thus similar to the situation for a graphene sheet for which the susceptibility per unit area is given by in which y = >/3"y0a0/2 and y0 is the nearest-neighbor tight binding overlap integral. For a graphene sheet, the magnetic susceptibility diverges (x -*■ oo) as T —>■ 0. Because of the large graphite interlayer spacing in comparison with the 2D lattice constant a0, the diamagnetic response of bulk graphite is dominated by that of the graphene layers. Since the interlayer coupling in bulk graphite is, however, nonvanishing, the susceptibility for 3D graphite is large, but not divergent, and is highly anisotropic, with small values of x for the magnetic field oriented in the plane and large values for the magnetic field parallel to the c-axis.

Since the diamagnetic susceptibility for H i. tubule axis is almost three orders of magnitude larger than that for H || tubule axis, the magnetic response of a carbon nanotubule is dominated by the field component perpendicular to its tube axis, except in the case that this axis is accurately aligned in the magnetic field direction. This difference in magnitude relates to the difference in magnetic flux L2H/4v for a field H parallel to the tubule axis and LaQH/*j2 for a perpendicular field. The factor 1/V2 arises from an angular average of the field component perpendicular to the circumference of the tubule. The susceptibility for tubules averaged over the magnetic flux as it changes by a flux quantum (see Fig. 19.37) is obtained by replacing EF in Eq. (19.35) by (0.2V3y0a0)(27r/L).

A more recent calculation has made use of the magnetic levels in a full tight binding formulation of the electronic structure [19.120]. Landau quantization of the 7r energy bands of a carbon nanotube is calculated within the tight binding approximation. The electronic energy bands in a magnetic field do not show explicit Landau levels, but they do have energy dispersion for all values of magnetic field. The energy bandwidth shows new oscillations (see Fig. 19.38) with a period that is scaled by a cross section of the unit cell of the tubule which is specified by the symmetry of the nanotube. The magnetic response of the electronic structure for such one-dimensional materials with a two-dimensional surface is especially interesting and should be relevant to recent magnetoresistance [19.121] and susceptibility [19.122-124] experiments.

In a two-dimensional cosine band with lattice constant a0, there is fractal behavior in the energy band spectra in a magnetic field H, depending on whether Hal/i>0 is a rational or irrational number, where <&0 = hc/e is a flux quantum [19.125], All energy dispersion relations are a periodic function of integer values of Hal/<t>(). However, the corresponding field is too large (~ 105 T) to observe this fractal behavior explicitly. In relatively weak fields 102 T), we observe only Landau levels, except for Landau subbands near E — 0 in the case of a 2D cosine band, since the wave functions near E = 0 are for extended orbits [19.126],

Ajiki and Ando [19.119] have shown an Aharonov-Bohm effect in carbon nanotubes for H || tubule axis and Landau level quantization for H J_ to the tubule axis. However, there is a limitation for using k p perturbation theory around the degenerate point K at the corner of the hexagonal Brillouin zone in the case of a carbon nanotube, since the k values taken in the direction of the tubule circumference can be far from the K point, leading to incorrect values of the energy, especially for small carbon nanotubes and for energies far from that at the K point [19.120].

The magnetic energy band structure for carbon nanotubes within the tight binding approximation is shown in Fig. 19.38, in which it is assumed that the atomic wave function is localized at a carbon site and the magnetic field varies sufficiently slowly over a length scale equal to the lattice constant. This condition is valid for any large magnetic field that is currently available, even if we adopt the smallest-diameter carbon nanotube observed

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