Chemistry Reference
In-Depth Information
6. Spontaneous Lattice Distortion
In the presence of strong electron-phonon interaction, metallic systems are
often unstable against lattice distortion causing band-gap opening. In fact,
Peierls distortion, or bond alternation, is known to spontaneously occur
in many organic conductors. Such lattice instability is expected to exist
also in graphene and carbon nanotubes. In fact, there have been several
theoretical studies on lattice instability in metallic nanotubes. Some con-
sidered an in-plane Kekule distortion corresponding to the zone-boundary
phonon
24,37-39
and effects of magnetic fields in metallic nanotubes and in
graphene.
40-42
Effects of out-of-plane distortion destroying the sublattice
symmetry were also considered,
24,43
and distortions corresponding long-
wavelength acoustic and optical modes were pointed out because they can
open up a gap in metallic nanotubes.
44-47
The lattice instability is certainly
the subject always attracting considerable interest.
48-55
7. Bilayer Graphene
We consider a bilayer graphene which is arranged in the AB (Bernal) stack-
ing. The upper layer is denoted as 1 and the lower layer denoted as 2. In
each layer, the unit cell contains two carbon atoms denoted by
A
1
and
B
1
in layer 1 and
B
2
in layer 2. For the inter-layer coupling, we include
only the coupling between vertically neighboring atoms. Then, electronic
states are described by the
k
A
2
and
p
Hamiltonian:
56,57
·
A
1
B
1
A
2
B
2
⎛
⎞
ˆ
0
γ
k
−
0
0
ˆ
⎝
⎠
,
γ
k
+
0
∆0
H
=
(16)
ˆ
0
∆0
γ
k
−
ˆ
0
0
γ
k
+
0
where ∆ (
≈
0
.
4 eV) represents the inter-layer coupling between sites B
1
and A
2
.
Let us define
)=
∆
2
2
+(
∆
2
)
2
,
ε
(
k
γk
γk
=
ε
(
k
)sin
ψ,
=
ε
(
k
)cos
ψ,
(17)
where
ψ
vanishes for
k
= 0 and approaches
π/
2 with increasing
k
. Then,
the eigen energies are given by
)sin
2
(
)cos
2
(
ε
±
1
(
k
)=
±
2
ε
(
k
ψ/
2)
,
±
2
(
k
)=
±
2
ε
(
k
ψ/
2)
.
(18)
