Energy-loss spectroscopy of ${\mathrm{C}}_{60}$ fullerenes with twisted electrons: Influence of orbital-angular-momentum transfer on plasmon generation

Recent experimental progress in creating and controlling singular electron beams that carry orbital angular momentum allows for new types of local spectroscopies. We investigate theoretically the twisted-electron-energy-loss spectroscopy (EELS) from the ${\mathrm{C}}_{60}$ fullerene. Of particular interest are the strong multipolar collective excitations and their selective response to the orbital angular momentum of the impinging electron beam. Based on ab initio calculations for the collective response, we compute EELS signals with twisted-electron beams, particularly in a transmission electron microscopy setup, and uncover the interplay between the plasmon polarity and the amount of angular momentum transfer.

Figure 1

Illustration of the plasmon modes by the area in the $x\text{−}z$ plane where the model fluctuation densities ${\rho }_{\nu L,M=0}\left(\mathbf{r}\right)>0$ (light green) and ${\rho }_{\nu L,M=0}\left(\mathbf{r}\right)<0$ (darker blue). (a) Volume plasmons $V1$ (top) and $V2$ (bottom), (b) symmetric surface plasmons for $L=1,2$, and (c) antisymmetric surface modes with $L=1,2$.

Figure 2

Multipeak fits of the frequency dependence ${\xi }_{\nu L}\left(\omega \right)$ of the density-density response function [cf. Eq. (14)] in terms of (a) and (b) the volume, (c) and (d) the symmetric surface, and (e) and (f) the antisymmetric surface plasmons. Solid gray lines show the TDDFT results ${\xi }_{\nu L}^{\mathrm{TDDFT}}\left(\omega \right)$ from Ref. [13]; solid red lines the fitted frequency dependence ${\xi }_{\nu L}^{\mathrm{fit}}\left(\omega \right)$ and dashed lines the individual peak contributions [see Eq. (17)].

Figure 3

Radial part $w_{{\ell }_{\mathrm{in}},{\ell }_{\mathrm{out}}}(q;R)$ of the effective potential for the typical value $q=0.01$ a.u., with $0\le {\ell }_{\mathrm{in}}\le 3$ and $0\le {\ell }_{\mathrm{out}}\le 3$.

Figure 4

Twisted-electron-energy-loss spectra when the beam axis coincides with the center of the ${\mathrm{C}}_{60}$ molecule (beam waist $W_0=30$ a.u.). (a) Fixed outgoing angular momentum $L_{\mathrm{out}}=0$ for different angular momenta $L_{\mathrm{in}}$ of the impinging beam. (b) Loss spectra for the same incoming angular momenta as in (a), but without fixing $L_{\mathrm{out}}$. The spectra have been normalized and shifted for better visibility.

Figure 5

Illustration of the (111) surface layer of ${\mathrm{C}}_{60}$ molecules (gray circles) along with the beam profile $F_{\ell }\left(R\right)$ (density plot) for (a) $\ell =0$ and (b) $\ell =1$. The direction of the phase variation is indicated by the black arrow.

Figure 6

Loss spectra for scattering from one layer of fcc ${\mathrm{C}}_{60}$ ($W_0=30$ a.u.) for different values of the OAM of the ingoing beam $L_{\mathrm{in}}$ and fixed outgoing OAM $L_{\mathrm{out}}=0$ (geometry as in Fig. 5). The curves have been normalized to the same frequency-integrated value. The region in the dashed rectangle is magnified in the bottom panel.

Figure 7

Loss spectra for scattering from one layer of fcc ${\mathrm{C}}_{60}$ as in Fig. 6, but without fixing the outgoing OAM $L_{\mathrm{out}}$.