Topological Aspects of Charge-Carrier Transmission across Grain

Letter pubs.acs.org/NanoLett

Topological Aspects of Charge-Carrier Transmission across Grain Boundaries in Graphene Fernando Gargiulo and Oleg V. Yazyev* Institute of Theoretical Physics, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland S Supporting Information *

ABSTRACT: Dislocations and grain boundaries are intrinsic topological defects of large-scale polycrystalline samples of graphene. These structural irregularities have been shown to strongly affect electronic transport in this material. Here, we report a systematic investigation of the transmission of charge carriers across the grain-boundary defects in polycrystalline graphene by means of the Landauer-Büttiker formalism within the tight-binding approximation. Calculations reveal a strong suppression of transmission at low energies upon decreasing the density of dislocations with the smallest Burgers vector b = (1,0). The observed transport anomaly is explained from the point of view of resonant backscattering due to localized states of topological origin. These states are related to the gauge field associated with all dislocations characterized by b = (n,m) with n − m ≠ 3q (q ∈ ). Our work identifies an important source of charge-carrier scattering caused by the topological defects present in large-area graphene samples produced by chemical vapor deposition. KEYWORDS: Polycrystalline graphene, electronic transport, topological defects, dislocations, grain boundaries, tight-binding model

S

ince its isolation in 2004,1 graphene has been attracting ever-increasing attention due to its extraordinary physical properties and potential technological applications.2−4 Early research experiments on graphene have been performed using single micrometer-scale samples obtained by micromechanical cleavage. However, technological applications require manufacturing processes that would allow for robust production at much larger scales, for example, the chemical vapor deposition technique.5−7 Recent experimental studies have shown that such extended graphene samples tend to be polycrystalline, that is, composed of micrometer-size single-crystalline domains of varying lattice orientation.8−10 Dislocations and grain boundaries are responsible for breaking the long-range order in polycrystals. These topological defects inevitably affect all physical properties of graphene.11−18 In particular, it has been demonstrated experimentally that grain boundaries dramatically alter the electronic transport properties of graphene.19−21 Understanding the effect of topological defects on chargecarrier transport in graphene is crucial for technological applications of this material in electronics, clean energy, and related domains. According to the Read−Shockley model,22 grain boundaries in two-dimensional crystals are equivalent to one-dimensional arrays of dislocations. These defects are characterized by the socalled Burgers vector that is defined nonlocally, and hence, does not change upon local modifications of the atomic structure.23 In other words, the Burgers vector is a structural topological invariant that makes dislocations distinct from local disorder such as point defects, impurities, adsorbates, and so forth. In graphene, the low-energy configurations of the cores of constituent dislocations are composed of pairs of pentagons © 2013 American Chemical Society

and heptagons with the resulting Burgers vector being dependent on their mutual positions.24−27 Remarkably, in certain highly ordered grain-boundary structures the conservation of momentum results in a complete suppression of the transmission of low-energy charge carriers.11 However, it is of paramount importance to understand the factors determining the charge-carrier transmission probability in a more general situation when no symmetry-related selection is present. For example, this is the case of grain boundaries with strongly perturbed periodic arrangement of dislocations, such as in typical samples of graphene grown by chemical vapor deposition.8,9 In this Letter, we report a systematic study of the chargecarrier transmission across grain boundaries in graphene by means of the Landauer-Büttiker approach. We find that the structural topological invariant of dislocations, the Burgers vector b, plays a crucial role in determining transport properties. In particular, for the case of grain boundaries formed by the minimal Burgers vector b = (1,0) dislocations, we find an unexpected suppression of the transmission of lowenergy charge carriers in the limit of small misorientation angles (or, equivalently, small dislocation densities). This counterintuitive behavior is explained from the point of view of resonant backscattering involving localized states of topological origin, which arise due to the gauge field created by dislocations characterized by b = (n,m) with n − m ≠ 3q (q ∈ ). The b = (1,1) dislocations are shown to behave as ordinary scattering Received: October 15, 2013 Revised: November 27, 2013 Published: December 2, 2013 250

dx.doi.org/10.1021/nl403852a | Nano Lett. 2014, 14, 250−254

Nano Letters

Letter

centers, resulting in very weak effects on the electronic transport. Our work provides an insight into the local transport properties of polycrystalline graphene samples. In our study we employ the nearest-neighbor tight-binding model Hamiltonian H = −t ∑ [ci†c j + h.c.] ⟨i , j⟩

