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Band structures of porous graphenes
Chemical Physics Letters 488 (2010) 187–192
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Band structures of porous graphenes Masashi Hatanaka * Department of Green and Sustainable Chemistry, School of Engineering, Tokyo Denki University, 2-2 Kanda Nishiki-cho, Chiyoda-ku, Tokyo 101-8457, Japan
a r t i c l e
i n f o
Article history: Received 13 January 2010 In final form 9 February 2010 Available online 12 February 2010
a b s t r a c t Band structures of porous graphenes are deduced by crystal orbital method. The dispersions suggest ca. 3.7 eV of band gaps. Bandwidths of the HOCOs and LUCOs are zero within Hückel approximation due to the nodal character of phenylene units. The graphene ribbons with n porous ladders also have singular electronic states, of which flat HOCOs and LUCOs are n-fold degenerate, respectively. The flat bands are useful for possible magnetic materials. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been increasing interest in graphitic materials as nanometer-scale carbons. In particular, single-layer graphitic structure, which is called graphene, is of great interest in view of their tunable electric properties. High-carrier mobility and quantum relativistic effects have been established experimentally [1,2]. Band structures of graphene ribbons, which are quasi onedimensional graphenes with two-sided cut ends, have also been well studied in view of their singular edge states [3–6]. Graphene ribbons can be regarded as quasi one-dimensional crystals, in which the number of the ladders is tunable. The simplest graphene ribbon is polyacene. The band structures have been studied as possible magnetic materials. Fujita and co-workers pointed out that there exists semi-flat bands in so-called acene (zigzag)-edged graphene ribbons. They found that the semi-flat bands appear at the wavenumber region |k| > 2p/3 [3,4]. The orbital phases have non-bonding characters, and thus, at least within one edge, ferromagnetic interactions are expected due to the degenerate states. Toward tunable electronic properties, modifications of graphenes have been widely suggested taking advantage of various edge states. For example, methylene-edged graphenes known as Klein edges [7,8] are promising candidates for new ferromagnets. Moreover, graphenes with one-sided dihydrogenated edges have been predicted to be ferromagnetic by the first principle calculations [9,10], and Wannier analysis for one-sided Klein edges predicted significant exchange integrals between the non-bonding electrons [11]. Recent advances in unzipping technique of carbon nanotubes also made it possible to realize well-characterized edge states of graphenes [12,13]. Experimentally, nanographene science has been drastically advanced by STM (scanning tunneling microscopy) observations of edge structures [14].
* Fax: +81 3 5280 3384. E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.02.014
In 2009, surface-catalytic synthesis of porous graphene 1 in Fig. 1 was reported by Bieri et al. [15]. This polymer has hexagonal pores in graphitic skeletons. They synthesized 1 by aryl–aryl coupling reaction of hexaiodocyclohexa-m-phenylene on Ag(1 1 1) surface. They also observed STM images of 1, and established the network structures, of which honeycombs have cyclohexa-m-phenylene (CHP) (2 in Fig. 1) unit. The inner carbon atoms are monohydrogenated so as to conjugate with whole the systems. The band structure of this material is of interest as a comparative example for conventional graphenes. The edge structures with porous ladders are also interesting. They are comparable with those of acene-edged graphene ribbons. In this Letters, band structure of 1 are predicted by crystal orbital method. Porous graphene ribbons are also examined as ladder polymers. First, the system is regarded as a two-dimensional polymer under the periodic boundary condition. Next, the porous ribbons are regarded as quasi one-dimensional polymers, in which the number of porous ladders n is tunable. The dispersion of 1 suggests a moderate band gap, and the HOCO (highest occupied crystal orbital) and LUCO (lowest unoccupied crystal orbital) have zero bandwidths within Hückel approximation. The porous graphene ribbons also have singular electronic states, of which flat HOCOs and LUCOs are n-fold degenerate, respectively. New types of magnetic graphenes are expected taking advantage of the flat bands. 2. Model compounds The unit cell of the porous graphene 1 is defined in Fig. 1. The lattice vectors lx and ly are also shown. This is regarded as a two-dimensional crystal. Figs. 2 and 3 show the unit cells and lattice vectors l for two types of porous graphene ribbons. These systems are classified by the number of the porous moieties n per unit. These are denoted as 1Xn and 1Yn with number of porous ladders n, of which limits at n ? 1 are identical to the two-dimensional system 1. In Fig. 2, porous graphene ribbons 1Xn (n = 1–5) cut out
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
1
µy
µx
n
2
Fig. 1. Molecular structures of porous graphene 1 and cyclohexa-m-phenylene (CHP) 2. lx and ly in 1 are lattice vectors.
