physics of graphene zero-mode anomalies and roles of...

Progress of Theoretical Physics Supplement No. 176, 2008 203

Physics of Graphene

Zero-Mode Anomalies and Roles of Symmetry

Tsuneya Ando

Department of Physics, Tokyo Institute of Technology,Tokyo 152-8551, Japan

A brief review is given on electronic and transport properties of monolayer graphene froma theoretical point of view. The topics include the effective-mass description of electronicstates, topological anomaly associated with Berry’s phase, singular diamagnetic susceptibil-ity, zero-mode anomalies and their removal due to level broadening effects, the symmetrycrossover among symplectic, unitary, and orthogonal due to the presence of special timereversal symmetry, and interaction with acoustic, optical, and zone-boundary phonons.

§1. Introduction

In an effective-mass approximation, an electron in a graphite monolayer orgraphene is described by Weyl’s equation for a massless neutrino.1)–4) Transportproperties in such an exotic electronic structure are quite intriguing, and the con-ductivity with/without a magnetic field including the Hall effect,5),6) quantum cor-rections to the conductivity,7) and the dynamical transport8) were investigated theo-retically. The results show that the system exhibits various characteristic behaviorsdifferent from conventional two-dimensional systems.9) Quite recently, this singlelayer graphene was fabricated using the so-called scotch-tape technique10) and themagnetotransport was measured including the integer quantum Hall effect, demon-strating the validity of the neutrino description of the electronic states.11),12) Sincethen the graphene became the subject of extensive theoretical and experimentalstudy. The purpose of this paper is to give a brief review on such electronic andtransport properties from a theoretical point of view.

Later experiments in higher magnetic fields showed spin splitting for severallow-lying Landau levels and additional splitting for level n = 0.13) Inter-bandmagneto-absorptions were reported for the first time14) in graphenes epitaxiallygrown on SiC substrates.15),16) The cyclotron resonance was reported also in scotch-tape graphene.17),18) The local density-of-states and carrier concentrations weremeasured using scanning single-electron-transistor.19) The negligibly small negativemagnetoresistance was observed20) and a weak positive magnetoresistance was re-ported.21) There were reports on optical phonons using the Raman scattering,22),23)

spin transport,24),25) angle-resolved photoemission spectroscopy,26),27) and so on.There has also been a rapid development in the theory of graphene.28)

The effective-mass description of electronic states is discussed in §2. The topo-logical anomaly associated with Berry’s phase and the resulting unique Landau-levelstructure are discussed in §3. The singular diamagnetic susceptibility is reviewedin §4. In §5, various zero-mode anomalies appearing in transport quantities and

204 T. Ando

their removal due to level broadening effects are discussed. The presence of specialtime-reversal symmetry and associated symmetry crossover are discussed in §6. Theelectron-phonon interactions are discussed in §7.

§2. Neutrino description

The structure of 2D graphite sheet is shown in Fig. 1. A unit cell containstwo carbon atoms which are denoted by A and B. We have the primitive transla-tion vectors a = a(1, 0) and b = a(−1/2,

√3/2), and the vectors connecting be-

tween nearest neighbor carbon atoms τ 1 = a(0, 1/√

3), τ 2 = a(−1/2,−1/2√

3), andτ 3 = a(1/2,−1/2

√3), where the lattice constant is given by a = 2.46 A. The cor-

responding reciprocal lattice vectors are given by a∗ = (2π/a)(1, 1/√

3) and b∗ =(2π/a)(0, 2/

√3). The first Brillouin zone is given by a hexagon with two corner points

K and K′. The corresponding wave vectors are given by K = (2π/a)(1/3, 1/√

3) andK ′ = (2π/a)(2/3, 0) for K and K′ points, respectively. Further, we have

exp(iK ·τ 1)=ω, exp(iK ·τ 2)=ω−1, exp(iK ·τ 3)=1,(2.1)

exp(iK ′ ·τ 1)=1, exp(iK′ ·τ 2)=ω−1, exp(iK ′ ·τ 3)=ω,

with ω = exp(2πi/3), satisfying 1+ω+ω−1 =0.Let φ(r) be a pz orbital localized around the origin. In a nearest-neighbor tight-

binding model, the wave function is written as

ψ(r) =∑RA

ψA(RA)φ(r−RA) +∑RB

ψB(RB)φ(r−RB), (2.2)

where, ψA(RA) and ψB(RB) are the amplitudes at RA = naa+nbb+τ 1 and RB =naa+nbb with integer na and nb. Taking into account only the hopping integral −γ0

(γ0∼3 eV) between nearest-neighbor atoms and neglecting overlapping integrals, wehave

εψA(RA) = −γ0

3∑l=1

ψB(RA−τ l), εψB(RB) = −γ0

3∑l=1

ψA(RB+τ l), (2.3)

Fig. 1. The lattice structure (a) and the first Brillouin zone (b) of graphene. The unit cell contains

two carbon atoms denoted by A and B. Two primitive translation vectors are denoted by a

and b and three vectors connecting neighboring A and B atoms are fi l (l = 1, 2, 3). Reciprocal

lattice vectors are given by a∗ and b∗.

Physics of Graphene 205

Fig. 2. The band structure obtained in a tight-binding model.

where we have chosen the energy origin at the level of an isolated pz orbital.Setting ψA(RA)=fA(k) exp(ik·RA) and ψB(RB)=fB(k) exp(ik·RB), we have

(0 hAB(k)

hAB(k)∗ 0

)(fA(k)fB(k)

)= ε

(fA(k)fB(k)

), hAB(k) = −γ0

3∑l=1

exp(−ik·τ l).(2.4)

Then, the energy bands become

ε±(k) = ±γ0

∣∣∣3∑l=1

exp(−ik·τ l)∣∣∣. (2.5)

According to Eq. (2.2), we have ε±(K)= ε±(K ′)=0, showing that there is no gapat K and K′ points. Figure 2 shows the obtained π bands.

