N. M. R. Peres

The electronic properties 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.

Fine structure constant defines visual transparency of graphene.

We show that the optical transparency of suspended graphene is defined by the fine structure constant, α = e/hc, the parameter that describes coupling between light and relativistic electrons and is traditionally associated with quantum electrodynamics rather than condensed matter physics. Despite being only one atom thick, graphene is found to absorb a significant (πα =2.3%) fraction of incident white light, which is a consequence of graphene’s unique electronic structure. This value translates directly into universal dynamic conductivity G =e/4h within a few % accuracy.There are few phenomena in condensed matter physics that are defined only by the fundamental constants and do not depend on material parameters. Examples are the resistivity quantum, h/e2 (h is Plancks constant and e the electron charge), that appears in a variety of transport experiments and the magnetic flux quantum, h/e, playing an important role in the physics of superconductivity. By and large, sophisticated facilities and special measurement conditions are required to observe any of these phenomena. We show that the opacity of suspended graphene is defined solely by the fine structure constant, a = e2/hc � 1/137 (where c is the speed of light), the parameter that describes coupling between light and relativistic electrons and that is traditionally associated with quantum electrodynamics rather than materials science. Despite being only one atom thick, graphene is found to absorb a significant (pa = 2.3%) fraction of incident white light, a consequence of graphenes unique electronic structure.

Field-Effect Tunneling Transistor Based on Vertical Graphene Heterostructures

Tunnel Barriers for Graphene Transistors Transistor operation for integrated circuits not only requires that the gate material has high-charge carrier mobility, but that there is also an effective way of creating a barrier to current flow so that the device can be switched off and not waste power. Graphene offers high carrier mobility, but the shape of its conduction and valence bands enables electron tunneling and makes it difficult to achieve low currents in an “off” state. Britnell et al. (p. 947, published online 2 February) have fabricated field-effect transistors in which a thin tunneling barrier created from a layered material—either hexagonal boron nitride or molybdenum disulfide—is sandwiched between graphene sheets. These devices exhibit on-off switching ratios of ≈50 and ≈10,000, respectively, at room temperature. Boron nitride or molybdenum disulfide layers sandwiched between graphene sheets act as tunneling barriers to minimize device leakage currents. An obstacle to the use of graphene as an alternative to silicon electronics has been the absence of an energy gap between its conduction and valence bands, which makes it difficult to achieve low power dissipation in the OFF state. We report a bipolar field-effect transistor that exploits the low density of states in graphene and its one-atomic-layer thickness. Our prototype devices are graphene heterostructures with atomically thin boron nitride or molybdenum disulfide acting as a vertical transport barrier. They exhibit room-temperature switching ratios of ≈50 and ≈10,000, respectively. Such devices have potential for high-frequency operation and large-scale integration.

Biased bilayer graphene: Semiconductor with a gap tunable by the electric field effect

We demonstrate that the electronic gap of a graphene bilayer can be controlled externally by applying a gate bias. From the magnetotransport data (Shubnikov-de Haas measurements of the cyclotron mass), and using a tight-binding model, we extract the value of the gap as a function of the electronic density. We show that the gap can be changed from zero to midinfrared energies by using fields of less, approximately < 1 V/nm, below the electric breakdown of SiO2. The opening of a gap is clearly seen in the quantum Hall regime.

Electronic properties of disordered two-dimensional carbon

Two-dimensional carbon, or graphene, is a semimetal that presents unusual low-energy electronic excitations described in terms of Dirac fermions. We analyze in a self-consistent way the effects of localized (impurities or vacancies) and extended (edges or grain boundaries) defects on the electronic and transport properties of graphene. On the one hand, point defects induce a finite elastic lifetime at low energies with the enhancement of the electronic density of states close to the Fermi level. Localized disorder leads to a universal, disorder independent, electrical conductivity at low temperatures, of the order of the quantum of conductance. The static conductivity increases with temperature and shows oscillations in the presence of a magnetic field. The graphene magnetic susceptibility is temperature dependent (unlike an ordinary metal) and also increases with the amount of defects. Optical transport properties are also calculated in detail. On the other hand, extended defects induce localized states near the Fermi level. In the absence of electron-hole symmetry, these states lead to a transfer of charge between the defects and the bulk, the phenomenon we call self-doping. The role of electron-electron interactions in controlling self-doping is also analyzed. We also discuss the integer and fractional quantum Hall effect in graphene, the role played by the edge states induced by a magnetic field, and their relation to the almost field independent surface states induced at boundaries. The possibility of magnetism in graphene, in the presence of short-range electron-electron interactions and disorder is also analyzed.