along the x axis of 1 is shown. The edges have bay areas consisting of seven carbon atoms. This type of edges resembles so-called acene edges in conventional graphene ribbons. In Fig. 3, porous graphene ribbons 1Yn (n = 1–5) cut out along the y axis of 1 is shown. The edge bay areas consist of 12 carbon atoms. This type of edges resembles so-called phenanthrene edges in conventional graphene ribbons. The dispersions were calculated by tight-binding method under simple Hückel approximation. To estimate the eigenvalues e, the Coulomb integral a and resonance integral b were set to be 7.2 and 3.0 eV, respectively [16]. Actual computations were done by Jacobi diagonalization of the Hückel Hamiltonian. For 1, the Brillouin zone was meshed by 21 21 points. For 1Xn and 1Yn, 21 points of wavenumbers were taken into account. The computation program was written by the author using FORTRAN77, and executed on a personal computer.
3. Results and discussion Fig. 4 shows the dispersion of the two-dimensional system 1. The dispersion is shown from C point (kx, ky) = (0, 0) to X point (kx, ky) = (p, 0), and to M point (kx, ky) = (p, p) in the Brillouin zone. The band energies E are represented in units of (a e)/b. The HOCO and LUCO are completely flat. The corresponding eigenvalues are a ± 0.618b. Although this is a closed-shell system, the interesting degeneracy in the frontier bands is akin to non-Kekulé extended systems [17,18], in which non-bonding crystal orbitals are degenerate. We note that the dispersion between X and M is completely zero for all the bands. This suggests that phenylene–phenylene bridge bonds along to the y axis inhibit the itinerant character of frontier electrons. The band gap becomes 1.236b (3.708 eV). Magnitude of this band gap is typical to conventional conducting polymers. The zero dispersions of HOCO and LUCO are due to no
mixings of HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of CHP units, as shown later. Flat bands at a ± 1.618b are also come from molecular orbitals of CHP units. These flat bands are also explained by orbital mixing considerations, also explained later. At a ± 1.000b, flat bands which come from orbital fragments of benzenes (e1g orbitals) coincidently appear. The character of these bands resembles that in poly-p-phenylene, in which e1g orbitals of the benzene fragments interact very weakly [19]. DOS (density of states) of 1 is also depicted in Fig. 4. It can be seen that the narrow frontier bands give significant contribution to DOS. Therefore, photoelectron spectroscopy experiments should give a major peak around at 9.1 eV, corresponding to the top of valence bands. The flatness of the HOCO and LUCO results from unique amplitude pattern of 2. Fig. 5 shows amplitude patterns of HOMO and LUMO of 2. The eigenvalues are a ± 0.618b (analytically a ± (1 51/2)b/2). These eigenvalues are identical to those of HOMO and LUMO of butadiene. Indeed, both in the HOMO and LUMO, the p amplitudes of butadiene’s HOMO and LUMO are spread at the peripheral bay areas separately. The peripheral amplitudes have nodes at the center of each phenylene unit, as emphasized by arrows in Fig. 5. Therefore, within the rough estimation, these orbitals do not mix with any other phenylene units at the nodal points. From perturbational molecular orbital theory [20], the HOMO–HOMO or LUMO–LUMO interactions between the nodal points become zero, and thus, flat dispersions of frontier bands are reasonably explained. Moreover, HOMO11 and LUMO+11 depicted in Fig. 5 also do not mix with any other orbital, similar to HOMO and LUMO. The eigenvalues of these orbitals are a ± 1.618b (analytically a ± (1 + 51/2)b/2). These eigenvalues are identical to those of the first and fourth p orbitals of butadiene. The amplitudes are also spread at the peripheral bay areas. These orbitals have nodes similar to the HOMO and LUMO of 2, and thus, these orbitals result in flat bands due to the node–node linking.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
