Single-layer behavior and its breakdown in twisted graphene layers

We report high magnetic field scanning tunneling microscopy and Landau level spectroscopy of twisted graphene layers grown by chemical vapor deposition. For twist angles exceeding ~3° the low energy carriers exhibit Landau level spectra characteristic of massless Dirac fermions. Above 20° the layers effectively decouple and the electronic properties are indistinguishable from those in single-layer graphene, while for smaller angles we observe a slowdown of the carrier velocity which is strongly angle dependent. At the smallest angles the spectra are dominated by twist-induced van Hove singularities and the Dirac fermions eventually become localized. An unexpected electron-hole asymmetry is observed which is substantially larger than the asymmetry in either single or untwisted bilayer graphene.     

Variational approach for the effects of periodic modulations on the spectrum of massless Dirac fermion

In the variational framework, we study the electronic energy spectrum of massless Dirac fermions of graphene subjected to one-dimensional oscillating magnetic and electrostatic fields centered around a constant uniform static magnetic field. We analyze the influence of the lateral periodic modulations in one direction, created by these oscillating electric and magnetic fields, on Dirac like Landau levels depending on amplitudes and periods of the field modulations. We compare our theoretical results with those found within the framework of non-degenerate perturbation theory. We found that the technique presented here yields energies lower than that obtained by the perturbation calculation, and thus gives more stable solutions for the electronic spectrum of massless Dirac fermion subjected to a magnetic field perpendicular to graphene layer under the influence of additional periodic potentials.     

Bound states of Dirac electrons in a graphene-based magnetic quantum dot

We investigate the magnetically confined states of the massless Dirac fermions in a graphene quantum dot formed by the inhomogeneous distributions of the magnetic fields inside and outside the dot. The calculated energy spectrum exhibits quite different features with and without the magnetic field inside the dot. It is found that the degeneracy of the relativistic Landau level with negative angular momenta can be lifted, and this degeneracy breaking can be modulated by the magnetic field inside the dot. Moreover, such a system can form the strongly localized states within the dot and along its boundary, especially with the magnetic field inside the dot.     

Scanning tunneling microscopy and spectroscopy of graphene layers on graphite

We report low temperature scanning tunneling microscopy and spectroscopy on graphene flakes supported on a graphite substrate. The experiments demonstrate that graphite is exceptionally well suited as a substrate for graphene because it offers support without disturbing the intrinsic properties of the charge carriers. The degree of coupling of a graphene flake to the substrate was recognized and characterized from the appearance of an anomalous Landau level sequence in the presence of a perpendicular magnetic field. By following the evolution of the Landau level spectra along the surface, we identified graphene flakes that are decoupled or very weakly coupled to the substrate. From the Landau level sequence in this flake, we extract the local Fermi velocity and energy of the Dirac point and find extremely weak spatial variation of these quantities confirming the high quality and non invasive nature of the graphite substrate.     

Electron-hole pair condensation in a graphene bilayer

The pairing of electrons and holes due to their Coulomb attraction in two parallel, independently gated graphene layers separated by a barrier is considered. At a weak coupling, there exists the BCS-like pair-condensed state. Despite the fact that electrons and holes behave like massless Dirac fermions, the problem of BCS-like electron—hole pairing in the graphene bilayer turns out to be rather similar to that in usual coupled semiconductor quantum wells. The distinctions are due to the Berry phase of electronic wavefunctions and different screening properties. We estimate the values of the gap in a one-particle excitation spectrum for different interlayer distances and carrier concentrations. The influence of the disorder is discussed. At a large enough dielectric susceptibility of the surrounding medium, the weak coupling regime holds at arbitrarily small carrier concentrations. Localized electron—hole pairs are absent in graphene, thus the behavior of the system versus the coupling strength is cardinally different from usual BCS—BEC crossover. The text was submitted by the authors in English.     

Reversal of Klein reflection by magnetic barriers in bilayer graphene

Massless Dirac fermions in monolayer graphene exhibit total transmission when normally incident on a scalar potential barrier, a consequence of the Klein paradox originally predicted by O Klein for relativistic electrons obeying the 3 + 1 dimensional Dirac equation. For bilayer graphene, charge carriers are massive Dirac fermions and, due to different chiralities, electron and hole states are not coupled to each other. Therefore, the wavefunction of an incident particle decays inside a barrier as for the non-relativistic Schr?dinger equation. This leads to exponentially small transmission upon normal incidence. We show that, in the presence of magnetic barriers, such massive Dirac fermions can have transmission even at normal incidence. The general consequences of this behavior for multilayer graphene consisting of massless and massive modes are mentioned. We also briefly discuss the effect of a bias voltage on such magnetotransport.     

