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

Quasiparticle scattering off phase boundaries in epitaxial graphene

We investigate the electronic structure of terraces of single layer graphene (SLG) by scanning tunnelling microscopy (STM) on samples grown by thermal decomposition of 6H-SiC(0001) crystals in ultra-high vacuum. We focus on the perturbations of the local density of states (LDOS) in the vicinity of edges of SLG terraces. Armchair edges are found to favour intervalley quasiparticle scattering, leading to the (√3 x √3)R30° LDOS superstructure already reported for graphite edges and more recently for SLG on SiC(0001). Using the Fourier transform of LDOS images, we demonstrate that the intrinsic doping of SLG is responsible for a LDOS pattern at the Fermi energy which is more complex than for neutral graphene or graphite, since it combines local (√3 x √3)R30° superstructure and long range beating modulation. Although these features have already been reported by Yang et al (2010 Nano Lett. 10 943-7) we propose here an alternative interpretation based on simple arguments classically used to describe standing wave patterns in standard two-dimensional systems. Finally, we discuss the absence of intervalley scattering off other typical boundaries: zig-zag edges and SLG/bilayer graphene junctions.

Graphene: Piecing it together

Graphene has a multitude of striking properties that make it an exceedingly attractive material for various applications, many of which will emerge over the next decade. However, one of the most promising applications lie in exploiting its peculiar electronic properties which are governed by its electrons obeying a linear dispersion relation. This leads to the observation of half integer quantum hall effect and the absence of localization. The latter is attractive for graphene-based field effect transistors. However, if graphene is to be the material for future electronics, then significant hurdles need to be surmounted, namely, it needs to be mass produced in an economically viable manner and be of high crystalline quality with no or virtually no defects or grains boundaries. Moreover, it will need to be processable with atomic precision. Hence, the future of graphene as a material for electronic based devices will depend heavily on our ability to piece graphene together as a single crystal and define its edges with atomic precision. In this progress report, the properties of graphene that make it so attractive as a material for electronics is introduced to the reader. The focus then centers on current synthesis strategies for graphene and their weaknesses in terms of electronics applications are highlighted.

Transport behavior and negative magnetoresistance in chemically reduced graphene oxide nanofilms

The electron transport behavior in chemically reduced graphene oxide (rGO) sheets with different thicknesses of 2, 3, and 5 nm was investigated. The four-probe method for the sheet resistance (R(S)) measurement on the intensively reduced graphene oxide samples indicates an Arrhenius characteristic of the electron transport at zero magnetic field B = 0, consistent with previous experimental results on well-reduced GO samples. The anticipated variable range hopping (VRH) transport of electrons in a two-dimensional electron system at low temperatures was not observed. The measured R(S) of the rGO samples are below 52 kΩ/square at room temperature. With the application of a magnetic field up to 4 T, negative magnetoresistance in the Mott VRH regime was observed. The magnetotransport features support a model based on the spin-coupling effect from the vacancy-induced midgap states that facilitate the Mott VRH conduction in the presence of an external magnetic field.

Magnetism-dependent transport phenomena in hydrogenated graphene: from spin-splitting to localization effects

Spin-dependent transport in hydrogenated two-dimensional graphene is explored theoretically. Adsorbed atomic hydrogen impurities can either induce a local antiferromagnetic, ferromagnetic, or nonmagnetic state depending on their density and relative distribution. To describe the various magnetic possibilities of hydrogenated graphene, a self-consistent Hubbard Hamiltonian, optimized by ab initio calculations, is first solved in the mean field approximation for small graphene cells. Then, an efficient order N Kubo transport methodology is implemented, enabling large scale simulations of functionalized graphene. Depending on the underlying intrinsic magnetic ordering of hydrogen-induced spins, remarkably different transport features are predicted for the same impurity concentration. Indeed, while the disordered nonmagnetic graphene system exhibits a transition from diffusive to localization regimes, the intrinsic ferromagnetic state exhibits unprecedented robustness toward quantum interference, maintaining, for certain resonant energies, a quasiballistic regime up to the micrometer scale. Consequently, low temperature transport measurements could unveil the presence of a magnetic state in weakly hydrogenated graphene.

