Pati S.K., Enoki T., Rao C.N. R. (eds.) Graphene and Its Fascinating Attributes
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164 T. Enoki, V. L. J. Joly and K. Takai
estimated at
2
/ 0.8
J J∆ ≈ , which is safely involved in the region
of a spin glass. The heat-treatment well above the I-M transition converts
the feature of the electrons from the localized to an itinerant one, where
the localized spins of edge states diminish as the fusion of nanographene
sheets takes place upon the elevation of heat-treatment temperature in the
metallic phase. However, a considerable amount of edge-state spins still
survive in the vicinity of the I-M transition. The inter-nanographene-
sheet interaction J
2
and inter-nanographite-domain interaction J
3
are
enhanced by strengthening the electron transfer paths, and works to
couple strongly the ferrimagnetic moments of nanographene sheets. This
causes the anomalous spin glass state appearing in the I-M transition
region.
4. Summary
The electronic structure of a nanographene sheet, which is considered to
be a condensed polycyclic hydrocarbon molecule extended to nano-
dimension, crucially depends on the shape of its edges, same to small
sized molecules in this family. The geometry of an arbitrary shaped
nanographene sheet is described in terms of a combination of zigzag and
armchair edges. In zigzag edges, the nonbonding
π-electron state
localized in the edge region is created, in contrast to the absence of such
state in the armchair edges. As the edge state is strongly spin polarized,
the localized edge-state spins contribute to form an unconventional
magnetic system in nanographene.
The electronic structure of graphene edges is investigated
experimentally to confirm the presence of edge state using UHV-
STM/STS techniques. The magnetic properties of the edge-state spins are
investigated using activated carbon fibers consisting of a 3D network of
nanographene sheets. The constituent individual nanographene sheet is
found to have a ferrimagnetic feature. Electron localization in the
electron transport between the nanographene sheets in the network
affects seriously the magnetism of the nanographene network.
Strengthening of the inter-nanographene-sheet interaction brings about
the formation of a spin glass state.
Magnetic Structures of Edge-State Spins in Nanographene 165
Acknowledgments
The present work is supported by the Grant-in-Aid for Scientific
Research No. 20001006 from the Ministry of Education, Culture, Sports,
Science and Technology, Japan.
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December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
Chapter 10
Electronic and Transport Properties of
Graphene Nanoribbons
Katsunori Wakabayashi
International Center for Materials Nanoarchitectonics (MANA),
National Institute for Materials Science (NIMS),
Namiki 1-1, Tsukuba 305-0044, Japan
ka.wakaba@gmail.com
Low-energy electronic states of graphene are described by the massless
Dirac Fermions. In nano-graphene, however, this description is strongly
modified due to the existence of edges. In this chapter, we discuss
the electronic transport properties of graphene nanoribbons, and clarify
the role of edge b oundary condition. Here we show that the graphene
nanoribbons with zigzag edges subjected to the long-ranged impurities
provide on-perfectly conducting channel, i.e. the absence of Anderson
localization, because in each valley the balance bet ween left- and right-
going modes is violated due to the central subband originating from the
edge states. We also study the electron transport through the graphene
junction structures, where the zero-conductance anti-resonances is
observed.
1. Introduction
Recently graphene, a single-layer hexagonal lattice of carbon atoms, has
emerged as a fascinating system for fundamental studies in condensed mat-
ter physics, as well as the promising candidate material for future appli-
cation in nanoelectronics and molecular devices.
1
The honeycomb crystal
structure of single layer graphene consists of two nonequivalent sublattices
and results in a unique band structure for the itinerant π-electrons near the
Fermi energy which behave as massless Dirac Fermion. The valence and
conduction bands touch conically at two nonequivalent Dirac points, called
K
+
and K
−
point, which form a time-reversed pair, i.e. opposite chiral-
ity. The chirality and a Berry phase of π at the two Dirac points provide
an environment for highly unconventional and fascinating two-dimensional
167
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
168 K. Wakabayashi
electronic properties,
2
such as the half-integer quantum Hall effect,
3
the
absence of backward scattering.
