Plasmon dispersion and damping in electrically isolated two-dimensional charge sheets

Using high-resolution reflection electron-energy-loss spectroscopy (HREELS), we compare experimental results for the wave-vector-dependent behavior of plasmons in a graphene sheet on SiC(0001) with that due to a filled band of surface states on semiconducting silicon. There are significant differences in behavior between the two systems and the behavior predicted for a classical two-dimensional sheet of electrons. In particular, the damping increases with wave vector independent of any obvious inelastic scattering channel. The results illustrate the importance of finite-momentum, nonlocal potential effects for the dynamical behavior of electrically isolated charge sheets.

Figure 1

(Color online) Sheet-plasmon loss peaks of graphene dispersing with increasing momentum transfer $q$. The lines in magnified spectra at large momentum transfer $\left(>0.5{\text{Å}}^{-1}\right)$ are smoothed by averaging 3–5 nearest-neighbor points. The incident electron-beam energy is 20.29 eV.

Figure 2

(Color online) Scattering cross-section intensity varying with $q$. The angular distribution profile (inset diagram, red line) of loss intensity is strongly peaked around $1.5°$ to specular direction (green line). Scattering angle $\left(0°\le {\theta }_s<5°\right)$ is magnified by three times in polar coordinates to ensure visibility.

Figure 3

(Color online) Plasmon dispersion for (a) graphene of similar charge density, $\left(E_F=0.57\text{eV}\right)$ $n\sim 2\times {10}^{13}{\text{cm}}^{-2}$ and (b) $\text{Si}\left(111\right)\text{−}\left(\sqrt{3}\times \sqrt{3}\right)R30°\text{-Ag}$ surface-state band $\left(E_F=0.19\text{eV}\right)$, $n\sim 2\times {10}^{13}{\text{cm}}^{-2}$. The respective band structures (inset diagrams) show the origin of SPEs (1) intraband and (2) interband. Graphene is ambipolar with electron and hole carriers in separate $\pi$, ${\pi }^{\ast }$ bands. $q_c$ indicates the onset where plasmon enters the damping region. The blue dashed curves correspond to $\sqrt{q}$ dispersion of free 2D electron gas with the same electron density, and the red full curves correspond to best fits to experimental results.

Figure 4

(Color online) Plasmon damping [full width at half maximum (FWHM) of loss peak, Fig. 1] plotted as a function of momentum transfer $q$ for (G) graphene and (SS) surface-state 2D systems.