Characteristics of level-spacing statistics in chaotic graphene billiards

A fundamental result in nonrelativistic quantum nonlinear dynamics is that the spectral statistics of quantum systems that possess no geometric symmetry, but whose classical dynamics are chaotic, are described by those of the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE), in the presence or absence of time-reversal symmetry, respectively. For massless spin-half particles such as neutrinos in relativistic quantum mechanics in a chaotic billiard, the seminal work of Berry and Mondragon established the GUE nature of the level-spacing statistics, due to the combination of the chirality of Dirac particles and the confinement, which breaks the time-reversal symmetry. A question is whether the GOE or the GUE statistics can be observed in experimentally accessible, relativistic quantum systems. We demonstrate, using graphene confinements in which the quasiparticle motions are governed by the Dirac equation in the low-energy regime, that the level-spacing statistics are persistently those of GOE random matrices. We present extensive numerical evidence obtained from the tight-binding approach and a physical explanation for the GOE statistics. We also find that the presence of a weak magnetic field switches the statistics to those of GUE. For a strong magnetic field, Landau levels become influential, causing the level-spacing distribution to deviate markedly from the random-matrix predictions. Issues addressed also include the effects of a number of realistic factors on level-spacing statistics such as next nearest-neighbor interactions, different lattice orientations, enhanced hopping energy for atoms on the boundary, and staggered potential due to graphene-substrate interactions.

Two-dimensional graphene with structural defects: elastic mean free path, minimum conductivity, and Anderson transition

Quantum transport properties of disordered graphene with structural defects (Stone-Wales and divacancies) are investigated using a realistic π-π* tight-binding model elaborated from ab initio calculations. Mean free paths and semiclassical conductivities are then computed as a function of the nature and density of defects (using an order-N real-space Kubo-Greenwood method). By increasing the defect density, the decay of the semiclassical conductivities is predicted to saturate to a minimum value of 4e2/πh over a large range (plateau) of carrier density (>0.5×10(14)  cm(-20). Additionally, strong contributions of quantum interferences suggest that the Anderson localization regime could be experimentally measurable for a defect density as low as 1%.

Disorder and electronic transport in graphene

In this review, we provide an account of the recent progress in understanding electronic transport in disordered graphene systems. Starting from a theoretical description that emphasizes the role played by band structure properties and lattice symmetries, we describe the nature of disorder in these systems and its relation to transport properties. While the focus is primarily on theoretical and conceptual aspects, connections to experiments are also included. Issues such as short- versus long-range disorder, localization (strong and weak), the carrier density dependence of the conductivity, and conductance fluctuations are considered and some open problems are pointed out.

Electronic properties of a biased graphene bilayer

We study, within the tight-binding approximation, the electronic properties of a graphene bilayer in the presence of an external electric field applied perpendicular to the system-a biased bilayer. The effect of the perpendicular electric field is included through a parallel plate capacitor model, with screening correction at the Hartree level. The full tight-binding description is compared with its four-band and two-band continuum approximations, and the four-band model is shown to always be a suitable approximation for the conditions realized in experiments. The model is applied to real biased bilayer devices, made out of either SiC or exfoliated graphene, and good agreement with experimental results is found, indicating that the model is capturing the key ingredients, and that a finite gap is effectively being controlled externally. Analysis of experimental results regarding the electrical noise and cyclotron resonance further suggests that the model can be seen as a good starting point for understanding the electronic properties of graphene bilayer. Also, we study the effect of electron-hole asymmetry terms, such as the second-nearest-neighbour hopping energies t' (in-plane) and γ(4) (inter-layer), and the on-site energy Δ.

