Geometric and Electronic Structure of New Carbon-Network Materials: Nanotube Array on Graphite Sheet

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1 Journal of the Physical Society of Japan Vol. 71, No. 11, November, 2002, pp #2002 The Physical Society of Japan Geometric and Electronic Structure of New Carbon-Network Materials: Nanotube Array on Graphite Sheet Takanori MATSUMOTO and Susumu SAITO Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo (Received July 1, 2002) We design a new class of carbon-network materials with a periodically modified graphite sheet. The modified part corresponds to (6,6) carbon-nanotube geometry. Their tube parts form triangular lattice on graphite sheet. On these systems each tube has six heptagons at the bottom, giving rise to a seamless sp 2 - C network with a negative curvature. We consider these nanotube arrays on graphite sheet with three kinds of tube-end geometries and various sizes for both graphite and tube parts. We report their electronic structures obtained by usinga realistic tight-bindingmodel, and for selected systems the density-functional theory. Interestingly, results show that most of them are semiconductors although both (6,6) tube and graphite are metallic. The difference in their tube-end geometries and the sizes of graphite and tube parts affect their electronic structures. Some have nearly flat band states around the Fermi level, showing a possibility of ferromagnetic behavior if hole or electron is doped. Some are direct-gap semiconductors whose interband transition is optically allowed. Their typical gap energies are about 1 ev. Therefore they should emit infrared light. KEYWORDS: graphite, nanotube, tight binding model, band structure, direct-gap semiconductor, flat-band ferromagnetism DOI: /JPSJ Introduction After the discovery and the macroscopic production of cage-structure C 60 cluster called fullerene, 1,2) attractive properties of this new form of carbon have been theoretically and experimentally studied intensively. Furthermore larger fullerenes, 3,4) multi-shelled fullerenes, 5) and graphitic nanotubes 6) have been synthesized. These materials consist of only sp 2 carbon atoms havingthree-fold coordination like a graphite sheet. A wide variety of materials derived from fullerenes and nanotubes, such as C 60 compound superconductors, 7) polymerized C 60 fullerites, 8 11) and carbonnanotube junctions, 12,13) have been also synthesized and studied. Their fascinatingphysical and chemical properties depend strongly on their network topologies. In the case of carbon nanotubes, for example, it is well known that the electronic transport properties of each nanotube depend sensitively on its diameter and chirality ) Fullerenes and nanotubes have zero-dimensional (0D) closed network and one-dimensional (1D) tubule network, respectively. In their solid phase, atoms constitute fullerenes and nanotubes, and these low-dimensional structures are constituent units to form three-dimensional (3D) materials. In this respect, carbon-nanostructure materials have geometrical hierarchy. For example, solid C 60 possesses the facecentered-cubic (fcc) lattice of atom-like C 60 clusters (0D! 3D). Two-dimensional triangular lattice of single-walled nanotubes (SWNTs) forms nanorope bundles (1D! 3D). Graphite can be regarded as the stacked material of planer graphite sheets (graphenes) having two-dimensional network (2D! 3D). Their cohesive mechanisms are considered mainly as weak van der Waals force. Although they all consist of sp 2 -C atoms, their different dimensionalities give rise to a variety of interestingphysical properties. In addition to these sp 2 -C materials, C 60 polymers should be considered as a new class of crystalline carbon having sp 2 sp 3 hybrid network. 17) They have covalent bonds between adjacent clusters. A formation of intercluster bonds gives rise to a formation of four membered rings. Furthermore, nanotubes encapsulatingvarious fullerenes (0D + 1D) have been synthesized ) In these materials, fullerenes are aligned in a chain. The graphite intercalation compounds in which fullerenes are arranged in the triangular lattice between graphene sheets (0D + 2D) have been designed and studied. 21) There can be other new forms of carbon with hybriddimensionalities to be explored in the future. Recently, by usingthe scanningtunnelingmicroscope (STM) techniques, the formation of tube structures from graphite was demonstrated. 