Electronic structure of a two-dimensional graphene monolayer in a spatially modulated
magnetic field: Peierls tight-binding model
Y. H. Chiu,1Y. H. Lai,2J. H. Ho,2 D. S. Chuu,1,
*
and M. F. Lin2,† 1Institute of Physics, National Chiao Tung University, Hsinchu, Taiwan 300 2Department of Physics, National Cheng Kung University, Tainan, Taiwan 701共Received 3 May 2007; revised manuscript received 18 August 2007; published 11 January 2008兲
Magnetoelectronic properties of a two-dimensional共2D兲 monolayer graphene are investigated by the Peierls tight-binding model. They are dominated by the period, strength, and direction of a spatially modulated magnetic field. Such a field could induce the reduction in dimensionality, change of energy dispersions, anisotropy at low energy, composite behavior in state degeneracy, extra band-edge states, and asymmetry of energy bands. There are partial flatbands at the Fermi level and one-dimensional parabolic bands at others. These make density of states exhibit delta-function-like structure and asymmetric prominent peaks, respec-tively. Energies of the extra band-edge states strongly depend on the period, while those of the original band-edge states exhibit little dependence. Both of them grow as the strength increases. The modulated and uniform magnetic fields differ from each other in energy dispersion, state degeneracy, and dimensionality. Important differences between a monolayer graphene and a 2D electron gas also exist.
DOI:10.1103/PhysRevB.77.045407 PACS number共s兲: 73.20.At, 73.22.⫺f, 81.05.Uw
I. INTRODUCTION
Condensed-matter systems, such as diamond, layered graphenes, carbon nanotubes, carbon tori, C60-related
fullerenes, and carbon onions, are purely made up of carbon atoms. Such systems have very special symmetric configura-tions, and their dimensionalities vary from three dimensions to zero dimension. They could exhibit rich electronic prop-erties, e.g., a wide-gap diamond, a semimetallic bulk graph-ite, a zero-gap monolayer graphene, a metallic armchair car-bon nanotube, and a small-gap nonarmchair carbon nanotube. Recently, few-layer graphenes with two-dimensional 共2D兲 hexagonal symmetry and nanoscaled thickness could be produced by controlling film thickness with single-atom accuracy.1 Much research has been ex-plored, such as growth,2 phonon,3 band structure,4–7 elec-tronic excitations,8–11 optical spectra,12,13 and transport properties.14–20
A 2D monolayer graphene owns linear bands intersecting at the Fermi level EF= 0. Energy bands are isotropic at low
energy 共ⱗ0.5 eV兲,21 and so are the low-frequency physical properties 共e.g., Coulomb excitations兲.8,10 They produce a vanishing density of states at EF= 0, which makes a
mono-layer graphene an exotic zero-gap semiconductor. The two important characteristics, isotropy and semiconductor, origi-nate from the hexagonal symmetric configuration. Electronic properties are completely changed by applying a uniform perpendicular magnetic field. Most of energy bands become the dispersionless Landau levels. The effective-mass model predicts that energies of the low Landau levels are propor-tional to the square root of field strength and quantum number.22These theoretical predictions have been verified by experimental measurements on transport properties16and op-tical spectra.12An inhomogeneous magnetic field might also strongly affect the essential physical properties. Haldane in-vestigated whether a 2D graphene could exhibit magnetocon-ductance in the presence of a vanishing net magnetic field.23 In this work, we focus mainly on the effects of a periodic magnetic field on electronic properties.
There have been numerous experimental24–27 and theoretical28–36 research for a two-dimensional electron gas 共2DEG兲 under a spatially modulated magnetic field. These works primarily analyze the transport properties,24–26,28,29 en-ergy bands,30–32 electronic excitations,33–36 and optical spectra.27The transport measurements24,25manifest the oscil-latory magnetoresistance. Energy bands of a 2DEG have parabolic energy dispersions. A periodic magnetic field leads to the drastic changes in electronic properties, e.g., the changes in state degeneracy, band-edge states, and curva-tures.
