Two-dimensional gas of massless Dirac fermions in graphene

Effect of disorder on transport in graphene

Quenched disorder in graphene is characterized by 5 constants and experiences the logarithmic renormalization even from the spatial scales smaller than the Fermi wavelength. We derive and solve renormalization group equations (RGEs) describing the system at such scales. At larger scales, we derive a nonlinear supermatrix sigma model completely describing localization and crossovers between different ensembles. The parameters of this sigma model are determined by the solutions of the RGEs.

Low-energy theory of disordered graphene

At low values of external doping, graphene displays a wealth of unconventional transport properties. Perhaps most strikingly, it supports a robust "metallic" regime, with universal conductance of the order of the conductance quantum. We here apply a combination of mean-field and bosonization methods to explore the large scale transport properties of the system. We find that, irrespective of the doping level, disordered graphene is subject to the common mechanisms of Anderson localization. However, at low doping a number of renormalization mechanisms conspire to protect the conductivity of the system, to an extend that strong localization may not be seen even at temperatures much smaller than those underlying present experimental work.

Friedel oscillations, impurity scattering, and temperature dependence of resistivity in graphene

We show that Friedel oscillations (FO) in grapehene are strongly affected by the chirality of electrons in this material. In particular, the FO of the charge density around an impurity show a faster (deltarho approximately r;{-3}) decay than in conventional 2D electron systems and do not contribute to a linear temperature-dependent correction to the resistivity. In contrast, the FO of the exchange field which surrounds atomically sharp defects breaking the hexagonal symmetry of the honeycomb lattice lead to a negative linear T dependence of the resistivity.

Raman scattering from high-frequency phonons in supported n-graphene layer films

Results of room-temperature Raman scattering studies of ultrathin graphitic films supported on Si (100)/SiO2 substrates are reported. The results are significantly different from those known for graphite. Spectra were collected using 514.5 nm radiation on films containing from n = 1 to 20 graphene layers, as determined by atomic force microscopy. Both the first- and second-order Raman spectra show unique signatures of the number of layers in the film. The nGL film analogue of the Raman G-band in graphite exhibits a Lorentzian line shape whose center frequency shifts linearly relative to graphite as approximately 1/n (for n = 1 omegaG approximately 1587 cm-1). Three weak bands, identified with disorder-induced first-order scattering, are observed at approximately 1350, 1450, and 1500 cm-1. The approximately 1500 cm-1 band is weak but relatively sharp and exhibits an interesting n-dependence. In general, the intensity of these D-bands decreases dramatically with increasing n. Three second-order bands are also observed (approximately 2450, approximately 2700, and 3248 cm-1). They are analogues to those observed in graphite. However, the approximately 2700 cm-1 band exhibits an interesting and dramatic change of shape with n. Interestingly, for n < 5 this second-order band is more intense than the G-band.

Energy gaps in graphene nanoribbons

Based on a first-principles approach, we present scaling rules for the band gaps of graphene nanoribbons (GNRs) as a function of their widths. The GNRs considered have either armchair or zigzag shaped edges on both sides with hydrogen passivation. Both varieties of ribbons are shown to have band gaps. This differs from the results of simple tight-binding calculations or solutions of the Dirac's equation based on them. Our ab initio calculations show that the origin of energy gaps for GNRs with armchair shaped edges arises from both quantum confinement and the crucial effect of the edges. For GNRs with zigzag shaped edges, gaps appear because of a staggered sublattice potential on the hexagonal lattice due to edge magnetization. The rich gap structure for ribbons with armchair shaped edges is further obtained analytically including edge effects. These results reproduce our ab initio calculation results very well.

Tunneling conductance of graphene NIS junctions

We show that, in contrast with conventional normal metal-insulator-superconductor (NIS) junctions, the tunneling conductance of a NIS junction in graphene is an oscillatory function of the effective barrier strength of the insulating region, in the limit of a thin barrier. The amplitude of these oscillations is maximum for aligned Fermi surfaces of the normal and superconducting regions and vanishes for a large Fermi surface mismatch. The zero-bias tunneling conductance, in sharp contrast to its counterpart in conventional NIS junctions, becomes maximum for a finite barrier strength. We also suggest experiments to test these predictions.

