Correlated Magnetic States in Extended One-Dimensional Defects in

Letter pubs.acs.org/NanoLett

Correlated Magnetic States in Extended One-Dimensional Defects in Graphene Simone S. Alexandre,† A. D. Lúcio,‡ A. H. Castro Neto,§,∥ and R. W. Nunes*,† †

Departamento de Física, ICEx, Universidade Federal de Minas Gerais, 31270-901, Belo Horizonte, MG, Brazil Departamento de Ciências Exatas, Universidade Federal de Lavras, 37200-000, Lavras, MG, Brazil § Graphene Research Centre and Department of Physics, National University of Singapore, 2 Science Drive 3, Singapore, 117542 ‡

ABSTRACT: Ab initio calculations indicate that while the electronic states introduced by tilt grain boundaries in graphene are only partially confined to the defect core, a translational grain boundary introduces states near the Fermi level that are very strongly confined to the core of the defect, and display a ferromagnetic instability. The translational boundary lies along a graphene zigzag direction and its magnetic state is akin to that which has been theoretically predicted to occur on zigzag edges of graphene ribbons. Unlike ribbon edges, the translational grain boundary is fully immersed within the bulk of graphene, hence its magnetic state is protected from the contamination and reconstruction effects that have hampered experimental detection of the magnetic ribbon states. Moreover, our calculations suggest that charge transfer between grain boundaries and the bulk in graphene is short ranged, with charge redistribution confined to ∼5 Å from the geometric center of the 1D defects. KEYWORDS: Graphene, defects, grain boundaries, quasi-1D electronic states, magnetic instability, calculations

C

resulting in charged defective lines surrounded by a doped graphene matrix. Our calculations show that while the electronic states introduced by several different types of tilt GBs in graphene hybridize with the bulk states and are only partially confined to the defect core, the translational GB [GB(2,0)|(2,0) in Figure 1a] introduces a sharp resonance just above the Fermi level (FL) in the density of states (DOS) of graphene, that is associated to electronic states that are very strongly confined to the core of the defect. Our results indicate that when a graphene sample containing translational GBs is doped, either chemically or by the application of external electric fields, and these quasi-1D states are populated, a ferromagnetic state is realized, which is strongly confined to the defect core and fully immersed within the graphene bulk. The quantum confinement of the electronic states induced by the presence of the boundary leads to an enhancement of the Coulomb interactions and to stronger electron−electron correlations. Because of their 1D nature, these correlated states do not show long-range order. Instead, they present power law or algebraic correlation functions. Moreover, given that these 1D states are immersed in a metallic environment, they are unique examples of open Luttinger liquids (OPL).14 (In what follows, we provide further discussion on the physical meaning of our results for a macroscopic graphene sample in connection

ontrolling electronic transport and tailoring magnetic states at the nanoscale in graphene rank among the main issues related to the prospective application of this material in nanoelectronics and spintronics. In particular, electronic and magnetic states of extended line defects in graphene have been considered as possible conducting one-dimensional (1D) electronic channels and platforms for tailored spin states for spintronic applications.1,2 Two recent developments highlight the focus on extended line defects: (i) the recognition that mass-scale production of graphene should inevitably lead to a polycrystalline material, containing one-dimensional tilt grain boundaries;3−6 and (ii) the recent theoretical prediction7 and experimental realization of a translational grain boundary in graphene by controlled deposition of the material on metallic substrates.2 In the case of tilt boundaries, a large volume of works addressing their electronic and structural properties has accumulated in the past few years,4−6,8 while for the translational boundary produced in Lahiri et al. experiment, valley-filter properties9 and its effects on the magnetic edge states of a graphene ribbon10 have been theoretically investigated. Therefore, the nature of the electronic states introduced by extended 1D defects in graphene is a topic that deserves close inspection. More specifically, whether such defects present quasi-1D electronic states immersed in the bulk of graphene is the question we seek to address in this work, employing ab initio calculations. We also consider the issue of self-doping in graphene induced by the presence of extended 1D defects,13 that occurs when the line defect attracts charge carriers, © 2012 American Chemical Society

