Graphene is one of the most interesting materials synthesized in the last years which two-dimensional nature along with the fascinating electronic properties attracts the great attention from the scientific community. Graphene has an unique electronic properties such as the linear dispersion law leading to zero effective mass for electrons and holes. It already applied as solar cell [1], liquid crystal device [2], molecular sensor [3] and nano-sized transistor prototype [4].

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.

Microfabrication of graphene devices used in many experimental studies currently relies on the fact that graphene crystallites can be visualized using optical microscopy if prepared on top of Si wafers with a certain thickness of SiO2. The authors study graphene’s visibility and show that it depends strongly on both thickness of SiO2 and light wavelength. They have found that by using monochromatic illumination, graphene can be isolated for any SiO2 thickness, albeit 300nm (the current standard) and, especially, ≈100nm are most suitable for its visual detection. By using a Fresnel-law-based model, they quantitatively describe the experimental data.

Low-field magnetoresistance is ubiquitous in low-dimensional metallic systems with high resistivity and well understood as arising due to quantum interference on self-intersecting diffusive trajectories. We have found that in graphene this weak-localization magnetoresistance is strongly suppressed and, in some cases, completely absent. The unexpected observation is attributed to mesoscopic corrugations of graphene sheets which can cause a dephasing effect similar to that of a random magnetic field.

Graphene is the two-dimensional building block for carbon allotropes of every other dimensionality. We show that its electronic structure is captured in its Raman spectrum that clearly evolves with the number of layers. The $D$ peak second order changes in shape, width, and position for an increasing number of layers, reflecting the change in the electron bands via a double resonant Raman process. The $G$ peak slightly down-shifts. This allows unambiguous, high-throughput, nondestructive identification of graphene layers, which is critically lacking in this emerging research area.

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.

We report free-standing atomic crystals that are strictly 2D and can be viewed as individual atomic planes pulled out of bulk crystals or as unrolled single-wall nanotubes. By using micromechanical cleavage, we have prepared and studied a variety of 2D crystals including single layers of boron nitride, graphite, several dichalcogenides, and complex oxides. These atomically thin sheets (essentially gigantic 2D molecules unprotected from the immediate environment) are stable under ambient conditions, exhibit high crystal quality, and are continuous on a macroscopic scale.

We report two-dimensional (2D) electron and hole gases induced at the surface of graphite by the electric field effect. The 2D gases reside within a few near-surface atomic layers and exhibit mobilities up to 15 000 and 60 000 cm(2)/V s at room and liquid-helium temperatures, respectively. The mobilities imply ballistic transport on mu m scale. Pronounced Shubnikov-de Haas oscillations reveal the existence of two types of charge carries in both electron and hole gases.

Quantum electrodynamics (resulting from the merger of quantum mechanics and relativity theory) has provided a clear understanding of phenomena ranging from particle physics to cosmology and from astrophysics to quantum chemistry 1 , 2 , 3 . The ideas underlying quantum electrodynamics also influence the theory of condensed matter 4 , 5 , but quantum relativistic effects are usually minute in the known experimental systems that can be described accurately by the non-relativistic Schrödinger equation. Here we report an experimental study of a condensed-matter system (graphene, a single atomic layer of carbon 6 , 7 ) in which electron transport is essentially governed by Dirac's (relativistic) equation. The charge carriers in graphene mimic relativistic particles with zero rest mass and have an effective ‘speed of light’ c * ≈ 10 6 m s -1 . Our study reveals a variety of unusual phenomena that are characteristic of two-dimensional Dirac fermions. In particular we have observed the following: first, graphene's conductivity never falls below a minimum value corresponding to the quantum unit of conductance, even when concentrations of charge carriers tend to zero; second, the integer quantum Hall effect in graphene is anomalous in that it occurs at half-integer filling factors; and third, the cyclotron mass m c of massless carriers in graphene is described by E = m c c * 2 . This two-dimensional system is not only interesting in itself but also allows access to the subtle and rich physics of quantum electrodynamics in a bench-top experiment.

The ability to control electronic properties of a material by externally applied voltage is at the heart of modern electronics. In many cases, it is the so-called electric field effect that allows one to vary the carrier concentration in a semiconductor device and, consequently, change an electric current through it. As the semiconductor industry is nearing the limits of performance improvements for the current technologies dominated by silicon, there is a constant search for new, non-traditional materials whose properties can be controlled by electric field. Most notable examples of such materials developed recently are organic conductors [1], oxides near a superconducting or magnetic phase transition [2] and carbon nanotubes [3-5]. Here, we describe another system of this kind - thin monocrystalline films of graphite - which exhibits a pronounced electric field effect, such that carriers in the conductive channel can be turned into either electrons or holes. The films remain metallic, continuous and of high quality down to a few atomic layers in thickness. The demonstrated ease of preparing such films of nearly macroscopic sizes and of their processing by standard microfabrication techniques, combined with submicron-scale ballistic transport even at room temperature, offer a new two-dimensional system controllable by electric-field doping and provide a realistic promise of device applications.

We describe monocrystalline graphitic films, which are a few atoms thick but are nonetheless stable under ambient conditions, metallic, and of remarkably high quality. The films are found to be a two-dimensional semimetal with a tiny overlap between valence and conductance bands, and they exhibit a strong ambipolar electric field effect such that electrons and holes in concentrations up to 1013 per square centimeter and with room-temperature mobilities of ∼10,000 square centimeters per volt-second can be induced by applying gate voltage.