Graphene sheet Armchair

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).

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-... 

In that nomenclature system, the center of a hexagon is chosen as the origin (0,0) and then it is superimposed with the center m,n) of another hexagon to form the nanotube. There are three types of carbon nanotubes. If the graphene sheet is rolled in the direction of the axis, it will produce either an armchair nanotube m = ) or a zig-zag nanotube m = 0). On the other hand, if the graphene sheet is rolled in any other m,n) direction it will produce a chiral nanotube and the chirality will depend on whether the sheet is rolled upwards or backwards. 

Let us recall that nanotubes can be considered as graphene sheets rolled up in different ways. If we consider the so-called chiral vectors c = nai + na2, in which a and a2 are the basis vectors of a 2D graphite lattice, depending on the value of the integers n and m, one can define three families of tubes armchair tubes (n = m), zig-zag tubes (n or m = 0), and chiral tubes (n m 0). Band structure calculations have demonstrated that tubes are either metallic compounds, or zero-gap semiconductors, or semiconductors [6,7]. More commonly, they are divided into metallic tubes (when n-m is a multiple of 3) or semiconducting ones. 

Fig. 2 Schematic representation of the folding of a graphene sheet into (a) zigzag, (b) armchair and (c) chiral nanotubes. 

Figure 3. (a) Schematic honeycomb structure of a graphene sheet. SWCNTs can be formed by folding the sheet along lattice vectors. The vectors ai and a2 are shown. Folding of the (8,8), (8,0), and (10,-2) vectors lead to armchair (b), zigzag (c), and chiral (d) tubes, respectively. From reference 1. 

Figure 3 Illustration of chiral vector C/, = n-a + m-ai using 2D graphene sheet with lattice vectors a and and with limiting achiral cases of zigzag (n, 0) and armchair (n, n) configuration...

Figure 6.48. Illustration of the honeycomb 2D graphene network, with possible unit cell vector indices n,m). The dotted lines indicate the chirality range of tubules, from 0 = 0 (zigzag) to = 30° (armchair). For 0 values between 0 and 30°, the formed tubules are designated as chiral SWNTs. The electrical conductivities (metallic or semiconducting) are also indicated for each chiral vector. The number appearing below some of the vector indices are the number of distinct caps that may be joined to the n,m) SWNT. Also shown is an example of how a (5,2) SWNT is formed. The vectors AB and A B which are perpendicular to the chiral vector (AA are superimposed by folding the graphene sheet. Hence, the diameter of the SWNT becomes the distance between AB and A B axes. Reprinted from Dresselhaus, M. S. Dresselhaus, G. Eklund, R C. Science ofFullerenes and Carbon Nanotubes. Copyright 1996, with permission from Elsevier.

Vm Fig. 28.22 (a) Vectors on a graphene sheet that define achiral zigzag and armchair carbon nanotubes. The angle 0 is defined as being 0° for the zigzag structure. The bold lines define the shape of the open ends of the tube, (b) An example of an armchair carbon nanotube, (c) An example of a zigzag carbon nanotube. 

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