Coexistence of Dirac cones and Kagome flat bands in a porous graphene

Accepted Manuscript Coexistence of Dirac cones and Kagome flat bands in a porous graphene Mina Maruyama, Nguyen Thanh Cuong, Susumu Okada PII:

S0008-6223(16)30745-X

DOI:

10.1016/j.carbon.2016.08.090

Reference:

CARBON 11276

To appear in:

Carbon

Received Date: 10 June 2016 Revised Date:

5 August 2016

Accepted Date: 29 August 2016

Please cite this article as: M. Maruyama, N.T. Cuong, S. Okada, Coexistence of Dirac cones and Kagome flat bands in a porous graphene, Carbon (2016), doi: 10.1016/j.carbon.2016.08.090. 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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Coexistence of Dirac cones and Kagome flat bands in a porous graphene Mina Maruyamaa,∗, Nguyen Thanh Cuongb , Susumu Okadaa a Graduate

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School of Pure and Applied Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8571, Japan b International Center for Young Scientist (ICYS) and International Center for Materials Nanoarchitectonics (MANA), National Institute for Materials Science (NIMS), Tsukuba, Ibaraki 305-0044, Japan

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Abstract

We investigate the geometric and electronic structures of porous graphene networks consisting of phenalenyl and phenyl groups, which are connected alternately with C3 symmetry via single bonds to form a honeycomb network with an internal degree of freedom. The networks possess both Dirac cones and Kagome flat bands near the Fermi level (EF ) because the phenalenyl and phenyl form hexagonal and Kagome lattices, respectively. Because of the large spacing be-

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tween phenalenyl units, the networks possess very slow massless electrons/holes at EF , the Fermi velocity of which is a hundred times lower than that of graphene, leading to spin polarization on the networks with antiferromagnetic (AF) and ferromagnetic (F) ordering as their stable states. Our calculations

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show the AF state is the ground state whose energy is lower by 14 meV than that of the F state. We also demonstrate that the electronic structure of the 2D networks is sensitive to the rotation of the phenyl unit connecting phenalenyl

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units by changing the π electron network.

1. Introduction Electronic structures of nanoscale and low-dimensional materials consist-

ing of three-fold-coordinated (sp2 ) carbon are sensitive to their dimensionality, ∗ Corresponding

author Email address: [email protected] (Mina Maruyama)

Preprint submitted to Carbon

August 3, 2016

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topology, and boundary condition. In contrast to graphene, which is a zerogap semiconductor, carbon nanotubes are either metallic or semiconducting

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depending on the periodic boundary condition along the circumference of the two-dimensional (2D) electronic energy band of graphene [1, 2, 3, 4]. An open

boundary condition results in a one-dimensional (1D) nanoribbon of graphene, leading to further variation of electronic structures [5, 6, 7, 8, 9, 10, 11, 12, 13].

The nanoribbons with armchair edges are metallic or semiconducting, and their

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band gap is inversely proportional to their width. In contrast, ribbons with zigzag or near-zigzag edges possess peculiar edge-localized states, namely edge

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states, with non-bonding nature of π electron states, causing magnetic ordering around the edge atomic sites. In addition to 1D boundary conditions, pores and 15

topological defects in hexagonal graphitic networks cause unusual flat dispersion bands at or near the Fermi level (EF ), leading to magnetic ordering throughout the networks in some cases [14, 15, 16, 17, 18].

Polymerization and oligomerization of hydrocarbon molecules can provide structurally well-defined π electron networks with various dimensionalities. These low-dimensional or porous graphene networks have attracted much attention

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because of their high surface areas and electronic structure modulation, which are promising for application in energy and electronic devices in the near future [19, 20, 21, 22]. The electronic structures of such polymers and oligomers

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strongly depend on the combination of hydrocarbon molecules and polymer chains, which allows us to tailor their physical and chemical properties by fabrication under optimum external conditions [23, 24, 25, 26]. This indicates that

