Info

Fig. 19.38. The energy at ky = 0 as a function of the inverse of a dimensionless magnetic length v = L/ltrl for (a) a zigzag tubule (10, 0) and (b) an armchair tubule (10, 10). Here I = (hc/eH)"2 and L = trd, [19.120],

experimentally, whose diameter is five times as large as ac_c = 1.421 A, the nearest-neighbor C-C distance [19.4].

The calculations described above indicate that susceptibility measurements on an array of carbon nanotubes will be dominated by the field component perpendicular to the axis of the tubule, unless the axes for all the tubules are accurately aligned with respect to each other, and the applied magnetic field is accurately aligned along the tubule axis direction. For the case H -L tubule axis, Fig. 19.37 shows that the magnitude of the differential susceptibility varies only weakly (< 10%) as a function of the magnetic flux, for both the semiconducting and metallic tubules, although the dependence on magnetic flux differs in phase between the metallic and semiconducting tubules, as seen in Fig. 19.37.

The Pauli susceptibility has also been measured by electron spin resonance (ESR) for a purified tube, using the controlled burning technique (see §19.2.5). Since the nanotubes burn less readily than nanoparticles, mainly carbon nanotubes remain as a result of this procedure [19.127],

19.6. Electronic Structure: Experimental Results

Experimental measurements to test the remarkable theoretical predictions of the electronic structure of carbon nanotubes are difficult to carry out because of the strong dependence of the predicted properties on tubule diameter and chirality. The experimental difficulties arise from the great experimental challenges in making electronic or optical measurements on individual single-wall nanotubes, and further challenges arise in making such demanding measurements on individual single-wall nanotubes that have been characterized with regard to diameter and chiral angle (d, and 6). Despite these difficulties, pioneering work has already been reported on experimental observations relevant to the electronic structure of individual nanotubes or on bundles of nanotubes, as summarized in this section.

19.6.1. Scanning Tunneling Spectroscopy Studies

The most promising present technique for carrying out sensitive measurements of the electronic properties of individual tubules is scanning tunneling spectroscopy (STS) because of the ability of the tunneling tip to probe the electronic density of states of either a single-wall nanotube [19.128], or the outermost cylinder of a multiwall tubule, or more generally a bundle of tubules. With this technique, it is further possible to carry out both STS and scanning tunneling microscopy (STM) measurements at the same location on the same tubule and therefore to measure the tubule diameter concurrently with the STS spectrum. It has also been demonstrated that the chiral angle 6 of a carbon tubule can be determined using the STM technique [19.12] or high-resolution TEM [19.1,14,33,38,42],

Several groups have thus far attempted STS studies of individual tubules. The first report of I-V measurements by Zhang and Lieber [19.14] suggested a gap in the density of states below ~200 meV and semiconducting behavior in the smallest of their nanotubes (6 nm diameter).

Although still preliminary, the study which provides the most detailed test of the theory for the electronic properties of the ID carbon nanotubes, thus far, is the combined STM/STS study by Oik and Heremans [19.129],

In this STM/STS study, more than nine individual multilayer tubules with diameters ranging from 1.7 to 9.5 nm were examined. STM measurements were taken to verify the exponential relation between the tunneling current and the tip-to-tubule distance in order to confirm that the tunneling measurements pertain to the tubule and not to contamination on the tubule surface. Barrier heights were measured to establish the valid tunneling range for the tunneling tip. Topographic STM measurements were made to obtain the maximum height of the tubule relative to the gold substrate, and these results were used for determination of the diameter of an individual tubule [19.129]. Then switching to the STS mode of operation, current-voltage (I-V) plots were made on the same region of the same tubule as was characterized for its diameter by the STM measurement. The I-V plots for three typical tubules are shown in Fig. 19.39. The results provide evidence for one metallic tubule with d, = 8.7 nm [trace (1)] showing ohmic behavior, and two semiconducting tubules [trace (2) for a tubule with d, = 4.0 nm and trace (3) for a tubule with d, = 1.7 nm] showing plateaus at zero current and passing through V = 0. The dl/dV plot in the upper inset provides a crude attempt to measure the density of states, the peaks in the dl/dV plot being attributed to (£„ - E)~1/2 dependent singularities in the ID density of states. Referring to Fig. 19.27, we see that, as the energy increases, the appearance of each new ID energy band is accompanied by a singularity in the density of states.

