Magnetic Ordering in Porous Graphenes - American Chemical Society

Article pubs.acs.org/JPCC

Magnetic Ordering in Porous Graphenes Masashi Hatanaka* Department of Green & Sustainable Chemistry, School of Engineering, Tokyo Denki University, 5 Senju-Asahi-cho, Adachi-ku, Tokyo 120-8551, Japan ABSTRACT: Magnetic ordering in porous graphenes is analyzed by crystal orbital methods. While in triangle- and parallelogram-pored graphenes, the frontier bands have wide band widths, and in hexagon-pored graphenes, the HOCOs (highest occupied crystal orbitals) and LUCOs (lowest unoccupied crystal orbitals) are completely flat at the Hückel approximation. The flat bands in the resultant honeycomb systems are disjoint/nondisjoint composite types, and ferromagnetic interactions in the cationic and anionic states are predicted by quantum-chemical calculations including electronic correlations. Possible isoelectronic systems with nitrogen and boron atoms are also investigated as spin doped materials. The origin of spin alignment and pore size effects on magnetism are clarified by Wannier analysis. These systems are promising candidates for two-dimensional organic ferromagnets.

1. INTRODUCTION There has been increasing interest in two-dimensional systems. Discovery of graphene has accelerated the research and development of new devices with tunable electronic structures.1 As is well-known, the band structure of graphene has a Dirac point, where the highest occupied crystal orbital (HOCO) and lowest unoccupied crystal orbital (LUCO) contact at the nonbonding level. This causes high carrier mobility of electrons and pseudorelativistic effects on the motion of electrons with very small effective masses. Graphene ribbons with singular edge states have also been studied as possible magnetic materials taking advantage of flat bands at the frontier level.2 These materials are related to carbon-based magnetism or organic ferromagnets, which have been observed in some graphitic materials such as HOPG (highly oriented pyrolytic graphite).3 In the field of polymer science, some porous materials have been known as conjugated microporous polymers (CMPs),4−10 and the amorphous powders have been investigated as gasstorage agents,4−7 light-harvesting materials,9 catalysts,5a,7a capacitors,6c and batteries.5e Much attention has been paid to the size of pores, surface area, and STM images. The gasstorage efficiency have also been analyzed by chemical and/or physical sorption isotherms for N2,4,5c,7a,b H2,4e,5b H2O,7b CO2,7c and CH4.7c As is revealed by molecular dynamics analysis,4d−f CMPs are not always planar. Nevertheless, in these materials, polyphenylene-based nanostructures with regular polygon pores are very interesting in that they are often obtained on some metal surfaces as nearly planar macromolecules, and the electronic states are comparative with those of graphenes. These materials can be categorized as “porous graphenes”, and geometrically classified into three groups by the shape of pores. For a given plane, there are only three possibilities of tessellations by regular polygons: triangles, parallelograms, and © 2012 American Chemical Society

hexagons. The simplest triangle pores in the graphene plane are formally realized by hydrogenation. As a model case, this was recently realized on the Cu(111) surface,8 and the STM images of the triangle pores were clearly observed. Parallelogram pores in the graphene plane were also realized by polyphenylene networks,9 and the effective pore size and optical properties were studied toward light-harvesting materials. In 2009, Bieri et al. synthesized a porous graphene with honeycomb polyphenylene rings.10a This important material was first synthesized by aryl−aryl coupling reactions on the Ag(111) surface. The Cu(111) and Au(111) surfaces also lead to the porous graphene assemblies.10b The hexagon pores were clearly observed by STM image. Taking advantage of the wellestablished porous structures, possible gas-separation agents have been theoretically suggested particularly toward helium gas.11 Immediately, the electronic structures of this porous graphene were analyzed by several workers.12 One of the remarkable properties of this porous graphene is flatness of the frontier bands.12a That is, both HOCOs and LUCOs are completely flat under the Hückel approximation. Though the neutral state of this porous graphene is a closed-shell system, we can expect flat-band ferromagnetism in the cationic or anionic states, if it is properly oxidized or reduced. This is reminiscent of flat-band ferromagnetism of non-Kekulé-type polymers.13−15 In general, porous graphenes with large pore sizes are probably not planar by themselves. Nevertheless, as Bieri et al. showed, some porous graphenes can be grown on metal surfaces by aryl−aryl coupling reactions, conserving the skeletal planarity.10 This is a very important technique to obtain porous graphenes with well-established electronic states. Thus, systematic studies on planar porous graphenes will serve as a Received: June 30, 2012 Revised: August 28, 2012 Published: August 30, 2012 20109

dx.doi.org/10.1021/jp306460m | J. Phys. Chem. C 2012, 116, 20109−20120

The Journal of Physical Chemistry C

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guiding principle for comprehensive understanding of their electronic states. Band structures of porous graphenes with an arbitrary size of pores have not been clarified yet. Topological dependence of the band structures is an interesting problem in that the dispersions probably depend on the tessellation pattern. The dispersions will depend on the shape and size of pores. In particular, whether or not flat bands appear near the frontier levels is an important criterion toward designing for novel organic ferromagnets. In this article, band structures of general porous graphenes are investigated by crystal orbital methods. It is shown that flat bands appear at frontier levels only when the pores are hexagonally tessellated. Then, flatness of the frontier bands is independent of the pore size. However, the band gap changes with the pore size, and converges to a nonzero limit. High-spin stabilities of some porous graphenes are shown by semiempirical and DFT (density-functional theory) calculations of the cationic and anionic oligomer models. Origin of the magnetic interactions is clarified by Wannier analysis. Isoelectronic systems containing nitrogen and boron atoms are also investigated as spin doped materials, which give promising strategies for realizing two-dimensional organic ferromagnets.