The Peierls tight-binding model is used to calculate the electronic structure of a 2D graphene in a spatially modu-lated magnetic field. The Hamiltonian is a huge Hermitian matrix for a large modulation period 共ⲏ1000 Å兲. The nu-merical techniques are developed to attain a bandlike Hamil-tonian matrix. The dependence of electronic properties on the direction, period, and strength of the modulated magnetic field will be investigated in detail, e.g., energy dispersions, state degeneracy, band-edge states, symmetry of energy bands, and density of states. A comparison with those of a uniform magnetic field is made. The important differences between a monolayer graphene and a 2DEG is also dis-cussed.
This paper is organized as follows. The bandlike Hamil-tonian matrix in a periodic magnetic field is derived in Sec. II. The main characteristics of the-electronic structures are discussed in Sec. III. Finally, Sec. IV contains concluding remarks.
II. PEIERLS HAMILTONIAN BAND MATRIX
The tight-binding model with nearest-neighbor interac-tions is used to calculate the -electronic structure of 2pz
orbitals. In the honeycomb structure of a 2D single-layer graphene in the absence of an external field, there are two kinds of carbon atoms, a and b, in a primitive unit cell. The wave function consisting of the two linear tight-binding
functions from periodic 2pz orbitals is expressed as 兩⌿k典
= Cak兩ak典+Cbk兩bk典, where 兩ak典=兺ieik·Ri兩aik典 and 兩bk典
=兺jeik·Rj兩bjk典. The Hamiltonian built from 兩ak典 and 兩bk典 is a
2⫻2 Hermitian matrix. The site energies are vanishing 共具aik兩H0兩aik典=具bik兩H0兩bik典=0兲, and the nearest-neighbor
hop-ping integral is given by
具bjk兩H0兩aik典 =␥0exp关ik · 共Ri− Rj兲兴, 共1兲
where␥0共=2.56 eV兲 共Ref. 21兲 is the atom-atom interaction
between two neighboring atoms at Riand Rj.
A monolayer graphene is assumed to exist in a spatially modulated magnetic field B = B sin共Kx兲zˆ along the armchair direction关the x axis in Fig.1共a兲兴, and the periodic length is
lB= 2/K=3b
⬘
RB, where parameter RB is useful indescrib-ing the dimensionality of the Hamiltonian matrix. The mag-netic flux, product of the field strength and the hexagonal area in the unit of flux quantum 共⌽0= hc/e=4.1356
⫻10−15T/m2兲, is ⌽=共3
冑
3Bb⬘
2/2兲/⌽0. b
⬘
= 1.42 Å is theC–C bond length. The modulated magnetic field that leads to the Peierls phase is characterized by the vector potential A = −关B cos共Kx兲兴/Kyˆ. The nearest-neighbor hopping integral becomes 具bjk兩HB兩aik典 =␥0exp
再
i冋
k·共Ri− Rj兲 + 2 ⌽0冕
Ri Rj A·dr册
冎
. 共2兲 For three nearest-neighbor atoms, their hopping integrals are, respectively, t1k共n兲=␥0exp关共ikxb⬘
/2+iky冑
3b⬘
/2兲+Gn兴, t2k共n兲=␥0exp关共ikxb⬘
/2−iky冑
3b⬘
/2兲−Gn兴, and t3k共n兲=␥0exp共−ikxb
⬘
兲, where Gn= −i关6共RB兲2⌽/兴cos关共n− 5/6兲/RB兴sin关共/6RB兲兴. The modulation period causes the
periodic boundary conditions along the x axis so that the corresponding Peierls phase is periodic in a period 2RB. An
enlarged rectangular unit cell includes 4RB carbon atoms.
