High-energy collective electronic excitations in free-standing single-layer graphene

In this joint experimental and theoretical work, we investigate collective electronic excitations (plasmons) in free-standing, single-layer graphene. The energy- and momentum-dependent electron energy-loss function was measured up to $50\mathrm{eV}$ along two independent in-plane symmetry directions ($\Gamma M$ and $\Gamma K$) over the first Brillouin zone by momentum-resolved electron energy-loss spectroscopy in a transmission electron microscope. We compare our experimental results with corresponding time-dependent density-functional theory calculations. For finite momentum transfers, good agreement with experiments is found if crystal local-field effects are taken into account. In the limit of small and vanishing momentum transfers, we discuss differences between calculations and the experimentally obtained electron energy-loss functions of graphene due to a finite momentum resolution and out-of-plane excitations.

Figure 1

Experimental momentum-dependent energy-loss function of free-standing single-layer graphene along the two nonequivalent symmetry directions $\Gamma M$ (a) and $\Gamma K$ (b). The experimental setup is sketched on the right: (c) inelastic scattering geometry in a TEM; (d) electron diffraction pattern of graphene with an illustration of the used (virtual) off-axis aperture geometry; (e) false-color image of a recorded $\omega$-$q$-map for single-layer graphene. Each line spectrum at position $q$ was extracted by binning (shaded area) over the binning width $\Delta q_x$, which corresponds to a (virtual) off-axis aperture $\Delta q_x\times \Delta q_y$ indicated in (d).

Figure 2

Ab initio calculations of the energy-loss function in graphene for different momentum transfers $q$ in the $\Gamma K$ direction. We compare theoretical results for different approximations (lines, see text for details) with corresponding experimental spectra (shaded area, normalized by a common factor).

Figure 3

Plot of contributing in-plane momentum transfers $|\overline{q}|=\sqrt{q_x^2+q_y^2}$ with their corresponding weights for two different-sized (virtual) off-axis apertures: $\Delta q_x=0.1{\AA }^{{}_{-1}}$ (solid line) and $\Delta q_x=0.01{\AA }^{{}_{-1}}$ (dashed line). We have used the slit width $\Delta q_y=0.25{\AA }^{{}_{-1}}$ and $q_E=0.006{\AA }^{{}_{-1}}$. The distributions are plotted at different positions $q$ along the axis $q_x$. Intensities have been scaled to 1 for all shown curves.

Figure 4

Effect of different-sized collection apertures on the experimental (a) and calculated (b) energy-loss function. Solid lines correspond to $\Delta q_x=0.1{\AA }^{{}_{-1}}$ and dashed lines to $\Delta q_x=0.01{\AA }^{{}_{-1}}$. At $q=0.1{\AA }^{{}_{-1}}$ a relative shift of the experimental plasmons peaks of $0.2\mathrm{eV}$ can be observed. This is accurately reproduced by corresponding calculations.

Figure 5

(a) Experimental energy-loss function of graphene along $\Gamma M$ extracted with a small binning width $\Delta q_x$ of $0.01{\AA }^{{}_{-1}}$. Intensities are plotted on a logarithmic scale. A peak between 30 and $45\mathrm{eV}$ is clearly visible for spectra below $q=0.1{\AA }^{{}_{-1}}$. (b) 2D intensity plot for the weights of contributing $q_{\text{tot}}$ with certain angles $\alpha$ at different aperture positions $q$ along the slit axis $q_x$. (c) Line profile extracted from (b) at position $q=0{\AA }^{{}_{-1}}$ for an aperture with a binning width $\Delta q_x$ of $0.01{\AA }^{{}_{-1}}$. The majority of contributing momentum transfers $q_{\text{tot}}$ are associated with an angle $\alpha$ smaller than ${45}^{\circ }$.