## Geometric and Electronic Structure of Carbon Nanotubes

In this section, we give an introduction to the structure and electronic properties of the single-walled carbon nanotubes (SWNTs). Shortly after the discovery of the carbon nanotubes in the soot of fullerene synthesis, single-walled carbon nanotubes were synthesized in abundance using arc discharge methods with transition metal catalysts [5,6,7]. These tubes have quite small and uniform diameter, on the order of one nanometer. Crystalline ropes of singlewalled nanotubes with each rope containing tens to hundreds of tubes of similar diameter closely packed have also been synthesized using a laser vaporization method [8] and other techniques, such as arc- discharge and CVD techniques. These developments have provided ample amounts of sufficiently characterized samples for the study of the fundamental properties of the SWNTs. As illustrated in Fig. 1, a single-walled carbon nanotube is geometrically just a rolled up graphene strip. Its structure can be specified or indexed by its circumferential periodicity [37]. In this way, a SWNT's geometry is completely specified by a pair of integers (n, m) denoting the relative position c = nai + ma2 of the pair of atoms on a graphene strip which, when rolled onto each other, form a tube.

Theoretical calculations [16,39,40,41] have shown early on that the electronic properties of the carbon nanotubes are very sensitive to their geometric structure. Although graphene is a zero-gap semiconductor, theory has predicted that the carbon nanotubes can be metals or semiconductors with different size energy gaps, depending very sensitively on the diameter and helicity of the tubes, i.e., on the indices (n, m). As seen below, the intimate connection between the electronic and geometric structure of the carbon nanotubes gives rise to many of the fascinating properties of various nanotube structures, in particular nanotube junctions.

The physics behind this sensitivity of the electronic properties of carbon nanotubes to their structure can be understood within a band-folding picture. It is due to the unique band structure of a graphene sheet, which has states crossing the Fermi level at only 2 inequivalent points in fc-space, and to the quantization of the electron wavevector along the circumferential direction. An isolated sheet of graphite is a zero-gap semiconductor whose electronic structure near the Fermi energy is given by an occupied n band and an empty n* band. These two bands have linear dispersion and, as shown in Fig. 2, meet at the Fermi level at the K point in the Brillouin zone. The Fermi surface of an ideal graphite sheet consists of the six corner K points. When forming a tube, owing to the periodic boundary conditions imposed in the circumferential direction, only a certain set of k states of the planar graphite sheet is allowed. The allowed set of fc's, indicated by the lines in Fig. 2, depends on the diameter and helicity of the tube. Whenever the allowed fc's include the point K, the system is a metal with a nonzero density of states at the Fermi level, resulting in a one-dimensional metal with 2 linear dispersing

MT K

Fig. 2. (Top) Tight-binding band structure of graphene (a single basal plane of graphite). (Bottom) Allowed fc-vectors of the (7,1) and (8,0) tubes (solid lines) mapped onto the graphite Brillouin zone

Fig. 2. (Top) Tight-binding band structure of graphene (a single basal plane of graphite). (Bottom) Allowed fc-vectors of the (7,1) and (8,0) tubes (solid lines) mapped onto the graphite Brillouin zone bands. When the point K is not included, the system is a semiconductor with different size energy gaps. It is important to note that the states near the Fermi energy in both the metallic and the semiconducting tubes are all from states near the K point, and hence their transport and other properties are related to the properties of the states on the allowed lines. For example, the conduction band and valence bands of a semiconducting tube come from states along the line closest to the K point.

The general rules for the metallicity of the single-walled carbon nanotubes are as follows: (n, n) tubes are metals; (n, m) tubes with n—m = 3j, where j is a nonzero integer, are very tiny-gap semiconductors; and all others are large-gap semiconductors. Strictly within the band-folding scheme, the n — m = 3j tubes would all be metals, but because of tube curvature effects, a tiny gap opens for the case where j is nonzero. Hence, carbon nanotubes come in three varieties: large-gap, tiny-gap, and zero-gap. The (n, n) tubes, also known as armchair tubes, are always metallic within the single-electron picture, being independent of curvature because of their symmetry. As the tube radius R increases, the band gaps of the large-gap and tiny-gap varieties decreases with a 1/R and 1/R2 dependence, respectively. Thus, for most experimentally observed carbon nanotube sizes, the gap in the tiny-gap variety which arises from curvature effects would be so small that, for most practical purposes, all the n — m = 3j tubes can be considered as metallic at room temperature. Thus, in Fig. 2, a (7,1) tube would be metallic, whereas a (8,0) tube would be semiconducting.

