Chemistry Reference
In-Depth Information
layers are in-phase and out-of-phase, respectively.
They are affected by
electron-phonon interactions in a different manner.
31
The symmetric mode
causes interband transitions between
) and therefore exhibits logarith-
mic singularity in a manner same as monolayer graphene when
ε
±
1
(
k
2.
On the other hand, this transition is not allowed for the antisymmetric
mode but interband transitions between
ε
F
=
ω
0
/
) contribute to
the phonon renormalization. Figure 7 shows calculated frequency shift and
broadening for two phonon modes.
One important feature is that the band structure can be strongly modi-
fied due to opening-up of a band gap by applied electric field.
58,59
Figure 8
shows a schematic illustration of the device structure, where
ε
+1
(
k
)and
ε
+2
(
k
eF d
represents
F
the potential difference between layers 1 and 2 (
is the effective electric
d
.
field and
=0
334 nm is the interlayer distance). The effective Hamiltonian
becomes
⎛
⎞
ˆ
eF d/
γ
k
−
2
0
0
⎝
⎠
.
ˆ
γ
k
+
eF d/
2∆
0
H
=
(19)
ˆ
0
∆
−eF d/
2
γ
k
−
ˆ
0
0
γ
k
+
−eF d/
2
Figure 9 shows the energy bands obtained by the diagonalization of this
Hamiltonian. The band gap appears and the minimum gap is located at a
nonzero value of
depending on the field.
The effective field is determined in a self-consistent manner because the
electron density distribution between two layers becomes different, giving
rise to interlayer potential difference. Some examples of the results of such
calculations are shown in Fig. 10.
60
k
n
s
g
v
g
s
∆
2
/
πγ
2
The unit
=
2
is the
n
s
≈
10
13
cm
−
2
electron concentration at
ε
F
=∆ for
eF d
=0. We have
2
.
5
×
for ∆
≈
0
.
4 eV. We can see
eF d
can become as large as
∼
∆
/
2 although
being dependent on
.
Figure 11 shows the frequency shift (top panel), broadening (middle
panel), and the spectral intensity (bottom panel) of the symmetric compo-
nent as a function of the electron concentration for
eF
ext
d
=0.
61
We have
eF
ext
d
assumed ∆
/
ω
0
= 2 corresponding to ∆
≈
0
.
4eVand
ω
0
≈
0
.
2eVand
.
δ/
ω
0
=0
1. The symmetric component shown in the figure describes the
relative intensity of the Raman scattering. At
n
s
=0 with
eF d
=0, the
optical phonons are exactly classified into symmetric and antisymmetric
modes. With the increase of
n
s
, they become mixed with each other, which
becomes particularly important when they cross each other.
