Defective Structure of BN Nanotubes: From Single Vacancies to

Figure 1 (a, b) Two HREM images of a bundle of boron nitride nanotubes taken after ... In the case of a zigzag (14,0) BN nanotube such conditions impl...
0 downloads 0 Views 527KB Size
Download PDF
NANO LETTERS

Defective Structure of BN Nanotubes: From Single Vacancies to Dislocation Lines

2006 Vol. 6, No. 9 1955-1960

A. Zobelli,*,†,‡ C. P. Ewels,†,§ A. Gloter,†,| G. Seifert,‡ O. Stephan,† S. Csillag,⊥ and C. Colliex† Laboratoire de Physique des Solides (UMR CNRS 8502), Baˆ t. 510 UniVersite´ Paris Sud, 91405 Orsay, France, Institu¨t fu¨r Physikalische Chemie und Elektrochemie, Technische UniVersita¨t Dresden, D-1062 Dresden, Germany, Institute des Materiaux, CNRS UMR6502, 2 rue de la Houssinie` re, 44322 Nantes, France, ICYS National Institute for Materials Science, Namiki 1-1, Tsukuba 305-0044, Japan, and Department of Physics, Stockholm UniVersity, Box 6730, S-113 85 Stockholm, Sweden Received May 12, 2006; Revised Manuscript Received June 19, 2006

ABSTRACT A combination of electron microscopy and theoretical calculations provides new insights into the structure, electronics, and energetics of point defects and vacancy lines in BN single-wall nanotubes (SWNT). We show that the point defects forming under electron irradiation in the BN SWNTs are primarily divacancies. Due to the partially ionic character of the BN bonding, divacancies behave like an associated Schottky pair, with a dissociation energy of around 8 eV. Clustering of multiple vacancies is energetically favorable and leads to extended defects which locally change the nanotube diameter and chirality. Nevertheless these defects do not alter significantly the band gap energy, and all of them have electronic structure similar to that of BN divacancies. We thus conclude that under irradiation BN SWNT may have a very stable alteration of its electronic and optical properties.

Introduction. Various studies have shown that atomic-scale intrinsic defects in carbon nanotubes can strongly influence the electronic and mechanical properties of such systems.1,2 Defects can also be seen as chemically active sites for tube side wall functionalization. Despite the clear importance of defects in nanotubes, the number of experimental reports is still limited, and notably no work has been performed to date linking the formation and transition from point to line defects in nanotubes. BN nanotubes represent an important new wide band gap nanomaterial, primarily due to their homogeneous electronic behavior: tubes of different chiralities are all semiconductors with almost the same theoretical optical band gap of 6.2 eV3,4 and experimentally determined at 5.8 eV.5 However, as for carbon nanotubes, boron nitride nanotubes are not defect free. Although there have been some theoretical studies of individual isolated native6-9 and mechanically induced defects,10,11 there is no experimental identification of point * Corresponding author. E-mail: [email protected]. † Laboratoire de Physique des Solides (UMR CNRS 8502), Ba ˆ t. 510 Universite´ Paris Sud. ‡ Institu ¨ t fu¨r Physikalische Chemie und Elektrochemie, Technische Universita¨t Dresden. § Institute des Materiaux, CNRS UMR6502. | ICYS National Institute for Materials Science. ⊥ Department of Physics, Stockholm University. 10.1021/nl061081l CCC: $33.50 Published on Web 07/28/2006

© 2006 American Chemical Society

or line defects in such nanotubes, or studies of their interaction and aggregation. Even now the important case of divacancies has not yet been treated. Transmission electron microscopy (TEM) is a suitable technique for imaging defective structures and also for inducing in situ the appearance of defects: the energetic electrons of the beam (energy of the order of magnitude 100 keV) are capable of removing single atoms from the lattice through direct knock-on collisions.12,13 Recent improvements in imaging techniques have demonstrated the possibility of direct visualization of single atomic defects in graphene layers14 and of time-monitoring their evolution under homogeneous electron irradiation. In this work we present the first TEM study of the evolution of boron nitride SWNTs under electron irradiation. Two types of defects for BN SWNTs are observed: isolated bright point defects and extended linear changes in tube diameter. The experimental results are compared to a theoretical study of the structure, energetics, and electronic properties for different types of defective structure (single vacancies, divacancies, vacancy lines). We will show that the most stable point defect in BN is the BN divacancy. This will also provide an explanation of the appearance of extended defective structures such as dislocation lines. A