This band-folding picture, which was first verified by tight-binding calculations [38,39,40], is expected to be valid for larger diameter tubes. However, for a small radius tube, because of its curvature, strong rehybridization among the a and n states can modify the electronic structure. Experimentally, nanotubes with a radius as small as 3.5 A have been produced. Ab initio pseudopotential local density functional (LDA) calculations [41] indeed revealed that sufficiently strong hybridization effects can occur in small radius nanotubes which significantly alter their electronic structure. Strongly modified low-lying conduction band states are introduced into the band gap of insulating tubes because of hybridization of the a* and n* states. As a result, the energy gaps of some small radius tubes are decreased by more than 50%. For example, the (6,0) tube which is predicted to be semiconducting in the band-folding scheme is shown to be metallic. For nanotubes with diameters greater than 1 nm, these rehybridization effects are unimportant. Strong a-n rehybridization can also be induced by bending a nanotube [42].

Energetically, ab initio total energy calculations have shown that carbon nanotubes are stable down to very small diameters. Figure 3 depicts the calculated strain energy per atom for different carbon nanotubes of various diameters [41]. The strain energy scales nearly perfectly as d-2 where d is the tube diameter (solid curve in Fig. 3), as would be the case for rolling a classical elastic sheet. Thus, for the structural energy of the carbon nano-tubes, the elasticity picture holds down to a subnanometer scale. The elastic constant may be determined from the total energy calculations. This result has been used to analyze collapsed tubes [11] and other structural properties of nanotubes. Also shown in Fig. 3 is the energy/atom for a (6,0) carbon strip. It has an energy which is well above that of a (6,0) tube because of the dangling bonds on the strip edges. Because in general the energy per atom of

a strip scales as d-1, the calculation predicts that carbon nanotubes will be stable with respect to the formation of strips down to below 4 A in diameter, in agreement with classical force-field calculations [43].

There have been many experimental studies on carbon nanotubes in an attempt to understand their electronic properties. The transport experiments [19,20,45,46,47] involved both two- and four-probe measurements on a number of different tubes, including multiwalled tubes, bundles of single-walled tubes, and individual single-walled tubes. Measurements showed that there are a variety of resistivity behaviors for the different tubes, consistent with the above theoretical picture of having both semiconducting and metallic tubes. In particular, at low temperature, individual metallic tubes or small ropes of metallic tubes act like quantum wires [19,20]. That is, the conduction appears to occur through well-separated discrete electron states that are quantum-mechanically coherent over distances exceeding many hundreds of nanometers. At sufficiently low temperature, the system behaves like an elongated quantum dot.

Figure 4 depicts the experimental set up for such a low temperature transport measurement on a single-walled nanotube rope from [20]. At a few degrees Kelvin, the low-bias conductance of the system is suppressed for voltages less than a few millivolts, and there are dramatic peaks in the conductance as a function of gate voltage that modulates the number of electrons in the rope (Fig. 5). These results have been interpreted in terms of single-electron charging and resonant tunneling through the quantized energy levels of the nanotubes. The data are explained quite well using the band structure of the conducting electrons of a metallic tube, but these electrons are confined to a small region defined either by the contacts or by the sample length, thus leading to the observed quantum confinement effects of Coulomb blockade and resonant tunneling.