The wave function and the Hamiltonian matrix element are, respectively, given by 兩⌿k典 =
兺
n=1 2RB Cak n 兩a nk典 + Cbk n 兩b nk典, 共3a兲 具bmk兩HB兩ank典 = 关t1k共n兲 + t2k共n兲兴␦m,n+ t3k共n兲␦m,n−1. 共3b兲 Cakn = Cakn+2RB and C bk n= Cbkn+2RB are derived because of the pe-riodical boundary condition. To solve the complicated calcu-lations of the huge Hamiltonian matrix, the base functions are chosen as the following sequence
兵兩a1k典,兩b2RBk典,兩b1k典,兩a2RBk典,兩a2k典,兩b2RB−1k典,兩b2k典,
兩a2RB−1k典, ... ,兩aRB−1k典,兩bRB+2k典,兩bRB−1k典,兩aRB+2k典, 兩aRBk典,兩bRB+1k典,兩bRBk典;兩aRB+1k典其.
The Hamiltonian matrix could be expressed as a 4RB⫻4RB
bandlike Hermitian matrix
冢
0 q* p1* 0 . . . 0 0 q 0 0 p2R B 0 . . . 0 p1 0 0 0 q 0 . . . 0 0 p2R B * 0 0 0 q* 0 0 ] q* 0 0 0 ] . . . q 0 pRB+1 0 ] ] 0 q 0 0 0 0 0 pRB+1 * q* 0冣
, 共4兲where pn⬅t1k共n兲+t2k共n兲 and q⬅t3k. Because the range of kx
is much smaller than that of kyfor a large RB, it is sufficient
just to consider one dimensional 共1D兲 energy dispersions along ky. That is to say, a modulated magnetic field could
effectively reduce the dimensionality by 1.
The-electronic structure strongly depends on the direc-tion of the modulated magnetic field, mainly owing to the anisotropic structure of a 2D monolayer graphene. For the zigzag direction 关Fig. 1共b兲兴, the similar calculations could also be done. By the detailed derivations, the three hopping integrals are t1k
⬘
共n兲=␥0exp关共ikx冑
3b⬘
/2+ikyb⬘
/2兲+Gn⬘
兴, t2k⬘
共n兲=␥0exp关共−ikx冑
3b⬘
/2+ikyb⬘
/2兲−Gn−1⬘
兴, and t3k⬘
共n兲=␥0exp关共−ikyb
⬘
兲+Gn⬙
兴, where Gn⬘
= −i关2共RB兲2⌽/3兴cos关共n− 1/2兲/RB兴sin共/2RB兲 and Gn
⬙
= −i关共2RB⌽/3兲cos关共n− 1兲/RB兴兴. The Hamiltonian matrix element is further given
by
具bmk兩HB兩ank典 = t1k
⬘
共n兲␦m,n+1+ t2k⬘
共n兲␦m,n−1+ t3k⬘
共n兲␦m,n.共5兲 With the base functions
FIG. 1. The primitive unit cell of a monolayer graphene in the spatially modulated magnetic field with period RB along 共a兲 the armchair direction and共b兲 zigzag direction.
兵兩a1k典,兩b2RBk典,兩b1k典,兩a2RBk典,兩b2k典,兩a2RB−1k典,兩a2k典,兩b2RB−1k典, ... ,兩bRB−1k典,兩aRB+2k典,兩aRB−1k典,兩bRB+2k典,兩aRBk典,兩bRB+1k典,兩bRBk典;兩aRB+1k典其,
the 4RB⫻4RB bandlike Hamiltonian matrix for the zigzag direction is
冢
0 u2R B * v1* 0 s1* 0 . . . 0 0 0 u2RB 0 0 v2RB 0 s2RB−1 0 0 0 0 v1 0 0 s2RB 0 0 0 v2R B * s 2RB * 0 0 ] s1 0 sR*B−1 0 0 s2R B−1 * 0 sRB+1 ] 0 0 sRB * vRB * 0 0 0 sRB 0 0 vRB+1 0 0 0 0 sRB−1 0 vRB 0 0 uRB 0 0 0 . . . 0 sRB+1 * 0 vRB+1 * uRB * 0冣
, 共6兲where sn⬅t1k
⬘
共n兲, un⬅t2k⬘
共n兲, and vn⬅t3k⬘
共n兲. TheHamil-tonian matrices in Eqs. 共4兲 and 共6兲, respectively, have two and three independent matrix elements.
