# Induced chiral Dirac fermions in graphene by a periodically modulated magnetic field

**ABSTRACT** The effect of a modulated magnetic field on the electronic structure of neutral graphene is examined in this paper. It is found that application of a small staggered modulated magnetic field does not destroy the Dirac-cone structure of graphene and so preserves its 4-fold zero-energy degeneracy. The original Dirac points (DPs) are just shifted to other positions in k space. By varying the staggered field gradually, new DPs with exactly the same electron-hole crossing energy as that of the original DPs, are generated, and both the new and original DPs are moving continuously. Once two DPs are shifted to the same position, they annihilate each other and vanish. The process of generation and evolution of these DPs with the staggered field is found to have a very interesting patten, which is examined carefully. Generally, there exists a corresponding branch of anisotropic massless fermions for each pair of DPs, resulting in that each Landau level (LL) is still 4-fold degenerate except the zeroth LL which has a robust $4n_t$-fold degeneracy with nt the number of pairs of DPs. As a result, the Hall conductivity $\sigma_{xy}$ shows a step of size $4n_te^2/h$ across zero energy. Comment: 6 pages, 6 figures

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**ABSTRACT:**Graphene-based superlattice (SL) formed by a periodic gap modulation is studied theoretically using a Dirac-type Hamiltonian. Analyzing the dispersion relation we have found that new Dirac points arise in the electronic spectrum under certain conditions. As a result, the gap between conduction and valence minibands disappears. The expressions for the position of these Dirac points in ${\bf k}$-space and threshold value of the potential for their emergence were obtained. At some parameters of the system, we have revealed interface states which form the top of the valence miniband.Physical review. B, Condensed matter 08/2012; 86(20). · 3.66 Impact Factor - SourceAvailable from: Le VAN Qui[Show abstract] [Hide abstract]

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Page 1

arXiv:0912.4104v1 [cond-mat.mes-hall] 21 Dec 2009

Induced chiral Dirac fermions in graphene by a periodically modulated magnetic field

Lei Xu,1Jin An,1and Chang-De Gong2,1

1National Laboratory of Solid State Microstructures and Department of Physics, Nanjing University, Nanjing 210093, China

2Center for Statistical and Theoretical Condensed Matter Physics,

and Department of Physics, Zhejiang Normal University, Jinhua 321004, China

(Dated: December 22, 2009)

The effect of a modulated magnetic field on the electronic structure of neutral graphene is exam-

ined in this paper. It is found that application of a small staggered modulated magnetic field does

not destroy the Dirac-cone structure of graphene and so preserves its 4-fold zero-energy degener-

acy. The original Dirac points (DPs) are just shifted to other positions in k space. By varying the

staggered field gradually, new DPs with exactly the same electron-hole crossing energy as that of

the original DPs, are generated, and both the new and original DPs are moving continuously. Once

two DPs are shifted to the same position, they annihilate each other and vanish. The process of

generation and evolution of these DPs with the staggered field is found to have a very interesting

patten, which is examined carefully. Generally, there exists a corresponding branch of anisotropic

massless fermions for each pair of DPs, resulting in that each Landau level (LL) is still 4-fold de-

generate except the zeroth LL which has a robust 4nt-fold degeneracy with nt the number of pairs

of DPs. As a result, the Hall conductivity σxy shows a step of size 4nte2/h across zero energy.

