Specific heat of single-walled carbon nanotubes (SWNTs)
The main contribution to the specific heat of nanotube systems is the vibrational one because the electronic one is negligible even at a few Kelvin[1]. The quasi-one-dimensionality of the nanotube systems has as a consequence the existence of four acoustic branches, which can result in a specific behaviour of the phonon specific heat at low temperature (LT) T. It can be analyzed using the non-interacting phonon picture and the quantum-statistical expression for the specific heat of a system of Bose particles
where 
is the phonon energy
and D(ω) is the phonon density of states (PDOS). The
high-temperature (or classical) limit of this expression does not depend on the
particular structure of the carbon system and is equal to 3kB/m ~ 2078 mJ/gK with m
being the atomic mass of carbon. The LT behavior of C is closely connected to the dimensionality of the system. For low
enough temperatures, when the population of the lowest optical branches can be
ignored, the specific heat is determined by the acoustic ones alone. If ωo is the frequency of the lowest-energy
optical phonon, then the optical phonons contribution to C can be ignored for temperatures below To ~ ħωo/6kB for which the factor
multiplying D(ω) becomes smaller than say 0.1. In
the interval below To, C(T) can be derived from the expression
above once the acoustic-phonon dispersion is known.
For the 3d system graphite, for any of the three acoustic branches ω ~ q, therefore, D(ω) ~ ω2 and C(T) ~ T3. For the 2d system graphene, for the in-plane longitudinal acoustic (LA) and transverse acoustic (TA) phonons ω ~ q, D(ω) ~ ω and C(T) ~ T2; for the out-of-plane acoustic (ZA) phonons ω ~ q2, D(ω) = const and C(T) ~ T.
For any nanotube, for the longitudinal acoustic (LA) and twist acoustic (TW) phonons ω ~ q, D(ω) = const and C(T) ~ T. For the transverse acoustic (TA) phonons ω ~ q2, D(ω) ~ ω–1/2 and C(T) ~ T1/2. The specific heat will follow these power laws below To which depends on the value of ωo. For example, for tubes (10,10) ωo ~ 20 cm-1 and To ~ 5 K. At very low temperature, C will increase as T1/2 while nearing To from below C will increase linearly with T. At temperatures higher than To, C(T) will be determined by the optical phonon branches. The peculiar T1/2 dependence has been corroborated by recent experimental data (for details, see [1]).
References
1.
V. N.
Popov, Phys. Rev. B 66 (2002) 153408-1/4.
Valentin Popov