Vv

band structure effects of NTs are also found for MWNTs. In most cases, however, the measured spectral features are not as sharp.

Furthermore, the prevailing dl/dV spectra display a pronounced zero-bias anomaly on a smaller energy scale of 1-10 meV. Such a tunneling spectra is shown in Fig. 21. These spectra are highly structured. The observed peaks are associated with broadened van Hove singularities due to the 1-D-band structure. This assignment is strongly supported by the observed peak-shifts in a parallel magnetic field B. The peaks are seen to move up and down with B in accordance with the Aharonov-Bohm effect. According to Fig. 17, the total peak shift amounts to E0/2, corresponding to 39meV for this MWNT (dt = 17 nm). The measured, shifts are somewhat smaller, « 22meV. With the exception of small voltages, i.e., for \V\ < 25meV, the mean peak spacing is « 25meV, in agreement with the maximum peak shift. In addition to the peaks, there is a pronounced zero-bias anomaly (ZBA). At V = 0 the

Fig. 21. Tunneling dl/dV spectra measured on a single MWNT using a high-ohmic contact for different magnetic fields B applied parallel to the tube [124]. Note, that the spectral features shift in a magnetic field (lines are a guide to the eyes for two well discernible peaks), and that there is a pronounced anomaly at V = 0. All curves are vertically displaced for clarity [124]

Sample bias (mV)

Fig. 21. Tunneling dl/dV spectra measured on a single MWNT using a high-ohmic contact for different magnetic fields B applied parallel to the tube [124]. Note, that the spectral features shift in a magnetic field (lines are a guide to the eyes for two well discernible peaks), and that there is a pronounced anomaly at V = 0. All curves are vertically displaced for clarity [124]

conductance is strongly suppressed, independent of B. This suggests that a pseudo-gap opens for low-energy quasi-particles. This finding must be related to the measured resistance increase at low temperature, as shown in Figs. 14 and 19.

Figure 22 shows a detailed analysis according to Bockrath et al. [125] of a zero-bias tunneling anomaly measured on a MWNT for different temperatures. The ZBA anomaly exhibits power-law scaling, i.e., dl/dV x Va if eV ^ kT and dl/dV x Ta if eV ^ kT. Such a dependence is in agreement with Luttinger liquid models [126]. Similar anomalies have recently been observed by Bockrath et al. for SWNTs [125]. Their measurement and analysis provide the first demonstration of LL behavior in carbon NTs.

The LL theory describes the interaction with a single parameter g [126]. The non-interacting Fermi-liquid case corresponds to g = 1 and 0 < g < 1 is valid for a LL with repulsive interaction. The parameter g is determined by the ratio of the single-electron charging energy to the single-particle level spacing and has been estimated to be g = 0.2-0.3 for SWNTs [120]. Because a MWNT consists of several shells, one might expect that the inner shells strongly screen the long-range Coulomb interaction leading to g ^ 1, i.e., to an effectively non-interacting Fermi liquid. However, it has recently been shown that g is only weakly modified and theoretically only scales with \/~N where N is the number of shells participating in screening [121]. Although there about 20 shells, the effective N is expected to be smaller and is only of order 1. LL behavior is therefore expected for MWNTs, too. Regarding the exponent a, one distinguishes 'bulk' from 'end' tunneling. The measurement in Fig. 22 corresponds to bulk tunneling for which a = (g-1 + g — 2)/8 [120].

Fig. 22. Normalized differential tunneling conductance dl/dV • T-a vs scaled voltage eV/kT measured on a single MWNT using a high-ohmic contact. The inset displays the equilibrium conductance as a function of temperature T on a log-log plot. The straight line corresponds to a = 0.36 [98]

eV/kT

Fig. 22. Normalized differential tunneling conductance dl/dV • T-a vs scaled voltage eV/kT measured on a single MWNT using a high-ohmic contact. The inset displays the equilibrium conductance as a function of temperature T on a log-log plot. The straight line corresponds to a = 0.36 [98]

From the experiment a « 0.36, which relates to an interaction parameter of g = 0.21. The same value was obtained by Bockrath et al. for SWNTs [125]. Currently, more evidence for LL liquid behavior in nanotubes (including MWNTs) is appearing [98,127,128,129].

