Core-level spectra from bilayer graphene
Bo E. Sernelius
FlatChem 1 (2016) 6–10
We derive core-level spectra for doped free-standing bilayer graphene. Numerical results are presented for all nine combinations of the doping concentrations 1012 cm-2, 1013 cm-2, and 1014 cm-2 in the two graphene sheets and we compare the results to the reference spectra for monolayer graphene. We furthermore discuss the spectrum of single-particle inter-band and intra-band excitations in the -plane, and show how the dispersion curves of the collective modes are modified in the bilayer system. In an earlier work we derived core-level spectra for single free-standing graphene sheets. Our derivations were based on a model used by Langreth for the core–hole problem for metals in the seventies. In a metal, shake-up effects involving collective excitations (plasmons) but also single-particle excitations can take place. The plasmons cause plasmon replicas of the main peak. The electron–hole pair excitations form a continuum, starting from zero frequency and upward. These excitations lead to a deformation, including a low-energy tail, of both the main peak and the plasmon replicas. The low-energy tailing is a characteristic of a metallic system, i.e. a system where the chemical potential is inside an energy band and not in a band-gap.
We addressed pristine and doped graphene in the earlier work. In the pristine case the chemical potential is neither inside an energy band nor in a band-gap. The Fermi surface is just two points in the Brillouin zone. This makes this system special. We found that there is still a low-energy tailing. In the doped case the chemical potential is inside an energy band and we would expect to find, and find, a tailing. However, the 2D character of the system means that the collective excitations are 2D plasmons with a completely different dispersion than in the ordinary 3D metallic systems. The 2D plasmons give contributions that start already from zero frequency and upward. This means that they contribute to the tail and no distinct plasmon replicas are distinguishable. Our theoretical results very well reproduced experimental results, both for pristine and doped graphene.
In the excitation process the photoelectron leaves the system and a core hole is left behind. The shape of the XPS spectrum depends on how fast the process is. One may use the adiabatic approximation if the process is very slow. In that approximation one assumes that the electrons in the system have time to relax around the core hole during the process. When we derive the XPS line shape we go to the other extreme and assume that the excitation process is very fast; we use the sudden approximation. In this approximation the core–hole potential is turned on instantaneously. The electrons have not time, during the process, to settle down and reach equilibrium in the potential caused by the sudden appearance of the core hole. This results in shake-up effects in the form of single particle excitations and collective excitations.
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