Magnetic confinement of massless Dirac fermions in graphene

Because of 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.     

Statistical measures and the Klein tunneling in single-layer graphene

Statistical complexity and Fisher–Shannon information are calculated in a problem of quantum scattering, namely the Klein tunneling across a potential barrier in graphene. The treatment of electron wave functions as masless Dirac fermions allows us to compute these statistical measures. The comparison of these magnitudes with the transmission coefficient through the barrier is performed. We show that these statistical measures take their minimum values in the situations of total transparency through the barrier, a phenomenon highly anisotropic for the Klein tunneling in graphene.     

Spin-polarized tunneling in a ferromagnetic graphene junction: Interplay between the exchange interaction and the orbital effect of the magnetic field

The influence of magnetic vector potential barrier (MVPB) on the spin-polarized transport of massless Dirac particles in ferromagnetic graphene is studied theoretically. The phenomenon of Klein tunneling of relativistic particles across a rectangular potential barrier prevents any of the massless fermions from being confined but they can be electrically confined by quantum dots with integrable dynamics (Bardarson et al., 2009) [36]. Utilization of only the in-plane exchange splitting in the ferromagnetic graphene cannot produce 100% spin polarization. This tunneling can be confined using the magnetic vector potential barrier, which leads to high degree of spin polarization. By combining the orbital effect and the Zeeman interaction in graphene junction, it is found that the junction mimics behavior of half-metallic tunneling junction, in which it acts as a metal to particles of one spin orientation but as an insulator or a semiconductor to those of the opposite orientation. The idea of the half-metallic tunneling junction can provide a source of ∼100% spin-polarized current, which is potentially very useful. Adjustment of the position of the Fermi level in ferromagnetic layer by placing a gate voltage on top of the ferromagnetic layer shows that reverse of the orientation of the completely spin-polarized current passing through the junction is controlled by adjusting the gate voltage. These interesting characteristics should lead to a practical gate voltage controlled spin filtering and spin-polarized switching devices as a perfect spin-polarized electron source for graphene-based spintronics.     

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.     

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.     

Resonant tunneling through double-barrier structures on graphene

Quantum resonant tunneling behaviors of double-barrier structures on graphene are investigated under the tightbinding approximation. The Klein tunneling and resonant tunneling are demonstrated for the quasiparticles with energy close to the Dirac points. The Klein tunneling vanishes by increasing the height of the potential barriers to more than 300 meV. The Dirac transport properties continuously change to the Schro¨dinger ones. It is found that the peaks of resonant tunneling approximate to the eigen-levels of graphene nanoribbons under appropriate boundary conditions. A comparison between the zigzag- and armchair-edge barriers is given.     

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.     

Tunneling of Dirac fermions through magnetic barriers in graphene

We have studied the tunneling of Dirac fermions through magnetic barriers in graphene. Magnetic barriers are produced via delta function-like inhomogeneous magnetic fields in which Dirac fermions in graphene experience the tunneling barrier in the real sense in contrast to Klein paradox caused by electrostatic barriers. The transmission through the magnetic barriers as functions of incident energy and angle of incoming fermions shows characteristic oscillations associated with tunneling resonances. We have also found the confined states in the magnetic barrier region which turn out to correspond to the total internal reflection in the usual optics.     

Goos-Hänchen-like shifts for Dirac fermions in monolayer graphene barrier

The quantum Goos-H?nchen effect in graphene is found to be the lateral shift of Dirac fermions on the total reflection at a single p-n interface. In this paper, we investigate the lateral shifts of Dirac fermions in transmission through a monolayer graphene barrier. Compared to the smallness of the lateral shifts in total reflection, the lateral shifts can be enhanced by the transmission resonances when the incidence angle is less than the critical angle for total reflection. It is also found that the lateral shifts, as the function of the barrier’s width and incidence angle, can be negative and positive in the cases of Klein tunneling and classical motion. The modulation of the lateral shifts can be realized by changing the electrostatic potential and induced gap, which gives rise to some applications in graphene-based devices.     

