Environmental Engineering Reference
In-Depth Information
In accordance withthe similar procedure in the NEGF formalism
for electronic transport [37], Eq. A.7 can be straightforwardly
rewritten as
2 π ω Tr L ( ω ) G S ( ω ) L ( ω ) G S ( ω )
d ω
=
J
(A.8)
0
where the quantities with underlines represent matrices with basis
in the scattering region. In Eq. A.8, G > , S ( ω ) is the greater/lesser
Green's function for the scattering region, and > , <
L , R ( ω )isthe
greater/lesser self-energy due to coupling to the left/right lead,
which is given by
L,R ( ω ) =− i f L,R ( ω ) +
1
2 ±
1
2
> , <
L,R ( ω )
(A.9)
Here,
i
)
r
a
L,R (
ω
)
=
L,R (
ω
)
L,R (
ω
(A.10)
r , a
L,R ( ω ) is the retarded/advanced self-energy due to the
coupling to the left/right lead.
In the case of coherent phonon transport, the lesser and greater
Green's functions G > , S ( ω ) satisfy the Keldysh equation:
G > , S ( ω ) = G S ( ω ) > , L ( ω ) + > , R ( ω ) G S ( ω ) (A.11)
while the retarded and advanced Green's functions G r , S ( ω )forthe
scattering region are expressed by
G r , S ( ω ) = ω
where
+ i 0 + D
R ( ω ) 1
2
r , a
r , a
L ( ω ) +
(A.12)
where D isthedynamicalmatrixderivedfromthesecondderivative
of the total energy with respect to the atom coordinates in the
scattering region.
Substituting Eqs. A.10 and A.11 into Eq. A.8, the thermal current
in Eq. A.8 is reduced to the Landauer formula of Eq. 2.5 in the
main text. The phonon transmission function ζ ( ω ) in Eq. 2.5 is also
expressed explicitly as
ζ
Tr
)
) G S (
) G S (
ω
=
R (
ω
ω
L (
ω
ω
(
)
)
(A.13)
This expression of ζ ( ω ) was derived by Mingo and Yang using
anatomisticGreen'sfunctionmethod[14]andmostrecentlybyDas
and Dhar using Lippmann-Schwinger theory [38].
 
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