3.8 Spectroscopy Using Scanning Tunneling Probes

The individual tubes in a MWNT have helicities, described by the (n, m) indexes, and the electronic structure calculations [28,47] for the SWNTs are also valid for the individual nanotubes in MWNTs. In principle, one can find metallic, or semimetallic nanotubes, if n — m = 0, modulo 3 is satisfied, or else the individual tubes are semiconducting, if n — m = 3q, q = 0,1, There seems to be some commensurability effect in play between the different layers, since the number of chiral angles observed in HRTEM is lower than the number of cylinders forming the MWNT [21]. Theoretically it has been predicted than even in the case of metallic layers, the nested nature of the tubes of different chiralities may introduce gaps or pseudo-gaps in the density of states in a similar way, as happens in the case of SWNTs organized in a bundle [130]. This is in contrast with graphite, where the interlayer interaction between the zero-gap semiconducting graphene layers creates a finite density of states at the Fermi level, making graphite metallic.

Energy (eV)

Fig. 23. Measured carbon nanotube differential conductances, proportional to the local density of states, along various positions near the tip (scans at B,C,D locations on the nanotube) and far from the tip (scan A) of a multi-walled carbon nanotube. The sharp peak near the Fermi level of the uppermost trace demonstrates the existence of a localized tip state [131]

Energy (eV)

Fig. 23. Measured carbon nanotube differential conductances, proportional to the local density of states, along various positions near the tip (scans at B,C,D locations on the nanotube) and far from the tip (scan A) of a multi-walled carbon nanotube. The sharp peak near the Fermi level of the uppermost trace demonstrates the existence of a localized tip state [131]

The electronic structure can be measured very reliably by STM. This technique was used to image the chirality and to measure the local density of states for SWNTs [63,64]. The lateral confinement of the electronic states due to the tubular structure causes symmetric singularities in the DOS, which also help to identify the tube chirality, since there is a reliable correspondence between the peak positions, diameter and chiral indexes. Carroll and coworkers have measured the Local Density Of States (LDOS) of multi-wall carbon nanotubes for different tube diameters scanning along the tube axis towards the tips (Fig. 23) [131]. Scanning far from the ends, the measurements show the outer layer of the tubes indicate a conducting character consistent with a graphitic-like density of states at low bias and symmetric singularities with respect to zero bias, due to 1-D quantum confinement. At the nanotube tip, the STM spectra show a localized state, whose position and width varies depending on the nanotube tip structure. Calculations have shown that the strength and the position of these states with respect to the Fermi level depend sensitively on the relative positions of the pentagons and their degree of confinement at the tube ends.

3.9 Discussion of the Main Issues

From the previous discussion, four basic issues emerge, which need further attention in the future: (1) Why does transport appear to be ballistic in one experiment and diffusive in another? (2) What is the origin of scattering? (3) What causes hole doping and what is the energy of the doping level in single MWNTs? (4) What is the ground state of carbon nanotubes?

Let us briefly discuss these questions. Electric transport measurements on single MWNTs give conflicting results. While Frank et al. [45] find ballistic transport, there is convincing evidence for diffusive transport from other experiments [44,46,68,69,97]. It is important to emphasize that the estimates for the mean-free path le from the latter experiments vary substantially, from le « 1nm to le > 100 nm. Taken this fact into account, the first question should be rephased to read: what is the reason for the large variation of scattering lengths that are observed in experiments on single MWNTs? This question immediately leads to the second question. What is the origin of the scattering? Are MWNTs defective, is the band structure modified by intertube hybridization, are there inclusions, or is the scattering caused by adsorbates? What is the role of the substrate on which the nanotubes resides in experiments using micro-fabrication? Arc-discharge grown MWNTs appear to be essentially defect-free in TEM micrographs. Furthermore, there is no evidence for bond defects from electron-spin resonance.

From the experiments and the discussion in this section the electronic intertube coupling is expected to be relatively weak. It is therefore believed that scattering is related (at least partly) to the issue of doping. MWNTs (and SWNTs) are found to be hole-doped in experiments on films and macro-bundles. Though the origin of the unintentional doping is not known yet, the apparent doping level can be changed in SWNTS if the nanotubes are heated to only 200 °C in high-vacuum [91]. This suggests that (part) of the doping might be caused by species present in air (for example by oxygen). Until now, the doping-level has not been specified in any experiment on a single MWNT. We need to find a way to quantify the doping level in the future.

The fourth question, finally, is a very intricate one: What is the ground-state of carbon nanotubes? A lot of measurements (e.g., magnetotransport) can be described by either non-interacting or only weakly interacting quasi-particles using theories like weak-localization based on the Fermi Liquid (FL) hypothesis. On the other hand, clear deviations from FL behavior has been observed too. The suppression of the quasi-particle density of states, observed in tunneling spectroscopy on single MWNTs (and SWNTs), suggests that nanotubes may develop a LL state. The ground-state question has become an even more exciting issue, because signatures for intrinsic superconductivity have recently appeared [132].

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