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Electronic Band Structure of Isolated and Bundled Carbon Nanotubes PHYSICAL REVIEW B, VOLUME 65, 155411 Electronic band structure of isolated and bundled carbon nanotubes S. Reich and C. Thomsen Institut fu¨r Festko¨rperphysik, Technische Universita¨t Berlin, Hardenbergstr. 36, 10623 Berlin, Germany P. Ordejo´n Institut de Cie`ncia de Materials de Barcelona (CSIC), Campus de la U.A.B. E-08193 Bellaterra, Barcelona, Spain ͑Received 25 September 2001; published 29 March 2002͒ We study the electronic dispersion in chiral and achiral isolated nanotubes as well as in carbon nanotube bundles. The curvature of the nanotube wall is found not only to reduce the band gap of the tubes by hybridization, but also to alter the energies of the electronic states responsible for transitions in the visible energy range. Even for nanotubes with larger diameters ͑1–1.5 nm͒ a shift of the energy levels of Ϸ100 meV is obtained in our ab initio calculations. Bundling of the tubes to ropes results in a further decrease of the energy gap in semiconducting nanotubes; the bundle of ͑10,0͒ nanotubes is even found to be metallic. The intratube dispersion, which is on the order of 100 meV, is expected to significantly broaden the density of states and the optical absorption bands in bundled tubes. We compare our results to scanning tunneling microscopy and Raman experiments, and discuss the limits of the tight-binding model including only ␲ orbitals of graphene. DOI: 10.1103/PhysRevB.65.155411 PACS number͑s͒: 71.20.Tx, 73.21.Ϫb, 71.15.Mb I. INTRODUCTION wall strongly alters the band structure by mixing the ␲* and ␴* graphene states. In contrast, Mintmire and White con- The electronic structure of carbon nanotubes is character- cluded from all-electron calculations of armchair tubes that ized by a series of bands arising from the confinement the differences between first principles and tight-binding cal- around the nanotube’s circumference. The critical points in culations are negligible for small enough energies.20 The ⌫ ␥ ϭ isolated nanotubes, which are at the point, the Brillouin- tight-binding overlap integral 0 2.5 eV, which they ex- Ϸ ␲ zone boundary, and sometimes also at kz 2 /3, give rise to tracted from their calculations, agrees nicely with scanning the square-root-like singularities in the density of states typi- tunneling microscopy ͑STM͒ measurements, but is 15 % 1,2 cal for one-dimensional systems. These singularities were smaller than the value found in optical experiments.9,10 Com- 3–5 directly studied by scanning tunneling experiments. Re- bined density-functional-theory ͑DFT͒ and parametrized cently more subtle structures like the secondary gap at the 6 6 techniques were used to calculate the electronic band struc- Fermi level were also observed. Ouyang et al. compared ture of bundles of ͑10,10͒ tubes.8,21 These studies focused on ͑ ͒ the density of states of an 8,8 nanotube isolated on a sub- valence- and conduction-band crossing at the Fermi level strate with one residing on top of a tube bundle. They and the secondary gap in bundled armchair tubes. showed that, indeed, a secondary gap opens up in bundled Here we report on a detailed study of the band structure of armchair tubes as predicted by Delaney et al.7,8 The second single walled nanotubes and nanotube bundles. We calcu- experimental method used widely to study the electronic dis- persion encompasses optical experiments and Raman lated the electronic dispersion for achiral as well as chiral scattering.9–13 Naturally, optical methods probe the elec- nanotubes to investigate the effects of curvature and inter- tronic dispersion only indirectly via absorption from the va- tube interaction on the electronic properties. The curvature- ␴ ␲ lence to conduction states. When the experimental findings induced - hybridization is found to have the most pro- are compared to a theoretical band structure, usually a tight- nounced effects on zigzag nanotubes, where we find a strong binding approximation of the graphite ␲ orbitals is downshift of the conduction bands, whereas the electronic employed.1,14–16 The electronic energies obtained in this ap- band structure is less affected in armchair and chiral nano- proximation are the same as those of the ␲ states of graphite tubes. Bundling of the nanotubes to ropes further decreases with that tight-binding model, when the boundary conditions the separation of the conduction and valence bands around around the circumference are considered. This simple picture the Fermi level. The dispersion perpendicular to the nano- Ͻ is frequently expanded to small tubes ͑diameter d Ϸ10 Å) tube axis is of similar strength as in other ␲-bonded carbon and to bundles of tubes, although Blase et al.17 showed con- material ranging from 200 meV, as calculated for bundles of vincingly that