22) In this experiment, graphite layers cover both the STM tip and the sample surface. This tip is pulled up after it contacts the graphite layers over the sample surface. Transmission electron microscopy (TEM) shows that the nanotube structure is found to grow between tip and graphite. Although a similar structure was previously proposed to model the growth process of nanotubes, 23) the TEM observation clearly indicates the possibility to produce the 1D + 2D hybrid sp 2 network materials. In the present work, we design and study similar 1D + 2D hybrid new carbon-network materials, i.e. the nanotube arrays on graphene, in which the short (6,6) nanotube lattices are periodically extracted from graphite sheet by introducing heptagons [Fig. 1(a)]. For this system, we consider various tube-end geometries of the extracted nanotubes shown in Figs. 1(b) 1(d). In the structure shown in Fig. 1(b), the extracted (6,6) nanotubes have open ends (OE). So, we call (b) the OE structure. Similar structure (c) with closed-end nanotubes is obtained by addingcaps to OE structure. The correspondingterminatingcap for (6,6) nanotubes is the hemisphere of C 84. The third structure (d) has double graphene sheets connected by (6,6) nanotubes. This structure is made from two OE structure layers. Therefore we call (c) and (d) CE (closed-end) structure and DS (doublesheet) structure, respectively. We optimized the geometry of 2765

2 2766 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 T. MATSUMOTO and S. SAITO Fig. 2. (a) Top view of the OE-C 162. A large hexagon denotes the Wigner Seitz cell of OE(5,1)-C 162. The longarrows a, b are primitive vectors of this material. The short arrows a 0, b 0 are primitive vectors of graphene (for C 2 unit cell). The top and side views of (b) OE(5,1)-C 162, (c) CE(5,1)-C 186, and (d) DS(3,0)-C 96 are also shown. Fig. 1. (a) Schematic picture of nanotube arrays on graphene. Shaded regions denote heptagons at the bottom of nanotubes. There are three kinds of materials studied in this work, (b) OE-structure, (c) CE-structure and (d) DS-structure (see text). these new carbon-network materials by total-energy minimization usingthe tight-binding(tb) model proposed by Xu et al. 25) Next their electronic structures are studied by using the Hamada Sawada TB model, 26) which has proven to show a good agreement with the result of the local-density approximation (LDA) for the electronic band structures of various carbon networks. It is known that graphene is a zerogap material, and that an infinite (6,6) carbon nanotube is metallic. The present study reveals that above three kinds of materials made from graphene and (6,6) nanotube are interestingly all semiconductors with a moderate energy gap, being different from both graphene and the (6,6) nanotube. The present paper is organized as follows: in 2, we explain the detailed geometry of the systems studied. In 3, the computational methods used in the electronic-structure calculation are described. In 4, results and discussion are given. Summary is given in Geometrical Structures We first consider the OE geometry with 162 C atoms per unit cell, OE-C 162, havingthe shortest tube part. The top view of the optimized geometry of this structure obtained is shown in Fig. 2(a). As an initial geometry before the optimization, we assume the two-dimensional ptriangular array of tube parts on a graphite sheet with a 5 ffiffi 3 times longer lattice constant than that of an original unit cell of the graphite sheet (2.456 A). The axis of each tube, perpendicular to a basal graphite sheet, corresponds to the six-fold symmetry axis of the system. The point group of OE-C 162 is C 6v. In Fig. 2(b), the Wigner Seitz cell of OE-C 162 is shown. There are six heptagons in the curved root area of each nanotube. The tube part above heptagons corresponds to the finite (6,6) nanotube with 36 C atoms. The remaining distorted graphitic part has 126 C atoms. Next, we study several CE and DS geometries. Among them, the Wigner Seitz cells of CE-C 186 and DS-C 96 are shown in Figs. 2(c) and 2(d), respectively. The closed-end nanotubes of CE-C 186 consist of OE-C 162 and the cap of C 24. The fullerene D 6h C 84 have one row of six hexagons along the equator normal to a six-fold axis. 24) The cap C 24 part with six pentagons corresponds to the portion of the D 6h C 84 above these six hexagons. The DS-C 96 studied is made by connectingtwo portions of OE-C 54 with removing12 C atoms at the middle. The stable geometries of these new sp 2 -C network materials are determined by total-energy minimization using the TB model proposed by Xu et al. 25) In this TB model total energy of a covalent system is given by the sum of electronic eigenvalues obtained by solving a TB Hamiltonian and a short-range repulsive energy given by the pairwise potential functions. This model can reproduce the energy volume curves of the local-density approximation in the density-