Therefore, as seen from the dispersion of 1, there appear flat bands not only at a ± 0.618b but also a ± 1.618b. In particular, flatness of the HOCO and LUCO of 1 is important. The two-dimensional network of 1 is synthesized by coupling reaction using hexaiodocyclohexa-m-phenylene [15]. In the actual compounds, there should be some incompleteness of couplings, which lead to various types of edge states. However, from the consideration above, the energy levels of HOMO and LUMO in any CHP oligomer are independent of the molecular weights or direction of couplings due to the nodal characters. Therefore, within the Hückel approximation, the resultant polymers should always have flat bands at the frontier levels. This situation implies that CHP oligomers with degenerate frontier orbitals are easily obtained by the coupling reactions, and lead to promising magnetic materials under proper dopants. Next, Fig. 6 shows the dispersions of 1Xn (n = 1–5). In 1X1, the HOCO and LUCO (a ± 0.618b) are completely flat, similar to 1. While the non-porous polyacene with similar skeleton has a degenerate semi-flat band due to the edge-localized states at |k| > 2p/3 [3,4], the porous graphene ribbon 1X1 has completely flat
µ
bands. Therefore, if proper dopants oxidize or reduce the materials, ferromagnetic interactions between frontier electrons are expected. The flat bands at a ± 1.618b are also seen, similar to the two-dimensional system 1. Another flat band coincidently appeared at a ± 1.000b is due to the degenerate e1g orbitals of benzene fragments. However, these bands do not always appear for all the n. In 1X2, the dispersion is qualitatively similar to 1X1. However, interestingly, the flat HOCOs and LUCOs are doubly degenerate, respectively, for all the k. This is due to increase of CHP units. In 1X3, similar dispersion with triply degenerate HOCOs and LUCOs are found, also due to the systematic increase of CHP units. In 1X4 and 1X5, the frontier orbitals are four- and five-fold degenerate, respectively. The flat bands at a ± 1.618b are also n-fold degenerate for all the k. The flat bands contribute to strong peaks in the DOS. DOS of 1X5 is shown in Fig. 8. Four major peaks of the DOS result from the flat bands at a ± 0.618b and a ± 1.618b. Fig. 7 shows the dispersions of 1Yn (n = 1–5). In these systems, the dispersions are qualitatively narrower than those of the 1Xn series due to the phenylene–phenylene bridge bonds, consistent
µ
n
1X1
µ
n
1X4 n
1X2 µ
µ
n
1X3
n
1X5 Fig. 2. Molecular structures of porous graphene ribbons 1Xn cut along x axis of 1.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
n
1Y1
n
1Y4
1Y2
n
n
1Y5 n
1Y3 Fig. 3. Molecular structures of porous graphene ribbons 1Yn cut along y axis of 1.
Node
Node
HOMO = +0.618
LUMO = -0.618 0.246 0.152
-0.246
-0.246 0.246
0.152 -0.152
-0.152
Node
Node
HOMO-11 = +1.618
0.152 0.246 -0.246
Fig. 4. Dispersion and DOS (density of states) of 1.
-0.152
LUMO+11 = -1.618
0.152 -0.152
-0.246 0.246
Fig. 5. Selected molecular orbitals of 2. The arrows denote the nodal points.
M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
Fig. 6. Dispersions of porous graphene ribbons 1Xn.
Fig. 7. Dispersions of porous graphene ribbons 1Yn.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
ferromagnetic interactions between the frontier electrons are highly expected. The doping effects will be also realized by introducing heteroatoms, for example, nitrogen or boron into the atomic sites of phenylene units. Toward modification of graphenes, these materials will be important in that their degenerate frontier crystal orbitals are available to design for organic ferromagnets. 4. Concluding remarks Band structures of porous graphenes are deduced by crystal orbital method. The dispersions predicted ca. 3.7 eV of band gaps. The HOCOs and LUCOs are flat due to nodal characters of HOMO and LUMO in cyclohexa-m-phenylene units. Graphene ribbons with n porous ladders also have flat HOCOs and LUCOs, which are n-fold degenerate, respectively. These compounds are promising precursors for new organic ferromagnets. References
Fig. 8. DOS of 1X5 and 1Y5.
