Environmental Engineering Reference
In-Depth Information
(
(
10
10
LA
TW
FL
5
5
0
0
0
0.1
0.2
0
4
8
Transmission ζ(ω)
Wave number
ka
/π
Figure 2.5
(a) The low-energy phonon dispersion relations. There are 4
acoustic modes: the longitudinal acoustic (LA), twisting (TW) modes, and
two flexure modes. Here,
a
denotes the length of the unit cell in the (10,
10)single-walledcarbonnanotube(SWNT).(b)Thecorrespondingphonon
transmission functionof the (10,10) SWNT, respectively.
low-energy dispersion relations, independent of the tube geometry
such as diameter or chirality.
In the case of the ballistic phonon transport, the phonon
transmission function is given as a number of phonon dispersion
branches, as shown in Eq. 2.8. In the low-temperature limit where
the thermal energy
k
B
T
is much lower than the excitation energy of
the lowest optical modes, the phonon-derived thermal conductance
of SWNTs exhibits 4
κ
0
because there are 4 conduction channels
as mentioned previously and seen in Fig. 2.5b [12]. In fact, the
quantization of thermal conductance of nanotubes was observed
experimentally [11].
2.3.2
Thermal Conductance Reduction by Defect Scattering
In this subsection, we apply the Landauer-NEGF formalism to the
thermal transport in defective CNTs [13]. Figure 2.6 represents the
phonon transmission function for the (8,8) SWNT with and without
defects. The dashed curve is the transmission function
ζ
p
(
ω
)for
the perfect (8,8) SWNT without any defects, and it displays a clear
stepwise structure that gives the number of phonon channels. In
