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.

This brief review discusses electronic properties of mesoscopic graphene-based structures. These allow controlling the confinement and transport of charge and spin; thus, they are of interest not only for fundamental research, but also for applications. The graphene-related topics covered here are: edges, nanoribbons, quantum dots, pn-junctions, pnp-structures, and quantum barriers and waveguides. This review is partly intended as a short introduction to graphene mesoscopics.

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We analyze nonperturbatively the behavior of a Josephson junction in which two BCS superconductors are coupled through an Anderson impurity. We recover earlier perturbative results which found that a δ = π phase difference is preferred when the impurity is singly occupied and the on-site Coulomb interaction is large. We find a novel intermediate phase in which one of δ = 0 and δ = π is stable while the other is metastable, with the energy E(δ) having a kink somewhere in between. As a consequence of the kink, the I − V characteristics of the junction are modified at low voltages.

We find a unitary operator which asymptotically diagonalizes the Tomonaga-Luttinger hamiltonian of one-dimensional spinless electrons. The operator performs a Bogoliubov rotation in the space of electron-hole pairs. If bare interaction of the physical electrons is sufficiently small this transformation maps the original Tomonaga-Luttinger system on a system of free fermionic quasiparticles. Our representation is useful when the electron dispersion deviates from linear form. For such situation we obtain non-perturbative results for the electron gas free energy and the density-density propagator.

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.

We investigate the Josephson coupling between two superconductors mediated through an infinite U Anderson impurity, adapting a variational wavefunction approach which has proved successful for the Kondo model. Unlike the Kondo problem, however, a crossing of singlet and doublet state energies may be produced by varying the ratio of Kondo energy to superconducting gap, in agreement with recent work of Clerk and Ambegaokar. We construct the phase diagram for the junction and discuss properties of different phases. In addition, we find the singlet and doublet state energies within a slave boson mean field approach. We find the slave boson mean field treatment is unable to account for the level crossing.

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.