3 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 T. MATSUMOTO and S. SAITO 2767 Fig. 3. Total energy per atom of OE(5,1)-C 162 (solid line) and graphene (dashed line) as a function of the lattice constant a. functional theory 27,28) for various carbon materials. These results show that all the designed materials are stable and that their cohesive energies are close to that of graphene. In Fig. 3, the calculated values of the total energies of OE-C 162 are shown as a function of the lattice constant a around a 0 ¼ 21:27 A, which is derived from the lattice constant of graphite (2.456 A) [Fig. 2(a)]. Energy is measured from the total energy of graphene optimized with the same procedure. As can be seen from Fig. 3, the cohesive energy per an atom of OE-C 162 is very close to that of graphene. The optimized lattice constant of OE-C 162 is A (¼ 0:988 a 0 ) while that of graphene are A (¼ 0:995 a 0 ). The height of optimized OE-C 162 is 4.46 A. The closest distance between the adjacent tube surfaces is A. This distance is much larger than the distance between layers in graphite, 3.35 A. Hence, the direct intertube interaction should be sufficiently small. The bond lengths between C atoms having dangling bonds at the end of each tube are about 1.27 A. Except these bonds, the optimized C C bond lengths are found to be between 1.40 A and 1.45 A. As is schematically shown in Fig. 4(a), we study each kind of materials with various sizes for both graphite part and the tube part. We examine how the difference of size influences the electronic band structures. On graphite sheet, we first assume the center-to-center distance between adjacent tubes to be n 4:254 A (n ¼ 3; 4; 5...) [Fig. 4(b)]. Here, the nanotube part is defined as the portion above the heptagons. Therefore the length of the finite (6,6) nanotube is to be m 1:228 A (m ¼ 0; 1; 2...). In the cases of DS structures, the distance between two graphenes is to be 4:95 þ 2:456 m A (m ¼ 0; 1; 2...). These values are slightly modified after the total-energy optimization. There are in principle an infinite number of different structures for these graphite-tube hybrid materials. All the structures we study have C 6v symmetry. We use the index (n; m) in order to represent the size of the graphite part and the tube part. In the following, OE-C 162 is relabeled as OE(5,1)-C 162 hereafter for convenience. For example, the lattice constant, the length of tube part, and the distance between two graphenes in DS(3,0)-C 96 [Fig. 2(d)] are A, 0 A, and 4.95 A, Fig. 4. (a) Schematic pictures showing the variation of the lattice constant and the tube length. Dotted lines denote tube axes. n and m values are the indexes of the in-plane lattice constant and the length of the tube (see text). (b) Comparison of the array of nanotubes to the basal graphene. a is one of the primitive vectors. Dots and dashed circles of index n denotes the axes and walls of (6,6) nanotube, respectively. respectively. We study the electronic band structures of OE- (5,1), CE-(n; m) (n ¼ 3; 4; 5 m ¼ 0;...5), and DS-(n; m) (n ¼ 3; 4; 5 m ¼ 0; 1; 2). 3. Computational Method As mentioned above, the electronic-structure calculation for each optimized structure is performed by usinganother TB model proposed by Hamada and Sawada. 26) This model includes not only the transfer integrals but also the overlap integrals of the 2s and 2p orbital of the carbon atom. These integrals have adequate interatomic-distance dependences. The parameter values have been chosen so that the model can reproduce well the electronic structure of graphite and C 60 given by the local-density approximation (LDA) in the density-functional theory. 27,28) For comparison, the band dispersion of graphite and (12,0) nanotube obtained by using the TB model and LDA are shown together in Fig. 5. The TB model reproduces the valence bands from 20 ev to 0 ev and the conduction bands up to 4 ev. Therefore this model describes well the bonding states, bonding states, and antibonding states. In addition, the present procedure has been applied successfully to other fullerenes, C 76 and C 84, 29) and to carbon nanotubes. 14) We also use LDA for some selected systems. The Troullier-Martins norm-conservingpsuedopotentials 30) is used to describe the electron-ion interaction, and the Kleinman Bylander 31) separable approximation is also used for psuedopotentials. We adopt the parameterized Ceperley Alder potential 32,33) for the exchange correlation potential. The wave functions are expanded in terms of the plane-wave basis set with a cutoff energy of 30 Ry.

