We study theoretically the quantum transport properties of the Dirac fermions with spin-orbit interactions (SOIs) in graphene by using the method of Schwinger proper time together with decomposition over Landau level poles and Kubo formula. The analytical expressions for both longitudinal and Hall conductivities are derived explicitly. It is found that, from some numerical examples, when the Rashba SOI is taken into account the Shubnikov-de Haas (SdH) oscillation peaks of the longitudinal conductivity versus the chemical potential are split, while the SdH oscillation of the longitudinal conductivity versus a external magnetic field exhibits a beating pattern. Furthermore, the Rashba SOI tends to suppress the quantum Hall effect in graphene.
The Dirac electrons of graphene, an intrinsic zero gap semiconductor, uniquely carry spin and pseudospin that give rise to many fascinating electronic and transport properties. While isolated zigzag graphene nanoribbons are antiferromagnetic semiconductors, we show by means of first-principles and tight-binding calculations that zigzag graphene nanoribbons supported on graphene are half-metallic as a result of spin- and pseudospin-symmetry breaking. In particular, half-metallic Dirac cones are formed at K (K') near the Fermi level. The present results demonstrate that the unique combination of…
New Dirac points appear when periodic potentials are applied to graphene, and there are many interesting effects near these new Dirac points. Here we investigate the $\textit{Zitterbewegung}$ effect of fermions described by a Gaussian wave packet in graphene superlattice near new Dirac points. The $\textit{Zitterbewegung}$ near different Dirac points has similar characteristics, while Fermions near new Dirac points have different group velocities in both $x$- and $y$-direction, which causes the different properties of the $\textit{Zitterbewegung}$ near new Dirac points. We also investigate the…
We propose that the strain induced effective pseudo-magnetic field in graphene can also be explained by a curl movement of the Dirac points, if the Dirac points can be regarded as a slowly varying function of position. We also prove that the Dirac points must be confined within two triangles, each one has 1/8 the area of the Brillouin zone.
We have mapped the Dirac point in exfoliated monolayer and bilayer graphene using spatially resolved scanning tunneling spectroscopy (STS) measurements at low temperature. The Dirac point shifts in energy at different locations in graphene. However, a cross correlation with the topography shows no correlation indicating that topographic features such as ripples are not the primary source of the variation. Rather, we attribute the shift of the Dirac point to random charged impurities located near the graphene. Our findings emphasize the need to advance exfoliated graphene sample preparation to …
It is highly desirable to integrate graphene into existing semiconductor technology, where the combined system is thermodynamically stable yet maintain a Dirac cone at the Fermi level. Firstprinciples calculations reveal that a certain transition metal (TM) intercalated graphene/SiC(0001), such as the strongly-bound graphene/intercalated-Mn/SiC, could be such a system. Different from free-standing graphene, the hybridization between graphene and Mn/SiC leads to the formation of a dispersive Dirac cone of primarily TM d characters. The corresponding Dirac spectrum is still isotropic, and the tr…
Massless Dirac fermions in graphene can acquire a mass through different kinds of sublattice-symmetry-breaking perturbations, and there is a growing need to determine this mass using a conventional method. We describe how the mass caused by a staggered sublattice potential is determined using Raman spectroscopy and explain the mechanism in terms of the pseudospin polarization of massive Dirac fermions.
Three dimensionally curved graphene with a wide range of curvature radii from 25 nm to 1000 nm demonstrates that nano-scale curvature is a new degree of freedom to tune the transport properties of graphene by manipulating 2D electron kinetics on 3D curved surfaces.
The adhesion of graphene on slightly lattice-mismatched surfaces, for instance of hexagonal boron nitride (hBN) or Ir(111), gives rise to a complex landscape of sublattice symmetry-breaking potentials for the Dirac fermions. Whereas a gap at the Dirac point opens for perfectly lattice-matched graphene on hBN, we show that the small lattice incommensurability prevents the opening of this gap and rather leads to a renormalized Dirac dispersion with a trigonal warping. This warping breaks the effective time reversal symmetry in a single valley. On top of this a new set of massless Dirac fermions …
We reexamine the electronic structure of graphene on SiC substrate by angle-resolved photoemission spectroscopy. We directly observed multiply cloning of Dirac cone, in addition to ones previously attributed to reconstruction. The locations, relative distances and anisotropy of Dirac cone replicas fully agree with the moiré pattern of graphene-SiC heterostructure. Our results provide a straightforward example of moiré potential modulation in engineering electronic structure with Dirac fermions.
