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Excitons of Edge and Surface Functionalized Graphene Nanoribbons Xi Zhu† and Haibin Su*,†,‡ DiVision of Materials Science, Nanyang Technological UniVersity, 50 Nanyang AVenue, Singapore 639798, and Institute of High Performance Computing, 1 Fusionopolis Way, Connexis 138632, Singapore ReceiVed: March 15, 2010; ReVised Manuscript ReceiVed: June 26, 2010
We apply density functional theory to study the optical properties of armchair graphene nanoribbons (AGNRs) functionalized from both edge and surface employing quasiparticle corrections and many body, that is, electron-hole, interactions. The variation in band gaps is scrutinized in terms of bonding character and the localization of wave functions. We have determined the family and functionalization dependence of quasiparticle correction, absorption spectrum, exciton binding strength, and its wave function’s spatial characteristics. In particular, all the excitons exhibit non-Frenkel character that results from the interplay among the extended π electron states and weakly screened columbic interactions. The functionalized AGNRs hold promising potential in optoelectronic applications. The successful synthesis of graphene,1-3 a single atomic carbon layer of graphite, has resulted in intensive investigations both on fundamental physics and promising application.4-10 When graphene is etched or patterned along one specific direction, a novel quasi-one-dimensional structure made of a strip of graphene of nanometers in width, can be obtained, which is named as graphene nanoribbon (GNR). Very recently, two beautifully controlled fabrication schemes of producing GNRs have been reported by unzipping carbon nanotubes through plasma etching and solution-based oxidation.11,12 The GNRs are predicted to exhibit various remarkable properties and may be a potential elementary structure for future carbon-based nanoelectronics.13-16 Particularly, GNRs are direct-bandgap materials desirable for generating and detecting light. Therefore, GNRs offer attractive potential for building integrated electronic and optoelectronic devices. In addition, GNRs are much more compatible to semiconductor architecture to manipulate than carbon nanotubes (CNTs) due to their flat structure. Carbon atoms on graphene edges are only bounded to two neighboring carbon atoms, and a dangling carbon bond offers a remarkable opportunity for tuning electronic properties due to the high ratio between edge and interior atoms. More experimental efforts are clearly needed to achieve precise control of edge modifications,17 since the physical and chemical properties of GNRs indeed exhibit strong dependence on detail atomic structures, that is, the shapes of edges in particular.18-24 In recent years, excitons of functional carbon nanostructures are becoming one of the most studied subjects in the low dimensional systems due to rich fundamental physics as well as its highly potential for optoelectronic device applications. The enhanced electron correlation in the low dimensional system becomes more important and is proved to play a key role in understanding electronic and optical properties of GNRs.25-28 One of the unresolved central issues is to understand the effects of edge and surface functionalization on the fundamentally important quantities, such as binding strength and wave function of excitons. In this paper, we simulate the electronic and optoelectronic properties of edge and surface modified armchair * To whom correspondence should be addressed. Email:
[email protected]. † Nanyang Technological University. ‡ Institute of High Performance Computing.
GNRs (AGNRs) including many-body effects explicitly. For simplicity, we choose the same nomenclature for GNRs as for CNTs in this paper. An unwrapped (n, m) CNT, for instance, is called a (n, m) GNR. Zigzag GNRs (n, n), which correspond to unwrapped armchair CNTs with zigzag edges, are still metallic if the spin degree of freedom is not considered. The electronic structure of armchair (n, 0) GNRs depends strongly on the nanoribbon width.16-23,29 This remarkable characteristic is very useful in making graphene-based nanoscale devices. In addition to an overall decrease of energy gap with increasing ribbon width as observed experimentally,17 theoretical studies predict a superimposed oscillation feature for AGNRs.16-23,29 According to this behavior, AGNRs are further classified into three distinct families, that is, m ) 3p, m ) 3p + 1, and m ) 3p + 2, with p being a positive integer, where m indicates the number of dimer lines across the ribbon width. This raises the opportunity to tailor the optoelectronic properties of AGNRs by appropriately manipulating the width of AGNRs. Here, we consider two families of AGNRs with m being 9 and 10. The notation is defined as follows: A-Wm-B where m stands for the width of AGNRs (the same meaning as above) and A and B denote the element used for edge passivation; Wm:H(n) where H(n) represents the location of hydrogen adsorbed on AGNRs. For instance, H-W9-F means that one edge of the AGNR with the width of 9 is passivated by H and the other edge by F; W10: H(5) refers to hydrogen absorbed at carbons at the fifth column of AGNR with the width of 10. The whole set of structures studied in this work are presented in Figure 1. In this paper, first, we introduce the method used for studying electronic and optical properties. Then, we investigate the electronic structures with the effect of quasiparticle correction, the optical absorption spectra including the electron-hole coupling, and the binding strength and wave function of excitons for both edge- and surface-functionalized AGNRs. Finally, we address the correlation between the binding energy and the spatial size of an exciton. The first-principles calculation of the optical excitations is carried out using a many-body perturbation theory approach, based on the three-step procedure.30 First, we obtain the groundstate electronic properties of the relaxed system within the local density approximation (LDA) using ABINIT31 code. The
