## Tk

Fig. 3. Measured specific heat of SWNTs, compared to predictions for the specific heat of isolated (10,10) tubes, SWNT ropes, a graphene layer and graphite [11]

contribution above 5 K, below which temperature the electronic contribution is also important.

The filled points in Fig. 3 represent the measured specific heat of SWNTs [11]. The measured C(T) agrees well with the predicted curve for individual (10,10) nanotubes. C(T) for the nanotubes is significantly smaller than that of graphene below ~50K, confirming the relative stiffness of the nanotubes to bending. On the other hand, the measured specific heat is larger than that expected for SWNT ropes. This suggests that the tube-tube coupling in a rope is significantly weaker than theoretical estimates [6,11].

Figure 4 highlights the low-temperature behavior of the specific heat. The experimental data, represented by the filled points, show a linear slope below 8K, but the slope does not extrapolate to zero at T = 0, as would be expected for perfectly isolated SWNTs. This departure from ideal behavior is most likely due to a weak transverse coupling between neighboring tubes. The measured data can be fit using a two-band model, shown in the inset. The dashed line in Fig. 4 represents the contribution from a single (four-fold degenerate) acoustic mode, which has a high on-tube Debye temperature ©D and a much smaller inter-tube Debye temperature ©¿. The dot-dashed line represents the contribution from the first (doubly-degenerate) subband, with minimum energy kB©subband. Because ©subband > ©¿, the subband is assumed to be essentially one-dimensional. The solid line represents the sum

of the two contributions, and fits the data quite well. The derived value for the on-tube Debye temperature is oJD = 960 K (80meV), which is slightly lower than the value of 1200 K (100 meV) which can be derived from the calculated phonon band structure. The transverse Debye temperature O^ is 13K (1.1 meV), considerably smaller than the expected value for crystalline ropes (5meV) or graphite (10meV). Finally, the derived value of Osubband is 50 K (4.3 meV), which is larger than the value of 30K (2.5meV) given by the calculated band structure [3].

Figure 5 shows the two reported measurements of the specific heat of MWNTs, along with the theoretical curves for graphene, isolated nanotubes, and graphite. Yi et al. [13] used a self-heating technique to measure the specific heat of MWNTs of 20-30 nm diameter produced by a CVD technique, and they find a linear behavior from 10 K to 300 K. This linear behavior agrees well with the calculated specific heat of graphene below 100 K, but is lower than all of the theoretical curves in the 200-300 K range. Agreement with the graphene specific heat, rather than that for graphite, indicates a relatively weak inter-layer coupling in these tubes. Because the specific heat of graphene at low T is dominated by the quadratic ZA mode, the authors postulate that this mode must also be present in their samples. As was discussed above (Sect. 1.1,1.2), such a band should should not exist in nanotubes. However, the phonon structure of large-diameter nanotubes, whose properties should approach that of graphene, has not been carefully studied. Mizel et al. report a direct measurement of the specific heat of arc-produced MWNTs [6]. The specific heat of their sample follows the theoretical curve for an isolated nanotube, again indicating a weak inter-layer coupling, but shows no evi dence of a graphene-like quadratic phonon mode. At present the origin of the discrepancy between the two measurements is unknown, although the sample morphologies may be different.

Fig. 5. Measured specific heat of MWNTs [6,13], compared to the calculated phonon specific heat of graphene, graphite, and isolated nanotubes

Fig. 5. Measured specific heat of MWNTs [6,13], compared to the calculated phonon specific heat of graphene, graphite, and isolated nanotubes

### 2 Thermal Conductivity

Carbon-based materials (diamond and in-plane graphite) display the highest measured thermal conductivity of any known material at moderate temperatures [14]. In graphite, the thermal conductivity is generally dominated by phonons, and limited by the small crystallite size within a sample. Thus the apparent long-range crystallinity of nanotubes has led to speculation [15] that the longitudinal thermal conductivity of nanotubes could possibly exceed the in-plane thermal conductivity of graphite. Thermal conductivity also provides another tool (besides the specific heat) for probing the interesting low-energy phonon structure of nanotubes. Furthermore, nanotubes, as low-dimensional materials, could have interesting high-temperature properties as well [16]. In this section, we will first discuss the phonon and electronic contributions to the thermal conductivity in graphite. Then we will examine the thermal conductivity of multi-walled and single-walled nanotubes.

The diagonal term of the phonon thermal conductivity tensor can be written as:

where C, v, and t are the specific heat, group velocity, and relaxation time of a given phonon state, and the sum is over all phonon states. While the phonon thermal conductivity cannot be measured directly, the electronic contribution Ke can generally be determined from the electrical conductivity by the Wiedemann-Franz law:

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