Electronic Properties and Applications of Single-Walled Carbon Nanotubes Nan Zheng Course: Solid State II Instructor: Elbio Dagotto Spring 2008 Department of Physics University of Tennessee Email: [email protected] Properties and Applications of Single-Walled Carbon Nanotubes Abstract: Carbon nanotubes have highly promising applications in future molecular electronics due to their unique electronic properties . This review begins with a brief introduction to experimental facts of structural and electronic properties of carbon nanotubes. The next section focuses on electronic structures of single walled carbon nanotube using the tight-binding model. Following that, applications of both semiconducting and metalic carbon nanotubes are presented. Finally the future developments of carbon nanotubes in both academic research industrial applicatons are discussed. 1. Introduction Carbon nanotubes are hollow cylinders composed of pure carbon with diameters of a few nanometers and lengths of many microns. A single-walled carbon nanotube (SWNT) may be conceived as a graphene, which is a single atomic layer of graphite, rolled into a seamless cylinder, whileMulti-walled carbon nanotubes (MWNT) consist of several layers of rolled graphite. Both SWNT and MWNT have similar mechanical and electronic properties. Because of the geometrical simpleness of the SWNT, this paper will mainly focuse on SWNT. Using modern microscopes like TEM and AFM, detailed structures of carbon nanotubes can be observed in experiments (see Fig. 1). The remarkable electronic properties of carbon nanotubes stem largely from the electronic structure of graphene, from which these nanotubes are derived. Graphene is a zero-gap semi-metal. While in most directions of the graphene sheet, energy gaps are not zero and electrons are not free to flow unless extra energy is provided, in certain special directions, energy gaps are zero and thus graphene displays the metallic property. In the band structure of graphene,conduction band and valance band contact each other at discrete points in k-space. However, when the graphene is rolled up into the nanotube, the direction along the axis of the nanotube is selected in the graphene sheet. Depending on geometric relation between the rolling direction and primitive vectors of the graphene sheet, the produced nanotube can be either metal or semiconductor. Since both metals and semiconductors can be made from the same all-carbon system, nanotubes are ideal candidates for molecular electronics technologies. The nanotube axis direction relative to the graphene is denoted by a pair of integers (n, m) [1]. Depending on the appearance of a belt of carbon bonds along the peripheral of the nanotube cross section, the nanotube is classified into either an armchair (n = m), or zigzag (n = 0 or m = 0, but not both zero), or chiral (any other n and m) structure. All armchair SWNTs are metals; those with n – m = 3k, where k is a nonzero integer, are semiconductors with a tiny band gap; and all others are semiconductors with a band gap that inversely depends on the nanotube diameter [1]. Figs. 1 (A),(B),(C) show the three different types of rolling-up pattern from a graphene sheet.Fig. 1: Schematic illustrations of the structures of (A) armchair, (B) zigzag, and( C) chiral SWNTs. Projections normal to the tube axis andperspective views along the tube axis are on the top and bottom, respectively. (D) Tunneling electron microscope image [2] showing the helical structure of a 1.3-nm-diameter chiral SWNT. (E) Transmission electron microscope ( TEM) image of a MWNT containing a concentrically nestedarray of nine SWNTs. (F) TEM micrograph [3] showing the lateral packing of 1.4-nm-diameter SWNTs in a bundle. (G) Scanning electron microscope (SEM) image of an array of MWNTs grown as a nanotube forest. [4] Fig. 2: (a) a1 and a2 are the lattice vectors of graphene. |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 graphene 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, in a manner similar to graphene. The fourth electron is a π orbital perpendicular to the cylindrical surface. [5]2. Electronic structures of carbon nanotubes Each carbon atom in the hexagonal lattice on a 2D graphene sheet possesses six valent electrons, among which are three 2sp2 electrons and one 2p electron. The three 22sp electrons form the three bonds in the plane of the graphene sheet, leaving an unsaturated π orbital [5] (Fig. 2(d)). These π orbitals, perpendicular to the graphene sheet and thus the nanotube surface when the graphene sheet is rolled, form a delocalized network on the surface of the nanotube, responsible for its electronic properties. First, it is beneficial to introduce the secular equation of tight-binding model, which is described below, ˆˆ()ii iHC E k SC=, (2.1) where ˆH and ˆSare called transfer integral matrices and overlap integral matrices respectively, which are defined by ˆ()jj j jHk H′′=Φ Φ, ()jj j jSk′′=Φ Φ (, 1, ,)jjn′= , (2.2) here jΦ denotes the atom wavefunction of position j which is interested in. A carbon atom at position sr has an unsaturated 2p orbital described by the wave function )(rsrχ. In the nearest neighbour approach of tight-binding model, the interaction between orbitals on different atoms vanishes unless the atoms are nearest neighbours. Mathematically, this can be written as ,0==BBAArrrrHHχχχχ ,arrrrrrBAABBAHH=−==γδχχχχ (2.3) here γ is the transfer integral constant. To calculate the electronic structure, the Bloch wavefunction for each of the sublattices is constructed as )()( rersssrrrkiksχφ∑⋅==312()()ik Rik R ik Reee fkγγ++ ≡iii (2.4) where s = A or B refers to each sublattice and rs refers to the set of points belonging to sublattice s, 1R, 2R,