Environmental Engineering Reference
In-Depth Information
where
i
L
[
ω
] is the thermal current spectral function of the left
thermal lead:
Tr
A
[
ω
]
L
[
ω
]
−
G
<
[
ω
]
L
[
ω
]
.
i
ω
2
π
i
L
[
ω
]
=
(3.56)
Similarly, the thermal current spectral function of the right
thermal lead is
Tr
A
[
R
[
ω
]
−
G
<
[
ω
]
R
[
ω
]
.
i
ω
2
π
i
R
[
ω
]
=
ω
]
(3.57)
The two thermal leads always give same total current for
steady states. However, their spectral functions can be different if
asymmetry exists. A thorough investigation of thermal rectification
requires suchinformation.
3.2.4
Thermal Conductance and Phonon Transmission
Thermal conductance is defined as
I
T
,
σ
=
lim
(3.58)
→
T
0
where
T
is the temperature difference between the two thermal
leads, assuming that the temperature of the left and right thermal
leads are
T
+
T
/
2and
T
−
T
/
2, respectively. We introduce
variationalderivative forGreen's function as
δ
G
δ
T
=
lim
F
(
T
L
,
T
R
)
−
F
(
T
,
T
)
T
L
−
T
R
T
→
0
+
/
−
/
−
F
(
T
T
2,
T
T
2)
F
(
T
,
T
)
=
lim
(3.59)
T
→
T
0
Similar variational derivative is introduced for other functions,
like self energy and thermal current. As variational derivative of
thermal current with respect to temperature, thermal conductance
can beexpressed as
∞
ω
2
π
ω
∂
d
f
˜
σ
=
ω
(
),
(3.60)
∂
T
0
where
˜
(
ω
) is the effectivephonon transmission function.
The effectivephonon transmission of the left thermal lead is
L
(
ω
)
=
Tr
G
r
(
R
+
L
,
1
˜
2
n
+
S
)
G
a
(3.61)
