Info

1000 3000 5000

Magnetic Field (Oe)

Fig. 19.47. Magnetic field dependence of the magnetic moment of carbon nanotubes at the indicated temperatures [19.122].

appears to be weakly dependent on H (see Fig. 19.48), but of course the magnitude of x varies strongly with temperature [19.122].

The temperature dependence of x for tubules is, however, remarkably different from that of graphite as shown in Fig. 19.48(i) where a large T dependence of x f°r carbon nanotubes is seen, in contrast to the much weaker temperature dependence for graphite. In fact, x f°r the tubules approaches the graphite value in the low-temperature limit [19.122,123]. Whereas the susceptibility is nearly constant as a function of magnetic field for 0.5 < H < 1.5 tesla, it is strongly temperature dependent, as shown in Fig. 19.48(ii), where the susceptibility x is plotted vs. temperature for several values of magnetic field. Anomalous behavior is observed at low fields, as seen in Fig. 19.45 [19.123] and in more detail in Fig. 19.46 [19.122], suggesting a strong field dependence of x at low H. No explanation of the anomalous low-temperature and low field dependences of x for carbon nanotubes has yet been published, and these phenomena await further study.

Referring to Fig. 19.46, it is seen that the magnetic field of 2 T used to acquire the data in Fig. 19.48(i) corresponds to the intermediate-field regime of Fig. 19.46. In this regime the Landau radius rL = (ch/eH)1/2 is becoming comparable to the largest tubule radius in the sample. Consequently, the data in Fig. 19.48(ii) show three different temperature-dependent behaviors [19.122], consistent with the three regimes of Fig. 19.46. Some possible explanation for the large temperature dependence of x at l°w T may arise from the following two effects. Since the band gaps and band overlaps for some of the tubules in the sample are small, the derivatives of the Fermi function become temperature dependent at low temperatures. In addition, carrier scattering effects may contribute to the observed x- In contrast, at high fields (e.g., 5 tesla), Fig. 19.48(ii) shows only a weak temperature dependence for x [19.122], At these high fields, rL has become comparable to the tubule size, and the electron orbits become spatially localized, thereby not being sensitive to the tubule curvature. Thus in the high-field regime, graphitic behavior is expected. The results in Fig. 19.48(ii) at 5 tesla are indeed similar to observations in graphite [19.141],

Using ESR techniques, the g-values for aligned carbon nanotubes have been measured [19.142], showing strong evidence for a g-factor given by g(d) = [(g„ cos 0? + (gx sin Off2 (19.38)

where g,, = 2.0137 and gx = 2.0103. These values for the g-factor yield an average value of gav = 2.012 that is close to the average value for graphite, which is gav = (gc + 2gab)/3 = 2.018, where gc = 2.050 and gab = 2.0026. Although carbon nanotubes are significantly less anisotropic than graphite, the average g-factor for carbon nanotubes is quite similar to

(i)

Fig. 19.48. Temperature dependence of the magnetic susceptibility, (i) Measured in a magnetic field of 2 T for various samples: (a) CM powder; (b) polycrystalline graphite anode; (c) gray-shell anode material; (d) a bundle of carbon nanotubes: H as perpendicular to the axis of the bundle, (e) a bundle of carbon nanotubes: H m parallel to the axis of the bundle [19.123], (ii) Measured at several magnetic fields for a bundle of carbon tubules [19.122].

Fig. 19.48. Temperature dependence of the magnetic susceptibility, (i) Measured in a magnetic field of 2 T for various samples: (a) CM powder; (b) polycrystalline graphite anode; (c) gray-shell anode material; (d) a bundle of carbon nanotubes: H as perpendicular to the axis of the bundle, (e) a bundle of carbon nanotubes: H m parallel to the axis of the bundle [19.123], (ii) Measured at several magnetic fields for a bundle of carbon tubules [19.122].

that for graphite. Both and gL exhibit a linear increase in magnitude as T is decreased from 300 K to 20 K, with low-temperature (20 K) values of gi( ~ 2.020 and gx ~ 2.016 [19.142], From measurements of the spin-lattice relaxation time a value of 10 3 s was estimated for the room temperature resistive scattering time, yielding an estimate of 10~3 il-cm for the resistivity of carbon nanotubes, consistent with transport measurements (see §19.6.2).

19.6.5. Electron Energy Loss Spectroscopy Studies

Electron energy loss spectroscopy (EELS) measurements on samples removed from the black ring material from a carbon arc discharge showed differences regarding the corresponding spectrum of graphite. These early measurements further show polarization effects depending on whether the incident electron beam is parallel or perpendicular to the axis of the tubule bundle contained in the deposit (see Fig. 19.49). It is not surprising that the EELS spectra for the nanotube soot deposit are different from that of graphite, considering the variety of lower-dimensional materials present in the nanotube soot deposit, and these materials have lower carrier densities and conductivities than graphite. The EELS measurements on the tubule bundle in Fig. 19.49 are sensitive to differences between the spectrum for the incident electron beam along the tubule axis of the bundle and perpendicular to this axis [Fig. 19.49(c)], These differences in the spectra show a shift of the EELS peaks for the nanotube soot to lower energies than in graphite, consistent with the lower carrier density of the as-prepared nano-tubule deposit [19.38]. Based on these preliminary results, quantitative EELS measurements on tubules with well-characterized d, and 9 would be very interesting.