Figure 2. Molecular structures of parallelogram-pored graphenes 5−8.

2. MODEL COMPOUNDS There are three types of regular polygon pores for tessellation of a given two-dimensional plane: triangles, parallelograms, and hexagons. Porous graphenes are constructed by linkage of phenylene units. We can characterize them by the shape of regular polygons and the number of phenylene units per edge of pore. A porous graphene 1 in Figure 1 has triangle pores

Figure 3. Molecular structures of hexagon-pored graphenes 9−12.

systems are shown by the dashed boxes. We note that the choice of unit cells is not unique. We here choose the unit cells so as to tessellate the two-dimensional planes with lattice vectors a1 and a2 with inner angle 60°, as shown in Figure 1. As for the honeycomb systems 9−12, Cartesian coordinates x and y (Figure 3) are also convenient to specify the orientation of the systems, as used later. The band calculations were done by the Hückel crystal orbital method, because flat bands in some porous graphenes essentially result from the nodal character of Hückel wave functions. The program for the Hückel band calculations was written by the author and performed on a personal computer. The Brillouin zone was meshed by 21 × 21 points, and the DOS (density of states) was obtained by counting all the eigenvalues. Semiempirical calculations were done under AM116a and PM316b approximation by the MOPAC program.16c DFT calculations were done by the GAMESS program17a with the B3LYP/3-21G and B3LYP/6-

Figure 1. Molecular structures of triangle-pored graphenes 1−4.

with three phenylene units per edge. 2 has triangle pores with four phenylene units per edge. 3 and 4 are similar extended systems of the triangle-tessellated systems. 5 in Figure 2 has parallelogram pores with three phenylene units per edge. Similarly, the size of pores is systematically increased in 6, 7, and 8. The porous graphene synthesized by Bieri et al. is 9 in Figure 3, which has two phenylene units per edge. The extended systems with larger pores are 10, 11, and 12. In Figures 1−3, the unit cells for the periodic two-dimensional 20110



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Figure 4. Dispersions and DOS for 1−4 at the Hückel level of theory.

Table 1. Summary of Band Calculations for 1−12a

31G(d) levels of theory.17b−d For simplicity, C−C and C−H bond lengths were fixed at 1.40 and 1.09 Å, respectively, and all the bond angles were fixed at 120°.

3. RESULTS AND DISCUSSION Band Structures of Triangle-Pored Systems. Figure 4 is the Hückel dispersions for 1−4 and the corresponding DOS. The dispersions are shown in the wavenumber k-space (k1, k2) from Γ(0,0) to X(π/a1, 0) and M(π/a1, π/a2). The band energy is represented by (α − ε)/β, where α is the Coulomb integral, β is the resonance integral, and ε is the eigenvalue of the Hückel matrix. All the bands are symmetric with respect to the nonbonding level α due to the pairing theorem. The frontier bands (HOCO and LUCO) are not flat. Thus, even if they are properly oxidized or reduced, the flat-band ferromagnetism is not expected. Table 1 shows a summary of the band calculations. The band widths of the HOCO and LUCO for 1−4 are 0.3639, 0.1540, 0.0813, and 0.0490 eV, respectively, which decrease with the pore size. On the other hand, we see that the band gaps between the top of the HOCO and bottom of the LUCO increase with the pore size. However, the band gap should be converged to a finite limit when the size of the pores becomes very large. Table 1 also shows the delocalization energy (DLE) per π electrons relative to the nonbonding level. These are the average of the total eigenvalues within the occupied bands, and thus close to the median of the total band energy. As a whole, the DLE of porous graphenes is very close to that of poly(p-phenylene) (PPP), and smaller than that of the pristine graphene. This is due to the quasi-onedimensionality of each edge of pores. Indeed, when the pore size increases, the DLE converges to that of PPP. The DOS of these materials shows very large peaks at ε = α ± 1.000β, as seen from Figure 4. These peaks result from phenylene− phenylene linkages of the skeletons, in which benzene e1g

polymers

frontier levels (unit in β)

band gap (unit in |β|)

band width of HOCO and LUCO (unit in |β|)

DLEb (unit in |β|)

1 2 3 4 5 6 7 8 9 10 11 12 PPP graphene

±0.2541 ±0.3111 ±0.3361 ±0.3499 ±0.2541 ±0.3111 ±0.3361 ±0.3499 ±0.6180 ±0.5392 ±0.5017 ±0.4797 ±0.4142 0.0000

0.5082 0.6222 0.6722 0.6998 0.5082 0.6222 0.6722 0.6998 1.2360 1.0784 1.0194 0.9594 0.8284 0.0000