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porous magnetic carbon nanomaterials may be synthesized by polymerizing hydrocarbon molecules with radical spins. Phenalenyl is one possible candidate as a constituent unit of such magnetic materials [27, 28, 29, 30, 31]. Dimerization

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of triphenylphenalenyl and cyclodehydrogenated phenalenyl leads to complexes containing two phenalenyls possessing singlet and triplet spin coupling as their stable states [32, 33, 34, 35]. Despite intensive experimental and theoretical work on two phenalenyls connected via a covalent π network, the possible electronic structures of 2D networks consisting of phenalenyl molecules have not 2

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yet been investigated in detail. The three-fold symmetry of the phenalenyl net-

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work causes honeycomb networks of radical spins or non-bonding state to be distributed on phenalenyls with appropriate interconnect units, leading to in-

teresting electron systems that are characterized by competition between slow massless electrons and radical spin at EF . 40

In this work, we aim to theoretically investigate the possible structures of

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2D porous hydrocarbon networks of sp2 C obtained by assembling phenalenyl molecules in a hexagonal manner with phenyl interconnects and their electronic

and magnetic properties using density functional theory. Our calculation shows

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that the sheet exhibits unusual electronic structure near EF : Radical spin states on phenalenyl form a Dirac cone at EF . The Fermi velocity of the Dirac cone is one hundred times lower than that of graphene because of the hexagonally arranged phenalenyl units with large interunit spacing. Simultaneously, flat dispersion bands emerge just below and above EF arising from the highest occupied (HO) and lowest unoccupied (LU) states of the phenyl units, respectively, which 50

form a Kagome lattice [36, 37]. Slow massless electrons at EF lead to magnetic

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ordering on the porous hydrocarbon sheet with antiferromagnetic (AF) and ferromagnetic (F) states even though it possesses zero density of states at EF . The spin exchange interaction between adjacent radical spins on phenalenyl is 10 meV per pair, preferring the singlet coupling. The electronic structure and spin state of the porous hydrocarbon network can be tuned by rotational angle

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of the phenyl connecting adjacent phenalenyls via single bonds.

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2. Calculation method

All calculations were performed within the framework of density functional

theory [38, 39] using the simulation tool of atom technology (STATE) code [40].

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To calculate the exchange-correlation energy among interacting electrons, we used the generalized gradient approximation with the Perdew-Burke-Ernzerhof functional [41]. To investigate the spin-polarized states of the porous hydrocarbon sheets, the spin degree of freedom was taken into account for all calculations.

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Vanderbilt ultrasoft pseudopotentials were used to describe the electron-ion interaction [42]. The valence wave functions and charge density were expanded

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in terms of the 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 [43]. To simulate an isolated porous

hydrocarbon sheet, the sheet was separated by a vacuum spacing of 10.58 ˚ A. 70

Atomic structures of the sheets were optimized until the force acting on each

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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 k points. To inject holes into

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the sheet, we used the effective screening medium (ESM) method to solve the Poisson equation under the field-effect-transistor structure with a counter gate 75

electrode situated above the sheet with vacuum regions of 5.29 ˚ A [44].

3. Results and discussion

Figure 1 shows the optimized structures of 2D porous hydrocarbon net-

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works consisting of phenalenyl connected via phenyl with various arrangements under the hexagonal cell parameter of 19.4 ˚ A. For all phenyl arrangements, 80

the phenalenyl units retained a planar structure as their stable conformation. As illustrated in Fig. 1a, two phenalenyl units per unit cell form a hexagonal lattice in which phenyl units connect adjacent phenalenyl units through cova-

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lent bonds like those of graphene. Simultaneously, three phenyl units form a Kagome lattice in which phenalenyl acts as an interunit bond. Because of the 85

planar conformations of both phenyl and phenalenyl, the porous hydrocarbon

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sheet possesses a 2D π electron system. Rotation of phenyl modulates the π electron system: By rotating one of three phenyl units, the π electrons form a 1D network in which the alternating phenalenyl and phenyl units form zigzag chains (1D structure in Fig. 1b). By rotating two phenyl units, π electrons are

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segmented into a phenalenyl dimer containing a phenyl unit (dimer structure in Fig. 1c). Finally, π electrons are localized on each phenalenyl and phenyl in the monomer structure depicted in Fig. 1d.

