Environmental Engineering Reference
In-Depth Information
Hot
(
T
H
)
Cold
(
T
C
)
ζ
m
(ω)
1
Quasi-1D phonon conductor
R
(ω)
R
m
(ω)
Figure 2.1
Landauer model for phonon transport in a quasi-1D conductor.
The conductor is connected to two ideal leads.
as phonon-phonon and electron-phonon scatterings, is connected
to ballistic quasi-1D leads without any scattering, which are in turn
connected to heat reservoirs. The left and right heat reservoirs
are assumed to be in thermal equilibrium with well defined
temperatures
T
H
and
T
C
(
T
H
), respectively. In this situation, the
relaxation of phonons occurs only in the heat reservoirs. Thus, a
phonon injected from the left and the right heat reservoirs follows
the Bose-Einstein distribution function
f
(
<
ω
ω
,
T
H
)and
f
(
,
T
C
),
respectively. Here,
ω
isthe frequency of the injected phonon.
Let us derive a general expression of thermal current
J
and
thermal conductance
κ
for the above situation. For simplicity, we
consider a symmetric system having the same leads for the left and
right regions. The thermal current carried by phonons with mode
m and frequency
ω
injected from the left heat reservoir toward the
central conductor is givenby
j
m
(
ω
)
=
ω
|
v
m
(
ω
)
|
D
m
(
ω
)
f
(
ω
,
T
H
)
=
1
2
π
ω
f
(
ω
,
T
H
)
(2.2)
while the one injected from the right heat reservoir toward the
central conductor is givenby
j
m
(
D
m
(
=
ω
|
|
ω
)
v
m
(
ω
)
ω
)
f
(
ω
,
T
C
)
1
2
π
ω
f
(
ω
,
T
C
)
=
(2.3)
In the first lines in Eqs. 2.2 and 2.3,
D
m
(
) denotes the phonon
density of states for a unit length in the left and the right leads
ω