PACS numbers: 73.43.Cd, 73.22.Pr, 73.61.Wp

Low energy physics of neutral graphene is character-

ized by the two inequivalent Dirac cones which is re-

lated by the time-reversal symmetry and described by

the relativistic massless Dirac equation.1,2,3,4Nearly all

important properties of neutral graphene is governed by

the chiral massless fermions around the two cones. For

example, the zero-energy anomaly due to the linear en-

ergy dispersion and the particle-hole symmetry of the

Dirac cones give rise to the anomalous quantum Hall

effect(QHE)3,4,5,6,7,8or the so-called half-integer QHE,

where the Hall conductivity is quantized to be half-

integer multiples of 4e2/h. When the Dirac-cone topol-

ogy is destroyed or replaced by other structures, the sys-

tem will undergo quantum phase transitions. In bilayer

graphene, each Dirac cone is replaced by two touching

parabolic bands,9,10,11which leads to the 8-fold degen-

eracy of zero-energy level,9giving rise to the quantized

Hall conductivity in bilayer graphene taken on values of

integer multiples of 4e2/h.9,10

Modulation of electronic structure in graphene has

already been experimentally realized, where periodic

electronic12,13or magnetic14,15potentials can be applied

to graphene by making use of substrate16,17,18,19or con-

trolled adatom deposition,20or by fabrication of periodic

patterned gate electrodes. This kind of graphene super-

lattice potential can change the Dirac-cone structure of

graphene dramatically,21,22which may lead to some new

phenomena, as well as potential application of graphene

materials.

In this paper, we present a study on the electronic

structure of monolayer neutral graphene and its un-

usual integer QHE under the influence of a periodically

modulated orbital magnetic field, which is schematically

shown in Fig.1. This kind of one-dimensional modula-

tion of magnetic field can be achieved in experiments

by applying an array of ferromagnetic stripes with al-

ternative magnetization on the top of a graphene layer,

or by making use of cold atoms in a honeycomb opti-

cal lattice,23,24,25or “artificial graphene” realized in a

nanopatterned two-dimensional electron gas.26Our anal-

ysis shows that generally the Dirac-cone structure can

not be smeared out by this time-reversal invariant mag-

netic field.Similar to the cases of periodic electronic

potential,21,22new DPs will be generated with varying

the amplitude of the field. What’s remarkable and dif-

ferent is that the newly generated DPs together with the

original DPs will move and evolve in k space with the

field. This leads to a series of quantum phase transi-

tions with each phase characterized by its unusual integer

QHE, which is expected to be observed by Hall measure-

ments.

We start with the tight-binding model on a honeycomb

lattice in the presence of a perpendicular, periodically

modulated orbital magnetic field. The Hamiltonian is

given by,

H = −t

?

<ij>

eiaijc†

icj+ H.c.,(1)

where c†

ator on site i, and < ij > denotes nearest-neighbor pairs

of sites. Here the spin index is suppressed since we do

not consider the Zeeman splitting. The magnetic flux

per hexagon (the summation of aij along the six bonds

around a hexagon) is given by?aij = φ ± δ, where φ

measures the uniform magnetic flux whereas δ the stag-

gered modulated flux, both of which are in units of φ0/2π

with φ0the flux quantum. Hereafter energy is measured

in unit of the nearest-neighbor hopping integral t.

To begin with, let us consider the effect of the two

simplest types of staggered magnetic fields (SMFs), in

order to extract the main physics behind graphene under

the influence of a modulated orbital magnetic field. The

i(ci) is an electron creation (annihilation) oper-

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(c) (d)

y

x

-? -?

? ?

-? -?

? ?

2 3

-? -?

? ?

-? -?

? ?

-? -?

(a) SMF-I (b) SMF-II

1 4

y

x

? ?

? ?

-? -?

? ?

? ?

-? -?

-? -?

? ?

-? -?

y

x

0

3

x

a L

y

x

graphene layer

array of ferromagnets

FIG. 1: (Color online) Illustration of the rectangular sample

of graphene under periodically modulated magnetic fields. (a)

and (b) represent two simplest SMFs, where each white and

yellow (grey) hexagon has a flux δ and −δ, respectively. The

numbers 1,··· ,4 represent the inequivalent atoms in a unit

cell. Each arrow represents a phase shift suffered by electrons

when hopping along the direction, which is δ/4 in case (a),

and δ/2 in case (b). (c) represents a long-period staggered

flux applied to graphene with lattice period 3a0Lx, where a0

is the lattice constant. (d) is a corresponding experimental

layout in which there is an array of ferromagnetic stripes with

alternative magnetization on the top of a graphene layer.

configurations of the two types are schematically shown

in Fig. 1(a) and (b), respectively, where proper gauge has

been chosen for each case.

