The Peierls Hamiltonian band matrix is developed to investigate magnetoelectronic properties of bilayer Bernal graphene. A uniform perpendicular magnetic field creates many dispersionless Landau levels ͑LLs͒ at low and high energies and some oscillatory LLs at moderate energy. State degeneracy of the low LLs is two times as much as that of the high LLs. Wave functions and state energies are dominated by the interlayer atomic interactions and field strength ͑B 0 ͒. The former induce two groups of LLs, more low LLs, the asymmetric energy spectrum about the Fermi level, and the change of level spacing. Two sets of effective quantum numbers, n 1 ef f 's and n 2 ef f 's, are required to characterize all the wave functions. They are determined by the strongest oscillation modes of the dominant carrier densities; furthermore, they rely on the specific interlayer atomic hoppings. The dependence of the quite low Landau-level energies on B 0 and n 1 ef f is approximatelylinear. An energy gap is produced by the magnetic field and interlayer atomic hoppings. E g grows with increasing field strength, while it is reduced by the Zeeman effect. The main features of magnetoelectronic structures are directly reflected in the density of states. The predicted electronic properties could be verified by the experimental measurements on absorption spectra and transport properties.

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The π-electronic structure of graphene in the presence of a modulated electric potential is investigated by the tight-binding model. The low-energy electronic properties are strongly affected by the period and field strength. Such a field could modify the energy dispersions, destroy state degeneracy, and induce band-edge states. It should be noted that a modulated electric potential could make semiconducting graphene semimetallic, and that the onset period of such a transition relies on the field strength. There exist infinite Fermi-momentum states in sharply contrast with two crossing points (Dirac points) for graphene without external fields. The finite density of states (DOS) at the Fermi level means that there are free carriers, and, at the same time, the low DOS spectrum exhibits many prominent peaks, mainly owing to the band-edge states.

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The-electronic excitations are studied for the AA-and AB-stacked bilayer graphites within the linear self-consistent-field approach. They are strongly affected by the stacking sequence, the interlayer atomic interactions, the interlayer Coulomb interactions, and the magnitude of the transferred momentum. However, they hardly depend on the direction of the transferred momentum and the temperature. There are three lowfrequency plasmon modes in the AA-stacked system but not the AB-stacked system. The AA-and AB-stacked plasmons exhibit the similar plasmons. The first low-frequency plasmon behaves as an acoustic plasmon, and the others belong to optical plasmons. The bilayer graphites quite differ from the monolayer graphite and the AB-stacked bulk graphite, such as the low-frequency plasmons and the small-momentum plasmons.