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 transitions 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 divergent polarization function within the random-phase-approximation, whose logarithmic singularities correspond to the discontinuities of the first derivative at momentum $q=2k_F$ in static case and indicates the topological phase transition point between the gapless semimetal and the gapped band insulator. We also obtain the power-law-dependent Friedel oscillation which can be enhanced by increasing the Rashba coupling. The Friedel oscillation contributes to the screened potential of the charged impurity which scale as $\sim r^{-1/2}$ in short distance from the impurity and scale as $\sim r^{-1/3}$ in long distance from the impurity. In single-particle excitation regime with electron–hole continuum, the interband and intraband transitions happen, and the plasmon dispersion starts to damped into the electron–hole pairs due to the nonzero imaginary part of the polarization function. We also observe the linear (weakly damped) plasmon model for the classical bilayer silicene which is similar to the high-energy $\pi$-plasmon or the case of conducting substrate which with strong metallic screening. The density dependence of the plasmon dispersion is also clarified in this paper. Our results can also be applied to the other low-energy Dirac models or the topological insulators.