##### From Nanoscale Transistors: Device Physics, Modeling and Simulation

## 5.4 Bandstructure of Carbon Nanotubes

### 5.4.1 Bandstructure of Graphene

A simple way to calculate the *E* vs. *k* relation of a carbon nanotube, is to quantize the two-dimensional *E* vs. *k* of the graphene sheet along the circumferential direction of the nanotube. The first step, therefore, is to understand the band structure of a graphene sheet.

The graphene lattice is shown in Fig. 5.9a. Note that each carbon atom does not see the same environment, but each unit cell of two carbon atoms does. The two-dimensional graphene lattice can be created by translating a unit cell by the vectors with integer combinations of ( *n*, *m*). Here and are the basis vectors of the lattice (also shown in Fig. 5.9),

(5.28a) | |

(5.28b) | |

where is the length of the basis vector, and *a _{CC}* ? 1.42

*is the nearest neighbor C-C bonding distance.*

Figure 5.9: (a) The graphene lattice in real space with the basis vectors and . (b) The first Brillouin zone of the reciprocal lattice with the basis vectors and .

The 2 *s*, 2 *p _{x}*, and 2

*p*orbitals give rise to the so-called

_{y}*?*bonds of graphene. The ? bonds are only weakly coupled to the 2

*p*orbitals, which produce the so-called ? bonds, which give rise to the electronic properties of graphene. The

_{z}*E*(

*k*) relation for graphene...

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