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necessary for the growth of the mechanism must involve the role of the Co or atoms. One proposal referred to as the “scootter mechanism” suggests that atoms of the metal catalyst attach to the dangling bonds at the open end of the tubes, and that these atoms scoot around the rim of the tube, absorbing carbon atoms as they arrive.

Generally when nanotubes are synthesized, the result is a mix of different kinds, some metallic and some semiconducting. A group at IBM has developed a method to separate the semiconducting the metallic nanotubes. The separation was accomplished by depositing bundles of nanotubes, some of which are metallic and some semiconducting, on a silicon wafer. Metal electrodes were then deposited over the bundle. Using the silicon wafer as an electrode, a small bias voltage was applied that prevents the semiconducting tubes from conducting, effectively making them insulators. A high voltage is then applied across the metal electrodes, thereby sending a high current through the metallic tubes but not the insulating tubes. This causes the metallic tubes to vaporize, leaving behind only the semiconducting tubes.

5.4.2. Structure

There are a variety of structures of carbon nanotubes, and these various structures have different properties. Although carbon nanotubes are not actually made by rolling graphite sheets, it is possible to explain the different structures by considera- tion of the way graphite sheets might be rolled into tubes. A nanotube can be formed when a graphite sheet is rolled up about the axis T shown in Fig. 5.14. The vector is called the circumferential vector, and it is at right angles to Three examples of nanotube structures constructed by rolling the graphite sheet about the vector

Figure 5.14. Graphitic sheet showing the basis vectors and of the two-dimensional unit cell, the axis vector about which the sheet is rolled to generate the armchair structure tube sketched in Fig. 5.1 la, and the circumferential vector at right angles to Other orienta- tions of T o n the sheet generate the zigzag and chiral structures of Figs. and respectively.

having different orientations in the graphite sheet are shown in Fig. 5.1 When Tis parallel to the C-C bonds of the carbon hexagons, the structure shown in Fig. 5.1 l a is obtained, and it is referred to as the "armchair" structure. The tubes sketched in Figs. 5.1 l b and 5.1 referred to respectively as the zigzag and the chiral structures,

are formed by rolling about a T vector having different orientations in the graphite plane, but not parallel to C-C bonds. Looking down the tube of the chiral structure, one would see a spiraling row of carbon atoms. Generally nanotubes are closed at both ends, which involves the introduction of a pentagonal topological arrangement on each end of the cylinder. The tubes are essentially cylinders with each end attached to half of a large structure. In the case of metal particles are found at the ends of the tubes, which is evidence for the catalytic role of the metal particles in their formation.

5.4.3. Electrical Properties

Carbon nanotubes have the most interesting property that they are metallic or semiconducting, on the diameter and chirality of the tube. Chirality refers to how the tubes are rolled with respect to the direction of the T vector in the graphite plane, as discussed above. Synthesis generally results in a mixture of tubes thirds of which are semiconducting and one-third metallic. The metallic tubes have the armchair structure shown in Fig. 5.1 la. Figure 5.15 is a plot of the energy gap of semiconducting chiral carbon nanotubes versus the reciprocal of the diameter, showing that as the diameter of the tube increases, the decreases. Scanning tunneling microscopy (STM), which is described in Chapter 3, has been used to

Figure 5.15. Plot of the magnitude of the energy band gap of semiconducting, chiral carbon nanotube versus the reciprocal of the diameter of the tube 1 nm). [Adapted from M. S. Dresselhaus et al., Mater. 4, 27

5.4. CARBON NANOTUBES

investigate the electronic structure of carbon nanotubes. In this measurement the position of the STM tip is fixed above the nanotube, and the voltage V between the tip and the sample is swept while the tunneling current Z is monitored. The measured conductance G = Vis a direct measure of the local electronic density of states. The density of states, discussed in more detail in Chapter 2, is a measure of how close together the energy levels are to each other. Figure 5.16 gives the STM data plotted as the differential conductance, which is versus the applied voltage between the tip and carbon nanotube. The data show clearly the energy gap in materials at voltages where very little current is observed. The voltage width of this region measures the gap, which for the semiconducting material shown on the bottom of Fig. 5.16 is 0.7

I I I I

VOLTAGE (V)

W

VOLTAGE (V)

Figure 5.16. Plot of differential conductance V ) obtained from scanning tunneling microscope measurements of the tunneling current of metallic (top figure) and semiconducting (bottom figure) nanotubes. [With permission from C. Dekker, Today 22 (May

At higher energies sharp peaks are observed in the density of states, referred to as van Hove singularities, and are characteristic of low-dimensional conducting materials. The peaks occur at the bottom and top of a number of subbands. As we have discussed earlier, electrons in the quantum theory can be viewed as waves. If the electron wavelength is not a multiple of the circumference of the tube, it will destructively interfere with itself, and therefore only electron wavelengths that are integer multiples of the circumference of the tubes are allowed. This severely limits the number of energy states available for conduction around the cylinder. The dominant remaining conduction path is along the axis of the tubes, making carbon nanotubes function as one-dimensional quantum wires. A more detailed discussion of quantum wires is presented later, in Chapter 9. The electronic states of the tubes do not form a single wide electronic energy band, but instead split into dimensional subbands that are evident in the data in Fig. 5.16. As we will see later, these states can be modeled by a potential well having a depth equal to the length of the nanotube.

Electron transport has been measured on individual single-walled carbon tubes. The measurements at a millikelvin K) on a single metallic nanotube lying across two metal electrodes show features in the

voltage measurements, as seen in Fig. 5.17. The steps occur at voltages which depend on the voltage applied to a third electrode that is electrostatically coupled to the nanotube. This resembles a field effect transistor made from a carbon nanotube, which is discussed below and illustrated in Fig. 5.2 1. The step like features in the curve are due to single-electron tunneling and resonant tunneling through single molecular orbitals. Single electron tunneling occurs when the capacitance of the nanotube is so small that adding a single electron requires an electrostatic charging

Figure 5.17. Plot of electron transport for two different gate voltages through a single metallic carbon nanotube showing steps in the curves. [With permission from C. Dekker,

5.4. CARBON NANOTUBES