We first show the evolution of DPs from an analytical

calculation for SMF-I shown in Fig. 1(a).

binding Hamiltonian in k-space can be written as

The tight-

H =

0 γk

γ†

k

0 ηk

η†

k

0

η†

0

ηk

0

γ†

−k

0

0

k

0 γ−k

√3

2ky+δ

(2)

where γk = −2te−ikx

The Hamiltonian H determines the energy spectrum of

electrons in graphene under SMF-I. The system has a pe-

riodicity of 2π as a function of δ due to gauge invariance,

so we restrict δ to range from 0 to 2π. The solution to

the DPs can be easily obtained: the original DPs located

at kx = 0, cos(√3ky) =

and the newly generated DPs located at kx = ±π/3,

cos(√3ky) = −1

An overall picture of the evolution of DPs in magnetic

Brillouin zone(MBZ) under SMF-I is shown in Fig. 2.

When δ = 0, the original pair of DPs (red filled circles)

are located at (0,±2π/3√3) [Fig. 2(a)]. As δ increases,

the two DPs move against each other along the ky di-

rection, and eventually they reach the center of MBZ at

2 cos(

4) and ηk = −teikx.

1

2− cosδ

2, for 0 < δ < 4π/3,

2− cosδ

2, for 2π/3 < δ < 2π.

ky ky ky (a) (b) (c) (d) ky

kx

?=0 ?=2?/3, critical 2?/3<?<? ?=?, critical

ky ky ky

(e) (f) (g) (h) ky

kx

?<?<4?/3 ?=4?/3, critical 4?/3<?<2? ?=2?

FIG. 2: (Color online) Schematic evolution of DPs in MBZ

with increasing staggered flux δ under SMF-I. “•” and “•”

represent the original DPs in pristine graphene and the in-

duced (additional) DPs respectively, while the arrows rep-

resent their moving directions.

isotropic Dirac cones whereas the green (grey) ellipses denote

anisotropic Dirac cones. The coordinates of the four corners

of MBZ are (±π/3,±π/√3).

The black circles denote

δ = 4π/3 [Fig. 2(f)] and thereafter disappear [Fig. 2(g)].

On the other hand, right at δ = 2π/3, one additional

pair of DPs are induced simultaneously at the four cor-

ners of MBZ (see the black filled circles in the figure).

Note that only two of the four induced DPs are inequiva-

lent and so only one pair contributes to the system, since

the four induced DPs are all located at the boundary

of MBZ. With increasing δ the two newly induced DPs

move along the lines kx= ±π/3 towards the edge cen-

ter respectively [Fig. 2(c)-(h)]. When δ = 2π, the two

DPs arrive at (±π/3,±2π/3√3). In this case, electrons

hopping along the arrows shown in Fig. 1(a) will suffer

an additional phase π/2, which is a pure gauge. Thus

the system corresponding to this value of δ [Fig. 2(h)] is

actually physical equivalent to that of Fig. 2(a), because

they differ only by a gauge transformation.

It is shown in Fig. 2 that DPs not only annihilate in

pairs but also emerge in pairs.22This is interpreted by

the fact that the two DPs in each pair are connected to

each other by the time reversal symmetry which is still

preserved by the SMF. Now we lay out our numerical

results to support these findings. We show for different

δ the energy dispersion near zero energy along the two

lines kx = 0 [Fig. 3(a)-(e)] and kx = ±π/3 [Fig. 3(f)-

(j)] where DPs reside. Note that δ = 2π/3 and 4π/3

are two critical values at which the pair of induced DPs

emerge and the original DPs completely superpose each

other, respectively. Apart from the two critical values,

as δ increases from 0 to 2π, the number of DPs changes

from one pair to two pairs and then back to one pair.

This interesting evolution of DPs will dramatically affect

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??

?

FIG. 3: (Color online) (a)-(e) Electron energy near the DPs

under SMF-I versus ky with kx = 0 for various values of

staggered flux δ. The dashed line represents the boundary of

MBZ. (f)-(j)The same as (a)-(e) but with kx = π/3. (k) Hall

conductivity σxy under SMF-I, with φ = 2π/768 for several

values of δ. Inset: The two renormalized factors β (β′) as

functions of staggered flux δ.

the degeneracy of the LLs which can be reflected by the

Hall conductivity.

