Progress of Theoretical Physics Supplement No. 176, 2008

253

Pseudospin and Deformation-Induced Gauge Field in Graphene
Ken-ichi Sasaki ∗) and Riichiro Saito
Department of Physics, Tohoku University, Sendai 980-8578, JapanThe basic properties of π-electrons near the Fermi level in graphene are reviewed from apoint of view of the pseudospin and a gauge ﬁeld coupling to the pseudospin. The applicationsof the gauge ﬁeld to the electron-phonon interaction and to the edge states are reported.

§1.

Introduction

The electronic properties of a single layer of graphite, graphene1)–4) have attracted much attention due to the “relativistic” character of π-electrons near the
Fermi level. The energy band structure of graphene exhibits a linear energy dispersion relation around the two inequivalent, hexagonal corners of the ﬁrst Brillouinzone in the k-space (the K and K points).5), 6) The wavefunction (Hamiltonian) of
π-electrons has two component (a 2 × 2 matrix) form due to the fact that the unit
cell of graphene consists of two carbon atoms (A and B atoms). The eﬀective-mass
Hamiltonian of π-electrons around the K point or the K point is given by linear
momentum operator, which is relevant to the linear energy dispersion relation ofgraphene. The eﬀective-mass equation is similar to the massless Dirac equation or
the Weyl equation.7)
The original Dirac equation for an electron has the form∂
−cσ · p̂ Imc 2
ψ(r, t),(1)i ψ(r, t) =Imc 2
cσ · p̂∂twhere c is the speed of light, m is the mass of electron, and p̂ (= −i∇) is themomentum operator. The Hamiltonian is a 4 × 4 matrix, which is written in termsof the Pauli matrices σ = (σx , σy , σz ) and 2 × 2 identity matrix I.∗∗) ψ(r, t) of
the Dirac equation is a four-component wavefunction, which naturally explains thespin degree of freedom for particle (electron) and the antiparticle (positron). In themassless limit (m = 0), the Dirac equation is split into two equations of a 2×2 matrix
form, that is, the Weyl equations for massless neutrinos. Setting ψ = t (ψL , ψR ) in
Eq. (1), we obtain the Weyl equation for ψR and ψL as∂
ψL (r, t) = −cσ · p̂ψL (r, t),∂t∂i ψR (r, t) = cσ · p̂ψR (r, t).
∂t
i

∗)

E-mail: sasaken@ﬂex.phys.tohoku.ac.jp„
0
We use the Pauli matrices of the form of σx =
1
„
«
1 0
The 2 × 2 identity matrix I is given by I =
.
0 1
∗∗)

«
„
1
0
, σy =
0
i

(2)
«
„
−i
1
, and σz =
0
0

«
0
.
−1

254

K. Sasaki and R. Saito

The two component of each Weyl equation represents the spin degree of freedom.
On the other hand, the eﬀective-mass equations for the K and K points ofgraphene are written as∂ K
ψ (r, t) = vF σ · p̂ψ K (r, t),∂t∂i ψ K (r, t) = vF σ  · p̂ψ K (r, t),
∂t
i

(3)

where vF is the Fermi velocity, σ = (σx , σy ), σ  = (−σx , σy ), and p̂ = (p̂x , p̂y ). The
equation for ψ K of Eq. (3) is similar to the Weyl equation for ψR of Eq. (2) withpz = 0 and by substituting c to vF . The equation for ψ K is similar to the Weyl
equation for ψL with pz = 0, the substitution of c to vF , and the negative sign infront of σy . Although the character of the two components of the eﬀective-massequations of graphene (A and B atoms) and of the Weyl equation (up spin and downspin) is diﬀerent from each other, the equation and the resulting solution for given
σx and σy are the same. Thus, it is appropriate that the two component structures
A ψ B ) and ψ  = t (ψ A , ψ B ) of the eﬀective-mass equations for graphene
ψK = t (ψK

K
K
K
K
near the Fermi level is referred to as the pseudospin. A pseudospin structure givesa rich variety of interesting physical phenomena of graphene and nanotubes. It is
known that absence of a backward scattering of an electron in graphene and carbon
nanotubes is relevant to the nature of pseudospin,8), 9) in which a 2π rotation of apseudospin wavefunction around the K point in the two-dimensional Brillouin zone
does not give the original wavefunction but gives minus sign to the wavefunction.
The pseudospin in the k-space behaves similarly to the real spin in the real space.
The interaction between an electron and an electromagnetic ﬁeld is given byreplacing p̂ in the Dirac equation with the kinematical momentum π̂ = p̂ − eA(r)where −e is the charge of electron and A(r) is a vector potential. The spin of anelectron is polarized by a magnetic ﬁeld, B(r) = ∇ × A(r), which can be shown
explicitly by the Dirac equation.7) In the non-relativistic limit, the Dirac equation
reduces to the Pauli equation in which the leading interaction between spin and amagnetic ﬁeld is reduced to the Zeeman term:e
σ · B(r).
(4)
2m
Since the equation for graphene is similar to the Dirac equation, the following question arises; What is the ﬁeld that polarizes the pseudospin? A magnetic ﬁeld is acandidate. The same procedure as that in Eq. (4) shows, however, that the pseudospin is not polarized by a magnetic ﬁeld because the Zeeman term appears withthe opposite sign at the K point and at the K point as−e
σ · B(r), (K point)2me
σ · B(r). (K point)(5)+2m
Thus the direction of the pseudospin polarization which is induced at the K pointby a magnetic ﬁeld is opposite to that at the K point by the same magnetic ﬁeld.
−

Pseudospin in Graphene

255

Thus the pseudospin of graphene is not polarized by B(r).∗) Mathematically, thisobservation leads us to assume a new gauge ﬁeld Aq (r) which has the opposite signof Aq (r) at the K point to the K point as∂
ψK (r, t) = vF σ · (p̂ + Aq (r)) ψK (r, t),∂t∂i ψK (r, t) = vF σ  · (p̂ − Aq (r)) ψK (r, t).
∂t
i

(8)

Then the corresponding ﬁeld deﬁned by B q (r) = ∇ × Aq (r) can polarize the pseudospin because the Zeeman term appears as the same sign for the K and K points
in this case.
There is an example of the pseudospin-polarized state, that is, a localized state
appearing near the zigzag edge of graphene, which is called the edge states.10) The
wavefunction of the edge state has a value only on A atoms when the zigzag edge
atoms consist of A atoms. Thus we expect that a ﬁeld B q (r) appears around theedge. We will show that the zigzag edge structure is relevant to a ﬁeld B q (r)which polarizes the pseudospin near the zigzag edge.11) A gauge ﬁeld Aq (r) anda ﬁeld B q (r) are important not only for the localized edge states but also for theextended states. For example, the electron-phonon (el-ph) interaction in graphene
can be expressed by Aq (r), which explains the chirality dependence of the el-phinteraction.12) In this paper we review our studies performed on the gauge ﬁeld
Aq (r) by showing that the el-ph interaction and the edge boundary are expressedby Aq (r).
Here, we would like to mention the relationship between our work and previously
published literature on deformation-induced gauge ﬁeld. Kane and Mele introduceda homogeneous deformation-induced gauge ﬁeld in the eﬀective-mass theory.13) The
gauge ﬁeld represents uniform lattice deformations such as uniform bend, twist andcurvature of a carbon nanotube. The uniform gauge ﬁeld for the curvature of a
nanotube changes the boundary condition around a tube axis which is given by ageneralized Aharanov-Bohm (AB) eﬀect and induces a small energy gap in a chiralmetallic carbon nanotube. The curvature-induced mini gap was observed by scanningtunneling spectroscopy (STS) experiment by Ouyang et al.14) A generalization of thegauge ﬁeld to a local ﬁeld is necessary for describing a local lattice deformation. In
the previous paper,15) we generalized the gauge ﬁeld introduced by Kane and Mele, to∗)

It is noted that even in the massless limit of Eq. (4), we obtain a similar coupling term bytaking the square of the Hamiltonian in the Weyl equation as(cﬀ · ı̂)2 = c2 I (p̂ − eA(r))2 − e~c2 ﬀ · B(r).

(6)

The ﬁrst term in the right-hand side of Eq. (6) does not concern the spin, but the second term in
the right-hand side of Eq. (6) is similar to the Zeeman term which shows that the spin is polarized
by a magnetic ﬁeld.
(vF ﬀ · ı̂)2 = vF2 I (p̂ − eA(r))2 − e~vF2 σz · Bz (r), (K point)´2`vF ﬀ  · ı̂ = vF2 I (p̂ − eA(r))2 + e~vF2 σz · Bz (r). (K point)
The opposite sign in front of e~vF2 σz · Bz (r) at the K point and at the Kpseudospin is not polarized by the magnetic ﬁeld.