To see the behavior in the vicinity of the K point, for example, we rewrite k asK+k and expand ε±(k) in terms of |k|a. To the lowest order, we have

−γ0

3∑l=1

exp[−i(K+k)·τ l] = −ω−1γ(kx−iky), (2.6)

with

γ =√

32aγ0. (2.7)

Therefore, when we define fA(K+k)= fA(k) and fB(K+k)=−ωfB(k), we have

γ

(0 kx−iky

kx+iky 0

)(fA(k)fB(k)

)= ε

(fA(k)fB(k)

). (2.8)

With the use of the Pauli spin matrices

σx =(

0 11 0

), σy =

(0 −ii 0

), σz =

(1 00 −1

), (2.9)

206 T. Ando

��

��

� �

��

��

Fig. 3. The energy dispersion and density of states in the vicinity of K and K′ points obtained in

a k · p scheme.

the above equation is rewritten as

γ(σ ·k)f(k) = εf(k), f(k) =(fA(k)fB(k)

), (2.10)

with σ=(σx, σy). In a similar manner, we obtain the equation for the K′ point, forwhich σ is replaced with σ∗.

Thus, in the vicinity of the K and K′ points, the energy dispersion is given by

εs(k) = sγ|k|, s=±1, (2.11)

and the density of states becomes

D(ε) =gvgsL2

∑s,k

δ(ε−sγ|k|) =gvgs|ε|2πγ2

, (2.12)

where L2 is the area of the system, gs = 2 is the spin degeneracy, and gv = 2 is thevalley degeneracy corresponding to the K and K′ points. These results are illustratedin Fig. 3. The density of states vanishes at ε = 0 and therefore graphene is oftencalled a zero-gap semiconductor. As will later become clear, the energy-dependenceof the conductivity indicates that graphene is a metal rather than a semiconductor.

In an effective-mass approximation or a k · p scheme near the K or K′ point,the electron motion is described by a Schrodinger equation in which k is replaced byoperator k=−i∇. We have

γ(σ ·k)FK(r) = εFK(r), γ(σ∗ ·k)FK′(r) = εFK′

(r), (2.13)

where FK(r) and FK′(r) are a two-component wave function

FK(r) =(FKA (r)FKB (r)

), FK′

(r) =(FK

′A (r)FK

′B (r)

). (2.14)

This is exactly the same as Weyl’s equation for a neutrino, except that the velocity,given by

v =γ

�, (2.15)

is much smaller than the light velocity (about 1/300). This equation of motionis quite useful for the description of characteristic features of electronic states ingraphene.


§3. Topological anomaly and Berry’s phase

The wave function near the K point is written as

F sk(r) =1L

exp(ik·r) F sk, (3.1)

where F sk is the spin function in which the spin is in the direction of k or −k,

F sk = exp[iφs(k)]R−1[θ(k)] |s). (3.2)

Here, φs(k) is an arbitrary phase, θ(k) is the angle between k and the ky axis, i.e.,kx+iky = +i|k|eiθ(k) and kx−iky =−i|k|e−iθ(k), R(θ) is the spin rotation operatoraround the z direction perpendicular to graphene sheet, and |s) is the spin functionwhen k is in the ky axis,

|s) =1√2

(−is1

). (3.3)

The spin rotation operator is given by

R(θ) = exp(iθ

2σz

)=

(exp(+iθ/2) 0

0 exp(−iθ/2)

). (3.4)

The rotation operator has the property

R(θ1)R(θ2) = R(θ1+θ2), R(−θ) = R−1(θ). (3.5)

Further, we have

R(θ±2π) = −R(θ), R(−π) = −R(+π). (3.6)

Therefore, for a phase φs(k) independent of the direction of k, wave function F sk

changes its sign under the 2π rotation of k or it has the opposite sign under +π and−π rotation of k.

The sign of the wave function itself depends on the choice of the phase. We canchoose φs(k)=−θ(k)/2, for example. Then, we have

F sk =1√2

(−i s e−iθ(k)

1

), (3.7)

which does not change the sign under θ(k)→θ(k)±2π. However, this wave functionchanges its sign if we consider Berry’s phase under the wave-vector rotation.29),30)

In fact, Berry’s phase e−iη under the wave-vector rotation is calculated as

η = −i∫ T

0dt

⟨sk(t)

∣∣∣ ddt

∣∣∣sk(t)⟩

= −π, (3.8)

where k(t) moves along a closed contour around k = 0 between time t = 0 andT . Therefore, the wave function acquires phase −π when k is rotated around theorigin.4),31),32) The sign change occurs only when the closed contour encircles the

208 T. Ando

origin k = 0 but not when the contour does not contain k = 0, showing the presenceof a topological singularity at k = 0. This topological singularity at k = 0 associatedBerry’s phase is the origin of the absence of backward scattering in metallic carbonnanotubes.4),31),32)

A singularity at ε = 0 manifests itself in magnetic fields even in classical me-chanics. The equation of motion in magnetic field B perpendicular to the system isgiven by

�k = −ecv×B. (3.9)

This gives the cyclotron frequency

ωc =eBv2

cε. (3.10)

The cyclotron frequency ωc diverges and changes its sign at ε = 0,6),33) showing thatε=0 is singular in a magnetic field.

In quantum mechanics kx and ky satisfy the commutation relation [kx, ky] =−i/l2, where l is the magnetic length given by l =

√c�/eB. Semiclassically, the

Landau levels can be obtained by the condition∮kxdky = ±2π

l2

(|n| + 1

2

), (3.11)

as εn = sgn(n)√|n| + 1/2 �ωB with integer n, where �ωB =

√2γ/l and sgn(n) =

n/|n| for n �= 0 and 0 for n = 0. Because of the uncertainty principle, k2 = 0 is notallowed and there is no Landau level at ε = 0. However, a full quantum mechanicaltreatment gives

εn = sgn(n)√

|n| �ωB , (3.12)

leading to the formation of Landau levels at ε = 0.1) This can be understood bythe cancellation of factor 1/2 in Eq. (3.11) by Berry’s phase corresponding to therotation in the k space.