10.1021/jp102341b 2010 American Chemical Society Published on Web 09/29/2010
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Figure 1. Structures of AGNRs with edge modifications (a) and hydrogen adsorption on the surfaces (b). Each structure has both side-view and top-view in (b). The red dot line indicates the location of the mirror-plane symmetry in W9:H(5). The white, gray, and cyan balls represent hydrogen, carbon, and fluorine atoms, respectively. The x-axis is along the periodic direction, and the y-axis is along the transverse direction from one edge to the other.
calculations are carried out using separable normconserving pseudopotentials and a plane-wave basis set. A kinetic energy cutoff of 50 Ry and 41 k-points are employed. Each atomic structure is fully relaxed, until forces acting on atoms are less than 0.01 eV/Å. Next, the quasiparticle corrections to the LDA eigenvalues are computed within the G0W0 approximation for the self-energy operator, where the LDA wave functions are used as good approximations for the quasiparticle ones and the screening is treated within the plasmon-pole approximation.32 The electron-hole interaction is included by solving the Bethe-Salpeter equation (BSE) in the basis set of quasielectron and quasihole states, where the static screening in the direct term is calculated within the random-phase approximation:33 S (Eck - EVk)AVck +
∑ AVS c k
S ) ΩSAVck
k V c
(1) where ASVck is the exciton wave function, Keh is the electron-hole coupling kernel, ΩS is the excitation energy, and Eck and EVk are the quasiparticle energies of the electron and hole states, respectively. We use total 10 bands with the Fermi level lies in between in our calculation. All the calculations are performed with the ABINIT and Yambo codes.31,34 The incident photon has the polarization along the principal axis of GNRs, that is, x-axis shown in Figure 1, so that the significant quenching due to the depolarization effect for the perpendicular polarization can be avoided.35,36 Since the supercell method is used in our calculations, a rectangular-shape truncated Coulomb interaction is applied to eliminate the image effect between adjacent supercells to mimic isolated GNRs.37 We have studied the band structures with GW corrections and optical absorptions of H-W9-H and H-W10-H. Our results (as tabulated in Table 1) are in accord with the previous studies.37
TABLE 1: Energy Gap Computed by LDA (ELDA-gap) (1st Column) and GW (EGW-gap) (2nd Column), Optical Transition Energy (ν11 and/or ν22) Including the Coupling Electron and Hole of Bound Excitons (3rd Column), and Exciton Binding Energy (Eb) Computed by EGW-gap - EBS of Edge Modified AGNRs W9 and W10 H-W9-H F-W9-H F-W9-F H-W10-H H-W10-F F-W10-F
ELDA-gap (eV)
EGW-gap (eV)
ν11, ν22
Eb (eV)
0.78 0.82 0.89 1.15 1.00 0.86
2.27 2.37 2.51 2.94 2.75 2.25
0.91 0.99 1.00 1.44, 1.76 1.30, 1.74 0.87, 1.45
1.36 1.38 1.51 1.50, 1.18 1.45, 1.01 1.38, 0.80
The electronic structures depend on both the width and passivation option of AGNRs as shown in Table 1. The band gap of W10 (3p + 1) is larger than that of W9 (3p) for the AGNRs whose both edges are hydrogen-passivated, which obeys the general family behavior in terms of energy gap as reported previously:23,29 Eg (3p + 1) > Eg (3p) > Eg (3p + 2). The noteworthy change happens to the AGNRs with fluorinepassivation at both edges. For instance, the F-passivation makes the band gap of W9 (3p) to be larger than that of W10 (3p + 1). Wang et al. have reported that energy gaps can be tuned by changing edge hopping parameters that couples with the variation of carbon-carbon bond lengths.29 Although the geometric deformations of carbon networks are almost the same between H-passivated GNRs and F-passivated ones, the bandgaps surprisingly show distinct dependence on the width of AGNRs. Note that the hopping integral πpp, representing the coupling between pz orbitals of two neighboring carbon atoms, increases as the C-C distance decreases. However, the πpp can decrease despite the reduction in C-C bond length, when the hybridization between two carbon atoms gets severely perturbed due to bonding to nearby atoms. As shown in Figure S1 of the
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Figure 2. Optical absorption spectra of functionalized AGNRs: edge modification H-Wm-H, H-Wm-F, and F-Wm-F, where m ) 9 (a) and m ) 10 (b); hydrogen adsorbed on the surfaces of W9 (W9:H(3) and W9:H(5)) (c) and W10 (W10:H(4) and W10:H(6)) (d). The substantial oscillator strength is transferred from interband to excitonic transition.