(7)

point shows that the256

K. Sasaki and R. Saito

include a local lattice deformation of graphene. Further, by deﬁning a deformationinduced magnetic ﬁeld, we explained the local modulation of the energy band gap,15)
which was observed in the STS measurement for pea-pod (C60 encapsulated carbon
nanotube) by Lee et al.16)
This paper is organized as follows. In §2, we derive the eﬀective-mass equations
for graphene and deﬁne the pseudospin. We show that the pseudospin determines
the elastic scattering amplitude of an electron by impurity potentials. In §3, wederive the eﬀective-mass equation in the presence of the lattice deformation, in which
gauge symmetry for Aq (r) is essential to the time-reversal properties of the el-phinteraction. In §4, the formulation explained in §3 is applied to the el-ph interaction
for optical and acoustic phonon modes. In §5, we will discuss the edge state by
Aq (r). In §6, summary of this paper is given.
§2. Eﬀective-mass theory

In this section, we derive an eﬀective-mass Hamiltonian from the nearest-neighbortight-binding model for a graphene. The nearest-neighbor tight-binding model isgiven by

  
† A
A † B
(cB
(9)
H0 = −γ0i+a ) ci + (ci ) ci+a ,i∈A a=1,2,3
A †where γ0 (≈ 2.7 eV) is the nearest-neighbor hopping integral, cAi ((ci ) ) is theannihilation (creation) operator of π-electron for A-atom at the position r i , and cBi+a† ) is that for B-atom at r
((cB
)
(≡
r
+
r
)
where
R
(a
=
1,
2,
3)
are
vectors
i+a
i
a
a
i+a
pointing to the three nearest-neighbor B-atoms from an A-atom (see Fig. 1).
We use the Bloch theorem to diagonalize Eq. (9). The Bloch wavefunction withwavevector k is deﬁned by

1  ik·ri s †e(ci ) |0,|Ψsk  = √Nu i∈s(s = A, B)(10)

where the sum on i is taken over the crystal, Nu is the number of the hexagonal unit
cells, and |0 denotes the state of carbon atoms without π-electrons. The oﬀ-diagonal
matrix element of H0 is given byfa (k) = −γ0 f (k),ΨAk |H0 |ΨBk  = −γ0
a=1,2,3

ΨBk |H0 |ΨAk  = −γ0fa (k)∗ = −γ0 f (k)∗ ,(11)

a=1,2,3

where fa (k) ≡ eik·Ra and f (k) is given by17)f (k) ≡fa (k).
a=1,2,3(12)

Pseudospin in Graphene

257

Fig. 1. A hexagonal unit cell of graphene consists of A (solid circle) and B (opencircle) atoms. a1√
and a2 are the unit vectors (a denotes the lattice constant: a ≡ |ai | = 3acc where acc is the
C-C bond length). Ra (a = 1, 2, 3) are vectors pointing to the nearest-neighbor B atoms
from an√
A atom. For the xyz coordinate
system,R
arewritten
as
R
=
a
e
,
R
=
−(
3/2)a
a
1
cc
y
2
cc ex −√
(1/2)acc ey , and R3 = ( 3/2)acc ex − (1/2)acc ey where ex (ey ) is the dimensionless unit vector
for the x-axis (y-axis). Local modulations of the hopping integral are deﬁned by δγ0a (ri ) (a =1, 2, 3) where r i (i ∈ A) is the position of an A-atom.

The diagonal matrix element of H0 , Ψsk |H0 |Ψsk , (s =A,B) can be taken to be zero.
The energy eigenequation is written by a 2 × 2 matrix form as
 k 

 k 
|ΨA 
0
f (k)
|ΨA 
=
−γ
.
(13)
E(k)
0
0
f (k)∗|ΨBk 
|ΨBk 
The energy band structure of graphene is obtained by solvingE(k)γ0 f (k)
= 0,
det
E(k)
γ0 f (k)∗(14)

whose solution is E(k) = +γ0 |f (k)|, (−γ0 |f (k)|) for the conduction (valence) energy
band. The conduction energy band and the valence energy band touch each other at
the K point, kF (= (4π/3a, 0)), and at the K point, −kF , where |f (k)| vanishes. By
expanding fa (k) in Eq. (11) around the wavevector of kF (the K point), we obtainfa (kF + k) = fa (kF ) + ifa (kF )k · Ra + · · · .

(15)

Here k (= (kx , ky )) is measured from the K point. Using kF = (4π/3a, 0), we get
2π
2π
f1 (kF ) = 1, f2 (kF ) = e−i 3 , and f3 (kF ) = e+i 3 . Substituting fa (kF ) into Eq. (15),
we obtain from Eq. (11) that3acc
(kx − iky ) + O(k 2 ),23acc
(kx + iky ) + O(k 2 ),ΨBkF +k |H0 |ΨAkF +k  = γ02

ΨAkF +k |H0 |ΨBkF +k  = γ0

where we used ΨAkF |H0 |ΨBkF  = −γ0 f (kF ) = 0.

(16)

258

K. Sasaki and R. Saito
From Eq. (16), we see that Eq. (13) becomes  kF +k 
3γ0 acc|ΨA0kx − iky|ΨAkF +k =.
E(kF + k)
kF +k
k
+k
F
kx + iky
0
2


|ΨB
|ΨB

(17)

By introducing the Fermi velocity as vF ≡ γ0 3acc /2 (∼ 106 m/s), the momentum
operator p̂ = −i∇, and the Pauli matrix σ = (σx , σy ), we obtain the eﬀective-mass
Hamiltonian near the K point as
H0K = vF σ · p̂.

(18)

H0K is a 2 × 2 matrix which operates on the two component wavefunction: ψ K (r) =t (ψ K (r), ψ K (r)) where ψ K (r) and ψ K (r) are the Bloch wavefunction for π-electrons
A
B
A
B
of the A and B atom in the unit cell. By introducing Θ(k) which is deﬁned
by an angle of k = (kx , ky ) measured from the kx -axis, then we get (kx , ky ) ≡|k|(cos Θ(k), sin Θ(k)), and0e−iΘ(k).
(19)
H0K = vF |k| +iΘ(k)e
0The energy eigenvalue of Eq. (19) is given by ±vF |p| and the energy dispersion
relation shows a linear dispersion relation near the K point as is known as the Diraccone. At the K (or K ) point, the valence and conduction bands touch to eachother and thus we call the degenerated point as the Dirac point. The eigenstates for
E = +vF |p| and E = −vF |p| are given by −iΘ(k)/2 
eik·r e−iΘ(k)/2
eik·r
e
K
K
, ψv,k (r) = √(20)
ψc,k (r) = √+iΘ(k)/2+iΘ(k)/2 ,
2S e
2S −e
which are a conduction band state and a valence band state with the wavevectork.∗) In Eq. (20), S denotes the surface area of graphene. In the case of single wall∗)

We note thateik·rK(r) = √
ψc,k
2S

„

1

e+iΘ(k)«eik·rK, ψv,k(r) = √
2S„

1−e+iΘ(k)

«
,

(21)

are also eigenstates of Eq. (18) with an additional phase factor of e±Θ(k)/2 . These wavefunctions
do not change their values under a 2π rotation in the k-space, Θ(k) → Θ(k) + 2π. This behavior is
diﬀerent from Eq. (20), in which the wavefunction of Eq. (20) changes the sign after the 2π rotation.
However, Berry’s phase for Eq. (21) which is deﬁned by
–„
«–
«»
„
»
Z 2π
´
∂
1
1
1 `
√
−i
= π,
(22)
Φ≡
dΘ √ 1, e−iΘ(k)+iΘ(k)∂Θ2
2 e
0
gives an extra phase shift of π. On the other hand, Berry’s phase for Eq. (20) vanishes because
«»
«–
„
»
–„
Z 2π
´
1 e−iΘ(k)/2
1 `
∂
√
= 0.
(23)
dΘ √ e+iΘ(k)/2 , e−iΘ(k)/2−iΦ≡+iΘ(k)/2∂Θ
22 e
0Thus, a 2π rotation of a pseudospin wavefunction around the K point in the two-dimensional Brillouin zone gives minus sign to the wavefunction in the both cases of Eqs. (20) and (21). It isconvenient to use the wavefunction of Eq. (20), since the eﬀect of Berry’s phase is included in the
wavefunction.