4 2768 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 T. MATSUMOTO and S. SAITO Fig. 5. Band structure of (a) graphite (b) (12,0) nanotube obtained by usingthe present TB model and LDA calculation. 4. Results and Discussion 4.1 Open-endstructures The electronic band structure of OE(5,1)-C 162 is shown in Fig. 6(a). Energy is measured from the 2p orbital level of an isolated carbon atom. Although this structure consists of zero-gap graphene and metallic (6,6) nanotubes, interestingly it is found to be a semiconductor. Both the top of the highest occupied (HO) band and the bottom of the lowest unoccupied (LU) band are located at the point. The OE(5,1)-C 162 is a direct-gap semiconductor and the gap value is 0.27 ev. As can be seem in Fig. 6(b) where several bands around the Fermi level are shown with a larger scale, both nearly flat (labeled ) and dispersive (labeled ) bands are found around the Fermi level. The HO band is the flat band havingvery small dispersion (about 0.05 ev). The LU band and the second highest occupied band are the dispersive bands havingthe substantial dispersion (about 0.26 ev). These dispersive bands show two-fold degeneracy at the point. Both the valence-band top and the conduction-band bottom are located at the point. The electronic band structure of graphene with the same lattice constant and primitive transition vectors as OE(5,1)- C 162 is also shown in Fig. 6(c). In this case, the correspondingwigner Seitz cell of the original graphene part should be C 150. It is well known that in the case of graphene the valence and conduction bands cross at the K point in hexagonal Brillouin zone (BZ) for the ordinary C 2 unit cell. Since the difference of the primitive translation vectors, at the zone foldingfor C 150 unit cell the K point of the C 2 original Brillouin zone is mapped to the point of the Fig. 6. (a) Band structure of OE(5,1)-C 162. Wigner Seitz cell is also shown. (b) Bands around Fermi level are shown. and denote the nearly flat band state and the dispersive band state, respectively. (c) Band structure of graphene for the C 150 unit-cell calculation. C 150 geometry is also shown. (d) Density of states (DOS) of OE(5,1)-C 162 (solid line) and graphene (dashed line). denotes the DOS peak associated with the flatband state. new C 150 Brillouin zone, and vise versa. The valence and conduction bands of graphene with the C 150 unit cell cross not at the K point but at the point. ComparingFig. 6(a) to Fig. 6(c), it is found that the energy band of OE(5,1)-C 162 has some features common to that of graphene. The calculated valence-band width of OE(5,1)-C 162 is ev (from 19:38 ev to 0.61 ev) while that of graphene is ev (from 19:47 ev to 0.46 ev). Their valence-band widths are almost the same. The behavior of some bands of OE(5,1)-C 162 resembles that of the correspondingbands of the graphene with the C 150 unit cell. The electronic structure around the Fermi level in OE(5,1)-C 162 is clearly different from that of graphene. The appearance of the nearly flat bands in OE(5,1)-C 162 is one of the important differences of the present hybrid-dimensionality systems from both graphene and nanotube. In Fig. 6(d), the calculated density of states (DOS) of OE(5,1)-C 162 (solid line) and graphene (dotted line) are shown. The DOS is calculated by using574 k-points in the whole Brillouin zone and broadened by Gaussian distribution functions with 0.04 ev width. There are sharp peaks around Fermi level in the DOS of OE(5,1)-C 162 in contrast to a linear behavior in the DOS of graphene. The downward arrow in Fig. 6(d) indicates the sharp peak generated from the flat band in Fig. 6(b). The existence of flat bands just below the Fermi level implies the possibility of ferromagnetic behavior if properly doped with holes. We have investigated the spatial distribution of each state obtained from the eigenvectors in the TB model. From this