(1)

where ci (c†i ) annihilates (creates) an electron at site i and ⟨i,j⟩ stands for pairs of nearest-neighbor atoms. The hopping integral t = 2.7 eV is assumed to be constant. The validity of the nearest-neighbor tight-binding model was verified by comparing with the results of calculations employing a more elaborate tight-binding parametrization28,29 which includes second- and third-nearest-neighbor hopping terms (Figure S1 of the Supporting Information). The tight-binding results are also in good agreement with first-principles calculations as shown previously.11 Coherent transport across grain boundaries in graphene is studied within the Landauer-Büttiker formalism, which relates the conductance G(E) at a given energy E to the transmission T(E) as G(E) = G0T(E) with G0 = 2e2/h being the conductance quantum.30 The transmission is evaluated by means of the nonequilibrium Green’s function approach using two-terminal device configurations with contacts represented by the semi-infinite ideal graphene leads T = Tr[ΓLGS† ΓR GS]

Figure 1. (a) A generic example of asymmetric periodic grain boundary composed of b = (1,0) dislocations. This grain-boundary structure is characterized by a pair of matching vectors (0,3)|(1,2). The periodicity vector d and the Burgers vector b are shown. Structures of symmetric grain boundaries formed by (b) b = (1,0) and (c) b = (1,1) dislocations. (d) Degenerate grain boundary (θ = 0°) with the Burgers vector of constituent dislocations oriented along the grain-boundary line (shown as dashed line).

(2)

The scattering region Green’s function GS is calculated as GS = [E +I − HS − Σ L − Σ R ]−1

(3)

employing the coupling matrices ΓL(R) for the left (right) lead given by ΓL(R) = i[Σ L(R) − Σ†L(R)]

use the matching-vectors notation in order to unambiguously identify the structures of our models. Figure 1a shows a generic example of an asymmetric grain boundary formed by the b = (1,0) dislocations. Figures 1b,c depict examples of symmetric (θL = θR) periodic grain boundaries formed by b = (1,0) and b = (1,1) dislocations, respectively. Figure 1d shows an example of a degenerate grain boundary (θ = 0°) with the Burgers vector of constituent dislocations oriented along the grain boundary line. In our study, we focus only on the models that do not result in transport gaps due to selection by momentum.32 We first focus on symmetric periodic grain boundaries formed by b = (1,0) (= 2.46 Å) dislocations. Such grain boundaries are defined by pairs of matching vectors belonging to the (l,l + 1)|(l + 1,l) series (l ∈ ). Hence, d = a0[3l(l + 1) + 1]1/2 where a0 = 2.46 Å is the lattice constant of graphene. Figure 2a shows the transmission probability T as a function of energy E and transverse momentum k∥ for the first member of this sequence (l = 1) characterized by d = 6.51 Å (Figure 1b). One clearly observes a projected Dirac cone in the irreducible half of the onedimensional Brillouin zone corresponding to the periodic grainboundary structure with T(k∥,E) ≲ 1, in agreement with previous calculations.11 Figure 2b shows T(k∥,E) for a grain boundary characterized by l = 8 and, hence, a larger periodicity d = 36.2 Å. The most evident difference between the two plots is the occurrence of multiple conductance channels as a result of band folding over smaller Brillouin zone. The striking feature, however, is the clear reduction of conductance close to the Dirac point energy E = 0. This counterintuitive decrease of

(4)