4,5
The successive miniaturization of the graphene electronic devices in-
evitably demands the clarification of edge effects on the electronic struc-
tures and electronic transport properties of nanometer-sized graphene. The
presence of edges in graphene has strong implications for the low-energy
spectrum of the π-electrons.
6
There are two basic shapes of edges, arm-
chair and zigzag which determine the properties of graphene ribbons. It
was shown that ribbons with zigzag edges (zigzag ribbon) possess local-
ized edge states with energies close to the Fermi level.
6
Theseedgestates
correspond to the non-bonding molecular orbitals,
6
and responsible for the
magnetism.
6–9
In contrast, edge states are completely absent for ribbons
with armchair edges. Recent experiments support the evidence of edge
localized states.
10,11
Also, graphene nanoribbons can experimentally be
produced by using lithography techniques and chemical techniques.
12–17
In this chapter, we focus on edge and geometry effects of the electronic
transport properties of graphene nanoribbons. In zigzag nanoribbons, for
disorder without inter-valley scattering a single perfectly conducting chan-
nel emerges associated with a chiral mode due to edge states, i.e. the ab-
sence of the localization.
18
Nano-graphene junctions are shown to have the
zero-conductance anti-resoances associated with the edge states.
19,20
The
relation between the condition of the resonances and geometry is discussed.
2. Electronic States of Graphene Nanoribbons
We describe the electronic states of graphene nanoribbons by the tight-
binding model. The Hamiltonian is written as,
H = −t
i,j
c
†
i
c
j
+
i
V
i
c
†
i
c
i
, (1)
where the operator c
†
i
creates an π-electrononthesitei. i, j denotes the
summation over the nearest neighbor sites. t the transfer integrals between
all the nearest neighbor sites are set to be unity for simplicity. This is
sufficient to show the intrinsic difference in the electronic states originating
from the topological nature of each system. The value of t is considered
to be about 2.75eV in a graphene system. The second term in Eq. (1)
represents the impurity potential, V
i
= V (r
i
) is the impurity potential at
a position r
i
. The effect of impurity potential on the electronic transport
properties will be discussed in the next section.
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
Electronic and Transport Properties of Graphene Nanoribbons 169
Fig. 1. Structure of graphene nanoribbon with (a) armchair edges (armchair ribbon)
and (b) zigzag edges (zigzag ribbon). The lattice constant is a and N defines the ribbon
width. The circles with dashed line indicate the missing carbon atoms for the edge
boundary condition of massless Dirac equation. The disordered region with randomly
distributed impurities lines in the shaded region and has the length L (see text in Sec. 3).
Randomly distributed circles schematically represent the long-ranged impurities.
The energy band structures of (a) armchair nanoribbons and (b) zigzag
nanoribbons for N = 30 are shown in Figs. 2(a)–(b), together with the
density of states. The wave number k is normalized by the length of the
primitive translation vector of each graphene nanoribbon, and the energy E
is scaled by the transfer integral t. For armchair nanoribbons, the system is
metallic when N =3M −1, where M is an integer. For the semiconducting
ribbons, the direct gap decreases with increasing ribbon width and tends to
zero in the limit of very large N. For narrow non-doped metallic armchair
nanoribbons, the energy gap can acquire due to Peierls instabilities toward
low-temperatures,
21
which is consistent with the recent density functional
theory calculation.
22
For zigzag ribbons, however, a remarkable feature
arises in the band structure, as shown in Fig. 2(b). We see that the highest
valence band and lowest conduction band are always degenerate at k = π.
It is found that the degeneracy of the center bands at k = π does not
originate from the intrinsic band structure of graphene sheet. These two
special center bands get flatter with increasing ribbon width. A pair of
partial flat bands appears within the region of 2π/3 ≤|k|≤π,wherethe
bands sit in the vicinity of the Fermi level. The electronic state in the
partial flat bands of the zigzag ribbons can be understood as the localized
state near the zigzag edge.
6
3. Electronic Transport Properties
We numerically discuss the electronic transport properties of the disordered
graphene nanoribbons. In general, electron scattering in a quantum wire
is described by using a scattering matrix.
