Graphene sheet unit vector

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).

In the above eqn, ID refers to the nanotubes whereas 2D refers to the graphene sheet, k is the ID wave vector, and t and Care unit vectors along the tubule axis and vector C, respectively, and p labels the tubule phonon branch. 

Figure 6.3a presents a unit cell of graphene sheet comprising two kinds of carbon atoms a and (1, where a = 2.46 A and x2 are the lattice constant and the vector connecting two carbon atoms, respectively. The corresponding Brillouin zone (BZ) in the reciprocal lattice is shown in Figure 6.3b. Assuming that only the nearest neighbors overlap and resonance integrals work in the system, the Jt-band energies are calculated from the secular equation as expressed by...

One way to define the structure of single-walled CNT is to think of each CNT as a result of rolling a graphene sheet specifying the direction of rolling and the circumference cross-section (Figure la). The unit vectors of the planar... 

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 9.17 The structure of a single graphene sheet denoted with (n,m) integer pairs (the base vector of the unit cell of the graphene sheet is also indicated)...

Figure 26.1 Physical stmcUire of a carbon nanotube starting from a graphene sheet, whose unit vectors j and Uj are shown. Every nanotube is uniquely determined by its chiral vector Cij = (n,m). The (11,4) nanotube is obtained by rolling the graphene sheet along the red shaded region.

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.

The carbon nanotubes are imagined as rolled up graphene sheets and the atoms on the graphene sheet are described with the help of two unit cell vectors ai = a(v/3,0) and a2 = a(, I). Here a is the inter-atomic distance and the centre of the hexagon (k, 1) is given by the translation vector R = kai -f lai with integers k and 1. As in the graphene each unit cell contains two atoms with the serial numbers r = 1 and r = 2,... 

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. 

Figure 13.9. Top a graphene sheet with the ideal lattice vectors denoted as ai, 02. The thicker lines show the edge profile of a (6,0) (zig-zag), a (4,4) (armchair), and a (4, 2) (chiral) tube. The tubes are formed by matching the end-points of these profiles. The hexagons that form the basic repeat unit of each tube are shaded, and the thicker arrows indicate the repeat vectors along the axis of the tube and perpendicular to it, when the mbe is unfolded. Bottom perspective views of the (8,4) chiral tube, the (7,0) zig-zag tube and the (7, 7) armchair tube along their axes.

SWCNT, which is a one-dimensional (ID) system, can be considered as the conceptual rolling of a section of two-dimensional (2D) graphene sheet into a seamless cylinder forming the nanotube. The structure of SWCNT is uniquely described by two integers ( , m), which refer to the number of 5i and U2 unit vectors of the 2D graphene lattice that are contained in the chiral vector, = ndi + ma.2. The chiral vector determines whether the nanotube is a semiconductor, metal, or semimetal. From the (n, m) indices, one can calculate the nanotube diameter dt), the chirality or chiral angle (0), the electronic energy bands, and the density of electronic states. The nanotube diameter dt) determines the munber of carbon atoms in the... 

Fig. 6 (a) The (n, m) nanotube naming scheme can be thought of as a vector (Ch) in an infinite graphene sheet that describes how to "roll up" the graphene sheet to make the nanotube. T denotes the tube axis, and ai and 2 are the unit vectors of graphene in real space, and (b) Different types of CNTs. 

The structure of a SWNT can be conceptualized by wrapping planar sheet of graphene into a seamless cylinder as shown in fig. 6. The way, the graphene sheet is wrapped is represented by a pair of indices (n, m). The integers, n and m, denote the number of unit vectors along two directions in the honeycomb crystal lattice of graphene. If m = 0, the nanotubes are called zigzag nanotubes, and if n = m, the nanotubes are called armchair nanotubes. Otherwise, they are called chiral nanotubes. Diameter of ideal CNT can be calculated from its unit vector (n, m) as follows ... 

A periodic sheet is a more consistent model, when a unit cell of graphite is translated n and m times across two lattice vectors (periodic sheets are marked as nxm, /r-layer graphene is denoted as nxmxk) to model a small zone on the sheet where the transition metal particle is bound (Figure 11.3). Similarly, some atoms may be cut off or substituted to model the defective or doped sheet. Hollow, bridge, and top adsorption sites may be assigned for atoms adsorbed on the sheet (Figure 11.3). 

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