Environmental Engineering Reference
In-Depth Information
and the superscript
+
(
−
) stands for the density of states for
phononswithpositive (negative) group velocity
v
m
(
ω
).Since
D
m
(
ω
)
is given by
D
m
(
ω
)
=
D
m
(
ω
)/2 with
D
m
(
ω
)
=
1/(
π
|
v
m
(
ω
)
|
) for quasi-
1D systems, it is canceled by the group velocity
v
m
(
ω
) in the second
linein Eqs. 2.2 and 2.3.
As shown in Fig. 2.1,
)] is the transmission (reflec-
tion) function for injected phonons with mode
m
and frequency
ζ
m
(
ω
)[
R
m
(
ω
ω
.
Thenetthermalcurrentattheleftlead,whichiscarriedbyphonons
with
m
and
ω
, isthus givenby
j
m
(
{
}
−
j
m
(
j
m
(
ω
)
=
ω
)
1
−
R
m
(
ω
)
ω
)
ζ
m
(
ω
)
)
j
m
(
)
j
m
(
=
ζ
m
(
ω
ω
−
ω
)
1
2
π
ωζ
m
(
ω
)
{
f
(
ω
,
T
H
)
−
=
f
(
ω
,
T
C
)
}
(2.4)
Similarly, we can obtain the same result as Eq. 2.4 for the
net thermal current at the right lead. This can be interpreted as
Kirchhoff'slawforphonontransport.UsingEq.2.4,thetotalthermal
current, referred to as Landauer's energy flux, can be expressed as
ω
max
m
J
=
j
m
(
ω
)
d
ω
ω
min
m
m
ω
max
m
1
2
π
=
ω
[
f
L
(
ω
)
−
f
R
(
ω
)]
ζ
m
(
ω
)
d
ω
(2.5)
ω
min
m
m
where
ω
min
m
are the maximum and minimum values
of frequency in the phonon dispersion curve for the mode
m
,
respectively.
Let us consider the linear response situation with respect
to the temperature difference
T
=
T
H
-
T
C
between hot and
cold heat reservoir. For a small temperature difference satisfying
T
ma
m
and
ω
<<
(
T
H
-
T
C
)/2, the thermal current is givenby
ω
max
m
T
2
π
df
(
ω
)
dT
ζ
m
(
ω
)
d
ω
J
=
ω
ω
min
m
m
∞
0
ω
ω
T
df
(
)
dT
ζ
ω
ω
, (2.6)
where
ζ
(
ω
)
=
m
ζ
m
(
ω
) is the total transmission function [2-4].
Notice that the thermal current in Eq. 2.6 is not proportional to the
=
(
)
d
2
π