0.3639 0.1540 0.0813 0.0490 0.5188 0.3069 0.2325 0.1284 0.0000 0.0000 0.0000 0.0000 0.5858c 3.0000

1.46 1.45 1.44 1.44 1.45 1.44 1.44 1.43 1.46 1.44 1.44 1.43 1.41 1.58

a

For references, data for PPP and pristine graphene are also noted. Delocalization energy per π electron relative to the nonbonding level. The calculations were done by numerical integral within the Brillouin zones meshed into 100 × 100. cBand crossing at the frontier levels is considered.

b

orbitals with nodal character are weakly interacted. Within the Hückel level, the bands at these energies are multifold degenerate, and contribute to the DOS with very large peaks. In 1−4, frontier bands are not degenerate. In principle, we cannot expect the flat-band ferromagnetism from these triangle tessellated systems. These systems are probably not planar due to the steric hindrance between the hydrogen atoms inside the pores. Treier et al. synthesized the smallest triangle-pored phenylene by catalytic aryl−aryl couplings on the Cu(111) surface, and observed its STM images.8 They also performed ab 20111



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eV) 8 or reaction heats of dehydrogenation into the corresponding peri-fused hydrocarbon (1.08 eV).8 Band Structures of Parallelogram-Pored Systems. Figure 5 shows dispersions for 5−8. Similar to the trianglepored systems 1−4, the HOCOs and LUCOs are not flat, and thus, flat-band ferromagnetism is also not expected from these systems. The band gaps also depend on the size of pores, as

initio calculations of the triangle oligophenylenes, and obtained the relaxed geometry with non-planarity. Nevertheless, the essential electronic states are approximately reproduced by the planar geometries, because the van der Waals interactions between the inner hydrogen atoms are at most 0.4 eV,8 which is smaller than energy barriers for dehydrogenations (1.4−1.8 20112



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Figure 7. The smallest units for triangle-, parallelogram-, and hexagon-pored polyphenylenes. The amplitude pattern of their frontier orbitals is also depicted.

Figure 8. Nodal character of hexagon-pored polyphenylenes. Molecular-orbital fragments of divinylbenzene (DVB) appear in each edge.

planarity and pore size are important parameters for the lightharvesting efficiency. In 6−8, the materials may also not be planar, particularly due to the steric hindrance between hydrogen atoms near the 1,2,4,5-connected phenylenes. Judging from change of band gaps and DLE, the essential character of frontier electrons of planar 6−8 is close to that of PPP. Also, flat bands appear at ε = α ± 1.000β due to the phenylene−phenylene linkages of the skeletons. These bands are also multifold degenerate, and contributions to the DOS are quite high. Though frontier levels and band gaps of 5−8 are coincidently identical to those of triangle-pored systems 1−4, frontier band widths of these compounds are larger than those of 1−4. This is due to the 1,2,4,5-phenylene units in 5−8, which allow significant bonding/antibonding interactions between each pore edge. Synthesis of these compounds by surface catalytic processes has not been done yet. However, judging from the band widths and non-planarity, magnetic interactions in the oxidized or reduced systems are probably very weak, if any. Band Structures of Hexagon-Pored Systems. Contrary to 1−4 and 5−8, hexagon-pored systems 9−12 can be obtained

seen from Table 1. Similar to 1−4, the band gaps increase with the size of pores, and converge to a finite limit when the system becomes large. Therefore, when the size of pores increases, the absorption of light should shift to the short-wavelength region. In 2010, parallelogram-pored graphene 5 was synthesized, and the N2 isotherm and some optical properties were investigated.9 The absorption peak appears at 363 nm, which is close to that of PPP (365 nm).9 This suggests non-planarity of 5, because full conjugation between polyphenylene units and 1,2,4,5phenylene units in 5 should cause a much narrower band gap than PPP. Thus, the absorption peak of the planar system should appear at the more long-wavelength visible region. Nonplanarity of the system causes a blue shift of electronic absorption, because the twisted phenylene moieties serve as quasi-independent phenyl groups. In the extreme case, frontier levels of each phenylene moiety are identical to those of doubly degenerate orbitals of benzene (α ± 1.000β). The excited state of 5 emits fluorescence at 443 nm.9 This is also considered to be blue-shifted by the non-planarity. 5 has already been applied to light-harvesting systems with coumarin dye, in which 5 serves as an energy donor to the dye.9 Thus, the degree of 20113



