Pati S.K., Enoki T., Rao C.N. R. (eds.) Graphene and Its Fascinating Attributes
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December 8, 2010 16:57 World Scientific Review Volume - 9in x 6in Chap8
144 T. Ando
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 A
2
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·p Hamiltonian:
56,57
H =
⎛
⎜
⎜
⎝
A
1
B
1
A
2
B
2
0 γ
ˆ
k
−
00
γ
ˆ
k
+
0∆0
0∆0γ
ˆ
k
−
00γ
ˆ
k
+
0
⎞
⎟
⎟
⎠
, (16)
where ∆ (≈ 0.4 eV) represents the inter-layer coupling between sites B
1
and A
2
.
Let us define
ε(k)=
∆
2
2
+(γk)
2
,γk= ε(k)sinψ,
∆
2
= ε(k)cosψ, (17)
where ψ vanishes for k = 0 and approaches π/2 with increasing k. Then,
the eigen energies are given by
ε
±1
(k)=±2ε(k)sin
2
(ψ/2),ε
±2
(k)=±2ε(k)cos
2
(ψ/2). (18)
December 8, 2010 16:57 World Scientific Review Volume - 9in x 6in Chap8
Phonons and Electron-Phonon Interaction in Graphene and Nanotube 145
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-0.5
0.0
0.5
1.0
1.5
2.0
2.5
Fermi Energy (units of hω
0
)
Frequency Shift and Broadening (units of λω
0
)
Shift
Broadening
Symmetric
∆/
hω
0
=2.0
δ/
hω
0
0.500
0.200
0.100
0.010
0.0 0.5 1.0 1.5 2.0 2.5 3.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Fermi Energy (units of hω
0
)
Frequency Shift and Broadening (units of λω
0
)
Shift
Broadening
Antisymmetric
∆/
hω
0
=2.0
δ/
hω
0
0.500
0.200
0.100
0.010
Fig. 7. Some examples of calculated frequency shift and broadening of the symmetric
(left) and antisymmetric mode (right) at the Γ point in a bilayer graphene. ∆/ω
0
=2.
The amount of the broadening due to disorder is denoted by δ.
Energy
1
2
z
eFd
Top Gate
Bottom Gate
Graphene
d
F
ext
V
TG
V
BG
Fig. 8. A schematic illustration of the
bilayer graphene with a top gate and a
bottom gate and the potential energy di-
agram. The distance between the layers
is given by d, the potential difference by
eF d,andF
ext
represents the field due to
the top gate.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
-0.5
0.0
0.5
1.0
1.5
Wave Vector (units of ∆/γ)
Energy (units of ∆)
Monolayer
eFd/∆
0.0
0.2
0.5
1.0
Fig. 9. The energy dispersion for vary-
ing values of the potential difference eF d.
The dot-dot-dashed lines show that of a
monolayer graphene.
The band ε
+1
(k) represents the lowest conduction band which touches the
highest valence band ε
−1
(k)atk = 0. The bands ε
±2
(k)aretheexcited
conduction and valence bands and ε
+2
(k)−ε
+1
(k)=δ independent of k.
In bilayer graphene, optical phonons are classified into symmetric and
antisymmetric modes in which the displacement of the top and bottom
December 8, 2010 16:57 World Scientific Review Volume - 9in x 6in Chap8
146 T. Ando
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 ε
±1
(k) and therefore exhibits logarith-
mic singularity in a manner same as monolayer graphene when ε
F
=ω
0
/2.
On the other hand, this transition is not allowed for the antisymmetric
mode but interband transitions between ε
+1
(k)andε
+2
(k) 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 eF d represents
the potential difference between layers 1 and 2 (F is the effective electric
field and d=0.334 nm is the interlayer distance). The effective Hamiltonian
becomes
H =
⎛
⎜
⎜
⎜
⎝
eF d/2 γ
ˆ
k
−
00
γ
ˆ
k
+
eF d/2∆ 0
0∆−eF d/2 γ
ˆ
k
−
00γ
ˆ
k
+
−eF d/2
⎞
⎟
⎟
⎟
⎠
. (19)
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 k 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
The unit n
0
s
= g
v
g
s
∆
2
/2πγ
2
is the
electron concentration at ε
F
=∆ for eF d=0. We have n
0
s
≈ 2.5×10
13
cm
−2
for ∆ ≈ 0.4 eV. We can see eF d can become as large as ∼ ∆/2 although
being dependent on eF
ext
d.