We consider the electronic structure of a slightly twisted graphene bilayer and show the coexistence of van Hove singularities (VHSs) and superlattice Dirac points in a continuum approximation. The graphene-on-graphene moiré pattern gives rise to a periodic electronic potential, which leads to the emergence of the superlattice Dirac points due to the chiral nature of the charge carriers. Owning to the distinguishing real and reciprocal structures, the sublattice exchange even and odd structures of the twisted graphene bilayer (the two types of commensurate structures) result in two different s…
We present results of tight binding calculations demonstrating existence of degenerate electronic shells of Dirac Fermions in narrow, charge neutral graphene quantum rings. We predict removal of degeneracy with finite magnetic field. We show, using a combination of tight binding and configuration interaction methods, that by filling a graphene ring with additional electrons this carbon based structure with half-filled shell acquires a finite magnetic moment.
Graphene is a 2-dimensional material with high carrier mobility and thermal conductivity, suitable for high-speed electronics. Conduction and valence bands touch at the Dirac point. The absorptivity of single-layer graphene is 2.3%, nearly independent of wavelength. Here we investigate the thermal radiation from biased graphene transistors. We find that the emission spectrum of single-layer graphene follows that of a grey body with constant emissivity (1.6 \pm 0.8)%. Most importantly, we can extract the temperature distribution in the ambipolar graphene channel, as confirmed by Stokes/anti-Sto…
We theoretically consider, comparing with the existing experimental literature, the electrical conductivity of gated monolayer graphene as a function of carrier density, temperature, and disorder in order to assess the prospects of accessing the Dirac point using transport studies in high-quality suspended graphene. We show that the temperature dependence of graphene conductivity around the charge neutrality point provides information about how close the system can approach the Dirac point although competition between long-range and short-range disorder as well as between diffusive and ballist…
In graphene devices with a varying degree of disorders as characterized by their carrier mobility and minimum conductivity, we have studied the thermoelectric power along with the electrical conductivity over a wide range of temperatures. We have found that the Mott relation fails in the vicinity of the Dirac point in high-mobility graphene. By properly taking account of the high temperature effects, we have obtained good agreement between the Boltzmann transport theory and our experimental data. In low-mobility graphene where the charged impurities induce relatively high residual carrier dens…
We analyze the zero-energy sector of the trigonal zigzag nanodisk and corner based on the Dirac theory of graphene. The zero-energy states are shown to be indexed by the edge momentum and grouped according to the irreducible representation of the trigonal symmetry group $C_{3v}$. Wave functions are explicitly constructed as holomorphic or antiholomorphic functions around the K or K' point. Each zero-energy mode is a chiral edge mode. We find a texture of magnetic vortices. It is intriguing that a vortex with the winding number $2$ emerges in the state belonging to the $E$ (doublet) representat…
We investigate generation of new Dirac cones in graphene under double-periodic and quasiperiodic superlattice potentials. We first show that double-periodic potentials generate the Dirac cones sporadically, following the Diophantine equation, in spite of the fact that double-periodic potentials are also periodic ones, for which previous studies predict consecutive appearance of the cones. The sporadic appearance is due to the fact that the dispersion relation of graphene is linear only up to an energy cutoff. We then show that quasiperiodic potentials generate the new Dirac cones densely with …
Graphene, as a promising material of post-silicon electronics, opens a new paradigm for the novel electronic properties and device applications. On the other hand, the 2D feature of graphene makes it technically challenging to be integrated into 3D transistors with a sufficient processor capacity. Although there are many attempts to assemble 2D graphene into 3D structures, the characteristics of massless Dirac fermions cannot be well preserved in these materials for transistor applications. Here we report a high-performance graphene transistor by utilizing 3D nanoporous graphene which is compr…
We have performed low temperature scanning tunneling spectroscopy measurements on exfoliated bilayer graphene on SiO2. By varying the back gate voltage we observed a linear shift of the Dirac point and an opening of a band gap due to the perpendicular electric field. In addition to observing a shift in the Dirac point, we also measured its spatial dependence using spatially resolved scanning tunneling spectroscopy. The spatial variation of the Dirac point was not correlated with topographic features and therefore we attribute its shift to random charged impurities.
