Environmental Engineering Reference
In-Depth Information
Figure 2.9
(a) The average phonon transmission
<ζ
(
ω
)
>
of the (5,5)
SWNT with 15%
13
Cfor
L
=
0.005
μ
m (circles), 0.05
μ
m (rectangles),
0.5
μ
m (triangles), and 5
μ
m (crosses). Averaging was performed over
200 random configurations. (b) The
L
-dependence of
M
(
ω
)
/<ζ
(
ω
)
>
-1 for
severalfrequencies,where
M
(
)isthenumberofphononmodesofpristine
SWNT. The dashed lines indicate linear fits for estimating the mean free
path. (c) The
L
-dependence of
ω
<
ζ>
for several frequencies. The dashed
lines represent linear fits for estimating the localization length. (d) The
meanfreepath(circles)andthelocalizationlength(rectangles)asfunctions
of
ln
ω
.
ξ
ω
localization regime for
L
(
) [24, 25]. Thus, we first determine
ω
ξ
ω
l
MFP
(
) for isotope-disordered SWNTs as follows.
Figure 2.9a shows the average phonon transmission
)and
(
<ζ
ω
>
of
the (5,5) SWNT with 15%
13
C for various
L
up to 5
μ
m. In the very
low-frequencyregion,
<ζ
(
ω
)
>
doesnotdecreaseandisalmostfour,
evenwhenisotopeimpuritiesarepresent.Perfecttransmission(i.e.,
ballistic transport) is realized because the wavelength of acoustic
phonons in the low-
ω
region is much longer than the length
L
.The
Landauer expression of thermal conductance eventually exhibits
universal quantization of 4
κ
0
at low temperatures even in the
presence of isotope impurities (4 corresponds to the number of
acoustic phonon modes).
In contrast,
<ζ
(
ω
)
>
decreases rapidly in the higher frequency
region with increasing
L
, as seen in Fig. 2.9a. There are two possible
mechanisms for the reduction of
<ζ
(
ω
)
>
: diffusive scattering and
Anderson localization. For diffusive scattering,
<ζ
(
ω
)
>
decreases
with
L
according to
(
)
ω
M
(
)
ζ
ω
)
=
(
)
,
(2.30)
+
/
ω
1
L
l
MFP
(