Lattice-induced double-valley degeneracy lifting in graphene by a magnetic field

We show that the recently discovered double-valley splitting of the Landau levels in the quantum Hall effect in graphene can be explained as the perturbative orbital interaction of intravalley and intervalley microscopic orbital currents with a magnetic field. This effect is facilitated by the translationally noninvariant terms that correspond to graphene's crystallographic honeycomb symmetry but do not exist in the relativistic theory of massless Dirac fermions in quantum electrodynamics. We discuss recent data in view of these findings.     

New generation of massless Dirac fermions in graphene under external periodic potentials

We show that new massless Dirac fermions are generated when a slowly varying periodic potential is applied to graphene. These quasiparticles, generated near the supercell Brillouin zone boundaries with anisotropic group velocity, are different from the original massless Dirac fermions. The quasiparticle wave vector (measured from the new Dirac point), the generalized pseudospin vector, and the group velocity are not collinear. We further show that with an appropriate periodic potential of triangular symmetry, there exists an energy window over which the only available states are these quasiparticles, thus providing a good system to probe experimentally the new massless Dirac fermions. The required parameters of external potentials are within the realm of laboratory conditions.     

Theory of magnetic oscillations in Weyl semimetals

Weyl semimetals are a new class of Dirac material that possesses bulk energy nodes in three dimensions, in contrast to two dimensional graphene. In this paper, we study a Weyl semimetal subject to an applied magnetic field. We find distinct behavior that can be used to identify materials containing three dimensional Dirac fermions. We derive expressions for the density of states, electronic specific heat, and the magnetization. We focus our attention on the quantum oscillations in the magnetization. We find phase shifts in the quantum oscillations that distinguish the Weyl semimetal from conventional three dimensional Schrödinger fermions, as well as from two dimensional Dirac fermions. The density of states as a function of energy displays a sawtooth pattern which has its origin in the dispersion of the three dimensional Landau levels. At the same time, the spacing in energy of the sawtooth spike goes like the square root of the applied magnetic field which reflects the Dirac nature of the fermions. These features are reflected in the specific heat and magnetization. Finally, we apply a simple model for disorder and show that this tends to damp out the magnetic oscillations in the magnetization at small fields.     

Electric field effect tuning of electron-phonon coupling in graphene

Gate-modulated low-temperature Raman spectra reveal that the electric field effect (EFE), pervasive in contemporary electronics, has marked impacts on long-wavelength optical phonons of graphene. The EFE in this two-dimensional honeycomb lattice of carbon atoms creates large density modulations of carriers with linear dispersion (known as Dirac fermions). Our EFE Raman spectra display the interactions of lattice vibrations with these unusual carriers. The changes of phonon frequency and linewidth demonstrate optically the particle-hole symmetry about the charge-neutral Dirac point. The linear dependence of the phonon frequency on the EFE-modulated Fermi energy is explained as the electron-phonon coupling of massless Dirac fermions.     

Bound states of Dirac fermions in monolayer gapped graphene in the presence of local perturbations

In graphene,conductance electrons behave as massless relativistic particles and obey an analogue of the Dirac equation in two dimensions with a chiral nature.For this reason,the bounding of electrons in graphene in the form of geometries of quantum dots is impossible.In gapless graphene,due to its unique electronic band structure,there is a minimal conductivity at Dirac points,that is,in the limit of zero doping.This creates a problem for using such a highly motivated new material in electronic devices.One of the ways to overcome this problem is the creation of a band gap in the graphene band structure,which is made by inversion symmetry breaking(symmetry of sublattices).We investigate the confined states of the massless Dirac fermions in an impured graphene by the short-range perturbations for "local chemical potential" and "local gap".The calculated energy spectrum exhibits quite different features with and without the perturbations.A characteristic equation for bound states(BSs) has been obtained.It is surprisingly found that the relation between the radial functions of sublattices wave functions,i.e.,f_m~+(r),g_m~+(r),and f_m~-(r),g_m~-(r),can be established by SO(2) group.     