Transition between electron localization and antilocalization in graphene

We show that quantum interference in graphene can result in antilocalization of charge carriers--an increase of the conductance, which is detected by a negative magnetoconductance. We demonstrate that depending on experimental conditions one can observe either weak localization or antilocalization of carriers in graphene. A transition from localization to antilocalization occurs when the carrier density is decreased and the temperature is increased. We show that quantum interference in graphene can survive at high temperatures, up to T approximately 200 K, due to weak electron-phonon scattering.

Regular conductance fluctuations indicative of quasi-ballistic transport in bilayer graphene

Quasi-periodic conductance fluctuations are observed in the low-temperature magneto-conductance of a bilayer graphene sample. The quasi-periodic nature of the fluctuations is confirmed by their Fourier power spectrum, which consists of just a small number of dominant frequency components. From an experimental study of these features, which are highly reminiscent of those reported previously for ballistic semiconductor quantum dots, we suggest that they are associated with the formation of an open quantum dot in the submicron graphene sample.

Carbon nanoelectronics: unzipping tubes into graphene ribbons

We report on the transport properties of novel carbon nanostructures made of partially unzipped carbon nanotubes, which can be regarded as a seamless junction of a tube and a nanoribbon. We find that graphene nanoribbons act at certain energy ranges as perfect valley filters for carbon nanotubes, with the maximum possible conductance. Our results show that a partially unzipped carbon nanotube is a magnetoresistive device, with a very large value of magnetoresistance. We explore the properties of several structures combining nanotubes and graphene nanoribbons, demonstrating that they behave as optimal contacts for each other, and opening a new route for the design of mixed graphene-nanotube devices.

Templated growth of graphenic materials

A novel strategy is proposed for the topologically controlled synthesis of extended graphenic sheets by additively reacting carbon into a pre-existing graphene sheet which is on top of a templating substrate. This concept is implemented and demonstrated using chemical vapor deposition (CVD). Novel morphological features observed in this study suggest unusual aspects of the CVD growth process. CVD results demonstrate the basic soundness of the synthesis strategy but highlight the sensitivity of the process to certain types of disruption and the need for alternative forms of embodiment.

The formation of atomic nanoclusters on graphene sheets

The formation of atomic nanoclusters on suspended graphene sheets has been investigated by employing a molecular dynamics simulation at finite temperature. Our systematic study is based on temperature-dependent molecular dynamics simulations of some transition and alkali atoms on suspended graphene sheets. We find that the transition atoms aggregate and make various size nanoclusters distributed randomly on graphene surfaces. We also report that most alkali atoms make one atomic layer on graphene sheets. Interestingly, the potassium atoms almost deposit regularly on the surface at low temperature. We expect from this behavior that the electrical conductivity of a suspended graphene doped by potassium atoms would be much higher than in the case doped by the other atoms at low temperature.

Localization and the Kosterlitz-Thouless transition in disordered graphene

We investigate disordered graphene with strong long-range impurities. Contrary to the common belief that delocalization should persist in such a system against any disorder, as the system is expected to be equivalent to a disordered two-dimensional Dirac fermionic system, we find that states near the Dirac points are localized for sufficiently strong disorder (therefore inevitable intervalley scattering) and the transition between the localized and delocalized states is of Kosterlitz-Thouless type. Our results show that the transition originates from bounding and unbounding of local current vortices.

Edge states in graphene: from gapped flat-band to gapless chiral modes

We study edge states in graphene systems where a bulk energy gap is opened by inversion symmetry breaking. We find that the edge bands dispersion can be controlled by potentials applied on the boundary with unit cell length scale. Under certain boundary potentials, gapless edge states with valley-dependent velocity are found, exactly analogous to the spin-dependent gapless chiral edge states in quantum spin Hall systems. The connection of the edge states to bulk topological properties is revealed.

