Chemistry Reference
In-Depth Information
which conserve the energy and the momentum and this is possible if
|E
F
| <
ω
G
/
2, to satisfy the Pauli principle
the real transitions are blocked, as a result life time increases or in other
words line-width decreases.
2. Now, at higher doping
|E
F
| >
ω
G
/
2D mode
The depencence of 2D peak position as a function of doping is very different
from the G peak. As mentioned earlier 2D-peak originates from a second
order, double resonant (DR) Raman scattering mechanism.
38,46,68
In this
process, the incoming laser radiation creates an electron-hole pair close to
the Fermi point
k
F
=
K
. The photo-excited electron is then scattered
towards the second inequivalent Fermi point
k
F
=
K
by a phonon of
energy
ω
q
and wavevector
q
. A scattering event with a second phonon, of
the same energy but opposite momentum, brings the electron back to its
original position in reciprocal space. The recombination of the electron-hole
pair finally results in the emission of a photon, whose energy is decreased
by 2
ω
q
with respect to the incoming laser radiation. The order of these
four events is not fixed, and all their combinations are possible and have to
be taken into account.
68
The position of the 2D-peak is calculated such that the the second-order,
double resonant scattering cross section is maximized. The trigonal warping
of the
π − π
∗
bands and the angular dependence of the electron-phonon
coupling (EPC) matrix elements,
38
results in phonons oriented along the
ΓKM
direction and with
q
>
K
giving a non-negligible contribution to
the 2D-peak. The exact value of
q
is determined by the constraint that the
energy of the incoming photons
ω
L
has to exactly match a real electronic
π
∗
,
q
)
q
), where
transition:
ω
L
=
(
−
(
π,
(
n,
k
) is the energy of an
and wave vector
k
,and
q
electron of band index
n
is measured from
K
q
+
andisinthe
ΓKM
direction. Recall,
. The D mode is associated
with the phonon near K-point on the TO branch. For laser energy of 2.41eV,
the magnitude of wave vector
q
is 0.844 in
2
a
0
q
=2
K
units,
a
0
being the lattice
parameter of graphene.
For the 2D peak measured at 514nm the influence of dynamic effects is
expected to be negligible, since the phonons giving rise to the 2D-peak are
far away from the Kohn anomaly at
K
. Therefore, the dynamic corrections
for 2D mode are small and hence the 2D-peak position as a function of dop-
ing is calculated from the lattice relaxation effect within a DFT framework
and has been given in Ref. [50]. The comparison between the theoretical
