Journal of the Physical Society of Japan
Vol. 75, No. 7, July, 2006, 074713#2006 The Physical Society of Japan

Gauge Field for Edge State in Graphene
Ken-ichi S ASAKI, Shuichi M URAKAMI1 y and Riichiro SAITO2 z
1

Institute for Materials Research, Tohoku University, Sendai 980-8577
Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656
2
Department of Physics, Tohoku University and CREST, JST, Sendai 980-8578(Received February 28, 2006; accepted May 1, 2006; published July 10, 2006)

By considering the continuous model for graphene, we analytically study a special gauge ﬁeld for theedge state. The gauge ﬁeld explains the properties of the edge state such as the existence only on the
zigzag edge, the partial appearance in the k-space, and the energy position around the Fermi energy. It isdemonstrated utilizing the gauge ﬁeld that the edge state is robust for surface reconstruction, and the next
nearest-neighbor interaction which breaks the particle-hole symmetry stabilizes the edge state.
KEYWORDS: graphene, edge state, Weyl equation, gauge field, next nearest-neighbor hopping, particle–holesymmetry
DOI: 10.1143/JPSJ.75.074713

1.

Introduction

A single layer of graphite or graphene is an element ofvarious carbon-based materials such as carbon nanotubes.1)
These materials are eminently suitable for future nanotechnology and thus fundamental understanding of theelectronic properties is indispensable. The electronic property of graphene is mainly determined by the -electrons
around the Fermi energy at the hexagonal corners (the K and
K0 points) of the two-dimensional Brillouin zone. It is known
that the eigenstates are classiﬁed into extended states andlocalized states. The edge state existing near the zigzag edge
is an example of the localized state.2) The tight-binding (TB)
lattice model has shown that the edge state exists near the
zigzag edge while no localized state exists near the armchairedge.2,3) Recently, by scanning tunneling microscopy andspectroscopy (STM/STS) experiments, Niimi et al.4) andKobayashi et al.,5) respectively, observed the edge state near
the zigzag edge and the energy position.
The edge shape determines the boundary condition for thewave function. It has been shown in the studies of singlewalled carbon nanotubes (SWNTs) that a boundary condition can be changed by an external magnetic ﬁeld anduniform lattice deformations. Using the continuous modelof graphene,6) Ajiki and Ando7) showed that a uniform
magnetic ﬁeld parallel to a SWNT axis aﬀects the energy
band structure of extended states. The magnetic ﬁeld isexpressed by a gauge ﬁeld which changes the boundary
condition around a tube axis through the Aharonov–Bohm(AB) eﬀect. Zaric et al.8) observed optical signatures of theAB eﬀect. Kane and Mele9) introduced another homogeneous gauge ﬁeld in the continuous model. The gauge ﬁeld
represents uniform lattice deformations such as uniform
bend, twist and curvature. The gauge ﬁeld changes theboundary condition around a tube axis through a generalized
AB eﬀect and makes a mini gap in chiral metallic SWNTs.
The curvature-induced mini gap was observed by STS
experiment by Ouyang et al.10) The perturbations above
modify the boundary condition for extended states and areE-mail: sasaken@imr.tohoku.ac.jp
E-mail: murakami@appi.t.u-tokyo.ac.jpzE-mail: rsaito@ﬂex.phys.tohoku.ac.jp
ysimilar in that they are represented by gauge ﬁelds in thecontinuous model. Is it possible to change the boundary
condition for the edge state? To answer this question, ageneralization of the gauge ﬁelds to local ﬁelds is necessary
since the edge states appear locally around the edge. In theprevious paper,11) we generalize the gauge ﬁeld introduced
by Kane and Mele, to include a local lattice deformation ofgraphene. The deformation-induced gauge ﬁeld accounts forthe local modulation of the energy band gap,11) which wasobserved in some pea-pod-like structures by Lee et al.12)
In this paper, using the continuous model, we show thatthe edge state can be formed by a deformation-induced‘‘magnetic’’ ﬁeld and we clarify why the edge state is formedin the zigzag edge, but not in the armchair edge. We explain
various properties of the edge states, and examine how theedge state are aﬀected by perturbations.
Here we brieﬂy introduce the continuous model. In an
undeformed graphene, the Hamiltonian for -electrons
around the Fermi point are given by two-component Weyl
equation.6) Two components of the wave function represent
the two sublattices in the unit cell. The Weyl equation gives
two-dimensional linear k energy dispersion relation aroundthe K and K0 points in the Brillouin zone. The Weyl equation
allows us to treat a variety of perturbations in a uniﬁed way
for studying transport properties, scattering process aroundimpurities, lattice defects and topological defects.13)
This paper is organized as follows. In §2, we deﬁne agauge ﬁeld for a locally deformed graphene system. Then wesolve the continuous model (Weyl equation with the gaugeﬁeld) to obtain the edge state. In §3, we present several
applications of the continuous model. The eﬀects of a lattice
deformation around the edge, the next nearest-neighborinteraction, and an external magnetic ﬁeld on the edge
state are examined. In §4, we discuss and summarize theresults. We derive the continuous model in Appendix A.
In Appendix B, we analyze a locally deformed graphenesystem using the TB model.
2.

Continuous Model

We consider an undeformed graphene as shown in
Fig. 1(a). The unit cell consists of A sublattice ( ) and B
sublattice ( ), and each sublattice is spanned by primitive
vectors a1 and a2 . j2a2  a1 j  2‘ is the unit length along
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K. SASAKI et al.