Supporting Information, the projected density of states (PDOS) of the H-passivated AGNRs clearly reveal that the peaks of the pz orbital of edge C atoms almost coincide with that of the inner C atoms below the Fermi level.38 The π bonding remains unchanged during the H-passivation of edge C atoms. However, the pz orbital of edge carbon in the F-passivated cases is shifted down to much lower energy. The pz orbitals of edge carbon connected to fluorine have three major peaks centered at -8.5, -6.5, and -2.0 eV below the Fermi energy. The center of the pz orbital spectrum of the edge carbon connected to fluorine decreases significantly to -6.0 eV in comparison with that of the edge carbon connected to hydrogen which is -4.0 eV below the Fermi energy. The fluorine shifts its connected carbon’s pz orbital spectrum out of the energy range of canonical π bonding. Consequently, despite the decrease of C-C bond lengths, the hopping parameters decrease instead, which indeed is the key to capture the systematic trend of gaps in F-passivated GNRs. This leads to the effective widths of F-W9-F and F-W10-F to be 7 (3p + 1) and 8 (3p + 2), respectively. Thus, the gap of F-W10-F is expected to be larger than that of F-W9-F according to the general family determined behavior. The effective width can be also applied to capture the relation of gaps between H-Wm-H and F-Wm-F. The main point is the effective width of AGNRs passivated by fluorine at both edges is reduced by two. For instance, the effective width of F-W9-F changes from 9 (3p) to 7 (3p + 1). Thus, the energy gap of H-W9-H is smaller than that of F-W9-F. The same analysis accounts for the larger gap of H-W10-H with respect to that of F-W10-F. The quasiparticle GW energy corrections in the order of 1 eV are somewhat larger than those found for bulk graphite or
diamond due to the ineffective screening for reduced dimensions as shown in Table 1.39 Clearly, the so-called “scissors operator”, in which the self-energy is approximated by a rigid shift of energy bands, cannot be applied legitimately here due to the complicated band and energy dependence of GW corrections. The GW corrections of the W10 series are larger than those of the W9 series. In addition, GW corrections of systems with bigger LDA gaps tend to be larger. For instance, the GW correction of F-W9-F is 1.62 eV compared to 1.49 (1.55) eV of H-W9-H (F-W9-H). Similarly, the GW correction of H-W10-H is 1.79 eV compared to 1.39 (1.75) eV of F-W10-F (F-W10H). The optical transition energies are computed by solving BSE numerically. The spectra are presented in Figure 2a and b with broadening parameter of 0.1 eV. Only the first (ν11) and/or second (ν22) optical transition energies associated with bound excitons are listed in the third column of Table 1. First, the major portion of oscillator strength is transferred from the interband to the excitonic transitions, which dominates the optical absorption spectra. Second, the order of optical energies correlate with energy gaps for both W9 and W10 series accordingly. The absolute values of optical energies are about half of GW gap values, which is the clear manifestation of strong coupling of the electron and hole in AGNRs. Moreover, the ratio of ν22/ν11 of the W10 series ranges from 1.25 to 1.40. The deviation from the ratio of 2 obtained from the π orbital only tight-binding model results from the delicate full band dispersion and many-body effects.28,40 The exciton binding energy is defined as the difference between GW gap and optical transition energies. It is well-known that the screening in one and quasione-dimensional materials is orders of magnitude weaker than
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in bulk materials. The free electron screening in the former case diverges logarithmically rather than in a power-law manner in the bulk metallic materials.41 Thus, the exciton binding strength is expected to be high in GNRs. The binding energies range from 1.36 to 1.51 eV for the W9 and W10 series. In general, the electron-hole interaction kernel has an attractive direct term by the screened Coulomb interaction and a repulsive exchange term mediated by the bare Coulomb interaction.33 The GW gap of H-W10-H is close to 3 eV, which results in a significantly heavier effective mass (larger kinetic energy)41 and fairly weaker screening to reduce the Coulomb interaction between electron and hole. Therefore, its exciton binding energy is the strongest in the W10 series. Similarly, the exciton binding energy of F-W9-F is the strongest in the W9 series due to the heavier effective mass and weaker screening. Next, we