LETTERS Unconventional quantum Hall effect and Berry's phase of 2 in bilayer graphene K.S.NOVOSELOV,E.McCANN2,S.V.MOROZOV13 V.I.FAL'KO2,M.I.KATSNELSON4,U.ZEITLER. D.JIANG1.F.SCHEDIN1 AND A.K.GEIM1* Manchester Centre for Mesoscience and Nanotechnology,University of Manchester,Manchester M139PL,UK 2Department of Physics,Lancaster University,Lancaster LA1 4YB.UK Institute for Microelectronics Technology,142432 Chemnogolovka,Russia Institute for Molecules and Materials,Radboud University of Nijmegen,Toernooiveld 1,6525 ED Nijmegen,The Netherlands *e-mail:geim@manchester.ac.uk Published online:26 February 2006;doi:10.1038/nphys245 here are two known distinct types of the integer quantum accumulate Berry's phase of 2x along cyclotron trajectories (here g Hall effect.One is the conventional quantum Hall is the energy of quasiparticles and p their momentum).The latter is effect,characteristic of two-dimensional semiconductor shown to be related to a peculiar quantization where the two lowest systems2,and the other is its relativistic counterpart observed Landau levels lie exactly at zero energy s,leading to the missing in graphene,where charge carriers mimic Dirac fermions plateau and double step shown in Fig.1b. characterized by Berry's phase which results in shifted Bilayer films studied in this work were made by the positions of the Hall plateaus Here we report a third type micromechanical cleavage of crystals of natural graphite,which was of the integer quantum Hall effect.Charge carriers in bilayer followed by the selection of bilayer flakes by using a combination graphene have a parabolic energy spectrum but are chiral of optical microscopy and atomic force microscopy as described and show Berry's phase 2 affecting their quantum dynamics. in refs 10,11.Multiterminal field-effect devices (see the inset in The Landau quantization of these fermions results in plateaus Fig.2a)were made from the selected flakes by using standard in Hall conductivity at standard integer positions,but the microfabrication techniques.As a substrate,we used an oxidized last (zero-level)plateau is missing.The zero-level anomaly heavily doped Si wafer,which allowed us to apply gate voltage Va is accompanied by metallic conductivity in the limit of low between graphene and the substrate.The studied devices showed concentrations and high magnetic fields,in stark contrast to an ambipolar electric field effect such that electrons and holes could the conventional,insulating behaviour in this regime.The be induced in concentrations n up to 1013 cm-2(n=aVs,where revealed chiral fermions have no known analogues and present a7.3x 1040cm-2V-for a 300 nm SiOz layer).For further details an intriguing case for quantum-mechanical studies. about microfabrication of graphitic field-effect devices and their Figure 1 provides a schematic overview of the quantum Hall measurements,we refer to earlier work34.0. effect(QHE)behaviour observed in bilayer graphene by comparing Figure 2a shows a typical QHE behaviour in bilayer graphene it with the conventional integer QHE.In the standard theory,each at a fixed Va (fixed n)and varying magnetic field B up to 30 T. filled single-degenerate Landau level contributes one conductance Pronounced plateaus are clearly seen in Hall resistivity Py in quantum e2/h towards the observable Hall conductivity (here e is high B,and they are accompanied by zero longitudinal resistivity the electron charge and h is Planck's constant).The conventional Px.The observed sequence of the QHE plateaus is described QHE is shown in Fig.la,where plateaus in Hall conductivity by py=h/4Ne2,which is the same sequence as expected for o make up an uninterrupted ladder of equidistant steps.In a two-dimensional(2D)free-fermion system with double spin bilayer graphene,QHE plateaus follow the same ladder but the and double valley degeneracy 212-15.However,a clear difference plateau at zero oy is markedly absent(Fig.1b).Instead,the Hall between the conventional and reported QHE emerges in the regime conductivity undergoes a double-sized step across this region. of small filling factors v<1 (see Figs 2b,c and 3).This regime In addition,longitudinal conductivity o in bilayer graphene is convenient to study by fixing B and varying concentrations of remains of the order of e2/h,even at zero o.The origin of electrons and holes passing through the neutrality point In0, the unconventional QHE behaviour lies in the coupling between where Py changes its sign and,nominally,v=0.Also,because two graphene layers,which transforms massless Dirac fermions, carrier mobilities u in graphitic films are weakly dependent on characteristic of single-layer graphene(Fig.1c),into a new n,measurements in constant B are more informative410.They type of chiral quasiparticle.Such quasiparticles have an ordinary correspond to a nearly constant parameter uB,which defines parabolic spectrum s(p)=p2/2m with effective mass m,but the quality of Landau quantization,and this allows simultaneous nature physicsIADVANCE ONLINE PUBLICATION www.nature.com/naturephysics 2006 Nature Publishing Group