A x (y)

y=0

y

B

δγ 1

A

a2
a1

z

vF AðxÞ ¼ ð2 ðxÞ; 0Þ for this case. Figures 1(b) and 1(c)correspond to the zigzag edge and the armchair edge if wetake 1 ¼ 0 and 2 ¼ 3 ¼ 0 , respectively. The diﬀerence between the two gauge ﬁelds becomes clear by deﬁning
the deformation-induced ‘‘magnetic’’ ﬁeld,(b) Bz (r) ≠ 0

unit cell

(a)

δγ 2

δγ 3

(c) Bz (r) = 0

Bz ðrÞ x
x=0

Ax (x)

Fig. 1. (a) Lattice structure of graphene. The unit cell consists of A
(closed circle) and B (open circle) sublattices, which leads to the pseudospin structure of the wave function. Local modulations of the hopping
integral is deﬁned by a ðrÞ (a ¼ 1; 2; 3). Two examples of local
deformation (borders) and resultant deformation-induced gauge ﬁelds
are shown in (b) and (c). In (b), the gauge ﬁeld (arrows) gives a ﬁnite
deformation-induced magnetic ﬁeld (ﬂux) illustrated by
and
while
no deformation-induced magnetic ﬁeld is present in (c).

the y-axis. A -electron hops from one site to the nearestneighbor sites with the hopping integral 0 ( 3 eV)
in each hopping process. Let vF ð 0 ‘=h Þ be the Fermi
velocity and p^  ð p^ x ; p^ y Þ the momentum operator. The
Hamiltonian for -electrons around the K point is thenwritten as6)
!
^
^
0

p

i
p
x
y
H 0K ¼ vF  0  p^ ¼ vF
; ð1Þ p^ x þ i p^ y0where  0  ðx ; y Þ and i (i ¼ x; y; z) are the Pauli matrices. The Hamiltonian operates on a two-component wave
K
function K ðrÞ ¼ t ð K
A ðrÞ; B ðrÞÞ. This two-component character is referred to as the pseudo-spin; hence, K
A ðrÞ andK
B ðrÞ are the pseudo-spin up and down states, respectively.
K
The energy eigenequation H 0K K
E ðrÞ ¼ E E ðrÞ is the Weyl
equation.
When the lattice is deformed locally, the nearest-neighbor
hopping integral depends on the position as 0 þ a ðrÞ(a ¼ 1; 2; 3) where a denotes three nearest-neighbor B sites
from an A atom [see Fig. 1(a)]. The Hamiltonian can be
expressed by
H K ¼ vF  0  ðp^  AðrÞÞ;ð2Þ

where the vector ﬁeld AðrÞ ¼ ðAx ðrÞ; Ay ðrÞÞ is the deformation-induced gauge ﬁeld. For a weak lattice deformation
satisfying ja ðrÞj  0 , the gauge ﬁeld is expressed by alinear combination of a ðrÞ as11) (see Appendix A)
1vF Ax ðrÞ ¼ 1 ðrÞ  ð2 ðrÞ þ 3 ðrÞÞ;
2
pﬃﬃﬃ
ð3Þ
3
vF Ay ðrÞ ¼
ð2 ðrÞ  3 ðrÞÞ:2
Here, we give two examples of local lattice deformation
on a line [Figs. 1(b) and 1(c)]. From now we call this type oflattice deformation border. The example of Fig. 1(b) is a
modiﬁcation of C–C bond lengths in a row at y ¼ 0, namely,1 ðrÞ 6¼ 0 at y ¼ 0 and 2 ðrÞ ¼ 3 ðrÞ ¼ 0. From eq. (3),
we obtain vF AðyÞ ¼ ð1 ðyÞ; 0Þ. The example of Fig. 1(c) isthat 1 ðrÞ ¼ 0 and 2 ðrÞ ¼ 3 ðrÞ 6¼ 0 at x ¼ 0. We obtain@Ay ðrÞ @Ax ðrÞ

;
@x
@y

ð4Þ

which is perpendicular to the graphene plane BðrÞ ¼ ð0; 0;
Bz ðrÞÞ. It should be noted that it is not a real magnetic ﬁeld.
It is called a magnetic ﬁeld only because it enters the
Hamiltonian in the similar way as the real magnetic ﬁeld.
We have a ﬁnite deformation-induced magnetic ﬁeld forFig. 1(b), and zero magnetic ﬁeld for Fig. 1(c). The gauge
ﬁeld which gives zero magnetic ﬁeld can be removed fromeq. (2) by a gauge transformation.11) As shown in Fig. 1(b),the magnetic ﬁeld appears locally around y ¼ 0; Bz ðyÞ isnegative at y < 0 as illustrated by and is positive at y > 0
as . The deformation-induced magnetic ﬁeld will accountfor the presence of edge states not at the armchair edge but atthe zigzag edge. The importance of the magnetic ﬁeld canbe seen by the fact that the energy eigenstate of eq. (2) also
2 K
satisﬁes H 2K K
E ðrÞ ¼ E
E ðrÞ:
v2F fðp^  AðrÞÞ2 þ h Bz ðrÞz g

K
2E ðrÞ ¼ E

KE ðrÞ:

ð5Þ

H 2K has only diagonal-components in the 2  2 matrix. One
can see that Bz ðrÞ in eq. (5) works as a potential and aﬀects
the electronic properties. This is because the diagonal element is similar to the non-relativistic Hamiltonian H nr ¼
ð1=2mÞðp^  AðrÞÞ2 VðrÞ where the sign in front of VðrÞcorresponds to the pseudo-spin and m is a mass.
We derive the edge states from the Weyl equation with thegauge ﬁeld of Fig. 1(b), corresponding to the zigzag edge.
Before going into detail, we outline the results here. We willshow that there are localized states in the energy spectrum
and the energy dispersion appears as the two solid lines at
px > 0 shown in Fig. 2(b). The velocity becomes small withincreasing the gauge ﬁeld and it becomes zero when thegauge ﬁeld is suﬃciently strong. The result of the continuous
model reproduces the ﬂat band of the edge states shown in
Fig. 2(e) which is calculated from the TB model2) with the
zigzag edge.
We assume that AðrÞ of Fig. 1(b) is quite localized within
jyj < g where g is a length of the order of lattice spacing,
namely, AðrÞ ¼ ðAx ðyÞ; 0Þ and Ax ðyÞ ¼ 0 for jyj g in eq.
(2). We parameterize the localized energy eigenstate as!

