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limits, defined by kT ^ |Ep| and kT ^ |EF|, respectively. For the latter (either strongly doped, or sufficient low temperature) a2D is given by (2) with E replaced by | EF | . Note, that the mean-free path is in general T-dependent, due to electron-phonon scattering, for example, thus le increases with decreasing T and approaches a finite zero-temperature value, only determined by static disorder. The equation therefore predicts true metallic behavior for the temperature-dependent resistance, i.e., the electric resistance should decrease with decreasing temperature due to the gradual suppression of electron-phonon scattering. In the high-temperature limit (kT ^ ^ i.e., low doping or ideal graphene) a2D is given (up to a numerical factor of order 1) by (2) with E replaced by kT. That a2D x T reflects the fact that the carrier density is proportional to T. Also this equation predicts a metallic temperature-dependence of the resistance provided that le x T-p with p > 1. This condition is always met, except for highly disordered samples for which le is a constant over a substantial temperature range. The resistance of such 'dirty' samples should display a R(T) x T-1 dependence. Unfortunately, there are no measurements for the electric sheet resistance of graphene, since graphene is a theoretical object. In contrast, there is an extensive source of literature for graphite (including graphitic fibers and tur-bostratic graphite) [15,16]. Graphite consists of a regular A-B-A-B stacking of graphene planes. There is a weak interplane coupling of order A ~ 10 meV, which results in semimetallic behavior and a band overlap of 40meV. Most importantly, graphite develops a finite DOS at EF and is therefore a metal. The sheet conductivity is approximately given by (2) with E replaced by A (if kT ^ A). We therefore expect a metallic temperature-dependent resistivity for crystalline graphite in the temperature range kT ^ A. Figure 9 shows the measured temperature-dependence of the electric resistivity p(T) for different forms of carbon fibers [15]. The upper curves correspond to disordered pregraphitic carbons while the ones with lower p correspond to highly graphi-tized samples. Because we will be discussing arc-discharge grown MWNTs in this chapter, arc-grown graphitic fibers (graphite whiskers [16,60]) provide a good reference material. Both arc-grown and single crystalline graphite display a metallic p(T), i.e., the resistivity decreases with decreasing temperature T. From the typical value of 5 x 10-6 fim (100K) for the resistivity, a sheet resistance Ra of Ra = 1.5 k^ is deduced. Taking (2) with E replaced by A ~ 10meV, we obtain a scattering length of le = 600 nm. Because the zero-temperature resistivity of single-crystalline graphite can be more than an order of magnitude lower, large mean-free paths for scattering le > 1|j.m are possible. Based on this consideration we can expect that carbon nanotubes may be 1-D ballistic conductors, provided that nanotubes of structural quality similar to single-crystalline graphite can be obtained. For the uppermost curve in Fig. 9, corresponding to highly disordered carbon fibers, the mean-free path is only of order 1.5 nm. In such disordered materials the graphene planes have a high density of structural defects as well as random stacking

Fig. 9. Electrical resistivity p as a function of temperature T for different form of graphitic fibers taken from [15]. The higher up the curves, the larger the disorder. The curve with the lowest resistivity corresponds to crystalline graphite

(turbostratic graphite) [16], and the interplane coupling is reduced to A « 0. The properties of disordered carbon fibers are therefore very well approximated by a set of independent sheets of 'dirty' graphene. For the comparison with MWNTs we should memorize the following two observations: 1) disordered graphite is characterized by a resistance upturn at low temperatures (localized behavior) with a typical sheet resistance of Ra ~ 100 k^, and 2) crystalline graphite is characterized by a metallic R(T) with Ra ~ 1kQ [61].