Pseudospin in Graphene259

carbon nanotube (SWNT), the kx -axis is taken in the circumferential direction on
the cylindrical surface and the ky -axis is deﬁned by the direction of a zigzag nanotube
axis (see the coordinate system in Fig. 1(a)). We see in Eq. (20) that the energy
K (r) is given by σ ψ K (r). This is because of aeigenstate for the valence band, ψv,kz c,kparticle-hole symmetry of the Hamiltonian: H0K σz = −σz H0K .∗)Similarly, the eﬀective-mass Hamiltonian for the K point is given by expandingfa (k) around −kF (the K point) in Eq. (11) as


H0K = vF σ  · p̂,


(24)


where σ  ≡ (−σx , σy ) and H0K operates on a two-component wavefunction: ψ K (r) =t (ψ K (r), ψ K (r)). The energy eigenstates for the conduction and valence energyABband are given, respectively, by








eik ·r e+iΘ(k )/2
eik ·r e+iΘ(k )/2KKK, ψv,k (r) = σz ψc,k (r) = √, (25)
ψc,k (r) = √
2S −e−iΘ(k )/2
2S e−iΘ(k )/2
where kx − iky ≡ |k |e−iΘ(k ) .
The linear energy dispersion relations near the K and K points, E = ±vF |p|, arecontrasted to the non-relativistic energy dispersion relation of |p|2 /2m where m isthe eﬀective-mass of the particle. The eﬀective-mass for graphene can be understood
to be zero from the deﬁnition of relativistic energy E = ± p2 c2 + m2 c4 = ±c|p|
for m = 0, where c is substituted to vF (∼ c/300). The wavefunction with twocomponents is deﬁned by the “pseudospin”. The pseudospin up (down) state t (1, 0)(t (0, 1)) corresponds to the wavefunction which has a value only on A (B) atoms. As
we see in Eq. (20), if we rotate k by 2π around the K point k = 0, that is, Θ(k) + 2π,KK(r) = −ψc,Θ(k)(r), which is the same structure for a real spin underthen ψc,Θ(k)+2π
a 2π rotation in the real space.
Here, we give an example to show that the pseudospin is relevant to a vanishing
matrix element for the lowest order backscattering amplitude.18) When a carrier inthe conduction band is denoted by |kc, then the matrix element of the backscattering
process is written as −kc|Himp |kc. When the impurity potential is long-range
compared with the lattice constant, the potential is modeled by a diagonal form andthe matrix element becomest 
  −iΘ(k)/2 
d2 r i2k·r e+iΘ(−k)/2
V (r)
0
e
e
= 0,
−kc|Himp |kc =−iΘ(−k)/20
V (r)ee+iΘ(k)/2
2S(26)
where we use Eq. (20) and Θ(−k) = Θ(k) ± π. This means that the interference
between the two component pseudospin makes the back scattering matrix element
vanish. In general, there are many impurities in a sample, which gives rise to the Anderson localization in the disordered systems. There is an interesting way to explain
the ballistic transport of carbon nanotube even in the many scattering events.8) For∗)
H0K σz = −σz H0K is obtained directly by Eqs. (18) and σz σi = −σi σz where i = x or y. Then
K
K
K
K
K
H0 (σz ψc,k(r)) = −σz H0K ψc,k(r) = −σz vF |p|ψc,k(r) = (−vF |p|)σz ψc,k
(r).

260

K. Sasaki and R. Saitoa back scattered wave, we can ﬁnd a time-reversal scattering wave whose wavefunction has an additional phase shift of π. The time reversal pair of scattered waves∗)
cancel with each other, which leads to the absence of the backward scattering.8)
This is contrasted with the standard concept that impurity scattering gives rise to
the Anderson localization for low-dimensional systems.
§3.

Gauge ﬁeld for a deformed graphene

The dynamics of the conducting electrons in graphene materials are diﬀerent
from those of ideal ﬂat graphene, because in the former case, there are shape ﬂuctuations (cylindrical shape, phonon vibration, etc.) that result in the modiﬁcationof the overlap matrix elements of nearest-neighbor π-orbitals and of the on-site potential energy. We refer the modiﬁcation of the nearest-neighbor hopping integral
as the oﬀ-site interaction and a shift of the on-site potential energy as the on-site
interaction. The modiﬁcation of the eﬀective-mass equations by the oﬀ-site (on-site)
interaction is discussed in §3.1 (§3.2). We discuss time-reversal symmetry propertyof the eﬀective-mass equations including the oﬀ-site and on-site interactions in §3.3.
3.1. Oﬀ-site interaction
First we consider the perturbation from the oﬀ-site interaction in which only
oﬀ-diagonal matrix element has a non-zero value. A lattice deformation induces a
local modiﬁcation of the nearest-neighbor hopping integral as −γ0 → −γ0 + δγ0a (r i )(a = 1, 2, 3) (see Fig. 1). We deﬁne the perturbation H1 as
H1 ≡

 



† A
A † B
δγ0a (r i ) (cB
)
c
+
(c
)
c
i+a
i
i
i+a .

(27)

i∈A a=1,2,3

The oﬀ-site matrix element of H1 with respect to the Bloch wave functions in Eq. (10)with k and k + δk is given by1  
δγ0a (r i )fa (k)e−iδk·r i ,Nui∈A a=1,2,3
 
1δγ0a (r i )fa (k)∗ e−iδk·(ri +Ra ) .
ΨBk+δk |H1 |ΨAk  =NuΨAk+δk |H1 |ΨBk  =(28)i∈A a=1,2,3

Here we consider the two possible cases for δk. When δk is suﬃciently small compared with the reciprocal lattice vector, k near the K (or K ) point is scattered to thek = k + δk within the region near the K (or K ) point, which we call the intravalley
scattering. On the other hand, if δk is comparable to the distance between the K
and K points, we expect scattering from K (K ) point to K (K) point, which wecall the intervalley scattering.
∗)

Here the time reversal pair of scattering is deﬁned for the scatterings within one valley of theK (K ) point. Later, we will use “time reversal symmetry” for the pair of k around the K point andk around the K point, when we consider the intervalley scattering.

Pseudospin in Graphene261

3.1.1. Intravalley scattering
In the case of intravalley scattering, when k is measured from kF , we get
1  
δγa (ri )fa (kF )e−iδk·ri + O(δkδγa ),ΨAkF +k+δk |H1 |ΨBkF +k  =Nui∈A a=1,2,31δγa (ri )fa (kF )∗ e−iδk·r i + O(δkδγa ), (29)ΨBkF +k+δk |H1 |ΨAkF +k  =Nui∈A a=1,2,3

The correction indicated by O(δkδγa ) in Eq. (29) is negligible when |δk||kF |.
2π−i 2π+i
Substituting f1 (kF ) = 1, f2 (kF ) = e 3 and f3 (kF ) = e 3 into Eq. (29), we getvF  qAx (ri ) − iAqy (r i ) e−iδk·ri ,ΨAkF +k+δk |H1 |ΨBkF +k  =Nui∈AvF  qkF +k+δkkF +k|H1 |ΨA=(30)
Ax (ri ) + iAqy (r i ) e−iδk·ri ,
ΨB
Nu
i∈A

where A (r) = (Aqx (r), Aqy (r)) is deﬁned by δγ0a (r) (a = 1, 2, 3) asq1δγ02 (r) + δγ03 (r) ,vF Aqx (r) = δγ01 (r) −2√
3
δγ02 (r) − δγ03 (r) .
vF Aqy (r) =
2

(31)



Since the diagonal term vanishes, that is, Ψsk |H1 |Ψsk  = 0 (s = A, B), Eq. (30)shows that H1 is expressed by vF σ · Aq (r) in the eﬀective-mass Hamiltonian. Thus,
the total Hamiltonian of a deformed graphene is expressed by
H0K + H1K = vF σ · (p̂ + Aq (r)).

(32)

Similarly, by expanding fa (k) in Eq. (28) around −kF of the K point, we ﬁnd H1
appears as −vF σ  · Aq (r). Thus, we obtain the eﬀective-mass Hamiltonian for theK point as

H0K + H1K = vF σ  · (p̂ − Aq (r)).