px x Gð yÞ eþgð yÞ
K
0
e
;
ð6Þ
E ðrÞ ¼ N exp ihegð yÞ
where N 0 is a normalization constant. The pseudo-spin
modulation is represented by gðyÞ, and the wave vector px
is a good quantum number because of the translational
symmetry along the border. Putting eq. (6) to the energy
Keigenequation, H K K
E ðrÞ ¼ E E ðrÞ, we obtaindEðGðyÞ þ gðyÞÞ ¼  eþ2gð yÞ ;px  Ax ðyÞ  h
dy
vF
ð7Þ
d
E 2gð yÞðGðyÞ  gðyÞÞ ¼  epx  Ax ðyÞ þ h:dyvF
By summing and subtracting the both sides of eq. (7), we
rewrite the energy eigenequation as

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K. SASAKI et al.

E

(a)

(b)
py
px

K

E

px

K

E

Localized stateKpx

e−G(y)

0

(b)

ξ

g(y)

ξg

− ξg

y

1 ξg
Ax (y)dyh̄ −ξg0yEdge state

(d)

c=0

(c)

(e)

c = 1/2

1

1

Fig. 3. (a) The amplitude of the wave function, expðGðyÞÞ, of a localizedstate whose localization length is . (b) The pseudo-spin modulation part,gðyÞ. From eq. (9), gðyÞ is a constant for jyj g , and abruptly changes
across the border (y ¼ 0).

c=1

1

K

0

E /γ0

0

E / γ0E / γ0Edge stateK0K sinh
-1
1

-1

2π/3

q

π

1

-1

2π/3

q

π

1

2π/3

q

π

Fig. 2. (a) Band structure ðpx ; EÞ around the K point of the Weyl equation
without the deformation-induced gauge ﬁeld and (b) that with the gauge
ﬁeld at the border. The shaded regions in (a) and (b) represent the
spectrum of extended states. The (two-dimensional) dispersion relation
around the K point for graphene without the border gives two cones(representing the linear k dispersion) whose apex is the K point as isshown in the inset of (a). The band structure of (a) is a one-dimensional
projection (onto py plane) of the linear k dispersion. In (b), the energy
dispersion for the localized state is represented by solid lines. When thebonds at y ¼ 0 become disconnected and zigzag edges appear, the energy
eigenvalues of the localized state converge to zero (E ! 0) for anypx (> 0) and forms a ﬂat energy band of edge state. Using the TB model,
we plot the band structure for an undeformed graphene in (c), for a zigzag
ribbon in (e), and for a graphene with the border (with weakened hopping
for the C–C bonds at y ¼ 0) in (d). q in (c), (d) and (e) is the momentum
parallel to the border and q ¼ 2=3 is the K point.

dgðyÞE¼  coshð2gðyÞÞ;px  Ax ðyÞ  h
dy
vF
dGðyÞ
E
¼ sinhð2gðyÞÞ:
h
dy
vF

ð8Þ

Since we are considering a localized solution aroundthe border, we assume GðyÞ jyj= where  ð> 0Þ is the
localization length. When GðyÞ jyj=, the solution of thesecond equation of eq. (8) is given by
8


1
>
 vF
1 h
>
>

sinh
ðy g Þ,
< 2
E


ð9Þ
gðyÞ ¼>
1hv>> þ sinh1  Fðy g Þ.
:2
E
The functions GðyÞ and gðyÞ are schematically shown in
Figs. 3(a) and 3(b), respectively. The sign of gðyÞ changes
across the border; this sign change means that the pseudospin ﬂips at the border. The ﬂip is induced by the gauge ﬁeld
Ax ðyÞ. To see this, we integrate the ﬁrst equation of eq. (8)from y ¼ g to g , and acquire
Z
Z g
dgðyÞ
1 g

dy ¼
Ax ðyÞ dy:ð10Þh gg dyWe have neglected other terms, since they are proportionalto g and become zero in the limit of g ¼ 0. By puttingeq. (9) to eq. (10), we ﬁnd

1



h vF
E



1
¼
h

Z g
Ax ðyÞ dy:

ð11Þ

g

When the right-hand side of eq. (11) is large, one obtainfrom eq. (9) that gðyÞ0 for y g and gðyÞ  0 fory g . In this case, the localized state is approximatelyta pseudo-spin-up state K
g and a
E ðrÞ / ð1; 0Þ for yKtpseudo-spin-down state E ðrÞ / ð0; 1Þ for y g . Hence, astrong gauge ﬁeld at the border makes pseudo-spin-polarized
localized states. Since the polarization of the pseudo-spinmeans that the wave function has amplitude only A (or B)
sublattice, this result agrees with the result by the TB modelfor the edge state.2)
Having described the wave function of the localized state,
let us now calculate E and . To this end, we use the ﬁrst
equation of eq. (8) for jyj g and obtain
E
px
ð12Þ
¼
 Z g:vF
1cosh
Ax ðyÞ dyh gMoreover, using eq. (11), we ﬁnd Z gh
1¼ px tanhAx ðyÞ dy :h gð13Þ

