Nonuniversal Transverse Electron Mean Free Path through Few-layer Graphene

In contrast to the in-plane transport electron mean-free path in graphene, the transverse mean-free path has received little attention and is often assumed to follow the “universal” mean-free path (MFP) curve broadly adopted in surface and interface science. Here we directly measure transverse electron scattering through graphene from 0 to 25 eV above the vacuum level both in reflection using low energy electron microscopy and in transmission using electronvolt transmission electron microscopy. From these data, we obtain quantitative MFPs for both elastic and inelastic scattering. Even at the lowest energies, the total MFP is just a few graphene layers and the elastic MFP oscillates with graphene layer number, both refuting the universal curve. A full theoretical calculation taking the graphene band structure into consideration agrees well with experiment, while the key experimental results are reproduced even by a simple optical toy model.

Figure 1

(a) Sketch of the ESCHER setup combining two electron guns for reflection (LEEM, red) and transmission (eV-TEM, blue) experiments. (b) Angle-resolved reflected-electron spectroscopy of exfoliated bulk graphite showing electronic bands as minima and band gaps as maxima in electron reflection in agreement with band structure calculations (black lines). Reproduced from Ref. [11]. (c) LEEM image of a free-standing membrane of 1, 2, and 3 layer graphene. (d) The same area imaged in eV-TEM. Electron energy (indicated in to top right) in both images is chosen for optimal contrast.

Figure 2

(a) Electron reflectivity $R(E_0)$ as a function of landing energy $E_0$ on 1–4 layer graphene. A general decreasing trend with strongly layer-dependent oscillations is observed. (b) The electron transmissivity $T(E_0)$ from the same areas also decreases with energy, but exhibits maxima where $R(E_0)$ has minima. (c), (d) Theoretical predictions of $R(E_0)$ and $T(E_0)$ obtained by ab initio methods [18, 19] reproduce the experimental data in all key features.

Figure 3

(a) Energy-dependent total MFP ${\lambda }_{\mathrm{tot}}$ for electrons impinging on 1–4 layer graphene obtained from the spectra in Fig. 2 using Eq. (1). It decreases with energy, but is generally much lower than the universal curve predicts. (b) The inelastic MFP ${\lambda }_{\mathrm{inel}}$ [Eq. (2)] shows a similar trend but the layer-dependent maxima are absent. (c) The elastic MFP ${\lambda }_{\mathrm{el}}$ [Eq. (3)] exhibits strong, layer-dependent peaks at the energies of high-transmission states where ${\lambda }_{\mathrm{el}}$ is very long. The enlarged inset shows that the ${\lambda }_{\mathrm{el}}$ is largest for monolayer graphene in the energy range from 6 to 12 eV. (d) An optical toy model based on transfer matrices describes position and shape of the layer dependent oscillations in the ${\lambda }_{\mathrm{el}}$ well. (e) When an energy-dependent absorption is taken into account, the experimental data in (c) is well described by this simple model over the full energy range. (f) The ${\lambda }_{\mathrm{el}}$ calculated from the ab initio spectra in Figs. 2, 2 also reproduces the experiment well over the full energy range.