There have also been high resolution low temperature Scanning Tunneling Microscopy (STM) studies, which directly probe the relationship between the structural and electronic properties of the carbon nanotubes [48,49]. Figure 6 is a STM image for a single carbon nanotube at 77 K on the surface of a rope. In these measurements, the resolution of the measurements allowed for the identification of the individual carbon rings. From the orientation of the carbon rings and the diameter of the tube, the geometric structure of the tube depicted in Fig. 6 was deduced to be that of a (11,2) tube. Measurement of the normalized conductance in the Scanning Tunneling Spectroscopy (STS) mode was then used to obtain the Local Density Of States (LDOS). Data on the (11,2) and the (12,3) nanotubes gave a constant density of states at the Fermi level, showing that they are metals as predicted by theory. On another sample, a (14, —3) tube was studied. Since 14 + 3 is not equal to 3 times an integer, it ought to be a semiconductor. Indeed, the STS measurement gives a band gap of 0.75 eV, in very good agreement with calculations.

The electronic states of the carbon nanotubes, being band-folded states of graphene, lead to other interesting consequences, including a striking geometry dependence of the electric polarizability. Figure 7 presents some results from a tight-binding calculation for the static polarizabilities of carbon nanotubes in a uniform applied electric field [50]. Results for 17 single-walled tubes of varying size and chirality, and hence varying band gaps, are given. The unscreened polarizability a0 is calculated within the random phase approximation. The cylindrical symmetry of the tubes allows the polarizability

Fig. 6. STM images at 77 K of a single-walled carbon nanotube at the surface of a rope [49]

tensor to be divided into components perpendicular to the tube axis, a0±_, and a component parallel to the tube axis, a0y. Values for a0± predicted within this model are found to be totally independent of the band gap Eg and to scale linearly as R2, where R is the tube radius. The latter dependence may be understood from classical arguments, but the former is rather unexpected. The insensitivity of a0± to Eg results from selection rules in the dipole matrix elements between the highest occupied and the lowest unoccupied states of these tubes. On the other hand, Fig. 7 shows that a0|| is proportional to R/Eg, which is consistent with the static dielectric response of standard insulators. Also, using arguments analogous to those for C60 [51,52], local field effects relevant to the screened polarizability tensor a may be included classically, resulting in a saturation of a± for large a0^, but leaving a| unaffected. Thus, in general, the polarizability tensor of a carbon nanotube is expected to be highly anisotropic with a| ^ a^. And the polarizability of small gap tubes is expected to be greatly enhanced among tubes of similar radii.

Just as in electronic states, the phonon states in carbon nanotubes are quantized into phonon subbands. This has led to a number of interesting phenomena [2,3] which are discussed elsewhere in this volume [53]. Here we mention several of them. It has been shown that twisting motions of a tube can lead to the opening up of a minuscule gap at the Fermi level, leading to the possibility of strong coupling between the electronic states and the twisting modes or twistons [54]. The heat capacity of the nanotubes is also expected to show a dimensionality dependence. Analysis [55] shows that the phonon contributions dominate the heat capacity, with single-walled carbon nano-tubes having a Cph ~ T dependence at low temperature. The temperature below which this should be observable decreases with increasing nanotube radius R, but the linear T dependence should be accessible to experimental investigations with presently available samples. In particular, a tube with a 100 A radius should have Cph ~ T for T < 7K. Since bulk graphite has Cph ~ T2-3, a sample of sufficiently small radius tubes should show a deviation from graphitic behavior. Multi-walled tubes, on the other hand, are expected to show a range of behavior intermediate between Cph ~ T and Cph ~ T2-3, depending in detail on the tube radii and the number of concentric walls.

In addition to their fascinating electronic properties, carbon nanotubes are found to have exceptional mechanical properties [56]. Both theoretical [57,58,59,60,61,62,63] and experimental [64,65] studies have demonstrated that they are the strongest known fibers. Carbon nanotubes are expected to be extremely strong along their axes because of the strength of the carboncarbon bonds. Indeed, the Young's modulus of carbon nanotubes has been predicted and measured to be more than an order of magnitude higher than that of steel and several times that of common commercial carbon fibers. Similarly, BN nanotubes are shown [66] to be the world's strongest large-gap insulating fiber.

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