Environmental Engineering Reference
In-Depth Information
∞
2
2
e
ω/
(
k
B
T
)
(
e
ω/
(
k
B
T
)
ω
σ
(
T
)
=
d
ω
−
1)
2
(
ω
),
(3.8)
2
π
k
B
T
2
0
where
k
B
is the Boltzmann constant.
Thenweshowthatmanyimportantfeaturesofmesoscopicther-
mal transport can be revealed by analyzing the Landauer formula
for thermal conductance (see Eq. 3.8). For ballistic transport,
ω
(
)
ω
isequaltothenumberofphononmodesoffrequency
withpositive
group velocity.
(
ω
) is an integer number and will not change
when the transport length
L
is increasing. Thus,
σ
(
T
) is constant
(independentof
L
)atagiventemperature.Thethermalconductivity
κ
∝
σ
L
isproportional to
L
and diverges for infinitelarge
L
.
Thermalconductanceisaweightedintegralofphonontransmis-
sion function, as shown by the Landauer formula. The weighting
factor is strongly dependent on the phonon frequency. Defining
x
=
ω/
(
k
B
T
),Eq.3.8 changes into
k
B
T
x
.
∞
k
B
T
h
x
2
e
x
(
e
x
σ
(
T
)
=
−
1)
2
dx
(3.9)
0
1)
2
)asa
function of
x
. It clearly shows that the weighting factor deceases
very fast with an increasing
x
. When
x
>
10, the weighting factor is
nearlyzero.Thisindicatesthatlower-frequencyphononshavemuch
larger contribution to thermal conductance than higher-frequency
phonons. The phonons of very high frequencies,
ω<
10
k
B
T
/
,
have negligible contribution to thermal conductance. At very low
temperatures (
T
close to zero), only those acoustic phonons of
nearly zero
ω
are excited by the temperature. They can give
quantizedthermal conductance aswewill discussnext.
Figure 3.2 presents the weighting factor (
x
2
e
x
/
(
e
x
−
3.1.5
Quantized Thermal Conductance
Whenthetransmissionfunction
(
ω
)
≡
1,Eq.3.9givesthequantum
of thermal conductance as
σ
0
=
π
2
k
B
T
3
h
.
(3.10)
10
−
13
W/K
2
)
T
represents the maximum possible
value of energy transported per phonon mode [5, 6]. Different from
σ
0
=
(9.456
×