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of the sheet is collapsed onto one dimension; because of the periodic boundary conditions on the tube, the circumferential wavevector is quantized and discrete 'subbands' develop. From a zone-folding picture, the splitting between the subbands at the r point is of order [1]

where R is the radius of the nanotube and v is the band velocity of the relevant graphene mode. The second effect of rolling the graphene sheet is to rearrange the low-energy acoustic modes. For the nanotube there are now four, rather than three, acoustic modes: an LA mode, corresponding to motion of the atoms along the tube axis, two degenerate TA modes, corresponding to atomic displacements perpendicular to the nanotube axis, and a 'twist' mode, corresponding to a torsion of the tube around its axis. The LA mode is exactly analogous to the LA mode in graphene. The TA modes in a SWNT, on the other hand, are a combination of the in-plane and out-of-plane TA modes in graphene, while the twist mode is directly analogous to the in-plane TA mode. These modes all show linear dispersion (there is no nanotube analogue to the ZA mode) and high phonon velocities: vla = 24km/s, vta = 9km/s, and vtwist = 15km/s for a (10,10) tube [3]. Because all of the acoustic modes have a high velocity, the splitting given in (4) corresponds to quite high temperatures, on the order of 100 K for a 1.4 nm-diameter tube. In the calculated band structure for a (10,10) tube, the lowest subband enters at ~2.5meV (30K), somewhat lower in energy than the estimate given by (4).

The inset to Fig. 1 shows the low-energy phonon density of states p(w) of a (10,10) nanotube (solid line), with p(w) of graphene (dot-dashed line) and graphite (dashed line) shown for comparison. In contrast to 2-D graphene and 3-D graphite, which show a smoothly-varying p(w), the 1-D nanotube has a step-like p(w), which has 1-D singularities at the subband edges. The markedly different phonon density of states in carbon nanotubes results in measurably different thermal properties at low temperature.

At moderate temperatures, many of the phonon subbands of the nanotube will be occupied, and the specific heat will be similar to that of 2-D graphene. At low temperatures, however, both the quantized phonon structure and the stiffening of the acoustic modes will cause the specific heat of a nanotube to differ from that of graphene. In the low T regime, only the acoustic bands will be populated, and thus the specific heat will be that of a 1-D system with a linear w(k). In this limit, T ^ hv/kBR, (1) can be evaluated analytically, yielding a linear T dependence for the specific heat [1]:

where pm is the mass per unit length, v is the acoustic phonon velocity, and R is the nanotube radius. Thus the circumferential quantization of the nanotube phonons should be observable as a linear C(T) dependence at the lowest temperatures, with a transition to a steeper temperature dependence above the thermal energy for the first quantized state.

Turning to the electron contribution, a metallic SWNT is a one-dimensional metal with a non-zero density of states at the Fermi level. The electronic specific heat will be linear in temperature [1]:

3hvFpm for T ^ hvF/kBR, where vF is the Fermi velocity and pm is again the mass per unit length. The ratio between the phonon and the electron contributions to the specific heat is [1]

Ce V

so that even for a metallic SWNT, phonons should dominate the specific heat all the way down to T = 0. The electronic specific heat of a semiconducting tube should vanish roughly exponentially as T ^ 0 [10], and so Ce will be even smaller than that of a metallic tube. However, if such a tube were doped so that the Fermi level lies near a band edge, its electronic specific heat could be significantly enhanced.

1.3 Specific Heat of SWNT Ropes and MWNTs

As was mentioned above, stacking graphene sheets into 3-D graphite causes phonon dispersion in the c direction, which significantly reduces the low-T specific heat. A similar effect should occur in both SWNT ropes and MWNTs. In a SWNT rope, phonons will propagate both along individual tubes and between parallel tubes in the hexagonal lattice, leading to dispersion in both the longitudinal (on-tube) and transverse (inter-tube) directions. The solid lines in Fig. 2 show the calculated dispersion of the acoustic phonon modes in an infinite hexagonal lattice of carbon nanotubes with 1.4 nm diameter [6]. The phonon bands disperse steeply along the tube axis and more weakly in the transverse direction. In addition, the 'twist' mode becomes an optical mode because of the presence of a nonzero shear modulus between neighboring tubes. The net effect of this dispersion is a significant reduction in the specific heat at low temperatures compared to an isolated tube (Fig. 3). The dashed lines show the dispersion relations for the higher-order subbands of the tube. In this model, the characteristic energy ^b^d of the inter-tube modes is

5meV), which is larger than the subband splitting energy, so that 3-D dispersion should obscure the effects of phonon quantization. We will address the experimentally-measured inter-tube coupling below.

The phonon dispersion of MWNTs has not yet been addressed theoretically. Strong phonon coupling between the layers of a MWNT should cause roughly graphite-like behavior. However, due to the lack of strict registry

Fig. 2. Calculated dispersion of the acoustic phonon modes in an infinite rope of 1.4 nm diameter SWNTs [6]. The phonon velocity is high in the longitudinal direction and lower in the transverse direction. The first two higher-order subbands (dashed lines) are shown for comparison between the layers in a MWNT, the interlayer coupling could conceivably be much weaker than in graphite, especially for the twist and LA modes, which do not involve radial motion. The larger size of MWNTs, compared to SWNTs, implies a significantly smaller subband splitting energy (4), so that the thermal effects of phonon quantization should be measurable only well below 1 K.

1.4 Measured Specific Heat of SWNTs and MWNTs

The various curves in Fig. 3 show the calculated phonon specific heat for isolated (10,10) SWNTs, a SWNT rope crystal, graphene, and graphite. The phonon contribution for a (10,10) SWNT was calculated by computing the phonon density of states using the theoretically derived dispersion curves [3] and then numerically evaluating (1); C(T) of graphene and graphite was calculated using the model of Al-Jishi and Dresselhaus [12]. Because of the high phonon density of states of a 2-D graphene layer at low energy due to the quadratic ZA mode, 2-D graphene has a high specific heat at low T. The low temperature specific heat for an isolated SWNT is however significantly lower than that for a graphene sheet, reflecting the stiffening of the acoustic modes due to the cylindrical shape of the SWNTs. Below ~ 5K, the predicted C(T) is due only to the linear acoustic modes, and C(T) is linear in T, a behavior which is characteristic of a 1-D system. In these curves, we can see clearly the effects of the interlayer (in graphite) and inter-tube (in SWNT ropes) dispersion on the specific heat. Below ~50K, the phonon specific heat of graphite and SWNT ropes is significantly below that of graphene or isolated (10,10) SWNTs. The measured specific heat of graphite [8,9] matches the phonon

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