In addition to this localized state, there is another localizedstate for the same px with the same  but with an opposite
sign of E. This results from a particle–hole symmetry of
the Hamiltonian; z H K z ¼ H K . By the particle–holesymmetry operation, the wave function is transformed asKKE ðrÞ ¼ z E ðrÞ.
The normalization condition of the wave function requiresthat  should be positive, which restricts the value of px in
eq. (13). Indeed, when Ax ðyÞ is positive, eq. (13) means thatthe localized states appear only at px > 0 around the K
point. This is the reason why the localized states appear in
the energy spectrum only in one side around the K point in
Fig. 2(b). On the other hand, the Hamiltonian around the K0
point is expressed by
H K0 ¼ vF   ðp^ þ AðrÞÞ;

ð14Þ

where   ðx ; y Þ. The diﬀerent signs in front of AðrÞ ineqs. (2) and (14) guarantee the time-reversal symmetry.
Because of the diﬀerent signs, a similar argument as we
used for the K point concludes that the localized state0
appears
R  px < 0 around the K point. Furthermore, when
ð1=h Þ g g Ax ðyÞ dy
0, E in eq. (12) becomes zero. The
zero energy eigenvalue between the K and K0 points in the074713-3

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K. SASAKI et al.

band structure corresponds to the ﬂat energy band of theedge state.2)
When Ax ðyÞ is negative (Ax ðyÞ  0), a ﬂat energy band
appears in the opposite side: px < 0 around the K point
and px > 0 around the K0 point. This condition, Ax ðyÞ  0,
corresponds to the Klein’s edges14) which are obtained byremoving A or B sites out of the zigzag edges having A or B
sites. Calculations on the TB model with the Klein’s edges14)
also agree with these results obtained here.
There are also extended states in addition to the edge
states. The energy dispersion relation of extended states
can be obtained as ðE=vF Þ2 ¼ p2x þ p2y , by setting GðyÞ
ipy y=h (jyj g ) in eqs. (6) and (8) where py is a real
number. The calculated energy bands are given by jEj >vF jpx j, shown as a shaded region in Fig. 2(b). It also agrees
with the TB calculation shown in Fig. 2(e). One can thenregard the localized state as a state with a complex wavenumber as follows. Since 1  tanh2 x ¼ 1= cosh2 x, we seethat the energy dispersion relation of the localized state
between E and  becomes
 2
 2
E
h
¼ p2x ; ðjyj g Þ;
ð15Þ
vF

which is the same as the linear dispersion relation ðE=vF Þ2 ¼
p2x þ p2y if one replaces py with iðh =Þ.
We have shown that three basic properties of the edge
states, i.e., the pseudo-spin polarization, the dependence on
the momentum, and the ﬂat energy band, obtained previously by the TB model,2) can be explained in terms of thegauge ﬁeld. In order to quantitatively compare the present
theory with the TB model, we have performed a TB
calculation for the geometry of Fig. 1(b) with changing 1 .
Here, we introduce an adiabatic parameter c by 1 ¼ c0 ,that is, c ¼ 0 and 1 correspond to no deformation and thezigzag edge, respectively. In Figs. 2(c), 2(d) and 2(e), weplot the band structure for c ¼ 0, 1/2, and 1. Comparing
these ﬁgures with the results of the continuous model, one
can ﬁnd a good correspondence between the TB modeland continuous model. Moreover, we analytically ﬁnd (seeAppendix B)
jEj
jpx j
þ Oðlp2x =h Þ;¼
vFcoshð lnð1  cÞÞ
ð16Þh¼ px tanhð lnð1  cÞÞ þ Oðlp2x =h Þ;for localized states around the K point. Thus, by comparingeq. (16) with eqs. (12) and (13), we conclude that the TB
model and the continuous model agree with each other near
the K point (px l=h  1), by the following relationshipZ
1 g
Ax ðyÞ dy ¼  lnð1  cÞ:ð17Þh g
The right-hand side diverges when c ! 1, which reproduces
the ﬂat energy band (E ! 0) in eq. (12) and gives =h ¼p1
x in eq. (13).
When a deformed graphene is located in an external
magnetic ﬁeld, the Hamiltonians for the K and K0 points are
given by replacing p^ in eqs. (2) and (14) with p^  eAem ðrÞ as
H K ¼ vF  0  ðp^  eAem ðrÞ  AðrÞÞ;H K0 ¼ vF   ðp^  eAem ðrÞ þ AðrÞÞ;

ð18Þ

where e denotes the electron charge and Aem ðrÞ is the gauge
ﬁeld for the external magnetic ﬁeld. The z-component of theexternal magnetic ﬁeld is given byBemz ðrÞ 

@Aem
y ðrÞ
@x



@Aem
x ðrÞ
:
@y

ð19Þ

The same signs in front of eAem ðrÞ in eq. (18) reﬂect theviolation of time-reversal symmetry. Due to the similarity
between the external magnetic ﬁeld and the deformationinduced magnetic ﬁeld in eq. (18), one may expect thatthe edge states can be induced also by Aem ðrÞ. However, as
we will see in §3, the deformation-induced magnetic ﬁeld
for the edge state corresponds approximately to one ﬂux
quantum in a hexagonal unit cell, and is thus extremely
strong ( 105 T), compared with the external magnetic ﬁeld.
3.

