Environmental Engineering Reference
In-Depth Information
where
M
(
ω
) is the number of phonon modes with
ω
[27]. On
the other hand, for phonon localization, the phonon-transmission
function decays exponentially with
L
accordingtothescalinglaw
[27]:
L
ξ
(
ω
)
.
ln
ζ
(
ω
)
=−
(2.31)
In other words, the localization length
ξ
(
ω
)isdefinedbythe
scaling law. To clarify the mechanism for the phonon-transmission
reduction,the
L
-dependencesofphonontransmissionareplottedin
Fig. 2.9b and c for the two mechanisms, respectively. As Fig. 2.9b
shows, the numerical data of
<ζ
(
ω
)
>
at
ω
=
34 cm
−
1
and 391
cm
−
1
are well fitted by the dashed lines. In particular, the slope of
thedashedlinefor
ω
=
34 cm
−
1
is almost zero, implying that the
mean free path
l
MFP
(
ω
) is very long and the phonon transport is
ballistic at this frequency, as has been discussed above. For
391
cm
−
1
, the slope is finite, which indicates that phonon transport
at this frequency is in the diffusive regime. In contrast, at higher
frequencies (
ω
=
1071, 1207, and 1513 cm
−
1
), the calculated
values deviate from the dashed lines with increasing
L
, although
they are well fitted in the short-
L
region. This deviation means that
the phonon-transmission reduction for high-
ω
phonons of a long-
L
SWNT cannot be explained by the diffusive scattering mechanism.
As seen in Fig. 2.9c, the data for
ω
=
1071, 1207, and 1513 cm
−
1
are well fitted by the dashed lines in the
<ζ
(
ω
)
>
plot. We can
thus conclude that the phonon-transmission reduction for high-
ω
phonons in a long-
L
SWNT is caused by Anderson localization of
phonons.
The mean free path
l
MFP
(
ω
) and the localization length
ξ
(
ω
)can
be estimated from the slope of dashed lines in Figs. 2.9b and c,
respectively. The estimated
l
MFP
(
ω
)and
ξ
(
ω
) for the (5,5) SWNT
with 15%
13
C are presented in Fig. 2.9d. Thus, the three distinct
regimes (ballistic, diffusive, and localization) could be clarified.
Similar results have been obtained for the (8,0) semiconducting
SWNT with 9.4%
14
C. In the following subsections (Section 2.3.3.2
and Section 2.3.3.3), we discuss the phonon transport properties of
the diffusiveand the Anderson localizationregimes, respectively.
ω
=