LETTERS Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene K. S. NOVOSELOV1, E. McCANN2, S. V. MOROZOV1,3, V. I. FAL’KO2, M. I. KATSNELSON4, U. ZEITLER4, D. JIANG1, F. SCHEDIN1 AND A. K. GEIM1* 1Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester M13 9PL, UK 2Department of Physics, Lancaster University, Lancaster LA1 4YB, UK 3Institute for Microelectronics Technology, 142432 Chernogolovka, Russia 4Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands *e-mail: geim@manchester.ac.uk Published online: 26 February 2006; doi:10.1038/nphys245 T here are two known distinct types of the integer quantum Hall effect. One is the conventional quantum Hall effect, characteristic of two-dimensional semiconductor systems1,2, and the other is its relativistic counterpart observed in graphene, where charge carriers mimic Dirac fermions characterized by Berry’s phase π, which results in shifted positions of the Hall plateaus3–9. Here we report a third type of the integer quantum Hall effect. Charge carriers in bilayer graphene have a parabolic energy spectrum but are chiral and show Berry’s phase 2π affecting their quantum dynamics. The Landau quantization of these fermions results in plateaus in Hall conductivity at standard integer positions, but the last (zero-level) plateau is missing. The zero-level anomaly is accompanied by metallic conductivity in the limit of low concentrations and high magnetic fields, in stark contrast to the conventional, insulating behaviour in this regime. The revealed chiral fermions have no known analogues and present an intriguing case for quantum-mechanical studies. Figure 1 provides a schematic overview of the quantum Hall effect (QHE) behaviour observed in bilayer graphene by comparing it with the conventional integer QHE. In the standard theory, each filled single-degenerate Landau level contributes one conductance quantum e2 /h towards the observable Hall conductivity (here e is the electron charge and h is Planck’s constant). The conventional QHE is shown in Fig. 1a, where plateaus in Hall conductivity σxy make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σxy is markedly absent (Fig. 1b). Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σxx in bilayer graphene remains of the order of e2 /h, even at zero σxy . The origin of the unconventional QHE behaviour lies in the coupling between two graphene layers, which transforms massless Dirac fermions, characteristic of single-layer graphene3–9 (Fig. 1c), into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ε(p) = p2 /2m with effective mass m, but accumulate Berry’s phase of 2π along cyclotron trajectories (here ε is the energy of quasiparticles and p their momentum). The latter is shown to be related to a peculiar quantization where the two lowest Landau levels lie exactly at zero energy ε, leading to the missing plateau and double step shown in Fig. 1b. Bilayer films studied in this work were made by the micromechanical cleavage of crystals of natural graphite, which was followed by the selection of bilayer flakes by using a combination of optical microscopy and atomic force microscopy as described in refs 10,11. Multiterminal field-effect devices (see the inset in Fig. 2a) were made from the selected flakes by using standard microfabrication techniques. As a substrate, we used an oxidized heavily doped Si wafer, which allowed us to apply gate voltage Vg between graphene and the substrate. The studied devices showed an ambipolar electric field effect such that electrons and holes could be induced in concentrations n up to 1013 cm−2 (n = αVg, where α≈7.3×1010 cm−2 V−1 for a 300 nm SiO2 layer). For further details about microfabrication of graphitic field-effect devices and their measurements, we refer to earlier work3,4,10,11. Figure 2a shows a typical QHE behaviour in bilayer graphene at a fixed Vg (fixed n) and varying magnetic field B up to 30 T. Pronounced plateaus are clearly seen in Hall resistivity ρxy in high B, and they are accompanied by zero longitudinal resistivity ρxx . The observed sequence of the QHE plateaus is described by ρxy = h/4N e2 , which is the same sequence as expected for a two-dimensional (2D) free-fermion system with double spin and double valley degeneracy1,2,12–15. However, a clear difference between the conventional and reported QHE emerges in the regime of small filling factors ν < 1 (see Figs 2b,c and 3). This regime is convenient to study by fixing B and varying concentrations of electrons and holes passing through the neutrality point |n| ≈ 0, where ρxy changes its sign and, nominally, ν = 0. Also, because carrier mobilities μ in graphitic films are weakly dependent on n, measurements in constant B are more informative3,4,10. They correspond to a nearly constant parameter μB, which defines the quality of Landau quantization, and this allows simultaneous nature physics ADVANCE ONLINE PUBLICATION www.nature.com/naturephysics 1 © 2006 Nature Publishing Group