Application of Continuous ModelIn this section, we apply the continuous model to thefollowing three examples.
First, we consider an eﬀect of surface reconstruction (SR)around the zigzag edge on the edge state. Since the SR isan additional lattice deformation around the edge, it can be
expressed by an additional gauge ﬁeld in the continuousSRmodel. Let ASR ðrÞ ¼ ðASR
x ; Ay Þ be the gauge ﬁeld for
SR. The perturbation is then represented by
!
SR
0
ASR
x þ iAySRH ðrÞ ¼ vF
: ð20Þ
SR
ASR
0x  iAy
To calculate the energy shift by H SR ðrÞ, we recall thatthe edge state ( KE¼0 ) is a pseudo-spin polarized state. Since
H SR ðrÞ only mixes the wave function on A and B
sublattices to each other, it is shown within the ﬁrst-order
perturbation theory that H SR ðrÞ does not shift the energy
SR Kof the edge state, namely, ESR  h K
0 jH j 0 i  0. This
result is consistent with the result of Fujita et al.,15) who
found that additional lattice distortion has little eﬀect on theenergies of the edge states.
Second, we consider next nearest-neighbor (nnn) hopping.
The nnn interaction decreases the energies of (i.e., stabilizes)
the edge states, as shown by the TB calculation.16) It
accounts for the STM/STS measurements,4,5) in which theedge state appears about 20 meV below the Fermi energy.
We note that the importance of the nnn on a localized statearound a defect is pointed out by Peres et al.17)
The reason for the stabilization is that the nnn interaction
mixes the wave function on the same sublattices, namely, it
gives diagonal terms in eq. (2). Here, we propose a simple
way to include the nnn hopping process into the continuous
model, and then calculate the energy shift of the edge states.
Let n ( 0:10 ) denote the nnn hopping integral.18) For
an undeformed graphene, the nnn interaction can be included
by adding n ðl=h Þ2 p^ 2 to eq. (1) as16)
 2‘
p^ 2 :
vF  0  p^  nð21ÞhThe border is incorporated by replacing p^ with p^  AðrÞ ineq. (21). As a result, we can write the total Hamiltonian as 2‘
H K ¼ H K  nðp^  AðrÞÞ2 :ð22Þh
It is noted that this replacement is not always correct in

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K. SASAKI et al.

general because the modulation of the nnn hopping integral
is generally independent of the nearest neighbor one. In
the present case, however, we adopt this replacement asa simplest approximation. Using ðH K =vF Þ2 ¼ ðp^  AðrÞÞ2 þh Bz ðrÞz [see eq. (5)] in eq. (22), we obtain


HK 2
‘2
HK ¼ H K  nþn Bz ðrÞz :
ð23Þ
0
h
Having calculated the energy spectrum of H K , we canevaluate the energy shift due to the nnn interaction using
eq. (23). Because the edge state satisﬁes H K K
0 ðrÞ ¼ 0
(E ¼ 0), the ﬁrst-order energy shift is given byZZ‘2
Bz ðrÞhz ðrÞi d2 r;
EKnnn  n
ð24Þ
h
y
K
where hz ðrÞi  ð K
0 ðrÞÞ z 0 ðrÞ is the pseudo-spin density.
It follows from eqs. (6), (9), (11) and (17) that8
1 þ2y=
>>Z
ðy < 0Þ,
<þ e

ð25Þ
hz ðrÞi dx ¼
1>>
:  e2y= ðy > 0Þ,which is nonzero only around the border, reﬂecting thepseudo-spin polarization. Since Bz ðyÞ is conﬁned near theborder as shown in Fig. 1(b), we assume Bz ðyÞ ¼ ðh =‘Þðy 
0þ Þ  ðh =‘Þðy  0 Þ. Putting eq. (25) into eq. (24), we
obtain‘EKnnn ¼ 2n ;ð26Þ

which reproduces the previous result of the TB model for theedge state.16)
An important feature of the nnn interaction is that theenergy shift is always negative and therefore it always
stabilizes the edge state. The stability comes from theinteraction between polarization of the pseudo-spin andthe deformation-induced magnetic ﬁeld. Another important
point is that the nnn interaction breaks the particle–holesymmetry. As discussed previously, the edge state appears inKpair ( K
E and z E ) at E ¼ 0 and their pseudo-spin densities
are the same. Thus, EKnnn is the same for the two states, andthey remain degenerate at the energy Ennn (< 0) though
the particle–hole symmetry is lost. It is clear in eq. (23) thatthe particle–hole symmetry for the edge state is broken
locally at the edge by the last term of the right-hand side.
Third, we consider an external magnetic ﬁeld. Since thesign in front of Aem ðrÞ is the same for the K and K0 points
as shown in eq. (18), the energy shift by a magnetic ﬁeld
becomes
ZZ
‘2
em
2
eBem
E  nð27Þ
z ðrÞhz ðrÞi d r:hThus, Bemz ðrÞ can shift the energy spectrum of the edge state.
When Bemz ðrÞ is uniform, we estimate the energy shift as2
Eem  pn ð‘=‘B Þ ﬃ, where ‘B is the magnetic length deﬁned
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
em
by ‘B ¼ h =eBz . Here, we assume that eq. (25) is stillvalid even in the presence of a weak uniform
magneticﬁeld.
pﬃﬃﬃﬃﬃﬃﬃ
ﬃ
The numerical value of ‘B (‘) is about 25= Bemz nm (0.2 nm)
where the magnetic ﬁeld is measured in the unit of tesla. Foremexample, Bem
 0:2 meV.
z  10 T gives a tiny shift E
This value is one-order smaller than the Zeeman splitting.

Thus, compared with the deformation-induced magnetic
ﬁeld, external magnetic ﬁeld has little eﬀect on the edge
state.
4.