We study the electronic structure of two Dirac electron gazes coupled by a periodic Hamiltonian such as it appears in rotated graphene bilayers. Ab initio and tight-binding approaches are combined and show that the spatially periodic coupling between the two Dirac electron gazes can renormalize strongly their velocity. We investigate in particular small angles of rotation and show that the velocity tends to zero in this limit. The localization is confirmed by an analysis of the eigenstates which are localized essentially in the AA zones of the Moire patterns.
We studied the square-octagonal lattice of the transition metal dichalcogenide MX$_2$ (with $M$=Mo, W; $X$=S, Se and Te), as an isomer of the normal hexagonal compound of MX$_2$. By band structure calculations, we observe the graphene-like Dirac band structure in a rectangular lattice of MX$_2$ with nonsymmorphic space group symmetry. Two bands with van Hove singularity points cross each at the Fermi energy, leading to two Dirac cones that locates at opposite momenta. Spin-orbit coupling can open a nontrivial gap at these Dirac points and induce the quantum spin Hall (QSH) phase, the 2D topolo…
We show that strongly photoexcited graphene monolayers with 35fs pulses quasi-instantaneously build up a broadband, inverted Dirac fermion population. Optical gain emerges and directly manifests itself via a negative optical conductivity for the first 200fs, where stimulated emission completely compensates absorption loss in the graphene layer. Our experiment-theory comparison with two distinct electron and hole chemical potentials reproduce absorption saturation and gain at 40fs, revealing, particularly, the evolution of the transient state from a hot classical gas to a dense quantum fluid wi…
The opening of a gap in single-layer graphene is often ascribed to the breaking of the equivalence between the two carbon sublattices. We show by angle-resolved photoemission spectroscopy that Ir- and Na-modified graphene grown on the Ir(111) surface presents a very large unconventional gap that can be described in terms of a phenomenological "massless" Dirac model. We discuss the consequences and differences of this model in comparison of the standard massive gap model, and we investigate the conditions under which such anomalous gap can arise from a spontaneous symmetry breaking.
The wavefunction of massless Dirac fermions is a two-component spinor. In graphene, a one-atom-thick film showing two-dimensional Dirac-like electronic excitations, the two-component representation reflects the amplitude of the electron wavefunction on the A and B sublattices. This unique property provides unprecedented opportunities to image the two components of massless Dirac fermions spatially. Here we report atomic resolution imaging of the two-component Dirac-Landau levels in a gapped graphene monolayer by scanning tunnelling microscopy and spectroscopy. A gap of about 20 meV, driven by …
We study the electronic and transport properties of a graphene-based superlattice theoretically by using an effective Dirac equation. The superlattice consists of a periodic potential applied on a single-layer graphene deposited on a substrate that opens an energy gap of $2Δ$ in its electronic structure. We find that extra Dirac points appear in the electronic band structure under certain conditions, so it is possible to close the gap between the conduction and valence minibands. We show that the energy gap $E_g$ can be tuned in the range $0\leq E_g \leq 2Δ$ by changing the periodic potential.…
First principles calculations, employed to address the properties of polycrystalline graphene, indicate that the electronic structure of tilt grain boundaries in this system displays a rather complex evolution towards graphene bulk, as the tilt angle decreases, with the generation of a new Dirac point at the Fermi level, and an anisotropic Dirac cone of low energy excitations. Moreover, the usual Dirac point at the {\bf K} point falls below the Fermi level, and rises towards it as the tilt angle decreases. Further, our calculations indicate that the grain-boundary formation energy behaves non-…
Massless Dirac fermions in graphene provide unprecedented opportunities to realize the Klein paradox, which is one of the most exotic and striking properties of relativistic particles. In the seminal theoretical work [Katsnelson et al., Nat. Phys. 2 620 (2006)], it was predicted that the massless Dirac fermions can pass through one-dimensional (1D) potential barriers unimpededly at normal incidence. Such a result seems to preclude confinement of the massless Dirac fermions in graphene by using 1D potential barriers. Here, we demonstrate, both experimentally and theoretically, that massless Dir…
Electron supercollimation, in which a wavepacket is guided to move undistorted along a selected direction, is a highly desirable property that has yet been realized experimentally. Disorder in general is expected to inhibit supercollimation. Here, we report a counter-intuitive phenomenon of electron supercollimation by disorder in graphene and related Dirac fermion materials. We show that one can use one-dimensional disorder potentials to control electron wavepacket transport. This is distinct from known systems where an electron wavepacket would be further spread by disorder and hindered in t…