Next, we will briefly discuss the electronic properties of the other reference compound, that is of a perfect single-wall carbon nanotube. A SWNT is obtained from a slice of graphene wrapped into a seamless cylinder. The periodic boundary condition around the nanotube circumference causes quantization of the transverse wavevector component. Let us denote the wavevector along the tube direction by k and the transverse component by k±. The allowed k± are spaced by 2/dt, where dt is the tube diameter. The 1-D band structure can now easily be constructed using for example the 2-D tight-binding band structure E(kx,ky) of graphene, shown in Fig. 8. Each k± within the first Brillouin zone gives rise to a 1-D subband (1-D dispersion relation) by expressing E as a function of k for the given k±. Hence, a set of subbands Ek± (k) is obtained. A large diameter tube will have many such subbands, while a small diameter tube has only a few. It turns out that both metallic and semiconducting nanotubes are possible. A nanotube is metallic if and only if the K points belong to the set of allowed k-vectors. In the k • p scheme, the eigenfunctions in the vicinity of the K points are approximated by a product of one of two fast oscillating graphene wavefunctions (r), changing sign on the scale of interatomic distances, and a slowly varying envelope function F(r) [58]. The periodic boundary condition leads to the following condition for [62]:

where L measures the length around the tube circumference and v = 0, ±1, depending on the wrapping or chiral vector [55]. Taking for the slowly varying function F a plane wave (free electrons) and noting that F should cancel the phase factor in (3), we obtain the condition 2

dt for the allowed transverse wavevectors. The approximate 1-D dispersion relations, valid in the vicinity of EF, can now be derived using E(k) = ±hvF\k\. Explicitly:

with E0 = 2ftvp/dt. These approximate dispersion relations are shown in Fig. 10. As can be seen, there are metallic (v = 0) and semiconducting (v = ±1) nanotubes. The bandgap of a semiconducting tube is Eg (dt) = 2E0/3, which is inversely proportional to the tube diameter. We obtain 0.68 eV for a tube with d = 1.3 nm. Since MWNTs have rather larger diameters of order 20 nm, the corresponding bandgaps are only Eg « 44 meV. We mention, that the set of 1-D subbands in Fig. 10 needs to be duplicated because the K and K' points which are equivalent in the absence of a magnetic field are both in the first Brillouin zone [47,55]. Furthermore, the crossing at the Fermi energy between the bonding and antibonding states need not necessarily be at k = 0 [47]. The Fermi level crossing for zig-zag (n, 0) tubes, which can be metallic or semiconducting, is at k = 0. In contrast, the level crossings are found at k = ±2n/(3a0) for armchair (n,n) tubes. Armchair tubes are always metallic. For all metallic nanotubes, the Fermi energy intersects two 1-D branches with positive velocity (right-movers) and two with negative velocity (left-movers) [47]. This corresponds to two 1-D modes (not counting spin) leading to a quantized conductance of G = 4e2/h for an ideal nanotube. Figure 10 also shows the corresponding DOS, which are characterized by sharp features (van Hove singularities) at the onset of subbands. The observations of these singularities in, for example, tunneling spectroscopy, is a direct proof for the presence of a 1-D band structure. This has indeed been observed in SWNTs using scanning-tunneling spectroscopy at low temperatures [63,64]. Since MWNTs consist of tubes with larger diameters (dt), we

Fig. 10. Approximate dispersion relations (left) for the 1-dimensional electronic energy bands and the corresponding density of state (right) of metallic (a), and semiconducting (b), single-wall carbon nanotubes close to the Fermi energy. The energy scale is in units of E0 = 2hvF/dt

need to understand how the van Hove singularities develop if dt approaches values comparable to the mean-free path le. The picture sketched above is valid if le ^ dt (1-D-ballistic transport). In contrast, if le ^ dt transport is 2-D-diffusive and the density of states should closely resemble that for graphene without any singularities. If, on the other hand, le is of the same order as dt, transport is neither fully 2-D-diffusive, nor 1-D-ballistic. In this regime, the characteristic subband features in the DOS are still present, albeit considerably broadened. One expects to see broadened peaks in the DOS with a mean level spacing of the 1-D subbands given by