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
assumed ∆/ω
0
= 2 corresponding to ∆ ≈ 0.4eVandω
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.
December 8, 2010 16:57 World Scientific Review Volume - 9in x 6in Chap8
Phonons and Electron-Phonon Interaction in Graphene and Nanotube 147
-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
2.0
Electron Density (units of g
v
g
s
∆
2
/2πγ
2
)
Energy Difference: eFd (units of ∆)
κ=2
eF
ext
d/∆
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Fig. 10. Calculated energy difference
eF d as a function of the electron concen-
tration for various values of eF
ext
d.The
static dielectric constant of the environ-
ment is chosen as κ =2.
-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5
0.0
0.5
1.0
Electron Density (units of g
v
g
s
∆
2
/2πγ
2
)
Intensity
0.0
0.5
1.0
1.5
Broadening (units of λω
0
)
∆/hω
0
=2
eF
ext
d/hω
0
=0.0
δ/
hω
0
=0.1
-1
0
1
2
Frequency Shift (units of λω
0
)
Fig. 11. Calculated frequency shift,
broadening, and strength of the symmet-
ric component. The solid and dashed
lines denote the high- and low-frequency
modes, respectively. The thin dotted lines
in the top panel show the frequencies for
symmetric and antisymmetric modes cal-
culated without inclusion of their mixing.
∆/ω
0
=2. δ/ω
0
=0.1. eF
ext
d/∆=0.
The appearance of two peaks at sufficiently high electron concentration
is in agreement with that of recent experiments showing double peaks in
a highly doped bilayer graphene.
62
In order to make detailed comparison
with experiments, we may have to consider a small frequency splitting of
symmetric and antisymmetric modes which is independent of the change in
the electron concentration.
Acknowledgments
This work was supported in part by Grant-in-Aid for Scientific Research
on Priority Area “Carbon Nanotube Nanoelectronics,” by Grant-in-Aid for
Scientific Research, by Global Center of Excellence Program at Tokyo Tech
December 8, 2010 16:57 World Scientific Review Volume - 9in x 6in Chap8
148 T. Ando
“Nanoscience and Quantum Physics” from Ministry of Education, Culture,
Sports, Science and Technology Japan.
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151
Chapter 9
Magnetic Structures of Edge-State Spins in Nanographene and
a Network of Nanographene Sheets
Toshiaki Enoki
*
, V. L. Joseph Joly and Kazuyuki Takai
Department of Chemistry, Tokyo Institute of Technology,
2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan
*
tenoki@chem.titech.ac.jp
The electronic structure of a nanographene sheet crucially depends on the shape
of its edges. In the zigzag-shaped edges, a nonbonding π-electron state (edge
state) is created in contrast to the absence of such state in the armchair edges.
Strong spin polarization of the edge state gives rise to unconventional
magnetism in nanographene. Ultra-high vacuum STM/STS experiments
confirm the presence of the edge state in graphene edges. The magnetism of a
3 dimensional disordered network of nanographene sheets is understood on the
basis of ferrimagnetic structure of the edge-state spins in individual constituent
nanographene sheet. Electron localization in the electron transport between
nanographene sheets seriously affects the dynamical magnetic feature of the
edge-state spins. The strengthening of the inter-nanographene-sheet magnetic
interaction brings about a spin glass state.
