Hofstadter Butterflies of Bilayer Graphene (original) (raw)

Experimental observation of the quantum Hall effect and Berry's phase in graphene

Nature, 2005

When electrons are confined in two-dimensional materials, quantum-mechanically enhanced transport phenomena such as the quantum Hall effect can be observed. Graphene, consisting of an isolated single atomic layer of graphite, is an ideal realization of such a two-dimensional system. However, its behaviour is expected to differ markedly from the well-studied case of quantum wells in conventional semiconductor interfaces. This difference arises from the unique electronic properties of graphene, which exhibits electron-hole degeneracy and vanishing carrier mass near the point of charge neutrality 1,2 . Indeed, a distinctive half-integer quantum Hall effect has been predicted 3-5 theoretically, as has the existence of a non-zero Berry's phase (a geometric quantum phase) of the electron wavefunction-a consequence of the exceptional topology of the graphene band structure 6,7 . Recent advances in micromechanical extraction and fabrication techniques for graphite structures 8-12 now permit such exotic two-dimensional electron systems to be probed experimentally. Here we report an experimental investigation of magneto-transport in a high-mobility single layer of graphene. Adjusting the chemical potential with the use of the electric field effect, we observe an unusual halfinteger quantum Hall effect for both electron and hole carriers in graphene. The relevance of Berry's phase to these experiments is confirmed by magneto-oscillations. In addition to their purely scientific interest, these unusual quantum transport phenomena may lead to new applications in carbon-based electronic and magneto-electronic devices.

Graphene physics via the Dirac oscillator in (2+1) dimensions

We show how the two-dimensional Dirac oscillator model can describe the dynamics of electrons in graphene. This model explains the origin of the left-handed chirality observed for electrons in monolayer and bilayer graphene. The relativistic dispersion relation observed for monolayer graphene is obtained directly from the energy spectrum, while the parabolic dispersion relation observed for the case of bilayer graphene is obtained in the non-relativistic limit. Additionally, if an external magnetic field is applied, the unusual Landau-level spectrum for monolayer graphene is obtained, but for bilayer graphene the model predicts the existence of a magnetic field-dependent gap. Finally, this model also leads to the existence of a chiral phase transition.

Properties of graphene: a theoretical perspective

Advances in Physics, 2010

The electronic properties of graphene, a two-dimensional crystal of carbon atoms, are exceptionally novel. For instance the low-energy quasiparticles in graphene behave as massless chiral Dirac fermions which has led to the experimental observation of many interesting effects similar to those predicted in the relativistic regime. Graphene also has immense potential to be a key ingredient of new devices such as single molecule gas sensors, ballistic transistors, and spintronic devices. Bilayer graphene, which consists of two stacked monolayers and where the quasiparticles are massive chiral fermions, has a quadratic low-energy band structure which generates very different scattering properties from those of the monolayer. It also presents the unique property that a tunable band gap can be opened and controlled easily by a top gate. These properties have made bilayer graphene a subject of intense interest.

The electronic properties of bilayer graphene

We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energy. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra-and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects. CONTENTS

Energy Spectrum and Quantum Hall Effect in Twisted Bilayer Graphene

Arxiv preprint arXiv:1202.4365, 2012

We investigate the electronic structure and the quantum Hall effect in twisted bilayer graphenes with various rotation angles in the presence of magnetic field. Using a low-energy approximation, which incorporates the rigorous interlayer interaction, we computed the energy spectrum and the quantized Hall conductivity in a wide range of magnetic field from the semi-classical regime to the fractal spectrum regime. In weak magnetic fields, the low-energy conduction band is quantized into electronlike and holelike Landau levels at energies below and above the van Hove singularity, respectively, and the Hall conductivity sharply drops from positive to negative when the Fermi energy goes through the transition point. In increasing magnetic field, the spectrum gradually evolves into a fractal band structure called Hofstadter's butterfly, where the Hall conductivity exhibits a nonmonotonic behavior as a function of Fermi energy. The typical electron density and magnetic field amplitude characterizing the spectrum monotonically decrease as the rotation angle is reduced, indicating that the rich electronic structure may be observed in a moderate condition.

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

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

Coexisting massive and massless Dirac fermions in symmetry-broken bilayer graphene

Charge carriers in bilayer graphene are widely believed to be massive Dirac fermions 1-3 that have a bandgap tunable by a transverse electric field 3,4 . However, a full transport gap, despite its importance for device applications, has not been clearly observed in gated bilayer graphene 5-7 , a longstanding puzzle. Moreover, the low-energy electronic structure of bilayer graphene is widely held to be unstable towards symmetry breaking either by structural distortions, such as twist 8-10 , strain 11,12 , or electronic interactions 7,13,14 that can lead to various ground states. Which effect dominates the physics at low energies is hotly debated. Here we show both by direct band-structure measurements and by calculations that a native imperfection of bilayer graphene, a distribution of twists whose size is as small as ∼0.1 • , is sufficient to generate a completely new electronic spectrum consisting of massive and massless Dirac fermions. The massless spectrum is robust against strong electric fields, and has a unusual topology in momentum space consisting of closed arcs having an exotic chiral pseudospin texture, which can be tuned by varying the charge density. The discovery of this unusual Dirac spectrum not only complements the framework of massive Dirac fermions, widely relevant to charge transport in bilayer graphene, but also supports the possibility of valley Hall transport 15 .

Two-Dimensional Gas of Massless Dirac Fermions in Graphene

Electronic properties of materials are commonly described by quasiparticles that behave as nonrelativistic electrons with a finite mass and obey the Schrödinger equation. Here we report a condensed matter system where electron transport is essentially governed by the Dirac equation and charge carriers mimic relativistic particles with zero mass and an effective "speed of light" c * ≈10 6 m/s. Our studies of graphene -a single atomic layer of carbon -have revealed a variety of unusual phenomena characteristic of two-dimensional (2D) Dirac fermions. In particular, we have observed that a) the integer quantum Hall effect in graphene is anomalous in that it occurs at halfinteger filling factors; b) graphene's conductivity never falls below a minimum value corresponding to the conductance quantum e 2 /h, even when carrier concentrations tend to zero; c) the cyclotron mass m c of massless carriers with energy E in graphene is described by equation E =m c c * 2 ; and d) Shubnikov-de Haas oscillations in graphene exhibit a phase shift of π due to Berry's phase.