Half-metallic graphene nanoribbons

Electrical current can be completely spin polarized in a class of materials known as half-metals, as a result of the coexistence of metallic nature for electrons with one spin orientation and insulating nature for electrons with the other. Such asymmetric electronic states for the different spins have been predicted for some ferromagnetic metals--for example, the Heusler compounds--and were first observed in a manganese perovskite. In view of the potential for use of this property in realizing spin-based electronics, substantial efforts have been made to search for half-metallic materials. However, organic materials have hardly been investigated in this context even though carbon-based nanostructures hold significant promise for future electronic devices. Here we predict half-metallicity in nanometre-scale graphene ribbons by using first-principles calculations. We show that this phenomenon is realizable if in-plane homogeneous electric fields are applied across the zigzag-shaped edges of the graphene nanoribbons, and that their magnetic properties can be controlled by the external electric fields. The results are not only of scientific interest in the interplay between electric fields and electronic spin degree of freedom in solids but may also open a new path to explore spintronics at the nanometre scale, based on graphene.

Intervalley scattering, long-range disorder, and effective time-reversal symmetry breaking in graphene

We discuss the effect of certain types of static disorder, like that induced by curvature or topological defects, on the quantum correction to the conductivity in graphene. We find that when the intervalley scattering time is long or comparable to tau(phi), these defects can induce an effective time-reversal symmetry breaking of the Hamiltonian associated to each one of the two valleys in graphene. The phenomenon suppresses the magnitude of the quantum correction to the conductivity and may result in the complete absence of a low-field magnetoresistance, as recently found experimentally. Our work shows that a quantitative description of weak localization in graphene must include the analysis of new regimes, not present in conventional two-dimensional electron gases.

Raman spectrum of graphene and graphene layers

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.

Interactions and phase transitions on graphene's honeycomb lattice

The low-energy theory of interacting electrons on graphene's two-dimensional honeycomb lattice is derived and discussed. In particular, the Hubbard model in the large-N limit is shown to have a semimetal-antiferromagnetic insulator quantum critical point in the universality class of the Gross-Neveu model. The same equivalence is conjectured to hold in the physical case N=2, and its consequences for various physical quantities are examined. The effects of the long-range Coulomb interaction and the magnetic field are discussed.

Weak-localization magnetoresistance and valley symmetry in 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 p-->-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.

Fractional quantum Hall States of Dirac electrons in graphene

We have investigated the fractional quantum Hall states of Dirac electrons in a graphene layer in different Landau levels. The relativistic nature of the energy dispersion relation of electrons in graphene significantly modifies the interelectron interactions. This results in a specific dependence of the ground state energy and the energy gaps for electrons on the Landau-level index. For the valley-polarized states, i.e., at nu=1/m, m being an odd integer, the energy gaps have the largest values in the n=1 Landau level. For the valley-unpolarized states, e.g., for the 2/3 state, the energy gaps are suppressed for n=1 as compared to those at n=0. For both n=1 and n=0, the ground state of the 2/3 system is fully valley-unpolarized.

Luttinger liquid at the edge of undoped graphene in a strong magnetic field

We demonstrate that an undoped two-dimensional carbon plane (graphene) whose bulk is in the integer quantum Hall regime supports a nonchiral Luttinger liquid at an armchair edge. This behavior arises due to the unusual dispersion of the noninteracting edge states, causing a crossing of bands with different valley and spin indices at the edge. We demonstrate that this stabilizes a domain wall structure with a spontaneously ordered phase degree of freedom. This coherent domain wall supports gapless charged excitations, and has a power law tunneling I-V with a nonintegral exponent. In proximity to a bulk lead, the edge may undergo a quantum phase transition between the Luttinger liquid phase and a metallic state.

Charge and spin Hall conductivity in metallic graphene

Graphene has an unusual low-energy band structure with four chiral bands and half-quantized and quantized Hall effects that have recently attracted theoretical and experimental attention. We study the Fermi energy and disorder dependence of its spin Hall conductivity sigma(xy)(SH). In the metallic regime we find that vertex corrections enhance the intrinsic spin Hall conductivity and that skew scattering can lead to sigma(xy)(SH) values that exceed the quantized ones expected when the chemical potential is inside the spin-orbit induced energy gap. We predict that large spin Hall conductivities will be observable in graphene even when the spin-orbit gap does not survive disorder.