For a metallic nanotube, two subbands are occupied at EF. Consequently, the electric conductance G is predicted to be G = 4e2/h, provided the complete wire is ballistic. We not only require le ^ dt, but in addition ballistic transport requires that le ^ L, where L is the length of the electrically contacted nanotube segment. More precisely, ballistic transport requires that backscattering must be absent altogether, taking into account both the nano-

tube together with the electric contacts. This condition is very hard to satisfy in the laboratory [44]. Important for MWNTs is the question of how the band structure might change due to inter-tube coupling. This question has only recently been addressed [65,66,67]. However, the main consequence of the low-energy properties can simply be guessed. It is convenient to consider only two tubes which couple only weakly to each other. If one is semiconducting and the other is metallic, it is obvious that the low-energy properties are just determined by the metallic one, with no modifications in the DOS around zero energy. If the two tubes are metallic, the situation is expected to be more complicated. However, in practice it is most likely that the two tubes have different chiralities. For example, one tube could be of the armchair type and the other a zig-zag one, to mention an extreme case. Because the zero-energy bands cross at different k points for the two tubes, hybridization is very weak around EF and the total DOS is just the sum of the two. Strong band structure modifications are only expected for tubes of similar chiralities [66]. We should keep in mind, however, that this picture is not valid, if the tubes are relatively strongly doped such that EF is shifted either into the valence or conduction bands. For a comparison of results obtained for SWNTs with MWNT studies, the following three aspects of SWNTs should be emphasized: 1) an ideal SWNT is either a metal or a semiconductor with a considerable band gap, 2) the band structure consists of a set of 1-D-subbands, leading to van Hove singularities in the density of states, and 3), the electrical conductance G is quantized in units of 4e2/h, provided that backscattering is absent altogether.

3.2 Electrical Transport in MWNTs: A Brief Review

A remarkable variety of physical phenomena have been observed in electrical transport. The first signature of quantum effects was found in the magnetoresistance (MR) of MWNTs. Song et al. studied bundles of MWNTs [68], while Langer et al. was able to measure the MR of a single MWNT for the first time [69]. In both cases a negative MR was observed at low temperatures indicative of weak localization [70,71,72,73]. From these MR experiments, the phase-coherence length l$ was found to be small, amounting to < 20 nm at 0.3 K, in strong contrast with the ballistic transport theoretically expected for a perfect nanotube. However, evidence for much larger coherence lengths in SWNTs was provided by the observation of zero-dimensional states in single-electron tunneling experiments [74,75], and in other experiments discussed below.

Recently, a pronounced Aharonov-Bohm resistance oscillation has been observed in MWNTs [46]. This experiment has provided compelling evidence that can exceed the circumference of the tube, so that large coherence lengths are possible for MWNTs too [76]. Since the magnetic-flux modulated resistance is in agreement with an Aharonov-Bohm flux of h/2e, this oscillatory effect is supposed to be caused by conventional weak-localization for which the backscattering of electrons is essential. In essence, as in the work of Langer et al. [69], 2-D-diffusive transport could explain the main observation reasonably well.

The Aharonov-Bohm experiment provides a convincing proof that the electric current flows in the outermost (metallic) graphene tube, at least at low temperatures T < 70 K. Presumably, this is a consequence of the way in which the nanotubes are contacted. In general, electrodes are evaporated over a MWNT, and the electrodes therefore contact the outermost tube preferentially. Since it is essentially only the outermost tube that carries the current, large diameter single graphene cylinders can now be investigated. Recently, proximity-induced superconductivity was found in weak links formed by a bundle of SWNTs in contact with two superconducting banks [77]. Furthermore, spin transport also has been considered. Here, a MWNT was contacted by two bulk ferromagnets (e.g., Co) and the electrical resistance was measured as a function of the relative orientation of the bulk magnetization in the two ferromagnetic contacts [78]. A resistance change of order 10% was found, from which the authors estimate the spin-flip scattering length to be > 130 nm. All the striking results mentioned above were obtained by contacting a single nanotube with the aid of micro- and nano-structured technologies.