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as completely planar polymers, as characterized by the STM image of 9.10 Moreover, XRD analysis on triazine based CMPs revealed the planar-stacked honeycomb networks.5b,d,e Thus, Hückel dispersions of 9−12 directly correspond to the recent research and development of nanoporous graphens. Figure 6 shows dispersions of 9−12. We see that frontier bands are completely flat in the whole wavenumber region. The flatness results from nodal characters of HOMOs and LUMOs in the cyclopolyphenylene units. Figure 7 shows HOMOs and LUMOs for the smallest cyclophenylenes for triangle- (13), parallelogram- (14), and hexagon-pored (15) systems at Hückel approximation. Frontier orbitals of 13 are doubly degenerate due to high symmetry. In 14 and 15, both HOMO and LUMO are not degenerate. While the frontier orbitals of 13 and 14 have few nodes at the peripheral sites, in 15, there are six nodes at each phenylene unit. We see that in 15 butadiene-like molecular orbital fragments are linked with no overlapping interactions. The flatness of HOCO and LUCO in 9 is attributed to the linkage of the orbital fragments at the nodal points. The nodal character is expanded for larger systems. Figure 8 shows the next large cyclopolyphenylene for hexagon-pored graphenes. There are also six nodes in the frontier orbitals, and the linkage between nodal points causes degenerate flat bands of the polymeric systems. We see that divinylbenzene (DVB)-like molecular orbital fragments are linked with no overlapping interactions. The nodal points always appear even if the size of pores becomes very large. Then, electronic states of each edge of pores are described by divinylpolyphenylene-like molecular orbital fragments. Thus, the flat bands always appear in hexagon-pored systems. As above, apart from the normalization factors, the amplitude patterns in each edge of cyclopolyphenylenes are identical to those of the HOMO and LUMO of butadiene or divinylpolyphenylenes. This coincidence is due to the 1,3linkage of m-phenylene moieties. As is well-known, mphenylene skeletons serve as a ferromagnetic coupler through the nonbonding character of the frontier level.14,15 In twodimensional systems such as 9−12, the 1,3,5-phenylene unit serves as a nonbonding coupler of the fragmental amplitude patterns, similar to the case in Mataga polymer.13 Though the hexagon-pored graphenes are not non-Kekulé molecules, the conservation of amplitude patterns in each edge also comes from the nonbonding interactions between each butadiene or divinylpolyphenylene unit, as seen from the nodal points. In other words, HOCOs and LUCOs of hexagon-pored graphenes are obtained by nonbonding orbital mixing between each butadiene or divinylpolyphenylene moiety (Figure 9a). However, we note that frontier levels of porous graphenes are not the nonbonding level α. Taking account of the fragmental property, frontier levels of large hexagon-pored systems are easily estimated from eigenvalues of the divinylpolyphenylenes. Figure 9b shows change of eigenvalues of HOCO and LUCO for large hexagon-pored systems with 1− 9 phenylene units per edge. Energy spectra of very long divinylpolyphenylenes should be identical to those of PPP, as shown in Figure 9c. Thus, when the hexagon pores become very large, the HOCO and LUCO levels of the systems converge to the upper and lower limits. These are ±0.4142β, as calculated by the Hückel dispersion in Figure 9c. The DOS of 9−12 (Figure 6) shows strong peaks at frontier levels corresponding to the flat bands. We note that the DOS at the frontier levels are relatively reduced with increase in the size of pores due to systematic increase in the number of

Figure 9. (a) Butadiene or divinylpolyphenylene fragments (bold lines) at each pore edge. (b) HOCO and LUCO levels of hexagonpored porous graphenes with n phenylene units per edge. (c) Dispersion of PPP.

nonfrontier electrons. Contrarily, the peaks at ε ± 1.000β resulting from e1g orbitals of benzene units become strong with an increase in the size of pores. However, as a whole, significant contributions of frontier flat bands to the DOS are expected to cause flat-band ferromagnetism when the systems are properly oxidized or reduced. Magnetic Ordering in Hexagon-Pored Dimer Models. Since hexagon-pored systems have flat HOCOs and LUCOs, we can expect ferromagnetic interactions between the frontier electrons, if they are properly oxidized or reduced. Existence of degenerate orbitals itself does not always result in ferromagnetic interactions. As is well-known in studies on organic biradicals, whether or not the half-filled orbitals span common atoms is an important criterion for spin alignment of the frontier electrons. This was pointed out by Borden and Davidson relating to conjugated biadicals.18 That is, when degenerate orbitals can be made to span common atoms, the system is ferromagnetic. On the other hand, when degenerate orbitals cannot be made to span common atoms, the system is antiferromagnetic. The former type of degenerate orbitals is called nondisjoint, and the latter type of orbitals is called disjoint.18 This has been applied as the Borden−Davidson rule, and many high-spin organic molecules were successfully synthesized.19,20 Nowadays, the Borden−Davidson rule is better than the conventional Heisenberg model21 in that some antiferromagnetic biradicals18b and polyradicals15a that are classically predicted to be ferrimagnetic are correctly classified into disjoint systems. The Borden−Davidson analysis has been mathematically established by unitary transformation 20114



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to make a pair of maximally localized orbitals, which minimize the exchange integral between them. This was proved by Aoki and Imamura,22a and the mathematical procedure has been well applied to designing for many organic ferromagnets.22b,c Here we consider a dimer model for 9. There are two types of dimers depending on the cut edges. One is the dimer 16 cut along the x axis, as shown in Figure 10. Another is the dimer 17

Figure 11. Frontier orbitals of 16. These orbitals are maximally localized within the freedom of degeneracy. These are nondisjoint orbitals, because they span common atoms at the bridge moieties.