The Hall conductivity can be calculated directly

through the standard Kubo formula27by numerical di-

agonalization of the Hamiltonian (1). In Fig. 3(k), the

resulting Hall conductivity σxynear zero energy is plot-

ted as a function of the Fermi energy EF. According to

the Hall plateaus steps in σxy, the system can be classi-

fied into three types. For 0 < δ < 2π/3 or 4π/3 < δ < 2π,

with spin degeneracy taken into account, σxyhas a step of

size 4e2/h, which is the same as that of pristine graphene,

shown in Fig. 3(k) for δ = 0.4π. For 2π/3 < δ < 4π/3,

σxy has a step of size 8e2/h across zero energy (neutral

filling) whereas a step of size 4e2/h in the other energy

range, which can be seen in Fig. 3(k) for δ = 0.8π. Re-

markably, right at δ = π, numerical results of Hall con-

ductivity show that all the steps have the same size of

8e2/h. We interpret these phenomena as follows.

The isotropic Dirac cones become anisotropic under

the influence of a modulated magnetic field (see Fig. 2).

The chiral fermions around an anisotropic Dirac cone

can be physically described by the anisotropic pseudospin

Hamiltonian,

H = vF

?

0ˆ p−

0ˆ p+

?

(3)

where ˆ p± = aˆ px± ibˆ py, vF = 3ta0/2? is the Fermi ve-

ky ky

(d) (e) (f)

ky

kx

?=0 ?=?/2, critical ?/2<?<2?/3

kx

ky ky ky

(a) (b) (c)

?=2?/3, critical 2?/3<?<? ?=?

FIG. 4: (Color online) Schematic evolution of DPs in MBZ

with increasing staggered flux δ under SMF-II. All symbols

used here have the same meanings as that in Fig. 2. The

coordinates of the four corners of MBZ are (±π/3,±π/2√3).

locity, and the two dimensionless coefficients a and b

measure the degree of anisotropy of the cone. In the

presence of a uniform magnetic field B this anisotropy

gives rise to a renormalized LLs En = ±β?vF

with β =

tor, and lB =

?φ0/4πB the magnetic length.

spin degeneracy taken into account, it is found that

for 0 < δ < 4π/3, the 4-fold degenerate LL spectrum

near the original Dirac cones have a β value given by

β = β(δ) = {2

δ = 0, β = 1), while for 2π/3 < δ < 2π, the 4-fold

degenerate LL spectrum near the induced Dirac cones

have another different β′value given by β′= β(2π − δ).

For a general δ between 2π/3 and 4π/3, the LLs for the

two branches are not degenerate except the zeroth LL,

which is exactly 8-fold degenerate. The zeroth LL is in-

dependent of the external uniform magnetic field so its

8-fold degeneracy cannot be removed, leading to a 8e2/h

Hall conductivity step at the zeroth LL and a 4e2/h step

at other LLs. However, when δ = π, the two factors

are equal to each other, i.e., β = β′, all LLs for these

cones (which are isotropic now) overlap and so are ex-

actly 8-fold degenerate. Therefore at δ = π, the Hall

conductivity can be expressed as σxy= 8(N + 1/2)e2/h,

with N LL index. We remark that actually, within the

range of δ where the two pairs of Dirac cones coexist,

there should exist a series of critical values of δ given by

β2/β′2= p/q, where p and q are two coprime integers.

At these critical values, besides the zeroth LL, the mqth

LL (with m = 0,1,2,... ) in the original pair of cones are

exactly degenerate with the mpth LL in the induced pair

of cones, giving rise to a 8e2/h Hall conductivity step at

these energies. Another critical value of δ is at δ = 4π/3,

where the zero point at kx = ky = 0 is not a DP, but

rather a semi-DP. Around the semi-DP, energy disper-

sion is found to be linear along kx, but parabolic along

ky. This peculiar feature can be compared with the elec-

tronic structure in VO2-TiO2nanoheterostructures.28

Now we turn to explore the second type of the SMF

shown in Fig. 1(b). Fig. 4 shows the schematic picture

?|n|/lB

With

√ab a dimensionless renormalization fac-

√3

1

1+cos(δ/2)(1+2 cos(δ/2)

