Fig. 5

Effect of electron beam geometry and number of layers studied using time-dependent density-functional-theory (TDDFT) in the random phase approximation (RPA) and compared to experimental monochromated LL-STEM-EELS. a EEL spectra of 1–3 layers of freestanding MoS₂ found using RPA. The q-resolved RPA spectra shown here are calculated for the q _|| in the Γ-M direction of the 2D Brillouin zone and normalised to the number of layers. RPA gives a good description of plasmons but not of excitons due to exclusion of electron–hole interactions, essential for excitons. For q _|| = 0 no difference was found with increasing N. Once the corrections are applied, the energy-loss is found to set in at ~ 3 eV (iii). For larger q, however, the spectra diverge for increasing N. Hence for non-zero q, peaks associated with plasmons come in at ~ 8.3 eV (iv) and ~ 14 eV (v). The peaks in the spectra associated with the plasmons also blueshift. Hence, the differences in the spectra with increasing number of layers as observed experimentally are due to beam geometry. The beam is probing a large range of q in plane across the sample, however, as the structure becomes 3D (with increasing number of layers) the beam is also increasingly probing out-of-plane excitations and as a consequence, the shape of experimental spectra changes with increasing number of layers. b The spectra summed over q contributions is shown here with q _max = 2.0A-1. When comparing the spectrum for q = 0 and the spectrum for summed q up to q _max, there is a peak at ~ 8.3 eV (iv) as well as at ~ 14 eV (v) that become prominent in the latter spectra. When compared to experimentally acquired spectra as shown in c, the peaks at ~ 3 eV (peak iii), the peak at ~ 8.3 eV (peak v) as well as the peak at ~ 14 eV (peak v) show a nice correspondence between experiment and calculations. In addition, the ~ 3 eV peak (iii) is present in all of the experimental as well as calculated spectra. The peak at ~ 8.3 eV is present in our calculated spectra for finite q as well as in some of the experimental spectra, where the peak appears and increases with increasing number of layers, supporting our earlier hypothesis about the beam probing an increasing contribution from q out-of-plane with increasing thickness (increasing number of layers N). d The spectra were also calculated for increasing number of layers N (for N = 1,2,3,5). The contribution from q = 0 spectrum to the summed spectra grows proportional with N. The plasmon peaks were also found to blueshift with increasing N