with the consideration above. In 1Y1, the HOCO and LUCO (a ± 0.618b) are completely flat, similar to 1 and 1X1. Therefore, under proper dopants, ferromagnetic interactions between frontier electrons are also expected in this series. The flat bands at a ± 1.618b are also seen, similar to 1 and 1X1. In 1Y2, the flat HOCOs and LUCOs are doubly degenerate, respectively, for all the k. Similarly, in 1Y3, 1Y4 and 1Y5, three-, four- and five-fold degenerate HOCOs and LUCOs are found, also due to the systematic increase of CHP units. The flat bands at a ± 1.618b are also n-fold degenerate for all the k. The DOS of 1Y5 is shown in Fig. 8. The four major peaks also result from the flat bands at a ± 0.618b and a ± 1.618b. Thus, in porous graphenes, flat bands always appear in frontier levels. Therefore, if the materials are properly oxidized or reduced,
[1] K.S. Novoselov et al., Science 306 (2004) 666. [2] K.S. Novoselov et al., Nature 438 (2005) 197. [3] M. Fujita, K. Wakabayashi, K. Nakada, K. Kusakabe, J. Phys. Soc. Jpn. 65 (1996) 1920. [4] K. Nakada, M. Fujita, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 54 (1996) 17954. [5] K. Wakabayashi, M. Sigrist, M. Fujita, J. Phys. Soc. Jpn. 67 (1998) 2089. [6] T. Hikihara, X. Hu, H.-H. Lin, C.-Y. Mou, Phys. Rev. B 68 (2003) 035432. [7] D.J. Klein, Chem. Phys. Lett. 217 (1994) 261. [8] D.J. Klein, L. Bytautas, J. Phys. Chem. A 103 (1999) 5196. [9] K. Kusakabe, M. Maruyama, Phys. Rev. B 67 (2003) 092406. [10] M. Maruyama, K. Kusakabe, J. Phys. Soc. Jpn. 73 (2004) 656. [11] M. Hatanaka, Chem. Phys. Lett. 484 (2010) 276. [12] D.V. Kosynkin, A.L. Higginbotham, A. Sinitskii, J.R. Lomeda, A. Dimiev, B.K. Price, J.M. Tour, Nature 458 (2009) 872. [13] L. Jiao, L. Zhang, X. Wang, G. Diankov, H. Dai, Nature 458 (2009) 877. [14] T. Enoki, K. Takai, Dalton Trans. (2008) 3773. [15] M. Bieri et al., Chem. Commun. (2009) 6919. [16] F.A. Matsen, J. Chem. Phys. 24 (1956) 602. [17] M. Hatanaka, R. Shiba, Bull. Chem. Soc. Jpn. 80 (2007) 2342. [18] M. Hatanaka, R. Shiba, Bull. Chem. Soc. Jpn. 81 (2008) 460. [19] M.-H. Whangbo, R. Hoffmann, R.B. Woodward, Proc. R. Soc. Lond. A 366 (1979) 23. [20] M.J.S. Dewar, R.C. Dougherty, The PMO Theory of Organic Chemistry, Plenum Press, New York, 1975, p. 73 (Chapter 3).