3−2cos(δ/2))

1

2}

1

2 (note when

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FIG. 5: (Color online) (a)-(d) Electron energy near the DPs

under SMF-II, as a function of ky with kx = 0 for various

values of staggered flux δ. The dashed lines represent the

boundary of MBZ. (e)-(h)The same as (a)-(d) but as a func-

tion of kx with ky = π/6√3, instead. (i) Hall conductivity

σxy under SMF-II with φ = 2π/768 for several values of δ.

of the evolution of DPs in the corresponding MBZ. Like

that of SMF-I, at the beginning of varying δ, the two DPs

of pristine graphene are located at (0,±π/3√3), and then

they move towards the origin along kydirection. When

δ = π/2, each DP of the pair changes into three DPs

at k=(0,±π/6√3) [Fig. 4(b)], but they completely su-

perpose each other and can not be distinguished there.

After that the original DPs go on moving along kydirec-

tion and eventually arrive at the origin at δ = 2π/3 and

then vanish, whereas the other two pairs of induced DPs

move along the ky= ±π/6√3 direction until δ = π they

reach the points (±π/6,±π/6√3) [Fig. 4(f)], and then

backtrack. The evolution of DPs from δ = π to 2π is just

the reverse of the above process.

Compared with the SMF-I, in a period of δ, there are

four critical values (δ = π/2,2π/3,4π/3, and 3π/2), at

which the DPs emerge or vanish. All the induced DPs

under SMF-I move parallel to the kyaxis while that un-

der SMF-II move parallel to the kx axis. This should

be associated with the configuration of the SMF, where

the induced DPs incline to move towards the periodic

direction of the SMF.

The electronic energy spectrum near the zero energy

are shown in Fig. 5(a)-(h). What is significant is that as

δ increasing from 0 to π, the number of DPs changes

from one pair to three pairs and then to two pairs.

So, the DPs indeed emerge and annihilate in pairs. In

Fig. 5(i), the Hall conductivity σxyis plotted as a func-

TABLE I: The evolution properties of DPs in MBZ at 0 <

δ < 2π/Lx for various Lx’s (Lx is from 2 to teens).

SMF δ

0 ≤ δ < δcd

Lx noa(pairs)

2

3

4

5

6

7

8

9

10

Le

2

3

4

5

6

7

8

9

10

L

nib(pairs)

0,2

0,2,3

0,2,4

0,2,4,5

0,2,4,6

0,2,4,6,7

0,2,4,6,8

0,2,4,6,8,9

0,2,4,6,8,10

0,2,··· ,L

2

3

4

5

6

7

8

9

10

L

ntc(pairs)

1,3

1,3,4

1,3,5

1,3,5,6

1,3,5,7

1,3,5,7,8

1,3,5,7,9

1,3,5,7,9,10

1,3,5,7,9,11

1,3,··· ,L + 1

2

3

4

5

6

7

8

9

10

L

1

1

1

1

1

1

1

1

1

1

0

0

0

0

0

0

0

0

0

0

δc < δ ≤ 2π/L

ano represents the number of pairs of DPs in MBZ with kx= 0.

bni represents the number of pairs of DPs in MBZ with kx =

±π/3Lx.

cnt (= no+ ni) represents the total number of pairs of DPs in

MBZ.

dδc is a critical value where the number of DPs at the line kx= 0

changes to zero. For Lx = 2,3,··· ,10, δc is approximately equal

to 0.835π, 0.595π, 0.465π, 0.375π,0.315π, 0.275π, 0.245π, 0.217π,

0.196π, respectively.

eL ≥ 2. For L is an odd number, ni= 0,2,4,··· ,L − 1,L; for L

is an even number, ni= 0,2,4,··· ,L.

tion of the Fermi energy EF.