23
Through the scattering matrix
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
170 K. Wakabayashi
Fig. 2. Energy band structure E(k) and density of states D(E) of (a) armchair nanorib-
bon and (b) zigzag nanoribbon. Here N = 30.
S, the an amplitude of scattered waves O are related to an amplitude of
incident waves I,
O
L
O
R
= S
I
L
I
R
=
rt
tr
I
L
I
R
. (2)
Here, r and r
are reflection matrices, t and t
are transmission matrices, L
and R denote the left and right lead lines. The Landauer-B¨uttiker formula
24
relates the scattering matrix to the conductance of sample. The electrical
conductance is calculated using the Landauer-B¨uttiker formula,
G(E)=
e
2
π
Tr(tt
†
)=
e
2
π
g(E). (3)
Here t(E) is the transmission matrix through the disordered region. For
simplicity, throughout this paper, we evaluate electronic conductance in
the unit of quantum conductance (e
2
/π), i.e. dimensionless conductance
g(E).
3.1. One-way excess channel system
In this subsection, we consider the conductance of zigzag nanoribbons in
the clean limit, which is simply given by the number of the conducting
channel. As can be seen in Fig. 3(a), there is always one excess left-going
channel in the right valley (K
+
) within the energy window of |E|≤1.
Analogously, there is one excess right-going channel in the left valley (K
−
)
within the same energy window. Although the number of right-going and
left-going channels are balanced as a whole system, if we focus on one of
two valleys, there is always one excess channel in one direction, i.e. a chiral
mode.
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
Electronic and Transport Properties of Graphene Nanoribbons 171
Now let us consider to inject electrons from left to right-side through
the sample. When the chemical potential is changed from E =0,the
quantization rule of the dimensionless conductance (g
K
+
) in the valley of
K
+
is given as
g
K
+
= n, (4)
where n =0, 1, 2, ···. The quantization rule in the K
−
-valley is
g
K
+
= n +1. (5)
Thus, conductance quantization of the zigzag nanoribbon in the clean limit
near E = 0 has the following odd-number quantization, i.e.
g = g
K
+
+ g
K
−
=2n +1. (6)
Fig. 3. (a) Energy dispersion of zigzag ribbon with N = 10. The valleys in the energy
dispersion near k =2π/3a (k = −2π/3a) originate from the Dirac K
+
(K
−
)-point
of graphene. The red-filled (blue-unfilled) circles denote the right (left)-moving open
channel at the energy E
0
(dashed horizontal line). In the left(right) valley, the degeneracy
between right and left moving channels is missing due to one excess right(left)-going
mode. The time-reversal symmetry under the intra-valley scattering is also broken. (b)
Schematic figure of scattering geometry at K
+
and K
−
points in zigzag nanoribbons,
where a single excess right-going mode exists for K
−
point. But a single excess left-going
mode exists for K
−
point. Here n
c
=0, 1, 2, ···.
Since we have an excess mode in each valley, the scattering matrix has
some peculiar features which can be seen when we explicitly write the valley
dependence in the scattering matrix. By denoting the contribution of the
right valley (K
+
)as+,andoftheleftvalley(K
−
)as−, the scattering
matrix can be rewritten as
⎛
⎜
⎜
⎜
⎝
O
+
L
O
−
L
O
+
R
O
−
R
⎞
⎟
⎟
⎟
⎠
=
rt
tr
⎛
⎜
⎜
⎜
⎝
I
+
L
I
−
L
I
+
R
I
−
R
⎞
⎟
⎟
⎟
⎠
. (7)
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
172 K. Wakabayashi
Here we should note that the dimension of each column vector is not identi-
cal. Let us denote the number of the right-going channel in the valley K
+
or the left-going channel in the valley K
−
as n
c
. For example, n
c
=1at
E = E
0
in Fig. 3(a). Figure 3(b) shows the schematic figure of scattering
geometry for K
+
and K− points. Thus, the reflection matrices have the
following matrix structures,
r =
n
c
n
c
+1
n
c
+1 r
++
r
+−
n
c
r
−+
r
−−
. (8)
The reflection matrices become non-square when the intervalley scattering
is suppressed, i.e. the off-diagonal submatrices (r
+−
, r
−+
and so on) are
zero.
