where the Lorenz number L0 = 2.45 x 10~8 (V/K)2. Thus it is in principle straightforward to separate the electronic and lattice contributions to k(T). In graphite, phonons dominate the specific heat above ~20K [17], while in MWNTs and SWNTs, the phonon contribution dominates at all temperatures.

In highly crystalline materials and at high temperatures (T > Od/ 10), the dominant contribution to the inelastic phonon relaxation time t is phonon-phonon Umklapp scattering. At low temperatures, however, Umklapp scattering disappears and inelastic phonon scattering is generally due to fixed sample boundaries or defects, yielding a constant t. Thus at low temperature (T < Od/10), the temperature dependence of the phonon thermal conductivity is similar to that of the specific heat. However, in an anisotropic material, the weighting of each state by the factor v2t becomes important. The thermal conductivity is most sensitive to the states with the highest band velocity and scattering time. In graphite, for instance, the a5-plane thermal conductivity can be closely approximated by ignoring the inter-planar coupling [17]. From this argument, we would expect that the temperature-dependence of the thermal conductivity of SWNT ropes and MWNTs should be close to that of their constituent tubes. However, bundling individual tubes into ropes or MWNTs may introduce inter-tube scattering, which could perturb somewhat both the magnitude and the temperature dependence of the thermal conductivity.

2.1 Thermal Conductivity of MWNTs

In highly graphitic fibers, k(T) follows a

temperature dependence until ~ 100 K, then begins to decrease with increasing T above ~150K [18]. This decrease in k(T) above 100 K is due to the onset of phonon-phonon Umklapp scattering, which grows more effective with increasing temperature as higher-energy phonons are populated. In less graphitic fibers, the magnitude of k is significantly lower, and the Umklapp peak in k(T) is not seen, because grain-boundary scattering dominates k(T) to higher temperatures.

Figure 6 shows the thermal conductivity of CVD-grown MWNTs, on a linear scale, from 4K to 300 K [13]. Because of the large diameter of these tubes, we expect them to act essentially as 2-D phonon materials. Indeed, at low temperature (T < 100K), k(T) increases as ~ T2, similar to the T2 3 behavior in graphite. The room-temperature thermal conductivity is small, comparable to the less-graphitic carbon fibers, and the MWNTs do not show a maximum in k(T) due to Umklapp scattering; both properties are consistent with a small crystallite size.

Fig. 6. Measured thermal conductivity of MWNTs [13] from 4K to 300 K

2.2 Thermal Conductivity of SWNTs

Figure 7 represents the measured k(T) of a bulk sample of laser-vaporization produced SWNTs, with ~1.4-nm diameter [20]. The different temperature-dependence of k(T) reflects the much smaller size of SWNTs compared to MWNTs. k(T) increases with increasing T from 8K to 300 K, although a gradual decrease in the slope above 250 K may indicate the onset of Umklapp scattering. Most striking is a change in slope near 35 K: below this temperature, k(T) is linear in T and extrapolates to zero at T=0. We will discuss the low-temperature behavior in detail below.

Although the temperature dependence of the thermal conductivity is the same for all 1.4 nm diameter SWNT samples, the magnitude of k(T) is sensitive to sample geometry. In disordered 'mat' samples, the the room-temperature thermal conductivity is ~35W/mK. However, in samples consisting of aligned SWNTs, the room-temperature thermal conductivity is

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