exchange integral is trivial within the zeroth order approximation. Thus, the spin gap (singlet−triplet energy gap) is considered to be very small, if any. The consideration above holds in not only 16 and 17 but also larger pored systems, because the nodal character is always conserved regardless of the pore size. Thus, as a rough estimation, porous graphenes with hexagon pores are expected to be ferromagnetic along the x axis and nearly nonmagnetic along the y axis. The spin gaps of 162+, 162−, 172+, and 172− can be confirmed by more precise calculations. Hartree−Fock (HF) wave functions with a single determinant are not suitable for description of these degenerate systems. Contrarily, CI (configuration interaction) calculations or DFT analysis are convenient for estimation of the spin gaps, because the lack of multiconfiguration effects in the HF wave functions is compensated through the electronic correlation included. We employed the AM1-CI, PM3-CI, and B3LYP methods, which have been well applied for estimation of spin gaps for many organic polyradicals. Table 2 shows a summary of the semiempirical calculations and spin gaps of 162+, 162−, 172+, and 172− at the AM1-CI(4,4) and PM3-CI(4,4) levels of theory. (4,4) denotes 4 electrons and 4 orbitals included in the configuration interactions. Spin gaps in 162+ are ca. +3 kcal/mol, which suggest enough stability of the triplet state. In 172+, the spin gaps are ca. +1.4 kcal/mol, which are smaller than those of 162+ due to the disjoint character of frontier orbitals. In the anionic systems 162− and 172−, singlet and triplet states are nearly degenerate within the AM1 and PM3 approximation. In disjoint systems, throughspace interactions between the spatially separated amplitudes lead to low-spin stabilities. However, when multicentered integrals are considered, exchange interactions between disjoint orbitals are not zero. To obtain more precise spin gaps, DFT analyses were performed under restricted B3LYP methods. Table 3 shows spin gaps of 162+, 162−, 172+, and 172− at the B3LYP/3-21G and B3LYP/6-31G(d) levels of theory. All the triplet states are stable in both nondisjoint and disjoint cases, which indicate that ferromagnetic interactions overcome spinpairing tendency of the through-space interactions. The frontier Kohn−Sham orbitals are nearly degenerate. Indeed, the frontier orbital gaps for singlet states of 162+, 162−, 172+, and 172− are 0.35, 0.43, 0.25, and 0.30 eV at the B3LYP/6-31G(d) level. The

Figure 10. Dimers and trimers of porous graphenes cut along the x and y axis.

cut along the y axis. It is easily shown that both dimers have doubly degenerate HOMOs and LUMOs at ±0.6180β. Figure 11 is the HOMOs and LUMOs of the dimer 16, which were obtained by a proper unitary transformation. These are maximally localized HOMOs and LUMOs, and the exchange integral is minimized within the degree of degeneracy. We note that the orbitals are not canonical but localized type, and the symmetry classification is not satisfied in the D2h point group. However, as is easily confirmed, simple linear combinations of the doubly degenerate components lead to b1u and a1u orbitals within the D2h point group. As seen from Figure 11, the localized orbitals of 16 are nondisjoint type. That is, the two orbitals span common atoms, particularly at the central bridge moiety. It is interesting that an amplitude pattern of the HOMO of butadiene appears at the peripheral carbon atomic sites, and the two degenerate components span common atoms as if they penetrate each other at the bridge. Thus, the dication and dianion of 16 are expected to have triplet states. On the other hand, the localized orbitals of 17 are disjoint type, as shown in Figure 12. The amplitude patterns of two orbitals span no common atoms. The localized orbitals are spatially separated into left and right pores, and thus, the 20115



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Figure 12. Frontier orbitals of 17. These orbitals are maximally localized within the freedom of degeneracy. These are disjoint orbitals, because they span no common atoms.

Table 2. Summary of Spin Gaps for 162+, 162−, 172+, and 172− at the AM1-CI and PM3-CI Levels of Theory AM1-CI(4,4) (kcal/mol) 162+ 162− 172+ 172−

PM3-CI(4,4) (kcal/mol)

singlet (kcal/mol)

triplet (kcal/mol)

ES−ET (kcal/mol)

singlet (kcal/mol)

triplet (kcal/mol)

ES−ET (kcal/mol)

774.199546 346.951316 830.459191 402.995230

771.196838 346.965112 829.048747 403.082433

+3.00 −0.014 +1.41 −0.087

717.455632 278.855424 763.500141 325.825368

714.611643 278.878614 762.091029 325.891538

+2.84 −0.023 +1.41 −0.066

Table 3. Summary of Spin Gaps for 162+, 162−, 172+, and 172− at the B3LYP/3-21G and B3LYP/6-31G(d) Levels of Theory B3LYP/3-21G 2+

16 162− 172+ 172−

B3LYP/6-31G(d)

singlet (hartree)

triplet (hartree)

ES−ET (kcal/mol)

singlet (hartree)

triplet (hartree)

ES−ET (kcal/mol)