rehybridization has a significant effect on the chiral tubes to 1 eV in armchair nanotube bundles. We com- electronic states. Also ignored in this approach are intramo- pare our findings to scanning tunneling and optical experi- lecular dispersions which are known to be quite large in solid ments. In particular, we discuss the limits of the tight-binding 18,19 C60 and graphite. On the other hand, first-principles stud- approximation, neglecting the curvature of the nanotube wall ies, which do not have these deficiencies, of the electronic and the coupling between the tubes in a nanorope. bands in ideal carbon nanotubes are extremely rare. Blase In Sec. II we describe the computational method used in et al.17 studied the rehybridization effects in small zigzag this work. The band structure of various isolated nanotubes is nanotubes. They showed that the curvature of the nanotube presented in Sec. III, and compared to zone-folding of 0163-1829/2002/65͑15͒/155411͑11͒/$20.0065 155411-1 ©2002 The American Physical Society S. REICH, C. THOMSEN, AND P. ORDEJO´ N PHYSICAL REVIEW B 65 155411 graphene and tight-binding calculations. In Sec. IV we dis- tubes might be rotated around their z axis to yield arrange- cuss the band structure of bundles of ͑10,0͒, ͑6,6͒, and ͑8,4͒ ments of particular high or low symmetry. In the calculations ͑ ͒ nanotubes along the kz direction and in the perpendicular of the 6,6 nanotube bundles we used the highest symmetry ͑ ͒ plane. We compare our results to two selected experiments— packing (D6h). The 8,4 tubes were placed according to D2 5 ͑ ͒ STM measurements by Odom et al. and Raman scattering instead of D56 in the isolated nanotube. Finally, the 10,0 by Jorio et al.22—in Sec. V. Section VI summarizes our work nanotube bundle was maximally disordered with respect to and contains our conclusions. rotation (C2h , i.e., no vertical planes or horizontal rotation axes͒. In the band structure calculations we used 10 to 30 k II. COMPUTATIONAL METHOD points along the z direction for the chiral ͑8,4͒ and ͑10,5͒ DFT calculations were performed with the SIESTA ab ini- nanotubes, 45 for the zigzag ͑10,0͒ and ͑19,0͒ nanotubes, and tio package.23 We used the local-density approximation pa- 60 for the armchair ͑6,6͒ tube. In the perpendicular direction rametrized by Perdew and Zunger24 and nonlocal norm- we included a total of 30 points for the ⌫M, MK, and K⌫ conserving pseudopotentials.25 The valence electrons were directions. This sampling was sometimes not fully sufficient described by localized pseudoatomic orbitals with a double- to describe accurately the crossing of the bands in the interior ␨ singly polarized ͑DZP͒ basis set.26 Basis sets of this size of the Brillouin zone. We recalculated parts of the Brillouin have been shown to yield structures and total energies in zone with a finer mesh, and found no level anticrossing when good agreement with those of standard plane-wave not explicitly stated in the text.31 Only for one of the tubes calculations.27 The cutoff radii for the s and p orbitals were discussed in this work were other first-principles band struc- 5.1 and 6.25 a.u., respectively. These radii correspond to a tures available. For the ͑10,0͒ tube and graphite we found an 50-meV energy shift ͑i.e., an increase in the energy of the excellent agreement with pseudopotential plane-wave electron energy levels by localization in the free atom͒,26 calculations.19,32 We also compared the band structure of the which was chosen as the one that minimizes the total energy ͑6,6͒ armchair tube to the all-electron calculations of Ref. 20, for a graphene sheet, at the given DZP basis level. Increasing where a series of larger diameter armchair tubes was inves- the cutoff of the orbitals only changes the total energies of tigated. The general features are reproduced in our calcula- the nanotubes considered here by less than 0.1 eV/atom, and tions as well. has virtually no effect on the structure and energies of the electronic states. Real-space integration was performed on a III. ISOLATED NANOTUBES regular grid corresponding to a plane-wave cutoff around 250 Ry, for which the structural relaxations and the elec- In this section we first discuss the band structure of two tronic energies are fully converged. For the total energy cal- small achiral nanotubes, paying particular attention to the culations of isolated tubes we used between 1 ͑chiral tubes͒ effects of rehybridization. The calculated band structure of ͑ ͒ and 30 armchair k points along the kz direction. The metal- two chiral nanotubes with a diameter dϷ8 Å are presented lic bundles ͓͑6,6͒ and ͑10,0͒ tubes͔ were sampled by a 10 in a separate subsection. Finally, the electronic dispersion in ϫ10ϫ30 Monkhorst-Pack28 k grid; only the ⌫ point was the optical range of a ͑19,0͒ tube is investigated.
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