Carbon nanotubes chirality

Table 6.1 Effect of carbon nanotube chirality on the conductivity.

Fig. 14. High resolution TEM observations of three multi-wall carbon nanotubes with N concentric carbon nanotubes with various outer diameters do (a) N = 5, do = 6.7 nm, (b) N = 2, do = 5.5 nm, and (c) N = 7, do = 6.5 nm. The inner diameter of (c) is d = 2.3 nm. Each cylindrical shell is described by its own diameter and chiral angle [151].

The circumference of any carbon nanotube is expressed in terms of the chiral vector = nai ma2 which connects two crystallographically equivalent sites on a 2D graphene sheet [see Fig. 16(a)] [162]. The construction in... 

The ID electronic energy bands for carbon nanotubes [170, 171, 172, 173, 174] are related to bands calculated for the 2D graphene honeycomb sheet used to form the nanotube. These calculations show that about 1/3 of the nanotubes are metallic and 2/3 are semiconducting, depending on the nanotube diameter di and chiral angle 6. It can be shown that metallic conduction in a (n, m) carbon nanotube is achieved when... 

Fig. 19. The energy gap FJ, for a general chiral single-wall carbon nanotube as a function of 100 kidt, where dt is the nanotube diameter in A [179].

Early transport measurements on individual multi-wall nanotubes [187] were carried out on nanotubes with too large an outer diameter to be sensitive to ID quantum effects. Furthermore, contributions from the inner constituent shells which may not make electrical contact with the current source complicate the interpretation of the transport results, and in some cases the measurements were not made at low enough temperatures to be sensitive to 1D effects. Early transport measurements on multiple ropes (arrays) of single-wall armchair carbon nanotubes [188], addressed general issues such as the temperature dependence of the resistivity of nanotube bundles, each containing many single-wall nanotubes with a distribution of diameters d/ and chiral angles 6. Their results confirmed the theoretical prediction that many of the individual nanotubes are metallic. 

In general, the number of phonon branches for a carbon nanotube is very large, since every nanotube has 6N vibrational degrees of freedom. The symmetry types of the phonon branches for a general chiral nanotube are obtained using a standard group theoretical analysis [194]... 

Fig. 3. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The pairs of integers ( , ) in the figure specify chiral vectors Cy, (see Table I) for carbon nanotubes, including zigzag, armchair, and chiral tubules. Below each pair of integers (n,m) is listed the number of distinct caps that can be joined continuously to the cylindrical carbon tubule denoted by (n,wi)[6]. The circled dots denote metallic tubules and the small dots are for semiconducting tubules.

Fig. 4. The relation between the fundamental symmetry vector R = p3] -1- qa2 and the two vectors of the tubule unit cell for a carbon nanotube specified by (n,m) which, in turn, determine the chiral vector C, and the translation vector T. The projection of R on the C, and T axes, respectively, yield (or x) and t (see text). After N/d) translations, R reaches a lattice point B". The dashed vertical lines denote normals to the vector C/, at distances of L/d, IL/d, 3L/d,..., L from the origin.

Although still preliminary, the study that provides the most detailed test of the theory for the electronic properties of the ID carbon nanotubes, thus far, is the combined STM/STS study by Oik and Heremans[13]. In this STM/STS study, more than nine individual multilayer tubules with diameters ranging from 1.7 to 9.5 nm were examined. The 7-Fplots provide evidence for both metallic and semiconducting tubules[13,14]. Plots of dl/dV indicate maxima in the ID density of states, suggestive of predicted singularities in the ID density of states for carbon nanotubes. This STM/ STS study further shows that the energy gap for the semiconducting tubules is proportional to the inverse tubule diameter l/<7, and is independent of the tubule chirality. 

These properties are illustrative of the unique behavior of ID systems on a rolled surface and result from the group symmetry outlined in this paper. Observation of ID quantum effects in carbon nanotubes requires study of tubules of sufficiently small diameter to exhibit measurable quantum effects and, ideally, the measurements should be made on single nanotubes, characterized for their diameter and chirality. Interesting effects can be observed in carbon nanotubes for diameters in the range 1-20 nm, depending... 

The existence of carbon nanotubes with diameters small compared to the de Broglie wavelength has been described by Iijima[l,2,3] and others[4,5]. The energy band structures for carbon nanotubes have been calculated by a number of authors and the results are summarized in this issue by M.S. Dresselhaus, G. Dres-selhaus, and R. Saito. In short, the tubules can be either metallic or semiconducting, depending on the tubule diameter and chirality[6,7,8]. The calculated density of states[8] shows singularities... 

Experimental measurements to test these remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because of the strong dependence of the predicted properties on tubule diameter and chirality. Ideally, electronic or optical measurements should be made on individual single-wall nanotubes that have been characterized with regard to diameter and chiral angle. Further ex-... 

Studies on the electronic structure of carbon nanotube (CNT) is of much importance toward its efficient utilisation in electronic devices. It is well known that the early prediction of its peculiar electronic structure [1-3] right after the lijima s observation of multi-walled CNT (MWCNT) [4] seems to have actually triggered the subsequent and explosive series of experimental researches of CNT. In that prediction, alternative appearance of metallic and semiconductive nature in CNT depending on the combination of diameter and pitch or, more specifically, chiral vector of CNT expressed by two kinds of non-negative integers (a, b) as described later (see Fig. 1). 

The structure of carbon nanotubes depends upon the orientation of the hexagons in the cylinder with respect to the tubule axis. The limiting orientations are zigzag and arm chair forms, Fig. 8B. In between there are a number of chiral forms in which the carbon hexagons are oriented along a screw axis, Fig. 8B. The formal topology of these nanotube structures has been described [89]. Carbon nanotubes have attracted a lot of interest because they are essentially onedimensional periodic structures with electronic properties (metallic or semiconducting) that depend upon their diameter and chirality [90,91]. (Note. After this section was written a book devoted to carbon nanotubes has been published [92], see also [58].)... 

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