Figure 1. (a, b) Two HREM images of a bundle of boron nitride nanotubes taken after irradiation times of t ) 0 s and t ) 20 s. Part b shows the appearance of single bright spots on the lower tube of the bundle associated with a small decrease in the tube diameter. As already shown in the case of carbon,1 such bright spots can be interpreted as the signature of point defects such as single vacancies. (c, e) Relaxed structures corresponding to single boron vacancy (c) and BN divacancy (e) for a (14,0) tube. (d, f) Corresponding HREM simulated images.

mechanism of formation of defect lines through preferential emission sites without any need from thermal migration is then proposed. We conclude with the effect of irradiation on the electronic structure of the tubes. Methodology. (A) High-Resolution Electron Microscopy. The microscopy observations were carried out in a FEI Tecnai F30 equipped with a field emission electron gun. The detection of single vacancies is a challenging task since the quality of high-resolution electron microscopy images (HREM) can be improved by increasing the beam intensity, the acceleration voltage, and the exposure time, but these parameters also affect the emission rate of atoms under irradiation. The experimental conditions have then to be set up in order to obtain a compromise between good images and a reasonable exposure time without destroying the tubes. The operating acceleration voltage has been set at 200 keV which combined with a spherical aberration coefficient of 1.2 mm gives a maximum contrast transfer function (CTF) at around 3.5 Å. In addition, the high coherency of the field emission gun allows contrast at higher frequency, and a hexagon-hexagon distance of 2.5 Å can also been distinguished with this equipment (see Figure 1a). At the chosen electron dose and voltage, knock-on processes are restricted to a rate which enables several images to be taken before defect creation alters the tube topology. We have then followed the evolution of the systems employing a multiexposure procedure, taking an image each second or half second. This allows us to correct the specimen drift and to reduce the image noise using an average image, obtained by aligning each frame on the basis of the previous one and summing the film each two or three frames. Furthermore, we have mostly imaged bundles of tubes instead of single isolated BN SWNTs, these latter being more sensitive to vibrations induced by the electron beam. (B) DFT-Based Calculations. We have studied several topologies of single-walled BN nanotubes. All the calculations were performed in periodic boundary conditions, with supercells 10 times the elementary nanotube cell along the axis tube direction. In the case of a zigzag (14,0) BN 1956

nanotube such conditions imply using a 560 atom unit cell for the perfect tube. This large supercell is required to avoid any coupling between defects in adjacent cells for the most extended defective structures investigated; dislocation edges in two adjacent cells are always separated by more than 22 Å. For a dislocation along the tube axis, the burger vector is orthogonal to the tube axis, meaning that the lattice contracts along the diameter of the tube and not along the tube axis. Due to the large size of the system, standard density functional techniques are too time demanding. Therefore structural optimizations were conducted in the framework of density functional theory in its tight binding approximation (DFTB)15,16 using the deMon 2004 code.17 This technique can be seen as an intermediate between ab initio density functional theory and classical empirical tight-binding methods. This approach, with the parameters developed for boron nitride,18,19 has been applied several times to solid-state BN and molecular systems, and it is known to give a realistic description of energetics and structure.20,21 After optimization, a more accurate value of the total energy was obtained by a single point calculation using density function theory in the local density approximation (DFT-LDA) as implemented in the AIMPRO code.22 Considering the large size of the unitary cell, just the Γ point has been used in the self-consistency cycles. Boron and nitrogen pseudopotentials are generated using the Hartwingster-Goedecker-Hutter scheme.23 Atomcentered Gaussians (five per atom) are used for the wave function basis, each multiplied by spherical harmonics up to a maximum angular momentum of l ) 1. For the Gaussian with the second smallest exponent the maximum is l ) 2. The formation energies of the defective systems are calculated as a function of the chemical potential of the boron and nitrogen species, the electronic chemical potential, and the total charge q of the system Ef(Nqi ) ) Et(Nqi ) - nBµB - nNµN + q(µe + e)