turn to the surface functionalized AGNRs. The surface adsorption with small atom or molecule can effectively modulate the electronic and magnetic properties of the GNR both experimentally and theoretically. The GNRs are proposed to be potential sensing units to probe NO, NO2, CO, and H2 species.42-45 Since the hydrogen adsorption on graphene is demonstrated experimentally,46-48 here, we focus on the optical properties of hydrogen adsorbed AGNRs. For the sake of continuity to the above edge modification study, we choose both W9 and W10 AGNRs and sample two different positions for adsorbing hydrogen atoms as shown in Figure 1b. When one hydrogen atom adsorbs on the top of the GNR surface, the other energetically prefers to be at the other side of the GNR surface as depicted in ref 46. The two carbon atoms bond with hydrogen atoms covalently; thus, the sp2 hybridization transforms into the sp3 one. Among the four figures in Figure 1b, only the W9: H(5) has a mirror plane (marked by red dash line) perpendicular to the GNR, while all the other three structures are asymmetrical in this regard. The sp3 carbons tilt out of plane (see Figure 1b) due to the hybridization, and the bond length between the sp3 and sp2 carbon atoms expand from 1.42 to 1.49 Å so as to form the 30° torsion configuration for all the adsorption cases. The PDOS clearly show that the energy of sp3 carbons’ pz orbital is shifted to the lower energy due to the formation of C-H bonds (as shown in Figure S1 in the Supporting Information). Since the gravity center of sp3 carbons’ pz orbital is out of the energy range of canonical π bonding, the width of Wm:H(n) decreases by two. Thus, the effective width of W9:H(5) changes from 9 (3p) to 7 (3p + 1). Since the general patterns of wave function remain the key features as of hydrogen-passivated AGNRs, the gap of W9:H(5) is expected to be larger than that of H-W9-H from the general observation of width-dependent gap behavior in AGNRs. The reflection symmetry is broken in W9:H(3) due to the fact that the hydrogens are not located on the mirror plane, which lifts the degeneracy of the second and third low conduction bands of H-W9-H as shown in Figure S3 in the Supporting Information. More importantly, the wave functions of both top valence and low conduction bands are localized to one edge as shown in Figure S2 in the Supporting Information. The delicate balance of these two competing factors, that is, the reduction of width versus the localization, results in the slightly larger gap in W9:H(3). Similarly, the symmetry breaking is manifested by the degeneracy-lifting of the two top valence and two low conduction bands, together with the localization of wave functions of both top valence and low conduction bands in both W10:H(4) and W10:H(6). Since their effective widths turn out to be 8 (3p + 2) due to the change in bonding character of two columns of carbons, both the reduction in width and
Zhu and Su TABLE 2: Energy Gap Computed by LDA (ELDA-gap) (1st Column) and GW (EGW-gap) (2nd Column), Optical Transition Energy (ν11 and/or ν22) Including the Coupling Electron and Hole of Bound Excitons (3rd Column), and Exciton Binding Energy (Eb) Computed by EGW-gap - EBS of Both W9 and W10 AGNRs with Hydrogen Functionalized on the Surfaces W9:H(3) W9:H(5) W10:H(4) W10:H(6)
ELDA-gap (eV)
EGW-gap (eV)
ν11, ν22
Eb (eV)
0.89 2.10 0.94 0.41
2.63 4.88 2.83 1.63
1.13, 1.34 2.96, 3.25 1.37, 1.68 0.54
1.50, 1.29 1.92, 1.63 1.46, 1.15 1.09
localization account for the decrease in band gaps of both W10: H(4) and W10:H(6) in comparison with H-W10-H as shown in Table 2. Similar to edge modified AGNRs, the quasiparticle GW energy corrections are in the order of electron volts as shown in Table 2 as well. The quasiparticle energy correction depends on the character of the electronic states. The material with a larger LDA gap usually has larger energy corrections within the same family of AGNRs. This agrees very well to the observation on the larger corrections of non-free-electron like bands.49,50 The optical transition energies computed by solving BSE numerically associated with bound excitons are listed in Table 2. Similarly, the oscillator strength transfer occurs from the interband to the excitonic transitions as presented in Figure 2c and d. The ratio of ν22/ν11 deviates significantly from 2 similar to the edge modified AGNRs, due to the delicate full band dispersion and many-body effects as well.40 The stronger exciton binding strength of W9:H(5) comparing with W9:H(3) originates in the relatively larger GW gap, which leads to heavier effective mass (larger kinetic energy) and fairly weaker screening to reduce the Coulomb interaction between electron and hole. Similarly, the exciton binding strength of W10:H(4) is stronger than that of W10:H(6). Lastly, in order to obtain insight into the effects of electron-hole interaction, we show the wave functions of excitons with lowest excitation energy in real space with one fixed hole position for all the functionalized GNR studied in this work. The exciton wave function is written as follows:
