Probing the topological phase transition via density oscillations in silicene and germanene

We theoretically investigated two kinds of density oscillations, the Friedel oscillation and collective excitation in the silicene and germanene within the random phase approximation, and found that the tunable spin-valley coupled band structure could lead to some exotic properties in these two phenomena. Based on an exact analytical and numerical analysis, we demonstrated that the beating of the screened potential as well as the undamped plasmon mode can be taken as fingerprints of a topological phase transition in doped silicene and doped germanene. Thus our proposal here establishes the connection between the topological phase transition and the density oscillations that can be accessed by a variety of experimental techniques.

Figure 1

The band structures at the $K_{+}$ valley for different $\gamma$'s. The solid blue line represents the spin-up band while the dashed red line corresponds to the spin-down band.

Figure 2

Static real part of the charge susceptibility (red line) in units of the density of states $D_0=2\mu /\pi {\hslash }^2{v}_F^2$, and its corresponding first derivative is schematically represented by the blue line.

Figure 3

The beating behavior of the screened potential (in units of $Ze{\Delta }_{\mathrm{so}}/\kappa \hslash v_F$) vs the ratios $\gamma$ for $\kappa =3$, $\mu =3{\Delta }_{\mathrm{so}}$, and $v_F=c/550$.

Figure 4

The numerical exact solution within RPA (red) and the long-wavelength plasmon frequency (blue), with the shaded areas representing the interband and the intraband SPE regions. (d) shows the effect of $\gamma$ on the plasmon frequency. In the inset of (a) or (c), there is a clear UPM in the SPE gap [34].