(1)

where Et is the total energy of the defective system, nB (nB) Nano Lett., Vol. 6, No. 9, 2006

and µN (µN) are respectively the total number of boron (nitrogen) atoms and the chemical potential. The electronic chemical potential µe is measured with respect to the top of the valence band (e). As reference for the nitrogen we have taken an N2 molecule and for the boron, a B12 cluster. The defective systems investigated are all neutrally charged. Emission of atoms occurs through direct collision between the relativistic electrons of the beam and the atoms of the tube. Formation energies of vacancies can be seen as a “static way”13 of estimating the knock-on energy threshold without taking into account the effects of relaxation of the system during the process of emission. For light elements and high electron energies, the displacement rate is a decreasing monotonic function of the displacement threshold energy.24 That means that atoms at sites with a lower vacancy formation energy have a higher probability of being ejected by the swift electron. It is possible to obtain a map of preferential sites for vacancy formation, looking at the formation energies of vacancies. Extended defect structures were determined by systematic removal of different atoms around a vacancy site. The atom with the lowest associated vacancy formation energy was then removed, and the process was repeated. Results and Discussion. Previous studies have suggested that single vacancies can exist in graphitic BN sheets,25 and similar defects have also been proposed in BN nanotubes.6 These calculations are confirmed by our DFT results. Single vacancy formation energies are a function of nanotube diameter. The formation energy for a neutral single boron vacancy increases from EVB ) 8.20 eV in a (8,0) nanotube (1.25 nm diameter) to 8.82 eV for a (14,0) tube (2.22 nm diameter), to 11.22 eV for planar h-BN. The equivalent figures for the nitrogen vacancy are EVN ) 7.02 eV ((8,0) tube), 7.24 eV ((14,0) tube), and 8.91 eV (h-BN). There is a qualitative structural difference between the monovacancy in a h-BN sheet and in a nanotube. In the planar structure the equilibrium geometry conserves 3-fold D3h symmetry around the missing atom for both boron and nitrogen single vacancies, exhibiting a small relaxation of the three neighboring atoms: 0.10 Å inward for the B atoms around the nitrogen vacancy (VN) and 0.09 Å outward for the N atoms around the boron vacancy (VB) with respect to their unrelaxed positions. In the case of a nanotube, the curvature induces sufficient lattice deformation to permit local bond reconstruction between two atoms neighboring the vacancy (see Figure 1c for the case of a boron vacancy). For VB in a (14,0) tube the reconstructed N-N bond length is as short as 1.47 Å. This behavior explains the diameter dependence of the vacancy formation energies. However a different picture emerges once we consider vacancy pairs. The interaction energy between two vacancies depends on their separation, and different vacancy combinations are possible: neighboring boron vacancies (VBB), neighboring nitrogen vacancies (VNN), or a next neighbor boron and nitrogen vacancy pair (VBN). Of these we found the most stable to be VBN. In a (8,0) tube separately VB and VN have a combined formation energy of around 17 eV, but on neighboring sites this drops to 9.2 eV; i.e., they have a Nano Lett., Vol. 6, No. 9, 2006