Received: May 8, 2012 Revised: August 20, 2012 Published: September 5, 2012 5097

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102

Nano Letters

Letter

Following ref 6, we use a (nL,mL)|(nR,mR) index notation to denote the GB defects by their translation vector T⃗ GB spanned in the basis of lattice vectors of the grains, respectively, on the left (a⃗′1 and a′2⃗ ) and on the right (a1⃗ and a2⃗ ) of the extended defects, as shown in Figure 1. In Figure 1a, we show the translational GB, which is a (2,0)|(2,0) defect, labeled as GB(2,0)|(2,0) in the following discussion. We consider three different tilt GB models: Figure 1b shows the GB(1,2)|(2,1) (tilt angle θ = 21.8° as defined in ref 8); Figure 1c shows the GB(5,0)|(3,3) (θ = 30°); and Figure 1d shows the GB(5,3)| (7,0) (θ = 21.8°). Note that the for the translational GB(2,0)| (2,0) the lattices of the two grains adjoining at the 1D defect are translated with respect to each other by one-third of the periodicity of the bulk lattice in the direction perpendicular to the boundary line.2 This 1D defect differs from the tilt GBs in the sense that the grains on the left and right of the boundary have the same orientation (i.e., θ = 0°). We start by examining the nature of the electronic bands introduced by GBs in graphene. The electronic structure of the GB(1,2)|(2,1) in this material has been amply discussed in ref 8, where the occurrence of an anisotropic Dirac cone on a Brillouin-zone (BZ) line along the defect direction was analyzed, and the GB(1,2)|(2,1) states near the FL were found to disperse in all directions, indicating hybridization with bulk states. The band structures for the three tilt GB models in our study are shown in Figure 2. The Brillouin zone that corresponds to ⃗ the GB supercells with the labeling of special k-points, is shown in Figure 5d below. The GB direction is along the y-axis of the supercell in all cases.

Figure 1. Geometries of one-dimensional periodic defects in graphene. Defect-core carbon atoms are shown in red. Lattice vectors of the grains on the left (a⃗1′ and a⃗2′ ) and on the right (a⃗1 and a⃗2) of the GBs are indicated by blue arrows. The translation vector T⃗ GB of defect is shown as a black arrow. The GBs are labeled by GB(nL,mL)|(nR,mR) where T⃗ GB = nLa⃗′1 + mLa⃗′2 = nRa⃗1 + mRa⃗2. (a) A translational grain boundary: GB(2,0)|(2,0). Tilt grain boundaries: (b) GB(1,2)|(2,1); (c) GB(5,0)|(3,3); and (d) GB(5,3)|(7,0).

with the absence of quantum fluctuations in our GGA-DFT mean-field results.) Furthermore, we find that for undoped systems, charge transfer between bulk graphene and the defects is essentially local with charge redistribution taking place within a region of ∼5 Å from the geometric center of the 1D defects. No selfdoping takes place, and both this charge-transfer region and the surrounding bulk matrix remain essentially neutral. The absence of a charge monopole moment means that in weakly doped samples these 1D defects should act as weak charge carrier scatterers, which is in agreement with recent transport experiments in graphene grown by chemical vapor deposition (CVD).15 Our calculations are performed in the framework of Kohn− Sham density functional theory (DFT), within the generalizedgradient approximation (GGA)16,17 and norm-conserving pseudopotentials in the Kleinman-Bylander factorized form.18,19 We use the LCAO method implemented in the SIESTA code20 with a double-ζ basis set plus polarization orbitals. A Mulliken charge partition is employed for the analysis of the charge-transfer between bulk and defects. Convergence tests were performed and our supercells sizes were chosen to ensure that all of our main conclusions and results are not affected by interaction between the defects and their periodic images. We consider the aforementioned types of extended 1D defects in graphene: (i) large-angle tilt grain boundaries,4−6 expected to occur commonly in mass-scale production of graphene; and (ii) translational grain boundaries produced by deposition of a graphene layer on a nickel substrate.2,7 We also consider oxidized forms of the defects with oxygen atoms bound to atoms at the defect core.

