Environmental Engineering Reference
In-Depth Information
function. They are equivalent to each other. A detailed discussion of
the NEGF formalisms is beyond the scope of this topic. The rigorous
descriptionoftheformalismscanbefoundinvariousliteratures[7-
9].Here,wewillbrieflyintroducethereal-timeGreen'sfunctionand
present related formula for thermal transport.
3.2.2.1 Six real-time Green's functions
There are six real-time Green's functions. They are defined as:
T
t
u
i
(
t
)
u
j
(
t
)
1. The time-ordered Green's function
G
ij
=−
i
2. The anti-time-ordered Green's function
G
¯
i
=−
T
¯
t
u
i
(
t
)
u
j
(
t
)
3. The “greater than” Green's function
G
ij
=−
ij
u
i
(
t
)
u
j
(
t
)
i
u
j
(
t
)
u
i
(
t
)
5. The retarded Green's function
G
ij
=−
4. The “less than”Green's function
G
ij
=−
i
θ
(
t
−
t
)
[
u
i
(
t
),
i
u
j
(
t
)
6. The advanced Green's function
G
ij
=+
θ
(
t
−
t
)
[
u
i
(
t
),
i
u
j
(
t
)
Here
i
,
j
=
1, 2, 3. . . , 3
N
are atomic indexes (
N
is number
of atoms in the system);
t
and
t
represent the real time;
T
t
and
T
¯
t
respectively denote the time-ordered and the anti-time-
ordered operators;
u
is the mass-weighted displacement as defined
previously;
θ
is the unit step function; the square brackets are the
commutators;
...
represents assembling average over the density
matrix.
Basedonthedefinition,therearesomerelationsbetweenthesix
real-time Green's functions.
+
G
¯
t
=
G
>
+
G
<
G
t
(3.19)
−
G
<
G
r
=
G
t
(3.20)
=
G
<
−
G
¯
t
G
a
(3.21)
Among the six Green's functions, only three of them are linearly
independent. Selecting which kind of Green's functions for suitable
for practical use is a little bit arbitrary. However, the mostly used
Green's functionsare: