19 Electronic Structure of Carbon Nanotube

5.6 Electronic Structure of Carbon Nanotube

5.6.1 Electronic Structure of Single-Wall Nanotubes

5.6.2 Energy dispersion of Armchair and Zig-Zag Nanotubes

5.6 Electronic Structure of Carbon Nanotube

5.6.1 Electronic Structure of Single-Wall Nanotubes

Figure 5.30 (a) a1 and a2 are the lattice vectors. |a1| = |a2| = √3 a, where a is the carbon–carbon bond length. There are two atoms per unit cell shown by A and B. SWNTs are equivalent to cutting a strip in the grapheme sheet (blue) and rolling them up such that each carbon atoms is bonded to its three nearest neighbours. The creation of a (n,0)   zigzag nanotube is shown. (b) Creation of a (n, n) armchair nanotube. (c) A (n,m) chiral nanotube. (d) The bonding structure of a nanotube. The n = 2 quantum number of carbon has four electrons. Three of these electrons are bonded to its three nearest neighbours by sp2 bonding. The fourth electron is a π orbital perpendicular to the cylindrical surface.

The electronic structure of carbon nanotubes can be    derived by a simple tight-binding calculation for the   -electrons of carbon atoms. It can be shown that  electronic structure of a carbon nanotube can be either metallic or semiconducting, depending on its diameter and chirality.  The electronic structure of a SWNT can be derived simply from that of two-dimensional graphite. It can be shown that by using periodic boundary conditions in the circumferential direction denoted by the chiral vector Ch, which becomes quantized, however the wave vector associated with the direction of the translational vector T (or along the nanotube axis) remains continuous. This results the energy bands as a set of one- dimensional energy dispersion relations which are cross sections of those for two-dimensional graphite (Figure 5.30).

Figure 5.31 (a) The Brillouin zone of a carbon nanotube is represented by the line segment WW’ which is parallel to K2. The vectors K1 and K2 are reciprocal lattice vectors corresponding to Ch and T, respectively. The figure corresponds to Ch =(4, 2), T =(4, -5), N = 28, K1 = (5bl +4b2)/28, K2 = (4bl – 2b2)/28. (b) The condition for metallic energy bands: if the ratio of the length of the vector YK to that of K1 is an integer, metallic energy bands are obtained.

When the energy dispersion relations of two-dimensional graphite, Eg2D(k) at line segments shifted from WW’ by μK1 ( =0, …N – 1) are folded so that the wave vectors parallel to K2 coincide with WW’ as shown in Figure 5.31, N pairs of 1D energy dispersion relations E (k) are obtained. These 1D energy dispersion relations are given by

which is the energy dispersion relations of a SWNT. The N pairs of energy dispersion curves given by Eq. 5.38 correspond to the cross sections of the two-dimensional energy dispersion surface shown in Figure 5.32, where cuts are made on the lines of kK2/|K2| + K1. If for a particular (n,m) nanotube, the cutting line passes through a K point of the 2D Brillouin zone (Figure 5.32, where the and * energy bands of two-dimensional graphite are degenerate, the one-dimensional energy bands have a zero energy gap. Hence, the density of states at the Fermi level has a finite value for these carbon nanotubes resulting metallic properties of SWNT. If, the cutting line does not pass through a K point, then the carbon nanotube is expected to show semiconducting behavior, with a finite energy gap between the valence and conduction bands, resulting semiconducting property.

Figure 5.32 The energy dispersion relations for 2D graphite are shown throughout the whole region of the Brillouin zone. The inset shows the energy dispersion along the high symmetry directions.

The condition for obtaining a metallic energy band is that the ratio of the length of the vector Y K to that of K1 in Fig. 1a is an integer. Since the vector Y K is given by,

the condition for metallic nanotubes is that (2n+m) or equivalently (n-m) is a multiple of 3. In particular,

the armchair nanotubes denoted by (n,n) are always metallic, and the zigzag nanotubes (n,0) are only metallic when n is a multiple of 3.

5.6.2 Energy dispersion of Armchair and Zig-Zag Nanotubes

Figure 5.33 Part of the unit cell and extended Brillouin zone of (a) armchair and (b) zigzag carbon nanotubes. ai and bi are unit vectors and reciprocal lattice vectors of two-dimensional graphite respectively. In the figure, the translational vector T and the corresponding reciprocal lattice vector K2 of the nanotube are shown.

To obtain the energy dispersion relation, the simplest cases to consider are the nanotubes having the highest symmetry. From Figure 5.33, we see the unit cells and Brillouin zones for the highly symmetric nanotubes, namely for (a) an armchair nanotube and (b) a zigzag nanotube.

Under appropriate periodic boundary conditions, the energy eigenvalues for the (n, n) armchair nanotube can be obtained from small number of allowed wave vectors kx,q in the circumferential direction

Substitution of the discrete allowed values for k, given by Eq.5.39 into Eq. 5.36 yields the energy dispersion relations ???(?) for the armchair nanotube, Ch = (n,n)

Where the superscript a refers to armchair and k is a one-dimensional vector in the direction of the vector K2=(b1– b2)/2 which corresponds to the Γ to K point vector in the two-dimensional Brillouin zone of graphite (Figure 5.32). The 1D dispersion relations (  ) for the (5,5) armchair nanotube are shown in Figure 5.34. The energy bands show a large degeneracy at the zone boundary in all armchair nanotube, where ka = , so that Eq. 5.36 becomes

for the 2D graphene sheet, independent of zone folding and independent of n. There are four carbon atoms in the unit cell of Figure 5.33, out of which two carbon atoms on the same sublattice of a graphene sheet are symmetrically equivalent, resulting degeneracy of the energy bands at the boundary of the Brillouin zone. As shown in Fig. 5.34, the valence and conduction bands for the armchair nanotube cross at a k point that is two thirds of the distance from k = 0 to the zone boundary at k = /a. The crossing at the Fermi level results the energy bands symmetric for ±   values. The degeneracy point between the valence and conduction bands at the band crossing leads to metallic property of (5,5) armchair nanotube. All (n,n) armchair nanotubes have a band degeneracy between the highest valence band and the lowest conduction band at = ± 2  ⁄(3  ), where the bands cross the Fermi level. Thus, all armchair nanotubes are expected to exhibit metallic conduction, similar to the behavior of 2D graphene sheets. The energy bands for the Ch =(n,0) zig-zag nanotube E_q²(K) can be obtained likewise from Eq. 5.36 by writing the periodic boundary condition on k_y as:

to yield the 1D dispersion relations for the 4n states for the (n,0) zigzag nanotube (denoted by the superscript z),

Figure 35: Bandstructure of (a) armchair and (b) zigzag band structure and corresponding density of states. The band gap Eg of a semiconducting tube is inversely proportional to the diameter and equal to Eg=2E0/3, where E0=2ℏvF/d.

References

1. Physical Properties of Carbon Nanotubes, by R. Saito, M .S. Dresselhaus and G. Dresselhaus, Imperial College Press, London, 1998.
2. Graphene: Carbon in Two Dimensions, by M. I. Katnelson, Cambridge University Press, New York, 2012.