Massive Dirac fermions in single-layer graphene

Motivated by the results of recent photoemission and tunneling studies, we discuss potential many-body sources of a finite gap in the Dirac fermion spectrum of graphene. Specifically, we focus on the putative Peierls- and Cooper-like pairing instabilities, which can be driven by sufficiently strong Coulomb and electron-phonon interactions, respectively. Our results compare favorably with the available experimental and Monte Carlo data.

Symmetry classes in graphene quantum dots: universal spectral statistics, weak localization, and conductance fluctuations

We study the symmetry classes of graphene quantum dots, both open and closed, through the conductance and energy level statistics. For abrupt termination of the lattice, these properties are well described by the standard orthogonal and unitary ensembles. However, for smooth mass confinement, special time-reversal symmetries associated with the sublattice and valley degrees of freedom are critical: they lead to block diagonal Hamiltonians and scattering matrices with blocks belonging to the unitary symmetry class even at zero magnetic field. While the effect of this structure is clearly seen in the conductance of open dots, it is suppressed in the spectral statistics of closed dots, because the intervalley scattering time is shorter than the time required to resolve a level spacing in the closed systems but longer than the escape time of the open systems.

Nonequilibrium effective vector potential due to pseudospin exchange in graphene

We show that exchange interactions in two-dimensional electron gases out of equilibrium can generate a fictitious vector potential with intriguing signatures in interference and Hall measurements. Detailed predictions are made for graphene, where the effect is enhanced by pseudospin exchange.

Phonon-mediated tunneling into graphene

Recent scanning tunneling spectroscopy experiments on graphene reported an unexpected gap of about +/-60 meV around the Fermi level [V. W. Brar, Appl. Phys. Lett. 91, 122102 (2007); 10.1063/1.2771084Y. Zhang, Nature Phys. 4, 627 (2008)10.1038/nphys1022]. Here we give a theoretical investigation explaining the experimentally observed spectra and confirming the phonon-mediated tunneling as the reason for the gap: We study the real space properties of the wave functions involved in the tunneling process by means of ab initio theory and present a model for the electron-phonon interaction, which couples the graphene's Dirac electrons with quasifree-electron states at the Brillouin zone center. The self-energy associated with this electron-phonon interaction is calculated, and its effects on tunneling into graphene are discussed. Good agreement of the tunneling density of states within our model and the experimental dI/dU spectra is found.

Proposal for a graphene-based current nanoswitch

We propose a graphene-based model nanoswitch for the control of current flow. The device is made up of a graphene nanoribbon with a longitudinal semi-infinite septum on its left side. The septum defines two terminals, where the current can be selectively directed by means of a high magnetic field and an adjustable potential step induced by a top gate. The device's working mechanism is based on the spatial chirality of currents in high magnetic fields and on the possibility of determining the electron-like or hole-like nature of the particles involved in transport by means of the adjustable potential step. The role of critical parameters such as the width of the ribbon and the presence of disorder are discussed.

Quantum Hall effect of massless dirac fermions in a vanishing magnetic field

The effect of strong long-range disorder on the quantization of the Hall conductivity sigma{xy} in graphene is studied numerically. It is shown that increasing Landau-level mixing progressively destroys all plateaus in sigma{xy} except the plateaus at sigma{xy}=-/+e{2}/2h (per valley and per spin). The critical state at the Dirac point is robust to strong disorder and belongs to the universality class of the conventional plateau transitions in the integer quantum Hall effect. We propose that the breaking of time-reversal symmetry by ripples in graphene can realize this quantum critical point in a vanishing magnetic field.

Shot noise in ballistic graphene

We have investigated shot noise in graphene field effect devices in the temperature range of 4.2-30 K at low frequency (f=600-850 MHz). We find that for our graphene samples with a large width over length ratio W/L, the Fano factor F reaches a maximum F ~ 1/3 at the Dirac point and that it decreases strongly with increasing charge density. For smaller W/L, the Fano factor at Dirac point is significantly lower. Our results are in good agreement with the theory describing that transport at the Dirac point in clean graphene arises from evanescent electronic states.