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The total energies of the porous hydrocarbon sheets with various phenyl con-

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formations are listed in Table 1. Among the four structures, the planar structure is the least stable with a total energy higher by 0.8 eV per cell than that

of the ground state conformation. The total energy monotonically decreases by increasing the number of rotated phenyl units, although the π conjugation de-

creases. The decrease of the total energy with respect to the rotation of phenyl

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units is caused by the steric hindrance between hydrogen atoms attached to the

phenyl and phenalenyl. Since the hydrogen atoms attached to the phenyl and phenalenyl are positively charged, the hydrogen atoms tend to separate each

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other to reduce the Coulomb’s repulsive interaction. The dynamical stability of the porous hydrocarbon sheets was investigated by ab initio molecular dynamics simulations conducted at a constant temperature up to 3000 K for simulation 105

times of 1 ps. Under all temperatures, all structures retained their initial network topologies. On the other hand, at the elevated temperatures, the phenyl units are tiled by angles of 20-40 degrees from the planar arrangement owing to the steric hindrance between hydrogen atoms attached to the phenalenyl and

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phenyl. The sheet hardly retains the planar conformation under the finite temperature. Thus, we confirm that the 2D networks of phenalenyl and phenyl proposed here are both statically and dynamically stable and are expected to be stable under ambient conditions once they are synthesized experimentally.

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Fig. 2 presents the electronic structures of 2D porous hydrocarbon sheets consisting of phenalenyl and phenyl. The planar structure possesses a char115

acteristic feature around EF (Fig. 2a). The porous hydrocarbon sheet is a

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zero-gap semiconductor with a pair of linear dispersion bands at EF . In addition to the linear dispersion bands, we find three bunched states just above and below the linear dispersion bands. One of the three branches exhibits perfect flat band nature while the remaining two have finite band dispersion, consistent

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with their Kagome band structure. These observations indicate that Dirac cone and Kagome flat bands coexist in the porous hydrocarbon sheet of phenalenyl and phenyl with flat conformation. The Dirac cone is robust while the Kagome bands are fragile against the rotation of phenyl: a Dirac cone is observed for all 5

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porous hydrocarbon sheets, regardless of the phenyl conformation. In contrast, the Kagome flat bands are absent for the sheets with 1D and dimer structures,

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because the electron system of the phenyl unit does not retain the symmetry of the Kagome lattice.

We evaluated the tight-binding parameters of the Dirac cone and Kagome band in the different sheet conformations from their bandwidth. The calculated

transfer integrals of the Dirac cone tD are 17, 16, 14, and 10 meV for the

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planar, 1D, dimer, and monomer conformations, respectively. Because of the narrow bandwidth, the Fermi velocities of the Dirac cone are 1.2×104 , 1.4×104 ,

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0.5×104 , 1.6×104 m/s for the planar, 1D, dimer, and monomer conformations, respectively. Therefore, the porous hydrocarbon sheets are possible fields for 135

strongly correlated electron systems that induce peculiar physical phenomena. A low Fermi velocity may cause large Fermi level instability even though the sheets possess zero density of states at EF . For the Kagome bands, we also evaluated the transfer integral tK . The calculated tK are 27 and 21 meV for the planar and monomer conformations, respectively.

