Integer quantum Hall conductivity and longitudinal conductivity in silicene under the electric field and magnetic field

We investigate the integer Hall conductivity and longitudinal conductivity of silicene under the magnetic field, electric field, and exchange field in this paper. We focus not only on the case of low-temperature and -function impurities (i.e., independent of the scattering momentum), which only exist the intra-Landau level transition, but also on the case of inter-Landau level transition which also with the non-elastic scattering. The resulting longitudinal conductivity is very different from the intra-Landau level one at low-temperature. The expressions of the Hall conductivity, longitudinal conductivity, valley contributed Hall conductivity, and the spin or valley Hall conductivity are deduced in this paper. We also compute the dynamical polarization under the magnetic field which is an important quantity related to many exciting and novel properties of the two-dimension materials, including the screened Coulomb scattering due to the charged impurities. The polarization function here is also related to the Landau level index under the magnetic field, and shows a step-like feature rather than the logarithmically divergent as which appears in the zero-magnetic field case, and it is also naturally related to the conductivity in silicene. The generalized Laguerre polynomial is used in both the longitudinal conductivity and dynamical polarization function under magnetic field. Our results are also valid for the silicene-like materials like the genmanene, MoS2.