Alternative approaches for contacting nanotubes have been developed as well. For example, Dai et al. [79] and Thess et al. [80] have measured the voltage drop along nanotubes using movable tips, and Kasumov et al. have developed a pulsed-laser deposition method [81]. Furthermore, scanningprobe manipulation schemes were developed [82,83,84], and recently is has been shown that SWNTs can directly be synthesized to bridge pre-patterned structures [85]. Still another elegant method allowing one to contact a single MWNT electrically has been used by Frank et al. [45], whereby a single nanotube extending out of a MWNT fiber (a macro-bundle) is contacted by gently immersing the fiber into a liquid metal (e.g., mercury). Immersing and pulling out the nanotube repeatedly is claimed to have a cleaning effect. In particular, graphitic particles are removed from the tubes. After some repeated immersions, an almost universal conductance step behavior is observed, with steps close to the quantized value G0 = 2e2/h. From these experiments, the researchers conclude that transport in MWNTs is ballistic over distances on the order of > 1|im. This is a very striking result because the experiments were conducted at room temperature. At present, it is not clear why the conductance is close to G0 instead of the theoretically expected value of 2G0.

3.3 Hall-Effect and Thermopower in Assembled Nanotubes

Already in 1994, Song et al. studied the electronic transport properties of macro-bundles of MWNTs with diameters tens of |im [68]. The authors measured the temperature-dependence of the resistance R, its change in magnetic field B and the Hall effect. Above 60 K, the MagnetoResistance (MR) is positive, i.e., R increases with magnetic field; they however also found that a pronounced negative MR peak develops at low temperatures. This negative MR-dependence is suggestive of Weak Localization (WL). WL is an interference correction to the Drude resistance of a metal. WL primarily lowers the diffusion coefficient D due to constructive interference of mutually time-reversed quasiclassical electron trajectories in zero magnetic field. Because only trajectories of lengths shorter than the phase-coherence length l$ can participate, sufficiently low temperatures are usually required. For T < 10 K, the conductivity a was found to show a ln(T) dependence. This is in agreement with 2-D-WL theory, so that < dt, where dt is the diameter of a typical MWNT within the bundle. This relatively short coherence length suggests a small mean-free path for scattering from static disorder, implying that the MWNTs are fairly disordered. This is further supported by the temperature dependence of the measured conductance G = R-1. This temperature dependence of the conductance, as well as of the Hall coefficient, are shown in Fig. 11, where G is shown to be proportional to T over a remarkably large temperature range of 50-300 K. Below « 50 K the T-dependence weakens to finally turn into a logarithmic dependence, characteristic of 2-D-WL. From the theoretical models, which have been developed for graphite in Sect. 3.1, a linear G(T) is predicted for the conductance of disordered graphene. In this case, le is mostly determined by scattering from static disorder, and therefore le is only weakly T-dependent. This yields G <x T because of the energy-dependent DOS of graphene. The cross-over at 50 K suggests metallic behavior which can be either due to intertube coupling (like in graphite) or due to doping, thus shifting EF away from the neutrality point. Because the change of the Hall coefficient RH from « 0 (high T) to a pronounced positive value (low T) coincides with the 'flattening' in

Fig. 11. The Hall coefficient RH (left scale) and the conductance (right scale) vs temperature T (taken from Song et al. [68])

G(T), the nanotubes in Fig. 11 are most likely slightly (hole) doped. From the cross-over temperature (« 50K), the Fermi energy is estimated to be « —4meV. Using the theoretical DOS for graphene, the sheet doping level is found to be n2D = 1.5 x 109 cm-2, which has to be compared to the bulk doping level of n3D = 6 x 1018 cm-3 obtained from RH. If the material would be homogeneous and densely packed then n2D = n3DdI , where dI = 3.4 A is the interplane distance of graphite. Under these assumptions, we obtain n2D = 2 x 1011 cm-3 from RH, which is more than two orders of magnitude larger than the previous estimate. Though appearing to be inconsistent, this disagreement can easily be resolved. The apparent inconsistency is caused by the assumption of complete filling. Moreover, we know today that most of the electric current in transport measurements is confined to the outermost metallic SWNT at low temperature [45,46]. Hence, the volume fraction cannot exceed dI /dt « 50, thus resolving this apparent inconsistency.