To clarify the physical origin of the Dirac cone in the planar system, we

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investigated the squared wave functions of the states at the K point (Fig. 3a). The wave functions are localized on phenalenyl units even though this porous hydrocarbon sheet possesses a 2D planar π electron network. By focusing on

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the wave function distribution within each phenalenyl, the state exhibits nonbonding nature in each phenalenyl unit. This indicates that the electron states associated with the Dirac cone can be ascribed to the singly occupied states of

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isolated phenalenyl monomers. In addition to the non-bonding nature within each phenalenyl unit, the wave function also exhibits non-bonding nature between adjacent phenalenyl units. Thus, the pair of linear dispersion bands at

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EF is ascribed to the hexagonally arranged singly occupied state or radical spin state of phenalenyl units. Similar non-bonding nature within and between phenalenyl units was also found for the porous hydrocarbon sheets with 1D, dimer, and monomer conformations (Figs. 3b-3d). Next, we examined the wave function associated with the Kagome band. 6

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Figure 4 shows the isosurfaces of the squared wave functions of Kagome bands

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just below EF at the Γ point. For the planar conformation, the wave functions are mainly distributed on the phenyl unit with the character of the HO states

of phenyl molecules and are extended throughout the networks. Therefore, the flat dispersion band is ascribed to the delicate balance of the electron transfer 160

among the HO state of the three phenyl units of the Kagome network, as in the

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case of the usual Kagome flat band state. In contrast, for the 1D and dimer

conformations that lack the Kagome flat band, the states below EF exhibit anisotropic nature, because of the orthogonality between the π states of the

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planar network and rotated phenyl units (Figs. 4b and 4c). For the monomer conformation, which possesses a Kagome flat band, the wave functions are more localized at the phenyl unit with the HO state nature and weakly extended throughout the network (Fig. 4d). This weak extended nature leads to the smaller tK of the monomer conformation than that of the planar conformation. The narrow band width of the Dirac cone of the porous hydrocarbon sheets 170

results in a large density of states near EF , which induces the Fermi level insta-

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bility, causing magnetic ordering on the sheet, even though the sheets possess zero density of states at EF because of the pair of linear dispersion bands at the K point. Figure 5 illustrates the spin density of the porous hydrocarbon sheets

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ρs (r) evaluated by the equation

ρs (r) = ρα (r) − ρβ (r)

where ρα (r) and ρβ (r) are the charge densities of α and β spin components,

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respectively. We found that all the porous hydrocarbon sheets show spin polarization. Polarized electrons are localized on the phenalenyl units; the α and β spins are distributed on one of two sublattices of the phenalenyl units, as in the case of the spin density of an isolated phenalenyl. Because of the spin

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density of the phenalenyl unit, each phenalenyl unit possesses S = 1/2 radical spin corresponding to the difference between the numbers of two sublattices. Furthermore, the S = 1/2 spin of the phenalenyl unit exhibits AF and F spin ordering throughout the sheets. 7

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The total energies of the porous hydrocarbon sheets with AF and F states are listed in Table 1. The total energy of each sheet with spin polarization is

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about 300 meV lower than that of the corresponding sheet with non-magnetic

states for all phenyl conformations. Furthermore, the total energy of the AF state is lower than that of the F state by 14, 9, and 4 meV for the planar, 1D,

and dimer conformations, respectively. These values indicate that the S = 1/2 spin on the phenalenyl unit interacts with its neighboring sites with weak sin-

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glet spin interaction J. The calculated spin interaction J per phenalenyl pair is about 10 meV. Note that the AF and F states are degenerate for the sheet with

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monomer conformation, because the total energies of theses states are the same within numerical error. Thus, this sheet does not exhibit long-range magnetic 195

ordering. In addition, the rotation of the phenyl unit may turn off the spin interaction between adjacent phenalenyl units because of the monotonic decrease of the energy difference between the AF and F states. Therefore, the magnetic states of the porous hydrocarbon sheets can be tuned by controlling the orientation of phenyl units with respect to the phenalenyl units: the planar, 1D, and dimer conformations are a S = 1/2 AF sheet, magnetically independent array

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of S = 1/2 AF chains, and magnetically isolated singlet-spin dimer network, respectively.