5 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 T. MATSUMOTO and S. SAITO 2769 analysis it has been confirmed that the eigenstate of dispersive bands at the point is primarily distributed over the graphite part. It may be expected that not only the bands but also other dispersive bands are the graphene origin bands. On the other hand, the eigenstate of flat band at the point is found to be mainly localized within the nanotube parts and heptagons, being consistent with a very small dispersion observed. 4.2 Closed-end structures Figure 7(a) shows the electronic band structure of CE(5,1)-C 186. It is also a direct-gap semiconductor having both valence-band maximum and conduction-band minimum values at the point. This energy band is different from that of OE(5,1)-C 162 in some respects. Although nearly flat bands also appear, they are away from the Fermi level in this case. The HO and LU are both dispersive bands and degenerate two-fold at the point. The gap value is 0.51 ev. The valance-band width is ev (from 19:57 ev to 0.25 ev). It is smaller than that of OE(5,1)-C 162 by 0.17 ev. ComparingCE(5,1)-C 186 to OE(5,1)-C 162, we can understand the effect of the C 24 cap part in the CE(5,1)-C 186 added to the OE(5,1)-C 162. Most importantly, the gap value of CE(5,1)-C 186 is larger by 0.24 ev than that of OE(5,1)-C 162, which gives rise to the above smaller valence-band width. In addition to the tube-end geometries, both the lattice constant and tube length are found to affect considerably their electronic energy bands near the Fermi level. Figure 7(b) shows the band structure of CE(5,3)-C 210 in which the length of the tube part is longer by A than that of CE(5,1)-C 186. The result shows that the flat band appears near the Fermi level. The flat and dispersive bands of CE(5,3)-C 210 almost degenerate at the valence-band top (0.241 ev). In the conduction-band region the eigenvalues of the flat and dispersive bands at the point are ev and ev, respectively. The HO band of CE(5,3)-C 210 have nearly flat dispersion whereas the LU band shows substantial dispersion. These results show that the length of the tube part affects the relative positions of and and bands. The dispersive bands being expected to be of graphene origin always appear around the Fermi level. The appearance of the flat band, however, is sensitive to their geometries. In CE structures, some are direct-gap semiconductors and some have possibility of flat-band ferromagnetism. Likewise, not only the change of the tube-part length but also the change of the graphite-part width is found to modify the electronic band structure in both OE and CE systems. Many direct-gap semiconductor CE structures have HO and LU bands which are doubly degenerate at the point. Their interband transition is found to be optically allowed according to the group-theory analysis. The gap value also depends on the index (n; m). The optical transition may occur at about 1.0 ev (correspondingto the wave length of about 1300 nm, i.e. infrared region) or at smaller transition energies. 4.3 Double-sheet structures Figure 8(a) shows the band structure of DS(3,0)-C 96. Both HO band and LU band possess two-fold degeneracy at the point. DS(3,0)-C 96 is also a direct-gap semiconductor whose interband transition is optically allowed. The gap value is 1.2 ev. DS(3,0)-C 96 is expected to emit infrared light of ¼ 1100 nm. The valence-band width is ev (from 19:60 ev to 0:01 ev). This value is considerably (by 0.34 ev) smaller than that of graphene since the top of valence band is shifted downward. In Fig. 8(b), DOS of DS(3,0)-C 96 (solid line) and graphene (dashed line) are shown. In a lower-energy range, DOS of DS(3,0)-C 96 is more different from that of graphene than that of OE(5,1)- C 162. DS(3,0)-C 96 have two pairs of dispersive bands degenerating at the point like bands of OE(5,1)-C 162 and CE(5,1)-C 186 since DS(3,0)-C 96 have two graphitic parts. The electronic energy bands of DS systems also depend on the (n; m) index. Some DS structures are directgap semiconductors which have dispersive bands degenerat- Fig. 7. C 210. Electronic band structures of (a) CE(5,1)-C 186 and (b) OE(5,3)- Fig. 8. (a) Electronic band structure and (b) density of states of DS(3,0)- C 96 usingtb model. (c) Its electronic band structure obtained in LDA is also shown.