From fits of the low-T conductance to WL theory, Song et al. could give estimates for several important parameters. For example, they obtained Ra ~ 6kQ for the sheet resistance, D « 50cm2/s for the diffusion coefficient and le « 5 nm for the mean-free path. The main results of this work, a positive RH suggesting hole doping and interference corrections at low T, have both been confirmed later by Baumgartner et al. with measurements on oriented MWNT films [86].

We have added the following discussion, to emphasize the question of doping in carbon nanotubes. This has recently received quite some attention, in particular, regarding research on SWNTs. The Hall coefficient clearly suggests hole doping (of unknown origin). Much more evidence for hole doping is emerging now from other work. The thermoelectric power S has been found to be positive and approximately linear in T at low T, both for SWNT and MWNT ropes [87,88,89]. The linear temperature dependence of the thermopower suggest metallic behavior. The substantial positive value of S of order 40-60 |V/ K at 300 K initially posed a serious problem, since S should vanish, if EF lies at the charge neutrality-point, as theoretically expected (for graphite S «—4 |V/ K). However, these findings of a substantial positive S are in agreement with the above-mentioned positive Hall coefficient, suggesting unintentional hole doping. If we assume that EF is considerably shifted into the valence band, then the large magnitude of S can easily be understood. Assuming, for simplicity, an energy-independent scattering length, but taking the DOS to be that of graphene, then S is interestingly exactly given by the standard textbook formula for free electrons, i.e., S = —n2k2T/(3eEF). From the measured value of S one derives —0. 14 eV for EF which amounts to a doping level of n2D = 5 x 1010cm-2. This level may appear to be large, but if we express the doping in terms of the elementary charge (e) per unit tube-length, there is on average only one e per 300 nm (assuming a SWNT with dt = 2nm). Most recently, it has been shown that the magnitude and even the sign of S can be changed by annealing the nanotubes in vacuum.

It is believed that such a treatment removes oxygen, which may act as the dopant [90,91,92]. Nanotubes might be more sensitive to enviromental conditions than initially believed. Clearly more work is needed to understand the nature of the doping and of the doping level.

3.4 Electrical Transport Measurement Techniques for Single MWNTs

Electric measurements on single MWNTs have been performed in three ways: (1) metallic leads are attached to a single tube supported on a piece of Si wafer with the aid of microfabrication technology [46,69,81,93]; (2) the end of a macrobundle of MWNTs, fixed on an moveable manipulator, is steered above a beaker containing a liquid metal (e.g., mercury) and the MWNTs are then gently lowered into the liquid metal. According to this method, a single MWNT makes contact to the liquid metal first, enabling conductance measurements to be made on a single nanotube [45], as shown in Fig. 12, and (3) a scanning-tunneling microscope can be used to measure the local electronic density of states by measuring the bias-dependent differential conductance while the tip is positioned above a single MWNT.

Because lithography is now widely used for contacting nanotubes, we discuss the lithographic approach in detail in the following. A droplet of a dispersion of nanotubes (NTs) is used to spread the NTs onto a piece of thermally oxidized Si. Then, a PMMA resist layer is spun over the sample. An array of electrodes, each consisting of two or more contact fingers together with their bonding pads, is exposed by electron-beam lithography. After development, a metallic film (mostly Au) is evaporated over the structure and then lifted off. The sample is now inspected by either Scanning-Electron Microscopy (SEM) or Scanning-Force Microscopy (SFM) and the structures that have one single

Fiber

Fiber

Fig. 12. (a) Schematic diagram of a contacting apparatus. The fiber is moved into the liquid metal, while the conductance G is measured. Taken from Frank et al. [45]. (b) Transmission electron micrograph of the end of a nanotube fiber with one single MWNT extending far out from the fiber [45]

nanotube lying under the electrodes are selected for electric measurements. An example of a single MWNT contacted by four Au fingers is shown in Fig. 13. Since the success of this contacting scheme works by chance, it is obvious that the yield is low. There are many structures which have either no or several NTs (NanoTubes) contacted in parallel. Since a large array of more than 100 structures can readily be fabricated, however, this scheme has turned out to be very convenient.