III. MAGNETOELECTRONIC PROPERTIES
The unoccupied conduction bands共Ec’s兲 are symmetric to
the occupied valence bands共Ev’s兲 about the Fermi level E F
= 0. Only the former are discussed in this work. We first look at the low-energy bands resulting from the modulated mag-netic field with period RB= 1000 along the armchair
direc-tion. At B = 0, most of energy bands are parabolic dispersions with the double degeneracy except two nondegenerate linear bands intersecting at EF= 0 关the solid circles in Fig. 2共a兲兴.
There is only one band-edge state in each energy band; fur-thermore, all the band-edge states are located at ky
pp
= 2/3
冑
3b⬘
共the original band-edge states兲. The modulated magnetic field leads to drastic changes in band-edge states and energy dispersions, as shown in Fig. 2共a兲 by the open circles at B = 20 T. The range of ky, where electronic statescould exist, becomes large. The linear bands are changed into partial flatbands at EF= 0. Also noted is that this result is
similar to that of carbon nanotubes in magnetic fields perpen-dicular to the symmetry axis.37The doubly degenerate para-bolic bands have weak energy dispersions or low curvatures at ky
pp
, and their number is largely reduced. Such effects sug-gest that a magnetic field could make electronic states flock together. The modulation effects of B on parabolic energy bands result in four extra band-edge states at ky
sp
’s, the strong energy dispersions close to ky
sp
’s, and the destruction of the double degeneracy. The two extra band-edge states at the left- and right-hand sites of ky
pp
might have different energies; that is, one side of the parabolic bands might be asymmetric to the other about the original band-edge states. Each para-bolic band exhibits the composite behavior in state
degen-eracy, the single and double degeneracies near ky sp
and ky pp
, respectively.
The number of subbands grows quickly as state energy Ec increases from zero. There are many middle energy bands
FIG. 2. Energy bands near 共a兲 Ec= 0, 共b兲 Ec=␥
0, and 共c兲 Ec
= 3␥0 for the armchair modulation direction at RB= 1000 and B = 20 T. Those without B are also shown for comparison.
near Ec⯝␥
0, as shown in Fig. 2共b兲. At B = 0, they include
complete flatbands at Ec=␥
0and parabolic bands at the
oth-ers. Both are doubly degenerate. The parabolic bands have a low curvature at ky
pp
=/2
冑
3b⬘
and a high curvature at ky sp= 0 共not shown兲. Moreover, in the small or large ky, the
modulated magnetic field could destroy double degeneracy and create extra band-edge states. It modifies the band cur-vatures at kypp, and makes the complete flatbands change into the partial flatbands.
The subband number decreases gradually with the further increase of state energy. The high-energy bands, as shown in Fig.2共c兲for B = 0, are parabolic dispersions with the double degeneracy and one band-edge state at ky
pp
= 0. All the ky pp
states remain unchanged in the presence of B, as seen in low and middle energy bands. However, the modulated magnetic field could reduce the number of subbands or widen the range of ky, produce the extra band-edge states at ky
sp⫽0, and
induce the composite behavior of the single and double de-generacies.
The strength, period, and direction of the modulated mag-netic field strongly affect the electronic structure, as shown in Figs.3共a兲and3共b兲for the low-energy bands. The range of partial flatbands increases with increasing B, while their number and curvatures exhibit the opposite behavior关Figs.