Figure 2. Band structure and density of states (DOS) of tilt grain boundaries in graphene. In each case, the total DOS is shown as a black curve and the partial DOS, projected onto the core atoms, is shown by the red curve. (a,b) GB(1,2)|(2,1); (c,d) GB(5,0)|(3,3); (e,f) GB(5,3)|(7,0). 5098

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102

Nano Letters

Letter

folds onto the point at 2π/3S along the Γ−Y line, where S is the period of the GB(2,0)|(2,0).23 Note that, to a large degree, the changes in the GB(2,0)|(2,0) band structure with respect to the bulk one are concentrated in a region of ±0.5 eV from the FL, and that within ∼0.2 eV above (below) the FL we find a band with flat sections along the Γ−Y (L−X) line (both are parallel to the defect direction). These states show little or no dispersion along the Γ−X and Y−L lines that are perpendicular to the defect, which is in contrast with what we observe in the tilt GB band structures in Figure 2a,c,e. The flat-band character of the states just above the FL, near the zone center, leads to a ferromagnetic instability for a bulk 1D defect in graphene consisting entirely of carbon atoms, as discussed below. The occurrence of extended van Hove regions in the band structure of the GB(2,0)|(2,0), shown in Figure 3a, shows up in the DOS in Figure 3b in the form of two sharp resonances within ∼± 0.1 eV from the FL. We observe that in one feature the DOS of the GB(2,0)|(2,0) differs markedly from those of the tilt GBs: the peak just above the FL shows a very strong concentration on the defect core with ∼85% of the total DOS concentrated on the basis orbitals centered on the core atoms. Adding the contributions of the atoms from the zigzag chains nearest to the core on both sides of the defect line, we already account for ∼96% of the total DOS for this peak. A strongly one-dimensionally confined empty state along the defect core is thus a characteristic of the GB(2,0)|(2,0). In our calculations, for the defects in a neutral charge state these states are empty, hence no spin polarization is induced. In order to investigate the formation of magnetic moments in gate-doped graphene systems containing GB(2,0)|(2,0) defects, we increased the electronic concentration in the supercell to a doping level that raises the FL by about ∼0.07 eV. In this case, a spin-polarized GGA-DFT calculation stabilizes a ferromagnetic state with a magnetic moment of 0.52 μB per defect unit, and a formation energy ∼40 meV (per defect unit) lower than that for the unpolarizeds-spin state. Figure 4 shows a

In order to understand the results in Figure 2, we note that the topological defects (TDs) contained in the core of the three tilt-GB defects consist of pentagons and heptagons only. Previous theoretical studies have found that isolated oddmembered rings (pentagons or heptagons), as well as a single pentagon-heptagon pair, do not introduce states at the FL (zero-mode states) in graphene, while an isolated evenmembered ring such as the octagon appearing in the core of the translational GB(2,0)|(2,0) gives rise to a finite DOS at the FL.21,22 Thus, the presence of zero-mode electronic states in our results for the translational GB(2,0)|(2,0), shown in Figure 3, is not unexpected, given the presence of octagonal rings in the core of this defect.

Figure 3. (a) Band structure and (b) density of states (DOS) of a translational grain boundary in graphene. The total DOS is shown as a black curve and the partial DOS, projected onto the core atoms, is shown by the red curve.

