The Concept of Cutting Lines in Carbon Nanotube Science

Abstract

A review is presented of one-dimensional cutting lines that are utilized to obtain the physical properties of carbon nanotubes from the corresponding properties of graphite by the zone-folding scheme. Quantization effects in general low-dimensional systems are brie y discussed, followed by a more detailed consideration of one-dimensional single-wall carbon nanotubes. The geometrical structure of the nanotube is described, from which quantum con ned states are constructed. These allowed states in the momentum space of graphite are known as cutting lines. Different representations of the cutting lines in momentum space are introduced. Electronic and phonon dispersion relations for nanotubes are derived by using cutting lines and the zone-folding scheme. The relation between cutting lines and singularities in the electronic density of states is considered. The selection rules for carbon nanotubes are shown to be directly connected with the cutting lines. Different experimental techniques are considered that con rm the validity of cutting lines and the zone-folding approach. JOURNAL OF NANOSCIENCE AND NANOTECHNOLOGY tum wells and quantum wires. We present the generic concept of cutting lines, which is here developed in detail for carbon nanotubes, as a prototype example of one-dimensional systems. Cutting lines are de ned as zone-folded one-dimensional (1D) Brillouin zone (BZ) line segments, expressed in relation to their two-dimensional (2D) BZ counterparts. Electronic and phonon dispersion relations and other properties of carbon nanotubes are discussed in terms of these cutting lines, which provide valuable physical insights. Experimental evidence for the validity of the cutting-line approach is also reviewed. Scope of This ReviewCarbon nanotubes play a fundamental role in the rapidly developing eld of nanoscience and nanotechnology because of their unique properties and high potential for applications. 1, 2 The use of carbon nanotubes in nanotechnology applications requires an understanding of their electronic band structure, vibrational spectra, and other relevant characteristics. All of these properties can be derived from the corresponding properties of the graphene layer by use of the zone-folding scheme, 2 though further 431

Thermodynamic properties of nanotubes: zone-folding approach

“…These N line segments defined by the wave vectors K 1 and K 2 represent the cutting lines in the unrolled reciprocal space of the nanotube [2]. The length and orientation of each cutting line in the reciprocal space are given by the wave vector K 2 , and the separation between two adjacent cutting lines is given by the wave vector K 1 .…”

“…The length and orientation of each cutting line in the reciprocal space are given by the wave vector K 2 , and the separation between two adjacent cutting lines is given by the wave vector K 1 . This representation of cutting lines is called [2] the fully K 1 -extended representation. In Fig.…”

“…The first one is formed by the three smallest translation vectors (up to sign) of the length a (a 1 , -a 2 , -a 1 + a 2 ). Using the orthogonality conditions [2,24,27] in the case of the hexagonal lattice with the angle γ = 60° between the primitive translation vectors…”

“…In some sense this procedure is equivalent to imposing of cyclic boundary conditions on the layer supercell in the direction which becomes the tube circumference. This circumstance is utilized in the so-called zone-folding (ZF) approach [2][3] by means of cutting the electron or phonon dispersion in the 2D reciprocal space of the layer used for NT rolling up. In such a way the properties of single-wall nanotubes (SWNTs) with sufficiently large diameters can be estimated from the results of the corresponding layer calculation.…”

“…In such a way the properties of single-wall nanotubes (SWNTs) with sufficiently large diameters can be estimated from the results of the corresponding layer calculation. The ZF approximation was widely used for the electron and phonon state calculations of the single-wall carbon nanotubes (SWCNTs) rolled up from the graphene layer [2][3][4][5][6][7]. Few zone-folding calculations are also known for the hexagonal boron nitride [8,9].…”