Ψ(rfe, rfh) )
∑ AVckφc,k(rfe)φV,k/ (rfh)
(2)
kVc
For the edge modified AGNRs, the overall exciton wave functions are very alike within the same family. Thus, only exciton wave functions of AGNRs with fluorine at both edges are presented in Figure 3a and b, since the distribution of exciton wave functions extends to fluorine atoms while not to hydrogen ones. For the AGNRs absorbed hydrogen atoms on the surfaces, the general patterns of exciton wave functions are very different. bh) is very skewed toward one edge in W9:H(3) (see The Ψ(r be,r Figure 3c), while is symmetric in W9:H(5) due the existence of one mirror plane in the structure (see Figure 3d). For both W10:H(4) and W10:H(6), both exciton wave functions are modulated to one edge. This special feature in spatial distribution can be understood by performing the direct product of wave functions of the top valence band (VB) and bottom conduction band (CB) (at Γ point) as shown in Figure S3 in the Supporting Information. For the structures where hydrogen atoms are not absorbed along the mirror plane of the surfaces, both VB and CB are localized at the same edge (for instance, Figure 3c for W9:H(3)). Hence, the exciton wave function is expected to be
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Figure 3. Representative exciton wave functions of selected edge and surface functionalized AGNRs: F-W9-F (a), F-W10-F (b), W9:H(3) (c), W9:H(5) (d), W10:H(4) (e), and W10:H(6) (f). The isovalue is 0.01 × RBohr-3, where RBohr is the Bohr radius.
located near the same edge. The binding energy (Eb) of each excitons provides important information of its electronic origin. For a Frenkel exciton, whose binding energy can be estimated according to EFb ∼ (2µ2)/(4πε0εrr3)51 in which µ is the transition dipole moment, ε0 is the vacuum permittivity, εr ∼2.5 is the static dielectric constant of AGNR, and r ∼1.4 Å is approximated by the interatomic distance. This yields µ with the order of magnitude of 1.02 × 10-29 C · m or 3 D which is an order of magnitude larger than the typical order for a Frenkelbe,r bh) is nontype transition dipole moment.52 Therefore, Ψ(r Frenkel along the AGNR x-axis28 and determined by the coulomb interaction in the AGNRs with one-dimensional character. The non-Frenkel feature is critical to the possible long lifetime of excitons. Similar to CNTs, multiphonon decay of an exciton is a possible mechanism in graphene nanoribbons when considering the optical phonon energy could be comparable to the optical band-gaps. If we choose 200 meV as the optical phonon energy, 20 meV as the broadening parameter, 1.4 eV as the exciton energy, and 1.0 and 2.0 as the interband and intraband Huang-Rhys factors, the estimated exciton lifetime due to the multiphonon decay mechanism is longer than 100 ps. To reconcile the observed fast decay rate in CNTs,53 another phonon-assisted indirect exciton ionization process was proposed, which led to additional fast decay channel with the presence of free charges. However, the finite widths of GNRs can limit the charge-trapping in the materials. In addition, the radiative lifetime at room temperature is typically at the order of a few nanoseconds.53,54 Thus, GNRs based light emitting devices can be expected to have high quantum yield owing to the possible long lifetime of excitons. In summary, we have investigated the optical properties of AGNRs functionalized from both edge and surface. The variation in band gaps is scrutinized by literally analyzing bonding character with PDOS and the localization of wave functions. We have determined the family and functionalization dependence of quasiparticle corrections, absorption edges, exciton binding energies, and its spatially features. We demonstrate the generality of family behavior in the trend of band gaps; however, this needs to be used with care with the presence of chemical modification. Moreover, all the excitons exhibit nonFrenkel character that results from the interplay among the extended π electron states and the poorly screened columbic interactions. Acknowledgment. We are grateful for the interesting discussions with H. Ren, Q. X. Li, and J. L. Yang at USTC; Y.
