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This article reviews the basic theoretical aspects of graphene, a one atom thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric…

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.

We report a novel kind of dodecagonal quasicrystal that has so far never been observed, nor theoretically predicted. It is composed of axially stacked hexagonal particle layers, with 12-fold rotational symmetry induced by 30 degrees…

We show that electronic materials with disallowed rotational symmetries that enforce quasiperiodic order can exhibit quantum oscillations and that these are generically associated with exotic "spiral Fermi surfaces." These Fermi surfaces…

We consider quantum rings realized in materials where the dynamics of charge carriers mimics that of two-dimensional (2D) Dirac electrons. A general theoretical description of the ring-subband structure is developed that applies to a range…

Electron fractionalization is intimately related to topology. In one-dimensional systems, fractionally charged states exist at domain walls between degenerate vacua. In two-dimensional systems, fractionalization exists in quantum Hall…

The physics of graphene is acting as a bridge between quantum field theory and condensed matter physics due to the special quality of the graphene quasiparticles behaving as massless two dimensional Dirac fermions. Moreover, the particular…

Stacking two atomic crystals with a twist between their crystal axes produces moir\'e potentials that modify the electronic properties. Here we show that double moir\'e potentials generated by superposing three atomic crystals create a new…

Graphene, a two dimensional (2D) carbon sheet, acquires many of its amazing properties from the Dirac point nature of its electronic structures with negligible spin-orbit coupling. Extending to 3D space, graphene networks with negative…

We report the observation by angle-resolved photoemission spectroscopy measurements of a highly anisotropic Dirac-cone structure in high quality SrMnBi2 crystals. We reveal a well-defined sharp quasi-particle, linearly dispersive with…

We present a phenomenological theory of the low energy moir\'e minibands of Dirac electrons in graphene placed on an almost commensurate hexagonal underlay with a unit cell pproximately three times larger than that of graphene.A slight…

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…

Exact stationary solutions of the electron-photon Dirac equation are obtained to describe the strong interaction between massless Dirac fermions in graphene and circularly polarized photons. It follows from them that this interaction forms…

Strong interaction between electrons in two-dimensional systems in the presence of a high magnetic field gives rise to fractional quantum Hall states that host quasiparticles with fractional charge and fractional exchange statistics. Here,…

The distinctive electronic properties of quasicrystals stem from their long range structural order, with invariance under rotations and under discrete scale change, but without translational invariance. d-dimensional quasicrystals can be…

A number of interesting properties of graphene and graphite are postulated to derive from the peculiar bandstructure of graphene. This bandstructure consists of conical electron and hole pockets that meet at a single point in momentum (k)…

Relativistic quantum theory of induced scattering of 2D Dirac particles by electrostatic field of impurity ion (in the Born approximation) in the doped graphene at the presence of an external electromagnetic radiation field (actually…

We explore the rotational degree of freedom between graphene layers via the simple prototype of the graphene twist bilayer, i.e., two layers rotated by some angle $\theta$. It is shown that, due to the weak interaction between graphene…

Two-dimensional (2D) materials, especially their most prominent member, graphene, have greatly influenced many scientific areas. Moreover, they have become a base for investigating the relativistic properties of condensed matter within the…

Graphene, the atomic-thin layer of carbon atoms, was first isolated on an insulating substrate in 2004 by two groups in Manchester University [1, 2] and Columbia [3]. Those milestone experiments established the Dirac nature of the charge…