6 2770 J. Phys. Soc. Jpn., Vol. 71, No. 11, November, 2002 T. MATSUMOTO and S. SAITO ingtwo-fold at the point near the Fermi level. In other DS structures, the flat band state sometimes appears just below the Fermi level. Hence, DS structures also have the possibilities of flat-band ferromagnetism. The band structure of the DS(3,0)-C 96 is studied also usingthe LDA [Fig. 8(c)]. ComparingFigs. 8(a) and 8(c), the result of TB model is found to be consistent with that of LDA. Two energy bands aligned at their valence-band top are very similar to each other. On the other hand, there are some quantitative differences between TB and LDA results. The valence-band width in the LDA is by 0.35 ev smaller than the TB value. Also the LDA gap value is 0.7 ev, which is by 0.5 ev smaller than that of TB result. It is well known that the LDA underestimates the fundamental-gap value of semiconductors in general. Therefore, it should be further studied whether the TB or the LDA gives a more reliable estimate of the energy-gap value in this system. 5. Summary We have designed a new class of all-carbon materials with hybriddimensionalities. They possess modified graphene from which the finite (6,6) nanotube portions are periodically extracted. As for the tube-end geometries, we consider three kinds, open-end, closed-end, and double-sheet structures. For each kind we further consider two variations, i.e. the variation of the graphite-part width and that of the tubepart length. It is confirmed from the total-energy calculation usingthe TB model that these hybrid dimensionality materials are energetically stable. We have studied their electronic structures usingthe TB model for carbon that are known to reproduce the results of the density-functional theory fairly well. In a selected case, we have also used the LDA. The results show that, although they are consisting of zero-gap graphene and metallic (6,6) nanotubes, they are novel semiconductor materials. The electronic structures obtained show that there are dispersive and flat bands around the Fermi level. The dispersive bands are expected to originate from the graphite part, while the flat-band states are distributed mainly within the tube parts. Profiles of the band structures are found to be rather sensitive to the tubeend geometries, the lattice constant, and the length of the tube part. Some structures havingflat bands around the Fermi level are expected to show ferromagnetism, if holes or electrons are properly doped. Some are direct-gap semiconductors whose transition is optically allowed. These direct-gap semiconductors may emit infrared light. It has been already shown that nanotube structure can be extracted from graphite sheets using STM. 22) Since the materials designed and studied here have interesting physically properties, they should be worth studyingfurther both theoretically and experimentally. Acknowledgement We would like to thank Professor A. Oshiyama, Dr. M. Saito, and Dr. O. Sugino for providing us with the program used in LDA calculations. We also would like to thank professor H. Hirayama, Dr. T. Miyake, Mr. K. Kanamitsu, and Mr. Y. Akai for useful discussion. Numerical calculations are performed on Fujitsu VPP and NEC SX5/R computers at the Research Center for Computational Science, Okazaki National Research Institute. 