Thermal Properties of Single-Walled Carbon Nanotubes
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Thermal Properties of Single-Walled Carbon Nanotubes J. Hone1, B. Batlogg2, Z. Benes3, M.C. Llaguno1, N.M. Nemes1, A.T. Johnson1, and J.E. Fischer3 1
Department of Physics and Astronomy and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia PA 19104-6272 2 Bell Laboratories, Lucent Technologies, Murray Hill, NJ 079743 3 Department of Materials Science and Engineering and Laboratory for Research on the Structure of Matter, University of Pennsylvania, Philadelphia PA 19104-6272 ABSTRACT The thermal properties of carbon nanotubes are strongly dependent on their unique structure and size, and show promise as an ideal material for thermal management on the micro- and macro-scale. The specific heat of nanotubes is similar to that of twodimensional graphene at high temperatures, but is sensitive to the effects of rolling the the graphene sheet into a small cylinder at low temperatures. Specifically, the acoustic phonon modes are stiffened due to the cylindrical geometry, and the phonon spectrum is quantized due to the small diameter of the tube. In bundles of single-walled nanotubes, the specific heat is a sensitive probe of inter-tube mechanical coupling. Measurements of the specific heat show that inter-tube coupling is relatively weak, and show direct evidence for quantum effects. The thermal conductivity of nanotubes should reflect the on-tube phonon structure. Aligned bundles of SWNTs show a high thermal conductivity (>200 W/m-K at room temperature), and possible quantization effects at low temperature. SPECIFIC HEAT1 A carbon nanotube can be thought of as a single graphene sheet that is wrapped into a cylinder. Therefore we will first examine the specific heat of such a single sheet. Than we will examine the effects of rolling the sheet into a nanotube. The expected effects of bundling tubes into a hexagonal array will be examined. Finally, the measured specific heat of bulk samples will be compared to the theoretical models. Introduction In graphite and related materials, including nanotubes, the specific heat is dominated by the phonons. The phonon specific heat of a material is determined by its phonon density of states (PDOS):
C ph = k B
∫ (e
x 2e x x
− 1)
2
ρ (ω )dω
x=
hω k BT
(1)
where ρ(ω) is the phonon density of states, as a function of the phonon frequency ω. The convolution factor in the integral is zero at x=0, and decreases to a value of ~0.1 at x=6. A17.1.1
Therefore a feature in the PDOS at an energy E will only begin to contribute to the specific heat (at the 10% level) at a temperature E/6kB. In general, it is not possible to evaluate Eq. (1) analytically. At low temperature, however, only the acoustic phonons (those with ω → 0 as q → 0) are generally occupied. For an acoustic mode with dispersion w ∝ qa, the low-temperature specific heat displays a power-law temperature dependence that depends on both the phonon dispersion and the dimensionality d:
C ph ∝ T
d
α
.
(2)
Graphene
Figure 1. Phonon bandstructure of a single 2D graphene sheet2
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