The thermal characteristics of carbon nanotubes are related to those of the two ) dimensional graphene, yet the rolling up induces a quantization of these properties. It shows most in nanotubes less than 2 nm in diameter.
The specific heat capacity of a substance comprises an electronic contribution Ce- and a contribution CPh ofthe phonons. The latter is dominant in carbon nanotubes, regardless of their structure. CPh is obtained from integration over the density function of phonon states and subsequent multiplication by a factor that considers the energy and the population of individual phonon levels.
Below the Debye temperature, only the acoustic modes contribute to heat capacity. It turns out that within a plane there is a quadratic correlation to the temperature, whereas linear behavior is observed for a perpendicular orientation. These assumptions hold for graphite, which indeed exhibits two acoustic modes within its layers and one at right angles to them. In carbon nanotubes, on the other hand, there are four acoustic modes, and they consequently differ from graphite in their thermal properties. Still at room temperature enough phonon levels are occupied for the specific heat capacity to resemble that of graphite. Only at very low temperatures the quantized phonon structure makes itself felt and a linear correlation of the specific heat capacity to the temperature is observed. This is true up to about 8 K, but above this value, the heat capacity exhibits a faster-than-linear increase as the first quantized subbands make their contribution in addition to the acoustic modes.
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