Environmental Engineering Reference
In-Depth Information
temperature gradient
dT
/
dx
in contrast to Fourier's law in Eq. 2.2
for diffusive phonons, but is in fact proportional to the temperature
difference
T
. Using Eq. 2.6, we obtain the linear response form of
thermal conductance asthe following expression
∞
0
ω
ω
J
1
2
df
(
)
dT
ζ
κ
≡
T
=
ω
ω
(
T
)
(
)
d
(2.7)
π
This is referred to as Landauer formula of phonon-derived
thermal conductance [2-4]. The thermal conductance of quasi-1D
systems can be estimated by calculating the phonon transmission
function
ζ
(
ω
) ofthe systems.
2.2.2
Ballistic Phonon Transport and Quantization of
Thermal Conductance
According to the Landauer formula in Eq. 2.7, the thermal conduc-
tance is expressed by the phonon transmission function
ζ
(
ω
). For
the moment, we consider an ideal situation where all phonons are
transmittedballisticallythroughaconductorwithoutanyscattering,
namely
ζ
m
(
ω
)
=
1 for all phonon modes. In this ballistic transport
case, the total transmission function isgiven by
ζ
(
ω
)
=
m
ζ
m
(
ω
)
=
M
(
ω
),
(2.8)
where
M
(
ω
) is the number of phonon modes with angular
frequency
ω
.
For quasi-1D systems such as nanowires, there are four acoustic
modes, namely a longitudinal, a twisting, and two flexure acoustic
modes (Fig. 2.2a). This is in a sharp contrast with usual 3D-bulk
materials which have three acoustic modes: a longitudinal and two
transverseacousticmodes.Inthelow-temperatureregionwherethe
opticalphononsarenotexcited,thethermalcurrentflowingthrough
a quasi-1D system is carried only by these four acoustic modes.
Substituting
4 into Eq. 2.7, the low-temperature thermal
conductance isquantized as
ζ
(
ω
)
=
∞
x
2
e
−
x
(
e
x
k
B
T
h
4
π
k
B
T
3
h
κ
=
×
=
≡
4
κ
0
,
(
T
)
4
−
1)
2
dx
(2.9)
0
where
h
isPlanck'sconstantand
κ
0
=
π
k
B
T
3
h
=
(9.4
×
10
−
13
W/K
2
)
×
T
(2.10)
