First-principles calculations of plasmon excitations in graphene, silicene, and germanene

Plasmon excitations in graphene, silicene, and germanene are studied using linear-response time-dependent density functional theory within the random phase approximation (RPA). In this work, we examine both the plasmon dispersion behavior and lifetime of extrinsic and intrinsic plasmons for these three materials. For extrinsic plasmons, we found that their properties are closely related to Landau damping. In the region without single-particle excitation (SPE), the plasmon dispersion shows a $\sqrt{q}$ behavior and the lifetime is infinite at the RPA level, while in the SPE region, the plasmon dispersion shows a quasilinear behavior and the lifetime is finite. Moreover, for intrinsic plasmons, unlike graphene, the plasmon dispersion behavior of silicene and germanene exhibits a two-peak structure, which can be attributed to the complex and hybridized band structure of these two materials.

Figure 1

The structure and the geometrical parameters of (a) graphene (a = 2.45 Å), (b) silicene (a = 3.87 Å, $\mathrm{\Delta }$ = 0.47 Å), and (c) germanene (a = 4.05 Å, $\mathrm{\Delta }$ = 0.67 Å).

Figure 2

The upper panels (a), (b), and (c) shows the electronic band structure for graphene, silicene, and germanene. The black lines represent the occupied bands and the red lines represent the unoccupied bands. The lower panels (d), (e), and (f) shows the total density of states (TDOS) and orbital projected density of states (PDOS) for graphene, silicene, and germanene, respectively.

Figure 3

The energy-loss spectra of graphene (a), (d), silicene (b), (e), and germanene (c), (f) under 0.05 electrons/cell concentration. The q points lies in the range from 0.014/Å to 0.14/Å and 0.025/Å to 0.25/Å along $\mathrm{\Gamma }-M$ and $\mathrm{\Gamma }-K$ directions, respectively.

Figure 4

The extrinsic plasmon dispersion behavior along $\mathrm{\Gamma }-M$ and $\mathrm{\Gamma }-K$ directions for graphene (a), silicene (b), and germanene (c) under 0.05 electrons/cell concentration. (d) The reciprocal space of graphenelike structures, red and blue lines represent the $\mathrm{\Gamma }-K$ and $\mathrm{\Gamma }-M$ directions starting from $\mathrm{\Gamma }$ or K point, respectively.

Figure 5

The 2D Dirac plasmon dispersions (2DP) of graphene (a), silicene (b), and germanene (c) under $\pm 0.05$ and $\pm 0.1$ dopings along $\mathrm{\Gamma }-M$ direction. The black, blue, red, and green dashed lines represent the single-particle excitation (SPE) boundaries under 0.05, −0.05, 0.1, and −0.1 electrons/cell doping concentrations, respectively. The pink and brown lines within no-SPE region correspond to $\sqrt{q}$-form dispersion of free 2D electron gas. One can clearly observe the nearly $\sqrt{q}$ hehavior without SPE and quasilinear behavior within SPE.

Figure 6

The 2D Dirac plasmon dispersions of graphene, silicene, and germanene under 0.05 (a), 0.1 (b), -0.05 (c), and -0.1 (d) electrons/cell doping concentrations.

Figure 7

The FWHM of 2D Dirac plasmon peaks for graphene (a), silicene (b), and germanene (c) under $\pm 0.05$ and $\pm 0.1$ doping concentrations. (d) The range of the lifetime transition point (red bar) compared to cutoff vectors $q_c$ (black dot) derived from Table 2. From left to right shows the result for graphene, silicene, and germanene, respectively, and in each part, the data are arranged in 0.05, 0.1, -0.05, and -0.1 electrons/cell order.

Figure 8

The FWHM of 2D Dirac plasmon peaks for graphene, silicene, and germanene under 0.05 (a), 0.1 (b), -0.05 (c), and -0.1 (d) doping concentrations.

Figure 9

The overall electron energy loss spectroscopy (EELS) of graphene (a), silicene (b), and germanene (c) along the $\mathrm{\Gamma }-M$ direction. The q points lies in the range from 0.014/Å to 0.14/Å.

Figure 10

The electron energy loss spectroscopy (EELS) of intrinsic graphene (a), silicene (b), and germanene (c) in smaller energy window along $\mathrm{\Gamma }-M$ direction. The corresponding transitions for peaks are labeled thereon.

Figure 11

Dispersion behavior of the peak positions as a function of q, extracted from the $\pi$-like peaks in $\mathrm{Im}{\chi }_{00}$ and $\mathrm{Im}{\chi }_{00}^0$ for silicene along $\mathrm{\Gamma }-M$ direction.

Figure 12

The real part of the dielectric function ($\mathrm{Re}\epsilon$) for silicene along $\mathrm{\Gamma }-M$ direction. From left to right, $\eta$ is set to be 0.005, 0.0005, and 0.00002 Ry, respectively. The upper panels (a), (b), and (c) show the result of using gradual form[Eq. (A2)] and the lower panels (d), (e), and (f) show the result of using the 2D form [Eq. (A1)]. The q points along the $\mathrm{\Gamma }-M$ direction lies in the range from 0.014/Å to 0.14/Å.

Figure 13

The peak positions of Im(${\epsilon }^{-1}$) and the Re($\epsilon$) = 0 locations with different $\eta$ values for silicene 2D Dirac plasmon at $q=0.0147$/Å (a), and for silicene $\pi$-like plasmon at $q=0.1341$/Å (b) along the $\mathrm{\Gamma }-M$ direction.

Figure 14

(a) The FWHM of the conventional 2D Dirac plasmon peaks as a function of q along the $\mathrm{\Gamma }-M$ direction for different $\eta$ values from 0.001 to 0.005 Ry. (b) The extrapolating result of FWHM values to the limit $\eta \to 0$ for a silicene Dirac plasmon at 0.05 electrons/cell doping concentration along the $\mathrm{\Gamma }-M$ direction.