Environmental Engineering Reference
In-Depth Information
However, for most of the realistic cases, the contact between
the thermal reservoirs and the quantum structure is generally
imperfect or the incoming phonons are always scattered due to
inhomogeneities. At this point,
τ
m
(
ω
)
<
1. Therefore, a central issue
in calculating the thermal conductance determined by Eq. 4.4 is
thentoobtainthetransmissioncoe
cient
),
several methods such as scattering or transfer matrices method
[54-61, 65, 66], boundary condition method [67], mode-matching
method [68], non-equilibrium Green's function method [42, 43, 62],
have been developed to obtain the transmission coe
cient. All
these methods are fundamentally equivalent, and in fact stem from
the single-particle elastic scattering theory of conduction. Among
these methods, the transmission coe
cient is generally written
as a sum of all the transmission coe
cient
τ
mn
(
ω
), where
τ
mn
(
ω
)
denotesthetransmissioncoe
cientfromthe
m
phononmodeinthe
incoming lead at frequency
ω
to the
n
phonon mode in the outgoing
lead, namely
τ
m
(
ω
).Tocalculate
τ
m
(
ω
n
τ
mn
(
ω
)
=
Trace(
σσ
†
),
τ
(
ω
)
=
(4.5)
m
where
σ
is a
N
O
×
N
I
scattering matrix. Here,
N
I
and
N
O
represent
the number of phonon channels at frequency
ω
that can be opened
in the incoming and outgoing leads, respectively. However, in real
calculations, we should use a su
ciently large
σ
to consider
the contributions from evanescent modes. Based on the non-
equilibrium Green's functions, the transmission coe
cient can be
also cast in anther equivalentform
τ
(
ω
)
=
Trace(
G
r
L
G
a
R
),
(4.6)
where
G
r
(
G
a
=
(
G
r
)
†
) is the retarded (advanced) Green's function
for the central scattering region and
L
(
R
)
represents the coupling
interaction between the central scattering region and the left
(right) electrode. This result is first derived for the electronic
transport. Recently, it has been repeatedly derived for thermal
transport based on atomistic models [69-73], and widely applied
in exploring the thermal properties in real materials [74, 75].
Here,wemainlyintroducethescattering-matrixmethodtocalculate
the transmission coe
cient
) of acoustic phonons within the
continuumelastic model, whichwillbediscussed later.
τ
m
(
ω
