Chiral angle

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 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... 

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. 1. The 2D graphene sheet is shown along with the vector which specifies the chiral nanotube. The chiral vector OA or Cf, = nOf + tnoi defined on the honeycomb lattice by unit vectors a, and 02 and the chiral angle 6 is defined with respect to the zigzag axis. Along the zigzag axis 6 = 0°. Also shown are the lattice vector OB = T of the ID tubule unit cell, and the rotation angle 4/ and the translation r which constitute the basic symmetry operation R = (i/ r). The diagram is constructed for n,m) = (4,2).

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-... 

Scanning tunneling spectroscopy (STS) can, in principle, probe the electronic density of states of a singlewall nanotube, or the outermost cylinder of a multi-wall tubule, or of a bundle of tubules. With this technique, it is further possible to carry out both STS and scanning tunneling microscopy (STM) measurements at the same location on the same tubule and, therefore, to measure the tubule diameter concurrently with the STS spectrum. No reports have yet been made of a determination of the chiral angle of a tubule with the STM technique. Several groups have, thus far, attempted STS studies of individual tubules. 

Fig. 1. Typical ED pattern of polychiral MWCNT. The pattern is the superposition of the diffraction patterns produced by several isochiral clusters of tubes with different chiral angles. Note the row of sharp oo.l reflexions and the streaked appearance of 10.0 and 11.0 type reflexions. The direction of beam incidence is approximately normal to the tube axis. The pattern exhibits 2mm planar symmetry [9].

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. 

The diffraction patterns due to different isochiral clusters are superimposed and well separated in a polychiral MWCNT diffraction pattern, suggesting that interference between waves scattered by tubes with different chiral angles can be neglected. It is therefore meaningful to discuss only isochiral clusters of tubes. Such clusters are only compatible with a constant intercylinder spacing c/2 for pairs of Hamada indices satisfying the condition = L +M +LM - (nc/a). Approximate solutions are for instance (8, 1) and (5, 5) [16,17]. 

The angular splitting of the ho.o (or hh.o) reflexions is a measure for the chiral angle T). However the observed splitting depends as the direction of incidence of the electron beam and must thus be corrected for tilt [20,25]. 

Using HRTEM the chiral angle can also be deduced from the moird or coincidence pattern formed in the central area of the tube image between "front" and "back" surfaces of the tube. 

The diffraction patterns of isochiral clusters of tubes with different chiral angles in MWCNTs are superimposed in the composite pattern, the different chiral angles can be measured separately by diffraction contrast imaging [26]. 

Fig. 16. (a) The chiral vector OA or Ch = nhi + md2 is defined on the honeycomb lattice of carbon atoms by unit vectors ai and a of a graphene layer and the chiral angle with respect to the zigzag axis (9 = 0°). Also shown are the lattice vector... 

Assuming the simplest but nontrivial form of the classical chiral angle such that... 

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