Chemistry Reference
In-Depth Information
2.0
15.0
δ
/h
ω
0
= 0.10
Frequency Shift
1/
ω
0
τ
4.0
0.5
0.2
1.5
0.1
0.0
10.0
3.0
1.0
2.0
0.5
5.0
1.0
0.0
Frequency Shift
Broadening
-0.5
0.0
0.0
0.0
0.5
1.0
1
.
5
2.0
-1
0
1
2
Fermi Energy (units of h
ω
0
)
Frequency (units of
λω
0
)
Fig. 4. (Left) The frequency shift and broadening of optical phonons in monolayer
graphene as a function of the Fermi energy.
τ
is a phenomenological relaxation time
characterizing the level broadening effect due to disorder. (Right) Calculated spectral
function of optical phonon for varying Fermi energy.
vanishing Fermi energy, we get the frequency shift due to virtual excita-
tions of all electrons in the
bands. However, this contribution is already
included in the definition of the frequency
π
ω
0
. Inordertoavoidsucha
problem, we have to subtract the contribution in the undoped graphene for
ω
= 0 corresponding to the adiabatic approximation.
Figure 4 shows the frequency shift and broadening for various values
of 1
/ω
0
τ
and an example of the spectral function, (
−
1
/π
)Im
D
(
q, ω
0
). For
nonzero
, the logarithmic singularity of the frequency shift and
the sharp drop in the broadening disappear, but the corresponding features
remain for 1
δ
or 1
/ω
0
τ
1. Similar results were reported independently
25
and
experiments giving qualitatively similar results were reported.
26-28
The calculation can easily be extended to the case in the presence of
magnetic field
/ω
0
τ
, where discrete Landau levels are formed and oscilla-
tions due to resonant interactions appear in the frequency shift and the
broadening.
23
The Landau-level energy is given by
B
)
ε
n
=sgn(
n
|n|
ω
B
ω
B
=
√
2
n
, ±
, ···
n
n
γ/l
(
=0
1
), where sgn(
) denotes the sign of
and
with
=
magnetic length
l
c
/eB
. Figure 5 shows calculated frequency shift
and broadening when
ε
F
/
ω
0
/
=1
4 and the corresponding phonon spec-
tral function, where
ε
F
is the Fermi energy in the absence of a magnetic
field. All resonant transitions from
−n
to
n
+1 a n d f r o m
−n−
1to
n
with
n>
0
