Chemistry Reference
In-Depth Information
Fig. 19. Phonon renormalization for BLG: (I)
E
F
<
ω
0
,(II)
ω
0
<E
F
<γ
1
, (III)
E
F
>γ
1
. Blue and red arrows correspond respectively to positive and negative contri-
butions to Π. Solid and dashed arrows correspond to interband and intraband processes
respectively.
terms. Let see what happen to Π with Fermi energy shift. We discuss three
different cases: (I),
|E
F
| <
ω
0
,(II)
ω
0
< |E
F
| <γ
1
, and (III)
|E
F
| >γ
1
.
Taking
|E
F
| <
ω
0
:the
positive contributions from interband transitions are suppressed, whereas
new negative intraband transitions are created, resulting in strong phonon
softening. Experimentally it has been observed at low temperatures in
Ref. [69]. At T=300K, these effects are blurred by the broadening of the
Fermi function, resulting in an almost doping independent phonon energy.
Case (II),
E
F
>
0 (the same applies for
E
F
<
0). Case (I),
E
F
decreases negative interband
contributions and creates new negative intraband transitions.
Taking into account their number and relative weight (given by
Φ
jj
/
ω
0
< |E
F
| <γ
1
:theshiftof
s,j,k
−
s
,j
,k
)), it has been shown that interband transitions out-
weight intraband ones, leading to in phonon hardening. Case (III),
(
|E
F
| >
γ
1
: this is similar to (II), with a significant difference that the second
subband filling suppresses negative intraband transitions at
k
K
,thereby
leading to the phonon hardening. Therefore, the kink in Fig. 14 is a direct
measurement of the interlayer coupling strength.
Bilayer graphene exhibits an inversion symmetry. As shown in Fig. 1,
the in plane lattice vibrations (G phonon) are classified into symmetric
(with displacement of two layers oscillating in-phase) and antisymmetric
(with displacement of two layers oscillating out-of-phase). The symmetric
mode is Raman active and the antisymmetric mode is IR active. Under
inversion operation through the inversion center of the bilayer structure, the
in- phase motion is invariant while the out-of-phase motion changes sign. In
an unperturbed bilayer graphene the in-phase phonon and the out-of-phase
∼