binding energy of around 8 eV. This shows that once a vacancy forms, the formation energy for a subsequent neighboring vacancy is close to zero; thus the probability of forming a second neighboring vacancy is higher than at any other site. This strong driving force suggests that vacancies in single-walled boron nitride tubes will tend to appear as boron-nitrogen pairs. That brings the appearance of homonuclear bonds, but it has already be shown that in BN systems this chemical frustration is more energetically favorable than the steric frustration introduced by lattice configurations that conserve the alternate boron-nitrogen.26 The effect is much stronger than, for example, the formation of VBB or VNN, although we also see strong binding in these cases. Two B vacancies in neighboring sites have a binding energy of over 6.43 eV. Interestingly the intermediate N atom moves to adopt the site of one of the missing B atoms, structurally equivalent to the stable divacancy in graphite. The N-N repulsion is overcome due to the energy saving associated with reducing the total dangling bonds by two. A similar binding energy of two N vacancies is seen. In parts a and b of Figure 1, we show two HREM images of a bundle of tubes taken after an irradiation time of t ) 0 s and t ) 20 s. Figure 1b shows clearly the appearance of single bright spots on the lower tube of the bundle, associated with a small decrease in the tube diameter. As already demonstrated in the case of carbon,14 such bright spots can be interpreted as the signature of point defects in the lattice of the tubes. HREM image simulations for the DFT optimized structure of a B single vacancy (VB, Figure 1c) and a BN divacancy (VBN, Figure 1.e) in a (14,0) BN nanotube can be seen in Figure 1d,f. Within the experimental conditions, it is not possible to decipher the local point group symmetry of the individual defects. In fact, VN, VB, and VBN all appear as a rounded bright spot for a large set of defocus values and geometries. The only noticeable difference is the higher contrast of a BN divacancy relative to monovacancies. Compared to carbon nanotubes or nanosheets,14 the contrast of single vacancies in BN nanotubes is reduced, due to the strong structural relaxation induced by the curvature and the heteronuclear nature of the materials. Both experimental and theoretical results agree that the observed individual defects are likely to be primarily BN divacancies. In parts a and b of Figure 2 we show two HREM images of a bundle of tubes taken after an irradiation time of t ) 0 s and t ) 20 s, the same conditions as the previous illustrated case. Starting from a perfect single-walled BN tube, after irradiation a kink appears in the side walls associated with a local diameter reduction necessarily associated with a change of chirality. The change of chirality can be explained by a line of neighboring vacancies running along the tube axis, with associated bond reconstruction (Figure 2c). The terminating edge dislocation cores introduce 5- and 7-fold member rings in the junction between the two parts of the tube (Figure 2e). We have investigated the energy of formation and possible routes for dislocation formation in a BN nanotube. There are potentially several ways to pass from a single vacancy to a dislocation line: (i) reorganization and agglomeration of vacancies due to thermal vacancy migra1957

Figure 2. (a, b) Two HREM images of a bundle of boron nitride nanotubes taken after an irradiation time of t ) 0 s and t ) 20 s. In part b the appearance of a kink on the wall of the tube is clearly shown. (c, d) A side view and a front view of the structure of a (14,0) with a dislocation line that reduce its diameter in the central part. In d it is possible to see the presence of a 5- and 7-fold member ring associated with the dislocation. (d, f) The corresponding HREM simulated images.

tion, (ii) electron beam induced migration, or (iii) preferential sites for vacancy formation. In graphite the energy for in-plane monovacancy migration was experimentally27 and theoretically28 estimated to be 1.61.8 eV. We have investigated through DFT-based calculations the migration path and activation barrier for both boron and nitrogen vacancies in a hexagonal boron nitride sheet.29 Vacancy migration to a first neighbor atomic site is not possible due to the heteronuclear nature of boron nitride; in that situation two homonuclear bonds would form and the structure obtained is not stable. The optimal migration path that avoids the formation of homonuclear bonds is migration to a second neighbor site. The activation barrier we calculate for this process is more than 3 eV for the boron vacancy and nearly 6 eV for the nitrogen vacancy. Migration of bivacancies is in the same range of energies. It is possible to give an upper estimation of the diffusion coefficient on the basis of the microscopy images. Under the hypothesis that the dislocation appears due to thermal vacancy migration, we can consider that during the exposure time of a single frame (around one second) vacancies would need to have migrated for a distance of the order of magnitude of several nanometers, giving an upper value of the diffusion coefficient of the order of 1 nm/s. Considering a simple Arrhenius formula for the diffusion coefficient, it is possible to obtain such a value with our calculated activation energies in the case of the migration of a boron vacancy for a temperature of around 1200 K (even higher for nitrogen). Since the experiments were conducted at room temperature, this is too low for thermal vacancy clustering within the characteristic time of our experiment. To conclude, thermal migration has to be excluded as an explanation of the appearance of dislocation lines. A similar argument excludes electron beam induced migration. The bright spots (representing point defects) which can be seen in both TEM images (Figure 1a,b) do not move over the 20 s between acquisitions; thus electron beam induced migration, if occurring at all, does not occur within the time scales required for vacancy line formation. The analysis of the formation energy of divacancies can be extended, by sequentially removing the least stable atom in the system. Without invoking vacancy migration, this leads 1958

