Graphene sheet Helicity

The quantity x is a dimensionless quantity which is conventionally restricted to a range of —-ir < x < tt, a central Brillouin zone. For the case yj = 0 (i.e., S a pure translation), x corresponds to a normalized quasimomentum for a system with one-dimensional translational periodicity (i.e., x s kh, where k is the traditional wavevector from Bloch s theorem in solid-state band-structure theory). In the previous analysis of helical symmetry, with H the lattice vector in the graphene sheet defining the helical symmetry generator, X in the graphene model corresponds similarly to the product x = k-H where k is the two-dimensional quasimomentum vector of graphene. 

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-... 

When an anode contains an appropriate amount of metals (or metal oxide), novel carbon materials such as SWNTs, metallofuJlerenes, filled nanocapsules, bam-boo -shaped tubes (23), nanochains (10), and MWNTs filled with metal carbides (24,25) are formed. Especially SWNTs are now attracting a great deal of interest from researchers in physics and materials science, because exotic electronic properties that vary between semiconducting and metallic states depending on how a graphene sheet is rolled (i.e., diameter and helical pitch of a tube) are predicted theoretically (26-28) and because unique quantum effects are revealed experimentally (29,30). 

Helical (or chiral) vector Ch defined from the director vectors (a-1) and (a2) of the graphene sheet by using a pair of integers (n, m) Ch = na-, + ma2 and chiral angle 0. Reprint from Carbon, vol. 33, No. 7, Dresselhaus M.S., Dresselhaus G., Saito R., Physics of carbon nanotubes, pages 883-891, Copyright (1995) with permission from Elsevier. 

In practice, defect-free coaxial nanotubes rarely occur in experimental preparations. The observed structures include the capped, bent, and toroidal SWNTs, as well as the capped and bent, branched, and helical MWNTs. Figure 14.1.11 shows the HRTEM micrograph of a helical multiwalled carbon nanotube which incorporates a small number of five- and seven-membered rings into the graphene sheets of the nanotube surfaces. 

Ideally, a SWCNT is made of a single perfect graphene sheet rolled up into a cylinder and closed by two caps (semi-fullerenes). Considering that there are different ways to roll up a section of a two-dimensional graphene sheet, the structure of SWCNTs is characterized by a roll-up vector, also called a helicity vector [11 and Chapter 14]. The diameter of these structures can vary between 0.4 and 2.5 nm and the length between a few micrometers and several millimeters. 

Generally, the electronic properties of an isolated defect-free SWCNT are dependent on the helicity (or chirality) of the CNT structure (i.e., the circumferential wavevector discussed above). The helicity is defined as the vector C, along which a graphene sheet is rolled up into a seamless hollow cylinder. The vector C can be expressed in terms of two integers (m, n) corresponding to the... 

The way in which these graphene sheets are rolled determines the atomic structure of the CNT, which is described in terms of chirality (helicity) of the tube, defined by the chiral vector Ch and chiral angle 6, Fig. 5.3a. The numbers (n, m) are integers and ai and a2 are the unit vectors of the hexagonal lattice of the graphene sheet. 

Creating a seamless cylinder from a graphene sheet can be done in three ways, each resulting in a tube that is said to have a distinct chirality or helicity [see Figure 1.4], The chirality of a specific SWCNT [which could be a single shell of a MWCNT as well] is described by the chiral or "roll-up" vector This vector is defined as the summation of multiples of the unit vector cells and 02 given in Equation 1.1. 

Because of the different ways of roUing a graphene sheet into a cylinder, achiral zigzag and armchair tubes and helical chiral nanotubes are distinguished [2]. Each individual NT is uniquely specified by a chiral vector, defined in terms of graphene... 

Carbon nanotubes have been studied extensively since their discovery [1] in 1991, because of the extraordinary physical properties they exhibit in electronic, mechanical, and thermal processes. A single-walled nanotube may be considered as a specific, one-dimensional giant molecule composed purely of carbon, whereas properties of multiwalled nanotubes are closest to those of graphite s in-plane properties, having sp hybridization of carbon bonds. To prepare closed-shell structures, one needs to insert topological defects into the hexagonal stmcture of graphene sheets. The extraordinary physical and chemical properties [2] and possible applications derived from these properties are attributed to the one-dimensionality and helicity of the nanotube structure. 

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