Interband and intraband transition, dynamical polarization and screening of the monolayer and bilayer silicene in low-energy tight-binding model

We investigate the interband and intraband transition of the monolayer and AB-stacked bilayer silicene in low-energy tight-binding model under the electric field, where we focus on the dynamical polarization function, screening due to the charged impurity, and the plasmon dispersion. We obtain the logarithmically divergen polarization function within the random-phase-approximation (RPA) whose logarithmic singularities corresponds to the discontinuities of the first derivative which is at the momentum ${\bf q}=2{\bf k}_{F}$ in static case and indicate the topological phase transition point from the gapless semimetal to the gapped band insulator. We also obtain the power-law-dependent Friedel oscillation which can be enhanced by increasing the Rashba-coupling, that can contribute to the screened potential of the charged impurity which scale as $\sim r^{-1/2}$ in the short distance from the impurity and scale as $\sim r^{-1/3}$ in the long distance from the impurity. In the single-particle excitation regime with the electron-hole continuum, the interband and intraband transition happen, and the plasmon dispersion, which we mainly focus on the optical plasmon (which $\sim\sqrt{{\bf q}}$ in long-wavelength limit) in this paper, start to damped into the electron-hole pairs due to the nonzero imaginary part of the polarization function.