(33)

The corresponding Schrödinger equations for Eqs. (32) and (33) are given in Eq. (3).15)
Thus, the oﬀ-site interaction can be included in the eﬀective-mass equations asa gauge ﬁeld, Aq (r). We call Aq (r) the deformation-induced gauge ﬁeld anddistinguish it from the electromagnetic gauge ﬁeld A(r).15) H0 + H1 does not breaktime-reversal symmetry even though the Aq (r) appears as a gauge ﬁeld, because thesign in front of Aq (r) is opposite to each other for the K and K points. This contrasts
with the fact that A(r) (magnetic ﬁeld) violates time-reversal symmetry because thesign in front of A(r) is the same for the K and K points as p̂ → p̂ − eA(r). The
time-reversal symmetry will be discussed in detail in §3.3. The rotation of Aq (r) or
the deformation-induced magnetic ﬁeld is deﬁned by B q (r) = ∇ × A(r) as
Bzq (r) =∂Aqy (r) ∂Aqx (r)
−
.
∂x
∂y

(34)

262

K. Sasaki and R. Saito

The direction of Bzq (r) is perpendicular to the graphene plane B q (r) = (0, 0, Bzq (r)).
The deformation-induced magnetic ﬁeld changes the energy band structure, which
will be shown in §5. The Bzq (r) couples with the pseudospin through σz term. Thisis shown by taking the square of the eﬀective-mass equations, as we see in Eqs. (6)and (7),E 2 ψ K (r) = vF2 (p̂ + Aq (r))2 + Bzq (r)σz ψ K (r),

E 2 ψ K (r) = vF2 (p̂ − Aq (r))2 + Bzq (r)σz ψ K (r),(35)

where Bzq (r)σz terms modify the direction of the pseudospin to the direction opposite to Bzq (r) in order to decrease the energy for the both K and K points. Here let
us stress a notable diﬀerence between Bzq (r) and the usual magnetic ﬁeld, Bz (r). For
Bzq (r), a uniform magnetic ﬁeld along the z-axis over a sample is possible to realize
the uniform spin polarization (the Zeeman term). However, since Bzq (r) is generated by the modiﬁcation of γ0 , the ﬁeld Bzq (r) appears only near the defect region
without periodic symmetry such as the zigzag edge11) which gives local polarizationof pseudospin. In general, when a graphene sample contains an equal number of A
and B atoms to each other, we haveS

Bzq (r)d2 r = 0,(36)where S denotes an integral for a whole sample.
3.1.2. Intervalley scattering
Next, we consider the matrix element of H1 for the intervalley scattering. Let us
deﬁne the wave vector of a Bloch state near the K point; kF + k and that of another|kF |), then the matrix element
Bloch state near the K point; −kF + k (|k|, |k |from the K to K points is written by using Eq. (28) as




1  
δγ0a (ri )fa (kF )∗ e−i2kF ·ri ei(k −k)·ri eik ·Ra
Nui∈A a=1,2,3

1  δγ0a (ri )fa (kF )∗ e−i2kF ·ri ei(k −k)·ri
Nui∈A a=1,2,3
vF  q=(37)
Ax (ri ) + iAqy (r i ) e−i2kF ·ri ei(k −k)·ri ,NuΨAkF +k |H1 |ΨB−kF +k  =

i∈A

and



1  
δγ0a (r i )fa (kF )∗ e−i2kF ·ri ei(k −k)·ri e−ik·Ra
Nui∈A a=1,2,3

1  δγ0a (r i )fa (kF )∗ e−i2kF ·ri ei(k −k)·ri
Nui∈A a=1,2,3
vF  q=(38)
Ax (r i ) + iAqy (ri ) e−i2kF ·ri ei(k −k)·ri .
NuΨBkF +k |H1 |ΨA−kF +k  =i∈APseudospin in Graphene

263

In Eqs. (37) and (38), from the ﬁrst line of the right-hand side to the second line, we
approximate eik ·Ra ≈ 1 and e−ik·Ra ≈ 1 since we consider the case of |k ||kF |and |k||kF |. Equations (37) and (38) show that the deformation-induced gauge
ﬁeld Aq (r) (Eq. (31)) can be applied to the matrix element for intervalley scattering,
too. It is noted that Eqs. (37) and (38) are identical to each other up to the leading
term. The matrix elements from the K point to the K point are given by replacing
kF by −kF in Eqs. (37) and (38) asvF  qΨA−kF +k |H1 |ΨBkF +k  =Ax (r i ) − iAqy (r i ) e+i2kF ·ri e−i(k −k)·ri ,Nui∈AvF  q−kF +kkF +k
|H1 |ΨA=Ax (r i ) − iAqy (r i ) e+i2kF ·ri e−i(k −k)·ri . (39)
ΨB
Nu
i∈A

In the eﬀective-mass model, the intervalley scattering appears as the oﬀ-diagonalterms in a 4 × 4 matrix:
 K K 
ψ (r)φq (r)σx
σ · (p̂ + Aq (r))
ψ (r)
E
= vF
,
(40)

q
q
∗

K
φ (r) σxσ · (p̂ − A (r))ψK (r)ψ (r)where we deﬁne a deformation-induced complex scalar ﬁeld φ(r)φq (r) ≡ (Aqx (r) + iAqy (r))e−i2kF ·r .

(41)

When Aq (r) is a smooth function of r, the eﬀect of intervalley scattering by the complex scalar ﬁeld φq (r) can be neglected because φq (r) becomes a rapidly oscillating
function of r due to the factor e−i2kF ·r , and the value of matrix element between
the K and K points becomes negligible, too. On the other hand, when Aq (r) is
a rapidly oscillating function of r, φq (r) becomes a smooth function of r and thematrix element between the K and K points is not negligible. For example, phononmodes at the K point contribute to the intervalley scattering. Since the vibrational
amplitude of the K point phonon modes are proportional to e−ikF ·ri , Aq (r i ) has thesame periodicity of e+2ikF ·ri where we use the fact e+3ikF ·ri = 1. Then, φq (r i ) would
behave as a constant in the real space.
3.2. On-site interaction
Now we consider the on-site interaction by a defect of the crystal. A lattice
deformation gives rise to not only a change of the transfer integral between A and B
atoms but also a change of the potential at the A (B) atom φA (φB ) which we callthe oﬀ-site and on-site deformation potential, respectively. We denote the on-site
deformation potential by a 2 × 2 matrix as
φA (ri )
0
.
(42)
Hon =
0
φB (r i + R1 )Let the displacement vector of A-atom at r i is uA (r i ) and that of B-atom at r j isuB (r j ), then the deformation potential of A-atom at r i , φA (r i ), is induced by the
relative displacements of three nearest neighbor B-atoms from the A-atom (uB (r i +Ra ) − uA (ri )) as

264

K. Sasaki and R. Saito

Fig. 2. Displacements of B-atoms at r i + Ra (a = 1, 2, 3), uB (ri + Ra ), give rise to a deformation
potential at A-atom of r i .

φA (r i ) =gon Ra · (uB (r i + Ra ) − uA (ri )) ,
acc

(43)

a=1,2,3

where gon denotes gradient of the atomic potential at r i , and denotes 3acc /2. Hereacc and that φA (ri ) depends linearly on
we assume that |uB (ri + Ra ) − uA (ri )|the relative displacement vector.
By expanding uB (r i +R2 ) as uB (r i +R2 ) = uB (r i +R1 )+((R2 −R1 )·∇)uB (r i +R1 ) + · · · and uB (r i + R3 ) as uB (r i + R3 ) = uB (ri + R1 ) + ((R3 − R1 ) · ∇)uB (ri +R1 ) + · · · , we see that Eq. (43) can be approximated byφA (r i ) = gon ∇ · uB (r i + R1 ) + · · · ,(44)where we have used a=1,2,3 Ra = 0. It is noted that a general expression for the
deformation potential, Eq. (44), is valid in the case that uB (r) is a smooth functionof r. When this is not the case, we have to use Eq. (43). In the continuous limit, we
may use r to represent the positions of both A and B atoms in the unit cell, thenwe have φA (r) = gon ∇ · uB (r) + · · · . Similarly, the deformation potential of B-siteof r i + R1 is given bygon −Ra · (uA (r i + R1 − Ra ) − uB (r i + R1 )) .
(45)
φB (r i + R1 ) =acc
a=1,2,3

By using uA (r i + R1 − R2 ) = uA (r i ) + ((R1 − R2 ) · ∇)uA (r i ) + · · · and uA (r i +R1 − R3 ) = uA (ri ) + ((R1 − R3 ) · ∇)uA (ri ) + · · · , we see that Eq. (45) can beapproximated byφB (r i + R1 ) = gon ∇ · uA (r i ) + · · · .