LETTERS a c,(ge21h例 h/4e2 2 h/8e2 h/12e2 nh/geB +5 10 20 B 3 (ge2/h) 6 20T 12T B=12T T=4K 3 a(ge2/h) nh/geB -2 0 n(102cm-3 Figure 2 Quantum Hall effect in bilayer graphene.a,Hall resistivitiesand measured as a function of B for fixed concentrations of electrons 2.5 x 1012 cm-2 induced by the electric field effect Inset:Scanning electron micrograph of one of more than ten bilayer devices studied in our work.The width of Figure 1 Three types of the integer quantum Hall effect.a,b,Schematic the Hall bar(dark central area)is approximately 1 um.The known geometry of our illustration of the conventional integer QHE found in 2D semiconductor systems(a), devices allowed us to convert the measured resistance into p with an accuracy of incorporated from refs 1,2,and the QHE in bilayer graphene described in the present better than 10%.b,c,and px are plotted as functions of n at a fixed B and paper(b).Plateaus in Hall conductivity occur at values(ge/h)N,where Nis an temperature T=4 K.Positive and negative n correspond to field-induced electrons integer,e2/h the conductance quantum and g the system degeneracy.The distance and holes,respectively.The Hall conductivity=(+)was calculated between steps along the concentration axis is defined by the density of states directly from experimental curves for py and p.allows the underlying gB/on each Landau level,which is independent of a 2D spectrum.Here,Bis sequences of QHE plateaus to be seen more clearly.crosses zero without any the magnetic field andh/e the flux quantum.The corresponding sequences of sign of the zero-level plateau that would be expected for a conventional 2D system. Landau levels as a function of carrier concentrations n are shown in blue and orange The inset shows the calculated energy spectrum for bilayer graphene,which is for electrons and holes,respectively.For completeness,c also shows the QHE parabolic at low s.Carrier mobilities in our bilayer devices were typically around behaviour for massless Dirac fermions in single-layer graphene. 3.000 cm2 V-s-1,which is lower than for devices made from single-layer graphene.This is surprising because one generally expects more damage and exposure in the case of single-layer graphene that is unprotected from the observation of several QHE plateaus during a single voltage sweep immediate environment from both sides. in moderate magnetic fields (Fig.2b).The periodicity An of quantum oscillations in px as a function of n is defined by the density of states gB/oo(where g is the degeneracy and oo is the Figure 2b shows that,although the Hall plateaus in bilayer flux quantum)on each Landau level-10(see Fig.1).In Fig.2c,for graphene follow the integer sequence o=(4e2/h)N for N>1, example,△n≈1.2×lo12cm-2atB=l2T,which yields g=4and there is no sign of the zero-N plateau at o=0,which is confirms the double-spin and double-valley degeneracy expected expected for 2D free-fermion systems2(Fig.1a).In this respect,the from band-structure calculations for bilayer graphene5 behaviour resembles the QHE for massless Dirac fermions(Fig.1c), 2 nature physics I ADVANCE ONLINE PUBLICATION I www.nature.com/naturephysics 2006 Nature Publishing Group
LETTERS xy (ge 2/h) – 3 –2 –1 – 3 –2 –1 – 3 –2 –1 1 2 3 1 2 3 1 2 3 nh/geB nh/geB nh/geB – 3 – 2 3 2 – 1 – 3 – 2 3 1 2 – 1 – 3 – 2 3 1 2 a b c σ xy (ge 2 σ /h) σxy (ge 2/h) Figure 1 Three types of the integer quantum Hall effect. a,b, Schematic illustration of the conventional integer QHE found in 2D semiconductor systems (a), incorporated from refs 1,2, and the QHE in bilayer graphene described in the present paper (b). Plateaus in Hall conductivity σxy occur at values (ge2 /h)N, where N is an integer, e2 /h the conductance quantum and g the system degeneracy. The distance between steps along the concentration axis is defined by the density of states gB/φ0 on each Landau level, which is independent of a 2D spectrum1–9. Here, B is the magnetic field and φ0 = h/e the flux quantum. The corresponding sequences of Landau levels as a function of carrier concentrations n are shown in blue