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89]. Thus if we consider finite temperatures or fluctuation effects, it is believed that such a Small Peierls gap can be neglected. As the tubule diameter increases, more wave vectors become allowed for the circumferential direction, the tubules become more two-dimensional, and the semiconducting band gap decreases, as illustrated in Fig. 19.30.

The effect of curvature of the carbon nanotubes has been considered within the tight binding approximation [19.102]. This complicates the calculation considerably, by introducing four tight binding parameters, with values given by Vppn = -2.77 eV, Vssa = -4.76 eV, Vsp„ = 4.33 eV, and Vppa = 4.37 eV [19.103,104], instead of the single tight binding parameter Vppv = —2.5 eV, which applies in the case where curvature is neglected. It should be mentioned that for both the armchair and zigzag tubules the band crossing at EF is between energy bands of different symmetry (tubule curvature has no effect on symmetry) so that no interaction or band splitting would be expected at EF. The effect of tubule curvature is further discussed in §19.8.

Some first principles calculations have been carried out for carbon nanotubes [19.92,102,105-107], yielding results in substantial agreement with the simple tight binding results which are described in this section. A comparison between the first principles local density functional determination of the band gap for the semiconducting tubules (open squares) in comparison with the tight binding calculation for (Vp v — —2.5 eV) is shown in Fig. 19.31 as the solid curve [19.102]. In the limit that we average the

Nanotube radius (nm)

Fig. 19.31. Band gap as a function of nanotube radius using a first principles local density functional method. The solid line shows estimates using a graphene sheet model with Vpp, = —y0 = 2.5 eV [19.102],

Nanotube radius (nm)

Fig. 19.31. Band gap as a function of nanotube radius using a first principles local density functional method. The solid line shows estimates using a graphene sheet model with Vpp, = —y0 = 2.5 eV [19.102], asymmetry of the bonding and antibonding energies in graphite, the average overlap integral y0 -> —2.5 eV [19.102],

Another local density approximation (LDA) calculation shows that the large curvature of small single-wall carbon nanotubes leads to significant hybridization of a•* and ir* orbitals, leading to the introduction of conduction band states into the energy gap. The effect of this hybridization is large for tubules of diameter less than that of C60 but is not so great for tubules in the range (d, >0.7 nm) that have been observed experimentally [19.105,108], The simple tight binding result for single-wall nanotubes that one third of the nanotubes are metallic and two thirds are semiconducting has a symmetry basis which is discussed in §19.4. There are several physical processes that tend to modify these simple considerations. A first principles LDA calculation [19.108] has determined that for low-diameter tubules the curvature of the graphene sheet results in band shifts which move their band edges into the semiconducting energy gap, hence suggesting that all small-diameter tubules should be metallic. The contrary conclusion could be reached if a Peierls distortion of the ID conductor produced an energy gap at the Fermi level. Several authors [19.46,89] have considered the Peierls distortion and have argued that it is suppressed in the carbon nanotube because of the large elastic energies that are associated with an in-plane lattice distortion. A splitting of the Appoint degeneracy associated with a symmetry lowering (Jahn-Teller splitting) has been considered by some authors as a mechanism for removal of the metallic degeneracy. At the present time the experimental evidence is not clear about the nature of the conductivity of single-wall carbon nanotubes.

LDA-based calculations have also been carried out for BN nanotubes, and the results suggest that BN nanotubes should be stable and should have a band gap of ~5.5 eV independent of diameter [19.108,109]. These calculations stimulated experimental work leading to the successful synthesis of pure multiwall BN nanotubes [19.110], with inner diameters of 1-3 nm and lengths up to 200 nm. The BN nanotubes are produced in a carbon-free plasma discharge between a BN-packed tungsten rod and a cooled copper electrode. Electron energy loss spectroscopy on individual nanotubes confirmed the BN stoichiometry [19.110,111],

