NANO LETTERS
Electronic Structures of Very Thin Carbon Nanotubes: Are They Still π-Electronic Materials?
2002 Vol. 2, No. 6 629-633
A. Ito, Y. Natsume, S. Ohmori, and K. Tanaka* Department of Molecular Engineering and AdVanced Research Institute of Nanoscale Science and Engineering, Graduate School of Engineering, Kyoto UniVersity, Sakyo-ku, Kyoto 606-8501, Japan Received March 6, 2002; Revised Manuscript Received April 8, 2002
ABSTRACT Electronic structures of single-walled carbon nanotubes with small diameters are examined by means of the ab initio self-consistent-field crystal orbital method. In particular, we focus on the tubes (3, 3), (4, 0), (5, 0), and (6, 0) as the candidates for ca. 4 Å-diameter CNT recently discovered. The highest occupied π and the lowest unoccupied π* bands of these tubes maintain π-character, irrespective of their large curvature, and hence, they are still considered as π-electronic materials. It is predicted that the armchair tube (3, 3) and the zigzag tube (4, 0) are metallic and that tubes (5, 0) and (6, 0) are semiconductive with narrow band gaps. These electronic behaviors of the zigzag tubes are interpreted in terms of the orbital interactions associated with the bond-alternant structures, rather than the σ−π hybridization, due to the simple curvature effect.
Introduction. A decade has passed since the discovery of carbon nanotubes (CNTs),1 and a considerable number of studies have been hitherto made on the intriguing electronic properties originating from their novel quasi-one-dimensional cylindrical structure.2 In particular, the most interesting property is that the electrical conduction behavior of the single-walled CNT (SWCNT) varies from wide-gap semiconductive to metallic with its diameter and helicity or, in other words, chiral vector.3-7 Moreover, CNTs manifest socalled quantum effects such as quantized conductance,8 the Coulomb blockade,9 and Aharonov-Bohm effect.10 Considering that a decrease of its diameter would cause more conspicuous quantum effects, the preparation of a CNT with a diameter as small as possible is of great interest. Although the small limit for the diameter of CNT has theoretically been predicted ∼4 Å,11 the thinnest CNT prepared so far had a diameter of 7 Å.12,13 Recently, however, a CNT with a diameter of about 5 Å has been discovered in the innermost shell of the multiwalled CNT (MWCNT).14,15 Moreover, Wang et al. have reported that the uniformly sized 4 Å-diameter SWCNT can be prepared by pyrolysis of tripropylamine absorbed in the channels of porous zeolite (AlPO4-5) single crystal.16-18 On the basis of local-density approximation (LDA) calculation in the framework of the density functional theory (DFT), Blase et al. have pointed out that when the diameter comes close to about 4 Å, a strong curvature effect can induce stabilization of the lowest unoccupied (LU) band due 10.1021/nl025546t CCC: $22.00 Published on Web 04/27/2002
© 2002 American Chemical Society
to strong σ-π hybridization.19 As a result, the electronic structures for such thin SWCNT can no longer be predicted from the simple criterion determined by their diameter and helicity.3-7 However, it is not yet necessarily clear whether such thin CNT is accompanied with C-C bond alternation that could severely cause change in electronic structures of those. Hence, it will be of importance from such a standpoint to revisit the electronic structures of SWCNTs with small diameter of ∼4 Å. In this work, we report the ab initio Hartree-Fock (HF) crystal orbital (CO) calculations for thin SWCNTs to examine the electronic structures of these very thin CNTs. Method of Calculation. All of the calculations of the thin CNTs were performed on the basis of the ab initio selfconsistent-field CO (SCF-CO) method implemented in CRYSTAL98 package.20 The 6-21G basis sets21 were employed for the carbon atoms. To obtain the geometry for the unit cell, the geometrical optimizations of the model clusters for the thin CNTs were first carried out by the ab initio molecular orbital (MO) method at the restricted Hartree-Fock (RHF)/3-21G level with the GAUSSIAN98 package.22 We adopted 5 and 7 unit cells as the model clusters for the armchair and zigzag tubes, respectively, and used the central part of the optimized model cluster as the unit cell for the thin CNTs (see enclosed area in Figure 1). Results and Discussion. The geometries for the unit cells of the thin CNTs studied here are shown in Figure 1. It is clearly seen that a tube with smaller diameter (d) has typical
Figure 1. Geometries of unit cells utilized for the ab initio SCF-CO calculations of (a) tube(3, 3) (d ) 4.09 Å), (b) tube(4, 0) (d ) 3.35 Å), (c) tube (5, 0) (d ) 4.08 Å), and (d) tube(6, 0) (d ) 4.77 Å). These are determined on the basis of the optimized structures of the model clusters at the RHF/3-21G level. Bond lengths are indicated in angstroms and angles in degrees.
