Environmental Engineering Reference
In-Depth Information
where
f
δ
n
∂
f
∂
T
−
1
δ
T
−
i
δ
n
S
=
.
(3.62)
δ
T
Similarly, the effective phonon transmission of the right thermal
lead is
R
(
ω
)
=
Tr
G
r
(
L
+
R
.
1
2
n
−
S
)
G
a
˜
(3.63)
The effective phonon transmission could be different due to
the existence of asymmetry. However, they give the same thermal
conductance.
3.2.5
The NEGF Method and the Landauer Formalism
Superficially, the NEGF method looks quite different from the
Landauer formalism. The NEGF method focuses on the scattering
region (i.e., the center part of the transport system), calculates its
Green'sfunctions,andthenusesthemtodescribethermal-transport
properties of the whole transport system. In contrast, the Landauer
formalismdoesnotcareaboutthedetailsofthecenterpart.Instead,
the key information for the Landauer formalism is the phonon
transmission function that is the probability of a phonon modes
transporting from one thermal lead to the other one.
The advantage of the Landauer formalism is rooted in its
simplicity. It is easy to use. At the same time, it provides a
clear physical picture for understanding mesoscopic transport
problems. However, the Landauer formalism itself cannot provide
the phonon transmission function, thus its application requires
input from other approaches. As a comparison, the NEGF method
is conceptually more complex. The NEGF method can deduce the
Landauer formalism as well as the phonon transmission. In this
sense, the NEGF method is more fundamental than the Landauer
formalism.
Then we show the relation between the NEGF method and the
Landauer formulism. Thermal conductance derived by the NEGF
method is presented in Eq. 3.60. This is actually of the same form
as the Landauer formalism. Meanwhile, the expression of phonon
transmission function is provided by Eqs. 3.60 and 3.63.