3共a兲 and 2共a兲兴. These results further demonstrate that the ability to flock electronic states is enhanced by the increasing field strength. The longer the period, the larger the effective range of ky关Figs.3共b兲and2共a兲兴. The period could alter state
energies and curvatures of extra band-edge states at ky sp
’s. It is also worth noting that ky
pp
= 2/3
冑
3b⬘
of the doubly de-generate parabolic bands is independent of period and strength. When the spatially modulated direction is along the zigzag structure, there are two partial flatbands at EF= 0 andmany parabolic bands at the others 关Fig.3共c兲兴. The former are doubly degenerate; the latter are fourfold degenerate near
ky pp
= 0 and doubly degenerate near ky sp
. That state degeneracy, subband number, ky’s of band-edge states, and range of
par-tial flatbands depend on the modulation direction, which di-rectly reflects the anisotropic characteristic of a graphene ge-ometry. In addition, the similar effects could also be found in moderate and high-energy bands.
Density of states共DOS兲, which is closely related to essen-tial features of the electronic structure, is defined as
D共兲 =
兺
,h=c,冕
1stBZ dkxdky 共2兲2 ⌫ 1 关Eh共k x,ky兲 −兴2+⌫2 . 共7兲 ⌫共=10−4␥0兲 is a phenomenological broadening parameter.
The integration on kxcould be roughly neglected because of
the very small range of kx. The low-frequency DOS at B = 0
is proportional to, as shown in Fig. 4共a兲. It vanishes at = 0 and has no special structures. However, the modulated magnetic field leads to a symmetric delta-function-like peak at = 0 关inset in Fig. 4共a兲兴 and considerable asymmetric square-root divergent peaks. The former comes from the two partial flatbands at EF= 0, and its height grows with the
in-FIG. 3. The low-energy bands along the armchair direction at 共a兲 RB= 1000, B = 40 T and共b兲 RB= 2000, B = 20 T, and those along
共c兲 the zigzag direction at RB= 1000, B = 20 T.
FIG. 4. The low-frequency density of states共a兲 along the arm-chair direction at RB= 1000 and different B’s and共b兲 at B=20 T and different RB’s or directions. The insets show those near EF= 0.
creasing field strength. The latter are dominated by the band-edge states of the 1D parabolic dispersions along kˆy 关Fig. 2共a兲兴. The asymmetric pronounced peaks could be further divided into weak subpeaks and strong principal peaks. They are, respectively, due to the band-edge states at ky
sp
’s and ky pp
. There are many pairs of subpeaks, and each pair of subpeaks is associated with the asymmetry of the 1D parabolic bands about the ky
pp
states关discussed earlier in Fig.2共a兲兴. The num-ber, frequencies, and heights of the asymmetric prominent peaks are sensitive to the changes in the strength, period, and modulation direction. The peak number decreases with the increase of the strength, while the peak frequencies exhibit a different behavior 关Fig. 4共a兲兴. The number of subpeaks in-creases as the period grows关Fig.4共b兲兴, while it is the other way around as the frequencies of subpeaks increase. The main features of principal peaks have the weak dependence on the period. When the modulation direction is orientated relatively close to the zigzag structure, more principal peaks with lower frequencies are observed 关comparison between the heavy and light solid curves in Fig. 4共b兲兴. Density of
states could display the high anisotropy even at very low frequency 关→0 in the inset of Fig. 4共b兲兴. However, the low-frequency physical properties without B are anisotropic only forⲏ0.25␥0, e.g., electronic excitations and absorp-tion spectra.9This result indicates that the anisotropy of the low-frequency electronic properties could be induced by means of a spatially modulated magnetic field.
The frequencies of prominent peaks in DOS deserve a closer investigation. Figure5共a兲shows the relation between the frequencies共sp’s兲 of the first six subpeaks and the
pe-riod at B = 20 T. These peaks correspond to the extra band-edge states at the left-hand neighborhood of ky
pp 关Fig. 2共a兲兴.
sp’s decline quickly as RB increases. As to the frequencies
of principal peaks共pp’s兲, their dependence on the period is
minor for a sufficient large RB 共ⲏ1000兲, as shown in Fig. 5共b兲. Both sp’s and pp’s are largely enhanced by the
in-creasing field strength关Figs. 5共c兲 and 5共d兲兴. There exists a special square-root relation between pp and B, i.e., pp
⬀
冑
B. In addition, the low-energy flat Landau levels due to auniform magnetic field共B0兲 also exhibit the square-root de-pendence on the field strength.22 The band-edge state ener-gies are closely related to the magneto-optical absorption frequencies. The predicted results could be verified by the optical spectroscopy.
