Electronic states and transport in carbon nanotubes

Tsuneya Ando ando@issp.u-tokyo.ac.jp    Institute for Solid State Physics, University of Tokyo 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8581, Japan

1 Introduction

Graphite needles called carbon nanotubes (CNs) were discovered recently [1,2] and have been a subject of an extensive study. A CN is a few concentric tubes of two-dimensional (2D) graphite consisting of carbon-atom hexagons arranged in a helical fashion about the axis. The diameter of CNs is usually between 20 and 300 Å and their length can exceed 1 μm. The distance of adjacent sheets or walls is larger than the distance between nearest neighbor atoms in a graphite sheet and therefore electronic properties of CNs are dominated by those of a single layer CN. Single-wall nanotubes are produced in a form of ropes [3,4]. The purpose of this article is to give a brief review of recent theoretical study on electronic and transport properties of carbon nanotubes.

Figure 1 shows a transmission micrograph image of multi-wall nanotubes and Fig. 2 a computer graphic image of a single-wall nanotube. Carbon nanotubes can be either a metal or semiconductor, depending on their diameters and helical arrangement. The condition whether a CN is metallic or semiconducting can be obtained based on the band structure of a 2D graphite sheet and periodic boundary conditions along the circumference direction. This result was first predicted by means of a tight-binding model ignoring the effect of the tube curvature [5–14].

These properties can be well reproduced in a k·p method or an effective-mass approximation [15]. In fact, the effective-mass scheme has been used successfully in the study of wide varieties of electronic properties of CN. Some of such examples are magnetic properties [16] including the Aharonov-Bohm effect on the band gap [15], optical absorption spectra [17,18], exciton effects [19], lattice instabilities in the absence [20] and presence of a magnetic field [21,22], and magnetic properties of ensembles of nanotubes [23].

Transport properties of CNs are interesting because of their unique topological structure. There have been some reports on experimental study of transport in CN bundles [24] and ropes [25,26]. Transport measurements became possible for a single multi-wall nanotube [27–31] and a single single-wall nanotube [32–36]. Single-wall nanotubes usually exhibit large charging effects presumably due to nonideal contacts [37–41].

In this article we shall mainly discuss electronic states and transport properties of nanotubes obtained theoretically in the k·p method combined with a tight-binding model. It is worth mentioning that several papers giving general reviews of electronic properties of nanotubes were published already [42–47].

In Sect. 2, electronic states are discussed first in a nearest-neighbor tight-binding model. Then, the effective mass equation is introduced and the band structure is discussed with a special emphasis on Aharonov-Bohm effects and formation of Landau levels in magnetic fields. In Sect. 3, optical absorption is discussed in the effective-mass scheme and the nanotube is shown to behave differently in light polarization parallel or perpendicular to the axis. The importance of exciton effects is emphasized and some related experiments are discussed.

In Sect. 4, effects of impurity scattering are discussed and the total absence of backward scattering is pointed out except for scatterers with a potential range smaller than the lattice constant. Further, the conductance quantization in the presence of lattice vacancies, i.e., strong and short-range scatterers, is also discussed. In Sect. 5, the transport across a junction of nanotubes with different diameters through a pair of topological defects such as five- and seven-member rings is discussed. In Sect. 6, a continuum model for phonons in the long-wavelength limit is introduced and effective Hamiltonian describing electron-phonon interaction is derived. A short summary is given in Sect. 7.

2 Electronic states

2.1 Two-dimensional graphite

The structure of 2D graphite sheet is shown in Fig. 3. We have the primitive translation vectors a = a(1, 0) and =a(−(1/2),3/2), and the vectors connecting between nearest neighbor carbon atoms →1=a(0,1/3), →2=a(−1/2,−1/23), and →3=a(1/2,−1/23). Note that a·b = – a2/2.

The primitive reciprocal lattice vectors a* and b* are given by *=(2π/a)(1,1/3) and *=(2π/a)(0,2/3). The K and K’ points are given as =(2π/a)(1/3,1/3) and K′ = (2π/a)(2/3, 0), respectively. We then have the relations, (iK⋅τ→1)=ω, (iK⋅τ→2)=ω−1, (iK⋅τ→3)=1, (iK′⋅τ→1)=1, (iK′⋅τ→2)=ω−1, and (iK′⋅τ→3)=ω, with ω = exp(2πi/3).

In a tight-binding model, the wave function is written...