Environmental Engineering Reference
In-Depth Information
Itiseasy to prove that
G 0 [
] 1 G 0 [
] G 0 [
] 1
= < [
ω
ω
ω
ω
].
(3.39)
Thus
G < [ ω ] = G r [ ω ] < [ ω ] + n [ ω ] G a [ ω ].
(3.40)
Asa shortsummary, we listbasicequationsof NEGF:
= (
] 1
G 0 [
) 2 I
r [
ω
ω +
δ
D CC
ω
]
i
G 0 [
G 0 [
< [
] G 0 [
ω
=
ω
ω
ω
]
G r [ ω ] = ( ω + i δ ) 2 I D CC
]
]
n [ ω ] 1
r [ ω ]
G < [ ω ] = G r [ ω ] < [ ω ] + n [ ω ] G a [ ω ]
is obtained by calculating
the surface Green's function. While the many-body self energy
n contributed by anharmonic interactions can be calculated by
many-bodyperturbationapproaches,forinstancebyusingFeynman
diagram techniques [12-14].
The self energy of thermal leads
3.2.2.4 Work flow of NEGF
Having introduced the NEGF formalism, we show the work flow
of NEGF in practical calculations, as presented in Fig. 3.4. Here
we assume that the Hamiltonian is known. How to obtain the
Hamiltonian will be discussed later. The first step is to calculate the
surface Green's function and the self energy of thermal leads. Then
applying Eq. 3.35 and Eq. 3.37, Green's functions without phonon-
phonon interactions ( G 0 and G 0 ) are obtained.
If anharmonic interactions are included in the calculation,
further calculations are required to obtain full Green's functions
( G r and G < ). There are two options for such calculations: non-self-
consistent calculation and self-consistent iteration. Both options
start from G 0 and G 0 , and use them to construct many-body self
energies(
n and n )byapplyingtheFeynmandiagramtechniques,
then calculate full Green's functions according to Eqs. 3.38 and
3.40. The non-self-consistent calculation terminates at this step.
Differently, the self-consistent iteration approach uses the newly
 
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