Chemistry Reference
In-Depth Information
10
10
ε
F
0
/h
ω
0
=0.250
δ
ε
F
0
/h
ω
0
=0.250
δ
/h
ω
0
= 0.10 0.05 0.02
/h
ω
0
= 0.05
Frequency Shift
1.5
Frequency Shift
Broadening
5
1.0
5
0
0.5
-1
+2
0
+
1
-2
+1
-1
0
-5
0
0.0
0.0
0.5
1.0
1.5
-5
0
5
Magnetic Energy: h
ω
B
(units of h
ω
0
)
Frequency (units of
λω
0
)
Fig. 5. (Left) The frequency shift and broadening of optical phonons in monolayer
graphene as a function of effective magnetic energy
ω
B
. Thin vertical lines show res-
onance magnetic fields.
0
F
/
ω
0
=1
/
4. The results for
δ/
ω
0
=0
.
1, 0.05, and 0.02 are
shown. (Right) The phonon spectral function for
ε
ε
F
/
ω
0
=1
/
4and
δ/
ω
0
=0
.
05. The
dotted line shows the peak position as a function of
ω
B
.
appear at the field where their energy difference becomes equal to
ω
0
.At
resonances, the phonon spectrum exhibits characteristic behavior. Recently
this magneto-phonon resonance was observed in Raman experiments.
29
The same tuning of the optical-phonon frequency and broadening due
to change in the Fermi level is also possible in carbon nanotubes. In fact,
effects of the electron-phonon interaction on the optical phonon in carbon
nanotubes were theoretically studied earlier than in graphene.
21
In nan-
otubes, the modes are classified into longitudinal and transverse, depending
on their displacement in the axis or circumference direction. Figure 6 shows
the results of the similar calculations in carbon nanotubes.
30
In semiconducting nanotubes, the imaginary part vanishes identically
because of the presence of a gap. The frequency of the longitudinal and
transverse modes is both shifted to higher frequency side and the shift is
smaller for the longitudinal mode for small
k
F
. The behavior of two modes
as a function of
k
F
is similar to that of “level crossing.” In metallic nano-
tubes, the transverse mode is not affected by the doping at all. For the lon-
gitudinal mode, the energy shift exhibits a downward shift and considerable
broadening.
21
For nonzero
k
F
, the self-energy has a logarithmic divergence
at
γk
F
=
ω
0
/
2 and increases logarithmically with
k
F
for
γk
F
>
ω
0
/
2for
