Low-energy electron reflectivity from graphene

Low-energy reflectivity of electrons from single- and multilayer 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 freestanding $n$-layer graphene slab produces $n-1$ 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.

Figure 1

Computed reflectivity for freestanding slabs of $n$-layer graphene. For each $n,$ a series of computations are performed with different vacuum widths; differently shaded (colored) data points are used for plotting the results for each width.

Figure 2

Wave functions at the energies of transmission resonances for $n=4$ layers of freestanding graphene. The real part of the wave function is shown by the thin solid line, the imaginary part by the thin dashed line, and the magnitude by the thick solid line (blue, red, and green, respectively, in the color version). The solid black dots indicate the positions of the graphene layers. The $+$, −, and 0 symbols indicate peaks in the wave functions that are concentrated between the graphene layers.

Figure 3

Measured electron reflectivity for single and multiple graphene layers (GL) on SiO${}_2$; curves are vertically offset for clarity. The LEEM image in the inset (acquired at energy 1.8 eV above the sample's vacuum level) shows the surface from where the reflectivity spectra were acquired.