Photoelectron  spectroscopy

In the method of photoelectron spectroscopy electrons are excited from the core levels or occupied valence band states and are analysed by their kinetic energy giving a replica of the electronic structure of solids. Analysis of the core levels (x-ray photoelectron spectroscopy, XPS) gives information about the chemical states of elements that might be influenced by the corresponding atomic coordination. In the case of the angle-resolved photoelectron spectroscopy (ARPES) the electrons emitted from the valence band are additionally analysed by their emission angle ($\,\theta\,$) and detected by the 2D detector giving photoemission intensity map for the particular k-direction in the Brillouin zone, which are used for the analysis of the electronic structure of the studied objects.

There are two straightforward ways to model XPS. The initial-state approximation assumes that the core electron transition is much faster that any following relaxation process. Thus the core state binding energy appears to be completely unscreened and (according to the Koopmans theorem) is equal to the Kohn-Sham orbital energy. This is realised in two cases: either when the photoelectron has a very high kinetic energy and leaves the sample so fast that the system does not have time for response or when there are no available electrons in the valence band to screen the core-hole. Therefore, the initial-state approximation is a good approach for isolators and semiconductors. The final-state approximation is an opposite limit case when the photon-electron interaction is adiabatically slow and the electronic subsystem transfers from the “old” ground state to the “new” one before the x-ray transition takes place.This approximation is a good approach for metallic systems where the conduction electrons screen the core-hole immediately. There is an intermediate approach (based on the Slater-Janak transition state equation) where only a half electron is excited and the obtained system is relaxed to its ground state.

In many cases experimentally obtained angle-resolved photoemission spectroscopy (ARPES) data can be directly compared with the respective results of the band-structure calculations. For example, in case of graphene and graphene/metal interfaces (if quasiparticle excitations, like phonons or low-energy plasmons, in the close vicinity to the Fermi level are not considered) one can perform a direct comparison of the experimental and DFT calculated energy dispersions for the graphene-related $\pi$ and $\sigma$ bands (electron correlations in graphene are very small due to the delocalased nature of the respective electrons). In the general case, because of a large mismatch between graphene and metal lattice constants, a supercell approach is used to model graphene-metal interfaces. The latter brings in an inconvenience: The folding of the bands into the smaller supercell Brillouin zone gives rise to complicated band structures. Thus, in order to make such comparison straightforward, the band unfolding procedure has to be applied to the graphene/metal long-range structures. Such procedure can be performed with the BandUP code that allows to unfold the band structure of the system on the $(1\times 1)$ primitive unit cells of the respective symmetry (graphene or metallic substrate).