Electrostatic force microscopy on oriented graphite surfaces: coexistence of insulating and conducting behaviors

We present measurements of the electric potential fluctuations on the surface of highly oriented pyrolytic graphite using electrostatic force and atomic force microscopy. Micrometric domainlike potential distributions are observed even when the sample is grounded. Such potential distributions are unexpected given the good metallic conductivity of graphite because the surface should be an equipotential. Our results indicate the coexistence of regions with "metalliclike" and "insulatinglike" behaviors showing large potential fluctuations of the order of 0.25 V. In lower quality graphite, this effect is not observed. Experiments are performed in Ar and air atmospheres.

Controlling the Electronic Structure of Bilayer Graphene

We describe the synthesis of bilayer graphene thin films deposited on insulating silicon carbide and report the characterization of their electronic band structure using angle-resolved photoemission. By selectively adjusting the carrier concentration in each layer, changes in the Coulomb potential led to control of the gap between valence and conduction bands. This control over the band structure suggests the potential application of bilayer graphene to switching functions in atomic-scale electronic devices.

Graphene-based composite materials

Graphene sheets--one-atom-thick two-dimensional layers of sp2-bonded carbon--are predicted to have a range of unusual properties. Their thermal conductivity and mechanical stiffness may rival the remarkable in-plane values for graphite (approximately 3,000 W m(-1) K(-1) and 1,060 GPa, respectively); their fracture strength should be comparable to that of carbon nanotubes for similar types of defects; and recent studies have shown that individual graphene sheets have extraordinary electronic transport properties. One possible route to harnessing these properties for applications would be to incorporate graphene sheets in a composite material. The manufacturing of such composites requires not only that graphene sheets be produced on a sufficient scale but that they also be incorporated, and homogeneously distributed, into various matrices. Graphite, inexpensive and available in large quantity, unfortunately does not readily exfoliate to yield individual graphene sheets. Here we present a general approach for the preparation of graphene-polymer composites via complete exfoliation of graphite and molecular-level dispersion of individual, chemically modified graphene sheets within polymer hosts. A polystyrene-graphene composite formed by this route exhibits a percolation threshold of approximately 0.1 volume per cent for room-temperature electrical conductivity, the lowest reported value for any carbon-based composite except for those involving carbon nanotubes; at only 1 volume per cent, this composite has a conductivity of approximately 0.1 S m(-1), sufficient for many electrical applications. Our bottom-up chemical approach of tuning the graphene sheet properties provides a path to a broad new class of graphene-based materials and their use in a variety of applications.

Specular Andreev reflection in graphene

By combining the Dirac equation of relativistic quantum mechanics with the Bogoliubov-de Gennes equation of superconductivity we investigate the electron-hole conversion at a normal-metal-superconductor interface in graphene. We find that the Andreev reflection of Dirac fermions has several unusual features: (1) the electron and hole occupy different valleys of the band structure; (2) at normal incidence the electron-hole conversion happens with unit efficiency in spite of the large mismatch in Fermi wavelengths at the two sides of the interface; and, most fundamentally: (3) away from normal incidence the reflection angle may be the same as the angle of incidence (retroreflection) or it may be inverted (specular reflection). Specular Andreev reflection dominates in weakly doped graphene, when the Fermi wavelength in the normal region is large compared to the superconducting coherence length.

Electron localization properties in graphene

We study localization properties of the Dirac-like electronic states in monolayers of graphite. In the framework of a general disorder model, we discuss the conditions under which such standard localization effects as the logarithmic temperature-dependent conductivity correction appear to be strongly suppressed, as compared to the case of a two-dimensional electron gas with parabolic dispersion, in agreement with recent experimental observations.

Charge carriers in few-layer graphene films

The nature of the charge carriers in 2D few-layer graphites (FLGs) has been recently questioned by transport measurements [K. S. Novoselov, Science 306, 666 (2004)10.1126/science.1102896] and a strong ambipolar electric field effect has been revealed. Our density functional calculations demonstrate that the electronic band dispersion near the Fermi level, and consequently the nature of the charge carriers, is highly sensitive to the number of layers and the stacking geometry. We show that the experimentally observed ambipolar transport is only possible for an FLG with a Bernal-like stacking pattern, whereas simple-carrier or semiconducting behavior is predicted for other geometries.

Strong suppression of weak localization in graphene

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.

