Armchair chirality

Additionally, the chirality of any nanotube is a big challenge for computational methods. In a normal periodic-boundary approach the periodicity of a chiral nanotube may be much larger than that of the non-chiral armchair and zigzag tubes. Therefore, any computational approach tackling chiral nanotubes is more prohibitive, and studies—especially using electronic-structure methods—are usually performed on armchair and zigzag tubes. 

According to the rolling angle of the graphene sheet, CNT have three chiralities armchair, zigzag, and chiral one. The chirality of nanotubes has significant impact on their electronic properties (Ma et al. 2010). 

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. 

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. 2. By rolling up a graphene sheet (a single layer of ear-bon atoms from a 3D graphite erystal) as a cylinder and capping each end of the eyiinder with half of a fullerene molecule, a fullerene-derived tubule, one layer in thickness, is formed. Shown here is a schematic theoretical model for a single-wall carbon tubule with the tubule axis OB (see Fig. 1) normal to (a) the 6 = 30° direction (an armchair tubule), (b) the 6 = 0° direction (a zigzag tubule), and (c) a general direction B with 0 < 6 < 30° (a chiral tubule). The actual tubules shown in the figure correspond to (n,m) values of (a) (5,5), (b) (9,0), and (c) (10,5).

Fig. 3. (a) Diffraction pattern of a well formed rope (superlattice) of armchair-like tubes. Note the presence of superlattice spots in the inset (b). The broadening of the streaks of 1010 type reOexions is consistent with a model in which the SWCNTs have slightly different chiral angles. 

When we compare the calculated Raman intensities for armchair, zigzag and chiral CNTs of similar diameters, we do not see large differences in the lower frequency Raman modes. This is because the lower frequency modes have a long... 

Fig. 10.7 Chirality vector and folding scheme for semiconducting and metallic nanotube (a). Zig-zag, armchair, and chiral nanotubes by rolling-up of the graphite lattice (b) (Reprinted from Terrones 2003. With permission from Annual Reviews)... 

If m = 0, the nanotubes are called zigzag nanotubes, if n = m, the nanotubes are defined as armchair nanotubes, and all other orientations are called chiral . The deviation of Cn from a is expressed by the inclination angle 0 and ranges from 0° ( armchair ) to 30° ( zigzag ) [17]. 

Fig. 1.1 (a) Schematic of unrolled SWCNT showing chiral vector Cn and the effect of m and n on the electronic properties of SWCNTs. (b, c, d) The direction of the chiral vector affects the appearance of the nanotube showing (b) (4,4) armchair, (c) (6,0) zigzag and (d) (5,3) exemplary chiral shape. With kind permission from [18],... 

Another complication in CNT applicability arises from the way the graphene sheet is rolled up to create the cylindrical structures, which is usually called helicity . Depending on the angle of the wrapping, three different structures (different helicities) can result (1) armchair, (2) chiral or (3) zigzag (Fig. 3.3). Such structures exhibit differ-... 

There are three general types of CNT structure (Figure 12.10). The zigzag nanotubes correspond to ( ,0) or (0,m) and have a chiral angle of 0°. The carbon-carbon position is parallel to the tube axis. Armchair nanotubes have (n,n) with a chiral angle of 30°. The carbon-carbon positions are perpendicular to the tube axis. Chiral nanotubes have general (n,m) values and a chiral angle of between 0° and 30°, and as the name implies, they are chiral. 

Figure 14.17 Idealized representation of defect-free (n,m)-SWNTs with open ends. Left a metallic conducting (10,10)-tube (armchair) middle a chiral, semiconducting (12,7)-tube right a conducting (15,0)-tube (zigzag). Armchair and zigzag tubes are achiral.

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