Environmental Engineering Reference
In-Depth Information
regime of (a) 625 nm-long (5,5) SWNT with 15%
13
C and (b) 210
nm-long(8,0)SWNTwith9.4%
14
C.AllthehistogramsinFig.2.11a,b
are well described by a Gaussian distribution function with the
universalfluctuation
ζ
(
ω
)
=
0.35
±
0.02inEq.2.33.Thisissimilar
to the fact that the electrical conductance histogram in the diffusive
regionisexpressedbyaGaussiandistributionfunctionwiththeUCF
[26]. In the localization regime,
P
(
ζ
) becomes an asymmetric non-
Gaussiandistributionwithalongtailtowardslarge
ζ
(
ω
).Byanalogy
withtheelectricalconductancehistograminthelocalizationregime
[27, 28], the asymmetric histogram is expected to be a lognormal
function of
ζ
.Infact,
P
(ln
ζ
) can be well described by a Gaussian
distribution, as seen in Fig. 2.11c,d. Unlike the diffusive regime, the
variance Var[ln
)
2
of
P
(ln
ζ
) decreases with increasing
ζ
]
≡
(
ln
ζ
ζ
<
as seen in the insets
of Fig. 2.11c,d, which is similar to the situation for electrons [27].
The phonon-transmission fluctuation in the localization regime is
material independent in the sense that the slope of Var[ln
ln
ζ>
according to Var[ln
ζ
]
=−
2
ln
ζ
] does
not depend on the tube geometry, isotope concentration, or the
difference between
13
Cand
14
C.
2.4 Concluding Remarks
Here,wefirstsummarizetheadvantagesofNEGFmethodasfollows:
(i) It is possible to calculate the fully-quantum phonon transport
by NEGF method, while it is di
cult to do that by non-equilibrium
molecular dynamics methods. (ii) Because the NEGF method is
excellent with the recursive algorithm as explained in Section 2.2.3,
the phonon transport in large system, which is comparable to
realistic materials, can be computed by NEGF method. (iii) The
many-bodyinteractions,suchasphonon-phononinteraction,canbe
systematically incorporated into the NEGFformalism [29].
In this chapter, as applications of NEGF method, we review
the influence of various defects on thermal-transport properties
of carbon nanotube. We show that the carbon nanotube exhibits
ballistic phonon transport at low temperature even in the presence
of defects and eventually the thermal conductance of carbon
nanotubes are quantized as the universal value at low temperature.