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Band structures of porous graphenes Masashi Hatanaka * Department of Green and Sustainable Chemistry, School of Engineering, Tokyo Denki University, 2-2 Kanda Nishiki-cho, Chiyoda-ku, Tokyo 101-8457, Japan
a r t i c l e
i n f o
Article history: Received 13 January 2010 In final form 9 February 2010 Available online 12 February 2010
a b s t r a c t Band structures of porous graphenes are deduced by crystal orbital method. The dispersions suggest ca. 3.7 eV of band gaps. Bandwidths of the HOCOs and LUCOs are zero within Hückel approximation due to the nodal character of phenylene units. The graphene ribbons with n porous ladders also have singular electronic states, of which flat HOCOs and LUCOs are n-fold degenerate, respectively. The flat bands are useful for possible magnetic materials. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction There has been increasing interest in graphitic materials as nanometer-scale carbons. In particular, single-layer graphitic structure, which is called graphene, is of great interest in view of their tunable electric properties. High-carrier mobility and quantum relativistic effects have been established experimentally [1,2]. Band structures of graphene ribbons, which are quasi onedimensional graphenes with two-sided cut ends, have also been well studied in view of their singular edge states [3–6]. Graphene ribbons can be regarded as quasi one-dimensional crystals, in which the number of the ladders is tunable. The simplest graphene ribbon is polyacene. The band structures have been studied as possible magnetic materials. Fujita and co-workers pointed out that there exists semi-flat bands in so-called acene (zigzag)-edged graphene ribbons. They found that the semi-flat bands appear at the wavenumber region |k| > 2p/3 [3,4]. The orbital phases have non-bonding characters, and thus, at least within one edge, ferromagnetic interactions are expected due to the degenerate states. Toward tunable electronic properties, modifications of graphenes have been widely suggested taking advantage of various edge states. For example, methylene-edged graphenes known as Klein edges [7,8] are promising candidates for new ferromagnets. Moreover, graphenes with one-sided dihydrogenated edges have been predicted to be ferromagnetic by the first principle calculations [9,10], and Wannier analysis for one-sided Klein edges predicted significant exchange integrals between the non-bonding electrons [11]. Recent advances in unzipping technique of carbon nanotubes also made it possible to realize well-characterized edge states of graphenes [12,13]. Experimentally, nanographene science has been drastically advanced by STM (scanning tunneling microscopy) observations of edge structures [14].
* Fax: +81 3 5280 3384. E-mail address: [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.02.014
In 2009, surface-catalytic synthesis of porous graphene 1 in Fig. 1 was reported by Bieri et al. [15]. This polymer has hexagonal pores in graphitic skeletons. They synthesized 1 by aryl–aryl coupling reaction of hexaiodocyclohexa-m-phenylene on Ag(1 1 1) surface. They also observed STM images of 1, and established the network structures, of which honeycombs have cyclohexa-m-phenylene (CHP) (2 in Fig. 1) unit. The inner carbon atoms are monohydrogenated so as to conjugate with whole the systems. The band structure of this material is of interest as a comparative example for conventional graphenes. The edge structures with porous ladders are also interesting. They are comparable with those of acene-edged graphene ribbons. In this Letters, band structure of 1 are predicted by crystal orbital method. Porous graphene ribbons are also examined as ladder polymers. First, the system is regarded as a two-dimensional polymer under the periodic boundary condition. Next, the porous ribbons are regarded as quasi one-dimensional polymers, in which the number of porous ladders n is tunable. The dispersion of 1 suggests a moderate band gap, and the HOCO (highest occupied crystal orbital) and LUCO (lowest unoccupied crystal orbital) have zero bandwidths within Hückel approximation. The porous graphene ribbons also have singular electronic states, of which flat HOCOs and LUCOs are n-fold degenerate, respectively. New types of magnetic graphenes are expected taking advantage of the flat bands. 2. Model compounds The unit cell of the porous graphene 1 is defined in Fig. 1. The lattice vectors lx and ly are also shown. This is regarded as a two-dimensional crystal. Figs. 2 and 3 show the unit cells and lattice vectors l for two types of porous graphene ribbons. These systems are classified by the number of the porous moieties n per unit. These are denoted as 1Xn and 1Yn with number of porous ladders n, of which limits at n ? 1 are identical to the two-dimensional system 1. In Fig. 2, porous graphene ribbons 1Xn (n = 1–5) cut out
188
M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
1
µy
µx
n
2
Fig. 1. Molecular structures of porous graphene 1 and cyclohexa-m-phenylene (CHP) 2. lx and ly in 1 are lattice vectors.
along the x axis of 1 is shown. The edges have bay areas consisting of seven carbon atoms. This type of edges resembles so-called acene edges in conventional graphene ribbons. In Fig. 3, porous graphene ribbons 1Yn (n = 1–5) cut out along the y axis of 1 is shown. The edge bay areas consist of 12 carbon atoms. This type of edges resembles so-called phenanthrene edges in conventional graphene ribbons. The dispersions were calculated by tight-binding method under simple Hückel approximation. To estimate the eigenvalues e, the Coulomb integral a and resonance integral b were set to be 7.2 and 3.0 eV, respectively [16]. Actual computations were done by Jacobi diagonalization of the Hückel Hamiltonian. For 1, the Brillouin zone was meshed by 21 21 points. For 1Xn and 1Yn, 21 points of wavenumbers were taken into account. The computation program was written by the author using FORTRAN77, and executed on a personal computer.
