Environmental Engineering Reference
In-Depth Information
and the total phonon DOSis
ω
π
2
Tr(Im
G
r
[
ρ
ω
=−
ω
(
)
]).
(3.49)
Although Eqs. 3.48 and 3.49 are obtained from a special case
of no phonon-phonon interactions, the same formulae work also
for cases including phonon-phonon interactions providing that the
quasi-particle picture does not break down. The phonon DOS and
LDOS gives the distribution of phonons in frequency space and in
real space. This kind of information is very helpful to the analysis of
quantumthermal-transport processes.
3.2.3.2 Thermal current
First, we would like to show two useful functions in the NEGF
formulism:
(1) the spectral function
A
[
ω
]
=
i
(
G
r
[
ω
]
−
G
a
[
ω
])
=
i
(
G
>
[
ω
]
−
G
<
[
ω
]).
(3.50)
(2) the
function describing scattering rate ofphonons
r
[
ω
]
−
a
[
ω
])
=
L
[
ω
]
+
R
[
ω
]
ω
=
[
]
i
(
n
[
ω
]
=
i
n
[
ω
]
r
a
n
[
ω
]
−
(3.51)
There isan important relation between them:
A
[
ω
]
=
G
r
[
ω
](
[
ω
]
+
n
[
ω
])
G
a
[
ω
].
(3.52)
For the case of no phonon-phonon interaction, this relation
changes into
A
0
[
ω
]
=
G
0
[
ω
]
[
ω
]
G
0
[
ω
]
(3.53)
The thermal current (for the left thermal lead) can be expressed
as
+∞
2
π
ω
Tr
G
r
[
]
. (3.54)
d
ω
L
[
+
G
<
[
a
I
L
=−
ω
]
ω
]
ω
]
L
[
ω
−∞
The formula, after symmetrization, can be written as
+∞
I
L
=
d
ω
i
L
[
ω
],
(3.55)
0