A step was taken toward achieving this goal when electron energy loss spectra were taken of individual carbon tubules, characterized for their outer diameter and number of cylindrical layers [19.143]. The results below 50 eV show a shift of the dominant electron energy loss peak due to collective a bond excitations from 27 eV in graphite to lower energies in carbon nanotubes. As the tubule diameter decreases, the it electron peak is quenched (see Fig. 19.49), consistent with a lower carrier density for decreasing tubule diameter d„ and consequently a smaller amount of screening. The greater average carrier localization, arising from the greater tubule curvature for the smaller-diameter tubules, may also contribute to the downshift of the electron energy loss peak [19.143]. Tubules of equal diameter, but having a greater number of layers, tend to have better structural order and therefore appear to have a more graphitic spectra. Physically, as the tubule diameter decreases, the curvature of the tubule surface increases, thereby reducing the interlayer correlation and changing the it

Fig. 19.49. (a) Low energy electron energy loss spectroscopy (EELS) traces at low electron energies for carbon nanotubes compared to those for graphite, (b) Core electron EELS spectra of graphite and carbon nanotubes in the case I geometry (the electron beam normal to the graphite c-axis). The spectra for graphite and the carbon nanotubes are virtually identical, (c) Core electron EELS spectra of graphite and carbon nanotubes in the case II geometry (the electron beam parallel to the c-axis of graphite and parallel to the tube axis). The additional cr* contribution to the nanotube spectrum is attributed to the curved nature of the nanotube graphene sheets [19.38], electron contribution to the interlayer coupling between concentric layers [19.143].

19.7. Phonon Modes in Carbon Nanotubes

In this section we summarize present knowledge of the phonon dispersion relations for carbon nanotubes, starting with a review of theoretical calculations of the dispersion relations (§19.7.1), followed by predictions for Raman and infrared activity in carbon nanotubes (§19.7.2). A summary is then given of the experimental observations of Raman spectra in car bon nanotubes (§19.7.3). No experiments have yet been reported on the infrared spectra of carbon nanotubes.

19.7.1. Phonon Dispersion Relations

The phonon modes for carbon nanotubes have been calculated based on zone folding of the phonon dispersion relations for a two-dimensional (2D) graphene layer [19.84], in analogy to the calculations previously described for the electronic energy band structure (see §19.5). Phonon dispersion relations have been obtained for nanotubes described by symmorphic and nonsymmorphic symmetry groups, and the results are discussed below.

The phonon dispersion relations for a 2D graphene sheet are well established (see Fig. 19.50) and both first principles [19.144] and phenomeno-logical [19.145,146] calculations are available. To obtain the general form of the dispersion relations, it is convenient to use a force constant model [19.84] reflecting the detailed symmetry of the carbon nanotubes. For such a force constant model calculation, the latest values for the force constants for graphite are used, thereby neglecting the effect of tubule curvature on the force constants. Values for the radial (4>in)) and tangential [4>("} (inplane) and <f>\f (out-of-plane)] bending force constants for graphite going out to fourth-neighbor (n) distances for displacements in the plane and normal to the plane are given in Table 19.7 [19.84].

Fig. 19.50. The phonon dispersion relations for a 2D graphene sheet plotted along high-symmetry directions, using the force constants in Table 19.7. The points from neutron scattering studies in 3D graphite [19.147] were used in the modeling to obtain the phonon dispersion relations throughout the Brillouin zone [19.84].

Fig. 19.50. The phonon dispersion relations for a 2D graphene sheet plotted along high-symmetry directions, using the force constants in Table 19.7. The points from neutron scattering studies in 3D graphite [19.147] were used in the modeling to obtain the phonon dispersion relations throughout the Brillouin zone [19.84].

Values of the force constant parameters for graphite out to fourth-neighbor interactions in 10" dyn-cm [19.84],

Radial Tangential tfrP = 36.50 ^ = 24.50 ^ = 9.82 0<2) = 8.80 = -3.23 = -0.40

0<4) = -1.92 <t>(? = 2.29 4>™ = -0.58

As discussed in §19.4.1, the unit cell of each carbon tubule is defined by the integers (n, m), which specify the tubule diameter and chirality [see Eqs. (19.2) and (19.3)] and also specify the ID unit cell for the tubule in terms of the chiral vector Ch and the lattice vector T along the cylindrical axis of the tubule [see Fig. 19.2(a)], The normal mode frequencies of the carbon tubules at the T point (k = 0) in the Brillouin zone can be determined by using the method of zone folding, i.e., by determining the frequencies at specific points in the hexagonal Brillouin zone of the 2D graphene sheet which are equivalent to the T point of the ID Brillouin zone of the tubule. Using this approach, the evaluation of the phonon mode frequencies in the tubule requires only the diagonalization of the dynamical matrix of a graphene sheet. This approach, however, neglects the effect of tubule curvature on the phonon dispersion relations.

The phonon dispersion relations in the tubule can thus be obtained from those of the 2D graphene sheet by using the relationship

Was this article helpful?

0 0

Post a comment