Discussion and Summary

By considering the edge state using the continuous model,
we found that the deformation-induced gauge ﬁeld [AðrÞ]and magnetic ﬁeld [Bz ðrÞ] explain basic properties of theedge state. It is summarized as follows:(1) The gauge ﬁeld can generate the edge state in energy
spectrum, depending on the gauge ﬁeld direction. Let
ek the unit vector along the border and Ak ðrÞ AðrÞ  ek , localized states (the edge state) appears if
the gauge ﬁeld has a component parallel to the border:
Ak ðrÞ 6¼ 0. The edge states are pseudo-spin polarized.
(2) The direction of the gauge ﬁeld, namely, Ak ðrÞ > 0 or
Ak ðrÞ < 0, is vital for the edge states, as it determines
the energy dispersion and wave vectors which allow
the edge states.
(3) The ﬂat energy band of edge states at zero energy
results from a divergence of the gauge ﬁeld; Ak ðrÞ !
1.
(4) The nnn interaction lowers the energy of the edge state
below the Fermi energy. This is because the nnn
induces a linear coupling between the pseudo-spin andBz ðrÞ.
(5) Bz ðrÞ for the edge state corresponds to an enormous
magnetic ﬁeld 105 T at the border. Thus a uniform
external magnetic ﬁeld has little eﬀect on the edge
state, compared with the deformation.
Here, we discuss impurity eﬀects on the edge states, as
impurity scattering may give rise to the Anderson localization for low-dimensional systems. However, it has been
proposed that the Anderson localization does not occur ingraphene, since graphene has a remarkable property, leading
to the absence of back-scattering.19) Because a rotation of
the wave vector around the Fermi point gives an additional
phase shift of  (Berry’s phase), a time-reversal pair of
scattered waves cancel with each other.19) However, in thepresence of deformations such as a vacancy, the wave
function gets an extra phase shift through the AB eﬀect ofthe deformation-induced gauge ﬁeld. The extra phase makes
an asymmetry between time-reversal pair of scatteringwaves and thus the back-scattering which contributes tothe localization will be recovered.
Finally, we point out a possible experiment to conﬁrm
the results of the present theory. A direct way of making
localized states is by applying a stress locally to an
undeformed graphene, thereby inducing a deformationinduced magnetic ﬁeld. The present theory predicts thata formation of localized states depends sensitively on the
direction of the stress, and such localized states can beprobed by a peak in the local density of states in STS
measurement.
In summary, we have formulated the edge state as alocalized solution of the continuous model for a deformedgraphene. We found that the deformation-induced gauge
ﬁeld at the border is relevant to the presence of the edge
state. The gauge ﬁeld can reproduce basic properties of theedge state; the pseudo-spin polarization, the dispersion, andthe ﬂat energy band. The gauge ﬁeld is induced by local

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J. Phys. Soc. Jpn., Vol. 75, No. 7

K. SASAKI et al.

lattice deformation in general and also can be simulatedby an external magnetic ﬁeld. Compared with an external
magnetic ﬁeld, local lattice deformations can generate astrong ﬁeld of order of 105 T. The gauge ﬁeld description
was extended to include the SR and the nnn interaction. In
terms of the gauge ﬁeld, we showed that the SR does notshift the energy of the edge state and the nnn stabilizes theedge state energy.
Acknowledgment
K.S. acknowledges support form the 21st Century COE
Program of the International Center of Research and
Education for Materials of Tohoku University. S.M. is
supported by Grant-in-Aid (No. 16740167) from the Ministry of Education, Culture, Sports, Science and Technology(MEXT), Japan. R.S. acknowledges a Grant-in-Aid (No.
16076201) from MEXT.
Appendix A:

Gauge Field for Weak Lattice
Deformation

First, we derive the gauge ﬁeld quoted in eq. (3). It can beobtained by considering the matrix element of the deformed
Hamiltonian between the Bloch wave functions. The deformed Hamiltonian is deﬁned by
X X
H deform a ðri ÞðcBiþa Þy cA
ðA:1Þ
i þ h.c.;i2A a¼1;2;3

cyi

where ci andare the canonical annihilation-creation
operators of the electrons at site i. The oﬀ-diagonal matrix
element of H deform is given byhkþkjH deform jkB i
A
1 X X
¼
a ðri Þ fa ðkÞeikri ;ðA:2Þ
Nu i2A a¼1;2;3
where fa ðkÞ  eikRa and Ra (a ¼ 1; 2; 3) are vectors pointing
to the nearest-neighbor B sites from an A site. Here, jks i is
the Bloch wave function deﬁned as
1 X ikri ye ci j0i; ðs ¼ A; BÞðA:3Þ
jks i ¼ pﬃﬃﬃﬃﬃﬃ
Nu i2swhere Nu denotes the number of graphene unit cells.
By expanding fa ðkÞ in eq. (A·2) around kF (wave vectorof the K point), we obtain fa ðkÞ ¼ fa ðkF Þ þ i fa ðkF Þðk  kF Þ Ra þ   . The second term and the higher order corrections
can be ignored if we consider the low-energy properties
near the Fermi points of a deformed graphene satisfyingja ðrÞj  0 . Then the dominant contribution is given by
1 X X
a ðri Þ fa ðkF Þeikri :ðA:4Þ
Nu i2A a¼1;2;3
The K point satisﬁes kF  a1 ¼ 4=3 and kF  ða2  a1 =2Þ ¼
0 (mod2Þ, and therefore
we obtain f1 ðkF Þ ¼ 1, f2 ðkF Þ ¼
2
2
eþi 3 and f3 ðkF Þ ¼ ei 3 . Substituting these into eq. (A·4),
we see by the deﬁnition of AðrÞ [eq. (3)]hkþkjH deform jkB i
AvF X
fAx ðri Þ þ iAy ðri Þgeikri :ðA:5Þ
Nu i2A
By the same calculation for hkþkjH deform jkA i, we obtainB
the matrix for H deform as vF  0  AðrÞ. Similarly, by expanding the nearest-neighbor Hamiltonian for an undeformed

py

q

J=0
a1

Border

J=N

Fig. B1. Geometry of graphene with the zigzag border. When c ¼ 1, thesystem becomes a graphene with zigzag edges since the hopping integral
across the border is ð1  cÞ0 .

graphene around the K point, we see vF  0  p^ . Thus, we
obtain vF  0  ðp^  AÞ, which is eq. (2).
Appendix B:

