Abstract
In the framework of dielectric theory, the static non-local self-energy of an electron near an ultra-thin polarizable layer has been calculated and applied to study binding energies of image-potential states near free-standing graphene. The corresponding series of eigenvalues and eigenfunctions have been obtained by numerically solving the one-dimensional Schrödinger equation. The image-potential state wave functions accumulate most of their probability outside the slab. We find that the random phase approximation (RPA) for the non-local dielectric function yields a superior description for the potential inside the slab, but a simple Fermi–Thomas theory can be used to get a reasonable quasi-analytical approximation to the full RPA result that can be computed very economically. Binding energies of the image-potential states follow a pattern close to the Rydberg series for a perfect metal with the addition of intermediate states due to the added symmetry of the potential. The formalism only requires a minimal set of free parameters: the slab width and the electronic density. The theoretical calculations are compared with experimental results for the work function and image-potential states obtained by two-photon photoemission.
Highlights
Graphene layers display a number of interesting properties and potential applications owing to the linearly dispersing bands found near the K point in the Brillouin zone [1, 2]
Using standard models for the dielectric response and the reflection of electromagnetic waves at a surface we have computed the static self-energy for an ultra-thin slab mimicking a graphene layer
For the purpose of obtaining image-potential state binding energies we find that FT makes an excellent and convenient approximation to the accurate random phase approximation (RPA)
Summary
Graphene layers display a number of interesting properties and potential applications owing to the linearly dispersing bands found near the K point in the Brillouin zone [1, 2]. It is useful and natural to search for simpler ways to obtain such an effective potential, which is the basic ingredient needed to understand the physics of image-potential states bound by an ultra-thin polarizable layer like graphene The simplest of these alternatives is to introduce a set of fitting parameters to continuously join solutions valid either inside or outside the solid. Two free parameters are introduced, i.e. the electronic polarizability of the thin slab, which is determined by the Fermi–Thomas (FT) screening wavelength inside the slab (electron density of the material), and a geometrical dimension given by the layer thickness This approach leads in a natural way to a potential with proper physical features: it is continuous and finite over the full spatial domain and it has the right asymptotic behavior toward the vacuum region. The experimental data from two-photon photoemission (2PPE) are in agreement with the calculated dependence
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