By the same token, our calculations for the three tilt GB models indicate that the absence of zero-mode states is also true for different continuum linear distributions of 5-fold and 7fold rings, such as in the simpler GB(1,2)|(2,1), where pentagon−heptagon pairs are separated from each other by hexagons along the core (Figure 2a,b), and also for the more complex structure of the GB(5,0)|(3,3) core, as shown in Figure 2c,d. The presence of a gap in the band structure of the GB(5,3)|(7,0), in Figure 2e,f, is a supercell-size effect, since in the limit of a sufficiently large cell the vanishing gap at the FL, characteristic of the DOS of the graphene bulk states, must be recovered. However, the finite gap we obtain for the GB(5,3)| (7,0) has a physical meaning, being indicative of a strong electronic antiresonance introduced by the GB(5,3)|(7,0) at the FL in graphene. This observation is also true for some of the oxidized GB systems we consider below. We shift our focus now to the question of the degree of hybridization between defect-related and bulk electronic states. Indication of a hybridized nature of the states for all three tiltGB models is provided by the analysis of the DOS shown in Figure 2b,e,f for the GB(1,2)|(2,1), the GB(5,0)|(3,3), and the GB(5,3)|(7,0), respectively. The figures show the total DOS and the partial DOS (PDOS) summed over the core atoms (shown in red in Figure 1). Generally, the contribution of the core atoms to the DOS peaks near the FL ranges from 37 to 55% in all three tilt GBs. These results will be contrasted with the case of the GB(2,0)|(2,0) in the following. The band structure and the DOS for the translational GB(2,0)|(2,0) are shown in Figure 3a,b, respectively. In Figure 3a, we also include the band structure of a 64-atom bulk supercell, obtained by removing the two atoms forming a dimer at the center of the defect core, from the geometry shown in Figure 1a. Given the periodicity of both defect and bulk supercells along the GB(2,0)|(2,0) direction (a zigzag direction of the bulk matrix), the Dirac point of the primitive bulk cell

Figure 4. Isosurface of spin polarization density for the domainboundary magnetic state.

representative isosurface of the difference between majority and minority spin densities in this system. The spin density is strongly concentrated on the zigzag chains on the two sides of the defect core, more specifically on the sublattice of atoms that bond to the dimers at the geometric center of the defect with negligible contributions from the atoms belonging to the other sublattice and from the dimer atoms themselves. We observe here the manifestation of a magnetic instability along a line defect that is fully immersed within the bulk of graphene. The fact that these states have a negligible contribution along the dimer atoms is perhaps the origin of the topological disruption in the electronic states that leads to the localization of the related π-orbital bands and hence to the magnetic state. The translational boundary lies along a 5099

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102

Nano Letters

Letter

graphene zigzag direction and its magnetic state is similar to that predicted to occur on zigzag edges of graphene ribbons. We speculate that the GB(2,0)|(2,0) magnetic moments should be more easily detectable experimentally than the one in graphene ribbons,23−26 since the zigzag chains along the GB are protected from reconstruction and contamination effects. We have also attempted to identify the occurrence of magnetic moments in all three tilt GB models described above for various different doping levels. At the GGA level, we find no magnetic instabilities associated with the tilt-GB resonances, which leads us to conclude that a very strong localization of the defectrelated electronic states is required for the instability to set in. We also consider the possibility of chemically doping the GB(2,0)|(2,0) states. This is important, since chemical reduction of a readily available material, such as graphene oxide, has been considered as a viable route for mass-scale graphene synthesis,11,12 and the presence of residual functional groups should affect the electronic and magnetic properties of the resulting material. For this, we consider only oxidized forms of the GB(1,2)|(2,1) and the GB(2,0)|(2,0), as shown in Figure 5. The corresponding band structures are shown in Figure 6.

Figure 6. Electronic bands for the oxidized GB models. Upper left: GB(1,2)|(2,1)+3O. Upper right: GB(2,0)|(2,0)+2O. Lower left: GB(2,0)|(2,0)+O, spin-unpolarized calculation. Lower right: GB(2,0)|(2,0)+O, spin-polarized calculation. Black (red) lines show majority (minority) spin bands.