Tight-binding approach to uniaxial strain in graphene

We analyze the effect of tensional strain in the electronic structure of graphene. In the absence of electron-electron interactions, within linear elasticity theory, and a tight-binding approach, we observe that strain can generate a bulk spectral gap. However, this gap is critical, requiring threshold deformations in excess of 20% and only along preferred directions with respect to the underlying lattice. The gapless Dirac spectrum is robust for small and moderate deformations and the gap appears as a consequence of the merging of the two inequivalent Dirac points only under considerable deformations of the lattice. We discuss how strain-induced anisotropy and local deformations can be used as a means to affect transport characteristics and pinch off current flow in graphene devices.

Optical conductivity of graphene in the visible region of the spectrum

We compute the optical conductivity of graphene beyond the usual Dirac-cone approximation, giving results that are valid in the visible region of the conductivity spectrum. The effect of next-nearest-neighbor hopping is also discussed. Using the full expression for the optical conductivity, the transmission and reflection coefficients are given. We find that even in the optical regime the corrections to the Dirac-cone approximation are surprisingly small a few percent. Our results help in the interpretation of the experimental results reported by Nair et al. Science 320, 1308 2008.

Colloquium: The Transport properties of graphene: An Introduction

An introduction to the transport properties of graphene combining experimental results and theoretical analysis is presented. In the theoretical description simple intuitive models are used to illustrate important points on the transport properties of graphene. The concept of chirality, stemming from the massless Dirac nature of the low energy physics of the material, is shown to be instrumental in understanding its transport properties: the conductivity minimum, the electronic mobility, the effect of strain, the weak (anti-)localization, and the optical conductivity.

Graphene bilayer with a twist: electronic structure.

We consider a graphene bilayer with a relative small angle rotation between the layers--a stacking defect often seen in the surface of graphite--and calculate the electronic structure near zero energy in a continuum approximation. Contrary to what happens in an AB stacked bilayer and in accord with observations in epitaxial graphene, we find: (a) the low energy dispersion is linear, as in a single layer, but the Fermi velocity can be significantly smaller than the single-layer value; (b) an external electric field, perpendicular to the layers, does not open an electronic gap.

Electron Tunneling through Ultrathin Boron Nitride Crystalline Barriers

We investigate the electronic properties of ultrathin hexagonal boron nitride (h-BN) crystalline layers with different conducting materials (graphite, graphene, and gold) on either side of the barrier layer. The tunnel current depends exponentially on the number of h-BN atomic layers, down to a monolayer thickness. Conductive atomic force microscopy scans across h-BN terraces of different thickness reveal a high level of uniformity in the tunnel current. Our results demonstrate that atomically thin h-BN acts as a defect-free dielectric with a high breakdown field. It offers great potential for applications in tunnel devices and in field-effect transistors with a high carrier density in the conducting channel.We investigate the electronic properties of heterostructures based on ultrathin hexagonal boron nitride (h-BN) crystalline layers sandwiched between two layers of graphene as well as other conducting materials (graphite, gold). The tunnel conductance depends exponentially on the number of h-BN atomic layers, down to a monolayer thickness. Exponential behaviour of I-V characteristics for graphene/BN/graphene and graphite/BN/graphite devices is determined mainly by the changes in the density of states with bias voltage in the electrodes. Conductive atomic force microscopy scans across h-BN terraces of different thickness reveal a high level of uniformity in the tunnel current. Our results demonstrate that atomically thin h-BN acts as a defect-free dielectric with a high breakdown field; it offers great potential for applications in tunnel devices and in field-effect transistors with a high carrier density in the conducting channel.