Depth nm

Fig. 16. Conductance G at room temperature measured in the apparatus shown in Fig. 12 as a function of depth of immersion of the nanotube fiber into liquid gallium. As the nanotube fiber is dipped into the liquid metal, the conductance increases in steps of Go = 2e2/h. The steps correspond to different nanotubes coming successively into contact with the liquid [104]

small. The latter is in contrast to recently published length-dependent intrinsic resistances of 4kQ/^,m [44] and & 10kQ/^,m [97], respectively.

To support ballistic transport over micrometer distances, Frank et al. came up with another interesting experimental observation. Large electric currents of order 1 mA can be driven through MWNTs without destroying them. Based on the electric power and bulk heat conductivity for graphite, the NT is expected to evaporate due to the large temperature rise. But no melting is observed. This observation of large current densities may however not be taken as a proof for ballistic transport. Rather, it shows that dissipation is largely absent (which is an exciting fact by itself). In fact, large currents of the same magnitude can be passed through lithographically contacted MWNTs without destroying them, although these MWNTs have been proven not to be quantum-ballistic [44]. An interpretation of this phenomena is difficult because it occurs in the nonlinear transport regime for applied voltages much larger than the subband separation [105]

3.6 Magnetotransport

We first discuss the MagnetoResistance (MR) in a parallel magnetic field B. This case is very appealing because the effect of B on the wavefunction can easily be described. The magnetic flux 0 through the nanotube gives rise to a Aharonov-Bohm phase modifying the boundary condition of the transverse wavevector k± (4) into:

is the magnetic flux quantum h/e and v = 0, ±1 [62,106]. Similarly, the index n in the approximate 1-D dispersion relation of (5) is changed into n + o. It then follows that the cut-off energy Env ($) for the subbands is:

This relation is shown in Fig. 17 for v = 0. The band structure is periodic in parallel magnetic field with the fundamental period given by . The drawing corresponds to a metallic tube, for which the cutoff energy is zero for $ = 0. Because v and the scaled flux appear as a sum in (8), a metallic tube is turned into a semiconducting one depending on the magnetic flux and vice versa. At half a flux quantum the band-gap reaches its maximum value of Eo, corresponding to 65meV for a SWNT with dt = 20nm. If, as claimed before, a single nanotube dominates transport in a MWNT, a periodic metal-insulator transition should be observed as a function of parallel field at low temperatures.

The dependence of the electric resistance of MWNTs in a parallel magnetic field has been studied by Bachtold et al. [46] and Fig. 18 shows results of a typical (MR) measurement. If we adhere to the notion that the measured resistance in MWNTs is due to the outermost metallic NT, R is expected to increase because of the appearance of a gap. However, on applying a parallel magnetic field B, the resistance rather sharply decreases. It is therefore clear that this effect must have another origin than the band structure modulation that we have just discussed. This decrease is associated with the phenomenon of WL [70,71,72,73].

Weak localization originates from the quantum-mechanical treatment of backscattering which contains interference terms, which add up construc-

En,o/E0

Fig. 17. Cutoff energy En,0 of 1-D-subbands for a (metallic) SWNT as a function of magnetic flux 0 through the tube. The band structure is periodic in parallel magnetic field with a period given by the magnetic flux quantum 0O = h/e. Both En,0 and 0 and plotted in dimensionless units
0 0

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