Figure 6 shows the electronic energy bands of the AF and F states of the

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porous hydrocarbon sheets with various phenyl conformations. In all cases, the spin polarization modulates the pair of linear dispersion bands at EF . For the AF state, flat dispersion bands emerge above and below EF instead of a Dirac

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cone because of the antiparallel coupling between two S = 1/2 radical spins of phenalenyl units. The energy gap between the upper and lower branches of the states associated with the spin polarization is 0.69-0.67 eV. For the F state,

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the Dirac cone is split into α and β spin states with a spin exchange energy of 0.65 eV, almost retaining its characteristic shape. In addition to the Dirac cone, electronic states near EF are also modulated by the spin polarization. For instance, the Kagome band is split into α and β spin states with an energy difference of 0.16 eV. 8

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The unusual electron systems near EF of the porous hydrocarbon sheets may

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exhibit further interesting physical properties upon hole injection into the Dirac cone and Kagome band. Because of the localized and extended natures of the

electronic states associated with the Dirac cone and Kagome band, respectively,

hole injection changes spin polarization in the porous hydrocarbon sheet. By 220

injecting holes into the porous hydrocarbon sheet with flat conformation by a

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counter planar electrode, we find that the number of polarized electron spins

strongly depends on the number of holes injected (Table 2). The spin density in the porous hydrocarbon sheet also strongly depends on the hole concentration

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(Fig. 7). For a low hole concentration corresponding to hole injection into the Dirac cone, the polarized electron spin exhibits qualitatively the same distribution as that of the undoped system. For a high hole concentration at which EF crosses the Kagome band, the spin density exhibits different characteristics to that of the undoped case: The polarized electron spin is distributed on not only the phenalenyl but also the phenyl units, reflecting the extended nature of the 230

wave functions associated with the Kagome band. Furthermore, in contrast to

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the non-doped network, the porous hydrocarbon sheet with a high hole concentration exhibits F spin ordering in its ground state. It should be noted that the carrier density to induce the magnetism associated with the Kagome flat band is about 0.8 × 1014 cm−2 . Thus, the magnetic state of the porous hydrocarbon 235

sheet is tunable using ionic gating by the formation of electrical double layers

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(EDLs) which can inject carrier density up to 8 × 1014 cm−2 [45].

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4. Discusion

Since the phenyl prefers the tiled/rotated conformation in their ground state

owing to the steric hindrance between hydrogen atoms attached to phenalenyl

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and phenyl, we further investigate the detailed energetics and spin interaction of a constituent unit of the porous hydrocarbon network. Here, we consider a phenalenyl dimer connected via phenyl as for the structural model of the the porous hydrocarbon network (Fig. 8a). Figure 8b show the total energy of the

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model molecule as a function of the rotational angle of phenyl with respect to the phenalenyl moieties. By rotating the phenyl, the total energy of the molecule

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monotonically decreases and possesses a minimum at the angle of about 35 degree. The total energy of the molecule with the rotation angle of 35 degree

is lower by about 120 meV than that of the flat arrangement. Then, by further increase the angle, the total energy monotonically increase. Therefore, the

phenyl connecting the phenalenyl units of the free-standing porous hydrocarbon

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sheet may be rotated by about 35 degree under the ambient condition as their ground state conformation. On the other hand, the porous hydrocarbon sheet

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may retain its flat conformation by adsorbing on appropriate substrates, such as graphene and h-BN, since the energy gain upon the interaction between the 255

sheet and substrates compensate the energy loss upon the rotation of phenyl. In addition to the energetics, it is worth to investigate how the spin interaction depends on the mutual rotational angle of the phenyl, because the spin interaction is terminated by the phenyl with the angle of 90 degree. Figure 8c shows the antiferromagnetic spin exchange energy J of the localized electron spin on phenalenyl moiety as a function of the rotational angle. The antifer-

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romagnetic spin exchange interaction is the largest at the planar conformation and monotonically decrease with increasing the rotational angle of phenyl. The spin exchange interaction at the stable angle of 35 degree is approximately half

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of that of the flat conformation, indicating that the spin interaction among the phenalneyl still extend throughout the sheet under the conformation with tilted phenyl. Thus, the sheet still exhibit the antiferromagnetic ordering under the

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ground state conformation.