−2294.452716 −2294.978635 −2753.715571 −2754.240458

−2294.462300 −2294.984403 −2753.725668 −2754.247833

+6.01 +3.62 +6.34 +4.63

−2307.187060 −2307.697003 −2768.993805 −2769.502458

−2307.196485 −2307.702200 −2769.003809 −2769.509483

+5.95 +3.26 +6.28 +4.41

spin pairs result from positive exchange integrals between the bridge carbons. In principle, positive exchange integrals are not realized unless the adjacent atoms are linked very weakly with zero overlapping or interacted in nondisjoint fashion by the skeletal topology. In 162+ and 162−, the parallel spin pairs are realized by nondisjoint amplitude of the frontier orbitals. In 172+ and 172−, two cyclophenylene moieties are linked at nodes to form disjoint frontier orbitals, and thus, one-centered exchange integrals are almost zero. Therefore, the parallel spin pairs are realized not by one-centered but two-centered or multicentered exchange integrals. Trimer Models. Higher oligomers of porous graphenes are also of interest as extended systems. Trimer models for 9 are shown as 18 and 19 in Figure 10. 18 is cut along the x axis, and 19 is cut along the y axis, similar to 16 and 17, respectively. We are interested in spin gaps of trimer cations and anions 183+, 183−, 193+, and 193− as extended systems of dimers. It is not difficult to show an important property that honeycomb porous graphene oligomers with N porous units have N-fold degenerate frontier orbitals. The frontier orbitals are half

triplet preference in disjoint systems is partially attributed to the large size of the system, because through-space interactions leading to the singlet states should be very small due to the widespread coefficients of the HOMO and LUMO. Thus, ferromagnetic interactions in hexagon-pored graphenes are supported by the DFT calculations. Figure 13 shows the spin density distribution of 162+, 162−, 172+, and 172− at the UHF/3-21G level of theory. Contrary to the restricted methods, we can observe both plus- and minussign spin densities by using UHF wave functions. It can be seen that spin alternation is realized at the adjacent carbon atomic sites, except for the bridge moieties. The spin alternation results from the effective exchange integral with a minus sign between the adjacent carbon atomic sites, which is deduced by classical valence bond description.21 At the bridge carbon atomic sites (marked by dashed boxes), we can see parallel spin pairs instead of spin alternation. The parallel spin pairs guarantee the high-spin stabilities of these species, because spin density distribution with fully spin-alternated structures leads to lowspin states due to cancellation of adjacent spins. The parallel 20116



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Figure 13. Spin density distribution for 162+, 162−, 172+, and 172− at the UHF/3-21G level of theory. Parallel spin pairs appear at the boxed bridge moieties.

Table 4. Summary of Spin Gaps for 183+, 183−, 193+, and 193− at the AM1-CI and PM3-CI Levels of Theory AM1-CI(6,6) (kcal/mol) 3+

18 183− 193+ 193−

PM3-CI(6,6) (kcal/mol)

doublet (kcal/mol)

quartet (kcal/mol)

ED−EQ (kcal/mol)

doublet (kcal/mol)

quartet (kcal/mol)

ED−EQ (kcal/mol)

1180.569527 528.269073 1284.604112 638.562430

1174.968783 527.674811 1284.338344 638.519486

+5.60 +0.59 +0.27 +0.043

1097.288084 433.047397 1179.310832 531.496558

1091.798500 433.040594 1178.784696 531.412285

+5.49 +0.0068 +0.53 +0.084

Figure 14. Boron and nitrogen doped dimers for porous graphenes.

183+ has quite a large spin gap (doublet−quartet), and the highspin stability is attributed to the nondisjoint character of three frontier orbitals. Trianion 183− is also predicted to be a quartet, but the spin gaps are small, similar to the case of dimer model 162−. As above, in the Hückel approximation, band widths of HOCO and LUCO in porous graphene 9 are zero. However, in LDA (local density approximation)12b or DFT levels,12c the bandwidth of LUCO (ca. 1 eV) is quite larger than that of

occupied when they are oxidized or reduced to cations or anions with charge ±N. Analogous to the dimer models, oligomers cut along the x axis have nondisjoint frontier orbitals and oligomers cut along the y axis have disjoint frontier orbitals. DFT calculations on these systems are difficult due to the large size and convergence problems, and we obtained only semiempirical solutions. Table 4 shows the spin gaps at the AM1-CI(6,6) and PM3-CI(6,6) levels of theory. We see that 20117



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Table 5. Summary of Spin Gaps for 20, 21, 22, and 23 at the B3LYP/3-21G and B3LYP/6-31G(d) Levels of Theory B3LYP/3-21G 20 21 22 23

B3LYP/6-31G(d)

singlet (hartree)

triplet (hartree)

ES−ET (kcal/mol)

singlet (hartree)

triplet (hartree)

ES−ET (kcal/mol)

