Low-energy electron reflectivity from graphene

Abstract

Low-energy reflectivity of electrons from single-and multi-layer graphene is examined both theoretically and experimentally. A series of minima in the reflectivity over the energy range of 0 -8 eV are found, with the number of minima depending on the number of graphene layers. Using first-principles computations, it is demonstrated that a free standing n-layer graphene slab produces 1  n reflectivity minima. This same result is also found experimentally for graphene supported on SiO 2 . For graphene bonded onto other substrates it is argued that a similar series of reflectivity minima is expected, although in certain cases an additional minimum occurs, at an energy that depends on the graphene-substrate separation and the effective potential in that space.

Layered Two-Dimensional Heterostructures and Their Tunneling Characteristics

“…For example, for 3 graphene layers there are 2 interlayer spaces and hence 2 re ectivity minima. [48] The theoretical spectrum including inelastic e ects shown in Fig. 3.5(b) is in good agreement with experiment.…”

“…signi cant interlayer character) are revealed by the relatively large symbol sizes. As is well known from prior work, [43,48] in graphite there is only a single band with interlayer character, the one labeled interlayer at the top of Fig. 3.5.…”

“…As rst discussed by Hibino et al [43,64] and extensively analyzed in our prior work, [48,58,59,65] the occurrence of minima in low-energy electron re ectivity spectra is associated with interlayer states that occur between the 2D planes of van der Waals (vdW) bonded materials. Such interlayer states arise from the image-potential states that exist on either side of a single 2D layer, [66,67] i.e.…”

“…(3.1) and that used previously in Ref. 48 is that the former refers to states with ±k z whereas the latter referred to even and odd states formed by linear combinations of the ±k z states. However, the resulting values for σ are identical for both cases.…”

“…Low-energy electron re ectivity (LEER) spectra of free-standing slabs of multilayer 2D materials are computed using a method described previously. [48,58] Inelastic e ects are included in the computations, [59] employing an imaginary part of the potential, V i . Following the detailed analysis of Krasovksii and co-workers, [60 62] in our prior work we employed the phenomenological expression V i = 0.4 eV + 0.06E where E is the energy of a state relative to the vacuum level.…”