Dynamical polarization function, plasmons, and screening in silicene and other buckled honeycomb lattices

We explore the dielectric properties of graphene-like two-dimensional Kane-Mele topological insulators manifest in buckled honeycomb lattices (such as silicene and germanene). The effect of an on-site potential difference (${\Delta }_z$) between sublattices is given particular attention. We present the results for the real and imaginary parts of the dynamical polarization function. We show that these results display features of three regimes (topological insulator, valley-spin polarized metal, and trivial band insulator) and may be used to extract information on the strength of the intrinsic spin-orbit coupling. We study the inverse dielectric function and provide numerical results for the plasmon branch. We discuss the behavior of the plasmon as a function of sublattice potential difference and show that the behavior of the plasmon branch as ${\Delta }_z$ is varied is dependent on the location of the chemical potential with respect to the gaps. The static polarization is discussed and numerical results for the screening of a charged impurity are provided. We observe a beating phenomenon in the effective potential which is dependent on ${\Delta }_z$.

Figure 1

Low-energy band structure of a buckled honeycomb lattice at the $K$ point for key values of sublattice potential difference (${\Delta }_z$). Top: (left) With no potential difference, the system is a TI with a band gap of $2{\Delta }_{\mathrm{so}}$. (right) With a finite ${\Delta }_z$ such that ${\Delta }_z<{\Delta }_{\mathrm{so}}$, the system remains a TI but now has two spin split bands the lower(upper) of which decreases(increases) with increased ${\Delta }_z$. Bottom: (left) When ${\Delta }_z={\Delta }_{\mathrm{so}}$, the lowest gap closes (${\Delta }_{\min }=0$) and the system is a VSPM. (right) As ${\Delta }_z$ is increased further such that ${\Delta }_z>{\Delta }_{\mathrm{so}}$, the gaps reopen and both increase with ${\Delta }_z$. The system is now a BI. In all cases, dashed blue lines correspond to spin up bands and solid red to spin down. The spin labels switch at $K^{\prime }$.

Figure 2

Regions for the (left) $K\uparrow /K^{\prime }\downarrow$ and (right) $K\downarrow /K^{\prime }\uparrow$ bands in which the polarization function has different expressions. Here, ${\Delta }_{\mathrm{so}}/\mu =0.3$ and ${\Delta }_z/{\Delta }_{\mathrm{so}}=1.5$.

Figure 3

Imaginary part of the polarization function scaled by $N\left(\mu \right)=2\mu /\pi$ (density of states of graphene at $\mu$) for ${\Delta }_{\mathrm{so}}/\mu =0.7$ and varying ${\Delta }_z$. When ${\Delta }_z=0$ (upper left frame), the Im$\Pi (\omega ,q)$ is the familiar case of gapped graphene. As ${\Delta }_z$ is applied it splits into the sum of two gapped systems with gaps given by ${\Delta }_{\min }$ and ${\Delta }_{\max }$. When ${\Delta }_z={\Delta }_{\mathrm{so}}$ (lower left frame), the polarization function is ungapped and can be understood as the sum of the ungapped graphene system and a system with gap ${\Delta }_{\max }$. The regions where Im$\Pi (\omega ,q)=0$ (white space) correspond to values of $q$ and $\omega$ for which there is no damping of a collective charge oscillation. Inset: The location of the chemical potential relative to the upper part of the band structure.

Figure 4

Real part of the polarization function for ${\Delta }_{\mathrm{so}}/\mu =0.7$ and ${\Delta }_z$ chosen to correspond to Fig. 3. Again, it behaves as the sum of two gapped systems. Inset: The location of the chemical potential relative to the gap(s).

Figure 5

Constant frequency cuts of the real part of the polarization function as a function of $q$ for ${\Delta }_{\mathrm{so}}/\mu =0.7$ and ${\Delta }_z/{\Delta }_{\mathrm{so}}=0.25$ [see Fig. 4]. (a) $\omega /\mu =0,0.5,1$, and 1.5. (b) $\omega /\mu =2,2.5$, and 3. Inset: The relative location of $\mu$ with respect to the band structure.

