Chemistry Reference
In-Depth Information
Equation (12) can be rewritten as:
α
∞
−∞
f
(
)
− f
(
−
)
Π(
E
F
)=
iδ
||d,
(13)
2
+
ω
0
+
α
=
A
uc
EPC
(Γ)
2
πMω
0
(
v
F
)
2
24 A
2
is the graphene unit-cell area,
where
A
uc
.
,
=5
M
is the carbon atom mass and EPC(Γ) is the electron-phonon coupling
constant.
[exp(
−E
F
k
B
T
δ
is a broadening factor accounting for charge inhomogeneity. By converting
E
F
f
(
)=1
/
) + 1] is the Fermi-Dirac distribution and
|v
F
|
√
πn
)
dyn
as
into electron density
n
(
E
F
=
) we can plot ∆
Pos
(
G
a function of
n
.
Lattice relaxation effect
The non-adiabatic effect alone does not explain the electron-hole asym-
metry seen in Fig. 11(a). To explain this we need to consider the effect of
doping on the phonons due to the doping induced change of the equilibrium
lattice parameter, termed as ∆
)
st
and has been calculated using the
Pos
(
G
relation:
30
)
st
=
n
2
−
n
3
−
|n|
3
/
2
Pos
G
−
.
n −
.
.
.
∆
(
2
13
0
0360
0
00329
0
226
(14)
, in units of 10
13
cm
−
2
, is positive and negative for electron and hole
doping, respectively.
where
n
Comparison between the experiment and theory (Single layer)
The theoretical trend (solid line) in Fig. 11(a) is generated from
∆
)
dyn
+∆
)
st
at T = 300K using the value of parameter
Pos
(
G
Pos
(
G
α
=4
α
is obtained from the density
functional theoretical (DFT) values of EPC(Γ) = 45.6 (eV)
2
/ A
2
and
10
−
3
and
.
4
×
δ
=0.1eV,where
δ
is obtained by fitting the experimental FWHM using Eq. (10), as shown
in Fig. 11(b). The quantitative agreement between the experiment and
theory (Fig. 11(a)) is poor for large doping, and requires to reconsider
the non-adiabatic calculations of Ref. [30] incorporating electron-electron
interactions.
Line-width
As mentioned in the previous section, the FWHM (solid line) in Fig. 11(b)
is plotted using the Eq. (10) to get the best fit with the experimental
data (open squares). The reduction of line-width at higher doping can be
understood from how the phonon decays into real electron-hole excitations