Figure 3. The red line in (a) represents the most energetically favorable dislocation path. The graph shows formation energies of single vacancies on the path of the dislocation line. The formation energies are reduced drastically compared to those of isolated vacancies; indeed vacancy formation becomes exothermic for oddalternate atom removal after N2.

to the formation of a vacancy line. The line reconstructs strong BN bonds along its length, terminating in two edge dislocation cores. On removal of an odd number of atoms, the tube relaxes in the same way as for a monovacancy, reconstructing a homonuclear bond and moving the doubly coordinated atom outward. This is equivalent to the end dislocation core lying in the shuffle plane. Each time an even number of atoms is removed, at the ends of the dislocation two pentagonal rings appear, one with a NN bond and the other one with a BB bond; i.e., the terminating dislocation cores both lie in the glide plane. Checking different orientations for a dislocation line, we find that in a zigzag tube the energetically most favorable configuration is for atoms to be removed along a sinusoidal line rolling around the tube (red line of atoms in Figure 3a). In the middle section the diameter decreases with a change of chirality; in the case considered, we pass from a (8,0) tube to a (7,0). In the graph of Figure 3 we show the evolution of the formation energies of a vacancy line, starting by removing a nitrogen atom. The graph shows that after having expelled a first BN unit, the energies of formation of the next nitrogen vacancies are almost constant to a value of 4 eV, less than half that of an isolated nitrogen vacancy. Remarkably, for the boron vacancies we found an almost constant negative value equal to -0.2 eV. That suggests that once a nitrogen Nano Lett., Vol. 6, No. 9, 2006

Figure 4. Electronic structure of a h-BN sheet and of a BN zigzag tube in the presence of boron (VB) and nitrogen (VN) single vacancies, of a boron-nitrogen vacancy pair (VBN) and of a dislocation line formed by five bivacancies (5 VBN). The gray filled regions represent the valence and conduction bands of the perfect systems. Filled squares indicate electrons, and open squares indicate holes.

atom is removed along the dislocation line, the next boron atom is spontaneously ejected. The same effect is seen if the line begins with a B vacancy; i.e., the atoms are always ejected in pairs independent of how the line commences. HREM image simulations are shown in parts d and f of Figure 2 for structures with a high number of vacancies, to reproduce a long dislocation line. They reproduce well the kink structure we commonly observed under TEM irradiation (see Figure 2b). The strong decrease of the formation energy shown in our calculations strongly supports the idea of preferential sites for vacancy formation under electron irradiation. The appearance of a long dislocation line can be interpreted as a “laddering” mechanism that expands a single vacancy to an extended defective structure. We turn now to the electronic structure of these defective systems. Hexagonal BN has extremely flat bands throughout k-space, and thus, unlike carbon, the calculated band gap of BN nanotubes remains almost fixed at the planar hexagonal BN value of ∼4.7 eV, with a small reduction as the nanotube diameter reduces.3,4 The intrinsic defects studied here introduce additional states into the gap. Single vacancies introduce half-filled shallow acceptor states and thus may be expected to act as electron acceptors (tube or bulk, see Figure 4). The main difference in vacancy states between the tube and bulk is for the boron vacancy level at -1.2 eV: the doubly degenerate state of the plane loses its degeneracy in the tube owing to the breaking of symmetry related to the N-N bond reconstruction described above. This degeneracy splitting decreases with increasing tube radius, demonstrating again that the bond reconstruction depends on the local curvature. The electronic structure of the neutral divacancy is different (Figure 4) with no half-filled shallow state available Nano Lett., Vol. 6, No. 9, 2006

