Abstract:
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.

Abstract:
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.

Abstract:
Charge carriers of graphene show neutrino-like linear energy dispersions as well as chiral behavior near the Dirac point. Here we report highly unusual and unexpected behaviors of these carriers in applied external periodic potentials, i.e., in graphene superlattices. The group velocity renormalizes highly anisotropically even to a degree that it is not changed at all for states with wavevector in one direction but is reduced to zero in another, implying the possibility that one can make nanoscale electronic circuits out of graphene not by cutting it but by drawing on it in a non-destructive way. Also, the type of charge carrier species (e.g. electron, hole or open orbit) and their density of states vary drastically with the Fermi energy, enabling one to tune the Fermi surface-dominant properties significantly with gate voltage. These results address the fundamental question of how chiral massless Dirac fermions propagate in periodic potentials and point to a new possible path for nanoscale electronics.

Abstract:
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 is generated, which are characterized by a group velocity that is half the one of pristine graphene.

Abstract:
Vacuum polarization of charged massless fermions is investigated in the superposition of Coulomb and Aharonov--Bohm (AB) potentials in 2+1 dimensions. For this purpose we construct the Green function of the two-dimensional Dirac equation with Coulomb and AB potentials (via the regular and irregular solutions of the radial Dirac equation) and calculate the vacuum polarization charge density in these fields in the so-called subcritical and supercritical regimes. The role of the self-adjoint extension parameter is discussed in terms of the physics of problem. We hope that our results will be helpful in the more deep understanding the fundamental problem of quantum electrodynamics and can be applied to the problems of charged impurity screening in graphene with taking into consideration the electron spin.

Abstract:
Graphene grown on Fe(110)by chemical vapor deposition using propylene is investigated by means of angle-resolved photoemission. The presence of massless Dirac fermions is clearly evidenced by the observation of a fully intact Dirac cone. Unlike Ni(111) and Co(0001), the Fe(110) imposes a strongly anisotropic quasi-one-dimensional structure on the graphene. Certain signatures of a superlattice effect appear in the dispersion of its \sigma-bands but the Dirac cone does not reveal any detectable superlattice or quantum-size effects although the graphene corrugation is twice as large as in the established two-dimensional graphene superlattice on Ir(111).

Abstract:
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.

Abstract:
The spectra of massless Dirac operators are of essential interest e.g. for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterised in terms of properties of the potentials, and we prove a Schnol'-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.

Abstract:
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.

Abstract:
There are two types of intrinsic surface states in solids. The first type is formed on the surface of topological insulators. Recently, transport of massless Dirac fermions in the band of "topological" states has been demonstrated. States of the second type were predicted by Tamm and Shockley long ago. They do not have a topological background and are therefore strongly dependent on the properties of the surface. We study the problem of the conductivity of Tamm-Shockley edge states through direct transport experiments. Aharonov-Bohm magneto-oscillations of resistance are found on graphene samples that contain a single nanohole. The effect is explained by the conductivity of the massless Dirac fermions in the edge states cycling around the nanohole. The results demonstrate the deep connection between topological and non-topological edge states in 2D systems of massless Dirac fermions.