5. Conclusion

In this work, we theoretically investigated the possible structures of 2D

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porous hydrocarbon networks of sp2 C consisting of phenalenyl and phenyl units. Phenalenyl formed a hexagonal lattice while phenyl formed a Kagome lattice because of their alternating arrangement with each phenalenyl connected to

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Table 1:

Relative total energies of porous hydrocarbon sheets consisting of phenalenyl and

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phenyl groups with antiferromagnetic (AF), ferromagnetic (F), and non-magnetic (NM) states. The energies are measured from that with the planar conformation without spin polarization.

AF state [meV]

F state [meV]

NM [meV]

2D

803

817

1116

Chain

301

310

612

Dimer

159

163

471

Monomer

0

0

313

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Structures

Table 2: Dependence of the number of polarized electron spins (∆ρ) on hole concentration.

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3

4

5

6

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∆ρ

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2

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Number of hole doping

three adjacent phenalenyls via phenyls. Our density functional theory calculations showed that the sheet exhibited unusual electronic structure near EF : 275

Radical electron states localized on phenalenyl units led to a Dirac cone at EF .

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The Fermi velocity of the Dirac cone was one hundred times lower than that of graphene. Simultaneously, flat dispersion bands emerge below and above the Dirac cone arising from the HO and LU states of phenyl, respectively, which form a Kagome lattice. Slow massless electrons at EF lead to magnetic ordering 280

on the porous hydrocarbon sheet with AF and F states. The spin exchange in-

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teraction between adjacent radical spins on phenalenyl units is J = 10 meV per pair, preferring the singlet coupling. We also demonstrated that the electronic structures and spin states of the porous hydrocarbon networks can be tuned

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by tilting the phenyl unit connecting adjacent phenalenyl via a single bond,

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lowering the dimensionality of the electron system near EF .

Acknowledgement M.M. acknowledges a Grant-in-Aid for Japan Society for the Promotion of Science (JSPS) Fellows. This work was supported in part by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and 11

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Figure 1: Optimized structures of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation. Black and pink balls denote carbon and hydrogen atoms, respectively. Black dotted lines show the unit

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cell of each network. Blue hexagons and red triangles show hexagonal and Kagome networks of phenalenyl and phenyl units, respectively.

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Energy [eV]

K

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K

L

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Figure 2: Electronic structures of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation. Energies are measured from that of EF .

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(d)

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(b)

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Figure 3: Isosurfaces of the wave function of the Dirac cone of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation at the K point.

Technology of Japan and the Joint Research Program on Zero-Emission Energy

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Research, Institute of Advanced Energy, Kyoto University. Computations were performed on a NEC SX-8/4B at the University of Tsukuba, a SGI ICE XA/UV at the Institute for Solid State Physics, The University of Tokyo, and a NEC

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SX-Ace at the Cybermedia Center, Osaka University.

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(b) HOMO-2

(c) HOMO-2

(d) HOMO-2

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Figure 4: Isosurfaces of the wave function of the Kagome flat dispersion band of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation at the Γ point.

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(a) F

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Figure 5: Spin densities of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation with antiferromag-

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Figure 6: Electronic structures of porous hydrocarbon sheets consisting of phenalenyl and phenyl groups with (a) planar, (b) 1D, (c) dimer, and (d) monomer conformation with antiferromagnetic and ferromagnetic states. EF is located at zero. Black lines and blue dotted

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lines denote majority and minority spin bands, respectively.

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Figure 7: Spin densities of a porous hydrocarbon sheet consisting of phenalenyl and phenyl groups with planar conformation and hole concentrations of 1h to 7h.

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(b)

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Figure 8: (a) A geometric structure of a phenalenyl dimer connected via phenyl (phenalenylphenyl-phenaleny). Black and pink circles denote carbon and hydrogen atoms, respectively. (b) Calculated total energy of phenalenyl-phenyl-phenaleny as a function of the rotational angle of phenyl unit. Energy is measured from that under the flat conformation. (c) Spin

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exchange energy of phenalenyl-phenyl-phenaleny as a function of the rotational angle of phenyl unit.

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