Bt

Fig. 18. Electrical resistance R as a function of parallel magnetic field B (see inset showing electron orbit). Arrows denote the resistance maxima corresponding to multiples of h/2e in magnetic flux through the nanotube taking the outer diameter [44]

tively in zero magnetic field. Backscattering is thereby enhanced, leading to a resistance larger than the classical Drude resistance. Because the interference terms cancel in a magnetic field of sufficient strength, WL results in a negative MR. However, for the specific geometry of a cylinder (or ring), the WL contribution is periodic in the magnetic flux through the cylinder, with half the AB (Aharonov-Bohm) period h/2e [107]. Indeed, in Fig. 18 the resistance has a second maximum at B = 8.2 T. From this field value, a diameter of dt = 18 nm is obtained for this MWNT. As was demonstrated by Bachtold et al. [46], the MR agrees with the Altshuler, Aharonov and Spi-vak (AAS) theory [108], only if the current is assumed to flow through one of outermost cylinders with a diameter corresponding to the independently measured outer diameter of the NT. It is therefore most likely that only one cylinder actually participates in transport. The conclusion that only one graphene cylinder carries the current can only unambiguously be drawn from the analysis of the low-temperature MR data (T < 20 K). We emphasize that it is not possible to relate the resistance maxima at ±8.2 T to a magnetic flux of h/e, because a tube diameter would then result which is larger than the actually measured outer diameter. The observation of a pronounced h/2e resistance peak proves that backscattering is present. The NTs therefore do not exhibit ballistic transport.

In a parallel magnetic field, resistance maxima should occur periodically. The onset of the second resistance peak, which is expected at B = 16.4T is clearly seen in Fig. 18. From this measurement, a phase coherence length of l^ « 200 nm is estimated in good agreement with the estimate given before [44].

Next, we will consider the MR in a perpendicular field, for which the MR has been extensively studied [44,68,69,76,109,110,111]. A negative MR is found for a transverse B field, in agreement with weak-localization theory [70,71,72,73]. Further support for the importance of interference contributions to the transport properties comes from the observation of non-local effects in multi-terminal devices [44],

A typical MR measurement is shown in Fig. 19. On applying a transverse magnetic field, the resistance decreases (negative MR), in agreement with WL theory. Aperiodic fluctuations are observed superimposed on the negative MR background, and these fluctuations are assigned to universal conductance fluctuations (UCFs). This MR can only be fitted with 1D-WL theory, i.e., l$ > ndt (dt = 23nm). Best fits to theory are shown in Fig. 19 by the dashed lines, and a very good agreement is found. As a cross-check, UCF amplitudes deduced using agree with the observation. An example is given by the vertical bar in Fig. 19 which corresponds to the expected UCF amplitude at 2.5 K. Let us estimate the phase coherence length at 2.5 K. The resistance peak at zero field corresponds the a conductance change of A G = -2.2 x 10~5 S. From A G « -(2e2/h)l0/L we obtain « 500 nm. This particular example shows a rather large and correspondingly a surprisingly large diffusion coefficient D and mean-free path le. These later parameters can be obatined from the vs T relation for dephasing by quasi-

Fig. 19. Four-terminal MR of a MWNT in a perpendicular magnetic field for different temperatures. The voltage probes are separated by 1.9|i.m. Dashed curves show fits using one-dimensional weak-localization (1D-WL) theory. Note, that the curves are not displaced vertically for clarity [44]

Fig. 19. Four-terminal MR of a MWNT in a perpendicular magnetic field for different temperatures. The voltage probes are separated by 1.9|i.m. Dashed curves show fits using one-dimensional weak-localization (1D-WL) theory. Note, that the curves are not displaced vertically for clarity [44]

elastic electron-electron scattering (Nyquist noise dephasing [112]) which is the dominant source of phase-randomization at low temperature [44]. One obtains D = 450-900cm2/s and le = 90-180nm. Again, this mean-free path is in agreement with the condition that lloc > l$ from which le > 100 nm follows.