§4. Singular diamagnetism

The unique Landau-level structure gives rise to a singular diamagnetism,1) whichis the origin of large diamagnetic susceptibility of bulk graphite.34),35) The thermo-dynamic function Ω is given by

Ω = −kBTgvgs2πl2

∑n

ϕ(εn), (4.1)

ϕ(ε) = ln{1+exp[β(ζ−ε)]}, (4.2)

where T is the temperature, kB is the Boltzmann constant, β = 1/kBT , and ζ thechemical potential. Let ns be the electron concentration. Then, the free energy iswritten as

F = nsζ +Ω. (4.3)


With the use of the relation ns = −(∂Ω/∂ζ)B, the magnetization is given by

M = −(∂F∂B

)ns

= −(∂Ω∂B

)ζ. (4.4)

Therefore, susceptibility χ, defined by M = χB for small B, can be obtained bycalculating Ω up to the order of B2.

The thermodynamic function is rewritten as

Ω = −kBTgvgs2πl2

∞∑n=0

g(�ωB√n)

(1− 1

2δn0

)ln

[1+2eβζ cosh(β�ωB

√n)+e2βζ

], (4.5)

where 1/2πl2 is the degeneracy of a Landau level and g(ε) is a cutoff function whichdecays smoothly but sufficiently rapidly in such a way that the summation converges.For example, we can choose

g(ε) =εαcc

εαc +εαcc, (4.6)

with cutoff energy εc and parameter αc (αc≥2).Consider a smooth function F (x) and its integral

∫ ∞

0F (x)dx =

∫ h/2

0F (x)dx+

∞∑j=1

∫ h/2

−h/2F (x+hj)dx, (4.7)

where h is a small positive number. By expanding this with respect to h, we have

h[12F (0) +

∞∑j=1

F (x+hj)]

=∫ ∞

0F (x)dx− 1

12h2

[F ′(0)+

12F ′(∞)

], (4.8)

up to the second order in h. Let h=(�ωB)2, x=nh, and

F (x) = g(√x) ln

[1+2 exp(βζ) cosh(β

√x)+exp(2βζ)

]. (4.9)

Then, we haveΩ = Ω0 +ΔΩ, (4.10)

where Ω0 is the thermodynamic potential in the absence of a magnetic field and

ΔΩ =112gvgs(�ωB)2

2πl2β exp(βζ)

[1+exp(βζ)]2=gvgsγ

2

12πl4

∫ ∞

−∞

(− ∂f(ε)

∂ε

)δ(ε)dε, (4.11)

where f(ε) is the Fermi distribution function. The susceptibility becomes

χ = −gvgsγ2

6π

( e

c�

)2∫ ∞

−∞

(− ∂f(ε)

∂ε

)δ(ε)dε. (4.12)

This singular susceptibility is characteristic to graphene and obtained first by Mc-Clure1) and later by many others.36)–38)

210 T. Ando

Consider the average contribution of states around εn (|n| 1) to the thermo-dynamic potential in the weak-field limit. We have∫ 1/2

−1/2sgn(n)�ωB

√|n|+t ϕ[

sgn(n)�ωB√

|n|+t]dt= ϕ(εn) +

β

96(�ωB)4[ε−3

n f(εn)−ε−2n f ′(εn)] + · · ·. (4.13)

Therefore, the change in the thermodynamic potential in a magnetic field becomes

ΔΩ =gvgs2πl2

(�ωB)2

48

∫ ∞

−∞sgn(ε)[ε−2f(ε)−ε−1f ′(ε)]dε. (4.14)

When the Fermi level lies well away from ε=0, the integral vanishes identically andtherefore the susceptibility vanishes.

At ε= 0, the integrand becomes singular and gives the “paramagnetic” contri-bution

ΔΩ = − gvgs2πl2

124

(�ωB)2∫ ∞

−∞

(− ∂f(ε)

∂ε

)δ(ε)dε. (4.15)

In fact, we have∫ ∞

0

(f(ε)ε2

− f ′(ε)ε

)dε−

∫ 0

−∞

(f(ε)ε2

− f ′(ε)ε

)dε = lim

ε→+0

(f(ε)ε

+f(−ε)−ε

)= 2f ′(0).

(4.16)For n=0, the average contribution to the thermodynamic potential in the weak-fieldlimit becomes∫ 1/2

−1/2ϕ[sgn(t)�ωB

√|t|]dt =

18(�ωB)2ϕ′′(0) + · · · = −β

8(�ωB)2f ′(0) + · · ·, (4.17)

giving

ΔΩ =gvgs2πl2

18(�ωB)2

∫ ∞

−∞

(− ∂f(ε)

∂ε

)δ(ε)dε. (4.18)

By adding the contribution from n �=0 given by Eq. (4.15), we have the result givenby Eq. (4.11). This shows clearly that the presence of the Landau level at ε = 0 isthe origin of the singular behavior of the susceptibility.

In a bilayer graphene with the so-called AB stacking as illustrated in Fig. 4,most important inter-layer interaction is given by the nearest-neighbor hopping γ1.Within this approximation, the effective Hamiltonian in the vicinity of the Fermilevel is written as39),40)

H =�

2

2m∗

(0 (kx+iky)2

(kx−iky)2 0

), (4.19)

with m∗ = �2γ1/2γ2. This gives the Landau level ε±n = ±�ωc

√n(n+1) with

n = 0, 1, · · · with ωc = eB/m∗c. In a similar but more complicated manner, thesusceptibility can be calculated as

χ = −( e�

2m∗c

)2 gvgsm∗

2π�2

∫ ∞

−∞g(ε) ln

εc|ε|

(− ∂f

∂ε

)dε, (4.20)


Fig. 4. The lattice structure of a bilayer graphene with the so-called AB stacking. Some represen-

tative hopping integrals are shown.

except for a constant term independent of energy. The above agrees with the re-sult obtained previously for a model of intercalated graphite.38),41) The energy-independent term depends on details of the energy cutoff.