and orange for electrons and holes, respectively. For completeness, c also shows the QHE behaviour for massless Dirac fermions in single-layer graphene. observation of several QHE plateaus during a single voltage sweep in moderate magnetic fields (Fig. 2b). The periodicity n of quantum oscillations in ρxx as a function of n is defined by the density of states gB/φ0 (where g is the degeneracy and φ0 is the flux quantum) on each Landau level1–10 (see Fig. 1). In Fig. 2c, for example, n ≈1.2×1012 cm−2 at B =12 T, which yields g =4 and confirms the double-spin and double-valley degeneracy expected from band-structure calculations for bilayer graphene14,15. –4 4 – 2 0 n (1012 cm – 2) 2 E p 20 T 12 T 0 30 B (T) 10 20 h/4e2 h/8e2 h/12e2 × 5 6 0 2 4 1 2 – 1 – 2 – 4 0 – 3 4 3 2 4 6 0 xy and xx (k Ω) xy (4e2/h) xx (k Ω) a b c ρ ρ σ ρ B = 12 T T = 4 K Figure 2 Quantum Hall effect in bilayer graphene. a, Hall resistivities ρxy and ρxx measured as a function of B for fixed concentrations of electrons n ≈ 2.5×1012 cm−2 induced by the electric field effect. Inset: Scanning electron micrograph of one of more than ten bilayer devices studied in our work. The width of the Hall bar (dark central area) is approximately 1 μm. The known geometry of our devices allowed us to convert the measured resistance into ρxx with an accuracy of better than 10%. b,c, σxy and ρxx are plotted as functions of n at a fixed B and temperature T = 4 K. Positive and negative n correspond to field-induced electrons and holes, respectively. The Hall conductivity σxy = ρxy/(ρ2 xy +ρ2 xx ) was calculated directly from experimental curves for ρxy and ρxx . σxy allows the underlying sequences of QHE plateaus to be seen more clearly. σxy crosses zero without any sign of the zero-level plateau that would be expected for a conventional 2D system. The inset shows the calculated energy spectrum for bilayer graphene, which is parabolic at low ε. Carrier mobilities μ in our bilayer devices were typically around 3,000 cm2 V−1 s−1 , which is lower than for devices made from single-layer graphene3,4. This is surprising because one generally expects more damage and exposure in the case of single-layer graphene that is unprotected from the immediate environment from both sides. Figure 2b shows that, although the Hall plateaus in bilayer graphene follow the integer sequence σxy =±(4e2 /h)N for N ≥ 1, there is no sign of the zero-N plateau at σxy = 0, which is expected for 2D free-fermion systems1,2 (Fig. 1a). In this respect, the behaviour resembles the QHE for massless Dirac fermions (Fig. 1c), 2 nature physics ADVANCE ONLINE PUBLICATION www.nature.com/naturephysics © 2006 Nature Publishing Group
LETTERS b theory,which attributes the finite metallic conductivity and 20T the absence of localization to the relativistic-like spectrum of 10T single-layer graphene.Bilayer graphene has the usual parabolic spectrum,and the observation of the maximum resistivity of approximately h/4e2 and,moreover,its weak dependence on B in this system is most unexpected.Note,however,that the quantization is less accurate than in single-layer graphene,as the T=4K peak value varied from 6 to 9 k for different bilayer devices. The unconventional QHE in bilayer graphene originates from peculiar properties of its charge carriers that are chiral fermions with a finite mass,as discussed below.First,we have calculated Bm) the quasiparticle spectrum in bilayer graphene by using the standard nearest-neighbour approximation.For quasiparticles d near the corners of the Brillouin zone known as K-points,we find 00K 8(p)=(1/2)(1/4)i+vip2,where v:=(v3/2)Yoa/h, a is the lattice periodicity,h=h/2 and yo and y are the ●● intra layer and inter layer coupling constants,respectively.This dispersion relation (plotted in Fig.2c)is in agreement with the first-principle band-structure calculations'4and,at low energies, B=10T becomes parabolic s=+p2/2m with m=y/2v (the sign+ refers to electron and hole states).Further analysis's shows that quasiparticles in bilayer graphene can be described by using the effective hamiltonian 100 200 T(K) i2=- 1 0 (分)2 2m元1 0 where元=p.