3. Results and discussion Fig. 4 shows the dispersion of the two-dimensional system 1. The dispersion is shown from C point (kx, ky) = (0, 0) to X point (kx, ky) = (p, 0), and to M point (kx, ky) = (p, p) in the Brillouin zone. The band energies E are represented in units of (a e)/b. The HOCO and LUCO are completely flat. The corresponding eigenvalues are a ± 0.618b. Although this is a closed-shell system, the interesting degeneracy in the frontier bands is akin to non-Kekulé extended systems [17,18], in which non-bonding crystal orbitals are degenerate. We note that the dispersion between X and M is completely zero for all the bands. This suggests that phenylene–phenylene bridge bonds along to the y axis inhibit the itinerant character of frontier electrons. The band gap becomes 1.236b (3.708 eV). Magnitude of this band gap is typical to conventional conducting polymers. The zero dispersions of HOCO and LUCO are due to no
mixings of HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) of CHP units, as shown later. Flat bands at a ± 1.618b are also come from molecular orbitals of CHP units. These flat bands are also explained by orbital mixing considerations, also explained later. At a ± 1.000b, flat bands which come from orbital fragments of benzenes (e1g orbitals) coincidently appear. The character of these bands resembles that in poly-p-phenylene, in which e1g orbitals of the benzene fragments interact very weakly [19]. DOS (density of states) of 1 is also depicted in Fig. 4. It can be seen that the narrow frontier bands give significant contribution to DOS. Therefore, photoelectron spectroscopy experiments should give a major peak around at 9.1 eV, corresponding to the top of valence bands. The flatness of the HOCO and LUCO results from unique amplitude pattern of 2. Fig. 5 shows amplitude patterns of HOMO and LUMO of 2. The eigenvalues are a ± 0.618b (analytically a ± (1 51/2)b/2). These eigenvalues are identical to those of HOMO and LUMO of butadiene. Indeed, both in the HOMO and LUMO, the p amplitudes of butadiene’s HOMO and LUMO are spread at the peripheral bay areas separately. The peripheral amplitudes have nodes at the center of each phenylene unit, as emphasized by arrows in Fig. 5. Therefore, within the rough estimation, these orbitals do not mix with any other phenylene units at the nodal points. From perturbational molecular orbital theory [20], the HOMO–HOMO or LUMO–LUMO interactions between the nodal points become zero, and thus, flat dispersions of frontier bands are reasonably explained. Moreover, HOMO11 and LUMO+11 depicted in Fig. 5 also do not mix with any other orbital, similar to HOMO and LUMO. The eigenvalues of these orbitals are a ± 1.618b (analytically a ± (1 + 51/2)b/2). These eigenvalues are identical to those of the first and fourth p orbitals of butadiene. The amplitudes are also spread at the peripheral bay areas. These orbitals have nodes similar to the HOMO and LUMO of 2, and thus, these orbitals result in flat bands due to the node–node linking.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
Therefore, as seen from the dispersion of 1, there appear flat bands not only at a ± 0.618b but also a ± 1.618b. In particular, flatness of the HOCO and LUCO of 1 is important. The two-dimensional network of 1 is synthesized by coupling reaction using hexaiodocyclohexa-m-phenylene [15]. In the actual compounds, there should be some incompleteness of couplings, which lead to various types of edge states. However, from the consideration above, the energy levels of HOMO and LUMO in any CHP oligomer are independent of the molecular weights or direction of couplings due to the nodal characters. Therefore, within the Hückel approximation, the resultant polymers should always have flat bands at the frontier levels. This situation implies that CHP oligomers with degenerate frontier orbitals are easily obtained by the coupling reactions, and lead to promising magnetic materials under proper dopants. Next, Fig. 6 shows the dispersions of 1Xn (n = 1–5). In 1X1, the HOCO and LUCO (a ± 0.618b) are completely flat, similar to 1. While the non-porous polyacene with similar skeleton has a degenerate semi-flat band due to the edge-localized states at |k| > 2p/3 [3,4], the porous graphene ribbon 1X1 has completely flat