Figure 6

Static polarization scaled by $N\left(\mu \right)$ for ${\Delta }_{\mathrm{so}}/\mu =0.7$ and varying ${\Delta }_z$. The graphene result (${\Delta }_{\mathrm{so}}=0$) is given by the solid black curve for comparison. Inset: Relative location of the chemical potential for finite ${\Delta }_z$ (not drawn to scale) [see text for a further discussion]. Like graphene, when the chemical potential is above both gaps (dashed red and dash-dotted blue curves), the polarization is constant. At $q/\mu =2\sqrt{1-{({\Delta }_{\max }/\mu )}^2}$ and $q/\mu =2\sqrt{1-{({\Delta }_{\min }/\mu )}^2}$, there are dips in the polarization after which the graphene-like interband behavior is retained. When ${\Delta }_{\min }<\mu <{\Delta }_{\max }$ so $\mu$ lies between the two gaps (solid green and dash-double-dotted orange curves), the $q=0$ polarization is half that of graphene and never remains constant. There is a small dip at $q/\mu =2\sqrt{1-{({\Delta }_{\min }/\mu )}^2}$ before a continual increase is observed. When $\mu$ sits in both gaps (dash-double-dotted purple and dotted magenta curves), $\Pi (0,0)=0$ and there are no cusps at higher $q$. For all cases, the polarization function at a given $q$ is always less than or equal to that of a lower ${\Delta }_z$ value.

Figure 7

Energy loss function for $\alpha =0.8$, ${\Delta }_{\mathrm{so}}/\mu =0.7$, and varying ${\Delta }_z$ chosen to correspond with Fig. 3. The red branch signifies the plasmon excitation. The plasmon is undamped in the regions that correspond to Im$\Pi (\omega ,q)=0$ (see Fig. 3). When the imaginary part of the polarization function is nonzero, the plasmon is quickly damped out in $(q,\omega )$ space. Due to the $1/[1-V(q\left)\Pi \right(\omega ,q\left)\right]$ dependence, the energy loss function is not simply the sum of two gapped systems. Inset: The location of the chemical potential relative to the band structure.

Figure 8

Energy loss function for $\alpha =0.8$, ${\Delta }_{\mathrm{so}}/\mu =0.375$, and varying ${\Delta }_z$. Inset: The location of the chemical potential relative to the band structure.

Figure 9

(a) Frequency cuts of $2\pi \alpha$Re$\Pi (\omega ,q)/\mu$ for $\alpha =0.8$, ${\Delta }_{\mathrm{so}}/\mu =0.375$ and ${\Delta }_z/{\Delta }_{\mathrm{so}}=1$. The intersection of $2\pi \alpha$Re$\Pi (\omega ,q)/\mu$ and the line of unit slope (solid orange line) represent solutions to Eq. (21). (b) Plasmon frequency as a function of $q$ corresponding to the solutions of Eq. (21). (c) Frequency cuts of the loss function in which we can see that there is one pole per frequency. (d) Zoomed-in version of (c).

Figure 10

Low-energy plasmon dispersion for $\alpha =0.8$, varying ${\Delta }_z$ and (left) ${\Delta }_{\mathrm{so}}/\mu =0.3$ and (right) ${\Delta }_{\mathrm{so}}/\mu =0.9$. Left: For ${\Delta }_{\mathrm{so}}/\mu =0.3$, the chemical potential is greater than ${\Delta }_{\max }$ for all values of ${\Delta }_z$ shown here. As ${\Delta }_z$ is increased, the plasmon branch at a given $q$ decrease in frequency. Right: For ${\Delta }_{\mathrm{so}}/\mu =0.9$, the nonzero ${\Delta }_z$ values correspond to ${\Delta }_{\min }<\mu <{\Delta }_{\max }$. Here the plasmon branch increases with ${\Delta }_z$ in the TI regime, reaches a maximum in the VSPM phase, and then decreases with increased ${\Delta }_z$ in the BI regime. Thus, the location of the plasmon branch in $(q,\omega )$ is highly dependent on the sublattice potential difference. Inset: The location of the band structure relative to the chemical potential (not drawn to scale). Note that the chemical potential is fixed and it is the band structure which is varying in energy.

Figure 11

Screened potential of a charged impurity in units of $Q/{\varepsilon }_0$ for $\alpha =0.8$, ${\Delta }_{\mathrm{so}}$, and varying ${\Delta }_z$. As a function of $r$, there are two primary decay rates: $1/r^3$ and $1/r^2$ for small and large $r$, respectively. With the inclusion of ${\Delta }_z$, a beating effect is observed with the distance between beats decreasing with increasing ${\Delta }_z$.