1) H. W. Kroto, J. R. Heath, S. C. O Briten, R. F. Curl and R. E. Smalley: Nature 318 (1985) ) W. Krätschmer, L. D. Lamb, K. Fostiropoulous and D. P. Hoffman: Nature 347 (1990) ) F. Diederich, R. Ettl, Y. Rubin, R. L. Whetten, R. Beck, M. Alvarez, S. Arz, D. Shensharma, F. Wudl, K. L. Khemani and A. Koch: Science 252 (1991) ) K. Kikuchi, N. Nakahara, T. Wakabayashi, S. Suzuki, H. Shiromaru, M. Miyake, K. Saito, I. Ikemoto, M. Kainosho and Y. Achiba: Nature 357 (1991) ) D. Ugarte: Nature 359 (1992) ) S. Iijima: Nature 354 (1991) 56. 7) A. F. Hebard, M. J. Rosseinsky, R. C. Haddon, D. W. Murphy, S. H. Glarum, T. T. M. Palstra, A. P. Ramirez and A. R. Kortan: Nature 350 (1991) ) A. M. Rao, P. Zhou, K. Wang, G. T. Hager, J. M. Holden, Y. Wang, W. Lee, X. Bi, P. C. Eklund, D. S. Cornett, M. A. Duncan and I. J. Amster: Science 259 (1993) ) Y. Iwasa, T. Arima, R. M. Fleming, T. Siegrist, O. Zhou, R. C. Haddon, L. J. Rothberg, K. B. Lyons, H. L. Cartere Jr., A. F. Hebatd, R. Tycko, G. Dabbagh, J. J. Krajewski, G. A. Thomas and T. Yagi: Science 264 (1995) ) M. Núñez-Regueiro, L. Marques, J.-L. Hodeau, O. Béhoux and M. Perroux: Phys. Rev. Lett. 74 (1995) ) G. Oszlanyi and L. Forro: Solid State Commun. 93 (1995) ) S. Iijima, T. Ichihara and Y. Ando: Nature 356 (1992) ) B. I. Dunlap: Phys. Rev. B 49 (1994) ) N. Hamada, S. Sawada and A. Oshiyama: Phys. Rev. Lett. 68 (1992) ) R. Saito, M. Fujita, G. Dresselhaus and M. S. Dresselhaus: Appl. Phys. Lett. 60 (1992) ) K. Tanaka, K. Okahara, M. Okada and T. Yamabe: Chem. Phys. Lett. 191 (1992) ) S. Okada and S. Saito: Phys. Rev. B 59 (1999) ) B. W. Smith, M. Monthioux and D. E. Luzzi: Nature 396 (1998) ) B. Burteaux, A. Claye, B. W. Smith, M. Monthioux, D. E. Luzzi and J. E. Fischer: Chem. Phys. Lett. 310 (1999) ) H. Kataura, Y. Kumazawa, Y. Maniwa, I. Umezu, S. Suzuki, Y. Ohtsuka and Y. Achiba: Synth. Met. 103 (1999) ) S. Saito and A. Ohiyama: Phys. Rev. B 49 (1994) ) J. Yamashita, H. Hirayama, Y. Ohshima and K. Takayanagi: Appl. Phys. Lett. 74 (1999) ) S. Sawada: Abstr. Int. Symp. Nanonetwork Materials: Fullerenes, Nanotubes, and Rerated Systems (Kamakura, January 15 18, 2001) p ) P. W. Fowler and J. Woolrich: Chem. Phys. Lett. 127 (1991) ) C. H. Xu, C. Z. Wang, C. T. Chan and K. M. Ho: J. Phys.: Condens. Matter 4 (1992) ) N. Hamada and S. Sawada: unpublished; S. Okada and S. Saito: J. Phys. Soc. Jpn. 64 (1995) ) W. Kohn and L. J. Sham: Phys. Rev. 140 (1965) A ) P. Hohengerg and W. Kohn: Phys. Rev. 136 (1964) B ) S. Saito, S. Sawada and N. Hamada: Phys. Rev. B 45 (1992) ) N. Troullier and J. L. Martins: Phys. Rev. B 43 (1990) ) L. Kleinman and D. M. Bylander: Phys. Rev. Lett. 48 (1992) ) D. M. Ceperley and B. J. Alder: Phys. Rev. Lett. 45 (1980) ) J. P. Perdew and A. Zunger: Phys. Rev. B 23 (1981) 5048.