Derivation of Eq. (16)Next, we consider a general border using the parameterc to obtain eq. (16). We write the Hamiltonian H 0 for an
undeformed periodic graphene shown in Fig. B·1 as,
X XðcBiþa Þy cA
ðB:1Þ
H 0  0i þ h.c.
a¼1;2;3 i2A

The Hamiltonian for graphene with zigzag edge is denotedby H zigzag , and we deﬁne H border by H 0  H zigzag . H border
describes the hopping across the border. From the deﬁnition,
H border is given byXy B
H border ¼ 0
ðcA
ðB:2Þ
I;J¼N Þ cI;J¼0 þ h.c.;
I

cyI;J

where cI;J and
are annihilation-creation operators of-electron at the edge site, ðI; JÞ (J ¼ 0 or N). In thefollowing, we consider HðcÞ  H 0  cH border where jcj 
1 corresponds to a weak deformation and c ! 1 leads to
H zigzag , i.e., zigzag edge formation. In addition to the zigzag
edge2) for c ¼ 1, HðcÞ describes the Klein’s edge14) whenc ! 1. When c ! 1, an electron cannot come to theborder sites, and the system reduces to a graphite ribbon with
the Klein’s edges which are made by removing the bordersites from the zigzag edges.
Before we examine localized state of HðcÞ analytically,
let us show the numerical result of the energy band structure around the K point including both localized state andextended state. In Figs. 2(c)–2(e), we plot the energy
spectrum of HðcÞ for N ¼ 10 with c ¼ 0, 1/2, and 1.
Because H border does not break the translational symmetry
along the border, the wave vector, q  k  a1 , remains to bea good quantum number. We consider the spectrum nearq ¼ 2=3 (the K point). In the energy spectrum of Hð0Þ
[Fig. 2(c)], the valence and conduction bands touch at q ¼2=3. Only extended states exist in the system and they are
doubly degenerate, which are supported by the (inversion)
symmetry under py $ py where py denotes the momentum perpendicular to the border. These degeneracies disappear as a result of the deformation, and the spacing ofenergy bands for Hð1Þ becomes about one half of that forHð0Þ. Apart from the change of energy spacing, we observeno signiﬁcant changes in the band structure of the extended
states.
On the other hand, the localized states have several
notable characteristics. First, localized state appears for2=3 < q  as we change c from 0 to 1, i.e., there is074713-6

J. Phys. Soc. Jpn., Vol. 75, No. 7

K. SASAKI et al.

an asymmetry about the K point in the momentum axis.
Second, the two localized states gradually depart from thevalence and conduction bands respectively as c is increased[Fig. 2(d)]. Third, those localized states merge when c ¼ 1
to become the ﬂat bands with zero energy [Fig. 2(e)]. In
the following, we will analyze these features by analyticallycalculating the energy eigenvalue and localization length of
localized state.
An energy eigenstate can be represented asA ðJÞ
0:
ðB:3Þ
E ðI; JÞ ¼ N expðiqIÞ
B ðJÞ
The energy eigenequation HðcÞE ¼E

E is written as

A ðJÞ ¼ B ðJ þ 1Þ  QB ðJÞ;B ðJ þ 1Þ ¼ A ðJÞ  QA ðJ þ 1Þ;

ðB:4Þ

where Q  2 cosðq=2Þ and  ( E=0 ) is the energy eigenvalue normalized by the hopping integral. The eigenstate isexpressed as
0
1
SJþ1




Qþ

B
C A ð0ÞA ðJÞSJC¼ SJ B;@
SJ1 A B ð0ÞB ðJÞ QþSJðJ ¼ 0; . . . ; NÞ ðB:5Þ
where SJ  sinðJÞ= sin . Here  is a wavenumber in the
direction of J and satisﬁes
2 ¼ Q2 þ 2Q cos  þ 1:

ðB:6Þ

The derivation of eq. (B·5) is as follows: ﬁrst, it is clear fromeq. (B·4) that eq. (B·5) holds for J ¼ 0 and 1. Second, let us
rewrite eq. (B·4) using eq. (B·6) as

 


A ðJ þ 1ÞQ þ 2 cos A ðJÞ
¼: ðB:7ÞB ðJ þ 1Þ
B ðJÞ
Q
Equation (B·5) can then be proved by induction.
The wave function is given by three parameters c, q and .
The parameter  is determined by the boundary condition:
the energy eigenequation at border sites, J ¼ 0 (B sites) andJ ¼ N (A sites),B ð0Þ ¼ ð1  cÞA ðNÞ  QA ð0Þ;
ðB:8Þ
A ðNÞ ¼ ð1  cÞB ð0Þ  QB ðNÞ:Equation (B·8) can be rewritten as
0
1
Q





A ðNÞ1 B
C A ð0Þ¼:@
2  ð1  cÞ2 A
1c
B ðNÞ
B ð0Þ
Q
ðB:9Þ
From eqs. (B·5) and (B·9), we obtain two equations relatingðA ð0Þ; B ð0ÞÞ to ðA ðNÞ; B ðNÞÞ. Then, by eliminatingA ðNÞ and B ðNÞ in these equations, we obtain the following
constraint equation for ,
0
1



SNþ1
Q
1
 1þ
þ
B Qþ S
C
AN
AN
N
B
C




detB
C ¼ 0;@1
SN1
 A
 1 þ
 Qþ

AN
AN
SN
ðB:10Þ

where
AN  ð1  cÞSN ;