+p, Figure 3 Resistivity of bilayer graphene near zero concentrations as a function A2 acts in the space of two-component Bloch functions(further of magnetic field and temperature.a-d,The peak in p remains of the order of referred to as pseudospins)describing the amplitude of electron h/4e2,independent of B(a,b)and T(c,d).This yields no gap in the Landau waves on weakly coupled nearest sites Al and B2 belonging to two spectrum at zero energy.b,For a fixedn0 and varying B,we observed only small nonequivalent carbon sublattices A and B and two graphene layers magnetoresistance.The latter varied for different devices and contact configurations marked as 1 and 2. (probably indicating the edge-state transport)and could be non-monotonic and of For a given direction of quasiparticle momentum p= random sign.However,the observed magnetoresistance (for bilayer devices without (pcos,psin),a hamiltonian H,of a general form chemical doping)never exceeded a factor of two in any of our experiments in (元) fields up to 20 T. 0 can be rewritten as where there is also no plateau but a step occurs when o passes the neutrality point.However,in bilayer graphene,this step has a 月，=e(p)on(p), (1) double height and is accompanied by a central peak in p which is twice as broad as all other peaks(Fig.2c).The broader peak yields where n =-(cosJo,sinJo)and vector o is made from Pauli that,in bilayer graphene,the transition between the lowest hole and matrices'5.For bilayer graphene,=2,but the notation I is electron Hall plateaus requires twice the number of carriers needed useful because it also allows equation (1)to be linked with for the transition between the other QHE plateaus.This implies the case of single-layer graphene,where =1.The eigenstates that the lowest Landau level has double degeneracy 2 x 4B/o, of H,correspond to pseudospins polarized parallel (electrons) which can be viewed as two Landau levels merged together at n0 or antiparallel (holes)to the 'quantization'axis n.An adiabatic (see the Landau level charts in Fig.1). evolution of such pseudospin states,which accompanies the Continuous measurements through v=0 as shown in Fig.2b,c rotation of momentum p by angle also corresponds to the have been impossible for conventional 2D systems where the rotation of axis n by angle As a result,if a quasiparticle encircles zero-level plateau in=(+)is inferred from a a closed contour in the momentum space (that is =2),a phase rapid (often exponential)increase in ph/e2 with increasing B shift=known as Berry's phase is gained by the quasiparticle's and decreasing temperature T for filling factors v<1,indicating wavefunction6.Berry's phase can be viewed as arising owing to an insulating state.To provide a direct comparison with the rotation of pseudospin,when a quasiparticle repetitively moves conventional QHE measurements,Fig.3 shows p in bilayer between different carbon sublattices (A and B for single-layer graphene as a function of B and T around zero v.Bilayer graphene graphene,and Al and B2 for bilayer graphene). shows little magnetoresistance or temperature dependence at the For fermions completing cyclotron orbits,Berry's phase neutrality point,in striking contrast to the conventional QHE contributes to the semiclassical quantization and affects the phase behaviour.This implies that o,,in bilayer graphene does not vanish of Shubnikov-de Haas oscillations (SdHOs).For single-layer over any interval of v and reaches zero only at one point,where graphene,this results in a -shift in SdHOs and a related px changes its sign.Note that p surprisingly maintains a peak 1/2-shift in the sequence of QHE plateaus,as compared with value of approximately h/ge in fields up to 20 T and temperatures the conventional 2D systems where Berry's phase is zero.For down to 1 K.A finite value of ph/4e2 in the limit of low bilayer graphene,=2 and there can be no changes in the carrier concentrations and at zero B was reported for single-layer quasiclassical limit (N1).One might also expect that phase 2 graphene.This observation was in qualitative agreement with cannot influence the QHE sequencing.However,the exact analysis nature physics I ADVANCE ONLINE PUBLICATION I www.nature.com/naturephysics 3 2006 Nature Publishing Group
LETTERS 20 T 0 T 10 T B = 10 T T = 4 K Vg (V) B (T) Vg (V) T (K) B T 100 K 4 K 50 K 6 9 3 0 6 9 3 0 3 6 9 3 6 9 – 50 50 0 0 5 10 15 – 50 0 50 0 100 200 300 xx (k Ω) a b c d ρ xx (k Ω) ρ xx (k Ω) ρ xx (k Ω) ρ Figure 3 Resistivity of bilayer graphene near zero concentrations as a function of magnetic field and temperature. a–d, The peak in ρxx remains of the order of h/4e2 , independent of B (a,b) and T (c,d). This yields no gap in the Landau spectrum at zero energy. b, For a fixed n ≈ 0 and varying B, we observed only small magnetoresistance. The latter varied for different devices and contact configurations (probably indicating the edge-state transport) and could be non-monotonic and of random sign. However, the observed magnetoresistance (for bilayer devices without chemical doping10) never exceeded a factor of two in any of our experiments in fields up to 20 T. where there is also no plateau but a step occurs when σxy passes the neutrality point. However, in bilayer graphene, this step has a double height and is accompanied