Integer Quantum Hall Effect in Trilayer Graphene

Physical Review B, 2012

Analytical expressions for the Hall conductivity σ yx and the longitudinal resistivity ρ xx are derived in gapped, single-layer graphene using linear response theory. The gap 2 , described by a mass term, is induced by a substrate made of hexagonal boron nitride (h-BN) and produces two levels at ±. It is shown that σ yx has the same form as for a graphene sample supported by a common substrate without a mass term. The differences are a shift in the energy spectrum, which is not symmetric with respect to the Dirac point for either valley due to the gap, the absence of a zero-energy Landau level, and the nonequivalence of the K and K valleys. In addition, the dispersion of the energy levels, caused by electron scattering by impurities, modifies mostly plateaus due to the levels at ±. It is shown that the resistivity ρ xx exhibits an oscillatory dependence on the electron concentration. The main difference with the usual graphene samples, on SiO 2 substrates, occurs near zero concentration, as the energy spectra differ mostly near the Dirac point.

Physical Review Letters, 2005

Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by 2 + 1 dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity σxy = −(2e 2 /h)(2n + 1) with n = 0, 1, . . ., that notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n = 0 Landau level and was discovered in recent experiments on ultrathin graphite films.

Physical Review B, 2010

Recent experiments indicate that AA-stacked bilayer graphenes (BLG) could exist. Since the energy bands of the AA-stacked BLG are different from both the monolayer and AB-stacked bilayer graphenes, different integer quantum Hall effect in the AA-stacked graphene is expected. We have therefore calculated the quantized Hall conductivity σxy and also longitudinal conductivity σxx of the AA-stacked BLG within the linear response Kubo formalism. Interestingly, we find that the AAstacked BLG could exhibit both conventional insulating behavior (theν = 0 plateau) and chirality for |μ| < t, whereν is the filling factor (ν = σxyh/e 2),μ is the chemical potential, and t is the interlayer hopping energy, in striking contrast to the monlayer graphene (MLG) and AB-stacked BLG. We also find that for |μ| = [(√ n2 + √ n1)/(√ n2 − √ n1)]t, where n1 = 1, 2, 3, • • •, n2 = 2, 3, 4, • • • and n2 > n1, the Hall conductivity is quantized as σxy = ± 4e 2 h n, n = 0, 1, 2, • • •, if |μ| < t and σxy = ± 4e 2 h n, n = 1, 2, 3, • • •, if |μ| > t. However, if |μ| = [(√ n1 + √ n2)/(√ n2 − √ n1)]t, theν = ±4(n1 + n2)n plateaus are absent, where n = 1, 2, 3, • • •, in comparison with the ABstacked BLG within the two-band approximation. We show that in the low-disorder and highmagnetic-field regime, σxx → 0 as long as the Fermi level is not close to a Dirac point, where Γ denotes the Landau level broadening induced by disorder. Furthermore, when σxy is plotted as a function ofμ, aν = 0 plateau appears acrossμ = 0 and it would disappear if the magnetic field B = πt 2 /N ehυ 2 F , N = 1, 2, 3, • • •. Finally, the disappearance of the zero-Hall conductivity plateau is always accompanied by the occurence of a 8e 2 /h-step atμ = t.

Physical Review B, 2006

We study the integer and fractional quantum Hall effect on a honeycomb lattice at half-filling (graphene) in the presence of disorder and electron-electron interactions. We show that the interactions between the delocalized chiral edge states (generated by the magnetic field) and Andersonlocalized surface states (created by the presence of zig-zag edges) lead to edge reconstruction. As a consequence, the point contact tunneling on a graphene edge has a non-universal tunneling exponent, and the Hall conductivity is not perfectly quantized in units of e 2 /h. We argue that the magnetotransport properties of graphene depend strongly on the strength of electron-electron interactions, the amount of disorder, and the details of the edges.