In these expressions HS is the Hamiltonian for the scattering region, ΣL(R) are the self-energies which couple scattering region to the leads and E+ = E + iηI (η → 0+). The dependence of T, GS, ΓL(R) and ΣL(R) on energy E and transverse momentum k|| is omitted for the sake of compact notation. We consider grain-boundary models constructed as periodic arrays of dislocations following the Read−Shockley model.22 Only dislocations formed by pentagons and heptagons are investigated as these structures preserve the 3-fold coordination of sp2 carbon atoms thus ensuring energetically favorable configurations of defects. This construction is consistent with experimental atomic resolution images of grain boundaries in polycrystalline graphene.8,9 Defect structures containing undercoordinated carbon atoms can also be realized but typically have significantly larger formation energies.31 The relative position of pentagons and heptagons define the structural topological invariant of dislocations, their Burgers vector. In turn, the Burgers vectors, their orientation with respect to the grain boundary line, and the distance between dislocation cores define the structural topological invariant of the grain boundary, its misorientation angle θ = θL + θR. This relation allows constructing arbitrary grain boundary models, as described in ref 24. Alternatively, periodic grain boundaries can be defined in terms of a pair of matching vectors (nL,mL)|(nR,mR), introduced in ref 11 (Figure 1a). In this paper we shall constrain ourselves to Burgers vectors b and interdislocation distance d in order to simplify the discussion. However, we will 251

dx.doi.org/10.1021/nl403852a | Nano Lett. 2014, 14, 250−254

Nano Letters

Letter

Figure 2. Electronic transport across periodic grain boundaries in graphene formed by the b = (1,0) dislocations. (a,b) Transmission probability as a function of energy E and transverse momentum k∥ across symmetric grain boundaries characterized by d = 6.51 Å and d = 36.2 Å, respectively. (c) Transmission probability close to the Dirac point (E = 10−3t) as a function of q∥ for different values of d. (d) Lowenergy transmission of the normally incident charge carriers as a function of interdislocation distance d for symmetric, asymmetric, and degenerate grain boundaries. The inset shows the logarithmic scale plot.

Figure 3. (a) Density of states (DOS) in the interface region computed for the grain boundaries formed by b = (1,0) dislocations with different values of d. (b) Positions of the DOS peaks as a function of d. The labels refer to peaks in panel (a) shown for the d = 23.5 Å grain boundary. (c) Local density of states (LDOS) at E = 10−3t for the d = 61.8 Å grain boundary. Circle areas are proportional to the LDOS. (d) LDOS as a function of distance from the defect core δ calculated at E = 10−3t for the d = 385 Å grain boundary. Solid lines indicate two trends consistent with the results of ref 45.

transmission, or equivalently enhancement of scattering upon decreasing the density of dislocations, suggests the topological origin of the discussed transport behavior. Figure 2c,d further investigate the details of charge-carrier transmission at very low energy (E = 10−3t). One clearly observes a monotonic decrease of T(q∥) (with q∥ = k∥ − (2π)/(3d) being the transverse momentum relative to the location of the projected Dirac point) as d increases (Figure 2c). The transmission probability of normally incident charge carriers (q∥ = 0) exhibits an inverse power scaling law T ∝ d−γ, with an exponent γ ≈ 0.5 (dashed lines in Figure 2d). Moreover, the observed scaling law is independent of the orientation of the Burgers vectors of dislocations relative to the grain-boundary line. This was explicitly demonstrated using several models of asymmetric and degenerate configurations (Figure 2d). The observed transport anomaly is further investigated by analyzing the local density of states (LDOS) calculated as

Figure 3c. The presence of localized states results in resonant backscattering at low energies. Such localized states are a common feature of many types of lattice defects in graphene. In particular, single-atom vacancies and covalent functionalization defects give rise to quasilocalized states at E = 0.33−38 Substitutional impurities as well as the Stone-Wales defect and structurally related point defects give rise to resonances at finite energies.28,29,39 However, localized states at dislocations emerge owing to a distinct physical mechanism as we explain below. The origin of the localized states is related to the topological nature of defects in polycrystalline graphene. Charge carriers of momentum k encircling a dislocation with Burgers vector b gain a phase φ = k·b.40 The aforementioned b = (1,0) dislocations thus give rise to φ = +2π/3 and φ = −2π/3 for the charge carriers in valleys τ = +1 and τ = −1, respectively. Starting from the Dirac equation for a massless particle H = (px − iAx )σx + (py − iA y )τσy