A uniform magnetic field differs from a spatially modu-lated magnetic field in the low-energy magnetoelectronic structures. In terms of the ability in flocking electronic states, the former is much stronger than the latter. A uniform mag-netic field could make linear or parabolic bands convert into the dispersionless Landau levels. Such levels are fourfold degenerate for each ky state. All the Landau states could be
regarded as the band-edge states. They would exhibit zero-dimensional features, but not one-zero-dimensional features. For example, the magneto-optical absorption spectra display the symmetric and asymmetric prominent peaks in cases B0and B, respectively.
The electronic structure of a 2DEG could be strongly af-fected by a spatially modulated magnetic field.30–32 It also displays the similar behaviors to a monolayer graphene, such
as the composite behavior in state degeneracy, creation of extra band-edge states, and change of curvatures. However, there are three significant differences between a 2DEG and a monolayer graphene. A 2DEG does not exhibit partial flat-bands at zero energy. Its magnetoelectronic structure is inde-pendent of the modulation direction. Moreover, the wave vectors of extra band-edge states are approximately close to
ky= 0 and hardly depend on the state energy. The
above-mentioned differences mainly come from the hexagonal structure of a monolayer graphene.
IV. CONDLUDING REMARKS
In summary, the magnetoelectronic structure of a 2D monolayer graphene is studied by the Peierls tight-binding model. The specific base functions are chosen to solve a huge Hamiltonian matrix. The strength, period, and direction of a spatially modulated magnetic field dominate the main features of electronic properties. Such a field could reduce dimensionality by 1, alter energy dispersions, cause aniso-tropy at low energy, induce composite behavior in state de-generacy 共the composite behavior of single and double
de-FIG. 5. Energies 共sp’s兲 of extra band-edge states at the
left-hand neighborhood of kypp and those 共pp’s兲 of the original band-edge states.共a兲 and 共b兲 are their dependences on the period; 共c兲 and 共d兲 correspond to the dependence on the strength.
generacies for the armchair direction兲, produce extra band-edge states, and destroy the symmetry of energy bands about the original edge states. Energies of the extra band-edge states strongly rely on the period, while the opposite is true for those of the original band-edge states. Both of them grow with the increase of the strength. Density of states owns many asymmetric prominent peaks, mainly owing to the band-edge states in 1D parabolic bands. The partial flat-bands also make DOS display delta-function-like structures at the Fermi level. A spatially modulated magnetic field con-trasts sharply with a uniform magnetic field in energy disper-sion, state degeneracy, and dimensionality. The important differences between a monolayer graphene and a 2DEG arise
from the hexagonal symmetry. They are the existence of the partial flatbands at zero energy, dependence on the modula-tion direcmodula-tion, and wave vectors of the band-edge states. The experimental measurements on the magneto-optical absorp-tion spectra could be utilized to examine the predicted elec-tronic properties.
ACKNOWLEDGMENTS
This work was supported by NSC and NCTS of Taiwan, under the Grant Nos. NSC 95-2112-M-006-028-MY3 and NSC 95-2119-M-009-030.
1K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang,
S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666共2004兲.
2C. Berger, Z. Song, T. Li, X. Li, A. Y. Ogbazghi, R. Feng, Z. Dai,
A. N. Marchenkov, E. H. Conrad, P. N. First, and W. A. de Heer, J. Phys. Chem. B 108, 19912共2004兲.
3A. H. Castro Neto and F. Guinea, Phys. Rev. B 75, 045404
共2007兲.
4C. P. Chang, C. L. Lu, F. L. Shyu, R. B. Chen, Y. K. Fang, and M.