It has been shown that in multi-layers the Hamiltonian can be decomposedinto that of mono-layer and those of bilayers.42) In fact, for odd layers 2M+1, thesystem is divided into the monolayer and M bilayers, and for even layers 2M , thesystem is decomposed into M bilayers. In each of the bilayer Hamiltonian, theeffective interlayer coupling is different from each other. This exact decompositionis quite useful in discussing electronic properties of multi-layer graphenes such asdiamagnetic susceptibility42) and optical properties both in the presence and absenceof a magnetic field.43)

§5. Zero mode anomalies

We consider, for example, a system with scatterers with a potential range muchsmaller than the typical electron wavelength (which is actually infinite at ε = 0) butlarger than the lattice constant of graphene.31) The relaxation time in the absenceof a magnetic field becomes

1τ0

=2π�

12〈ni|ui|2〉 |ε|

2πγ2=

2π�|ε|W, (5.1)

with W being a dimensionless parameter characterizing the scattering strength givenby

W =〈niu

2i 〉

πγ2, (5.2)

where ui and ni are the strength and the concentration of scatterers, respectively,and 〈· · · 〉 means the average over impurities. We usually have W � 1. With theuse of the Boltzmann transport equation, the transport relaxation time becomes

212 T. Ando

τ(ε) = 2τ0(ε) and the conductivity

σ0 = gvgse2

4π2�

1W, (5.3)

independent of the Fermi level.5) This can also be derived from the Einstein relation

σ0 = e2D∗D(ε), (5.4)

where the diffusion coefficient is given by D∗ = v2τ(ε)/2 = γ2τ(ε)/2�2. This shows

that graphene should be regarded as a metal rather than a zero-gap semiconductor.It is interesting to note that the conductivity is nonzero even at ε = 0 (when the limit|ε| → 0 is taken). However, there can be singularity at ε = 0 because the density ofstates vanishes and strictly speaking, the above discussion is not applicable.

In the presence of a magnetic field, the conductivity tensor σμν with μ = x, yand ν = x, y is given by

σxx = σyy =σ0

1 + (ωcτ)2, σxy = −σyx =

σ0(ωcτ)1 + (ωcτ)2

. (5.5)

Using the explicit expressions for ωc and τ , we have σxx = σ0ξ4/(1 + ξ4) and σxy =

−σ0ξ2/(1 + ξ4), with ξ =

√2πW (εF/�ωB). Because τ(εF)−1 ∝ |εF|, the dependence

on the Fermi energy εF is fully scaled by �ωB . Therefore, the conductivity is singularat (�ωB , εF) = (0, 0). For example, σxx exhibits a singular jump to zero at εF = 0from σ0 for nonzero εF in the limit of the vanishing magnetic field �ωB→0.

A singular behavior also appears in the dynamical conductivity.8) In a relaxation-time approximation, the dynamical conductivity is given by

σ(ω) = gvgse2

16�

[ 4π

iεF�ω + i[�/τ(εF)]

+ 1 +i

πln

�ω + i[�/τ(�ω/2)] − 2εF�ω + i[�/τ(�ω/2)] + 2εF

], (5.6)

where the first term in the bracket represents the Drude conductivity determinedby states in the vicinity of the Fermi level and the second and third terms representinterband optical transitions. It is interesting to note that the interband conductivityis given by a universal value e2/8� for �ω > 2εF in the case of weak scattering.Because �/τ(ε) ∝ |ε|, the frequency dependence is completely scaled by the Fermienergy, i.e., σ(ω, εF) = σ(�ω/εF). This scaling shows that σ(ω, εF) exhibits a singularbehavior at the point (ω, εF) = (0, 0). The correct way may be to let ω → 0 at eachεF, leading to a singular jump of the static conductivity to gvgse

2/16� at εF = 0from σ0 for nonzero εF. Further, for εF = 0, σ(ω) = gvgse

2/16� independent of ω.A more refined treatment has been performed for the density of states and the

static conductivity in a self-consistent Born approximation,44)–46) in which level-broadening effects are properly taken into account.5) Figure 5 shows an example.The density of states becomes nonzero at ε = 0 because of level broadening and alsoenhanced due to level repulsion effect near ε = 0.

Further, the conductivity at εF = 0 is given by

σmin =gvgse

2

2π2�, (5.7)


Fig. 5. Some examples of the density of states (a) and the conductivity (b) of a monolayer graphene

calculated in the self-consistent Born approximation. ε0 is an arbitrary energy unit and εc is

the cutoff energy corresponding to the half of the π-band width. The conductivity exhibits a

sharp jump in the limit of weak scattering (W � 1) from the Boltzmann result σ0 for ε �= 0

down to σ = e2/π2~ at εF = 0.

which is universal and independent of the scattering strength. The resulting conduc-tivity varies smoothly across εF = 0 but exhibits a sharp jump in the limit of weakscattering (W�1) from the Boltzmann result σ0 for ε �= 0 down to σmin at εF = 0.A similar calculation for a bilayer graphene shows much smoother variation withenergy with minimum value σ ≈ gvgse

2/2π2� slightly dependent on the scattering

strength.40)

The Boltzmann transport equation, as well as the Einstein relation, gives σ ∝e2τ(εF)v(εF)2D(εF). In conventional systems, we have the relation εF = m∗v(εF)2/2,where m∗ is an effective mass. With the use of the fact ns ∝ εFD(εF ), this leads tothe usual expression σ = neeμ with mobility μ = eτ(εF)/m∗, where ns is the electronconcentration. In the present system, however, the velocity v is independent of εFor m∗ ∝ εF if we assume εF = m∗v2/2 as above. Further, the scattering probability�/τ(εF) is proportional to the final-state density of states with a coefficient indepen-dent of εF. Because the density of states is proportional to εF, the relaxation timeis inversely proportional to εF. As a result the mobility μ becomes proportional toε−2F ∝ n−1

s , leading to the conductivity σ = nseμ independent of the Fermi energyand the electron concentration.