π /d d Im(GS(E , k ))n , n dk 2 (5) π 0 Figure 3a shows the density of states (DOS) calculated by summing the LDOS over atoms located in the grain-boundary region of 40 Å width. One clearly observes the presence of sharp van Hove singularities both in the valence and conduction bands superimposed on the linear contribution of pristine graphene. The peak positions converge to the Dirac point energy as the distance between dislocations d increases (Figure 3b). These DOS peaks can be attributed to the electronic states localized at the dislocations, as corroborated by

LDOSn(E) = −

∫

(6)

the effect of a dislocation is accounted for by means of a gauge field A ∝ k·b.41−44 Using this continuum model Mesaros et al. predicted that an isolated b = (1,0) dislocation gives rise to quasi-localized modes at E = 0.45 The continuous model for b = (1,0) has two low-energy solutions with the LDOS decaying as ∝ δ − 2/3 and ∝ δ−4/3, where δ is the distance from the defect core. Our numerical calculations for the grain-boundary model with a very large distance between dislocations confirm the analytical result showing the two solutions coexisting on different sublattices of the graphene lattice (Figure 3d). 252

dx.doi.org/10.1021/nl403852a | Nano Lett. 2014, 14, 250−254

Nano Letters

Letter

process upon transmission. The LDOS calculated for these grain boundary configurations show no localized states at low energies (not shown here). To conclude, our study reveals an intriguing aspect of chargecarrier transport in topologically disordered graphene. Predicted anomalous scattering is especially pertinent to lowangle grain boundaries (d ≫ |b|) composed of dislocations with the minimal Burgers vector b = (1,0). These dislocations are dominant in realistic samples due to the reduced elastic response23 and may even occur within seemingly singlecrystalline domains of graphene.46 The prevalence of the minimal Burgers vector dislocations has also been confirmed by the recent transmission electron microscopy investigations.47,48 Unlike covalently bound adatoms that also act as resonant scattering centers,34−38 dislocations cannot be easily eliminated from the sample due to their topological nature and high diffusion barriers at normal conditions.49 Our work thus identifies an important source of charge-carrier scattering in large-area samples of graphene produced by high efficiency techniques, such as the chemical vapor deposition.

In order to gain a qualitative understanding of the dependence of transmission on d (Figure 2c,d), one has to appreciate the fact that the finite distance between dislocations forming a grain boundary allows for the hybridization of localized states. As a result, the LDOS peaks at positive and negative energies emerge in lieu of the E = 0 peak for an isolated dislocation. As the distance between dislocations d increases, the hybridization diminishes, thus reducing the peak energies (Figures 3a,b) and resulting in the progressive decrease of the transmission close to E = 0. At finite energies, however, the minimum of transmission is achieved at certain distance between dislocations. Moreover, the energy-dependent profiles clearly show the emergence of suppressed transmission in the low-energy region upon increasing d (see Figure S2 of the Supporting Information). It is worth stressing that although our conclusions are based on periodic models of dislocations, there is no strict requirement of periodicity, in contrast to the case of suppressed conductivity due to momentum conservation.11 This has been explicitly verified by means of supercell calculations that show that transmission is insensitive to perturbation of the periodic arrangement of dislocations in grain boundaries. More generally, all dislocations characterized by Burgers vectors b = (n,m) with n − m ≠ 3q (q ∈ ) have a similar effect on charge carriers in graphene because of equal values for the k·b product for the two valleys. However, dislocations with n − m = q are expected to behave as ordinary (topologically trivial) scatterers since k·b = 0. We verify this statement by investigating the transmission through the grain boundaries formed by b = (1,1) (= 4.23 Å) dislocations (Figure 1c). Following the convention defined earlier, these grain boundaries are defined by the pairs of matching vectors belonging to (1, l + 1)|(l + 1,1) series (l ∈ ). Figure 4a shows

■

ASSOCIATED CONTENT

S Supporting Information *

The results of validation tests for the nearest-neighbor tightbinding model, investigation of the energy dependence of charge-carrier transmission probability, and the reference data files for Figures 2c and 4a. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: oleg.yazyev@epfl.ch. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We would like to thank G. Autès, A. Cortijo, M. I. Katsnelson, L. S. Levitov, and A. Mesaros for discussions. This work was supported by the Swiss National Science Foundation (Grant PP00P2_133552).