by a central peak in ρxx , which is twice as broad as all other peaks (Fig. 2c). The broader peak yields that, in bilayer graphene, the transition between the lowest hole and electron Hall plateaus requires twice the number of carriers needed for the transition between the other QHE plateaus. This implies that the lowest Landau level has double degeneracy 2 × 4B/φ0, which can be viewed as two Landau levels merged together at n ≈0 (see the Landau level charts in Fig. 1). Continuous measurements through ν = 0 as shown in Fig. 2b,c have been impossible for conventional 2D systems where the zero-level plateau in σxy = ρxy/(ρ2 xy + ρ2 xx ) is inferred1,2 from a rapid (often exponential) increase in ρxx h/e2 with increasing B and decreasing temperature T for filling factors ν < 1, indicating an insulating state. To provide a direct comparison with the conventional QHE measurements, Fig. 3 shows ρxx in bilayer graphene as a function of B and T around zero ν. Bilayer graphene shows little magnetoresistance or temperature dependence at the neutrality point, in striking contrast to the conventional QHE behaviour. This implies that σxy in bilayer graphene does not vanish over any interval of ν and reaches zero only at one point, where ρxy changes its sign. Note that ρxx surprisingly maintains a peak value of approximately h/g e2 in fields up to 20 T and temperatures down to 1 K. A finite value of ρxx ≈ h/4e2 in the limit of low carrier concentrations and at zero B was reported for single-layer graphene3 . This observation was in qualitative agreement with theory, which attributes the finite metallic conductivity and the absence of localization to the relativistic-like spectrum of single-layer graphene3 . Bilayer graphene has the usual parabolic spectrum, and the observation of the maximum resistivity of approximately h/4e2 and, moreover, its weak dependence on B in this system is most unexpected. Note, however, that the quantization is less accurate than in single-layer graphene, as the peak value varied from 6 to 9 k for different bilayer devices. The unconventional QHE in bilayer graphene originates from peculiar properties of its charge carriers that are chiral fermions with a finite mass, as discussed below. First, we have calculated the quasiparticle spectrum in bilayer graphene by using the standard nearest-neighbour approximation12. For quasiparticles near the corners of the Brillouin zone known as K-points, we find ε(p) = ±(1/2)γ1 ± (1/4)γ2 1 +v2 F p2, where vF = ( √3/2)γ0 a/h¯, a is the lattice periodicity, h¯ = h/2π and γ0 and γ1 are the intra layer and inter layer coupling constants, respectively13. This dispersion relation (plotted in Fig. 2c) is in agreement with the first-principle band-structure calculations14 and, at low energies, becomes parabolic ε = ±p2 /2m with m = γ1 /2v2 F (the sign ± refers to electron and hole states). Further analysis15 shows that quasiparticles in bilayer graphene can be described by using the effective hamiltonian Hˆ 2 = − 1 2m 0 (πˆ +)2 πˆ 2 0 where πˆ = pˆ x +ipˆ y . Hˆ 2 acts in the space of two-component Bloch functions (further referred to as pseudospins) describing the amplitude of electron waves on weakly coupled nearest sites A1 and B2 belonging to two nonequivalent carbon sublattices A and B and two graphene layers marked as 1 and 2. For a given direction of quasiparticle momentum p = (pcosϕ,psinϕ), a hamiltonian Hˆ J of a general form 0 (πˆ +)J πˆ J 0 can be rewritten as Hˆ J = ε(p)σ·n(ϕ), (1) where n = −(cosJϕ,sin Jϕ) and vector σ is made from Pauli matrices15. For bilayer graphene, J = 2, but the notation J is useful because it also allows equation (1) to be linked with the case of single-layer graphene, where J = 1. The eigenstates of Hˆ J correspond to pseudospins polarized parallel (electrons) or antiparallel (holes) to the ‘quantization’ axis n. An adiabatic evolution of such pseudospin states, which accompanies the rotation of momentum p by angle ϕ, also corresponds to the rotation of axis n by angle Jϕ. As a result, if a quasiparticle encircles a closed contour in the momentum space (that is ϕ = 2π), a phase shift Φ = Jπ known as Berry’s phase is gained by the quasiparticle’s wavefunction16. Berry’s phase can be viewed as arising owing to rotation of pseudospin, when a quasiparticle repetitively moves between different carbon sublattices (A and B for single-layer graphene, and A1 and B2 for bilayer graphene). For fermions completing cyclotron orbits, Berry’s phase contributes to the semiclassical quantization and affects the phase of Shubnikov–de Haas oscillations (SdHOs). For single-layer graphene, this results in a π-shift in SdHOs and a related 1/2-shift in the sequence of QHE plateaus3–9, as compared with the conventional 2D systems where Berry’s phase is zero. For bilayer graphene, Φ = 2π and there can be no changes in the quasiclassical limit (N 1). One might also expect that phase 2π cannot influence the QHE sequencing. However, the exact analysis nature physics ADVANCE ONLINE PUBLICATION www.nature.com/naturephysics 3 © 2006 Nature Publishing Group