Imaging mechanical vibrations in suspended graphene sheets

We carried out measurements on nanoelectromechanical systems based on multilayer graphene sheets suspended over trenches in silicon oxide. The motion of the suspended sheets was electrostatically driven at resonance using applied radio frequency voltages. The mechanical vibrations were detected using a novel form of scanning probe microscopy, which allowed identification and spatial imaging of the shape of the mechanical eigenmodes. In as many as half the resonators measured, we observed a new class of exotic nanoscale vibration eigenmodes not predicted by the elastic beam theory, where the amplitude of vibration is maximum at the free edges. By modeling the suspended sheets with the finite element method, these edge eigenmodes are shown to be the result of nonuniform stress with remarkably large magnitudes (up to 1.5 GPa). This nonuniform stress, which arises from the way graphene is prepared by pressing or rubbing bulk graphite against another surface, should be taken into account in future studies on electronic and mechanical properties of graphene.

Weak localization in graphene flakes

We show that the manifestation of quantum interference in graphene is very different from that in conventional two-dimensional systems. Because of the chiral nature of charge carriers, it is not only sensitive to inelastic, phase-breaking scattering, but also to a number of elastic scattering processes. We study weak localization in different samples and at different carrier densities, including the Dirac region, and find the characteristic rates that determine it. We show how the shape and quality of graphene flakes affect the values of the elastic and inelastic rates and discuss their physical origin.

Weak localization in monolayer and bilayer graphene

We demonstrate quantitative experimental evidence for a weak localization correction to the conductivity in monolayer and bilayer graphene systems. We show how inter- and intra-valley elastic scattering control the correction in small magnetic fields in a way which is unique to graphene. A clear difference in the forms of the correction is observed in the two systems, which shows the importance of the interplay between the elastic scattering mechanisms and how they can be distinguished. Our observation of the correction at zero-net carrier concentration in both systems is clear evidence of the inhomogeneity engendered into the graphene layers by disorder.

Transport length scales in disordered graphene-based materials: strong localization regimes and dimensionality effects

We report on a numerical study of quantum transport in disordered two dimensional graphene and graphene nanoribbons. By using the Kubo and the Landauer approaches, transport length scales in the diffusive (mean free path and charge mobilities) and localized regimes (localization lengths) are computed, assuming a short range disorder (Anderson-type). The electronic systems are found to undergo a conventional Anderson localization in the zero-temperature limit, in agreement with localization scaling theory. Localization lengths in weakly disordered ribbons are found to strongly fluctuate depending on their edge symmetry, but always remain several orders of magnitude smaller than those computed for 2D graphene for the same disorder strength. This pinpoints the role of transport dimensionality and edge effects.

Valley-contrasting physics in graphene: magnetic moment and topological transport

We investigate physical properties that can be used to distinguish the valley degree of freedom in systems where inversion symmetry is broken, using graphene systems as examples. We show that the pseudospin associated with the valley index of carriers has an intrinsic magnetic moment, in close analogy with the Bohr magneton for the electron spin. There is also a valley dependent Berry phase effect that can result in a valley contrasting Hall transport, with carriers in different valleys turning into opposite directions transverse to an in-plane electric field. These effects can be used to generate and detect valley polarization by magnetic and electric means, forming the basis for the valley-based electronics applications.

Topological delocalization of two-dimensional massless Dirac fermions

The beta function of a two-dimensional massless Dirac Hamiltonian subject to a random scalar potential, which, e.g., underlies theoretical descriptions of graphene, is computed numerically. Although it belongs to, from a symmetry standpoint, the two-dimensional symplectic class, the beta function monotonically increases with decreasing conductance. We also provide an argument based on the spectral flows under twisting boundary conditions, which shows that none of the states of the massless Dirac Hamiltonian can be localized.