■

REFERENCES

(1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Electric Field Effect in Atomically Thin Carbon Films. Science 2004, 306, 666−669. (2) Geim, A. K.; Novoselov, K. S. The rise of graphene. Nat. Mater. 2007, 6, 183−191. (3) Katsnelson, M. I. Graphene: carbon in two dimensions. Mater. Today 2007, 10, 20−27. (4) 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, 109−162. (5) Kim, K. S.; Zhao, Y.; Jang, H.; Lee, S. Y.; Kim, J. M.; Kim, K. S.; Ahn, J.-H.; Kim, P.; Choi, J.-Y.; Hong, B. H. Large-scale pattern growth of graphene films for stretchable transparent electrodes. Nature 2009, 457, 706−710. (6) Li, X.; Cai, W.; An, J.; Kim, S.; Nah, J.; Yang, D.; Piner, R.; Velamakanni, A.; Jung, I.; Tutuc, E.; Banerjee, S. K.; Colombo, L.; Ruoff, R. S. Large-Area Synthesis of High-Quality and Uniform Graphene Films on Copper Foils. Science 2009, 324, 1312−1314. (7) Bae, S.; Kim, H.; Lee, Y.; Xu, X.; Park, J.-S.; Zheng, Y.; Balakrishnan, J.; Lei, T.; Ri Kim, H.; Song, Y. I.; Kim, Y.-J.; Kim, K. S.; O, B.; Ahn, J.-H.; Hong, B. H.; Iijima, S. Roll-to-roll production of 30-

Figure 4. Electronic transport across grain boundaries composed of b = (1,1) dislocations. (a) Transmission probability close to the Dirac point (E = 10−3t) as a function of q∥ for the grain boundaries characterized by different values of d within the l ≠ 3p family. (b) Lowenergy transmission of the normally incident charge carriers as a function of interdislocation distance d. Two families of grain boundaries are distinguished.

that already the first members of this family exhibit transmission probabilities close to 1, which further increase as the interdislocation distance d increases. Furthermore, one can distinguish two families of grain-boundary structures characterized by l = 3p and l ≠ 3p (p ∈ ℕ) (Figure 4b). Within both families, the effect of dislocations can be described in terms of scattering cross sections, σl=3p ≈ 0.5 Å and σl≠3p ≈ 0.01 Å. The first value is significantly larger since both Dirac points correspond to k∥ = 0, thus enabling the intervalley scattering 253

