Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene

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 σ_{x y} make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σ_{x y} is markedly absent (Fig. 1b). Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σ_{x x} in bilayer graphene remains of the order of e²/h, even at zero σ_{x y}. The origin of the unconventional QHE behaviour lies in the coupling between two graphene layers, which transforms massless Dirac fermions, characteristic of single-layer graphene^{3,4,5,6,7,8,9} (Fig. 1c), into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ɛ(p)=p²/2m with effective mass m, but accumulate Berry’s phase of 2π along cyclotron trajectories (here ɛ 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 ɛ, 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 V_g 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=α V_g, where α×ばつ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 work^3,4,10,11 .

Figure 2a shows a typical QHE behaviour in bilayer graphene at a fixed V_g (fixed n) and varying magnetic field B up to 30 T. Pronounced plateaus are clearly seen in Hall resistivity ρ_{x y} in high B, and they are accompanied by zero longitudinal resistivity ρ_{x x}. The observed sequence of the QHE plateaus is described by ρ_{x y}=h/4N e², which is the same sequence as expected for a two-dimensional (2D) free-fermion system with double spin and double valley degeneracy^{1,2,12,13,14,15} . 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 ρ_{x y} 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 informative^3,4,10 . They correspond to a nearly constant parameter μ B, which defines the quality of Landau quantization, and this allows simultaneous observation of several QHE plateaus during a single voltage sweep in moderate magnetic fields (Fig. 2b). The periodicity Δn of quantum oscillations in ρ_{x x} as a function of n is defined by the density of states g B/φ₀ (where g is the degeneracy and φ₀ is the flux quantum) on each Landau level^{1,2,3,4,5,6,7,8,9,10} (see Fig. 1). In Fig. 2c, for example, Δn×ばつ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 graphene^14,15 .

Figure 2b shows that, although the Hall plateaus in bilayer graphene follow the integer sequence σ_{x y}=±(4e²/h)N for N≥1, there is no sign of the zero-N plateau at σ_{x y}=0, which is expected for 2D free-fermion systems^1,2 (Fig. 1a). In this respect, the behaviour resembles the QHE for massless Dirac fermions (Fig. 1c), where there is also no plateau but a step occurs when σ_{x y} passes the neutrality point. However, in bilayer graphene, this step has a double height and is accompanied by a central peak in ρ_{x x}, 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 ×ばつ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 σ_{x y}=ρ_{x y}/(ρ_{x y}²+ρ_{x x}²) is inferred^1,2 from a rapid (often exponential) increase in ρ_{x x}≫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 ρ_{x x} 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 σ_{x y} in bilayer graphene does not vanish over any interval of ν and reaches zero only at one point, where ρ_{x y} changes its sign. Note that ρ_{x x} 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 ρ_{x x}≈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 kΩ 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 , where , a is the lattice periodicity, ħ=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 ɛ=±p²/2m with m=γ₁/2v_F² (the sign ± refers to electron and hole states). Further analysis¹⁵ shows that quasiparticles in bilayer graphene can be described by using the effective hamiltonian

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 of a general form

can be rewritten as

where n=−(cosJ φ,sinJ φ) and vector 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 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 J φ. As a result, if a quasiparticle encircles a closed contour in the momentum space (that is φ=2π), a phase shift Φ=J π 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 plateaus^{3,4,5,6,7,8,9} , 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 (see the Supplementary Information) of the Landau-level spectra for hamiltonian showing Berry’s phase J π 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 ɛ_N=ħω_c(N+1/2) and the lowest state lies at finite energy ħω_c/2, where cyclotron frequency ω_c=e B/m. For single-layer graphene (J=1, Φ=π), and there is a single state ɛ₀ at zero energy^5,6,7,8,9 . For bilayer graphene (J=2, Φ=2π), and the two lowest states ɛ₀=ɛ₁ 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 graphene^{3,4,5,6,7,8,9} . 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.

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).

Authors and Affiliations

1. Manchester Centre for Mesoscience and Nanotechnology, University of Manchester, Manchester, M13 9PL, UK

   K. S. Novoselov, S. V. Morozov, D. Jiang, F. Schedin & A. K. Geim

2. Department of Physics, Lancaster University, Lancaster, LA1 4YB, UK

   E. McCann & V. I. Fal’ko

3. Institute for Microelectronics Technology, 142432, Chernogolovka, Russia

   S. V. Morozov

4. Institute for Molecules and Materials, Radboud University of Nijmegen, Toernooiveld 1, 6525, ED Nijmegen, The Netherlands

   M. I. Katsnelson & U. Zeitler

Corresponding author

Correspondence to A. K. Geim.

Competing interests

The authors declare no competing financial interests.

Reprints and permissions