2  ð1  cÞ2:
Q

ðB:11Þ

Equation (B·10) has 2N solutions of  for given N. From thenumerical calculation, we ﬁnd that for jQj > 1, namely, for
0 < q < 2=3 or 4=3 < q < 2, the equation has 2N realvalue solutions of , and for jQj < 1 (2=3 < q < 4=3), it
has 2ðN  1Þ real value solutions (extended states) and twocomplex value solutions (localized states). As for localizedstate,  can be parameterized as  ¼ i‘= or  ¼  þ i‘=depending on 1 < Q < 0 and 0 < Q < 1, respectively.
Here,  is the localization length and thus when  ! 1 thestate becomes an extended state. From eq. (B·6), we have
2 ¼ ðQ 1Þ2 for  ¼ 0 and . Thus, one can understandthat there are two points in ðq; Þ plane satisfying  ¼ 0,
where a localized state turns into an extended state.20) One
point is located at ð2=3; Þ (Q ¼ 1) which is the K point
and the other is located at ð4=3; 0Þ (Q ¼ 1) which is theK0 point.
When N
1, by solving eq. (B·10) for localized state, we
acquirepﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !
‘cð2  cÞ þ c2 ð2  cÞ2 þ 4Q2 ð1  cÞ2
þ ;¼ ln
2jQj
ðB:12Þ
where    denotes correction of OðeN‘= Þ and therefore
it is negligible when N
=‘. To obtain eq. (B·12), we
neglected A2ineq.
(B·10)
since
AN is large as OðeN‘= Þ. In
N
this approximation, we get QðQ þ e‘= Þ þ ðQ þ e‘= Þ ¼
22 for Q < 0. Using eqs. (B·6) and (B·11), we rewrite thiscondition asðe‘= Þ2 cðc  2Þ ‘=e  ð1  cÞ2 ¼ 0;
Q

ðB:13Þ

which gives eq. (B·12). The energy eigenvalue is obtainedby inserting eq. (B·12) into eq. (B·6). It is to be noted that
not only c ¼ 0 but also c ¼ 2 give no localized state sinceeq. (B·13) gives ‘= ¼ 0 in these cases. For c ¼ 2, thehopping integral across the border is þ0 , while other bonds
are 0 . By a gauge transformation, this diﬀerence in signof the hopping integral on the border can be absorbed
into an AB phase  around the nanotube (to the y-directionin Fig. B·1). Thus the system is equivalent to a SWNT
without border, with a  ﬂux (a half of the ﬂux quantum)
parallel to the NT axis. Because it has no border, it has noedge state.
Next, to understand E and  for localized state near the K
point, we deﬁne the wave vector kx measured from K point
as q ¼ 2=3 þ ja1 jkx . Because Q ¼ 2 cosðq=2Þ, we have
Q ¼ 1  ‘kx þ ð‘kx Þ2 =6 þ Oðð‘kx Þ3 Þ. Then, from eq. (B·6)
with  ¼  þ i‘=, we obtain
 2
 2
E
h
2
¼ px 
þ;ðB:14Þ
vF
where    indicates higher order corrections such as ‘p3x =h .
The ﬁrst two dominant terms in the right-hand side reproduce eq. (15). By expanding eq. (B·12) in terms of px(¼ h kx ), we obtain eq. (16).
Finally, we check whether E and  of the continuous

074713-7

J. Phys. Soc. Jpn., Vol. 75, No. 7

K. SASAKI et al.

(b)

(a)

10

TB
Continuous

0.6
8π
9

10
10

10

0.2
q=

0

0

0.2

3

q=

7π
q=
9

0.4

TB
Continuous

4

10

ξ/

E /γ 0

q=

5

13 π
18

10

2

q =

c

0.6

0.8

1

10

7π
9

1
0

q=

0.4

13 π
18

-1

0

8π
9

0.2

0.4

c

0.6

0.8

1

Fig. B2. (a) Energy eigenvalue E=0 and (b) localization length =‘ as a function of c. The solid and dashed curves represent the TB model and thecontinuous model, respectively. Three wave vectors q ¼ 13=18, 7=9, and 8=9 are chosen as examples. When c ¼ 1, each curve in (a) converges tozero, which agrees with the ﬂat band of edge states, and each solid curve in (b) reproduces previous results of the localization length.2,20)

model reproduce the results of the TB model for localizedstates satisfying kx ‘  1. First, using eqs. (B·6) and (B·12),
we plot E and  for q ¼ 13=18, 7=9, and 8=9 as solid
curves in Figs. B·2(a) and B·2(b), respectively. In Fig.
B·2(a), E decreases slowly at small c. However, every curve
decreases almost linearly as a function of c for large c andconverges to zero. In Fig. B·2(b) we see that  decreases
quickly with increasing c and converge at =‘ ¼ 1= ln jQj[eq. (B·12) with c ¼ 1]. The convergence agrees with thepreviously published literatures.2,20) The dashed lines in
Figs. B·2(a) and B·2(b) are E and  derived from thecontinuous model. The wave vectors q ¼ 13=18, 7=9, and
8=9, which we used in the plot for the TB model,
correspond to ja1 jkx ¼ =18, =9, and 2=9, in the continuous model. Using eqs. (12) and (17), we plot E=0 (¼ )
as dashed curves in Fig. B·2(a). One can see that the solidand dashed curves are almost identical for q ¼ 13=18.
However, their diﬀerence is visible for q ¼ 7=9 and not
negligibly small for q ¼ 8=9. Using eqs. (13) and (17), weplot =‘ as the dashed curves in Fig. B·2(b). The continuousand the TB models agree with each other with c near zero.
For general c and q, while the trend of  agrees for the twomodels, their diﬀerence becomes large as the state is locatedapart from the K point. The discrepancy between the TB andcontinuous models comes from the assumption of the linear
k energy dispersion relation of the Weyl equation. The
diﬀerence increases as the state is located apart from theFermi point, which represents the deviation of the energy
band from the linear energy dispersion. The energy dispersion relation may be improved to reproduce the TB
model by adding some higher order terms to the Weyl
equation.

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