Thermal Conduction of Graphene - Nanoscale Energy Transport and Harvesting - page 104

Case (1): H n = 0
Define G 0 as the Green's function of the center part without
anharmonic interactions. When the center part is decoupled with
thermal leads, itsGreen's function g satisfies the equation
g r [ ω ] = ( ω + i δ ) 2 I − D CC − 1 . (3.31)
The influence of thermal leads is projected into the center part
in the form of self energy . Each thermal lead contributes a self
energy term

= L + R

(3.32)

The self energy of thermal leads is written as
α = D C α g αα D α C ( α = L, R), (3.33)
where the surface Green's function g αα is the Green's function of
isolated semi-infinitethermal lead. Itsatisfies
g r ) 2 I − D αα − 1 ( α = L, R). (3.34)

The surface Green's function can be calculated by a simple
iteration method [10] or by a more e cient decimation approach
[11].
Applying the Dysonequation, G 0 can becalculated from g :
G 0 [ ω ] =

[ ω ] = (

ω + i δ

αα

r [ ω ] − 1
= ( ω + i δ ) 2 I − D CC −

( g r [ ω ]) − 1

−

r [ ω ] − 1

(3.35)

G 0 [ ω ] = G 0 [ ω ] ( g r [ ω ] ) − 1 g < [ ω ] ( g a [ ω ] ) − 1 G 0 [ ω ]
+ G 0 [ ω ] < [ ω ] G 0 [ ω ] (3.36)
For equilibrium states, fluctuation-dissipation theorem is satis-
fied and Eq. 3.36 can befurther simplified into
G 0 [ G 0 [

< [

] G 0 [

ω = ω ω ω

] ] ]. (3.37)

Case (2): H n = 0

Define G as the Green's function of the center part with anharmonic
interactions. The anharmonic interaction H n gives rise to a many-
bodyselfenergyterm n . G can beobtainedfrom G usingtheDyson
equation.

G 0 [ ω ] − 1

n [ ω ] − 1

G r [ ω ] =

−

= (

] − 1

) 2 I

r [ r

ω +

i

δ −

D CC −

ω

]

−

n [

ω

(3.38)