magnetic moment of 1.00 μB/cell. For networks with longer interconnects, the sheet is a zero gap semiconductor with a Dirac cone at the Fermi level of which band width decreases with increasing length of the interconnect despite the network containing pentagonal rings.

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1. Introduction

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The electronic structures of graphitic sp2 C networks are sensitive to both the local and global network topologies of the π electrons [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. Graphene is the starting material to consider such nanoscale sp2 C materials [16, 17]. Open boundary condition on graphene cause graphene nanoflakes which possess either closed-shell or open-shell electronic structures depending on their molecular shapes [8, 9, 18]. For instance, phenalenyl and triangulene, which are triangular molecular shapes, have radical spins of S = 1∕2 and S = 1, respectively, corresponding to the sublattice imbalance [8, 9, 18, 19, 20, 21]. For the molecules with hexagonal shape, (e.g. benzene and coronene), the band gap monotonically decreases with increasing molecular sizes [8, 9]. For the case of onedimensional boundary conditions, tubular and striped graphitic network materials can be synthesized under periodic and open boundary conditions, respectively. The tubes can be metals or semiconductors which are determined by the chirality of the hexagonal network along their circumference [1, 2, 10, 11, 22]. In the case of graphene nanoribbons, a peculiar edge localized state is induced by the delicate balance among the electron transfer at the zigzag edges, in addition to the metallic and semiconducting electronic structures of ribbons with armchair edges [12, 13, 14, 15, 23, 24, 25, 26]. In addition to these boundary conditions, topological defects in sp2 C networks, such as pentagonal and heptagonal rings, cause further versatile electronic structures: Flat band states and Kagome bands are induced at or near the Fermi level leading to spin polarization around them or throughout the networks [4, 5, 6, 7, 27, 28].

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In our previous work, we investigated a sp2 C network composed of hexagonally arranged phenalenyls units with phenyl interconnects that exhibits unusual electronic structures around the Fermi level [29]. The radical spin state of phenalenyl causes the Dirac cone with extremely narrow band width, leading to the ferromagnetic and antiferromagnetic spin ordering in the radical spin localized on the phenalenyl units. In addition to the Dirac cone associated with the radical spin on the phenalenyl unit, the electron states associated with the phenyl interconnects cause Kagome flat bands just below and above the Dirac cone, because the phenyl interconnects form a Kagome network by phenalenyl units [30, 31]. This result raises the question whether the Kagome flat band and Dirac cone can coexists at the Fermi level by selecting appropriate hydrocarbon interconnects between the phenalenyl units.

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In this study, we theoretically investigated the geometric and electronic structures of a twodimensional (2D) network of sp2 C atoms composed of phenalenyl units connected by oligoacene interconnects as a potential candidate for a porous graphitic network with an unusual electronic structure. Our first-principles total-energy calculations show that the porous graphitic networks simultaneously possess a Dirac cone and a Kagome flat band at the Fermi level for a particular oligoacene interconnect. The partially filled Kagome flat band at the Fermi level induces spin polarization with ferromagnetic ordering throughout the network. Furthermore, although the networks contain pentagonal rings between phenalenyl units and oligoacene interconnects that deviate from a bipartite network, the networks possess a Dirac cone that has a wave function that exhibits a nonbonding nature as the edge state of the graphene nanoribbon with zigzag edges [12, 13]. As the length of the oligoacene interconnect increases, the width of the Kagome band monotonically decreases, so that the flat band state associated with the Kagome band shifts downward and is filled by electrons, and thus spin ordering disappears. This indicates that the electronic and magnetic properties of the porous graphitic network can be controlled by the size of the pores.

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2. Calculation method

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All of the calculations were performed within the framework of the density functional theory [32, 33] using the simulation tool of atom technology (STATE) code [34]. To calculate the exchange–correlation energy among the interacting electrons, we used the generalized gradient approximation with the Perdew–Burke–Ernzerhof functional [35]. To investigate the spin-polarized states of the porous hydrocarbon sheets, the spin degree of freedom was taken into account in all of the calculations. Vanderbilt ultrasoft pseudopotentials were used to describe the electron–ion interactions [36]. The valence wave functions and deficit charge density were expanded in terms of plane waves with cutoff energies of 25 and 225 Ry, respectively, which sufficiently describe the electronic structure and energetics of hydrocarbon molecules and graphene-related materials [37]. To simulate an isolated porous hydrocarbon sheet, the sheet was separated by a vacuum spacing of 10.58 Å. The atomic structures of the sheets were fully optimized until the force acting on each atom was less than 1.33 × 10−3 HR/au. Integration over the Brillouin zone was carried out using an equidistance mesh of 2 × 2 × 1 \Vec{k} points. To ensure the thermal stability of the networks, we performed an ab initio molecular dynamics (MD) simulation using the velocity scaling method to keep the temperature constant during the simulation.

3. Results and discussions

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Figure 1 shows the optimized structures of porous hydrocarbon sheets consisting of phenalenyl units and oligoacene interconnects. They contain not only hexagonal rings but also pentagonal rings between phenalenyl units and oligoacene interconnects. C2, benzene, naphthalene, and anthracene are representative interconnects, which characterize the size of the pores of the networks. By focusing on the phenalenyl units, the network can be regarded as a honeycomb network with structural hierarchy or an internal degree of freedom arising from the hexagonally arranged phenalenyl units. The network can also be regarded as a Kagome lattice in terms of the oligoacene interconnects. Under the equilibrium lateral lattice constant of each network, these networks retain a planar conformation. The optimized bond lengths belonging of the phenalenyl units is range from 1.36 to 1.42 Å, irrespective to of the oligoacene length, which are similar values to those of an isolated phenalenyl molecule. For the oligoacene interconnects, the optimum bond lengths are 1.43–1.47 and 1.36–1.42 Å for the atoms without and with H atoms, respectively.

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Figure 1: Optimized structures of porous hydrocarbon sheets consisting of phenalenyl connected by (a) C2, (b) benzene, (c) naphthalene, and (d) anthracene. Black and pink balls denote carbon and hydrogen atoms, respectively.

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The calculated total energies, 𝜖, of the networks consisting of phenalenyl units and oligoacenes are given in Table 1. The total energy, 𝜖, was determined by the following equation:

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\begin{equation*} \epsilon=\frac{E_{total} \mu_{C} \times N_{C} -\mu_{H} \times N_H}{N_{C}}, \end{equation*}

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where Etotal, NC, NH, μC, and μH are the total energy of the network, number of C atoms, number of H atoms, chemical potential of the C atoms in graphene, and chemical potential of the H atoms in benzene, respectively. The total energies are lower than that of the isolated phenalenyl and oligoacene molecules, but higher than that of the polyacene. The total energy monotonically decreases with increasing length of the oligoacene. The network containing anthracene is the most stable structure with the total energy of 129 meV among the networks studied here. The relative energies of the networks with C2, benzene, and naphthalene are 120, 43, and 14 meV higher than the network with anthracene, respectively. Therefore, it is concluded that the network prefer the longer oligoacene interconnects or larger vacancies, because of the itinerant nature of π electrons which is distributed throughout the networks. We also determined the bulk modulus of the networks. The calculated bulk modulus for the networks with C2, benzene, naphthalene, and anthracene are 0.96, 0.45, 0.32, and 0.20 GPa, respectively. The results indicate that increasing the oligoacene interconnect length monotonically softens the network.

Table 1: Total energy of networks consisting of phenalenyl units and oligoacene interconnect (ε). Interconnects

Total energy (ε) [meV]

C2

250

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Benzene Naphthalene Anthracene

172 143

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129

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The stability of the 2D networks was further investigated by first-principles MD simulations. The MD simulations were performed under constant temperatures of 2000 and 3000 K for 10 ps simulation times. Under both temperatures, all of structures retain their initial network topologies and planar structures. Thus, we have confirmed that the 2D networks composed of phenalenyl units and oligoacene interconnects are both statically and dynamically stable, and they are expected to be stable under ambient conditions once they are synthesized using appropriate synthesis procedures.

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Figure 2 shows the electronic structures of the porous graphitic networks consisting of phenalenyl units and oligoacenes. All of the networks are metals with the two or more electronic energy bands crossing the Fermi level. Furthermore, the electronic structure is sensitive to the length of the oligoacene interconnect. All of the networks have a Dirac cone at the Fermi level, and the band width ranges from 1.5 to 0.8 eV depending on the oligoacene length. The calculated band widths of the Dirac cone for the networks containing C2, benzene, naphthalene, and anthracene are 1.54, 1.40, 1.00, and 0.81 eV, respectively (Table 2). In addition to the Dirac cone, characteristic dispersion bands emerge at or just below/above the Fermi level. The energy band consists of three branches: Two branches form the Dirac cone at the Brillouin zone boundary of the K point and the remaining branch exhibits a flat band nature throughout the Brillouin zone, indicating Kagome band nature. For the networks containing C2, the Kagome flat band crosses the Fermi level. Thus, interplay between the Kagome flat band and the Dirac cone may cause the unusual physical properties in this network. With increasing oligoacene length, the Kagome flat band gradually shifts downward owing to a decrease of the band width of the Kagome band. Accordingly, the Kagome flat band is fully filled with electrons for the networks with oligoacenes with the C6 or longer. The calculated band widths of the Kagome band for the networks containing C2, benzene, naphthalene, and anthracene are 1.85, 0.60, 0.55, and 0.54 eV, respectively. Figure 2: Electronic structures of porous hydrocarbon sheets consisting of phenalenyl connected by (a) C2, (b) benzene, (c) naphthalene, and (d) anthracene. The energies are relative to EF.

Table 2: Band widths and filling of the Dirac cone and the Kagome band of networks consisting of phenalenyl units and

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oligoacene interconnect. PF, HF, and FF indicate the Dirac cone and the Kagome band which are partially filled, half filled, and fully filled by electrons, respectively.

