Graphene is a rapidly rising star on the horizon of materials science and condensed-matter physics. This strictly two-dimensional material exhibits exceptionally high crystal and electronic quality, and, despite its short history, has already revealed a cornucopia of new physics and potential applications, which are briefly discussed here. Whereas one can be certain of the realness of applications only when commercial products appear, graphene no longer requires any further proof of its importance in terms of fundamental physics. Owing to its unusual electronic spectrum, graphene has led to the emergence of a new paradigm of 'relativistic' condensed-matter physics, where quantum relativistic phenomena, some of which are unobservable in high-energy physics, can now be mimicked and tested in table-top experiments. More generally, graphene represents a conceptually new class of materials that are only one atom thick, and, on this basis, offers new inroads into low-dimensional physics that has never ceased to surprise and continues to provide a fertile ground for applications.

The rise of graphene

This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

/pdf/the-electronic-properties-of-graphene-2siyzsnf5w.pdf

The electronic properties of graphene

Graphene devices on standard SiO(2) substrates are highly disordered, exhibiting characteristics that are far inferior to the expected intrinsic properties of graphene. Although suspending the graphene above the substrate leads to a substantial improvement in device quality, this geometry imposes severe limitations on device architecture and functionality. There is a growing need, therefore, to identify dielectrics that allow a substrate-supported geometry while retaining the quality achieved with a suspended sample. Hexagonal boron nitride (h-BN) is an appealing substrate, because it has an atomically smooth surface that is relatively free of dangling bonds and charge traps. It also has a lattice constant similar to that of graphite, and has large optical phonon modes and a large electrical bandgap. Here we report the fabrication and characterization of high-quality exfoliated mono- and bilayer graphene devices on single-crystal h-BN substrates, by using a mechanical transfer process. Graphene devices on h-BN substrates have mobilities and carrier inhomogeneities that are almost an order of magnitude better than devices on SiO(2). These devices also show reduced roughness, intrinsic doping and chemical reactivity. The ability to assemble crystalline layered materials in a controlled way permits the fabrication of graphene devices on other promising dielectrics and allows for the realization of more complex graphene heterostructures.

Boron nitride substrates for high-quality graphene electronics

/pdf/honeycomb-carbon-a-review-of-graphene-bjl5wl6ai8.pdf

Honeycomb Carbon: A Review of Graphene

Raman spectroscopy is an integral part of graphene research. It is used to determine the number and orientation of layers, the quality and types of edge, and the effects of perturbations, such as electric and magnetic fields, strain, doping, disorder and functional groups. This, in turn, provides insight into all sp(2)-bonded carbon allotropes, because graphene is their fundamental building block. Here we review the state of the art, future directions and open questions in Raman spectroscopy of graphene. We describe essential physical processes whose importance has only recently been recognized, such as the various types of resonance at play, and the role of quantum interference. We update all basic concepts and notations, and propose a terminology that is able to describe any result in literature. We finally highlight the potential of Raman spectroscopy for layered materials other than graphene.

/pdf/raman-spectroscopy-as-a-versatile-tool-for-studying-the-54mhwmcfzi.pdf

Raman spectroscopy as a versatile tool for studying the properties of graphene

There 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 plateaus3,4,5,6,7,8,9. 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.

/pdf/unconventional-quantum-hall-effect-and-berry-s-phase-of-2p-57wjgzjh0u.pdf

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

We derive an effective two-dimensional Hamiltonian to describe the low-energy electronic excitations of a graphite bilayer, which correspond to chiral quasiparticles with a parabolic dispersion exhibiting Berry phase $2\ensuremath{\pi}$. Its high-magnetic-field Landau-level spectrum consists of almost equidistant groups of fourfold degenerate states at finite energy and eight zero-energy states. This can be translated into the Hall conductivity dependence on carrier density, ${\ensuremath{\sigma}}_{xy}(N)$, which exhibits plateaus at integer values of $4{e}^{2}/h$ and has a double $8{e}^{2}/h$ step between the hole and electron gases across zero density, in contrast to $(4n+2){e}^{2}/h$ sequencing in a monolayer.

http://staff.ulsu.ru/moliver/ref/graphite/mcca06.pdf

Landau-level degeneracy and quantum Hall effect in a graphite bilayer.

A tight-binding model is used to calculate the band structure of bilayer graphene in the presence of a potential difference between the layers that opens a gap $\ensuremath{\Delta}$ between the conduction and valence bands. In particular, a self-consistent Hartree approximation is used to describe imperfect screening of an external gate, employed primarily to control the density $n$ of electrons on the bilayer, resulting in a potential difference between the layers and a density dependent gap $\ensuremath{\Delta}(n)$. We discuss the influence of a finite asymmetry gap $\ensuremath{\Delta}(0)$ at zero excess density, caused by the screening of an additional transverse electric field, on observations of the quantum Hall effect.

/pdf/asymmetry-gap-in-the-electronic-band-structure-of-bilayer-4u0n55413e.pdf

Asymmetry gap in the electronic band structure 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 energies. 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.

/pdf/the-electronic-properties-of-bilayer-graphene-1ewz99z0rw.pdf

The electronic properties of bilayer graphene.

Because of the chiral nature of electrons in a monolayer of graphite (graphene) one can expect weak antilocalization and a positive weak-field magnetoresistance in it. However, trigonal warping (which breaks $\mathbf{p}\ensuremath{\rightarrow}\ensuremath{-}\mathbf{p}$ symmetry of the Fermi line in each valley) suppresses antilocalization, while intervalley scattering due to atomically sharp scatterers in a realistic graphene sheet or by edges in a narrow wire tends to restore conventional negative magnetoresistance. We show this by evaluating the dependence of the magnetoresistance of graphene on relaxation rates associated with various possible ways of breaking a ``hidden'' valley symmetry of the system.

/pdf/weak-localization-magnetoresistance-and-valley-symmetry-in-5a8tfi205w.pdf

Edward McCann

Papers

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

Landau-level degeneracy and quantum Hall effect in a graphite bilayer.

Asymmetry gap in the electronic band structure of bilayer graphene

The electronic properties of bilayer graphene.

Weak-localization magnetoresistance and valley symmetry in graphene.