Another finding in Fig. 19 worth mentioning is that weak localization results in an increase of the electric resistance at low temperatures, but this is not the only contribution to the resistance increase. For large magnetic fields SGWL ^ 0 but the resistance is still seen to be strongly temperature dependent. Here we note that the curves in Fig. 19 are not displaced for clarity, but rather a temperature-dependent background resistance is observed and this effect is usually associated with electron-electron interaction. While WL primarily enters as a correction to the diffusion coefficient, the carrier-carrier interaction suppresses the single-particle DOS. For a diffusive wire for which the coherence length (the thermal length) is larger than the width but smaller than the length, theory predicts a temperature dependence for the conductance correction SGee oc T"1/2 [113,114,115], Knowing SGWL(T), one can then plot G(T) — 5Gwl{T) as a function of \/T.

It turns out, however, that this relation does not hold [70,71,72,73]. The interference and interaction corrections SGWL and SGee are derived by WL perturbation theory and are therefore assumed to be small. But exactly this assumption is not valid here. One therefore needs to treat the interaction exactly.

This is indeed possible in 1-D, using bozonization techniques. Carbon nanotubes are predicted to become so-called Luttinger Liquids (LL) in which a pseudo-gap in the quasi-particle DOS opens at low energy [116,117,118,119,120]. A perfect ballistic LL cannot be distinguished from the perfect Fermi liquid quantum wire in an equilibrium transport measurement whereby G is just quantized to 2e2/h (for one mode). However, with backscattering, G is renormalized. In the so-called strong backscat-tering limit (low T limit) G <x Ta with a weaker T-dependence at higher temperatures. For a SWNT a « 0.3-0.8, while a can be reduced to lower values in MWNTs due to the enhanced screening by multiple shells [121]. Though R(T) in Fig. 14 does not follow a power law, the temperature dependence of R is not inconsistent with LL theory. This dependence is expected for a LL in the weak-backscattering limit. LL theory includes the electron-electron interaction. In addition, scattering has been incorporated into the theory. However, the dependence on magnetic field has not yet been treated. It would be very interesting to see whether such a 'complete' theory could explain the measured MR, which can surprisingly be fitted quite well with theoretical results from WL perturbation theory based on Fermi-liquid quasi-particles [44,46,69,76]

3.7 Spectroscopy on Contacted MWNTs

Figure 20 shows a differential current-voltage characteristic (dl/dV) measured on a single MWNT, using an inner contact which by chance was high-ohmic. This particular tunneling contact had a contact resistance of 300 kQ, whereas the other contacts had resistances ^ 10 k Q. The measured spectrum agrees surprisingly well with predicted spectra based on simple tight-binding calculations for a metallic NT [106,122,123]. Firstly, there is a substantial DOS at the Fermi energy, i.e., at V = 0, so that the NT is metallic. Secondly, the almost symmetric peak structure, appearing as a pseudo-gap is caused by the additional 1D-subbands in the valence band (V < 0 ) and conduction band (V > 0) with threshold energies of order « 50meV. At the onset of the subbands, van Hove singularities are expected. The spectrum in Fig. 20 agrees remarkably well with scanning-tunneling measurements of Wildoer et al. for SWNTs [63,64]. But because of the difference in tube diameter, the energy scales are quite different. These subbands should be spaced by 2E0 = 4hvF/dt = 150meV for the MWNT in Fig. 20 which has dt = 17nm. The measured spacing of 110 mV is in reasonable agreement with this estimate. The observation of van Hove features in dI/dV demonstrates that the mean-free path le cannot be much shorter than the nanotube circumference. If le < ndt, all 1-D band structure features would be expected to be washed out. The observed spectrum in Fig. 20 nicely demonstrates that the unusual

Fig. 20. Differential (tunneling) conductance dI/dV measured on a single MWNT using a high-ohmic contact (300 kO) at T = 4.2K [44]. This spectrum qualitatively confirms the DOS expected for a metallic nanotube in which the wave vector is quantized around the tube circumference leading to lD-subbands. Positive (negative) voltages correspond to empty (occupied) nanotube states

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