bond-alternant structure for the both armchair and zigzag tubes. In addition, one of the two kinds of bond angles decreases with decreasing diameter. The bond-alternant structures for the SWCNTs have also been found in our previous work.6,7 Another bond angle remains at ∼120°, irrespective of the diameter. Moreover, it was found that optimization of the cluster models for the tube (3, 0) with a possible diameter of 2.35 Å results in the collapse of the tube shape. Thus, from theoretical point of view, the smallest limit of the tube diameter would probably be around 3 Å. In Figure 2, the band structures and density of states (DOS) of the four kinds of CNTs, with the unit cell employing the optimized geometry of the model clusters in Figure 1, are shown. The band structure of armchair-type tube (3, 3) (d ) 4.09 Å) still shows metallic property similar to the ordinary armchair tubes with larger diameters despite its extraordinarily thin structure. This result is in good agreement with the recent LDA study.18 The zigzag tubes (5, 0) and (6, 0) were found to be semiconductive with band gaps of 2.4 and 1.7 eV, respectively, whereas the tube (4, 0) (d ) 3.35 Å) turned out to be metallic from the band structure shown in Figure 2b. On the other hand, according to the previous LDA calculations,19 it has been predicted that the tube (6, 0) becomes metallic. Note that the HF method tends to overestimate the band gap of condensed materials, whereas the LDA method to underestimate that. Hence, it can be safely said that the very thin zigzag CNT becomes narrowgap or nearly metallic for tube (5, 0). Moreover, the recent LDA calculation also supported the metallic property of tube (5, 0).18 630
What is more important is the reason that a semiconductive zigzag tube changes into a metallic one with decreasing diameter as found in tube (4, 0). In ref 19, the hybridization of the σ*- and π*-bands due to the curvature effect in the tubes with diameters smaller than 5 Å was emphasized, leading to the conclusion that the π*-band is strongly stabilized and finally crosses the Fermi level. A similar feature takes place in tube (4, 0) in our present study (see Figure 2b) as well. However, there seems to be some more material for consideration, in addition to the enhanced hybridization effect. At the Γ point, the bands for tubes (n, n) and (n, 0), whose unit cell always contains 4n carbon atoms, are classified into degenerate and nondegenerate bands. Considering the bands derived from 2s and 2p atomic orbitals (AOs), the bands calculated for the tubes (n, n) and (n, 0) ought to include 16 nondegenerated ones, and the remainders doubly degenerated. These 16 specific bands are closely related to those of a graphene sheet. That is, in the tubes (n, n) and (n, 0) abovementioned, the unit cell containes n units each consisting of four carbon atoms as shown in Figure 3a, that corresponds to the double size of the unit cell of a graphene sheet. In the band structure of a graphene sheet having this double-sized unit cell in Figure 3a, 16 nondegenerate (nd) bands (twelve pseudo-σ- and four pseudo-π-bands) similarly appear, as shown in Figure 3b. It is thus apparent that the band structures along the Γ-K and the Γ-X directions are similar to the 16 nondegenerate bands of the tubes (n, n) and (n, 0), respectively. Of these 16 nondegenerate bands, the highest occupied (HO) and lowest unoccupied (LU) pseudo-π-bands, Nano Lett., Vol. 2, No. 6, 2002
Figure 2. Calculated band structures and density of states (DOS) for (a) tube(3, 3), (b) tube(4, 0), (c) tube(5, 0), and (d) tube(6, 0). Zero on the ordinate designates the Fermi level.