Fig. 19.39. Current-voltage I vs. V traces taken with scanning tunneling spectroscopy (STS) on individual nanotubes of various diameters: (1) d, = 87 A, (2) d, = 40 A, and (3) d, = 17 A. The top inset shows the conductance vs. voltage plot for data taken on the 17 A nano-tube. The bottom inset shows an I-V trace taken on a gold surface under the same conditions [19.129],

Fig. 19.39. Current-voltage I vs. V traces taken with scanning tunneling spectroscopy (STS) on individual nanotubes of various diameters: (1) d, = 87 A, (2) d, = 40 A, and (3) d, = 17 A. The top inset shows the conductance vs. voltage plot for data taken on the 17 A nano-tube. The bottom inset shows an I-V trace taken on a gold surface under the same conditions [19.129],

Fig. 19.40. Experimental points for the energy gap vs. 100/d„ the inverse nanotube diameter where d, is in A, for nine semiconducting carbon nanotubes. The dashed line is a fit to these points; the full line corresponds to a calculation [19.86,98] for semiconducting zigzag nanotubes [19.129],

Finally, results for all the semiconducting tubules measured in this study by Oik and Heremans [19.129] are shown in Fig. 19.40, where the energy gaps determined for the individual tubules are plotted vs. 100/d„ the reciprocal tubule diameter [19.129]. Although the band gap vs. 100/dt data in Fig. 19.40 are consistent with the predicted functional form shown in Fig. 19.30, the experimentally measured values of the band gaps are about a factor of 2 greater than that estimated theoretically on the basis of a tight binding calculation [19.86,98]. Further experimental and theoretical work is needed to reach a detailed understanding of these phenomena.

19.6.2. Transport Measurements

The most detailed transport measurements that have been reported thus far have been done on a single bundle of carbon nanotubes to which two gold contacts were attached by lithographic techniques [19.121,124], In this way, the temperature dependence of the resistance R(T) was measured from 300 K down to 200 mK, and the results of R vs. T are shown in Fig. 19.41 on a log T scale. The temperature dependence of the resistance for the tubule bundle was well fit from 2 to 300 K by a simple two-band semimetal model [19.130-132], yielding the electron and hole concentrations n and p as a function of temperature T

n = CnkBT ln[l + exp(EF/kBT)], p = CpkBT ln{l + exp[(A - EF)/kBT]}, (19.36)

where Cn and Cp are fitting parameters and A is the band overlap. The results of this fit are shown as a solid curve in Fig. 19.41, yielding a band

TEMPERATURE [K]

Fig. 19.41. Electrical resistance R as a function of temperature at the indicated magnetic fields for a single bundle of carbon nanotubes covering a range of tubule diameters and chiral angles. The two vertical lines separate the three temperature regimes: (I) showing that R(T,H) for a tubule bundle is consistent with a two-band model for a semimetal; (II) showing saturation effects in R(T,H); and (III) showing anomalous low-T behavior in R(T,H). The continuous curve is a fit using the two-band model for graphite with a band overlap energy of A = 3.7 meV and a Fermi level in the middle of the band overlap. The inset shows a schematic representation of the geometry of the sample and its electrical contacts (not to scale). The carbon nanotube bundle is electrically connected at both ends to two predefined gold pads [19.121].

TEMPERATURE [K]

Fig. 19.41. Electrical resistance R as a function of temperature at the indicated magnetic fields for a single bundle of carbon nanotubes covering a range of tubule diameters and chiral angles. The two vertical lines separate the three temperature regimes: (I) showing that R(T,H) for a tubule bundle is consistent with a two-band model for a semimetal; (II) showing saturation effects in R(T,H); and (III) showing anomalous low-T behavior in R(T,H). The continuous curve is a fit using the two-band model for graphite with a band overlap energy of A = 3.7 meV and a Fermi level in the middle of the band overlap. The inset shows a schematic representation of the geometry of the sample and its electrical contacts (not to scale). The carbon nanotube bundle is electrically connected at both ends to two predefined gold pads [19.121].