3π/2 < δ < 2π, the Hall conductivity can be expressed

as σxy = 4(N + 1/2)e2/h, which is the same as that of

pristine graphene. This means that the Dirac-cone topol-

ogy is preserved within this range without new induced

DPs. For π/2 < δ < 2π/3 or 4π/3 < δ < 3π/2, σxyhas

a 12e2/h step across the zeroth LL and 4e2/h or 8e2/h

step at the other LLs, implying the zeroth LL is 12-fold

degenerate. Interestingly, for 2π/3 < δ < 4π/3, all LLs

are 8-fold degenerate and the Hall conductivity can be

expressed as σxy = 8(N + 1/2)e2/h.

with quantized values of half-integer multiples of 8e2/h

is robust and is interpreted by the fact that the original

pair of DPs has disappeared and the induced two pairs

of DPs are located symmetrically in MBZ giving rise to

exact 8-fold degeneracy of their corresponding LLs.

Thus far we have discussed two simplest SMFs where

the magnetic flux alternates along the armchair chains or

zigzag chains, respectively. Now we generalize our theory

to the cases of SMF with long spatial period, shown in

Fig. 1(c). Taking SMF-I for example, we make the mag-

netic flux alternate every Lx zigzag chains. Numerical

analysis shows that, as δ increases from 0 to 2π/Lx,30

the number of pairs of original DPs no(at kx= 0) de-

For 0 < δ < π/2 or

This expression

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FIG. 6: (Color online) The case with Lx = 6. Electron energy

near the DPs under a long-period SMF shown in Fig. 1 (c) , as

a function of ky with (a)-(d) kx = 0 and (e)-(f) kx = π/18 for

various values of staggered flux δ. The figures in (b),(f) and

(c),(g) have been shifted to the left by 0.144π and 0.093π,

respectively. The total number of pairs of DPs is 1,3,5,6,

respectively for the four δ values.

creases from one to zero, while that of the induced DPs ni

(exactly located at the boundary of MBZ kx= ±π/3Lx)

increases from zero to Lxgradually obeying a sequence

of 0,2,4,··· ,Lx. Accordingly, the total number of pairs

of DPs is nt= 1,3,5,··· ,Lx+ 1,Lx. For the detail, see

Table I. In Fig. 6, we take Lx= 6 for example. When

0 < δ < 2π/Lx, the number of pairs of the induced DPs

shows a sequence of ni = 0,2,4,6 [Fig. 6(e)-(h)], while

that of the original pair is always no= 1 or 0 [Fig. 6(a)-

(d)]. Once again, the DPs emerge and annihilate in pairs.

After application of such a long-period SMF, each pair

of Dirac cones has generally different anisotropy, i.e., has

different renormalization factor β in their corresponding

LL spectrum, so all LLs except zeroth LL are still 4-fold

degenerate. However, the zeroth LL is exact 4nt-fold de-

generate, leading to a step of size 4nte2/h in σxyat the

zeroth LL,31as δ increasing from 0 to 2π/Lx. Therefore,

under a modulated orbital magnetic field, the property

of neutral graphene is actually governed by the number

of pairs of DPs. This is a significant signature and can be

detected by Hall measurements. So long as Lxis no more

than the magnetic length of the systems, the physics con-

tained in them is similar. But for graphene system with

the magnetic length much less than Lx, the DPs will be-

come more and more dense and will finally be merged

into the zeroth LL of graphene.29This deserve further

study and will be discussed elsewhere.

In summary, we have investigated the electronic struc-

ture in neutral graphene under periodically modulated

magnetic fields. It is found the modulated magnetic field

can induce additional DPs in graphene and the evolution

of these DPs can be manipulated by the magnitude and

period of the field. These induced DPs add additional

degeneracy to the LLs, especially a 4nt-fold degeneracy

at the zeroth LL, leading to an unusual integer QHE near

neutral filling. These phenomena are expected to be ob-

served by Hall measurements.

Acknowledgments

L. X. thanks Y. Zhou and Y. Zhao for useful dis-

cussion.This work was supported by NSFC Projects

10504009, 10874073 and 973 Projects 2006CB921802,

2006CB601002.

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30Here we only consider the weak magnetic flux case δ <

2π/Lx for two reasons. One is the weak magnetic flux

can be easily achieved in experiment; the other is that

for strong magnetic flux, it is hard to find a good rule to

describe the evolution of DPs.

31The Hall plateaus are a little different from that discussed

in Ref.21. In particular, when Lx is an even number, there

is a 4Lxe2/h Hall conductivity step.