In the case of the strongly deformed graphene, gapless graphene may turn to gapped graphene at the critical deformation. We find that, like semi-massless fermions, electrons in the deformed graphene, at the critical point, mimic the dispersions of the massless fermions in one direction and the massive fermions in the other. Our predicted dispersion formula is the generalization of the previously predicted formula. The behavior of the particle-like semi-massless fermions tunneling through a gate barrier is contrasted with that of the (pure) massless fermions tunneling through a gate barrier in …
A graphene superlattice is formed by a one-dimensional periodic potential and is characterized by the emergence of new Dirac points in the electronic structure. The group velocity of graphene's massless Dirac fermions at the new points is drastically reduced, resulting in a measurable effect in the conductance spectroscopy. We show here that tunnel spectroscopy using a superconducting hybrid junction is more sensitive to the formation of Dirac points in the spectrum of graphene superlattices due to the additional contribution of Andreev processes. We examine the transport properties of a graph…
Recent studies show that periodic potentials can generate superlattice Dirac points at energies in graphene (is the Fermi velocity of graphene and G is the reciprocal superlattice vector). Here, we perform scanning tunneling microscopy and spectroscopy studies of a corrugated graphene monolayer on Rh foil. We show that the quasi-periodic ripples of nanometer wavelength in the corrugated graphene give rise to weak one-dimensional (1D) electronic potentials and thereby lead to the emergence of the superlattice Dirac points. The position of the superlattice Dirac point is space-dependent and show…
Due to Klein tunneling, electrostatic potentials are unable to confine Dirac electrons. We show that it is possible to confine massless Dirac fermions in a monolayer graphene sheet by inhomogeneous magnetic fields. This allows one to design mesoscopic structures in graphene by magnetic barriers, e.g. quantum dots or quantum point contacts.
The second-order Dirac equation (DE) and its velocity operator of graphene electrons in an electromagnetic field are obtained according to tight-binding k.p method. With extra terms included, they demonstrate the motion of graphene electrons more completely through a more complete Ehrenfest theorem and present finer properties of graphene electrons. Eigen-energy given by the second-order DE for field-free graphene indicates that extra terms may affect the trembling motion of graphene electrons. For graphene in a magnetic field, eigen-energy given by the second-order DE suggests that graphene e…
At an interface between contacts and graphene, the mathematical equation that governs the propagation of electrons transforms from the Schrodinger to the Dirac equation. The condition of current probability conservation at such an interface does not determine uniquely the boundary conditions for the quantum wavefunction. We discuss the possible form of boundary conditions, determine its influence on the transmission coefficient of a contacted graphene structure and suggest that optical experiments on photonic crystals with Dirac points can help identifying, under certain circumstances, the pro…
To answer this question, we discuss the properties of electronic viscosity in deformed graphene by introducing strain and velocity gradient as pseudo-magnetic and pseudo-electric fields, respectively, into the Dirac model. We found that viscosity decreases with applied strain, simultaneously leading to a rather significant increase of the Reynolds number and enabling a real possibility for manifestation of noticeable turbulent effects in strained graphene.
There is evidence for existence of massless Dirac quasi-particles in graphene, which satisfy Dirac equation in (1+2) dimensions near the so called Dirac points which lie at the corners at the graphene's brilluoin zone. We revisit the derivation of Dirac equation in (1+2) dimensions obeyed by quasiparticles in graphene near the Dirac points. It is shown that parity operator in (1+2) dimensions play an interesting role and can be used for defining "conserved" currents resulting from the underlying Lagrangian for Dirac quasi-particles in graphene which is shown to have U_{A}(1)*U_{B}(1) symmetry.…
Graphene and topological insulators (TI) possess two-dimensional Dirac fermions with distinct physical properties. Integrating these two Dirac materials in a single device creates interesting opportunities for exploring new physics of interacting massless Dirac fermions. Here we report on a practical route to experimental fabrication of graphene-Sb2Te3 heterostructure. The graphene-TI heterostructures are prepared by using a dry transfer of chemical-vapor-deposition grown graphene film. ARPES measurements confirm the coexistence of topological surface states of Sb2Te3 and Dirac π bands of grap…
We report measurements of the cyclotron mass in graphene for carrier concentrations n varying over three orders of magnitude. In contrast to the single-particle picture, the real spectrum of graphene is profoundly nonlinear so that the Fermi velocity describing the spectral slope reaches ~3x10^6 m/s at n <10^10 cm^-2, three times the value commonly used for graphene. The observed changes are attributed to electron-electron interaction that renormalizes the Dirac spectrum because of weak screening. Our experiments also put an upper limit of ~0.1 meV on the possible gap in graphene.