(46)Thus, for the intravalley scattering, we may rewrite Eq. (42) using Eqs. (44) and(46) as
0∇ · uB (r)+ ··· .
(47)
Hon = gon0
∇ · uA (r)Pseudospin in Graphene

265

According to the result of density-functional theory by Porezag et al.,19), 20) we will
use the parameter for gon (=17 eV). For the later discussion of el-ph interaction of
acousticuA (r) + uB (r)
,
2

(48)

u(r) ≡ uB (r) − uA (r),(49)s(r) ≡and opticalphonon modes, we rewrite Eq. (47) using the Pauli matrices as
Hon =

gon
gon
I∇ · (uA (r) + uB (r)) +σz ∇ · (uB (r) − uA (r)).
2
2

(50)

Comparing with the expressions for Aq (r) and φq (r) in Eqs. (31) and (41), and for
Eq. (50), the eﬀective-mass Hamiltonian of the defect of the crystal is given by
vF φq (r)σx
HK
,
(51)
H = vFvF φq (r)∗ σxHK
where we deﬁnegon
σz ∇ · u(r),2gon
σz ∇ · u(r).
HK = vF σ  · (p̂ − Aq (r)) + gon I∇ · s(r) +2
HK = vF σ · (p̂ + Aq (r)) + gon I∇ · s(r) +(52)

3.3. Time-reversal and pseudospin symmetries
Here we show that Aq (r) keeps time-reversal symmetry of graphene system.
The time-reversal operation, T , is deﬁned by exchanging the K point and the Kpoint, and taking a complex conjugation of the four component wavefunction as  K ∗ 


 K
ψ (r)
0 I
ψ (r)
=
, where T =
K
(53)
T

I 0
ψ K (r)∗ψ K (r)and K is the complex conjugate operator. In order to check that Eq. (53) is timereversal operation, it is useful to introduce the vector potential for electro-magnetism,
A(r), into the eﬀective-mass Hamiltonian asφq (r)σx
σ · (p̂ + Aq (r) − eA(r)).
(54)
H = vFφq (r)∗ σxσ  · (p̂ − Aq (r) − eA(r))
Then the electromagnetic current operator, J (r), is given by


∂H
σ 0
= −evF.
J (r) =
0 σ
∂A(r)

(55)

Using T in Eq. (53), we can show thatT −1 J (r)T = −J (r),

(56)

266

K. Sasaki and R. Saito

where we have used K −1 σK = −σ  and K −1 σ  K = −σ. The negative sign on theright-hand side of Eq. (56) shows that Eq. (53) is time-reversal operation.
When we apply T to the wavefunction of Eq. (40), we get K ∗   K ∗ 
φq (r)σx
σ · (p̂ + Aq (r))
ψ (r)
ψ (r)
= vF
.
(57)
E
φq (r)∗ σxσ  · (p̂ − Aq (r))
ψ K (r)∗ψ K (r)∗By taking the complex conjugation (c.c.) of the ﬁrst row of Eq. (57), we getEψ K (r)∗ = vF σ · (p̂ + Aq (r))ψ K (r)∗ + vF φq (r)σx ψ K (r)∗
c.c.





→ Eψ K (r) = vF σ  · (p̂ − Aq (r))ψ K (r) + vF φq (r)∗ σx ψ K (r),(58)

where we have used σ ∗ = −σ  and p̂∗ = −p̂. Similarly, by taking the complex
conjugation of the second row of Eq. (57), we get
Eψ K (r)∗ = vF φq (r)∗ σx ψ K (r)∗ + vF σ  · (p̂ − Aq (r))ψ K (r)∗c.c.

→ Eψ K (r) = vF φq (r)σx ψ K (r) + vF σ · (p̂ + Aq (r))ψ K (r).

(59)

The equations in the last lines of Eqs. (58) and (59) are nothing but the second andﬁrst row of Eq. (40), respectively. Therefore, Eq. (54) (or Eq. (40)) is symmetric
under the time-reversal transformation (when A(r) = 0).
When φq (r) = 0, it is useful to deﬁne an operation S that transforms the electronof k to that of −k within the same valley. The operation S is deﬁned by
 



 K
σy ψ K (r)∗
ψ (r)
I 0
=
(60)
, where S =
σy K.
S


0 I
σy ψ K (r)∗
ψ K (r)This operation is referred to as eﬀective time-reversal symmetry or special timereversal symmetry because we obtain S −1 J (r)S = −J (r) for the special case ofφq (r) = 0, Aq (r) = 0, and A(r) = 0. By applying Eq. (60) to the wavefunction of
Eq. (40), we get for the ﬁrst row asEσy ψ K (r)∗ = vF σ · (p̂ + Aq (r))σy ψ K (r)∗ + vF φq (r)σx σy ψ K (r)∗σy→ Eψ K (r)∗ = vF σ  · (p̂ + Aq (r))ψ K (r)∗ − vF φq (r)σx ψ K (r)∗c.c.

→ Eψ K (r) = vF σ · (p̂ − Aq (r))ψ K (r) − vF φq (r)∗ σx ψ K (r).

(61)

Here, from the ﬁrst line to the second line, we multiplied σy to both sides and usedσy σσy = σ  . To get the last line, we took the complex conjugation of the second line
and used (σ  )∗ = −σ and p̂∗ = −p̂. The special operation S becomes a symmetrywhen Aq (r) = 0. Note that Aq (r) = 0 results in φq (r) = 0 because of Eq. (41).
We comment on the deﬁnition of the time-reversal operation and the on-site
interaction. The transformation deﬁned by Eq. (53) does not contain any Pauli spin
matrix since time-reversal symmetry has nothing to do with the pseudospin degreeof freedom. It is also noted that the pseudospin degree of freedom (σy in Eq. (60))
is necessary to deﬁne the special symmetry of Eq. (60). Even in the presence of theon-site deformation potential of Eq. (42), the time-reversal symmetry of Eq. (53) isPseudospin in Graphene267

valid since Eq. (53) does not contain the Pauli spin matrix. However, the special
symmetry of Eq. (60) is lost when Eq. (42) is not symmetric about the sublattice.21)
Namely, when we take into account a non-vanishing σz term of Eq. (50) for Eq. (61),the symmetry of Eq. (60) is broken even when Aq (r) = 0.
§4.

Electron-phonon interaction

In this section, we apply the Aq (r) for phonon mode to describe the el-phinteraction. There are six phonon modes in graphene: in-plane longitudinal optical/acoustic mode (LO/LA), in-plane tangential optical/acoustic mode (iTO/iTA),
and out-of-plane tangential optical/acoustic mode (oTO/oTA).17) We consider longwavelength in-plane optical phonon modes; the LO and iTO phonon modes near theΓ point. Using Eqs. (31) and (50), we will derive Aq (r) and Hon for the el-ph interaction of the LO/iTO modes. The material in this section has been used to analyze
the LO/iTO phonon frequency shift in metallic SWNTs.12) The phonon frequency
shift is observed as a function of the Fermi energy (the Kohn anomaly) in the Raman
spectroscopy experiments.22)–27)
4.1. Γ point optical phonon modes
The el-ph interaction originates from a change of the atomic potential due toa vibration of a carbon atom. In Fig. 3, we show a change of the atomic potential
whereby δγ0a (r) is induced. The atomic potential whose origin is located at r isdenoted by solid curves and the shifted potential whose origin is located at r + Δr isplotted by dashed curves. The deformation potential is given by Φ(r + Δr) − Φ(r) =Δr · ∇Φ(r) + · · · , which gives rise to matrix element between the π-electron at Aatom and the π-electron at B-atom as ΨB (r + Ra )|∇Φ(r)|ΨA (r) · (Ra /acc ). In thispaper we denote this matrix element as goﬀ / , that is, ΨB (r + Ra )|∇Φ(r)|ΨA (r) ·(Ra /acc ) = goﬀ / . According to density functional calculation by Porezag et al.,19)
we have goﬀ / = 3.0 eV/Å20), 28) where ≡ 3acc /2 = 2.13 Å.

Fig. 3. The atomic potential of a carbon atom at r is denoted by solid curves and the shifted
potential of the carbon atom at r + Δr is plotted by dashed curves. The deformation potential
at the B atom is given by Φ(r + Δr) − Φ(r) = Δr · ∇Φ(r) + · · · . The deformation potential gives
rise to a scattering of a π-electron from A-atom to the B-atom as ΨB (r + Ra )|∇Φ(r)|ΨA (r) ·(Ra /acc ).