dx.doi.org/10.1021/nl403852a | Nano Lett. 2014, 14, 250−254

Nano Letters

Letter

in. graphene films for transparent electrodes. Nat. Nanotechnol. 2010, 5, 574−578. (8) Huang, P. Y.; Ruiz-Vargas, C. S.; van der Zande, A. M.; Whitney, W. S.; Levendorf, M. P.; Kevek, J. W.; Garg, S.; Alden, J. S.; Hustedt, C. J.; Zhu, Y.; Park, J.; McEuen, P. L.; Muller, D. A. Grains and grain boundaries in single-layer graphene atomic patchwork quilts. Nature 2011, 469, 389−392. (9) Kim, K.; Lee, Z.; Regan, W.; Kisielowski, C.; Crommie, M. F.; Zettl, A. Grain Boundary Mapping in Polycrystalline Graphene. ACS Nano 2011, 5, 2142−2146. (10) An, J.; Voelkl, E.; Suk, J. W.; Li, X.; Magnuson, C. W.; Fu, L.; Tiemeijer, P.; Bischoff, M.; Freitag, B.; Popova, E.; Ruoff, R. S. Domain (Grain) Boundaries and Evidence of “Twinlike” Structures in Chemically Vapor Deposited Grown Graphene. ACS Nano 2011, 5, 2433−2439. (11) Yazyev, O. V.; Louie, S. G. Electronic transport in polycrystalline graphene. Nat. Mater. 2010, 9, 806−809. (12) Grantab, R.; Shenoy, V. B.; Ruoff, R. S. Anomalous Strength Characteristics of Tilt Grain Boundaries in Graphene. Science 2010, 330, 946−948. (13) Gunlycke, D.; White, C. T. Graphene Valley Filter Using a Line Defect. Phys. Rev. Lett. 2011, 106, 136806. (14) Ferreira, A.; Xu, X.; Tan, C.-L.; Bae, S.-K.; Peres, N. M. R.; Hong, B.-H.; Ö zyilmaz, B.; Castro Neto, A. H. Transport properties of graphene with one-dimensional charge defects. Europhys. Lett. 2011, 94, 28003. (15) Wei, Y.; Wu, J.; Yin, H.; Shi, X.; Yang, R.; Dresselhaus, M. The nature of strength enhancement and weakening by pentagon-heptagon defects in graphene. Nat. Mater. 2012, 11, 759−763. (16) Rodrigues, J. N. B.; Peres, N. M. R.; Lopes dos Santos, J. M. B. Scattering by linear defects in graphene: A continuum approach. Phys. Rev. B 2012, 86, 214206. (17) Van Tuan, D.; Kotakoski, J.; Louvet, T.; Ortmann, F.; Meyer, J. C.; Roche, S. Scaling Properties of Charge Transport in Polycrystalline Graphene. Nano Lett. 2013, 13, 1730−1735. (18) Radchenko, T. M.; Shylau, A. A.; Zozoulenko, I. V.; Ferreira, A. Effect of charged line defects on conductivity in graphene: Numerical Kubo and analytical Boltzmann approaches. Phys. Rev. B 2013, 87, 195448. (19) Yu, Q.; Jauregui, L. A.; Wu, W.; Colby, R.; Tian, J.; Su, Z.; Cao, H.; Liu, Z.; Pandey, D.; Wei, D.; Chung, T. F.; Peng, P.; Guisinger, N. P.; Stach, E. A.; Bao, J.; 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, 443−449. (20) Tsen, A. W.; Brown, L.; Levendorf, M. P.; Ghahari, F.; Huang, P. Y.; Havener, R. W.; Ruiz-Vargas, C. S.; Muller, D. A.; Kim, P.; Park, J. Tailoring Electrical Transport Across Grain Boundaries in Polycrystalline Graphene. Science 2012, 336, 1143−1146. (21) Koepke, J. C.; Wood, J. D.; Estrada, D.; Ong, Z.-Y.; He, K. T.; Pop, E.; Lyding, J. W. Atomic-Scale Evidence for Potential Barriers and Strong Carrier Scattering at Graphene Grain Boundaries: A Scanning Tunneling Microscopy Study. ACS Nano 2013, 7, 75−86. (22) Read, W. T.; Shockley, W. Dislocation Models of Crystal Grain Boundaries. Phys. Rev. 1950, 78, 275−289. (23) Nelson, D. R. Defects and geometry in condensed matter physics; Cambridge University Press: Cambridge, 2002. (24) Yazyev, O. V.; Louie, S. G. Topological defects in graphene: Dislocations and grain boundaries. Phys. Rev. B 2010, 81, 195420. (25) Liu, Y.; Yakobson, B. I. Cones, Pringles, and Grain Boundary Landscapes in Graphene Topology. Nano Lett. 2010, 10, 2178−2183. (26) Carpio, A.; Bonilla, L. L.; Juan, F. d.; Vozmediano, M. A. H. Dislocations in graphene. New J. Phys. 2008, 10, 053021. (27) Carlsson, J. M.; Ghiringhelli, L. M.; Fasolino, A. Theory and hierarchical calculations of the structure and energetics of [0001] tilt grain boundaries in graphene. Phys. Rev. B 2011, 84, 165423. (28) Lherbier, A.; Dubois, S. M. M.; Declerck, X.; Roche, S.; Niquet, Y.-M.; Charlier, J.-C. Two-Dimensional Graphene with Structural Defects: Elastic Mean Free Path, Minimum Conductivity, and Anderson Transition. Phys. Rev. Lett. 2011, 106, 046803.