LETTERS (see the Supplementary Information)of the Landau-level spectra Received 22 December 2005:accepted 2 February 2006;published 26 February 2006. for hamiltonian H,showing Berry's phase I shows that there is an associated J-fold degeneracy of the zero-energy Landau level References (that is Berry's phase of 2n leads to observable consequences in 1.Prange,R.E.Girvin,S.M.The Quantum Hall Effect (Springer,New York,1990). 2.Macdonald,A.H.Quantm Hall Effect:A Perspective (Kluwer Academic,Dordrecht,1990). the quantum limit N=0).For the free-fermion QHE systems 3.Novoselov.K S.etal Two-dimensional gas of massless Dirac fermions in graphene.Nature 438. 197-200(2005). (no Berry's phase),the energy is given by Ew =ho(N+1/2) 4.Zhang.Y,Tan.I W.Stormer,H.L.Kim,P.Experimental observation of the quantum Hall effect and the lowest state lies at finite energy ho/2,where cyclotron and Berry's phase in graphene.Nature 438,201-204(2005). frequency.=eB/m.For single-layer graphene(U=l,Φ=π， 5.MeClure,L W.Diamagr nctism of graphite.Phys.Rev 104,666-671(1956). 6.Haldane.E D.M.Model for a g m Hall effect without Landau leves:Condensed-matter ew=±咋√2 ehBN and there is a single state so at zero energy. realization of the'parity anomaly.Phys.Rev.Lett.61,2015-2018(1988). For bilayer graphene (J=2,Φ=2r,ew=±ho.√N(N-I)and 7.Zheng.Y.Ando,T.Hall conductivity of a two-dimensional graphite system.Phys.Rev.B65. 245420(20021. the two lowest states lie at zero energy!5. 8.Gusynin,V.P.&Sha ov.S.G.Unc onventional integer quantum Hall effect in graphene.Phys.Rev The existence of a double-degenerate Landau level explains the Lc.95,146s01(2D05 9.Peres,N.M.R.Guinea,F Castro Neto,A.H.Electronic properties of two-dimensional carbon. unconventional QHE found in bilayer graphene.This Landau level Preprint at (2005). lies at the border between electron and hole gases and,taking into 10.Novoselov,K.S.et al.Electric field effect in atomically thin carbon films.Science 306,666-669 (2004). 11.Novoselov.K.S.et al.Two din account the quadruple spin and valley degeneracy,it accommodates crystals.Proc.Natl Acad.Sci.USA102 10451-10453(2005). carrier density 8B/o.With reference to Fig.1,the existence of such 12.Wallace.P.R.The band theory of graphite.Phys.Rev:71,622-634(1947). a Landau level implies that there must be a QHE step across the 13.Dresselhaus,M.S.Dresselhaus,G.Intercalation compounds of graphite.Ad:Phrys.51. 1-18620021 neutrality point,similarly to the case of single-layer graphene 14.Trickey.S.B..Muller-Plathe,F,Die en,G.H.F&Bocttger,IC.Interplanar binding and lattice Owing to the double degeneracy,it takes twice the number of relaxation in a graphite delayer.Phys.Rev.B 45,4460-4468(1992). 15.McCann,E Falko.V.I.Landau level degeneracy and quantum Hall effect in a graphite bilayer. carriers to fill it (as compared with all other Landau levels),so that Preprint at (2005). the transition between the corresponding QHE plateaus must be 16.Berry.M.V.Quntal phase factor accompanying adiabatic change.ProcR Soc LondA9. 45-57(1984 twice as wide (that is 8B/o as compared with 4B/o).Also,the step between the plateaus must be twice as high,that is 8e2/h as Acknowledgements compared with 4e2/h for the other steps at higher carrier densities. We thank the High Field M gnet Labora tory (Nijmegen)for their hospitality.U.Z and K.S.N.were This is exactly the behaviour observed experimentally. partially supported by EuroMagNET of the th FrameworkStructuring the European Research Area. Research Infrastructures Action'and by the Leverhulme Trust.S.V.M.acknowledges support from the In conclusion,bilayer graphene adds a new member to the Russian Academy of Sciences.This research was funded by the EPSRC (UK). small family of QHE systems,and its QHE behaviour reveals the Correspondence and requests for materials should be addressed to A.K.G. ntary Inform n accompanies this paper on www w.nature.com/naturephysics. existence of massive chiral fermions with Berry's phase 2,which are distinct from other known quasiparticles.The observation Competing financial interests of a finite metallic conductivity of approximately e2/h for such The authors declare that they have no competing financial interests fermions poses a serious challenge for theory. Reprints and permission information is available online at http://npg-nature.com/reprintsandpermissions nature physics I ADVANCE ONLINE PUBLICATION I www.nature.com/naturephysics 2006 Nature Publishing Group