19.5.3. Multiwall Nanotubes and Arrays

Many of the experimental observations on carbon nanotubes thus far have been made on multiwall tubules [19.10,11,98,112,113]. This has inspired a number of theoretical calculations to extend the theoretical results initially obtained for single-wall nanotubes to observations in multilayer tubules. These calculations for multiwall tubules have been informative for the interpretation of experiments and influential for suggesting new research directions. Regarding the electronic structure, multiwall calculations have been done predominantly for double-wall tubules, although some calculations have been done for a four-wall tubule [19.10,11,112] and also for nanotube arrays [19.10,11]

The first calculation for a double-wall carbon nanotube [19.98] was done with a tight binding calculation. The results showed that two coaxial zigzag nanotubes that would each be metallic as single-wall nanotubes yield a metallic double-wall nanotube when a weak interlayer coupling between the concentric nanotubes is introduced. Similarly, two coaxial semiconducting tubules remain semiconducting when a weak interlayer coupling is introduced [19.98]. More interesting is the case of coaxial metal-semiconductor and semiconductor-metal nanotubes, which also retain their metallic and semiconducting identities when a weak interlayer interaction is introduced. On the basis of this result, we conclude that it might be possible to prepare metal-insulator device structures in the coaxial geometry, as has been suggested in the literature [19.35,114].

A second calculation was done with density functional theory in the local density approximation to establish the optimum interlayer distance between an inner (5,5) armchair tubule and an outer armchair (10,10) tubule. The result of this calculation yielded a 3.39 A interlayer separation [19.10,11], with an energy stabilization of 48 meV/carbon atom. The fact that the interlayer separation is somewhat less than the 3.44 A separation expected for turbostratic graphite may be explained by the interlayer correlation between the carbon atom sites both along the tubule axis direction and circumferentially. A similar calculation for double-layered hyperfullerenes has been carried out, yielding an interlayer spacing of 3.524 A between the two shells of [email protected], with an energy stabilization of 14 meV/atom (see §19.10) [19.78], In addition, for two coaxial armchair tubules, estimates for the translational and rotational energy barriers of 0.23 meV/atom and 0.52 meV/atom, respectively, were obtained, suggesting significant translational and rotational interlayer mobility for ideal tubules at room temperature. Of course, constraints associated with the cap structure and with defects on the tubules would be expected to restrict these motions. Detailed band calculations for various interplanar geometries for the two coaxial armchair tubules confirm the tight binding results mentioned above [19.10,11].

Inspired by experimental observations on bundles of carbon nanotubes, calculations of the electronic structure have also been carried out on arrays of (6,6) armchair nanotubes to determine the crystalline structure of the arrays, the relative orientation of adjacent nanotubes, and the spacing between them. Figure 19.32 shows one tetragonal and two hexagonal arrays that were considered, with space group symmetries P42/mmc (D9Ah), P6/mmm (D\h), and P6/mcc (D\h), respectively [19.10,11,106]. The calculation shows that the hexagonal P6/mcc (D\h) space group is the most stable, yielding an alignment between tubules that relates closely to the ABAB stacking of graphite, with an intertubule separation of 3.14 A at closest approach, showing that the curvature of the tubules lowers the minimum interplanar distance (as is also found for fullerenes where the corresponding distance is 2.9 A). The calculations further show that the hexagonal intertubule arrangement results in a gain in cohesive energy of 2.2 meV/carbon atom, while the atomic matching between adjacent tubules results in a gain in cohesive energy of 3.4 meV/carbon atom. The importance of the intertubule interaction can be seen in the reduction in the intertubule closest approach distance to 3.14 A for the P6/mcc (Djh) structure, from 3.36 A and 3.35 A, respectively, for the tetragonal P42/mmc (D94h) and P6/mmm (D\h) space groups. A plot of the electronic dispersion relations for the most stable P6/mmm (D\h) structure is given in Fig. 19.33 [19.10,11,106], showing the metallic nature of this tubule array. It is expected that further calculations will consider the interactions between nested nanotubes having different symmetries, which on physical grounds should have a weaker interaction, because of a lack of correlation between near neighbors. Fur-

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