In the case of charged-impurity scattering, however, the matrix element itself isproportional to the inverse of the Fermi energy both in the presence and absence ofscreening.33),47) Consequently, the low-temperature mobility becomes independent

214 T. Ando

Fig. 6. Some examples of the dynamical conductivity of a monolayer graphene calculated in the

self-consistent Born approximation. (a) W−1 = 50. (b) W−1 = 20. The frequency dependence

is nearly scaled by εF as long as εF �= 0. This scaling becomes less valid when εF is close to 0

as is more clear in (b).

of the electron concentration and the conductivity increases in proportion to the elec-tron or hole concentration ns.33),47) Therefore, the dependence of the conductivityon the electron concentration can be sensitive to the nature of dominant scatterersin the system. Note, however, that this does not change the situation that grapheneshould be regarded as a metal rather than a semiconductor.

Calculations in the self-consistent Born approximation were performed also forthe dynamical conductivity.8) Figure 6 shows some examples of the results. Thefrequency dependence is scaled completely by εF as long as εF �= 0. When εF is veryclose to 0, however, the conductivity at ω = 0 becomes small and the discrete jumppresent in the Boltzmann conductivity is removed. The energy scale causing thiscrossover behavior becomes exponentially small for weaker W , i.e., ∝ exp(−W−1/2),leading to a singular behavior of the dynamical conductivity in the weak scatteringlimit.

The singular diamagnetic susceptibility discussed in the previous section is alsoaffected by disorder. When a constant broadening Γ independent of the energy isassumed, the delta function is broadened into a Lorentzian, i.e., δ(ε) → Γ/π(ε2 +Γ 2).48) A more appropriate treatment in the self-consistent Born approximationgives a more singular result with much sharper peak and long tail decaying slowlyin proportion to |ε|−1.49) Figure 7 shows an example of the results. In a bilayergraphene disorder effects can simply be included by replacing |ε| with

√ε2+Γ 2 with

a phenomenological parameter Γ because the susceptibility exhibits only a weaker


Fig. 7. Some examples of the diamagnetic susceptibility obtained in the self-consistent Born ap-

proximation. The susceptibility has a sharp peak at ε = 0 and a long tail decaying in proportion

to |ε|−1.

logarithmic singularity as mentioned above and the density of states is independentof energy near ε = 0.

The calculation within the self-consistent Born approximation can be made fortransport quantities in high magnetic fields. In the high-field limit where Landau lev-els are separated from each other, the density of states and the diagonal conductivityare given by5)

D(ε) =gvgs2πl2

1πΓn

√1−

(ε−εnΓn

)2, (5.8)

σxx(ε) =gvgs2π2�

(|n|+δn0)[1−

(ε−εnΓn

)2], (5.9)

Γn = �ωB√

2W (1+δn0). (5.10)

The conductivity at ε = 0 is independent of a magnetic field and is given by Eq.(5.7). Further, when the Fermi level lies in the gap between adjacent Landau levels,the Hall conductivity is quantized into gvgs(j+1/2)(e2/h) with integer j.6) Althoughthe electron localization effects at tail regions of Landau levels were not explicitlyincluded in this quantized conductivity, it is obvious that states are exponentiallylocalized except in the vicinity of the center of each Landau level and the Hallconductivity is quantized into the same value.50)–54) The recent experiments11),12)

are in agreement with the theoretical prediction. Figure 8 shows the density of states,the diagonal conductivity, and the Hall conductivity σxy as a function of energy ε.

The observed minimum conductivity at zero energy in the absence of a magneticfield seems to be 3 ∼ 4 times as large as σmin given by Eq. (5.7) predicted in theself-consistent Born approximation. It is difficult to discuss the exact value of theconductivity in the vicinity of zero energy, including σmin, because a self-consistentdetermination of the screening and the density of states is necessary and because

216 T. Ando

Fig. 8. (a) Some examples of the density of

states (a), the diagonal conductivity σxx

(b), and the Hall conductivity σxy (c) of

a monolayer graphene calculated in the

self-consistent Born approximation. The

broadening of the Landau level n becomes

~ωB

p2W (1 + δn0) and the peak value of

σxx becomes (|n|+ δn0)(e2/π2

~). The con-

ductivity at ε = 0 remains e2/π2~ indepen-

dent of the magnetic field. The Hall con-

ductivity in the gap between adjacent Lan-

dau levels is quantized into 4(j+1/2)(e2/h)

with integer j.

effects of higher-order scattering55) are likely to be important. The behavior near ε≈0 in vanishing magnetic field is expected to remain as an interesting and challengingsubject both theoretically and experimentally.


§6. Symmetry crossover

The Weyl equation (2.13) is invariant under a special time-reversal operation S,

F S = KF ∗, (6.1)

where F ∗ represents the complex conjugate of the wave function F and K is ananti-unitary matrix K = −iσy satisfying K2 = −1. The corresponding operationfor an operator P is given by PS = K tPK−1, where tP stands for the transpose ofP . This corresponds to the time reversal in systems with spin-orbit interaction andleads to

F S2 ≡ (F S)S = −F . (6.2)

The system belongs to the symplectic universality class when only S constitutes arelevant symmetry.56)

This symmetry prevails even in the presence of impurities unless their potentialrange is smaller than the lattice constant a. In fact, for such scatterers, the effectivepotential is the same for the A and B sites and does not cause any mixing betweenthe K and K′ points.31) In this case a quantum correction or a weak-localizationcorrection to the Boltzmann conductivity becomes positive and diverges logarith-mically.7) This so-called anti-localization behavior is the same as that appears insystems with strong spin-orbit interaction.57)

The operation S is not the real time-reversal in graphene. Actually, the Blochfunctions at the K and K′ points are mutually complex conjugate and therefore, theK point is converted into the K′ point and the K′ point into the K point under thereal time reversal. In the present k · p scheme, this operation T is expressed as