Topological aspects of graphene

Topological aspects of the electronic properties of graphene, including edge effects, with the tight-binding model on a honeycomb lattice and its extensions to show the following: (i) Presence of the pair of massless Dirac dispersions, which is the origin of anomalous properties including a peculiar quantum Hall effect (QHE), is not accidental to honeycomb, but is generic for a class of two-dimensional lattices that interpolate between square and π-flux lattices. Topological stability guarantees persistence of the peculiar QHE. (ii) While we have the massless Dirac dispersion only around E=0, the anomalous QHE associated with the Dirac cone unexpectedly persists for a wide range of the chemical potential. The range is bounded by van Hove singularities, at which we predict a transition to the ordinary fermion behaviour accompanied by huge jumps in the QHE with a sign change. (iii) We establish a coincidence between the quantum Hall effect in the bulk and the quantum Hall effect for the edge states, which is another topological effect. We have also explicitly shown that the E=0 edge states in honeycomb in zero magnetic field persist in magnetic field. (iv) We have also identified a topological origin of the fermion doubling in terms of the chiral symmetry.     

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.     

Recent progresses of quantum confinement in graphene quantum dots

Graphene quantum dots (GQDs) not only have potential applications on spin qubit, but also serve as essential platforms to study the fundamental properties of Dirac fermions, such as Klein tunneling and Berry phase. By now, the study of quantum confinement in GQDs still attract much attention in condensed matter physics. In this article, we review the experimental progresses on quantum confinement in GQDs mainly by using scanning tunneling microscopy (STM) and scanning tunneling spectroscopy (STS). Here, the GQDs are divided into Klein GQDs, bound-state GQDs and edge-terminated GQDs according to their different confinement strength. Based on the realization of quasi-bound states in Klein GQDs, external perpendicular magnetic field is utilized as a manipulation approach to trigger and control the novel properties by tuning Berry phase and electron–electron (e–e) interaction. The tip-induced edge-free GQDs can serve as an intuitive mean to explore the broken symmetry states at nanoscale and single-electron accuracy, which are expected to be used in studying physical properties of different two-dimensional materials. Moreover, high-spin magnetic ground states are successfully introduced in edge-terminated GQDs by designing and synthesizing triangulene zigzag nanographenes.     

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.     

Graphene-based electronic spin lenses

We theoretically demonstrate the capability of a ferromagnetic-normal interface in graphene to focus an electron wave with a certain spin direction. The essential feature is the negative refraction Klein tunneling, which is spin resolved when the exchange energy of ferromagnetic graphene exceeds its Fermi energy. Exploiting this property, we propose a graphene normal-ferromagnetic-normal electronic spin lens through which an unpolarized electronic beam can be collimated with a finite spin polarization. Our study reveals that magnetic graphene has the potential to be the electronic counterpart of the recently discovered photonic chiral metamaterials that exhibit a negative refractive index for only one direction of the circular polarization of the photon wave.     

Atomic collapse,Lorentz boosts,Klein scattering,and other quantum-relativistic phenomena in graphene

Electrons in graphene, which behave as massless relativistic Dirac particles, provide a new perspective on the relation between condensed matter and high-energy physics. We discuss atomic collapse, a phenomenon in which discrete energy levels of superheavy atoms are transformed into resonant states. Charge impurities in graphene provide a convenient condensed matter system in which this effect can be explored. Relativistic dynamics also manifests itself in graphene p–n junctions. We show how the transport problem in the presence of a magnetic field can be solved with the help of a Lorentz transformation, and use it to investigate magnetotransport in p–n junctions. Finally, we review a recent proposal to use Fabry–Pérot resonances in p–n–p structures as a vehicle to investigate Klein scattering, another hallmark phenomenon of relativistic dynamics.     

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.     

Scattering of massless Dirac particles by oscillating barriers in one dimension

We study the scattering of massless Dirac particles by oscillating barriers in one dimension. Using the Floquet theory, we find the exact scattering amplitudes for time-harmonic barriers of arbitrary shape. In all cases the scattering amplitudes are found to be independent of the energy of the incoming particle and the transmission coefficient is unity. This is a manifestation of the Klein tunneling in time-harmonic potentials. Remarkably, the transmission amplitudes for arbitrary sharply-peaked potentials also become independent of the driving frequency. Conditions for which barriers of finite width can be replaced by sharply-peaked potentials are discussed.