Magnetic Properties

The magnetic properties of carbon nanotubes can be effectively monitored by the Electron Spin Resonance (ESR) technique which has been intensively used to study the electronic properties of graphitic and conjugated materials [133]. This method has several advantages: it has a very high sensitivity, it responds only to the paramagnetic signal (unless static measurements are made, which are usually dominated by the large diamagnetic response), and it can distinguish between different spin species, e.g., localized and conduction electron spins. Three different quantities are determined by ESR: (1) the g-factor, which depends on the chemical environment of the spins via spin-orbit coupling (and also the hyperfine interaction); (2) the linewidth, which is governed by the spin relaxation mechanism; and (3) the intensity of the signal, which is proportional to the static susceptibility.

For MWNTs we are mainly interested in the ESR response of the conduction electrons. The conduction electron spin g-factor is determined by the spin-orbit splitting of the energy levels in the presence of a magnetic field [134,135]. In the case of degenerate bands (as in graphite at the K point), theory predicts that spin-orbit coupling, which removes the degeneracy, induces a large g-shift which varies inversely with temperature. This is exactly what is observed in graphite when the magnetic field is perpendicular to the plane, allowing large orbital currents, thereby increasing the spin-orbit coupling. When the magnetic field is parallel to the planes, orbital currents are suppressed, and a small g-shift, nearly independent of temperature, is observed. When the Fermi level is shifted away from the K point by doping, the K point degeneracy and the g-shift anisotropy disappear. (Despite this understanding of the variation of the g-shift, a rigorous theory is still missing in graphite due to the complicated band structure near the point of degeneracy.)

Using this qualitative description for graphite, one can speculate about the g-factor for carbon nanotubes. When the magnetic field is parallel to the tube axis, nearly the same value is obtained for the g-factor as is found in graphite when the field is in the plane, except at a field for which the cyclotron radius equals the geometrical radius of the nanotubes. Such fields (typically 1 T and greater) are higher than those used in X-band spectrometers (-9 GHz).

When the magnetic field is perpendicular to the tube axis, orbital currents cannot completely close, as in a plane, and we can therefore expect a smaller g-shift than in graphite. A decrease of this g-shift with decreasing tube diameter is then expected. Some of these predictions are indeed realized in MWNTs [136]. The average observed g-value in MWNTs is 2.012, as compared to 2.018 in graphite, and the g-factor anisotropy is lower in MWNTs than in graphite.

Spin relaxation in metals and semimetals depends also on the spin-orbit coupling. More precisely it is the modulation of the spin-orbit coupling by lattice vibrations that causes spin relaxation. It can be shown, that in the framework of the Elliott theory [135], and when the spin relaxation time is proportional to the momentum relaxation time, and is governed by phonon scattering, the linewidth increases with increasing temperature. Opposite to this expectation, in graphite the linewidth increases when the temperature decreases. This behavior in graphite is attributed to motional narrowing over the g value distribution. Because of the semimetallic nature of graphite, the density of states at the Fermi level is low, and hence the spin susceptibility is low, and is in the 10~8 emu/g range at room temperature [133]. We have seen that the g-factor in MWNTs, a local property, is very close to that of graphite. In the following subsections we will see how the spin relaxation and spin susceptibility are modified in MWNTs relative to graphite. Changing the dimensionality from 2-D to 1-D is expected to significantly modify the principal ESR characteristics. First, we have to understand what kind of ESR signal should be expected in a 1-D system.

4.1 Spin Relaxation in Quasi-1-D Systems

Extensive studies of the Conduction Electron Spin Resonance (CESR) line-width (A H) in isotropic metals have shown that the dominant process in the spin-lattice relaxation time (Ti) is the spin-flip scattering of conduction electrons by acoustic phonons. The same scattering process gives the momentum relaxation time tr measured by the electrical resistivity. Elliott has derived [134,135] a relation between the two relaxation times:

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