along the Γ−Y line that is ∼0.1 eV below the FL in the pure GB(2,0)|(2,0) neutral system is now deeper in energy. More importantly, a significant change is observed in the flat band of confined 1D states near Γ that now crosses the FL. As a result, a magnetic ground state is stabilized with a net magnetic moment of 1.1 μB per unit cell, as shown in the lower left panel of Figure 6, where majority and minority spin bands are plotted. Note that spin-polarization is strongly localized on the 1D defect with significant exchange splitting restricted to the quasi-1D bands crossing the FL. This ferromagnetic ground state is 43 meV (per defect unit) lower in energy than the nonpolarized state. This value is very close to what we found for the doped GB(2,0)|(2,0) system above. The ferromagnetic instability of these flat-band states is further confirmed by a calculation we performed for a doped GB(2,0)|(2,0)+2O system with extra electronic charge added to the defect supercell. From Figure 6, it can be seen that in the undoped case the flat band for this system lies entirely above the FL. Again, with doping the FL shifts into the band of quasi-1D states, and the system stabilizes a ferromagnetic state. Hence, chemical doping offers another way of stabilizing this ferromagnetic state. The experimental observation of extended magnetic textures within the nonmagnetic bulk of graphene can be done by local spin probes. One possible method is the fabrication of C13-enriched samples and the use of nuclear magnetic resonance.27 Other more direct methods with very high resolution would be the use of spin-polarized scanning tunneling microscopy or magnetic exchange force microscopy.1 Regarding the physical interpretation of our GGA-DFT results, we note that, as stated above, these correlated quasi-1D states should not show any long-range order. The fact that we obtain a ferromagnetic state for an infinite system in our calculations is connected with the fact that, being a mean-field theory, the GGA-DFT approach we employ does not include the quantum fluctuations that inhibit the formation of a longrange-order state. Hence, for graphene samples presenting doped translation-boundary defects the magnetic instability we identify here should manifest itself in the formation of finitesize magnetic domains with a vanishing average magnetization over the macroscopic sample. Thus, given the strong 1D

Figure 5. Oxidized versions of grain boundaries in graphene. Defectcore carbon atoms are shown in red, and oxygen atoms are shown in blue. (a) GB(1,2)|(2,1)+3O with three oxygen atoms per defect unit. (b) GB(2,0)|(2,0)+2O with two oxygen atoms per defect unit. (c) GB(2,0)|(2,0)+O with one oxygen atom per defect unit. (d) Brillouinzone and special points for GB-supercell electronic bands.

Note that for the oxidized forms of the GB(2,0)|(2,0), just as in the case without O doping, the presence of extended van Hove regions is clearly seen in the band structures in Figure 6. For both the GB(2,0)|(2,0)+2O (two oxygen atoms per defect unit) and GB(1,2)|(2,1)+3O (three oxygen atoms per defect unit), where oxygen-induced unzipping of the C−C bonds takes place, we observe the opening of small gaps of 55 and 190 meV, respectively. We do not find magnetic ground states in the neutral system in either of these two cases. In the case of the GB(2,0)|(2,0)+O system with one oxygen atom per defect unit, the oxygen atom bonds in the bridge position with the two carbon atoms in the center of the defect core. In this case, as shown in Figure 6, the occupied band 5100

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102

Nano Letters

Letter

In summary, we find that the electronic states introduced by several different tilt grain boundaries in graphene are only partially confined to the 1D defect core, while a translational grain boundary introduces unoccupied electronic states near the FL that are very strongly confined to the core. When populated by doping, these confined quasi-1D electronic states display the formation of ferromagnetic moments in an extended 1D defect that is fully contained within the graphene bulk matrix and consists entirely of carbon atoms. Being fully bulkimmersed, this ferromagnetic state is protected from reconstruction and contamination and should be more easily detectable experimentally than magnetic states predicted to exist along the edges of graphene ribbons. Furthermore, our calculations indicate that charge transfer between bulk graphene and grain-boundary cores is strongly localized with charge redistribution confined to ∼5 Å from the geometric center of the defect, implying that in the neutral charge state these 1D defects should act as weak charge scatterers in graphene.

confinement of the translational GB states, we expect the correlation functions for these states to fulfill the requirements for the formation of OPL14 states immersed in the graphene matrix. We turn our attention now to the charge transfer between the 1D defects and the surrounding graphene bulk, since this may give important qualitative information about the nature of the scattering of charge carriers by such 1D defects. We consider the GB(2,0)|(2,0) and the GB(1,2)|(2,1) in their neutral charge state. In Figure 7a, we show the profile of the

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: rwnunes@fisica.ufmg.br. Notes

The authors declare no competing financial interest. ∥ On leave from Boston University, U.S.A.