Band structures of bilayer graphene superlattices

We formulate a low energy effective Hamiltonian to study superlattices in bilayer graphene (BLG) using a minimal model which supports quadratic band touching points. We show that a one dimensional (1D) periodic modulation of the chemical potential or the electric field perpendicular to the layers leads to the generation of zero-energy anisotropic massless Dirac fermions and finite energy Dirac points with tunable velocities. The electric field superlattice maps onto a coupled chain model comprised of "topological" edge modes. 2D superlattice modulations are shown to lead to gaps on the mini-Brillouin zone boundary but do not, for certain symmetries, gap out the quadratic band touching point. Such potential variations, induced by impurities and rippling in biased BLG, could lead to subgap modes which are argued to be relevant to understanding transport measurements.     

Band structures of symmetrical graphene superlattice with cells of three regions

We study the electronic band structures of massless Dirac fermions in symmetrical graphene superlattice with cells of three regions. opening gaps and additional Dirac points. Finally, we inspect the potential effect on minibands, the anisotropy of group velocity and the energy bands contours near Dirac points. We also discuss the evolution of gap edges and cutoff region near the vertical Dirac points.     

Observation of magnetophonon resonance of Dirac fermions in graphite

Coherent coupling of Dirac fermion magnetoexcitons with an optical phonon is observed in graphite as marked magnetic-field dependent splittings and anticrossing behavior of the two coupled modes. The sharp magnetophonon resonance occurs in regions of the graphite sample with properties of superior single-layer graphene having enhanced lifetimes of Dirac fermions. The greatly reduced carrier broadening to values below the graphene electron-phonon coupling constant explains the appearance of sharp resonances that reveal a fundamental interaction of Dirac fermions.     

Fock-Darwin states of dirac electrons in graphene-based artificial atoms

We investigate the Fock-Darwin states of the massless chiral fermions confined in a graphitic parabolic quantum dot. In light of Klein tunneling, we analyze the condition for confinement of the Dirac fermions in a cylindrically symmetric potential. New features of the energy levels of the Dirac electrons as compared to the conventional electronic systems are discussed. We also evaluate the dipole-allowed transitions in the energy levels of the dots. We propose that in the high magnetic field limit, the band parameters can be accurately determined from the dipole-allowed transitions.     

Conductance quantization in graphene nanoribbons: adiabatic approximation

A theory of electron states for graphene nanoribbons with a smoothly varying width is developed. It is demonstrated that the standard adiabatic approximation allowing to neglect the mixing of different standing waves is more restrictive for the massless Dirac fermions in graphene than for the conventional electron gas. For the case of zigzag boundary conditions, one can expect a well-pronounced conductance quantization only for highly excited states. This difference is related to the relativistic Zitterbewegung effect in graphene.     

Realization of Graphene Physics Through a Fully Optical System

The two-dimensional form of carbon known as graphene awaken the scientific community interest due to its exotic electronic properties, emerging from the behavior of electrons near the Fermi level as massless Dirac fermions in a (1+2)-dimensional “relativistic” space-time, which renders a bridge between condensed matter and relativistic quantum field theory. Optical systems are also prodigal in providing analogues of complex quantum mechanical systems. Here, it is proposed an optical realization capable of capturing the essential physics of the Dirac equation in (1+2)-D dimensions, simulating the properties of graphene through the use of lightwave technology.     

Onset of landau-level formation in carbon-nanotube-based electronic Fabry-Perot resonators

We report on the onset of Landau-level formation in a carbon nanotube-based Fabry-Perot resonator. Supported by excellent agreement between calculated and measured magnetoconductance patterns, the applied perpendicular magnetic field is shown to modulate the Fabry-Perot conductance oscillations consistently with the formation of a Landau level in the 1D massless Dirac fermions particle excitations.     

Two-dimensional quantum ring in a graphene layer in the presence of a Aharonov–Bohm flux

In this paper we study the relativistic quantum dynamics of a massless fermion confined in a quantum ring. We use a model of confining potential and introduce the interaction via Dirac oscillator coupling, which provides ring confinement for massless Dirac fermions. The energy levels and corresponding eigenfunctions for this model in graphene layer in the presence of Aharonov–Bohm flux in the centre of the ring and the expression for persistent current in this model are derived. We also investigate the model for quantum ring in graphene layer in the presence of a disclination and a magnetic flux. The energy spectrum and wave function are obtained exactly for this case. We see that the persistent current depends on parameters characterizing the topological defect.