268

K. Sasaki and R. Saito

When Δr = u(r) (u(r) is deﬁned in Eq. (49)), the change of the C-C bondlength is given by u(r) · (Ra /acc ). Thus δγ0a (r) for the LO and TO modes is given
by
Ra
goﬀ
u(r) ·.
(62)δγ0a (r) =acc
By putting Eq. (62) into Eq. (31), we obtaingoﬀ
R2 + R3q,u(r) · R1 −vF Ax (r) =
acc
2
(63)
√
goﬀ 3
q
vF Ay (r) =u(r) · (R2 − R3 ) .
acc 2√
Because R1 − (R2 + R3 )/2 = ey and 3/2(R2 − R3 ) = − ex (see the caption ofFig. 1), we can rewrite Eq. (63) by Aq (r) as
goﬀ(uy (r), −ux (r)),(64)
vF (Aqx (r), Aqy (r)) =acc
where ux (r) ≡ u(r) · ex and uy (r) ≡ u(r) · ey .29) For the on-site el-ph interaction of
Eq. (50), we can neglect the term which is proportional to σ0 since the LO and TO
optical phonon modes satisfy uA (r) = −uB (r). Then, we getgon
σz ∇ · u(r).
(65)
Hon =
2
The resulting eﬀective-mass Hamiltonian for the K point and K point becomesgon
σz ∇ · u(r),HK = 6vF σ · (p̂ + Aq (r)) +2gon
σz ∇ · u(r),(66)
HK = vF σ  · (p̂ − Aq (r)) +2
where Aq (r) is given by Eq. (64). Putting Eq. (64) into Eq. (34), we see in Eq. (66)that ∇ · u(r) is proportional to the deformation induced magnetic ﬁeld sincevF acc q
B (r).
(67)∇ · u(r) = −goﬀ z
Thus, if Bzq (r) = 0, the energy spectrum for an electron has a local energy gap
around EF because the terms proportional to σz in the Hamiltonian Eq. (66) work
as time dependent mass term.∗) However, the optical phonon mode does not generatea static (time independent) energy gap because u(r) is oscillating as a function oftime as u(r) cos(ωt) so that the time average of Bz (r) ﬁeld vanishes. However, theenergy gap can be oscillating as a function of time when the vibration of atoms iscoherent in the space.
Next we consider an out-of-plane TO (oTO) phonon mode. The oTO phonon
eigenvector is pointing in the direction perpendicular to the 2D plane and does notgive rise to a ﬁrst order, in-plane bond-length change. Thus, the oﬀ-site and on-site
el-ph couplings for the oTO mode are negligible as compared with those for in-plane
LO and TO modes.
∗)

The mass term is deﬁned by mσz in the eﬀective-mass Hamiltonian where m produces anenergy gap in the energy spectrum.

Pseudospin in Graphene269

4.1.1. Phonon softening for q = 0 LO/iTO phonon mode
For a uniform (ux , uy ) in the real space, that is, for the LO and TO modes withthe phonon wave vector q = 0, the el-ph interaction for the LO and TO phononmodes is given only by Aq in Eq. (66) since ∇ · u(r) = 0. For the Γ point optical phonon modes, we consider a virtual electron-hole pair creation by the el-phinteraction, which contributes to the phonon frequency shift in second-order perturbation theory (see Fig. 4(a)). The phonon frequency shift as a function of the Fermi
energy has been observed in Raman spectroscopy for graphene22), 23) and metallic
nanotubes.24)–27)
The el-ph matrix element for the electron-hole pair generation near the K point
is given from Eqs. (20) and (64) byeh(k)|H1K |ω =

K
K
(ψc,k
(r))∗ vF σ · Aop ψv,k(r)d2 r Θ(k) t   −i Θ(k) 2
goﬀ e+i 2e0uy + iux
.
=
Θ(k)
Θ(k)
+i 2
uy − iux
02acc e−i 2
−e

(68)

Let us introduce the angle α between vector u = (ux , uy ) and the x-axis, then byputting ux = u cos α and uy = u sin α (ux + iuy = ueiα ) into Eq. (68) we obtaineh(k)|H1K |ωα  = −igoﬀucos(Θ(k) − α).
acc

(69)

The el-ph matrix element for the electron-hole pair generation near the K point,eh(k)|H1K |ωα , is the same as Eq. (69). In Eq. (69), α depends on the LO and TO
modes in the case of SWNTs, while α can not be deﬁned uniquely in the case of ﬂatgraphene. Let us consider the case of a zigzag SWNT. Then, we denote x (y) as acoordinate around (along) the axis as shown in Fig. 1(b). In this case, u = (ux , 0)
(α = 0) is assigned to the TO phonon mode while u = (0, uy ) (α = π/2) is assigned
to the LO phonon mode. By calculating Eq. (69) for the TO mode with α = 0 andfor the LO mode with α = π/2, we getucos Θ(k),accusin Θ(k).
eh(k)|H1K |ωLO  = −igoﬀacceh(k)|H1K |ωTO  = −igoﬀ(70)

For a “metallic” zigzag SWNT without the curvature eﬀect,30), 31) we obtain
Θ(k) = ±π/2 for a metallic energy band with kx = 0 (see Fig. 4(b)). Then, Eq. (70)
tells us that only the LO mode gives rise to a non-vanishing el-ph matrix element andthe TO mode does not contribute to the electron-hole pair creation. The amplitude
for an electron-hole pair creation depends strongly on the curvature eﬀect whichshifts the relative position of the cutting line from kx = 0 (see Fig. 4(b)). Whenthe curvature eﬀect is taken into account, cos Θ(k) = kx /(kx2 + ky2 )1/2 is nonzero due
to kx = 0. Thus, the low energy electron-hole pair can couple to the TO phononmode and the matrix element becomes maximum when ky = 0. On the other hand,
the high energy electron-hole pair still decouples to the TO phonon mode since270

K. Sasaki and R. Saito

Fig. 4. (a) An electron-hole pair on the cutting line near the K point is excited by the Γ point
phonon mode (q = 0). The intermediate electron-hole pair state contributes to the phonon
frequency shift. (b) The kx (ky ) axis corresponds to the zigzag SWNT circumferential (axis)
direction. The amplitude for an electron-hole pair creation (Eq. (70)) depends strongly on therelative position of the cutting line from the K point. If the cutting line crosses the K point,then the angle Θ(k) (≡ arctan(ky /kx )) takes π/2 (−π/2) values for ky > 0 (ky < 0). In thiscase, the LO mode strongly couples to an electron-hole pair, while the TO mode is decoupledfrom the electron-hole pair.

cos Θ(k) → 0 for |ky ||kx |. This leads to a phonon frequency hardening for the
TO phonon mode of a zigzag nanotube when the Fermi energy is located near the
Dirac point.12)
The description of deformation induced gauge ﬁeld for a lattice deformation(Eq. (31)) is useful to show the appearance of the curvature-induced mini energy
gap in metallic zigzag SWNTs.13) For a zigzag SWNT, we have δγ01 = 0 and δγ02 =δγ03 = 0 from the rotational symmetry around the tube axis (see Fig. 1). Then,
using Eqs. (31), we get that Aqx = −δγ02 /vF and Aqy = 0 as the curvature eﬀect. The
cutting line of kx = 0 for the metallic zigzag nanotube is shifted by a ﬁnite constantvalue of Aqx because of the AB eﬀect for the deformation-induced gauge ﬁeld Aq (r).
This explains the appearance of the curvature-induced mini energy gap,
Egap =

60(meV · nm2 )
,
d2t

(71)

where dt is the diameter of a metallic zigzag SWNT.
4.1.2. q = 0, LO/iTO mode
The description of the el-ph interaction as a gauge ﬁeld Aq (r) can be extended
to show the decoupling between the TO mode with q = 0 and the electrons using
a gauge symmetry argument.12) The TO phonon mode with q = 0 does not change
the area of the hexagonal lattice but instead gives rise to a shear deformation for the
hexagonal unit cell of graphene. Thus, the TO mode (uTO (r)) satisﬁes∇ · uTO (r) = 0,

∇ × uTO (r) = 0.