■

ACKNOWLEDGMENTS S.S.A., R.W.N., and A.L.D. acknowledge support from Brazilian agencies CNPq, FAPEMIG, and Instituto do Milênio em Nanociências-MCT. AHCN thanks the financial support of the NRF-CRP award Novel 2D Materials with Tailored Properties: Beyond Graphene (R-144-000-295-282), U.S./DOE Grant DEFG02-08ER46512, and U.S./ONR Grant MURI N00014-09-11063.

Figure 7. (a) Charge density distribution for the GB(2,0)|(2,0) (blue circles) and the GB(1,2)|(2,1) (red squares). (b) Integrated charge IQ(r) (see text) for the GB(2,0)|(2,0) and the GB(1,2)|(2,1).

charge distribution around the 1D defects, and in Figure 7b we show the integrated charge per atom, both as a function of the distance to the geometric center of the defect, for the GB(2,0)| (2,0) and the GB(1,2)|(2,1). The charge distribution is shown as linear charge densities λ(r) representing the net charge summed over atoms at the same distance from the 1D defect, divided by the period of the defect unit. The integrated charge is the integral of this linear charge density profile, computed as total net charge per atom, and defined as follows: R IQ (R ) = [1/N (R )]∑r = 0 λ(ri) × Sd , where λ(ri) is the linear i charge density for the line of atoms at a distance ri from the defect, Sd is the period of the 1D defect, and N(R) is the number of atoms summed over all lines with ri ≤ R. Figure 7 shows that the charge redistribution around the 1D defects is nonmonotonic with the linear charge distributions alternating in sign on both sides of the defect, as seen in Figure 7a. Note also that charge transfer between the 1D defects and the graphene matrix occurs in a range of ∼3−5 Å from the defect line with the graphene matrix becoming neutral beyond this range, as shown in the plots for IQ(R) in Figure 7b. Since the charge distribution recovers neutrality within ∼5 Å from the defect center, we expect no long-range Coulomb scattering of charge carriers from both 1D defects in their neutral state. In our calculations, oxidation of the 1D defects is found not to affect this localized character of the charge balance between the defects and the graphene bulk. In the neutral charge state, such extended 1D defects should then play a minor role as a source of carrier scattering in graphene.

■

REFERENCES

(1) Castro Neto, A. H.; Guinea, F.; Peres, N. M. R.; Novoselov, K. S.; Geim, A. K. The electronic properties of graphene. Rev. Mod. Phys. 2009, 81 (1), 109−162. (2) Lahiri, J.; Lin, Y.; Bozkurt, P.; Oleynik, I. I.; Batzill, M. An extended defect in graphene as a metallic wire. Nat. Nanotechnol. 2010, 5 (5), 326−329. (3) Yu, Q. K.; Jauregui, L. A.; Wu, W.; Colby, R.; Tian, J. F.; Su, Z. H.; Cao, H. L.; Liu, Z. H.; Pandey, D.; Wei, D. G.; Chung, T. F.; Peng, P.; Guisinger, N. P.; Stach, E. A.; Bao, J. M.; Pei, S. S.; Chen, Y. P. Control and characterization of individual grains and grain boundaries in graphene grown by chemical vapour deposition. Nat. Mater. 2011, 10 (6), 443−449. (4) Simonis, P.; Goffaux, C.; Thiry, P. A.; Biro, L. P.; Lambin, P.; Meunier, V. STM study of a grain boundary in graphite. Surf. Sci. 2002, 511 (1−3), 319−322. (5) Cervenka, J.; Flipse, C. F. J. Structural and electronic properties of grain boundaries in graphite: Planes of periodically distributed point defects. Phys. Rev. B 2009, 79 (19), 195429. (6) Yazyev, O. V.; Louie, S. G. Nat. Mater. 2010, 9 (10), 806−809. (7) Kahaly, M. U.; Singh, S. P.; Waghmare, U. V. Carbon Nanotubes with an Extended Line Defect. Small 2008, 4 (12), 2209−2213. (8) Araújo, J. S.; Nunes, R. W. Complex evolution of the electronic structure from polycrystalline to monocrystalline graphene: Generation of a new Dirac point. Phys. Rev. B 2010, 81 (7), 073408. (9) Gunlycke, D.; White, C. T. Graphene Valley Filter Using a Line Defect. Phys. Rev. Lett. 2011, 106 (13), 136806. (10) Lin, X. Q.; Ni, J. Half-metallicity in graphene nanoribbons with topological line defects. Phys. Rev. B 2011, 84 (7), 075461.