C2 Benzene Naphthalene Anthracene

1.536 1.401 1.009 0.807

PF HF HF HF

1.853 0.610 0.542 0.543

PF FF FF FF

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Dirac cone Kagome band Interconnects Band width [eV] Band filling Band width [eV] Band filling

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To elucidate the physical origin of the unusual electronic structures of the porous graphitic networks composed of phenalenyl units and oligoacene interconnects, Figure 3 shows the isosurfaces of the squared wave function of the porous graphitic networks at the Γ point and around the Fermi level. For all of the networks, the electron states associated with the Dirac cone exhibit a nonbonding nature even though the networks contain pentagonal rings. The higher branch of the Dirac cone is distributed on one of two sublattice sites of the phenalenyl unit. The lower branch is distributed on the other sublattice states and C2 belonging to the pentagonal ring. This indicates that the Dirac cone can be ascribed to the nonbonding state of the phenalenyl units, which are hexagonally arranged and connected by oligoacene interconnects. This effectively satisfies the bipartite symmetry, except for the pentagonal ring between the phenalenyl and oligoacene units. In contrast, the electronic states associated with the Kagome band exhibit usual bonding and antibonding nature of the π electrons and are extended throughout the networks. Thus, the flat dispersion band at or below the Fermi level is ascribed to the delicate balance between π electron transfer in these covalent networks.

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Figure 3: Isosurfaces of the wave functions of porous hydrocarbon sheets consisting of phenalenyl connected by (a) C2, (b) benzene, (c) naphthalene, and (d) anthracene at the Γ point.

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Figure 4 shows the energy levels and wave functions of phenalenyl dimers connected by oligoacene interconnects [38, 39]. All of the phenalenyl dimers have closed-shell electronic structures. The energy gaps between the highest occupied and lowest unoccupied states are 0.67, 0.66, 0.50, and 0.39 eV for C2, benzene, naphthalene, and anthracene interconnects, respectively. For all of the dimers, the electronic states around the gap exhibit qualitatively the same nature as that in the 2D network, indicating that phenalenyl dimers with oligoacene interconnects are the constituent units of the porous graphitic network. Figure 4: Energy levels and isosurfaces of the wave functions of phenalenyl connected by (a) C2, (b) benzene, (c) naphthalene, and (d) anthracene. The energies are relative to the highest occupied state. Because the Kagome flat band of the sheet consisting of phenalenyl units and C2 interconnects crosses the Fermi level, the sheet exhibits spin polarization owing to the Fermi level instability arising from the large density of states at the Fermi level because of splitting into the electronic states of the α and β spin components. Figure 5 shows the spin density of the network with phenalenyl units and C2 interconnects, which is defined as the difference

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between the electron densities of the α and β spin components. The polarized electron spin is distributed on the phenalenyl part and C2 interconnects with approximately ferromagnetic ordering throughout the network with a magnetic moment of 1.00 μB per unit cell. The distribution of polarized electron spin qualitatively corresponds with the wave function distribution of the Kagome flat band states, verifying that the magnetic ordering is induced by the Kagome band at the Fermi level. The small spin density at the center of the phenalenyl unit with the opposite spin component indicates that the electron states associated with the Dirac cone also affect the spin density distribution. Thus, the interplay between the Kagome band and the Dirac cone leads to the unusual spin density in the porous graphitic network consisting of phenalenyl units and C2 interconnects. Figure 5: Spin densities of porous hydrocarbon sheets consisting of phenalenyl dimers connected by C2. Blue and yellow isosurfaces indicate the sign of the polarized electron spin.

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4. Conclusion

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In summary, using density functional theory with the generalized gradient approximation, we have investigated the geometric and electronic structures of 2D networks of sp2 C atoms composed of phenalenyl units connected by oligoacene interconnects as the representative structure of porous graphitic networks. The calculations show that the porous graphitic networks simultaneously possess a Dirac cone and a Kagome flat band at or near the Fermi level. For the network with the shortest interconnect (C2), the partially filled Kagome flat band at the Fermi level induces spin polarization with ferromagnetic ordering throughout the network with a magnetic moment of 1.00 μB per unit cell. Furthermore, although the networks contain pentagonal rings between phenalenyl units and oligoacene interconnects, they possess a Dirac cone with a wave function that exhibits nonbonding nature as the edge state of graphene nanoribbon with zigzag edges. As the length of oligoacene interconnect increases, the width of the Kagome band monotonically decreases, so the flat band state associated with the Kagome band shifts downward and is completely filled with electrons, and thus spin ordering disappears. This indicates that electron filling and magnetic ordering of the porous graphitic network can be controlled by the size of the oligoacene interconnect.

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Acknowledgement

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This work was supported by JST-CREST Grant Number JPMJCR1532 from the Japan Science and Technology Agency, JSPS KAKENHI Grant Numbers JP17H01069, JP16H00898, and JP16H06331 from Japan Society For the Promotion of Science, and the Joint Research Program on Zero-Emission Energy Research, Institute of Advanced Energy, Kyoto University. Part of the calculations was performed on an NEC SX-Ace at the Cybermedia Center at Osaka University and on an SGI ICE XA/UV at the Institute of Solid State Physics, The University of Tokyo.

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