F TK = e−iψσzF ∗

K′ , F TK′ = e−iψσzF ∗

K , (6.3)

where ψ is an arbitrary phase factor and FK and FK′ are the wave functions at theK and K′ points, respectively.7),58) This immediately gives

F T 2

K ≡ (F TK)T = FK , F T 2

K′ ≡ (F TK′)T = FK′ , (6.4)

characteristic of the conventional orthogonal symmetry. When we can neglect mixingbetween the K and K′ points and confine ourselves to states in each valley, however,the T symmetry is irrelevant and the special S symmetry becomes relevant. In thepresence of short-range scatterers causing mixing between K and K’ points, the Ssymmetry is violated, but the T symmetry prevails. As a result the system nowbelongs to the orthogonal class.7),59)

The actual equi-energy line deviates from the circle and has trigonal warpingwhen the energy becomes nonzero. This effect can be included by a higher orderk · p term. For the K point it is given by

H1 = γ

(0 h1(kx, ky)

h1(kx, ky)† 0

), h1(kx, ky) = α

a

4√

3(kx + iky)2, (6.5)

where α is a constant of the order of unity. This higher order term gives rise to atrigonal warping of the dispersion. This expression with α = 1 has been derived from

218 T. Ando

Fig. 9. Energy bands of a metallic carbon nanotube with circumference L obtained in a k·p scheme.

a nearest-neighbor tight-binding model,60) but is much more general if we regard α asan adjustable parameter. In the presence ofH1, the special time reversal symmetry isdestroyed because HS

1 = −H1.61) As a result the system now belongs to the unitaryclass. For the K′ point, the effective Hamiltonian is obtained by the replacement ofh1 by −h†1.

The time reversal symmetry is known to manifest itself as a quantum correctionto the conductivity. If the system has the S symmetry, quantum correction Δσ givenby so-called maximally crossed diagrams exhibits an anti-localization behavior, i.e.,the conductivity increases logarithmically with the decrease of the temperature.57)

If the system has the T symmetry, Δσ exhibits a weak-localization behavior and theconductivity decreases logarithmically with the temperature decrease. The crossoverbetween the weak- and anti-localization behavior caused by short-range scattererswas demonstrated.7) Suppression of weak localization observed in graphene20) wasanalyzed by the crossover among symplectic, orthogonal, and unitary due to vari-ous perturbations mentioned above.62) A weak positive magnetoresistance was alsoexperimentally observed in epitaxial graphenes.21)

The symmetry plays important roles in carbon nanotubes.4) In nanotubes, thewave vector along the circumference direction is quantized into κν(n) = (2π/L)[n−(ν/3)] by periodic boundary conditions due to their cylindrical form, where L is thecircumference, n is an integer, and ν = 0 or ±1 determined by the structure.4) Ananotube becomes a metal when ν = 0 and a semiconductor when ν = ±1. Figure 9shows the energy bands in a metallic nanotube. In the presence of the S symmetry,the reflection coefficients satisfy the symmetry relation rαβ = −rβα, where α and β

denote in-coming channels and α and β their time reversal or out-going channels.Metallic nanotubes satisfy this symmetry and possess an odd number of currentcarrying channels for each of the K and K′ points. When only a single conductingchannel is present, we have rαα = 0 and the conductance is given by an ideal value2e2/π�.31),32)

The determinant of an antisymmetric matrix with odd dimension is known tovanish identically. By definition, rβα represents the amplitude of a reflective wave βwith wave function ψβ(r) corresponding to an in-coming wave α with wave functionψα(r). The vanishing determinant of r shows that there exists at least one nontrivialsolution for the equation

∑ncα=1 rβαaα = 0. Then, there is no reflected wave for the

incident wave∑

α aαψα(r). Therefore, this becomes a perfectly conducting channel


which transmits through the system without being scattered back.63)

An interesting feature of nanotubes is the Aharonov-Bohm (AB) effect on theband structure.64),65) Recently, the splitting of optical absorption and emission peaksdue to flux was observed.66)–68) In the presence of a magnetic flux the time-reversalsymmetry is absent and therefore the perfectly conducting channel is destroyed.Explicit numerical calculations69) showed that the perfect channel is quite fragileand disappears for flux less than 1% of the flux quantum φ0 = ch/e, while theabsence of backscattering in linear bands is quite robust against such perturbations.

When the flux is a half of the flux quantum, the time-reversal symmetry is re-covered but the channel number is even. In this case, the determinant of an antisym-metric matrix does not vanish and the perfect channel is destroyed. Therefore, theflux dependence in nanotubes is quite different from a φ0/2 oscillation in metallicsystems on a cylinder surface, predicted theoretically70) and observed experimen-tally.71) This AAS oscillation arises due to the symmetry change caused by the fluxwith period φ0/2. The symmetry crossover due to magnetic field,69) short-rangescatterers,59) and trigonal warping61) manifests itself in the flux dependence of thelocalization effect in metallic carbon nanotubes.58)

Quantum correction to the conductivity was also studied in metallic carbon nan-otubes,72) but turned out to be not so successful in describing the features demon-strated in the numerical results. This absence of backscattering and the presence of aperfect channel lead also to intriguing behavior in the dynamical conductivity.73)–76)

Further, strong short-range scatterers like lattice vacancies77)–80) and topologicaldefects like five- and seven-membered rings81),82) and a Stone-Wales defect83) alsocause a singular scattering behavior.

§7. Electron-phonon interaction

There are three different in-plane phonon modes having significant contributionsto interaction with electrons, long-wavelength acoustic phonons, zone-center opticalphonons, and zone-boundary phonons. Out-of-plane modes, strongly affected bysubstrates and therefore not well understood yet, do not strongly interact with in-plane electron motion and therefore can usually be neglected.