1. Introduction
After the discovery of graphene (a single layer of graphite) in 2004,
1-4
graphene has attracted rapidly growing attention not only as a central
issue in basics science but also as a target in cutting-edge
nanotechnology. What is most interesting in graphene is the
unconventional physics behind the electronic behavior observed
experimentally. Indeed, the electronic structure of graphene is described
in terms of massless Dirac fermion, which plays in giving the
unprecedented electronic phenomena, such as anomalous quantum Hall
152 T. Enoki, V. L. J. Joly and K. Takai
effect,
5
spin Hall effect,
6
etc. When an infinite graphene sheet is cut into
nano-sized graphene (nanographene) sheets or semi-infinite graphene
sheets, another important issue arises related to the creation of graphene
edges, as evidenced by early theoretical
7-10
and experimental works,
12-15
which have been performed several years before the discovery of
graphene.
Depending on the direction of the cutting line of a graphene sheet,
there can be two types of edges created; that is, armchair and zigzag
edges.
15
What is important here is that the electronic structure of
nanographene depends crucially on the geometrical shape of the edges.
Indeed, according to the theoretical and experimental studies,
7-15
a
nonbonding π-electron state called “edge state” is created along the
zigzag edges, whereas no such state is present in the armchair edges.
In connection to the presence/absence of edge state, we should remind
the issue on the aromaticity in condensed polycyclic hydrocarbon
molecules.
16-20
The nonbonding π-electron state is created also in
these molecules with the geometrical dependence same to that in
nanographene and graphene edges. The edge state well localized in the
vicinity of the zigzag edge is therefore a kind of surface state and plays
an important role in giving electronic features that are specific to
nanographene sheet. In addition, the edge state, which is singly occupied
and strongly spin polarized, contributes to create strong spin magnetism.
Interestingly, the localized spins of the edge state are ferromagnetically
arranged through a strong ferromagnetic intra-zigzag-edge exchange
interaction (~10
3
K).
21
The circumference of a nanographene sheet can be
described in terms of a combination of zigzag and armchair edges.
Therefore, depending on the shape of a nanographene sheet, the interplay
between intra-zigzag-edge ferromagnetic interaction and inter-zigzag-
edge ferromagnetic/antiferromagnetic exchange interaction is expected to
gives rise to a variety of magnetic structures, taking into account that the
magnetic feature of edge state is sensitive to the detailed edge geometry
and chemical modifications of edges.
In this paper, we present the electronic structure of the edge state and
the magnetic properties of a disordered network of nanographite domains,
each of which consists of a stack of 3-4 nanographene sheets.
Magnetic Structures of Edge-State Spins in Nanographene 153
2. Electronic Structure of Edge State and STM/STS Observations
Let us start with the relationship between aromaticity in small
hydrocarbon molecules and the issue on edge state in
nanographene/graphene edges. Benzene, which is a typical Kekulé
molecule and the primary building block of graphene, consists of three
bonding π- and three antibonding π*-orbitals split mutually with a large
HOMO-LUMO gap, which corresponds to energy stabilization of
aromaticity. When benzene rings are fused to each other to form
condensed polycyclic hydrocarbon molecules, the Kekulé structure with
a large HOMO-LUMO gap is usually conserved, as evidenced in
naphthalene, anthracene, etc. However, there are exceptional molecules
called non-Kekulé molecules, in which the energy stabilization is
failed.
16-20
A typical example is phenalenyl radical consisting of three
benzene rings fused in a triangle shape as shown in Fig. 1(b). In this
molecule, an additional π-electron state is present at the Fermi level in
the energy gap as a nonbonding π-electron state, which works to
destabilize the molecule. A similar situation in the electronic structure is
present in triangle molecules having larger sizes, as shown in Figs. 1(c)
and (d), where the triangle molecules consisting of 6 and 10 benzene
rings have 2 and 3 nonbonding π-electron states, respectively. Here,
Lieb’s theorem
17
can tell us how large number of nonbonding states is
present in a molecule concerned. According to the Lieb’s theorem, the
(a)
(b)
(c)
(d)
(e)
Fig. 1. (a) Kekulé molecules, (b), (c), (d) triangle-shaped non-Kekulé molecules consisting
of 3, 6 and 10 benzene rings, (e) spatial distribution of the nonbonding π-electron state.