F. Lin, Carbon 42, 2975共2004兲.
5F. Guinea, A. H. Castro Neto, and N. M. R. Peres, Phys. Rev. B
73, 245426共2006兲.
6S. Latil and L. Henrard, Phys. Rev. Lett. 97, 036803共2006兲. 7E. McCann, Phys. Rev. B 74, 161403共R兲 共2006兲.
8F. L. Shyu and M. F. Lin, J. Phys. Soc. Jpn. 69, 607共2000兲. 9J. H. Ho, C. P. Chang, and M. F. Lin, Phys. Lett. A 352, 446
共2006兲.
10J. H. Ho, C. L. Lu, C. C. Hwang, C. P. Chang, and M. F. Lin,
Phys. Rev. B 74, 085406共2006兲.
11X. F. Wang and T. Chakraborty, Phys. Rev. B 75, 041404共R兲
共2007兲.
12M. L. Sadowski, G. Martinez, M. Potemski, C. Berger, and W. A.
de Heer, Phys. Rev. Lett. 97, 266405共2006兲.
13L. Joly, L. Tati-Bismaths, and W. Weber, Phys. Rev. Lett. 97,
187404共2006兲.
14J. S. Bunch, Y. Yaish, M. Brink, K. Bolotin, and P. L. McEuen,
Nano Lett. 5, 287共2005兲.
15K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko, M. I.
Katsnelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Nat. Phys. 2, 177共2006兲.
16K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I.
Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature共London兲 438, 197 共2005兲.
17Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim, Nature共London兲
438, 201共2005兲.
18K. S. Novoselov, Z. Jiang, Y. Zhang, S. V. Morozov, H. L.
Stormer, U. Zeitler, J. C. Maan, G. S. Boebinger, P. Kim, and A. K. Geim, Science 315, 1379共2007兲.
19V. P. Gusynin and S. G. Sharapov, Phys. Rev. Lett. 95, 146801
共2005兲.
20E. McCann and V. I. Fal’ko, Phys. Rev. Lett. 96, 086805共2006兲. 21P. R. Wallace, Phys. Rev. 71, 622共1947兲.
22J. W. McClure, Phys. Rev. 104, 666共1956兲. 23F. D. M. Haldane, Phys. Rev. Lett. 61, 2015共1988兲.
24H. A. Carmona, A. K. Geim, A. Nogaret, P. C. Main, T. J. Foster,
M. Henini, S. P. Beaumont, and M. G. Blamire, Phys. Rev. Lett.
74, 3009共1995兲.
25P. D. Ye, D. Weiss, R. R. Gerhardts, M. Seeger, K. von Klitzing,
K. Eberl, and H. Nickel, Phys. Rev. Lett. 74, 3013共1995兲.
26Mayumi Kato, Akira Endo, Makoto Sakairi, Shingo Katsumoto,
and Yasuhiro Iye, J. Phys. Soc. Jpn. 68, 1492共1999兲.
27S. Cina, D. M. Whittaker, D. D. Arnone, T. Burke, H. P. Hughes,
M. Leadbeater, M. Pepper, and D. A. Ritchie, Phys. Rev. Lett.
83, 4425共1999兲.
28P. Vasilopoulos and F. M. Peeters, Superlattices Microstruct. 7,
393共1990兲.
29F. M. Peeters and P. Vasilopoulos, Phys. Rev. B 47, 1466共1993兲. 30I. S. Ibrahim and F. M. Peeters, Phys. Rev. B 52, 17321共1995兲. 31Andrey Krakovsky, Phys. Rev. B 53, 8469共1996兲.
32A. Y. Rom, Phys. Rev. B 55, 11025共1997兲. 33J. M. Luttinger, Phys. Rev. 84, 814共1951兲.
34S. M. Stewart and C. Zhang, Semicond. Sci. Technol. 10, 1541
共1995兲.
35S. M. Stewart and C. Zhang, Phys. Rev. B 52, R17036共1995兲. 36S. M. Stewart and C. Zhang, J. Phys.: Condens. Matter 8, 6019
共1996兲.
