Abstract:
The tight-binding model of bilayer graphene is used to find the gap between the conduction and valence bands, as a function of both the gate voltage and as the doping by donors or acceptors. The total Hartree energy is minimized and the equation for the gap is obtained. This equation for the ratio of the gap to the chemical potential is determined only by the screening constant. Thus the gap is strictly proportional to the gate voltage or the carrier concentration in the absence of donors or acceptors. In the opposite case, where the donors or acceptors are present, the gap demonstrates the asymmetrical behavior on the electron and hole sides of the gate bias. A comparison with experimental data obtained by Kuzmenko et al demonstrates the good agreement.

Abstract:
The tight-binding model of a graphene bilayer is used to find the gap between the conduction and valence bands, as a function of both the gate voltage and as the doping by donors or acceptors. The total Hartree energy is minimized and the equation for the gap is obtained. This equation for the ratio of the gap to the chemical potential is determined only by the screening constant. Thus the gap is strictly proportional to the gate voltage or the carrier concentration in the absence of donors or acceptors. In the opposite case, where the donors or acceptors are present, the gap demonstrates the asymmetrical behavior on the electron and hole sides of the gate bias.

Abstract:
We calculate the classic Hall conductivity and mobility of the undoped and doped (or in the gate voltage) graphene as a function of temperature, magnetic field, and carrier concentration. Carrier collisions with defects and acoustic phonons are taken into account. The Hall resistivity varies almost linearly with temperature. The magnetic field dependence of resistivity and mobility is anomalous in weak magnetic fields. There is the square root contribution from the field in the resistivity. The Hall mobility diverges logarithmically with the field for low doping.

Abstract:
We analyze the features of the graphene mono- and multilayer reflectance in the far-infrared region as a function of frequency, temperature, and carrier density taking the intraband conductance and the interband electron absorbtion into account. The dispersion of plasmon mode of the multilayers is calculated using Maxwell's equations with the influence of retardation included. At low temperatures and high electron densities, the reflectance of multilayers as a function of frequency has the sharp downfall and the subsequent deep well due to the threshold of electron interband absorbtion.

Abstract:
Taking into account the constraints imposed by the lattice symmetry, the phonon dispersion is calculated for graphene with interactions between the first and second nearest neighbors in the framework of the Born-von Karman model. Analytical expressions are obtained for the out-of-plane (bending) modes determined only by two force constants as well as for the in-plane modes with four force constants. Values of the force constants are found in fitting to elastic constants and Raman frequencies observed in graphite.

Abstract:
Raman scattering on phonon--plasmon coupled modes in high magnetic fields is considered theoretically. The calculations of the dielectric function were performed in the long-wave approximation for the semiclassical and ultra-quantum magnetic fields taking into account the electron damping and intrinsic lifetime of optical phonons. The Raman scattering has resonances at the frequencies of coupled modes as well as at multiples of the cyclotron frequency. The dependence of the Raman cross section on the carrier concentration is analyzed.

Abstract:
The optical conductivity of graphite in quantizing magnetic fields is studied. Both the dynamical conductivities, longitudinal as well as Hall's, are analytically evaluated. The conductivity peaks are explained in terms of electron transitions. We have shown that the trigonal warping in graphite can be considered within the perturbation theory at the strong magnetic field larger than 1 T approximately. The main optical transitions obey the selection rule with $\Delta n=1$ for the Landau number $n$, however the $\Delta n=2$ transitions due to the trigonal warping with the small probability are also essential. The Kerr rotation and reflectivity in graphite in the quantizing magnetic fields are calculated. Parameters of the Slonczewski--Weiss--McClure model are used in the fit taking into account the previous dHvA measurements and correcting some of them for the case of the strong magnetic fields.

Abstract:
The optical conductivity of graphene, bilayer graphene, and graphite in quantizing magnetic fields is studied. Both dynamical conductivities, longitudinal and Hall's, are analytically evaluated. The conductivity peaks are explained in terms of electron transitions. We have shown that trigonal warping can be considered within the perturbation theory for strong magnetic fields larger than 1 T and in the semiclassical approach for weak fields when the Fermi energy is much larger than the cyclotron frequency. The main optical transitions obey the selection rule with \Deltan = 1 for the Landau number n, however the \Deltan = 2 transitions due to the trigonal warping are also possible. The Faraday/Kerr rotation and light transmission/reflection in the quantizing magnetic fields are calculated. Parameters of the Slonczewski-Weiss-McClure model are used in the fit taking into account the previous dHvA measurements and correcting some of them for the case of strong magnetic fields.

Abstract:
Reflectance and transmittance of graphene in the optical region are analyzed as a function of frequency, temperature, and carrier density. We show that the optical graphene properties are determined by the direct interband electron transitions. The real part of the dynamic conductivity in doped graphene at low temperatures takes the universal constant value, whereas the imaginary part is logarithmically divergent at the threshold of interband transitions.

Abstract:
Taking into account the constraints imposed by the lattice symmetry, we calculate the phonon dispersion for graphene with interactions between the first, second, and third nearest neighbors in the framework of the Born--von Karman model. Analytical expressions obtained for the dispersion of the out-of-plane (bending) modes give the nonzero sound velocity. The dispersion of four in-plane modes is determined by coupled equations. Values of the force constants are found in fitting with frequencies at critical points and with elastic constants measured on graphite.