Probing mesoscopic crystals with electrons: One-step simultaneous inelastic and elastic scattering theory

Inelastic scattering of the medium-energy ($\sim 10–100$ eV) electrons underlies the method of the high-resolution electron energy-loss spectroscopy (HREELS), which has been successfully used for decades to characterize pure and adsorbate-covered surfaces of solids. With the emergence of graphene and other quasi-two-dimensional (Q2D) crystals, HREELS could be expected to become the major experimental tool to study this class of materials. We, however, identify a critical flaw in the theoretical picture of the HREELS of Q2D crystals in the context of the inelastic scattering only (“energy-loss functions” formalism), in contrast to its justifiable use for bulk solids and surfaces. The shortcoming is the neglect of the elastic scattering, which we show is inseparable from the inelastic one, and which, affecting the spectra dramatically, must be taken into account for the meaningful interpretation of the experiment. With this motivation, using the time-dependent density functional theory for excitations, we build a theory of the simultaneous inelastic and elastic electron scattering at Q2D crystals. We apply this theory to HREELS of graphene, revealing an effect of the strongly coupled excitation of the $\pi +\sigma$ plasmon and elastic diffraction resonances. Our results open a path to the theoretically interpretable study of the excitation processes in crystalline mesoscopic materials by means of HREELS, with its supreme resolution on the meV energy scale, which is far beyond the capacity of the now overwhelmingly used EELS in transmission electron microscopy.

Figure 1

Illustration of the reflection (a) and transmission (b) geometries of the EELS experiment on a Q2D crystal.

Figure 2

Calculated EEL reflection spectrum of monolayer graphene [thick black line: Eq. (5)], the one calculated with the use of the surface loss function [dashed line: Appendix pp1, Eq. (A3)], and the coefficient of reflection (thin blue line plotted against the right $y$ axis). For better visualization, EEL spectra are split into two parts, the low-energy one scaled by 0.1.

Figure 3

Calculated EEL transmission spectrum of monolayer graphene [thick line: Eq. (5)], the one calculated with the use of the loss function [dashed line: Eq. (4)], and the coefficient of transmission.

Figure 4

Calculated EEL spectrum of monolayer graphene (solid lines) and the experimental EELS in TEM (circles). Experimental data are digitized from Ref. [30].

Figure 5

Calculated EEL spectrum of monolayer graphene (solid line) and the experimental EELS in TEM (circles). Experimental data are of Ref. [29] (digitized from Ref. [32]). Inset shows the spectrum calculated using the loss function of Eq. (4) (neglect of the elastic scattering).

Figure 6

Calculated EEL reflection spectrum of bilayer graphene (thick black line), loss function (dashed line), and the coefficient of reflection (blue). In contrast to the monolayer case, $\pi$ plasmon is influenced too by the inclusion of the elastic scattering.

Figure 7

Calculated EEL transmission spectrum of bilayer graphene (thick black line), loss function (dashed lines), and the coefficient of transmission (red). In contrast to the monolayer case, $\pi$ plasmon is influenced too by the inclusion of the elastic scattering.

Figure 8

Calculated EEL spectrum of graphene (smooth solid lines) and the experimental EELS in TEM (noisy lines). Experimental data are from Ref. [31].