Chemistry Reference
In-Depth Information
Optical phonons are represented by the relative displacement of two
sub-lattice atoms A and B,
u
(
r
)=
q
,µ
b
†
−
q
µ
)
e
µ
(
q
)
e
i
q
·
r
,
(
b
q
µ
+
(8)
2
NMω
0
N
M
ω
0
where
is the number of unit cells,
is the mass of a carbon atom,
q
x
,q
y
) is the wave vector,
is the phonon frequency at the Γ point,
q
=(
b
q
µ
µ
denotes the modes ('t' for transverse and 'l' for longitudinal), and
and
b
q
µ
are the creation and destruction operators, respectively. Define
q
x
=
q
cos
ϕ
q
and
q
y
=
q
sin
ϕ
q
with
q
=
|
q
|
.
Then, we have
e
l
(
q
)=
i
ϕ
q
).
The interaction between optical phonons and an electron in the vicinity
of the K and K' points is given by
21
(cos
ϕ
q
,
sin
ϕ
q
)and
e
t
(
q
)=
i
(
−
sin
ϕ
q
,
cos
−
√
2
β
Γ
γ
=+
√
2
β
Γ
γ
, H
K
int
H
int
b
2
σ
∗
×
u
(
r
)
=
b
2
σ×
u
(
r
)
,
(9)
where the vector product for vectors
a
=(
a
x
,a
y
)and
b
=(
b
x
,b
y
)in2Dis
defined by
a
a
x
b
y
−a
y
b
x
. This means that the lattice distortion gives
rise to a shift in the origin of the wave vector or an effective vector potential,
i.e.,
×
b
=
direction. The interaction strength
is characterized by the dimensionless coupling parameter
u
x
in the
y
direction and
u
y
in the
x
36
√
3
π
β
2
2
2
1
ω
0
λ
Γ
=
.
(10)
2
Ma
2
2)
2
.
This shows that the interaction is not strong and therefore the lowest order
perturbation gives suciently accurate results.
The phonon Green's function is written as
10
−
23
gand
10
−
3
(
For
M
=1
.
993
×
ω
0
=0
.
196 eV, we have
λ
Γ
≈
3
×
β
Γ
/
2
ω
0
D
µ
(
q
,ω
)=
)
.
(11)
(
ω
)
2
−
(
ω
0
)
2
−
2
ω
0
Π
µ
(
q
,ω
The phonon frequency is determined by the pole of
D
µ
(
q
,ω
). In the case
of weak interaction, the shift of the phonon frequency, ∆
ω
µ
, and the broad-
ening, Γ
µ
,aregivenby
1
1
∆
ω
µ
=
Re Π
µ
(
q
,ω
0
)
,
Γ
µ
=
−
Im Π
µ
(
q
,ω
0
)
.
(12)
When we calculate the self-energy of optical phonons starting with the
known phonon modes in graphene, its direct evaluation causes a problem
of double counting.
24
In fact, if we apply the above formula to the case of