In this paper we analyse the electronic properties of Dirac electrons in finite-size ribbons and in circular and hexagonal quantum dots made of graphene.
By solving two-component spinor equation for massless Dirac Fermions, we show that graphene under a periodic external magnetic field exhibits a unique energy spectrum: At low energies, Dirac Fermions are localized inside the magnetic region with discrete Landau energy levels, while at higher energies, Dirac Fermions are mainly found in non-magnetic regions with continuous energy bands originating from wavefunctions analogous to particle-in-box states of electrons. These findings offer a new methodology for the control and tuning of massless Dirac Fermions in graphene.
This is a short review of two-dimensional Dirac fermions in graphene and similar systems such as boron nitride, quasi-2D organic salts $α$-(BEDT-TTF)$_2$I$_3$, artificial graphene with cold atoms in optical lattices, etc. The emphasis is on magnetic field properties (semi-classical quantization of cyclotron orbits, Landau levels, quantum Hall effect, magneto-plasmons, magneto-phonon resonance, magneto-transport, etc.) and on topological properties (Berry phase, winding number, inter-band effects, merging transition of Dirac points, etc.).
We present a tight binding theory of the Dirac point resonances due to adsorbed atoms and molecules on an infinite 2D graphene sheet based on the standard tight binding model of the graphene p-band electronic structure and the extended Huckel model of the adsorbate and nearby graphene carbon atoms. The relaxed atomic geometries of the adsorbates and graphene are calculated using density functional theory. Our model includes the effects of the local rehybridization of the graphene from the sp^2 to sp^3 electronic structure that occurs when adsorbed atoms or molecules bond covalently to the grap…
Electronic properties of two-dimensional allotropes of carbon, such as graphene and its bilayer, multi-layer epitaxial graphene, few-layer Bernal-stacked graphene, as well as of three-dimensional bulk graphite are reviewed from the viewpoint of recent optical spectroscopy studies. Attention is focused on relativistic-like character of quasi particles in these systems, which are referred to as massless or massive Dirac fermions.
Effects of disorder on the electronic transport properties of graphene are strongly affected by the Dirac nature of the charge carriers in graphene. This is particularly pronounced near the Dirac point, where relativistic charge carriers cannot efficiently screen the impurity potential. We have studied time-dependent conductance fluctuations and magnetoresistance in graphene in the close vicinity of the Dirac point. We show that the fluctuations are due to the quantum interference effects due to scattering on impurities, and find an unusually large reduction of the relative noise power in magn…
In contrast to recent description [Phys. Rev. Lett. 106 (2011)116803], we show that pseudo spin in graphene is not completely a real angular momentum. The pseudo spin only in the direction perpendicular to graphene sheet is real angular momentum, while the pseudo spin parallel to graphene plane is still not real angular momentum. Interestingly, it is also shown that the Newtonian-like force and pseudo spin torque of massive Dirac electrons in graphene under strain field mimic gravitomagnetic force and gravitomagnetic spin torque, respectively. This is due to the equivalence of pseudo spin and …
We study how low-energy charge carriers scatter off periodic and linear graphene grain boundaries oriented along the zigzag direction with a periodicity three times greater than that of pristine graphene. These defects map the two Dirac points into the same position, and thus allow for intervalley scattering to occur. Starting from graphene's first-neighbor tight-binding model we show how can we compute the boundary condition seen by graphene's massless Dirac fermions at such grain boundaries. We illustrate this procedure for the 3-periodic pentagon-only grain boundary, and then work out the l…
The low energy physics of both graphene and surface states of three-dimensional topological insulators is described by gapless Dirac fermions with linear dispersion. In this work, we predict the emergence of a "heavy" Dirac fermion in a graphene/topological insulator hetero-junction, where the linear term almost vanishes and the corresponding energy dispersion becomes highly non-linear. By combining {\it ab initio} calculations and an effective low-energy model, we show explicitly how strong hybridization between Dirac fermions in graphene and the surface states of topological insulators can r…
Starting from an engineered periodic optical structure formed by waveguide arrays comprised of two interleaved lattices, we simulate a deformed Dirac equation. We show that the system also simulate graphene nano ribbons under strain. This optical analogue allows us to study the phenomenon of Zitterbewegung for the modified Dirac equation. Our results show that the amplitude of Zitterbewegung oscillations changes as the deformation parameter is changed.