Alternatively, it is also possible to first structure a regular pattern of alignment marks on the substrate. After adsorbing the NTs, the sample is first imaged with SEM or SFM in order to locate suitable NTs. Having noted the coordinates of the designated nanotubes aided by the alignment marks, the electrodes can be structured directly onto the respective NTs with high precision. This improves the yield at the cost of an additional lithography step. Let us emphasize that the Au electrodes are evaporated directly onto the nanotubes. The reverse scheme is also possible. Here, electrode structures are made first and the NTs are adsorbed thereafter. In this scheme (nano-tube over the contacts), the contact resistances are typically found to be large (> 1MO). It was only with the aid of local electron exposure directly onto the NT-Au contacts that this resistance could be lowered to acceptable values [95]. In contrast, the contact resistance Rc can be surprisingly small in the former scheme (nanotube under the contacts). Rc is of order 0.1... 20k^ with an average of 4kQ [44]. Rc has been determined by comparing the 2-terminal (R2t) with the 4-terminal (R4t) resistance according to

An 'ideal' contact is defined to have no backscattering and to inject electrons in all modes equally. Electrons incident from the NT to a contact will then be adsorbed by the contact with unit probability. Because the contact couples to both right and left propagating modes equally, Ohm's law should

Fig. 13. SEM of a single multi-wall nanotube electrically contacted by four Au fingers from above. The separation between the inner two contacts is 2|i.m [94]

be valid in this limit. It is important to realize that for ideal contacts, R4t cannot contain any nonlocal contributions, i.e., contributions to the resistance that arise from a NT segment not located in between the inner two contacts. Any sign of non-locality points to the presence of non-ideal contacts. Conceptually, it is very hard to image that the Au-electrodes provide an ideal contact to a metallic nanotube, because the electronic properties change abruptly from something which is more like a semimetal (well within the nanotube) to a high-carrier density metal well within the metal contact.

In order to confirm the ballistic 1-D nature of an ideal metallic nanotube in transport measurements, ideal contacts (no backscattering) are required. Only then, will the conductance be quantized and equal to G = 4e2/h [96]. Since the metallic electrodes are extended and it is very likely that the coupling from the metal to the tube is not uniform, metal-tube tunneling occurs at many different spots simultaneously. Unfortunately, models for such contacts are not yet available. The interpretation of the measured resistance would be much easier, if we were able to intentionally choose the strength of the coupling at the contacts. If this were possible, one could fabricate 4-terminal devices with low-ohmic contacts for the outer two contacts (placed as far apart as possible) and very weak-coupling (non-perturbing) contacts for the inner two. R4t of such a device would then give the real intrinsic nanotube resistance caused by scattering in the tube. In particular, R4t = 0 for the ballistic ideal metallic NT. Furthermore, a high-ohmic contact (a true tunneling contact) can be used for tunneling spectroscopy from which the density of states in the tube can be derived.

Until now, lithographically fabricated contacts are only accidentally high-ohmic. However, an elegant non-invasive measurement of the electrostatic potential is possible with scanning-probe microscopy in which a tip is used to probe the potential along a biased nanotube [97]. In addition to allowing for more than just two contacts, electrostatic gates can be added to supported NTs enabling one to tune the carrier concentration (Fermi energy). A metal electrode (not in direct contact with the NT) or the substrate itself can serve as a gate. In the later case, which is preferred by most researchers, a degenerately-doped Si substrate is used and the isolation between the gate and the NT is provided by a thin oxide layer with a thickness of 0.1-1 ^m.

