This article reviews the theoretical and experimental work related to the electronic properties of bilayer graphene systems. Three types of bilayer stackings are discussed: the AA, AB, and twisted bilayer graphene. This review covers single-electron properties, effects of static electric and magnetic fields, bilayer-based mesoscopic systems, spin-orbit coupling, dc transport and optical response, as well as spontaneous symmetry violation and other interaction effects. The selection of the material aims to introduce the reader to the most commonly studied topics of theoretical and experimental research in bilayer graphene.

We study the electronic properties of twisted bilayers graphene in the tight-binding approximation. The interlayer hopping amplitude is modeled by a function, which depends not only on the distance between two carbon atoms, but also on the positions of neighboring atoms as well. Using the Lanczos algorithm for the numerical evaluation of eigenvalues of large sparse matrices, we calculate the bilayer single-electron spectrum for commensurate twist angles in the range 1• . We show that at certain angles θ greater than θc ≈ 1.89• the electronic spectrum acquires a finite gap, whose value could be as large as 80 meV. However, in an infinitely large and perfectly clean sample the gap as a function of θ behaves non-monotonously, demonstrating exponentially-large jumps for very small variations of θ. This sensitivity to the angle makes it impossible to predict the gap value for a given sample, since in experiment θ is always known with certain error. To establish the connection with experiments, we demonstrate that for a system of finite sizeL the gap becomes a smooth function of the twist angle. If the sample is infinite, but disorder is present, we expect that the electron mean-free path plays the same role asL. In the regime of small angles θ < θc, the system is a metal with a well-defined Fermi surface which is reduced to Fermi points for some values of θ. The density of states in the metallic phase varies smoothly with θ.

Tight-binding calculations predict that the AA-stacked bilayer graphene has one electron and one hole conducting band, and that the Fermi surfaces of these bands coincide. We demonstrate that as a result of this degeneracy, the bilayer becomes unstable with respect to a set of spontaneous symmetry violations. Which of the symmetries is broken depends on the microscopic details of the system. For strong on-site Coulomb interaction we find that antiferromagnetism is the most stable order parameter. For an on-site repulsion energy typical for graphene systems, the antiferromagnetic gap can exist up to room temperature.

Exploiting the peculiar properties of proximity-induced superconductivity on the surface of a topological insulator, we propose a device which allows the creation of a Majorana fermion inside the core of a pinned Abrikosov vortex. The relevant Bogolyubov-de Gennes equations are studied analytically. We demonstrate that in this system the zero-energy Majorana fermion state is separated by a large energy gap, of the order of the zero-temperature superconducting gap $\Delta$, from a band of single-particle non-topological excitations. In other words, the Majorana fermion remains robust against thermal fluctuations, as long as the temperature remains substantially lower than the critical superconducting temperature. Experimentally, the Majorana state may be detected by measuring the tunneling differential conductance at the center of the Abrikosov vortex. In such an experiment, the Majorana state manifests itself as a zero-bias anomaly separated by a gap, of the order of $\Delta$, from the contributions of the nontopological excitations.Comment: 9 pages, 2 eps figures, new references are added, several typos are correcte

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The vortex patterns stabilized by the square array of artificial pinning sites with a tunable pinning strength are studied by using a phenomenological approach in the London limit. The transitions between pinned and deformed triangular lattices are analyzed as a function of the amplitude of the vortexpinning site interaction and the characteristic length-scale of this interaction. The critical current and different phases of vortex lattice are studied in presence of external transport current.

We analyze the phase diagram for a system of weakly-coupled electrons having an electron-and a hole-band with imperfect nesting. Namely, both bands have spherical Fermi surfaces, but their radii are slightly different, with a mismatch proportional to the doping. Such a model is used to describe: the antiferromagnetism of chromium and its alloys, pnictides, AA-stacked graphene bilayers, as well as other systems. Here we show that the uniform ground state of this model is unstable with respect to electronic phase separation in a wide range of model parameters. Physically, this instability occurs due to the competition between commensurate and incommensurate antiferromagnetic states and could be of importance for other models with imperfect nesting.

We study a heterostructure which consists of a topological insulator and a superconductor with a hole. The hole pins a vortex. The system supports a robust Majorana fermion state bound to the vortex core. We investigate the possibility of using scanning tunneling spectroscopy (i) to detect the Majorana fermion in the proposed setup and (ii) to study excited states bound to the vortex core. The Majorana fermion manifests itself as a magnetic-field dependent zero-bias anomaly of the tunneling conductance. Optimal parameters for detecting Majorana fermions have been obtained. In the optimal regime, the Majorana fermion is separated from the excited states by a substantial gap. The number of zero-energy states equals the number of flux quanta in the hole; thus, the strength of the zero-bias anomaly depends on the magnetic field. The lowest energy excitations bound to the core are also studied. The excited states spectrum differs from the spectrum of a typical Abrikosov vortex, providing additional indirect confirmation of the Majorana state observation.