We develop a microscopic theory of the weak localization of two-dimensional massless Dirac fermions which is valid in the whole range of classically weak magnetic fields. The theory is applied to calculate magnetoresistance caused by the weak localization in graphene and conducting surfaces of bulk topological insulators.
Superlattice in graphene generates extra Dirac points in the band structure and their number depends on the superlattice potential strength. Here, we have created a lateral superlattice in a graphene device with a tunable barrier height using a combination of two gates. In this Letter, we demonstrate the use of lateral superlattice to modify the band structure of graphene leading to the emergence of new Dirac cones. This controlled modification of the band structure persists up to 100 K.
In the previous publications (E. Kogan and V. U. Nazarov, Phys. Rev. B {\bf 85}, 115418 (2012) and E. Kogan, Graphene {\bf 2}, 74 (2013)) we presented point group symmetry classification of energy bands in graphene. In the present note we generalize classification by considering the space group symmetry. Also in the framework of group theory we describe destruction of the Dirac points in graphene by a supercell potential.
Motivated by recent graphene transport experiments, we have undertaken 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.
Application of secondary quantized self-consistent Dirac -- Hartree -- Fock approach to consider electronic properties of monolayer graphene with accounting of spin-polarized states allows to coherently explain experimental results on energy band minigaps and charge carrier asymmetry in graphene, propose a description of valent and conduction zones shifts and gives a nice theoretical estimation of electron and holes cyclotron masses which is in very good agreement with known experimental data.
Using first principles pseudopotential density functional theory calculations, we find that terminating zigzag graphene nanoribbons (ZGNR) with monovalent alkali atoms at a reduced concentration has a dramatic impact on their properties. In particular, using sodium atoms for the saturation of ZGNR edges at half the concentration of edge-carbon atoms make it a one dimensional, perfect semimetal, where the valance and conduction bands meet at only a single, Dirac-like point. Unlike pristine graphene, the Dirac-"cones" of Na-ZGNR is not symmetric with respect to wave vector, but rather it is tilt…
Electrons moving in graphene behave as massless Dirac fermions, and they exhibit fascinating low-frequency electrical transport phenomena. Their dynamic response, however, is little known at frequencies above one terahertz (THz). Such knowledge is important not only for a deeper understanding of the Dirac electron quantum transport, but also for graphene applications in ultrahigh speed THz electronics and IR optoelectronics. In this paper, we report the first measurement of high-frequency conductivity of graphene from THz to mid-IR at different carrier concentrations. The conductivity exhibits…
A remarkable manifestation of the quantum character of electrons in matter is offered by graphene, a single atomic layer of graphite. Unlike conventional solids where electrons are described with the Schrodinger equation, electronic excitations in graphene are governed by the Dirac Hamiltonian. Some of the intriguing electronic properties of graphene, such as massless Dirac quasiparticles with linear energy-momentum dispersion, have been confirmed by recent observations. Here we report an infrared (IR) spectromicroscopy study of charge dynamics in graphene integrated in gated devices. Our meas…
Energy band structure of the bilayer graphene superlattices with $δ$-function magnetic barriers and zero average magnetic flux is studied within the four-band continuum model, using the transfer matrix method. The periodic magnetic potential effects on the zero-energy Dirac point of pristine bilayer graphene are exactly analyzed. Magnetic potential is shown also to generate finite-energy Dirac points at the edges of Brillouin zone the positions of which and the related dispersions are determined in the case of symmetric potentials.
We report a thermoelectric study of graphene in both zero and applied magnetic fields. As a direct consequence of the linear dispersion of massless particles, we find that the Seebeck coefficient Sxx diverges with 1 /, where n2D is the carrier density. We observe a very large Nernst signal Sxy (~ 50 uV/K at 8 T) at the Dirac point, and an oscillatory dependence of both Sxx and Sxy on n2D at low temperatures. Our results underscore the anomalous thermoelectric transport in graphene, which may be used as a highly sensitive probe for impurity bands near the Dirac point.
