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Based on the Kubo formula, we have studied the electron transport properties of a gapped graphene in the presence of a strong magnetic field. By solving the Dirac equation, we find that the Landau level spectra in two valleys differ from each other in that the n = 0 level in the K valley is located at top of the valence band, whereas it is at the bottom of the conduction band in the K' valley. Thus, in an individual valley, the symmetry between conduction and valence bands is broken by the presence of a magnetic field. By using the self-consistent Born approximation to treat the long range potential scattering, we formulate the diagonal and the Hall conductivities in terms of the Green function. To perform the numerical calculation, we find that a large bandgap can suppress the quantum Hall effect, owing to the enhancement of the bandgap squeezing the spacing between the low-lying Landau levels. On the other hand, if the bandgap is not very large, the odd integer quantum Hall effect experimentally, observed in the gapless graphene, remains in the gapped one. However, such a result does not indicate the half integer quantum Hall effect in an individual valley of the gapped graphene. This is because the heights of the Hall plateaux in either valley can be continuously tuned by the variation of the bandgap. More interestingly, we find that the height of the diagonal conductivity peak corresponding to the n = 0 Landau level is independent of the bandgap if the scattering is not very strong. In the weak scattering limit, we demonstrate analytically that such a peak takes a universal value e(2)/(hpi), regardless of the bandgap.

The RKKY interaction between two magnetic impurities absorbed on the surface layer of half-filled AB-stacked multilayer graphene (ABSMLG) is theoretically studied based on the lattice Green's function technique. In comparison with the case of monolayer graphene, the RKKY interaction in such multilayer graphene presents distinct properties in some aspects. Firstly, from the numerical results, we find that the thickness of the ABSMLG influences the RKKY interaction in a complicated manner, depending on the odd/even parity of the number of layers and the sublattice attribution of the positions of the two magnetic impurities. Then, we derive the asymptotic expressions of the RKKY interactions in ABSMLG in the long-distance limit. For even-layered ABSMLG, we find that the RKKY interactions of the 1A-1A, 1B-1A and 1B-1B couplings fall off as 1/R(2), 1/R(4) and 1/R(6) (1A and 1B stand for, respectively, the sublattice points in the surface layer, which are positioned directly on the plaquette and on a lattice point of the layer underneath). On the other hand, in odd-layered ABSMLG, the decays of these interactions follow the 1/R(2), 1/R(3) and 1/R(3) power laws respectively. In addition, we also find that these analytical expressions are quantitatively valid to describe the RKKY interaction in ABSMLG when the distance between the two magnetic impurities is larger than the lattice constant of graphene by one order of magnitude.