−2268.507158 −2327.993360 −2727.749697 −2787.235938

−2268.533971 −2328.021007 −2727.777630 −2787.264378

+16.8 +17.3 +17.5 +17.8

−2281.051423 −2340.885158 −2742.838057 −2802.671975

−2281.078115 −2340.912479 −2742.865895 −2802.700019

+16.7 +17.1 +17.5 +17.6

exchange integrals increases with the one-centered integral on the active sites. However, the one-centered integral on each site does not always increase by doping. Indeed, (rr|rr) on the proper valence carbon, boron, and nitrogen are estimated to be 11.13,26a,b 6.36,26c and 16.63 eV,26a,b respectively. Therefore, the reason for the large high-spin stabilities in the doped systems lies in other effects. In heteroatom containing systems, disjoint/nondisjoint classification is not rigorously applied. In general, degeneracy of the frontier orbitals is lifted, and the unitary transformation procedure to find the best localized orbitals is not precisely applied. In such cases, the half-occupied orbitals are described by nearly original (canonical) type, of which amplitude patterns are spread widely. Similar to biradicals, canonical orbitals span common atoms to a greater extent than localized orbitals. Therefore, the effective exchange integrals are much larger than those of localized type. This is the reason why the ferromagnetic interactions are enhanced in the doped systems. In this sense, the heteroatom systems are quasi-nondisjoint systems, in which canonical orbitals widely span common atoms to lead to large exchange integrals. If the orbital gaps are smaller enough than exchange integrals between canonical orbitals, high-spin states are reasonably preferred. Indeed, frontier orbital gaps for singlet states of 20, 21, 22, and 23 are 0.21, 0.20, 0.15, and 0.15 eV at the B3LYP/6-31G(d) level, which are smaller enough than the spin gaps. Thus, doped porous graphenes are promising candidates for two-dimensional ferromagnets. In actual synthesis, a small amount of boron and/or nitrogen sources may be embedded into the skeletons from scratch to avoid the clustering of B/N. Spin injection into neutral porous graphenes will also be realized by doping of metals or metal ions, similar to the case of fullerenes. Wannier Analysis of Porous Graphenes. Dimer models in the hexagon-pored graphene are convenient for description of the local spin preference of oligomers cut along the peculiar axis. However, for prediction of global spin preference, Wannier analysis is a more powerful tool to grasp the whole ferromagnetic interactions of the polymeric systems.15 It has been shown that one-dimensional porous graphene ribbons always have flat bands at the frontier levels, regardless of the number of porous ladders.12a In principle, Wannier functions can be constructed from corresponding Bloch functions at any dimension. However, in the present case 9, pore-to-pore orbital interactions along the y axis are nearly zero. Thus, for simplicity, we only consider Wannier functions of onedimensional polymer cut along the x axis, similar to 16. Wannier functions of one-dimensional polymer cut along the y axis are almost evident from the amplitude pattern of the frontier orbitals in oligomer 17. We can show from variational principle that the Wannier functions should be symmetric with respect to the lattice vector.15a Figure 15 shows Wannier functions of HOCO and LUCO of the one-dimensional porous polymer cut along the x axis. The parentheses represent the Wannier centers, and the integer τ represents the difference

HOCO. The small spin gaps in these anionic species are related to the nontrivial bandwidth of the LUCO of 9. In 193+ and 193−, spin gaps are also positive but small. This is attributed to the disjoint character of the three frontier orbitals. As a whole, we can conclude that polycationic systems deduced by porous graphene oligimers cut along the x axis are highly expected to show ferromagnetic interactions due to the itinerant character of the frontier electrons. Doped Porous Graphenes. Spin injection into porous graphenes is a promising strategy for realization of ferromagnetism based on the neutral honeycomb skeletons. This is realizable by the doping technique. That is, the cationic and anionic states of porous graphenes are isoelectronic with boron or nitrogen doped systems, and the resultant systems are expected to be stable and synthesizable compounds as twodimensional polyradicals. It is noteworthy that porous BN and BC2N with closed shells also have semiflat bands, particularly in HOCOs.12d Recent studies on half metallicity and magnetism in bridged-triphenylamine23a or carbon-nitride23b porous sheets also encourage us toward spin doped systems. Possible doped models for the dimer cut along the x axis are shown in Figure 14. 20 is a boron doped biradical isoelectronic with 162+, and 21 is a nitrogen doped biradical isoelectronic with 162−. Boron or nitrogen per pore leads to one-electron oxidization or reduction and produces spin 1/2 at the unit cell. This situation resembles that of hetero fullerenes. For simplicity, heteroatoms are placed on each pore ring so that the molecular geometry satisfies C2h symmetry. Similarly, possible doped models isoelectronic with 172+ and 172− are shown as 22 and 23, which are boron and nitrogen doped systems cut along the y axis. Table 5 shows the spin gaps of these doped systems at the B3LYP/3-21G and B3LYP/6-31G(d) levels of theory. For all the doped molecules, triplet states are preferred, as expected above. In particular, disjoint systems 22 and 23 as well as nondisjoint systems 20 and 21 are predicted to be ferromagnetic, and the spin gaps are much larger than the carbon-based oligomers (Table 3). If half-occupied orbitals are described by the best localized type (nondisjoint), increase in the ferromagnetic interactions should be explained by enhancement of one-centered exchange integrals. For a given set of nondisjoint orbitals, the exchange integral is approximately proportional to the one-centered electron-repulsion integral at the active sites. One-centered electron-repulsion integrals on the rth site are semiempirically estimated as (rr |rr ) ≅ Ip − Ea