Fu and H. Ågren at KTH; and M. Reiher at ETH Zu¨rich. The advice from L. E. McNeil is particularly instructive. Work at NTU was supported in part by a MOE AcRF-Tier-1 grant (no. M52070060) and A*STAR SERC grant (no. M47070020). The manuscript was partially prepared while H.B.S. was visiting the Institute for Mathematical Sciences at National University of Singapore in 2009. Supporting Information Available: The PDOS, wave functions of top valence band and bottom conduction band (at the Γ point), and band structures of the edge and surface functionalized AGNRs studied in this work: H-W9-H, H-W9F, F-W9-F; H-W10-H, H-W10-F, F-W10-F; W9:H(3), W9:H(5); W10:H(4), and W10:H(6). This material is available free of charge via the Internet at http://pubs.acs.org. References and Notes (1) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Zhang, Y.; Dubonos, S. V.; Grigorieva, I. V.; Firsov, A. A. Science 2004, 306, 666. (2) Novoselov, K. S.; Geim, A. K.; Morozov, S. V.; Jiang, D.; Katsnelson, M. I.; Grigorieva, I. V.; Dubonos, S. V.; Firsov, A. A. Nature 2005, 438, 197. (3) Zhang, Y. B.; Tan, Y. W.; Stormer, H. L.; Kim, P. Nature 2005, 438, 201. (4) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; Geim, A. K. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, 10451. (5) Novoselov, K. S.; Jiang, Z.; Zhang, Y.; Morozov, S. V.; Stormer, H. L.; Zeitler, U.; Maan, J. C.; Boebinger, G. S.; Kim, P.; Geim, A. K. Science 2007, 315, 1379. (6) Zhang, Y. B.; Small, J. P.; Pontius, W. V.; Kim, P. Appl. Phys. Lett. 2005, 86, 073104. (7) Berger, C.; Song, Z. M.; Li, T. B.; Li, X. B.; Ogbazghi, A. Y.; Feng, R.; Dai, Z. T.; Marchenkov, A. N.; Conrad, E. H.; First, P. N.; de Heer, W. A. J. Phys. Chem. B 2004, 108, 19912. (8) Peres, N. M. R.; Guinea, F.; Neto, A. H. C. Phys. ReV. B 2006, 73, 125411. (9) Kane, C. L.; Mele, E. J. Phys. ReV. Lett. 2005, 95, 226801. (10) Pereira, V. M.; Guinea, F.; dos Santos, J.; Peres, N. M. R.; Neto, A. H. C. Phys. ReV. Lett. 2006, 96, 036801. (11) Kosynkin, D. V.; Higginbotham, A. L.; Sinitskii, A.; Lomeda, J. R.; Dimiev, A. B.; Price, K.; Tour, J. M. Nature 2009, 458, 872. (12) Jiao, L.; Zhang, L.; Wang, X.; Diankov, G.; Dai, H. Nature 2009, 458, 877. (13) Son, Y. W.; Cohen, M. L.; Louie, S. G. Nature 2006, 444, 347. (14) Wakabayashi, K. Phys. ReV. B 2001, 64, 125428. (15) Barone, V.; Hod, O.; Scuseria, G. E. Nano Lett. 2006, 6, 2748. (16) Areshkin, D. A.; Gunlycke, D.; White, C. T. Nano Lett. 2007, 7, 204. (17) Han, M. Y.; Ozyilmaz, B.; Zhang, Y. B.; Kim, P. Phys. ReV. Lett. 2007, 98, 206805.
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