Graphene sheet Zigzag

Closely related to the ID dispersion relations for the carbon nanotubes is the ID density of states shown in Fig. 20 for (a) a semiconducting (10,0) zigzag carbon nanotube, and (b) a metallic (9,0) zigzag carbon nanotube. The results show that the metallic nanotubes have a small, but non-vanishing 1D density of states, whereas for a 2D graphene sheet (dashed curve) the density of states... 

Fig. 20. Electronic 1D density of states per unit cell of a 2D graphene sheet for two (n, 0) zigzag nanotubes (a) the (10,0) nanotube which has semiconducting behavior, (b) the (9, 0) nanotube which has metallic behavior. Also shown in the figure is the density of states for the 2D graphene sheet (dotted line) [178].

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).

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-... 

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.

Figure 13.1 Construction of the (5, 3) SWNT from a graphene sheet. First step the chiral vector Ch is obtained by starting from (0,0) and moving 5 units in the ai direction and then 3 units in the U2 direction. Second step the (dashed) sheet determined by the length of the chiral vector Ch is rolled up along the chiral vector, so the origin (0,0) coincides with the end of Ch, in this example (5, 3). For this nanotube, the chiral angle formed between C and the a direction is 21.8 degrees (calculated from Equation 13.2). The inset illustrates the rolled-up structure of a typical SWNT, this is a (14,0) nanotube with zigzag structure. In this case, the chiral angle is 0.

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. 

---