µ
bands. Therefore, if proper dopants oxidize or reduce the materials, ferromagnetic interactions between frontier electrons are expected. The flat bands at a ± 1.618b are also seen, similar to the two-dimensional system 1. Another flat band coincidently appeared at a ± 1.000b is due to the degenerate e1g orbitals of benzene fragments. However, these bands do not always appear for all the n. In 1X2, the dispersion is qualitatively similar to 1X1. However, interestingly, the flat HOCOs and LUCOs are doubly degenerate, respectively, for all the k. This is due to increase of CHP units. In 1X3, similar dispersion with triply degenerate HOCOs and LUCOs are found, also due to the systematic increase of CHP units. In 1X4 and 1X5, the frontier orbitals are four- and five-fold degenerate, respectively. The flat bands at a ± 1.618b are also n-fold degenerate for all the k. The flat bands contribute to strong peaks in the DOS. DOS of 1X5 is shown in Fig. 8. Four major peaks of the DOS result from the flat bands at a ± 0.618b and a ± 1.618b. Fig. 7 shows the dispersions of 1Yn (n = 1–5). In these systems, the dispersions are qualitatively narrower than those of the 1Xn series due to the phenylene–phenylene bridge bonds, consistent
µ
n
1X1
µ
n
1X4 n
1X2 µ
µ
n
1X3
n
1X5 Fig. 2. Molecular structures of porous graphene ribbons 1Xn cut along x axis of 1.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
n
1Y1
n
1Y4
1Y2
n
n
1Y5 n
1Y3 Fig. 3. Molecular structures of porous graphene ribbons 1Yn cut along y axis of 1.
Node
Node
HOMO = +0.618
LUMO = -0.618 0.246 0.152
-0.246
-0.246 0.246
0.152 -0.152
-0.152
Node
Node
HOMO-11 = +1.618
0.152 0.246 -0.246
Fig. 4. Dispersion and DOS (density of states) of 1.
-0.152
LUMO+11 = -1.618
0.152 -0.152
-0.246 0.246
Fig. 5. Selected molecular orbitals of 2. The arrows denote the nodal points.
M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
Fig. 6. Dispersions of porous graphene ribbons 1Xn.
Fig. 7. Dispersions of porous graphene ribbons 1Yn.
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M. Hatanaka / Chemical Physics Letters 488 (2010) 187–192
ferromagnetic interactions between the frontier electrons are highly expected. The doping effects will be also realized by introducing heteroatoms, for example, nitrogen or boron into the atomic sites of phenylene units. Toward modification of graphenes, these materials will be important in that their degenerate frontier crystal orbitals are available to design for organic ferromagnets. 4. Concluding remarks Band structures of porous graphenes are deduced by crystal orbital method. The dispersions predicted ca. 3.7 eV of band gaps. The HOCOs and LUCOs are flat due to nodal characters of HOMO and LUMO in cyclohexa-m-phenylene units. Graphene ribbons with n porous ladders also have flat HOCOs and LUCOs, which are n-fold degenerate, respectively. These compounds are promising precursors for new organic ferromagnets. References
Fig. 8. DOS of 1X5 and 1Y5.
with the consideration above. In 1Y1, the HOCO and LUCO (a ± 0.618b) are completely flat, similar to 1 and 1X1. Therefore, under proper dopants, ferromagnetic interactions between frontier electrons are also expected in this series. The flat bands at a ± 1.618b are also seen, similar to 1 and 1X1. In 1Y2, the flat HOCOs and LUCOs are doubly degenerate, respectively, for all the k. Similarly, in 1Y3, 1Y4 and 1Y5, three-, four- and five-fold degenerate HOCOs and LUCOs are found, also due to the systematic increase of CHP units. The flat bands at a ± 1.618b are also n-fold degenerate for all the k. The DOS of 1Y5 is shown in Fig. 8. The four major peaks also result from the flat bands at a ± 0.618b and a ± 1.618b. Thus, in porous graphenes, flat bands always appear in frontier levels. Therefore, if the materials are properly oxidized or reduced,
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