Acoustic phonons are described well by a continuum mode.84) The potentialenergy functional for displacement u = (ux, uy) is written as

U [u] =1

2L2

∫dxdy

(B(uxx−uyy)2 + S[(uxx−uyy)2 + 4u2

xy]), (7.1)

where B is the bulk modulus, S is the shear modulus, and the strain tensor is givenby

uxx =∂ux∂x

, uyy =∂uy∂y

, uxy =12

(∂ux∂y

+∂uy∂x

). (7.2)

The kinetic energy is written as

K[u] =M

2L2

∫dxdy[(ux)2+(uy)2], (7.3)

220 T. Ando

where M is the carbon mass. This immediately leads to a transverse mode ωT = vT qand longitudinal mode ωL = vLq with velocity vT =

√S/M and vL =

√B/M ,

respectively.The effective interaction Hamiltonian is given by

H2 = g1(uxx+uyy) + g2[(uxx−uyy)σx−2uxyσy], (7.4)

for the K point.84) The interaction strength g1 is called the deformation potential(∼ 20 eV in bulk graphite) and g2 ≈ βΓγ0/4 appears due to change in the bondlength, where the dimensionless parameter βΓ is given by

βΓ = −d ln γ0

d ln b, (7.5)

where b is the bond length of nearest-neighbor carbon atoms given by b = a/√

3.84)

The Hamiltonian for the K′ point is obtained by replacing σ by −σ∗. The secondterm proportional to g2 is usually not important because g1 g2, except in metalliccarbon nanotubes in which the deformation potential cannot give rise to backscat-tering and therefore does not contribute to the resistance. It is straightforward tocalculate the transport relaxation time or the mobility using this Hamiltonian.

The long-wavelength optical phonons in the two-dimensional graphite was dis-cussed previously based on a valence-force-field model,84),85) and in the following weshall limit ourselves to the long-wavelength limit. Let u = (ux, uy) be the relativedisplacement of two sub-lattice atoms A and B, i.e.,

u(R) =1√2[uA(R)−uB(R)], (7.6)

where R denotes a coordinate specifying a unit cell. In the long-wavelength limit Rcan be replaced by a continuous coordinate r. Then we have

u(r) =∑q,μ

√�

2NMωΓ(bqμ + b†−qμ)eμ(q)eiq·r, (7.7)

where N is the number of unit cells, ωΓ is the phonon frequency at the Γ point,q = (qx, qy) is the wave vector, μ denotes the modes (t for transverse and l for lon-gitudinal), and b†qμ and bqμ are the creation and destruction operators, respectively.Define qx = q cosϕ(q) and qy = q sinϕ(q) with q= |q|. Then, we have

el(q) = i(cosϕ(q), sinϕ(q)), et(q) = i(− sinϕ(q), cosϕ(q)). (7.8)

The optical phonon modifies the distance between neighboring carbon atoms andtherefore the band structure through the change in the resonance integral betweencarbon atoms. The interaction Hamiltonian for the K point is given by85)

HKint = −

√2βΓγ

b2σ×u(r), (7.9)

where the vector product for vectors a=(ax, ay) and b=(bx, by) in two dimension isdefined by a×b=axby−aybx. For the K′ point, we should replace σ by −σ∗.


Next, we consider the phonon around the K and K′ point, or the zone-boundaryphonon. In a manner similar to the extraction of the envelope functions for electrons,the lattice displacement vector is written as86)

uA(RA) = uKA (RA)eiK·RA + uK′

A (RA)eiK′·RA ,

uB(RB) = uKB (RB)eiK·RB + uK′

B (RB)eiK′·RB .

(7.10)

We neglect the dispersion also for the zone-boundary phonon. In general, there existfour independent eigenmodes for each wavevector. However, after straightforwardcalculations, we can see that only one mode with the highest frequency contributesto the electron-phonon interaction. This mode is known as a Kekule type distortiongenerating only bond-length changes. The interaction Hamiltonian is given by

HKK′int = 2

βKγ

b2

(0 ω−1ΔK′(r)σy

ωΔK(r)σy 0

), (7.11)

where βK is another appropriate parameter, which is equal to βΓ for the tight-bindingmodel, and ΔK(r) and ΔK′(r) are defined by the projection of the displacement tothe Kekule mode,

ΔK(r) = ωf †uKA (r) + (f∗)†uKB (r),

ΔK′(r) = (f∗)†uK′

A (r) + f †uK′

B (r),f =

12

(1i

), (7.12)

and they are complex conjugate to each other, ΔK(r) = ΔK′(r)∗. In the secondquantized form,

ΔK(r) =∑

q

√�

2NMωK(bKq + b†K′−q)e iq·r,

ΔK′(r) =∑

q

√�

2NMωK(bK′q + b†K−q)e iq·r, (7.13)

where ωK is the frequency of the Kekule mode. It is worth noting that ΔK cannotbe given by a simple summation over the K and K′ modes. We should take a properlinear combination of the K and K′ modes in order to make the lattice displacementa real variable. We can easily understand the operator form of ΔK and ΔK′ in theinteraction Hamiltonian by considering the momentum conservation with the factthat 2K − K ′ and K − 2K ′ are reciprocal lattice vectors. The Hamiltonian is thesame as that obtained previously for a static Kekule distortion.87)–89)

We can calculate the electron lifetime τ due to emission and absorption of anoptical phonon. The lifetime of an electron with energy ε is given by the scatteringprobability from the initial state to possible final states via emission and absorptionof one phonon. For the zone-center phonon, the final states are limited to the samevalley to which the initial electron state belongs. Further, we should evenly evaluatecontributions from the longitudinal and transverse modes.