Figure 4. Changes in the energy levels of the πnd and πnd* bands at the Γ point for zigzag tubes (n, 0) (n ) 4-9) as a function of the tube diameter. These calculations were carried out at the RHF/ STO-5G//RHF/STO-3G level according to the size of system considered.
Figure 3. Schematic drawings of (a) the primitive unit for armchair and zigzag tubes and (b) the Brillouin zone of a graphene sheet with the primitive unit in (a) as the unit cell, and the calculated band structure along the Γ-X and Γ-K directions of the Brillouin zone.
simply indicated as πnd and πnd*, show remarkable change in energy, parallel with decreasing diameter, as found in Figure 2. In particular, the energy gap between the πnd- and the πnd*-bands at the Γ point decreases dramatically in going from tube (6, 0) to tube (5, 0), as plotted in Figure 4. On the other hand, the doubly degenerate bands are rather insensitive Nano Lett., Vol. 2, No. 6, 2002
to the decrease of diameter. For instance, although both the HO and the LU bands in tube (6, 0) are doubly degenerate (Figure 2d), these pseudo-π-bands remain almost unchanged for tubes (4, 0) and (5, 0). Hereafter, we focus on the πnd- and πnd*-bands, and investigate changes of the π-character and the energy levels with respect to the zigzag tubes (n, 0). As an index of π-character for the pseudo-π-bands, we define the quantity P given by P)
Fπ (Fπ + Fσ)
(1)
where Fπ and Fσ represent the electron densities normal and 631
Figure 5. Index of π-character (P) of the COs accompanied with the πnd and πnd* bands at the Γ point for zigzag tubes (n, 0) (n ) 4-9) as a function of the tube diameter. These calculations were carried out at the RHF/STO-5G//RHF/STO-3G level according to the size of system considered.
Figure 6. CO patterns of (a) a graphene sheet, (b) zigzag tube(n, 0), and (c) armchair tube (n, n) at the Γ point. Atoms surrounded by the broken line construct the primitive unit.
tangent to the surface of the tube, respectively. The change in P values for the two bands in Figure 5 signifies that the σ-π hybridization proceeds with an increase in curvature, that is, a decrease in diameter of the tube. However, this change is not abrupt enough to explain the appearance of a metallic property in tube (4, 0). Let us look closely here at the COs of the CNTs in Figures 6b in order to answer the question raised above. The CO patterns accompanied with the πnd and πnd* bands at the Γ point well reflect those of a graphene sheet (Figure 6a). The CO pattern of πnd-band specifically shows out-of-phase combinations along the direction of the tube axis and inphase combinations along that of the tube circumference. In contrast, that of the πnd*-band is contrary to the πnd-band. What is to be noticed furthermore is the remarkable shorten632
Figure 7. Ratio of the bond length along the tube axis (r1) and the tube circumference (r2) for zigzag tubes (n, 0) (n ) 4-9) as a function of the tube diameter. These calculations were carried out at the RHF/STO-5G//RHF/STO-3G level according to the size of system considered.