Quantum criticality and minimal conductivity in graphene with long-range disorder

We consider the conductivity sigma of graphene with negligible intervalley scattering at half filling. We derive the effective field theory, which, for the case of a potential disorder, is a symplectic-class sigma model including a topological term with theta=pi. As a consequence, the system is at a quantum critical point with a universal value of the conductivity of the order of e(2)/h. When the effective time-reversal symmetry is broken, the symmetry class becomes unitary, and sigma acquires the value characteristic for the quantum Hall transition.

Weak localization in bilayer graphene

We have performed the first experimental investigation of quantum interference corrections to the conductivity of a bilayer graphene structure. A negative magnetoresistance--a signature of weak localization--is observed at different carrier densities, including the electroneutrality region. It is very different, however, from the weak localization in conventional two-dimensional systems. We show that it is controlled not only by the dephasing time, but also by different elastic processes that break the effective time-reversal symmetry and provide intervalley scattering.

Randomness-induced XY ordering in a graphene quantum hall ferromagnet

Valley-polarized quantum Hall states in graphene are described by a Heisenberg O(3) ferromagnet model, with the ordering type controlled by the strength and the sign of the valley anisotropy. A mechanism resulting from electron coupling to the strain-induced gauge field, giving a leading contribution to the anisotropy, is described in terms of an effective random magnetic field aligned with the ferromagnet z axis. We argue that such a random field stabilizes the XY ferromagnet state, which is a coherent equal-weight mixture of the K and K' valley states. The implications such as the Berezinskii-Kosterlitz-Thouless ordering transition and topological defects with half-integer charge are discussed.

Weak antilocalization in epitaxial graphene: evidence for chiral electrons

Transport in ultrathin graphite grown on silicon carbide is dominated by the electron-doped epitaxial layer at the interface. Weak antilocalization in 2D samples manifests itself as a broad cusplike depression in the longitudinal resistance for magnetic fields 10 mT<B<5 T. An extremely sharp weak-localization resistance peak at B=0 is also observed. These features quantitatively agree with graphene weak-(anti)localization theory implying the chiral electronic character of the samples. Scattering contributions from the trapped charges in the substrate and from trigonal warping due to the graphite layer on top are tentatively identified. The Shubnikov-de Haas oscillations are remarkably small and show an anomalous Berry's phase.

The rise of graphene

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.

Bipolar supercurrent in graphene

Graphene--a recently discovered form of graphite only one atomic layer thick--constitutes a new model system in condensed matter physics, because it is the first material in which charge carriers behave as massless chiral relativistic particles. The anomalous quantization of the Hall conductance, which is now understood theoretically, is one of the experimental signatures of the peculiar transport properties of relativistic electrons in graphene. Other unusual phenomena, like the finite conductivity of order 4e(2)/h (where e is the electron charge and h is Planck's constant) at the charge neutrality (or Dirac) point, have come as a surprise and remain to be explained. Here we experimentally study the Josephson effect in mesoscopic junctions consisting of a graphene layer contacted by two closely spaced superconducting electrodes. The charge density in the graphene layer can be controlled by means of a gate electrode. We observe a supercurrent that, depending on the gate voltage, is carried by either electrons in the conduction band or by holes in the valence band. More importantly, we find that not only the normal state conductance of graphene is finite, but also a finite supercurrent can flow at zero charge density. Our observations shed light on the special role of time reversal symmetry in graphene, and demonstrate phase coherent electronic transport at the Dirac point.

Quantum transport of massless Dirac fermions

Motivated by recent graphene transport experiments, we undertake a numerical study of the conductivity of disordered two-dimensional massless Dirac fermions. Our results reveal distinct differences between the cases of short-range and Coulomb randomly distributed scatterers. We speculate that this behavior is related to the Boltzmann transport theory prediction of dirty-limit behavior for Coulomb scatterers.

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.