When the electrons are injected from the left lead of the sample and
the intervalley scattering is suppressed, a system with an excess channel is
realised in the K
−
-valley. Thus, for single valley transport, the r
−−
and
r
−−
are n
c
×(n
c
+1) and (n
c
+1)×n
c
matrices, respectively, and t
−−
and
t
−−
are (n
c
+1)×(n
c
+1)andn
c
×n
c
matrices, respectively. Noting the
dimensions of r
−−
and r
−−
, we find that r
†
−−
r
−−
and r
−−
r
†
−−
have
a single zero eigenvalue. Combining this property with the flux conserva-
tion relation (S
†
S = SS
†
= 1), we arrive at the conclusion that t
−−
t
†
−−
has an eigenvalue equal to unity, which indicates the presence of a per-
fectly conducting channel (PCC) only in the right-moving channels. Note
that t
−−
t
†
−−
does not have such an anomalous eigenvalue. If the set of
eigenvalues for t
−−
t
†
−−
is expressed as {T
1
,T
2
, ··· ,T
n
c
},thatfort
−−
t
†
−−
is expressed as {T
1
,T
2
, ··· ,T
n
c
, 1}, i.e. a PCC. Thus, the dimensionless
conductance g for the right-moving channels is given as
g
K
−
=
n
c
+1
i=1
T
i
=1+
n
c
i=1
T
i
, (9)
while that for the left-moving channels is
g
K
−
=
n
c
i=1
T
i
. (10)
We see that g
K
−
= g
K
−
+ 1. Since the overall time reversal symmetry
(TRS) of the system guarantees the following relation:
g
K
+
= g
K
−
,
g
K
−
= g
K
+
,
(11)
December 8, 2010 17:39 World Scientific Review Volume - 9in x 6in Chap10
Electronic and Transport Properties of Graphene Nanoribbons 173
the conductance g = g
K
+
+ g
K
−
(right-moving) and g
= g
K
+
+ g
K
−
(left-moving) are equivalent.
3.2. Perfectly conducting channel
The dimensionless electrical conductance, g(E)=Tr(tt
†
)(t(E)isthetrans-
mission matrix through the disordered region), is calculated by means of
the recursive Green function method.
25
As shown in Fig. 1, the impurities
are randomly distributed with a density n
imp
in the nanoribbons. In our
model we assume that the each impurity potential has a Gaussian form of
a range d
V (r
i
)=
r
0
(random)
u exp
−
|r
i
− r
0
|
2
d
2
(12)
where the strength u is uniformly distributed within the range |u|≤u
M
.
Here u
M
satisfies the normalization condition:
u
M
(full space)
r
i
exp
−r
2
i
d
2
/(
√
3/2) = u
0
. (13)
In this work, we set n
imp.
=0.1, u
0
=1.0andd/a =1.5forLRIand
d/a =0.1 for SRI. Since the momentum difference between two valleys
is rather large, ∆k = k
+
− k
−
=4π/3a, only short-range impurities (SRI)
with a range smaller than the lattice constant causes inter-val ley scattering.
Long-range impurities (LRI), in contrast, restrict the scattering processes
to intra-valley scattering.
4
We focus first on the case of LRI using a potential with d/a =1.5which
is already sufficient to avoid inter-valley scattering. Figure 4(a) shows the
averaged dimensionless conductance as a function of L for different inci-
dent energies(Fermi energies), averaging over an ensemble of 40000 sam-
ples with different impurity configurations for ribbons of width N = 10.
The potential strength and impurity density are chosen to be u
0
=1.0and
n
imp.
=0.1, respectively. As a typical localization effect we observe that
g gradually decreases with growing length L (Fig. 4). However, g con-
verges to g = 1 for LRIs (Fig. 4(a)), indicating the presence of a single
perfe ctly c onducting channel. It can be seen that g(L) has an exponential
behavior as
g−1 ∼ exp(−L/ξ) (14)