ing of bond length along the tube axis (r1) as well as the lengthening along the tube circumference (r2) due to bond alternation observed for the zigzag tubes with diameter smaller than 5 Å (Figure 7). This leads to an increase in the overlap integral along the tube axis and a decrease in that along the tube circumference. That is, the bond shortening of r1 enhances the in-phase overlap between the adjacent π AOs stabilizing πnd*-orbital interaction, and the bond lengthening of r2 also reduces the destabilizing πnd*-orbital interaction, both of which lift down the position of the πnd*band. On the other hand, the πnd-band is lifted up due to destabilization of the concerning π-orbital interaction by shortening of r1 effective to enhancement of the out-of-phase overlap and by lengthening of r2 effective to reduction of the in-phase overlap. All of these factors rationalize the band behavior of the tube (4, 0). As is evident from Figure 7, the bond alternation becomes conspicuous in tubes (4, 0) and (5, 0) in accordance with the abrupt change in the energy gap between the πnd- and πnd*-bands at the Γ point (Figure 4). For tube (5, 0), although the πnd*-band becomes LU owing to decrease in the energy gap between the πnd- and πnd*-bands, there is no drastic change in the entire band structure (Figure 2c). As a result, the band gap for tube (5, 0) is similar to that for tube (6, 0) despite decrease in the energy gap between the πnd- and πnd*-bands. On the other hand, in the course of this inevitable lowering of the πnd*band, it crosses with the doubly degenerate pseudo-π HO band to make tube (4, 0) metallic. Thus, very thin zigzag CNTs can abruptly change from narrow band gap materials (tubes (6, 0) and (5, 0)) to a metallic one (tube (4, 0)) while rather maintaining the π character of these two bands. Conclusion. We have examined the electronic structures of the very thin CNTs with small diameters of about 4 Å using the ab initio SCF-CO calculations. It has been shown that these tubes (tubes (3, 3) and (4, 0)) are still qualified as π-electronic materials. There are three major findings in the present study: (1) The bond-alternant structures become more conspicuous with decreasing tube diameter. Nano Lett., Vol. 2, No. 6, 2002
(2) The armchair tube (3, 3) remains metallic irrespective of its large curvature. The zigzag tubes, (5, 0) and (6, 0) are semiconductive with narrow band gap. However, the tube (4, 0) is predicted to be metallic, contrary to the rule to classify the electronic properties of tubes with the ordinary diameters. (3) The reason tube (4, 0) is metallic has been interpreted in terms of the crossing of the πnd* band and an occupied doubly degenerate π one due to energetical stabilization of the former. Acknowledgment. This work was supported by a Grantin-Aid for Scientific Research (B) from Japan Society for the Promotion of Science (JSPS). Numerical calculations were partly carried out at the Supercomputer Laboratory of the Institute for Chemical Research of Kyoto University. References (1) Iijima, S. Nature 1991, 354, 56. (2) Carbon Nanotubes: Synthesis, Structure, Properties, and Applications; Dresselhaus, M. S., Dresselhaus, G., Avouris, P., Eds.; Springer-Verlag: Berlin, 2001. (3) Tanaka, K.; Okahara, M.; Okada, M.; Yamabe, T. Chem. Phys. Lett. 1992, 191, 469. (4) Saito, R.; Fujita, M.; Dresselhaus, G.; Dresselhaus, M. S. Phys. ReV. B 1992, 46, 1804. (5) Hamada, N.; Sawada, S.; Oshiyama, A. Phys. ReV. Lett. 1992, 68, 1579. (6) Tanaka, K.; Ago, H.; Yamabe, T.; Okahara, M.; Okada, M. Int. J. Quantum Chem. 1997, 63, 637. (7) Science and Technologies of Carbon Nanotubes; Tanaka, K., Yamabe, T., Fukui, K., Eds.; Elsevier Science: Oxford, 1999; p 40. (8) Frank, S.; Poncharal, P.; Wang, Z. L.; de Haar, W. A. Science 1998, 280, 1744. (9) Tans, S. J.; Devoret, M. H.; Dai, H.; Thess, A.; Smalley, R. E.; Geerigs, L. J.; Dekker, C. Nature 1997, 386, 474.
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