In contrast, only inter-valley components exist in the interaction HamiltonianHKK′

int given by Eq. (7.11) for the zone-boundary phonon. Any scattering processes

222 T. Ando

are classified into two types: One is the transition between “one K-electron with oneK-phonon” and “one K′-electron,” and the other is between “one K-electron” and“one K′-electron with one K′-phonon.” For example, an electron around the K pointcan be scattered to the K′ point accompanied by absorption of one phonon aroundthe K point, and this belongs to the former process. The electron scattering fromthe K to K′ point can also be induced by the emission of one phonon around the K′

point, while this is classified into the latter one.It appears that there is no remarkable similarity between zone-center and zone-

boundary phonons, but calculated scattering probabilities for both phonons are givenby the same formula,

�

τ= πλα

[nα|ε+ �ωα| + (nα + 1)|ε− �ωα|

], (7.14)

where α represents Γ or K, nα is the Planck distribution function for phonon of thefrequency ωα, and λα denotes the dimensionless coupling constant,

λα =36

√3

π

�2

2Ma2

1�ωα

(βα2

)2

. (7.15)

On the assumption that βα is independent of α as that in the nearest-neighbortight-binding model, the phonon frequency is the unique parameter which deter-mines the electron lifetime. For zone-center phonons, �ωΓ = 196 meV and λΓ =2.9 × 10−3(βΓ /2)2, while �ωK = 161.2 meV and λK = 3.5 × 10−3(βK/2)2 forzone-boundary phonons. As a consequence, the contributions from zone-boundaryphonons are larger than those from zone-center phonons.

Moreover, it is reasonable to assume nα = 0 since optical phonons are hardlyexcited even at room temperature. Namely, the phonon emission process becomesdominant and we have

�

τ= πλα|ε− �ωα|. (7.16)

This simply shows that the electron lifetime is inversely proportional to the couplingparameter λα and to the density of states at the energy of the final state. Whatshould be stressed here is that the phonon emission is possible only when the energyof the initial electron is larger than that of the phonon to be emitted. Otherwise,the final states are fully occupied at zero temperature and the phonon emissionnever takes place. In this sense, the zone-boundary phonon has another advantageover the zone-center phonon. Therefore, the zone-boundary phonon gives dominantscattering for high-field transport in graphenes owing to its smaller frequency andlarger coupling constant.

The electron-phonon interaction also modifies phonon spectra.90) Consider thelong-wavelength optical phonon at the Γ point. The phonon spectrum is describedby Green’s function, written as

Dμ(q, ω) =2�ωΓ

(�ω)2−(�ωΓ )2−2�ωΓΠμ(q, ω). (7.17)


Fig. 10. The frequency shift and broadening

of optical phonons as a function of the

Fermi energy.

Fig. 11. Calculated spectral function of opti-

cal phonon for varying Fermi energy.

The phonon frequency is determined by the pole of Dμ(q, ω). In the present case, thephonon self-energy is sufficiently small, leading to the shift of the phonon frequencyΔωμ and broadening Γμ,

Δωμ =1�ReΠμ(q, ωΓ ), Γμ = −1

�ImΠμ(q, ωΓ ). (7.18)

To the lowest order in the interaction, the phonon self-energy is calculated to be

Πμ(ω) = −λΓ∫ ∞

0ε dε [f(−ε)−f(ε)]

( 1�ω+2ε+i0

− 1�ω−2ε+i0

), (7.19)

independent of the phonon mode. We are calculating the self-energy of opticalphonons starting with the known phonon modes in graphene. Therefore, the directevaluation of the above self-energy causes a problem of double counting.87) In fact,if we apply the above formula to the case of vanishing Fermi energy, we get thefrequency shift due to virtual excitations of all electrons in the π bands. However,this contribution is already included in the definition of the frequency ωΓ . In order toavoid such a problem, we have to subtract the contribution in the undoped graphenefor ω=0 corresponding to the adiabatic approximation.

This contribution can be obtained from the above by putting ω=0 and f(ε)=f0(ε), with f0 = 0 for ε>0 and 1 for ε<0 as

Π2Dμ (ω) = −λΓ

∫ εc

0ε dε

( 12ε+i0

+1

2ε−i0)

= −λΓ εc, (7.20)

where we have introduced the cutoff energy εc of the order of the half of the π-band

224 T. Ando

width. Thus, the redefined self-energy becomes

Πμ(ω) = λΓ

∫ ∞

0dε [1−f(−ε)+f(ε)]+λΓ

∫ ∞

0dε [f(−ε)−f(ε)]

(�ω+iδ)2

(�ω+iδ)2−4ε2, (7.21)

where a phenomenological imaginary part δ = �/τ has been introduced as a pa-rameter characterizing scattering of electrons due to disorder. We have the relationf−ζ(ε) = 1− fζ(−ε), which leads to the conclusion that the self-energy is symmetricbetween ζ >0 and ζ <0, i.e., the electron-hole symmetry.

At zero temperature, in particular, we have

Πμ(ω) = λΓ εF − 14λΓ (�ω+iδ)

(ln

�ω+2εF+iδ�ω−2εF+iδ

+ πi), (7.22)

where εF is the Fermi energy. In the clean limit δ→0, we have

Πμ(ω) = λΓ εF − 14λΓ�ω

[ln

∣∣∣�ω+2εF�ω−2εF

∣∣∣ + πiθ(�ω−2εF)], (7.23)

where θ(t)=1 for t>1 and 0 for t<1. The frequency shift, i.e., the real part, divergeslogarithmically to −∞ at εF = �ωΓ /2. Apart from this logarithmic singularity, thephonon frequency increases roughly in proportion to εF for εF>�ωΓ . The broadeningis nonzero only for εF<�ωΓ /2.

Figure 10 shows the frequency shift and broadening for various values of δ. Fornonzero δ, the logarithmic singularity of the frequency shift and the sharp drop inthe broadening disappear, but the corresponding features remain for δ�1. Figure 11shows the corresponding spectral function (−1/π)ImD(q, ωΓ ). Similar results werereported independently91) and experiments giving qualitatively similar results werereported recently.22),23)

The calculation can easily be extended to the case in the presence of a mag-netic field, where discrete Landau levels are formed and oscillations due to resonantinteractions appear in the frequency shift and the broadening.92) It has also beenperformed in bilayer graphene.93)

Acknowledgements

The author would like to thank Hidekatsu Suzuura and Mikito Koshino forcollaborations in many of the works presented here. This work was supported inpart by Grants-in-Aid for Scientific Research and for Scientific Research on PriorityArea “Carbon Nanotube Nanoelectronics” from the Ministry of Education, Culture,Sports, Science and Technology, Japan.

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