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Fig. 19.33. Self-consistent E(k) band structure (48 valence and 5 conduction bands) for the hexagonal II [P6/mcc (D^)] arrangement of single-wall carbon nanotubes, calculated along different high-symmetry directions in the Brillouin zone. The Fermi level at Er — 0 is positioned at the degeneracy point appearing between K and H, indicating metallic behavior for this tubule array [19.11].

a metallic inner tubule covered by a semiconducting (or insulating) outer tubule. This concept could be extended to the design of tubular metal-semiconductor devices without introducing any doping impurities [19.6]. From a conceptual standpoint, there are many possibilities for arranging arrays of metallic and semiconducting carbon nonotubes.

Modification of the conduction properties of semiconducting carbon nanotubes by B (p-type) and N (n-type) substitutional doping has also been discussed [19.116]. Modifications to the electronic structure by filling the capillaries of the tubes (see §19.9) have also been proposed [19.9]. Exohedral doping of the space between nanotubes in a tubule bundle could provide yet another mechanism for doping the tubules.

No superconductivity has yet been found in carbon nanotubes or nano-tube arrays (see §19.11). Despite the prediction that ID electronic systems cannot support superconductivity [19.117,118], it is not clear that such theories are applicable to carbon nanotubes, which are tubular with a hollow core and have only a few unit cells around the circumference. It may perhaps be possible to align an ensemble of similar tubules to form a 3D solid, consisting of ID constituents. In such a case we would expect the possibility of superconductivity to be highly sensitive to the density of states at the Fermi level. The appropriate doping of tubules may provide another possible approach to superconductivity for carbon nanotube systems.

19.5.4. ID Electronic Structure in a Magnetic Field

The electronic structure of single-wall carbon nanotubes in a magnetic field has attracted considerable attention because of the intrinsic interest of the phenomena and the ability to make magnetic susceptibility and magnetoresistance measurements on bundles of carbon nanotubes. There are two cases to be considered, each having a different physical meaning, since the specific results for the magnetization and the magnetic susceptibility depend strongly on whether the magnetic field is oriented parallel or perpendicular to the tubule axis [19.119,120]. Calculations for the magnetization and susceptibility have been carried out for these two field orientations using two different approaches: k • p perturbation theory and a tight binding formulation [19.119,120]. In the case of the magnetic field parallel to the tubule axis, no magnetic flux penetrates the cylindrical graphene plane of the tubule, so that only the Aharonov-Bohm (AB) effect occurs.

Because of the periodic boundary conditions around a tubule, it is expected that the electronic states in the presence of a magnetic field parallel to the tubule axis will show a periodicity in the wave function

where </> = <i>/i>() defines a phase factor, i> is the magnetic flux passing through the cross section of the carbon nanotube, and 4>0 is the magnetic flux quantum 4>0 = ch/e [19.119], Thus, as the flux $ is increased, Eq. (19.34) predicts a periodicity in magnetic properties associated with the phase factor exp(2iri(j>). This periodic behavior as a function of magnetic flux is expected for all types of tubules (metallic and semiconducting), independent of tubule diameter d, and chiral angle 0, as shown in Fig. 19.34. At small magnetic fields, Fig. 19.34 shows that the magnetic moment is oriented along the direction of magnetic flux for metallic tubules (solid curve) and is opposite to the flux direction for the case of the semiconducting tubules. In the Aharonov-Bohm effect, the magnetic field changes the periodic boundary conditions which describe how to cut the 2D energy bands of the graphene layer and generate the ID bands for the tubules [19.119]. When the magnetic field increases, the cutting line for the 2D bands is shifted in a direction perpendicular to the tubule axis. As a result, oscillations in the metallic energy bands periodically open an energy gap as a function of increasing magnetic field. Likewise, the energy band gap for the semiconducting tubules oscillates, thus closing the gap periodically [19.119], For the magnetic field parallel to the tubule axis and (f> = 1/6, calculation of the magnetic moment and magnetic susceptibility for the metallic tubules as a function of tubule circumference (Fig. 19.35) shows almost no dependence on circumference, for tubule diameters d, greater than about

Fig. 19.34. The magnetic moment for h || tubule axis calculated in a tight binding model vs. magnetic flux for a metallic (solid line, v = 0) and semiconducting (dotted line, v = ±1) carbon nanotube of diameter 16.5 nm. Here L is the circumference of the nanotube and a0 = 0.246 nm [19.119],

Magnetic Flux (units of ch/el

Magnetic Flux (units of ch/el

Fig. 19.34. The magnetic moment for h || tubule axis calculated in a tight binding model vs. magnetic flux for a metallic (solid line, v = 0) and semiconducting (dotted line, v = ±1) carbon nanotube of diameter 16.5 nm. Here L is the circumference of the nanotube and a0 = 0.246 nm [19.119], o <

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