overlap of A = 3.7 meV, in contrast to R(T) for 3D graphite, which corresponds to a band overlap of A = 40 meV [19.130]. A smaller value of A for the tubules is expected, since the turbostratic stacking of the adjacent layers within a multiwall tubule greatly reduces the interlayer C-C interaction relative to graphite. Since this interlayer interaction is responsible for the band overlap in graphite, the carrier density for the nanotubes in the limit of low-temperature (T 0) is expected to be smaller by an order of magnitude than to that of graphite. Since the resistance measurement emphasizes the contribution of the metallic tubules in an ensemble of metallic and semiconducting tubules, this experiment provides very strong evidence for the presence of metallic tubules. The presence of metallic tubules is confirmed by numerical estimates of the magnitude of the room temperature resistance, assuming that one third of the layers of a multiwall tubule are metallic [19.121]. The corresponding resistivity value of 10~2 to 10"3 ÎÎ-cm was calculated from the data in Fig. 19.41, in rough agreement with the best conductivity of ~100 S-cm reported for bulk arc deposited carbon material (containing about two thirds nanotubes and one third nanoparti-cles) [19.2], and with independent estimates of the resistivity of the black core material from the arc deposit of 10~2 il-cm [19.2], Preliminary resistivity measurements using four current-voltage probes on a single nanotube 20 nm in diameter show a decrease in room temperature resistivity by an order of magnitude (to 10"4 ft-cm) relative to a microbundle of carbon nanotubes 50 nm in diameter [19.133]. Although resistivity measurements are sensitive to the transport properties of the conducting nanotubes, such measurements provide little information about the semiconducting tubules, which hardly contribute to the resistivity measurements.

19.6.3. Magnetoresistance Studies

In addition to the temperature dependence of the zero-field resistance, Fig. 19.41 shows results for the temperature dependence of the resistivity in a magnetic field. According to the theory of Ajiki and Ando [19.119] described in §19.5.4, the application of a magnetic field normal to the tubule axis is expected to introduce a Landau level at the degeneracy point between the valence and conduction bands, thus increasing the density of states at the Fermi level. This increase in the density of states is expected to reduce the resistance of the carbon nanotubes, consistent with the experimental observations, which quantify the magnitude of the negative magnetoresistance. At low-temperatures below 2 K, an unexpected and unexplained behavior of large magnitude is observed both in the zero-field and high-field temperature-dependent resistance. A summary of these results for the magnetoresistance AR/R = [R(H) - R(0)]/R(0) at low tempera-

Fig. 19.42. Magnetic field dependence of the magnetoresistance AR/R at the indicated temperatures for the same sample geometry as shown in the inset to Fig. 19.41. While the magnetoresistance shows a quadratic dependence on magnetic field above T = 1 K, a more complicated behavior is found at lower temperatures [19.121],

Fig. 19.42. Magnetic field dependence of the magnetoresistance AR/R at the indicated temperatures for the same sample geometry as shown in the inset to Fig. 19.41. While the magnetoresistance shows a quadratic dependence on magnetic field above T = 1 K, a more complicated behavior is found at lower temperatures [19.121],

MAGNETIC FIELD [T]

MAGNETIC FIELD [T]

ture is shown in Fig. 19.42, and may be related to the universal conductance fluctuations discussed below.

In contrast to the two-terminal transport experiments described above for a single bundle of nanotubes (< 1 /xm diameter), four-terminal transport measurements, including p(T), transverse magnetoresistance, and Hall effect experiments, have been reported on samples much larger in size (60 /i,m diameter and 350 /¿m in length between the two potential contacts) [19.134], The magnetoresistance results of Fig. 19.43 are explained in a preliminary way in terms of a semimetal model. Regarding the transverse magnetoresistance and Hall effect measurements (where the current is along the axis of the nanotube bundles and the magnetic field is normal to the current), complicated behavior is observed as a function of temperature and magnetic field. At high temperature, the magnetoresistance can be described classically, yielding magnitudes for the electron and hole contributions to the conductivity and for the relative mobilities of holes and electrons (np/ixn ~ 0.7 and p300 K = 6.5 x 10"3 iî-cm). As T decreases, ixp increases and the contributions to the transport properties of holes increase relative to that for electrons, so that the low-temperature transport is dominated by holes, as is common in disordered carbon-based systems [19.60].

Negative magnetoresistance is found at low T (<77 K) and low H (see Fig. 19.43), while positive magnetoresistance contributions dominate the measurements at high temperatures and also at high fields. The authors had some success in fitting the experimental results for the temperature-

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