Chemistry Reference
In-Depth Information
(b)
(c)
(a)
ω
ω
ω
Fig. 15. (a). Phonon decay into real (solid) and virtual (dashed) electron-hole excita-
tions. The gap between the vertical arrows corresponds to the phonon energy
ω
G
.(b)
Electron doped: (E
F
>
ω
G
/
2) no real transitions, only virtual transitions. (c) Hole
doped: (E
F
<
ω
G
/
2) again only virtual transitions.
resulting in upshift of the phonon frequency with respect to the undoped
case. This effect will be symmetric for both the electron and hole doping.
∆
)
dyn
is calculated from the phonon self energy Π.
30,51,65,66
Pos
(
G
)
dyn
=Re[Π(
∆
Pos
(
G
E
F
)
−
Π(
E
F
=0)]
.
(9)
The electron-phonon coupling (EPC) contribution to FWHM(G) is given
by:
FWHM(G)
EPC
=2Im[Π(
E
F
)]
(10)
The self energy corrections are calculated using the time dependent pertur-
bation (TDPT) theory and can be written for phonon with wave vector
q
as:
E
F
)=
f
sk
)
− f
s
k
)]
[
(
(
k
,
k
,
s
,
s
|W
kq
|
2
Π(
(11)
s
k
−
s
k
+
ω
q
+
iδ
W
kq
f
where
is the strength of the electron-phonon interaction,
(
)=
[exp(
−E
F
k
B
T
1
/
) + 1] is the Fermi-Dirac distribution,
k
=
k
+
q
and
δ
is
a broadening factor accounting for charge inhomogeneity. Thus, the self
energy for the
E
2
g
)modeat
Γ
in SLG is:
30,66
q
= 0 phonon (
α
∞
0
s,s
φ
ss
f
(
sk
)
− f
(
s
k
)
γ
2
kdk
Π(
E
F
)=
(12)
sk
−
s
k
+
ω
0
+
iδ
Here
φ
ss
is weightage of the electron-phonon coupling (EPC) for transitions
between conduction and valence bands in a bilayer graphene and
s
=+1and
s
= -1 are band indices for the conduction and valence bands respectively.
