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. The ideas underlying quantum electrodynamics also influence the theory of condensed matter, 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) 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* approximately 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 recent discovery of graphene has sparked much interest, thus far focused on the peculiar electronic structure of this material, in which charge carriers mimic massless relativistic particles. However, the physical structure of graphene--a single layer of carbon atoms densely packed in a honeycomb crystal lattice--is also puzzling. On the one hand, graphene appears to be a strictly two-dimensional material, exhibiting such a high crystal quality that electrons can travel submicrometre distances without scattering. On the other hand, perfect two-dimensional crystals cannot exist in the free state, according to both theory and experiment. This incompatibility can be avoided by arguing that all the graphene structures studied so far were an integral part of larger three-dimensional structures, either supported by a bulk substrate or embedded in a three-dimensional matrix. Here we report on individual graphene sheets freely suspended on a microfabricated scaffold in vacuum or air. These membranes are only one atom thick, yet they still display long-range crystalline order. However, our studies by transmission electron microscopy also reveal that these suspended graphene sheets are not perfectly flat: they exhibit intrinsic microscopic roughening such that the surface normal varies by several degrees and out-of-plane deformations reach 1 nm. The atomically thin single-crystal membranes offer ample scope for fundamental research and new technologies, whereas the observed corrugations in the third dimension may provide subtle reasons for the stability of two-dimensional crystals.

No abstract

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 impermeable to all gases and liquids 1-3 , and even such a small atom as hydrogen is not expected to penetrate through graphene's dense electronic cloud within billions of years 3-6 . Here we show that monolayers of graphene and hexagonal boron nitride (hBN) are unexpectedly permeable to thermal protons, hydrogen ions under ambient conditions. As a reference, no proton transport could be detected for a monolayer of molybdenum disulfide, bilayer graphene or multilayer hBN. At room temperature, monolayer hBN exhibits the highest proton conductivity with a low activation energy of 0.3 eV but graphene becomes a better conductor at elevated temperatures such that its resistivity to proton flow is estimated to fall below 10 -3 Ohm per cm 2 above 250°C. The proton barriers can be further reduced by decorating monolayers with catalytic nanoparticles. These atomically thin proton conductors could be of interest for many hydrogen-based technologies.Graphene has recently attracted renewed attention as an ultimately thin membrane that can be used for development of novel separation technologies (for review, see refs. 7,8). If perforated with atomic or nanometer accuracy, graphene may provide ultrafast and highly selective sieving of gases, liquids, ions, etc. 2,9-19 However, in its pristine state, graphene is absolutely impermeable for all atoms and molecules moving at thermal energies [1][2][3][4][5][6][7] . Theoretical estimates for the kinetic energy E required for an atom to penetrate through monolayer graphene vary significantly, depending on the employed model, but even the smallest literature value of 2.4 eV for atomic hydrogen 3-6 is 100 times larger than typical k B T which ensures essentially an impenetrable barrier (k B is the Boltzmann constant and T the temperature). Therefore, only accelerated atoms are capable of penetrating through the one atom thick crystal 20,21 . The same is likely to be valid for other two dimensional (2D) crystals 22,23 , although only graphene has so far been considered in this context. Protons can be considered as an intermediate case between electrons that tunnel relatively easily through atomically thin barriers 24 and small atoms. It has been calculated that E decreases by a factor of up to 2 if hydrogen is stripped of its electron 4,5 . Unfortunately,

We discuss a general approach to a realistic theory of the electronic structure in materials containing correlated d-or f-electrons. The main feature of this approach is the taking into account the energy dependence of the electron self-energy with the momentum dependence being neglected (local approximation). It allows us to consider such correlation effects as the "non-Fermi-step" form of the distribution function, the enhancement of the effective mass including "Kondo resonances", the appearance of the satellites in the electron spectra etc. To specify the form of the self-energy, it is useful to distinguish (according to the ratio of the on-site Coulomb energy U to the bandwidth W) three regimes -strong, moderate and weak correlations. In the case of strong interactions (U/W > 1 -"rare-earth system") the Hubbard-I approach is the most suitable. Starting from an exact atomic Green function with the constrained density matrix n mm ′ the band structure problem is formulated as the functional problem on n mm ′ for f-electrons and the standard LDA-functional for delocalized electrons. In the case of moderate correlations (U/W∼1 -"metal -insulator regime, Kondo systems") we start from the d= ∞ dynamical mean field iterative perturbation scheme (IPS) of G. Kotliar et. al. and also make use of our multiband atomic Green function for constrained n mm ′ . Finally for the weak interactions (U/W < 1 -"transition metals") the self-consistent diagrammatic fluctuationexchange (FLEX)-approach of N. Bickers and D. Scalapino is generalized to the realistic multiband case. We presents two-band, two-dimensional model calculations for all three regimes. A realistic calculation in Hubbard-I scheme with the exact solution of the on-site multielectron problem for f(d)-shells was performed for mixed-valence 4f compound TmSe, and for the classical Mott insulator NiO. 71.10.+x,74.20.Mn,74.72.Bk

To obtain an effective many-body model of graphene and related materials from first principles we calculate the partially screened frequency dependent Coulomb interaction. In graphene, the effective on-site (Hubbard) interaction is U00 = 9.3 eV in close vicinity to the critical value separating conducting graphene from an insulating phase emphasizing the importance of non-local Coulomb terms. The nearest-neighbor Coulomb interaction strength is computed to U01 = 5.5 eV. In the long wavelength limit, we find the effective background dielectric constant of graphite to be ǫ = 2.5 in very good agreement with experiment.

Density functional calculations of electronic structure, total energy, structural distortions, and magnetism for hydrogenated single-layer, bilayer, and multi-layer graphene are performed. It is found that hydrogen-induced magnetism can survives only at very low concentrations of hydrogen (single-atom regime) whereas hydrogen pairs with optimized structure are usually nonmagnetic. Chemisorption energy as a function of hydrogen concentration is calculated, as well as energy barriers for hydrogen binding and release. The results confirm that graphene can be perspective material for hydrogen storage. Difference between hydrogenation of graphene, nanotubes, and bulk graphite is discussed.