3.5 Electrical Measurements on Single MWNTs

Figure 14 shows the characteristic dependence of the equilibrium electric resistance R on the temperature T for a single MWNT with low-ohmic contacts, where it is shown that R(T) increases with decreasing T. This decrease is rather moderate and distinctly different to what was found in macrobundles and films, discussed in Sect. 3.3, see Fig. 11, where it was found that R-1 = G(T) a: T. In contrast, R(T) in Fig. 14 cannot be expressed as a simple power-law, so that if we would try to write G <x. Ta,

352 Laszlo Forra and Christian Schonenberger 20

Fig. 14. Four-terminal equilibrium electric reistance R vs T of a single arc-grown MWNT with a contact separation of 300 nm [98]

then a would range from 0.07 (100K) to 0.19 (0.4K). Hence, the T dependence for a single MWNT is weaker as compared to macro-bundles. According to our previous discussion (Sect. 3, 3.1), the non-metallic looking behavior of Fig. 14 could be taken as a sign that MWNTs are similar to highly disordered graphite (Fig. 9). This is, however, not true. Taking R(300K) = 6kQ, the sheet resistance and the resistivity are found to be Ra = 1.3kQ and p = 0.4 |ifim, respectively. Here, we have used again the fact that R is determined by the outermost metallic SWNT [46]. Only arc-grown graphitic fibers or single-crystalline graphite have shown such low resitivities. But contrary to MWNTs, these 'clean' materials display a metallic R(T)-dependence: the resistance decreases with decreasing temperature. Based on Ra we conclude that single arc-grown MWNT are highly graphi-tized with a low degree of disorder, comparable to that of single-crystalline graphite, and, that the resistance increase must be a characteristic feature for this tubular one-dimensional form of graphite. The low-T resistance of MWNTs differs also from SWNTs. Measurements on SWNTs almost always display Coulomb Blockade (CB) behavior below « 10 K [74,75]. In this regime of single-electron tunneling, the whole NT acts as an island weakly coupled to the environment, i.e., with contact resistances larger than the resistance quantum R0 = h/2e2 = 12.9kQ [99]. For low applied voltages, R is exponentially suppressed once kT ^ Ec, where Ec = e2/2C is the single-electron charging energy of the NT with capacitance C with respect to the environment. A typical value is Ec « 1meV for a 1|im long nanotube segment. Figure 14 is representative for all measured MWNTs with low-ohmic contacts. Coulomb blockade is not observed. According to theory, 2/3 of the tubes should be semiconducting with a substantial band gap (even for MWNTs) of order Eg « 40meV for a 20 nm diameter tube. This band gap should give rise to an exponential temperature-dependence, according to R(T) a: exp(2Eg/kT) which has never been observed. MWNTs have many tubes in parallel, from which it is clear that the metallic ones will take over at low temperatures.

352 Laszlo Forra and Christian Schonenberger 20

Because the electrodes are in contact with the outermost shell, the outermost metallic NT dominates. If the outermost tube happens to have a gap, a larger contact resistance might be expected because the electrons would have to tunnel from the electrode through the semiconducting tube into the first metallic NT. This also has not been observed until now. Of all MWNTs contacted from above (i.e., with metal evaporated over the NT), appreciably high-ohmic contacts are rare (only 10%) [44]. A possible explantion could be that the NTs are doped, either intrinsically or due to charge transfer from the metallic contact.

A metallic gate allows study of the dependence of the nanotube conductance on the carrier concentration. Gate sweeps are also important for providing a convincing proof of CB. If G(T) is determined by CB at low T, then G should be periodic in the gate voltage Vg with a period given by e/C. Figure 15 shows two such gate-sweeps for a single MWNT at 0.4K, one for zero magnetic field, the other in a transverse field of B = 2T. Here G is strongly modulated by Vg with rms-fluctuations of order 0.3G0 around an average value of 0.8G0. Coulomb blockade would result in a periodic modulation with a period in Vg estimated to be 50 mV. But the fluctuations in Fig. 15 occur rather on a scale of 1 V. One might argue that the NT splits into a sequence of smaller islands at low T, which would give rise to a fluctuation pattern with a larger characteristic voltage scale. This picture is quite similar to strong localization for which an exponential resistance increase would be expected at low T, which is not observed. We therefore conclude that CB is not relevant for MWNTs with low-ohmic contacts down to 300 mK.