(29) Lherbier, A.; Dubois, S. M. M.; Declerck, X.; Niquet, Y.-M.; Roche, S.; Charlier, J.-C. Transport properties of graphene containing structural defects. Phys. Rev. B 2012, 86, 075402. (30) Büttiker, M.; Imry, Y.; Landauer, R.; Pinhas, S. Generalized many-channel conductance formula with application to small rings. Phys. Rev. B 1985, 31, 6207−6215. (31) Akhukov, M. A.; Fasolino, A.; Gornostyrev, Y. N.; Katsnelson, M. I. Dangling bonds and magnetism of grain boundaries in graphene. Phys. Rev. B 2012, 85, 115407. (32) Only periodic structures characterized by marching vectors (nL,mL)|(nR,mR) such that either both nL − mL = 3p and nR − mR = 3q, or both nL − mL ≠ 3p and nR − mR ≠ 3q (p,q ∈ ) are considered.11 (33) Yazyev, O. V.; Helm, L. Defect-induced magnetism in graphene. Phys. Rev. B 2007, 75, 125408. (34) Titov, M.; Ostrovsky, P. M.; Gornyi, I. V.; Schuessler, A.; Mirlin, A. D. Charge Transport in Graphene with Resonant Scatterers. Phys. Rev. Lett. 2010, 104, 076802. (35) Wehling, T. O.; Yuan, S.; Lichtenstein, A. I.; Geim, A. K.; Katsnelson, M. I. Resonant Scattering by Realistic Impurities in Graphene. Phys. Rev. Lett. 2010, 105, 056802. (36) Yuan, S.; De Raedt, H.; Katsnelson, M. I. Modeling electronic structure and transport properties of graphene with resonant scattering centers. Phys. Rev. B 2010, 82, 115448. (37) Ferreira, A.; Viana-Gomes, J.; Nilsson, J.; Mucciolo, E. R.; Peres, N. M. R.; Castro Neto, A. H. Unified description of the dc conductivity of monolayer and bilayer graphene at finite densities based on resonant scatterers. Phys. Rev. B 2011, 83, 165402. (38) Radchenko, T. M.; Shylau, A. A.; Zozoulenko, I. V. Influence of correlated impurities on conductivity of graphene sheets: Timedependent real-space Kubo approach. Phys. Rev. B 2012, 86, 035418. (39) Choi, H. J.; Ihm, J.; Louie, S. G.; Cohen, M. L. Defects, Quasibound States, and Quantum Conductance in Metallic Carbon Nanotubes. Phys. Rev. Lett. 2000, 84, 2917−2920. (40) Iordanskii, S. V.; Koshelev, A. E. Interaction of excitations with dislocations in a crystal. J. Exp. Theor. Phys 985, 63, 820. (41) Lammert, P. E.; Crespi, V. H. Topological Phases in Graphitic Cones. Phys. Rev. Lett. 2000, 85, 5190−5193. (42) Cortijo, A.; Vozmediano, M. A. Effects of topological defects and local curvature on the electronic properties of planar graphene. Nucl. Phys. B 2007, 763, 293−308. (43) Mesaros, A.; Sadri, D.; Zaanen, J. Berry phase of dislocations in graphene and valley conserving decoherence. Phys. Rev. B 2009, 79, 155111. (44) Vozmediano, M.; Katsnelson, M.; Guinea, F. Gauge fields in graphene. Phys. Rep. 2010, 496, 109−148. (45) Mesaros, A.; Papanikolaou, S.; Flipse, C. F. J.; Sadri, D.; Zaanen, J. Electronic states of graphene grain boundaries. Phys. Rev. B 2010, 82, 205119. (46) Coraux, J.; N’Diaye, A. T.; Busse, C.; Michely, T. Structural Coherency of Graphene on Ir(111). Nano Lett. 2008, 8, 565−570. (47) Warner, J. H.; Margine, E. R.; Mukai, M.; Robertson, A. W.; Giustino, F.; Kirkland, A. I. Dislocation-Driven Deformations in Graphene. Science 2012, 337, 209−212. (48) Lehtinen, O.; Kurasch, S.; Krasheninnikov, A. V.; Kaiser, U. Atomic scale study of the life cycle of a dislocation in graphene from birth to annihilation. Nat. Commun. 2013, 4, 2098. (49) Banhart, F.; Kotakoski, J.; Krasheninnikov, A. V. Structural Defects in Graphene. ACS Nano 2011, 5, 26−41.

254

dx.doi.org/10.1021/nl403852a | Nano Lett. 2014, 14, 250−254