LETTERS (see the Supplementary Information) of the Landau-level spectra for hamiltonian Hˆ J showing Berry’s phase Jπ shows that there is an associated J-fold degeneracy of the zero-energy Landau level (that is Berry’s phase of 2π leads to observable consequences in the quantum limit N = 0). For the free-fermion QHE systems (no Berry’s phase), the energy is given by εN = h¯ωc (N + 1/2) and the lowest state lies at finite energy h¯ωc/2, where cyclotron frequency ωc = eB/m. For single-layer graphene (J = 1, Φ = π), εN = ±vF √2ehBN ¯ and there is a single state ε0 at zero energy5–9. For bilayer graphene (J = 2, Φ = 2π), εN = ±h¯ωc √N(N −1) and the two lowest states ε0 = ε1 lie at zero energy15. The existence of a double-degenerate Landau level explains the unconventional QHE found in bilayer graphene. This Landau level lies at the border between electron and hole gases and, taking into account the quadruple spin and valley degeneracy, it accommodates carrier density 8B/φ0. With reference to Fig. 1, the existence of such a Landau level implies that there must be a QHE step across the neutrality point, similarly to the case of single-layer graphene3–9. Owing to the double degeneracy, it takes twice the number of carriers to fill it (as compared with all other Landau levels), so that the transition between the corresponding QHE plateaus must be twice as wide (that is 8B/φ0 as compared with 4B/φ0). Also, the step between the plateaus must be twice as high, that is 8e2 /h as compared with 4e2 /h for the other steps at higher carrier densities. This is exactly the behaviour observed experimentally. In conclusion, bilayer graphene adds a new member to the small family of QHE systems, and its QHE behaviour reveals the existence of massive chiral fermions with Berry’s phase 2π, which are distinct from other known quasiparticles. The observation of a finite metallic conductivity of approximately e2 /h for such fermions poses a serious challenge for theory. Received 22 December 2005; accepted 2 February 2006; published 26 February 2006. References 1. Prange, R. E. & Girvin, S. M. The Quantum Hall Effect (Springer, New York, 1990). 2. Macdonald, A. H. Quantum Hall Effect: A Perspective (Kluwer Academic, Dordrecht, 1990). 3. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fermions in graphene. Nature 438, 197–200 (2005). 4. Zhang, Y., Tan, J. W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005). 5. McClure, J. W. Diamagnetism of graphite. Phys. Rev. 104, 666–671 (1956). 6. Haldane, F. D. M. Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ‘parity anomaly’. Phys. Rev. Lett. 61, 2015–2018 (1988). 7. Zheng, Y. & Ando, T. Hall conductivity of a two-dimensional graphite system. Phys. Rev. B 65, 245420 (2002). 8. Gusynin, V. P. & Sharapov, S. G. Unconventional integer quantum Hall effect in graphene. Phys. Rev. Lett. 95, 146801 (2005). 9. Peres, N. M. R., Guinea, F. & Castro Neto, A. H. Electronic properties of two-dimensional carbon. Preprint at (2005). 10. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films. Science 306, 666–669 (2004). 11. Novoselov, K. S. et al. Two dimensional atomic crystals. Proc. Natl Acad. Sci. USA 102, 10451–10453 (2005). 12. Wallace, P. R. The band theory of graphite. Phys. Rev. 71, 622–634 (1947). 13. Dresselhaus, M. S. & Dresselhaus, G. Intercalation compounds of graphite. Adv. Phys. 51, 1–186 (2002). 14. Trickey, S. B., Muller-Plathe, F., Diercksen, G. H. F. & Boettger, J. C. Interplanar binding and lattice ¨ relaxation in a graphite delayer. Phys. Rev. B 45, 4460–4468 (1992). 15. McCann, E. & Falko, V. I. Landau level degeneracy and quantum Hall effect in a graphite bilayer. Preprint at (2005). 16. Berry, M. V. Quantal phase factor accompanying adiabatic change. Proc. R. Soc. Lond. A 392, 45–57 (1984). Acknowledgements We thank the High Field Magnet Laboratory (Nijmegen) for their hospitality. U.Z. and K.S.N. were partially supported by EuroMagNET of the 6th Framework ‘Structuring the European Research Area, Research Infrastructures Action’ and by the Leverhulme Trust. S.V.M. acknowledges support from the Russian Academy of Sciences. This research was funded by the EPSRC (UK). Correspondence and requests for materials should be addressed to A.K.G. Supplementary Information accompanies this paper on www.nature.com/naturephysics. Competing financial interests The authors declare that they have no competing financial interests. Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/ 4 nature physics ADVANCE ONLINE PUBLICATION www.nature.com/naturephysics © 2006 Nature Publishing Group
