Structure

Carbon nanotubes (CNT) and fullerenes are large molecules constructed entirely of carbons. Single-walled carbon nanotubes (SWNT) can be viewed as a strip cut from an infinite graphene sheet rolled up into a tube (see Figure 15.1). The diameter and helicity of a SWNTare uniquely defined by the roll-up vector Ck = na1 + ma2 that connects crystallographically equivalent sites on the graphene lattice, where a1 and a2 are the graphene lattice vectors and n and m are integers. Translation vector T is along the tube axis and, thus, orthogonal to Ck. In terms of such definitions, integers n and m characterize the rolling directions, chirality and diameter d = |Ck|/π of a particular carbon nanotube, therefore, SWNTs are usually defined by these two numbers as (n, m).

Electronic properties in 1D systems

Translation symmetry is the main feature of solid states, which permits classification of the wavefunctions of any electronic states. According to the so-called Bloch theorem, the wavefunctions of a periodic system ψ(r) are given as products of a periodic function unk(r) and the exponential phase function exp(ikr), ψnk(r) = exp(ikr)unk(r). The quantum number n is a property of the unit cell. The wavenumber, k, is the main quantum number of the periodic systems, which satisfies the translation symmetry (Peierls, 1995). Thus, the state corresponding to any k-number is the stationary state with the energy eigenvalue E(k).