for doping. Nonetheless, the neutral divacancy creates one discrete empty level in the band gap (two for the h-BN), flat throughout k-space and independent of tube diameter. Since we believe the dominant intrinsic point defect species in our sample to be divacancies, this demonstrates that the possibility of doping BN SWNTs with vacancies via irradiation is unlikely. Nonetheless the optical properties and conductivity may be altered by the presence of divacancies, and these levels should be readily detectable via optical spectroscopy. The BN divacancy is actually an intimate vacancy pair example of a Schottky defect pair, i.e., a pair of oppositely charged defect centers. Although in general this pair of charged defects may be spatially distributed as long as their total net concentrations are the same, in practice there is strong electrostatic interaction between the two oppositely charged centers which will be maximized when they form in neighboring sites. This effect is augmented in mixed ionic-covalent materials such as BN where an intimate vacancy pair has fewer dangling bonds than its separated vacancy pair counterpart. Coherent interaction of vacancies may be weak in the case of the bulk material but may play an important role in irradiated nanostructures where a high defect density is obtained and there is a high local energy transfer during electron collision. We suggest that this is why we see intimate divacancy formation in our system, and indeed this may also be common in other ionic materials under irradiation damage. At this point, we can mention that electron paramagnetic resonance (EPR) experiments provide evidence that paramagnetic centers on γ-ray irradiated h-BN31 can be interpreted as the effect of single nitrogen vacancies. More recently similar paramagnetic defects have been reported on multiwalled boron nitride nanotubes at an extremely low concentration, certainly less than one per tube, and with no difference with the case of the bulk BN.30 Nevertheless BN divacancies are nonparamagnetic and might have been undetected in these previous experiments. The extended defects have a band structure very similar to that of the divacancy. As discussed above a dislocation line induces a change in the chirality of the tube, but since the electronic structure of BN nanotubes is not sensitive to changes of the chirality, that aspect just reduces slightly the band gap but it keeps practically unchanged the band structure. Much more significant is the fact that at the two ends of the dislocation line the presence of 5-fold member rings induces two homonuclear bonds, giving rise to this empty state in the gap. This explains the fundamental electronic similarity between a BN divacancy and a long dislocation line as shown in Figure 4 and suggests that the additional empty states are spatially located at the two ends of the dislocation. Thus although in principle we have an extremely complex system (multiple nanotubes with a range of diameters and chiralities, and a mixture of point and line defects), remarkably our experimental and theoretical study suggests that these essentially reduce to a single simple, uniform electronic system, namely, nanotubes with wide, flat bands separated by ∼4.4 eV, with one empty intrinsic defect level in the gap 1959

due to homopolar B-B and N-N bonds associated with divacancies and edge dislocation cores. This electronic simplicity makes BN SWNTs almost unique as a nanomaterial and suggests they may have an extremely bright future in fields such as photonics and optoelectronics. To conclude, our study reveals that electronic irradiation of BN nanotubes gives rise primarily to two defect species: neighboring BN divacancies and extended defect lines. The vacancy pairs effectively self-compensate, giving rise to one empty state in the gap. Vacancy formation energies are drastically reduced in the vicinity of other vacancies, with spontaneous atom emission predicted in some cases. The appearance of the extended defect structures can be explained by the creation of a series of neighboring single vacancies through a mechanism of “laddering” the tube lattice. This proposed mechanism for vacancy clustering has a particularly strong driving force and may be general to a large variety of materials, particularly with heteronuclear structure. For the same reason such materials may also exhibit a tendency to form divacancies under electron irradiation. Similar effects to those observed here in BN may then be expected for other heteronuclear fullerene-like materials such as WS2 nanotubes, MoS2 fullerenes, or BN nanohorns, where such defects may also control the reactivity properties. Acknowledgment. We thank H. Pascard for providing the BN tube samples. Thanks also to E. Coronel for the support given on the microscopic observations. A.Z. thanks the New Fullerene like materials network, Contract HPRNCT-2002-00209, for financial support. C.P.E. acknowledges financial support from a Marie Curie Individual Research Fellowship of the European Community under Contract Number MCFI-2002-01436. References (1) Biel, B.; Garcı`a-Vidal, F. J.; Rubio, A.; Flores, F. Phys. ReV. Lett. 2005, 95, 266801. (2) Sammalkorpi, M.; Krashenninikov, A.; Kuronen, A.; Nordlund, K.; Kaski, K. Phys. ReV. B 2004, 70, 245416.

