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.

The phase diagram for doped manganites and related compounds is analyzed in terms of the Kondo-lattice model taking into account an interplay between electrons localized due to lattice distortions and those in the band states. It is shown that the number of itinerant charge carriers can be significantly lower than that implied by the doping level. The competition between the homogeneous (ferromagnetic or antiferromagnetic) and phase-separated states is discussed and a strong tendency to the phase separation was revealed for a wide doping range.

The two-band Hubbard model is used to analyze a possibility of a non-uniform charge distribution in a strongly correlated electron system with two types of charge carriers. It is demonstrated that in the limit of strong on-site Coulomb repulsion, such a system has a tendency to phase separation into the regions with different charge densities even in the absence of magnetic or any other ordering. This tendency is especially pronounced if the ratio of the bandwidths is large enough. The characteristic size of inhomogeneities is estimated accounting for the surface energy and the electrostatic energy related to the charge imbalance.Comment: 14 pages, 5 figures, RevTe

In the framework of a two-band model, we study the phase-separation regime of different kinds of strongly correlated charge carriers as a function of the energy splitting between the two sets of bands. The narrow (wide) band simulates the more localized (more delocalized) type of charge carriers. By assuming that the internal chemical pressure on the CuO2 layer due to interlayer mismatch controls the energy splitting between the two sets of states, the theoretical predictions are able to reproduce the regime of phase separation at doping higher than 1/8 in the experimental pressure-doping- Tc phase diagram of cuprates at large microstrain as it appears in overoxygenated La2 CuO4. © 2008 The American Physical Society

Specific features of orbital and spin structure of transition metal compounds in the case of the face-sharing MO6 octahedra are analyzed. In this geometry, we consider the form of the spin-orbital Hamiltonian for transition metal ions with double (e σ g ) or triple (t2g) orbital degeneracy. Trigonal distortions typical of the structures with face-sharing octahedra lead to splitting of t2g orbitals into an a1g singlet and e π g doublet. For both doublets (e σ g and e π g ), in the case of one electron or hole per site, we arrive at a symmetric model with the orbital and spin interaction of the Heisenberg type and the Hamiltonian of unexpectedly high symmetry: SU(4). Thus, many real materials with this geometry can serve as a testing ground for checking the prediction of this interesting theoretical model. We also compare general trends in spin-orbital ("Kugel-Khomskii") exchange interaction for three typical situations: those of MO6 octahedra with common corner, common edge, and the present case of common face, which has not been considered yet.

The arrested nanoscale phase separation in a two-band Hubbard model for strongly correlated charge carriers is shown to occur in a particular range in vicinity of the topological Lifshitz transition, where the Fermi energy crosses the bottom of the narrow band and a new sheet of the Fermi surface related to the charge carriers of the second band comes into play. We determine the phase separation diagram of this two-band Hubbard model as a function of two variables, the charge carrier density and the energy shift between the chemical potential and the bottom of the second band. In this phase diagram, we first determine a line of quantum critical points for the Lifshitz transition and find criteria for the electronic phase separation resulting in an inhomogeneous charge distribution. Finally, we identify the critical point in presence of a variable long-range Coulomb interaction where the scale invariance of the coexisting phases with different charge densities appears. We argue that this point is relevant for the regime of scale invariance of the nanoscale phase separation in cuprates like it was first observed in La2CuO4.1.