5101

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102

Nano Letters

Letter

(11) Gomez-Navarro, C.; Meyer, J. C.; Sundaram, R. S.; Chuvilin, A.; Kurasch, S.; Burghard, M.; Kern, K.; Kaiser, U. Nano Lett. 2010, 10 (4), 1144−1148. (12) Bagri, A.; Mattevi, C.; Acik, M.; Chabal, Y. J.; Chhowalla, M.; Shenoy, V. B. Structural evolution during the reduction of chemically derived graphene oxide. Nature Chem. 2010, 2 (7), 581−587. (13) Peres, N. M. R.; Guinea, F.; Neto, A. H. C. Electronic properties of disordered two-dimensional carbon. Phys. Rev. B 2006, 73 (12), 125411. (14) Neto, A. H. C.; Chamon, C. D.; Nayak, C. Open Luttinger liquids. Phys. Rev. Lett. 1997, 79 (23), 4629−4632. (15) Ferreira, A.; Xu, X. F.; Tan, C. L.; Bae, S. K.; Peres, N. M. R.; Hong, B. H.; Ozyilmaz, B.; Neto, A. H. C. Transport properties of graphene with one-dimensional charge defects. EPL 2011, 94 (2), 28003. (16) Kohn, W.; Sham, L. J. Self-consistent equations including exchange and correlation effects. Phys. Rev. 1965, 140 (4A), A1133− A1138. (17) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett. 1996, 77 (18), 3865− 3868. (18) Troullier, N.; Martins, J. L. Efficient pseudopotentials for planewave calculations. Phys. Rev. B 1991, 43 (3), 1993−2006. (19) Kleinman, L; Bylander, D. M. Efficacious form for model pseudopotentials. Phys. Rev. Lett. 1982, 48 (20), 1425−1428. (20) Soler, J. M.; Artacho, E.; Gale, J. D.; Garcia, A.; Junquera, J.; Ordejon, P.; Sanchez-Portal, D. The SIESTA method for ab initio order-N materials simulation. J. Phys.: Condens. Matter 2002, 14 (11), 2745−2779. (21) Cortijo, A.; Vozmediano, M. A. H. Effects of topological defects and local curvature on the electronic properties of planar graphene. Nucl. Phys. B 2007, 763 (3), 293−308. (22) da Silva-Araújo, J; Chacham, H; Nunes, R. W. Gap-opening in topological defect networks in graphene. Phys. Rev. B 2010, 81 (19), 193405. (23) Nakada, K.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Edge state in graphene ribbons: Nanometer size effect and edge shape dependence. Phys. Rev. B 1996, 54 (24), 17954−17961. (24) Lee, H.; Son, Y. W.; Park, N.; Han, S. W.; Yu, J. J. Magnetic ordering at the edges of graphitic fragments: Magnetic tail interactions between the edge-localized states. Phys. Rev. B. 2005, 72 (17), 174431. (25) Soon, Y. W.; Cohen, M. L.; Louie, S. G. Half-metallic graphene nanoribbons. Nature 2006, 444 (7117), 347−349. (26) Alexandre, S. S.; Mazzoni, M. S. C.; Chacham, H. Edge states and magnetism in carbon nanotubes with line defects. Phys. Rev. Lett. 2008, 100 (14), 146801. (27) Balasubramanian, G.; Neumann, P.; Twitchen, D.; Markham, M.; Kolesov, R.; Mizuochi, N.; Isoya, J.; Achard, J.; Beck, J.; Tissler, J.; Jacques, V.; Hemmer, P. R.; Jelezko, F.; Wrachtrup, J. Ultralong spin coherence time in isotopically engineered diamond. Nat. Mater. 2009, 8 (5), 383−387.

5102

dx.doi.org/10.1021/nl3017434 | Nano Lett. 2012, 12, 5097−5102