Gate Voltage (V)

Fig. 15. Conductance of a 300 nm single MWNT segment at T = 0.4 K as a function of gate voltage applied to the degenerately doped Si substrate. The nanotube is spaced from the back-gate by a Si-oxide layer 400 nm in thickness. Lower (upper) curve is measured in zero (2 T) magnetic field. The upper curve is offset vertically by e2/h for clarity [98]

Gate Voltage (V)

Fig. 15. Conductance of a 300 nm single MWNT segment at T = 0.4 K as a function of gate voltage applied to the degenerately doped Si substrate. The nanotube is spaced from the back-gate by a Si-oxide layer 400 nm in thickness. Lower (upper) curve is measured in zero (2 T) magnetic field. The upper curve is offset vertically by e2/h for clarity [98]

It is remarkable that G > G0 for certain Vg in Fig. 15. The observed pattern reflects the quantum states in this nanotube segment approaching the 0D limit. The pattern is aperiodic because of random scattering, and the NT behaves like a chaotic cavity. The fluctuation is suggestive of so-called Universial-Conductance Fluctuations (UCF) [100,101,102].

The upper curve in Fig. 15 demonstrates that the pattern is completely changed in a magnetic field of 2 T. Because the two curves show no correlations, the so-called correlation field is estimated for be Bc < 2 T. This allows us to estimate the phase-coherence length l$ which must be > 100 nm. Hence, this 300 nm NT segment is certainly 1-D with respect to quantum coherence and even close to 0D (quantum dot). Let us estimate the expected mean level spacing A E0D for an ideal NT quantum dot of length L. If spin-degeneracy is removed, then A E0D = hvF/(4L), amounting to 3meV for L = 300nm. One now needs to know the leverage factor A Vg /A EF which we estimate to be 100. Hence, 0D features should show up in the gate-dependence on a voltage scale of order 0.3 V, which is indeed the case.

These measurements also allow us to estimate the mean-free path le. Because strong localization is not observed, we conclude that the localization length is lloc > With lloc ~ Nle, where N « 4 is the number of modes, we see that le > 20 nm, so that the mean-free path is of the same order as the diameter of the MWNTs. Hence, there is elastic scattering leading to the interference patterns at low temperature, but this scattering is not strong enough to localize the electron states in the MWNT. This result, that MWNTs are not free of elastic scattering (there is some disorder), has to be contrasted with work of Frank et al. [45]. Figure 16 shows the measured conductance G of a fiber of MWNTs while lowering this fiber continuously into a liquid metal (see also Fig. 12). In Fig. 16 we see that G increases in steps of magnitude close to the quantized conductance G0 = 2e2/h. Each step is due to an additional MWNT coming into contact with the liquid. The nearly equal conductance of « G0 for each MWNT, that makes contact with the liquid Ga, has been taken as evidence for quantum ballistic transport. This effect is very striking, since the experiments in Fig. 16 were done at room temperature. Why the measured G is only one-half of the expected quantized conductance for an ideal metallic NT is not understood at present [103]. The measured G « G0 is also in disagreement with other transport measurements on nominally similar MWNTs. For example, the NT in Fig. 14 has a room temperature resistance of only 6 k Q which corresponds to « 4e2/h. Values for the two-terminal conductance of up to 4G0 have been found at room temperature [44]. This is not unexpected because higher subbands contribute to the conductance at room temperature, too. Recall, that the mean-level spacing is only 33meV for dt = 20 nm. The result of Frank et al. can be summarized as follows: the measured resistance is a contact resistance, which is quantized and the intrinsic NT resistance, determined from the depth-dependence, is

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