1960

(3) Wirtz, L.; Marini, A.; Rubio, A. Phys. ReV. Lett. 2006, 96, 126104. (4) Park, C. H.; Spataru, C. D.; Louie, S. G. Phys. ReV. Lett. 2006, 96, 126105. (5) Arenal, R.; Stephan, O.; Kociak, M.; Taverna, D.; Loiseau, A.; Colliex, C. Phys. ReV. Lett. 2005, 95, 127601. (6) Schmidt, T.; Baierle, R.; Piquini, P.; Fazzio, A. Phys. ReV. B 2003, 67, 113407. (7) Piquini, P.; Baierle, R. J.; Schmidt, T.; Fazzio, A. Nanotechnology 2005, 16, 827. (8) Moon, W. H.; Hwang, H. Phys. Lett. A 2004, 320, 446. (9) Miyamoto, Y.; Rubio, A.; Berber, S.; Yoon, M.; Toma`nek, D. Phys. ReV. B 2004, 69, 121413. (10) Bettinger, H.; Dumitrica, T.; Scuseria, G.; Yakobson, B. Phys. ReV. B 2002, 65, 41406. (11) Dumitrica, T.; Bettinger, H.; Scuseria, G.; Yakobson, B. Phys. ReV. B 2003, 68, 85412. (12) Banhart, F. Rep. Prog. Phys. 1999, 62, 1181. (13) Banhart, F.; Li, J.; Krasheninnikov, A. Phys. ReV. B 2005, 71, 241408. (14) Hashimoto, A.; Suenaga, K.; Gloter, A.; Urita, K.; Iijima, S. Nature 2004, 430, 870. (15) Porezag, D.; Frauenheim, T.; Ko¨hler, T.; Seifert, G.; Kaschner, R. Phys. ReV. B 1995, 51, 12947. (16) Seifert, G.; Porezag, D.; Frauenheim, T. Int. J. Quantum Chem. 1996, 58, 185. (17) Ko¨ster, A.; et al. deMon 2004: NRC: Ottawa, Canada, 2004. (18) Fowler, P.; Heine, T.; Michell, D.; Schmidt, R.; Seifert, G. J. Chem. Soc., Faraday Trans. 1996, 92, 2197. (19) Widany, J.; et al. Phys. ReV. B 1996, 53, 4443. (20) Seifert, G.; Fowler, P. W.; Porezag, D.; Frauenheim, T. Chem. Phys. Lett. 1997, 268, 352. (21) Fowler, P.; Rogers, K.; Seifert, G.; Terrones, M.; Terrones, H. Chem. Phys. Lett. 1999, 299, 359. (22) Jones, R.; Briddon, P. Semicond. Semimet. 1998, 51A, 287. (23) Hartwingster, C.; Goedecker, S.; Hutter, J. Phys. ReV. B 1998, 58, 3641. (24) McKinsley, W.; Feshbach, H. Phys. ReV. 1948, 74, 1759. (25) Orellana, W.; Chacham, H. Phys. ReV. B 2001, 63, 125205. (26) Rogers, K. M.; Fowler, P. W.; Seifert, G. Chem. Phys. Lett. 2000, 332, 43-50. (27) Hinman, G. W.; Haubold, A.; Gardner, J. O.; Layton, J. K. Carbon 1970, 8, 341. Asari, E.; Kitajima, M.; Nakamura, K. G.; Kawabe, T. Phys. ReV. B 1993, 47, 11 143. (28) El-Barbary, A. A.; Telling, R.; Ewels, C.; Heggie, M.; Briddon, P. Phys. ReV. B 2003, 68, 144107. (29) Zobelli, A.; et al. To be published. (30) Panich, A. M.; Shames, A. I.; Froumin, N.; Tang, C. C.; Bando, Y. Phys. ReV. B 2005, 72, 85307. (31) Fanciulli, M. Philos. Mag. B 1997, 76, 363.

NL061081L

Nano Lett., Vol. 6, No. 9, 2006