(72)

Thus, the on-site deformation potential is zero in Eq. (66). Using Eqs. (64) and (72),
we see that the deformation-induced magnetic ﬁeld, Bzq (r), becomes zero instead thePseudospin in Graphene271

divergence of Aq (r) exists, which are(∇ × Aq (r)) · ez = Bzq (r) = −∇ · Aq (r) =goﬀ∇ · uTO (r) = 0,vF accgoﬀ(∇ × uTO (r)) · ez = 0.
vF acc(73)

In this case, Aq (r) can be represented by the gradient of a scalar function, Ψa (r),as Aq (r) = ∇Ψa (r). Since we can choose a Aq (r) = 0 gauge in Eq. (66) by a
redeﬁnition of the phase of the wavefunction as ψ K (r) → exp(−iΨa (r)/)ψ K (r) andψ K (r) → exp(+iΨa (r)/)ψ K (r),15) the Aq (r) ﬁeld in Eq. (66) disappears for the
TO mode with q = 0. This explains why the TO mode with q = 0 completely
decouples from the electrons and that only the TO mode with q = 0 couples withelectrons. In this sense, the TO phonon mode at the Γ point is anomalous since
the el-ph interaction for the TO mode can not be eliminated by a phase of the
wavefunction. On the other hand, the LO phonon mode with q = 0 changes the areaof the hexagonal unit cell while it does not give rise to a shear deformation. Thus,
the LO mode (uLO (r)) satisﬁes∇ · uLO (r) = 0,

∇ × uLO (r) = 0.

(74)

Using Eqs. (64) and (74), we see that the LO mode gives rise to a deformationinduced magnetic ﬁeld as
Bzq (r) = 0,

∇ · Aq (r) = 0.

(75)

Since a magnetic ﬁeld changes the energy band structure of electrons through themass term, the LO mode near the Γ point (with q = 0) is important for electronicenergy spectrum.
§5.

Edge states of graphene

Finally, we give two examples of lattice deformation along a line in graphene
as shown in Figs. 5(a) and (b) whose ﬁeld Bzq (r) can polarize the pseudospin. In
Fig. 5(a), we modify the hopping integral at the dotted line of y = 0, so that δγ01 (r) =0 at y = 0 and δγ02 (r) = δγ03 (r) = 0. From Eq. (31), we obtain vF Aq (y) = (δγ01 (y), 0).
In Fig. 5(b), the hopping integral along the dotted line at x = 0 (precisely, x = 0+or x = 0− ) is changed as δγ01 (r) = 0 and δγ02 (r) = δγ03 (r) = 0 at x = 0. We
obtain vF Aq (x) = (−δγ02 (x), 0) for this case. Figures 5(a) and (b) correspond to
the generation of the zigzag edge and the armchair edge in the limit of a strong
perturbation, respectively, if we take δγ01 = γ0 and δγ02 = δγ03 = γ0 for the hopping
integral at the two lines. The behavior of the electronic structure for the two cases arecompletely diﬀerent from each other, which we show by calculating the deformationinduced magnetic ﬁeld. In the case of Fig. 5(a), we have a ﬁnite deformation-induced
magnetic ﬁeld Bzq (r) = 0 near the line at y = 0. The deformation-induced magnetic
ﬁeld Bzq (y) is negative at y < 0 as illustrated by ⊗ and is positive at y > 0 as
. On the other hand, in the case of Fig. 5(b), the deformation-induced magnetic

272

K. Sasaki and R. Saito

Fig. 5. Local deformation along a line and corresponding deformation-induced gauge ﬁeld Aq (r) are
shown by arrows in (a) zigzag and (b) armchair edge. In (a), Aq (r) gives a ﬁnite deformationinduced magnetic ﬁeld Bzq (r) (ﬂux) illustrated by and ⊗ while no deformation-induced magnetic ﬁeld is present in (b).

ﬁeld is zero. The gauge ﬁeld which gives zero magnetic ﬁeld can be removed from
Hamiltonian by a gauge transformation, which is discussed in the previous section.
The deformation-induced magnetic ﬁeld accounts for the presence of so-called edge
states only at the zigzag edge, which is shown as below. As we show in §1, the edge
state is a localized wavefunction near the zigzag edge in which the pseudospin of theedge state is perfectly polarized. This means that the amplitude of the wavefunction
has a value only on either A or B atoms.
We will derive the edge states from the eﬀective-mass model with the gauge ﬁeldof Fig. 5(a).11) Before going into the details, let us ﬁrst outline the story. We willshow that there are localized pseudospin-polarized states in the energy spectrum andthe energy dispersion appears at px > 0 shown as the two solid lines in Fig. 6(b).
The velocity for the energy dispersion becomes small with increasing the gauge ﬁeld
and it becomes zero when the gauge ﬁeld is suﬃciently strong which corresponds tothe zigzag edge. This result of the eﬀective-mass model can reproduce the result of
the tight-binding (TB) lattice model as shown in Figs. 6(c), (d) and (e),11) where
the TB lattice model is deﬁned by an adiabatic parameter c as δγ01 (y = 0) = cγ0 in
Eq. (27). Here c = 0 and c = 1 correspond to no deformation and the zigzag edge,
respectively.
We assume that Aq (r) of Fig. 5(a) is quite localized within |y| < ξg , that is,q
A (r) = (Aqx (y), 0) and Aqx (y) = 0 for |y| ≥ ξg in Eq. (33), where ξg is a length ofthe order of lattice spacing. We parameterize the localized energy eigenstate asK(r) = N  exp(ikx x)e−G(y)ψE

 +g(y) e,e−g(y)(76)

where N  is a normalization constant. The pseudospin polarization is represented byg(y), and the wave vector kx is a good quantum number because of the translational
symmetry along the x-axis. In the direction of y, we assume a localized nature of
the wavefunction which was obtained by TB calculation. Putting Eq. (76) to thePseudospin in Graphene

273

Fig. 6. (a) Band structure (kx , E) around the K point of the eﬀective-mass model without the
deformation-induced gauge ﬁeld and (b) that with the gauge ﬁeld along the x-axis. The shaded
regions in (a) and (b) represent the spectrum of extended states for diﬀerent ky . The (twodimensional) dispersion relation around the K point for graphene without the deformation
gives two cones (representing the linear k dispersion) whose apex is the K point as is shown in
the inset of (a). The band structure of (a) is a one-dimensional projection (onto ky plane) of
the linear k dispersion. In (b), the energy dispersion for the localized state is represented bysolid lines. When the bonds at y = 0 become disconnected (c = 1) and zigzag edges appear,
the energy eigenvalues of the localized state converge to zero (E → 0) for any kx (> 0) andforms a ﬂat energy band of edge state. Using the TB model, we plot the band structure foran undeformed graphene in (c), for a zigzag ribbon in (e), and for a graphene with weakened
hopping integral for the C-C bonds at y = 0 in (d). kx a in (c), (d) and (e) is the wave vector
parallel to the x-axis and kx a = 2π/3 is the K point.


K (r) = Eψ K (r), we obtainenergy eigenequation, HK ψEE

E
d(G(y) + g(y)) = − e+2g(y) ,
dy
vF
E
d
px − Aqx (y) +  (G(y) − g(y))) = − e−2g(y) .
dyvFpx − Aqx (y) − (77)

By summing and subtracting the both sides of Eq. (77), we rewrite the energy

274

K. Sasaki and R. Saito

eigenequation aspx − Aqx (y) − 


E
dg(y)
=−
cosh(2g(y)),
dy
vF

E
dG(y)
=
sinh(2g(y)).
dy
vF

(78)

Since we are considering a localized solution around the x-axis, we assume G(y) ∼|y|/ξ where ξ (> 0) is the localization length. When G(y) ∼ |y|/ξ, the solution of
the second equation of Eq. (78) is given by⎧⎨ − 1 sinh−1 ~vF , (y ≤ −ξg )2
 ξE 
(79)
g(y) =⎩ + 1 sinh−1 ~vF . (y ≥ ξg )2
ξE
The functions G(y) and g(y) are schematically shown in Figs. 7(a) and (b), respectively. The sign of g(y) changes across the x-axis; the sign change of g(y) means thatthe pseudospin direction changes at the x-axis. The change of pseudospin is inducedby the gauge ﬁeld Aqx (y). To see this, we integrate the ﬁrst equation of Eq. (78) fromy = −ξg to ξg , and we get the following relation:
−

ξg

1
dg(y)
dy =

−ξg dyξg−ξg

Aqx (y)dy.