Quantum Hall ferromagnetism in graphene

Graphene is a two-dimensional carbon material with a honeycomb lattice and Dirac-like low-energy excitations. When Zeeman and spin-orbit interactions are neglected, its Landau levels are fourfold degenerate, explaining the 4e2/h separation between quantized Hall conductivity values seen in recent experiments. In this Letter we derive a criterion for the occurrence of interaction-driven quantum Hall effects near intermediate integer values of e2/h due to charge gaps in broken symmetry states.

Unusual microwave response of dirac quasiparticles in graphene

Recent experiments have proven that the quasiparticles in graphene obey a Dirac equation. Here we show that microwaves are an excellent probe of their unusual dynamics. When the chemical potential is small, the intraband response can exhibit a cusp around zero frequency Omega and this unusual line shape changes to Drude-like by increasing the chemical potential |mu|, with width linear in mu. The interband contribution at T=0 is a constant independent of Omega with a lower cutoff at 2mu. Distinctly different behavior occurs if interaction-induced phenomena in graphene cause an opening of a gap Delta. At a large magnetic field B, the diagonal and Hall conductivities at small Omega become independent of B but remain nonzero and show a structure associated with the lowest Landau level. This occurs because in the Dirac theory the energy of this level, E0 = +/-Delta, is field independent in sharp contrast to the conventional case.

Sub-Poissonian shot noise in graphene

We calculate the mode-dependent transmission probability of massless Dirac fermions through an ideal strip of graphene (length L, width W, no impurities or defects) to obtain the conductance and shot noise as a function of Fermi energy. We find that the minimum conductivity of order e2/h at the Dirac point (when the electron and hole excitations are degenerate) is associated with a maximum of the Fano factor (the ratio of noise power and mean current). For short and wide graphene strips the Fano factor at the Dirac point equals 1/3, 3 times smaller than for a Poisson process. This is the same value as for a disordered metal, which is remarkable since the classical dynamics of the Dirac fermions is ballistic.

Electronic Confinement and Coherence in Patterned Epitaxial Graphene

Ultrathin epitaxial graphite was grown on single-crystal silicon carbide by vacuum graphitization. The material can be patterned using standard nanolithography methods. The transport properties, which are closely related to those of carbon nanotubes, are dominated by the single epitaxial graphene layer at the silicon carbide interface and reveal the Dirac nature of the charge carriers. Patterned structures show quantum confinement of electrons and phase coherence lengths beyond 1 micrometer at 4 kelvin, with mobilities exceeding 2.5 square meters per volt-second. All-graphene electronically coherent devices and device architectures are envisaged.

Spin-filtered edge states and quantum Hall effect in graphene

Electron edge states in graphene in the quantum Hall effect regime can carry both charge and spin. We show that spin splitting of the zeroth Landau level gives rise to counterpropagating modes with opposite spin polarization. These chiral spin modes lead to a rich variety of spin current states, depending on the spin-flip rate. A method to control the latter locally is proposed. We estimate Zeeman spin splitting enhanced by exchange, and obtain a spin gap of a few hundred Kelvin.

Functionalized single graphene sheets derived from splitting graphite oxide

A process is described to produce single sheets of functionalized graphene through thermal exfoliation of graphite oxide. The process yields a wrinkled sheet structure resulting from reaction sites involved in oxidation and reduction processes. The topological features of single sheets, as measured by atomic force microscopy, closely match predictions of first-principles atomistic modeling. Although graphite oxide is an insulator, functionalized graphene produced by this method is electrically conducting.

Landau-level splitting in graphene in high magnetic fields

The quantum Hall (QH) effect in two-dimensional electrons and holes in high quality graphene samples is studied in strong magnetic fields up to 45 T. QH plateaus at filling factors nu = 0, +/-1, +/-4 are discovered at magnetic fields B > 20 T, indicating the lifting of the fourfold degeneracy of the previously observed QH states at nu = +/-4(absolute value(n) + 1/2), where n is the Landau-level index. In particular, the presence of the nu = 0, +/-1 QH plateaus indicates that the Landau level at the charge neutral Dirac point splits into four sublevels, lifting sublattice and spin degeneracy. The QH effect at nu = +/-4 is investigated in a tilted magnetic field and can be attributed to lifting of the spin degeneracy of the n = 1 Landau level.

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

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 2pi. 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, sigma(xy)(N), which exhibits plateaus at integer values of 4e2/h and has a double 8e2/h step between the hole and electron gases across zero density, in contrast to (4n + 2)e2/h sequencing in a monolayer.