Environmental Engineering Reference
In-Depth Information
twothermalleadsfromeachother.Themass-reducedforceconstant
is the mass-weighted second derivative of energy with respect to
displacement at the equilibriumpositions.
2
E
∂
u
i
∂
u
j
|
0
.
∂
[
D
αβ
]
ij
=
D
αβ
=
(3.15)
ij
Theterm
V
describestheharmoniccouplingbetweenthecenter
part with the two thermal leads.
V
=
(
u
L
)
T
D
LC
u
C
+
(
u
C
)
T
D
CR
u
R
.
(3.16)
The term
H
n
represents the nonlinear interactions or the many-
body interactions. Here it denotes the anharmonic interactions
among the center part. It contains the high order (higher than 2)
derivatives of energy with respective to displacement at the
equilibriumpositions.
1
n
!
H
(
n
)
H
n
=
=
D
i
1
,
i
2
,
···
,
i
n
u
i
1
u
i
2
···
u
i
n
(3.17)
=
···
=
···
···
n
3,4,
n
3,4,
i
1
,
i
2
,
,
i
n
where
n
E
∂
D
i
1
,
i
2
,
···
,
i
n
=
u
i
n
|
0
.
(3.18)
∂
u
i
1
∂
u
i
2
···
∂
Iftheinteractionsareharmonic,therewillbenophonon-phonon
interactions. The presentence of
H
n
will cause phonon-phonon
scatterings. The
n
th order term
H
(
n
)
corresponds to interactions
involving
n
phonons. Most theoretical treatments, if considering
anharmonic interactions, will include
H
n
till the 4th term. This
is because the higher order terms usually have negligible effects
on thermal transport and dealing with those terms is extremely
troublesome. While it is important to including the 4th term that
helps stabilizing the system. The inclusion of the 3rd anharmonic
term only sometimes causes the system energetically unstable (i.e.,
absence of lower boundary for the energy).
3.2.2
The NEGF Formalism
The NEGF formalism is a powerful tool for dealing with many-
body interactions. Two types of formalisms are widely used:
the contour-ordered Green's function and the real-time Green's