(1)

where Ip and Ea represent the ionization potential and electron affinity of the commonly spanned atomic site.24 On the other hand, the exchange integral between purely disjoint orbitals is approximately zero, and the residual interactions are related to the two-centered integral. As is easily deduced from the Mataga−Nishimoto expression,25 two-centered integrals at a given geometry also increase with an average of one-centered integrals at each center. Therefore, the absolute value of 20118



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of the ith and jth degenerate orbitals. These indexes are common in HOCO (HOMO) and LUCO (LUMO) due to the pairing theorem. Without complicated calculations, we can deduce a semiquantitative tendency of size effects on the exchange integrals. When the size of pores n increases (as above, n is the number of phenylene units per the pore edge; for porous graphenes 9, 10, 11, and 12, n is 2, 3, 4, and 5, respectively), each square of Wannier coefficient decreases like 1/n. Therefore, the exchange integral decreases like n·(1/n)2 = 1/n. That is, each exchange integral is approximately proportional to 1/n. When n = 2, the exchange integral is estimated by using the Wannier coefficients and (rr|rr) (≑11 eV) to be ca. 1.5 kcal/mol, and spin gaps for the dimer models become 2K ≑ 3.0 kcal/mol. This is well consistent with the DFT results on the dimer models above. In the present cases, 2∑rar(0)2ar(1)2 should be close to the Lij index of the dimers with two pores, because the nondisjoint orbitals span common atoms mainly on the bridge moieties. Since explicit calculations of Wannier functions with large n are very complex, we use the Lij index of the dimer models instead of eq 2 to estimate the exchange integrals. Figure 16 shows the pore-size dependence of the

Figure 15. Wannier functions for a one-dimensional porous graphene ribbon. The parentheses represent the Wannier centers.

from the Wannier center in unit of the period. The nodal characters at the peripheral sites are conserved even in this extended system. The Wannier functions are mainly spread at the cell center, and each Wannier function always spans common atoms at the bridge moieties. Since any Wannier function is obtained by proper parallel translations, Wannier functions of the porous graphene are always nondisjoint along the x axis. This situation guarantees ferromagnetic interations of polycations and polyanions of the porous graphene. Thus, the Wannier functions are nondisjoint along the x axis and disjoint along the y axis. As a whole, the honeycomb porous graphene becomes a two-dimensional system with disjoint/ nondisjoint composite bands. From quantum-chemical calculations above, ferromagnetic interactions along the y axis are also positive, as well as those along the x axis. Such an analysis of course holds when the size of the pores changes. Thus, hexagon-porous graphenes are disjoint/nondisjoint composite systems with ferromagnetism. Their polycations and polyanions are promising candidates for organic two-dimensional ferromagnets. Pore Size Effects on Magnetism. Finally, we pay attention to the pore-size dependence of the magnetism. Pore size effects on the ferromagnetic interactions are interesting problems for evaluating the exchange integral. Here we again consider the one-dimensional porous polymer above. When only one-centered integrals are taken into account, exchange integral K between the adjacent Wannier functions is expressed as15a

Figure 16. Pore-size effect on the exchange integral in the onedimensional porous graphene ribbon.

exchange integral between the adjacent Wannier functions based on the Lij index. The exchange integral between the adjacent cells decreases with n, and decays like 1/n. The order of (rr|rr) is ca. 10 eV, and thus, when n is larger than 4, highest and lowest spin states are nearly degenerate at room temperature. Thus, we can conclude from the Wannier analysis that porous graphene 9 is the best precursor for twodimensional organic ferromagnets based on CMPs. The planarity and well-established electronic states will contribute to a new dimension to flat-band ferromagnetism.

4. CONCLUSIONS Magnetic ordering in porous graphenes was theoretically investigated by crystal orbital methods. Only hexagonal pores cause flat bands at the frontier levels, and both HOCOs and LUCOs form disjoint/nondisjoint composite bands. The Wannier functions localized at each pore span common atoms between the adjacent pores along one of the principal axes, and the exchange interactions are positive, particularly in the polycationic systems. Semiempirical and DFT calculations supported the ferromagnetic interactions in polycations and polyanions of hexagon pored systems. Boron and nitrogen doped isoelectronic systems are also predicted to be ferromagnetic due to the quasi-nondisjoint property of nearly degenerate frontier orbitals. Exchange interaction per pore is antiproportional to the pore size, and thus, porous graphenes

cell

K ≅ 2 ∑ ar (0)2 ar (1)2 (rr |rr ) r

(2)

where ar(τ) is the Wannier coefficient at the τth cell. That is, τ = 0 corresponds to the Wannier center, and τ = ±1 corresponds to the adjacent cells. Usually, ar(τ) with |τ| ≥ 2 can be neglected due to rapid decay of Wannier coefficients. r is the index of atomic sites within the unit cell. The factor 2 results from the symmetry of the Wannier functions. Since (rr| rr) is almost constant, the magnitude of exchange integrals is estimated by 2∑rar(0)2ar(1)2. This is akin to the Lij index ∑rCri2Crj2 of organic biradicals introduced by Aoki and Imamura,22a where Cri and Crj are the rth Hückel coefficients 20119



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with small hexagon pores are promising precursors for twodimensional organic ferromagnets.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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Magnetic Ordering in Porous Graphenes - American Chemical Society

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