Measuring unoccupied electron states in van der Waals materials

We study van der Waals (vdW) systems using low-energy electron microscopy (LEEM) and spectroscopy. Specifically, reflection vs. energy, i.e. R(E) spectra, give us direct information on the local layer structure with ~2 nm resolution. For graphene and hBN an unoccupied interlayer state is added with each additional layer. These interlayer states hybridize, such that the number of states scales with the number of layers (See Fig. 1). If an incoming electron is resonant with a state, it can be transmitted into the sample, leading to a minimum in R(E). Hence, the local R(E) curve gives direct information on the number of layers and their stacking [1-3]. We have taken this concept one step further and study the 2D-dispersion relations of these unoccupied bands. For that, we have developed ‘angle-resolved reflected-electron spectroscopy’ (ARRES) [2,3]. 
Interestingly, whereas at resonance reflection minima are expected, maxima are anticipated in transmission. To test this, we created eV-TEM, i.e. transmission EM at very low energies (0-100 eV) [1]. Indeed, the transmission vs. energy T(E) curves for freestanding graphene show maxima at the interlayer state energies. Moreover, the combination of T(E) and R(E) allows us to study inelastic path lengths for electrons of these energies [1].
Summarizing, we are able to measure the unoccupied band structure of vdW materials (above Evac) directly, and we can do that at the nanoscale. The latter is a necessity when studying heterogeneous and/or twisted vdW systems, such as twisted bilayer graphene [4,5].