Science (New York, N.Y.), 2014

The nature of fractional quantum Hall (FQH) states is determined by the interplay between the Coulomb interaction and the symmetries of the system. The distinct combination of spin, valley, and orbital degeneracies in bilayer graphene is predicted to produce an unusual and tunable sequence of FQH states. Here, we present local electronic compressibility measurements of the FQH effect in the lowest Landau level of bilayer graphene. We observe incompressible FQH states at filling factors ν = 2p + 2/3, with hints of additional states appearing at ν = 2p + 3/5, where p = -2, -1, 0, and 1. This sequence breaks particle-hole symmetry and obeys a ν → ν + 2 symmetry, which highlights the importance of the orbital degeneracy for many-body states in bilayer graphene.

Physical Review Letters, 2006

Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-like lowenergy excitations. When Zeeman and spin-orbit interactions are neglected its Landau levels are four-fold degenerate, explaining the 4e 2 /h separation between quantized Hall conductivity values seen in recent experiments. In this paper we derive a criterion for the occurrence of interactiondriven quantum Hall effects near intermediate integer values of e 2 /h due to charge gaps in broken symmetry states.

Physical Review B, 2008

Recent successes in manufacturing of atomically thin graphite samples [1] (graphene) have stimulated intense experimental and theoretical activity . The key feature of graphene is the massless Dirac type of low-energy electron excitations. This gives rise to a number of unusual physical properties of this system distinguishing it from conventional two-dimensional metals. One of the most remarkable properties of graphene is the anomalous quantum Hall effect . It is extremely sensitive to the structure of the system; in particular, it clearly distinguishes single-and double-layer samples. In spite of the impressive experimental progress, the theory of quantum Hall effect in graphene has not been established. This theory is a subject of the present paper. We demonstrate that the Landau level structure by itself is not sufficient to determine the form of the quantum Hall effect. The Hall quantization is due to Anderson localization which, in graphene, is very peculiar and depends strongly on the character of disorder . It is only a special symmetry of disorder that may give rise to anomalous quantum Hall effects in graphene. We analyze the symmetries of disordered singleand double-layer graphene in magnetic field and identify the conditions for anomalous Hall quantization.

Physical Review B, 2011

The quantum anomalous Hall effect can occur in single and few layer graphene systems that have both exchange fields and spin-orbit coupling. In this paper, we present a study of the quantum anomalous Hall effect in single-layer and gated bilayer graphene systems with Rashba spin-orbit coupling. We compute Berry curvatures at each valley point and find that for single-layer graphene the Hall conductivity is quantized at σxy = 2e 2 /h, with each valley contributing a unit conductance and a corresponding chiral edge state. In bilayer graphene, we find that the quantized anomalous Hall conductivity is twice that of the single-layer case when the gate voltage U is smaller than the exchange field M , and zero otherwise. Although the Chern number vanishes when U > M , the system still exhibits a quantized valley Hall effect, with the edge states in opposite valleys propagating in opposite directions. The possibility of tuning between different topological states with an external gate voltage suggests possible graphene-based spintronics applications.

Nature Physics, 2006

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 systems 1,2 , 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 plateaus 3-9 . 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. 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 e 2 /h towards the observable Hall conductivity (here e is the electron charge and h is Planck's constant). The conventional QHE is shown in , where plateaus in Hall conductivity σ xy make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σ xy is markedly absent . Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σ xx in bilayer graphene remains of the order of e 2 /h, even at zero σ xy . 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 3-9 , into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ε(p) = p 2 /2m with effective mass m, but

Modern Physics Letters B, 2012

We investigate the quantum Hall effect in graphene. We argue that in graphene in presence of an external magnetic field there is dynamical generation of mass by a rearrangement of the Dirac sea. We show that the mechanism breaks the lattice valley degeneracy only for the n = 0 Landau levels and leads to the new observed ν = ±1 quantum Hall plateaus. We suggest that our result can be tested by means of numerical simulations of planar Quantum Electro Dynamics with dynamical fermions in an external magnetic fields on the lattice.