(80)

We have neglected other terms, since they are proportional to ξg and become zeroin the limit of ξg = 0. By putting Eq. (79) to Eq. (80), we ﬁnd
1 ξg q−1 vF=A (y)dy.
(81)− sinhξE −ξg xWhen the right-hand side of Eq. (81) is large, we obtain from Eq. (79) that g(y)00 for y ≥ ξg . In this case, the localized state is a pseudospinfor y ≤ −ξg and g(y)K (r) ∝ t (1, 0) for y ≤ −ξ and a pseudospin-down state ψ K (r) ∝ t (0, 1) forup state ψEgEy ≥ ξg . Thus, a strong gauge ﬁeld at the x-axis makes pseudospin-polarized localizedstates. Since the polarization of the pseudospin means that the wave function has
amplitude only A (or B) atoms, this result is consistent with the result by the TB
model for the edge state.10) We understand that the edge state is a pseudospin
polarized and localized state in the real space which is induced by the deformationinduced magnetic ﬁeld of the Aq (r) along the x-axis.
Let us now calculate E and ξ. To this end, we use the ﬁrst equation of Eq. (78)for |y| ≥ ξg and obtain
E
=
vF


cosh

Moreover, using Eq. (81), we get= px tanh
ξ

−px
1




1


ξg
−ξg

ξg
−ξg

.

(82)

Aqx (y)dy
Aqx (y)dy.

(83)

Pseudospin in Graphene

275

Fig. 7. (a) The amplitude of the wave function, exp(−G(y)), of a localized state whose localization
length is ξ. (b) The pseudo-spin modulation part, g(y). From Eq. (79), g(y) is a constant for|y| ≥ ξg , and abruptly changes across the x-axis (y = 0).

In addition to this localized state, there is another localized state for the same px
with the same ξ but with an opposite sign of E. This results from a particle-holesymmetry of the Hamiltonian; σz HK σz = −HK . By the particle-hole symmetryK (r) = σ ψ K (r).
operation, the wave function is transformed as ψ−E
z E
The normalization condition of the wave function requires that ξ should bepositive, which restricts the value of px to px > 0 in Eq. (83). In fact, when Aqx (y) is
positive, Eq. (83) means that the localized states appear only at px > 0 around theK point. This is the reason why the localized states appear in the energy spectrum
only in one side around the K point as shown in Fig. 6(b). On the other hand, inthe case of the K point the Hamiltonian is expressed by Eq. (32). Because of thediﬀerent signs in front of Aq (r) of Eqs. (32) and (33), a similar argument gives px < 0ξ0,
for the edge states around the K point. Furthermore, when (1/) −ξg g Aqx (y)dy
E in Eq. (82) becomes zero. The zero energy eigenvalue between the K and K
points in the band structure corresponds to the ﬂat energy band of the edge state.10)
When Aqx (y) is negative (Aqx (y)0), a ﬂat energy band appears in the opposite
side: px < 0 around the K point and px > 0 around the K point. This condition,
0, corresponds to the Klein’s edges32) which are obtained by removing A
Aqx (y)or B atoms out of the zigzag edges having A or B atoms. Calculated result by the
TB model with the Klein’s edges32) is thus analytically explained by Aq (r).
There are also extended states in addition to the edge states. The energy dispersion relation of extended states can be obtained as (E/vF )2 = p2x + p2y , by setting
G(y) ∼ −ipy y/ (|y| ≥ ξg ) in Eqs. (76) and (78) where py is a real number. The
calculated energy bands are given by |E| > vF |px |, shown as a shaded region in
Fig. 6(b). It also agrees well with the TB calculation shown in Fig. 6(e). Since
1 − tanh2 x = 1/ cosh2 x, we see that the energy dispersion relation of the localizedstate between E and ξ becomes
 2
 2

E
2
= px −, (|y| ≥ ξg )
(84)
vF
ξ
which is the same as the linear dispersion relation (E/vF )2 = p2x + p2y if one replaces
py with i(/ξ). Thus one can regard the localized state as a state with a complex
wavenumber.

276

K. Sasaki and R. Saito

We have shown that three basic properties of the edge states, i.e., the pseudospin
polarization, the dependence on the momentum, and the ﬂat energy band, obtained
previously by the TB model,10) can be explained analytically in terms of the gaugeﬁeld. In order to compare the present theory with the TB model quantitatively, we
have performed a TB calculation for the geometry of Fig. 5(a) with setting δγ01 = cγ0 .
In Figs. 6(c), (d) and (e), we plot the band structure for c = 0, c = 1/2, and c = 1.
Comparing these ﬁgures with the results of the continuous model, one can ﬁnd an
exact correspondence between the TB model and continuous model. Moreover, weanalytically ﬁnd11)
|px |
|E|
+ O(lp2x /),=vF
cosh(− ln(1 − c))= px tanh(− ln(1 − c)) + O(lp2x /),ξ(85)

for localized states around the K point. Thus, by comparing Eq. (85) with Eqs. (82)and (83), we conclude that the TB model and the continuous model agree with each1), by the following relationship:
other near the K point (px l/
1


ξg
−ξg

Aqx (y)dy = − ln(1 − c).

(86)

The right-hand side diverges when c → 1, which reproduces the ﬂat energy band
(E → 0) in Eq. (82) and gives ξ/ = p−1
x in Eq. (83). When c → 0, we haveq1 ξgc, which conﬁrms Eq. (31) that is derived by assuming a weak~ −ξg Ax (y)dy

perturbation for δγ0a (r).∗)By considering the edge state using the eﬀective-mass model, we found that the
deformation-induced gauge ﬁeld (Aq (r)) and magnetic ﬁeld (Bzq (r)) explain basic
properties of the edge state. It is summarized as follows:(1) The gauge ﬁeld can generate the edge state in energy spectrum, depending on thegauge ﬁeld direction. Let e the unit vector along the edge and Aq (r) ≡ Aq (r) · e ,the edge state appears if the gauge ﬁeld has a component parallel to the edge:Aq (r) = 0. This explains the presence (absence) of the edge state near the zigzag(armchair) edge.
(2) The edge states are pseudospin polarized states. The direction of the pseudospin
polarization is determined by the direction or sign of Bzq (r).
(3) The direction of the gauge ﬁeld, namely, Aq (r) > 0 or Aq (r) < 0, is vital for theedge states, since it determines the energy dispersion and wave vectors which allow
the edge states.
(4) The ﬂat energy band of edge states at zero energy is obtained by the limit;Aq (r) → ±∞.
∗)It is interesting to note that Bzq (r) for the edge state corresponds to an enormous magnetic
ﬁeld ∼ 105 T at the zigzag edge.11), 33) Thus a uniform external magnetic ﬁeld has little eﬀect on theedge state, compared with the deformation.

Pseudospin in Graphene
§6.

277

Discussion and summary

By formulating the eﬀective-mass equation for a graphene with a lattice deformation, we have shown that the deformation can be represented by the deformationinduced gauge ﬁeld, Aq (r). We formulate the el-ph interactions and represent theshape of edge by Aq (r). The appearance of the gauge ﬁeld in the eﬀective-massequation is reasonable because the Feynman diagram for the el-ph interaction isbasically the same as that for the electromagnetic interaction of A(r). The only difference between the ﬁelds Aq (r) and A(r) is that Aq (r) does not break time-reversal
symmetry as a whole system while Aq (r) breaks time-reversal symmetry locally in
k-space. Thus, we think that a time-reversal symmetric gauge ﬁeld is useful not only
for graphene system but also other lattice systems that have an internal degree of
freedom like the pseudospin.
Here we would like to mention the extension of the eﬀective-mass model forthe edge states. It is known that the Coulomb interaction makes the real spin(not pseudospin) of the edge states polarized.10) We considered the eﬀective-massmodel for the Hubbard model with a mean ﬁeld approximation, and found that the
Hubbard term appears as a mass term whose sign depends on the spin of the edge
states. The mass term shifts the energy of the edge states up or down relative to
E = 0 depending on the sign of the mass so that the ground state exhibits the spinpolarized state. The gauge ﬁeld and the mass term of the eﬀective-mass model giverise to a parity anomaly phenomenon. We have shown that ferromagnetism of theedge states closely relates to the parity anomaly.33)
In summary, the lattice deformation of graphene is modeled in the eﬀectivemass theory by the deformation-induced gauge ﬁeld which keeps the time-reversal
symmetry but breaks the special symmetry deﬁned within each valley. The formalismof the gauge ﬁeld and the gauge symmetry is useful for understanding anomalous
physical properties of graphene.
Acknowledgements
The authors are grateful to Prof. S. Murakami (Tokyo Institute of Technology)
for his outstanding instruction for our collaborating works. We also wish to thankProf. Mildred Dresselhaus, Prof. Jing Kong, and Dr. Hootan Farhat for sharing
experimental data on Raman spectroscopy of carbon nanotube. R. S. acknowledges
a Grant-in-Aid (Nos. 16076201 and 20241023) from MEXT.
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