Diploma - Max Planck Institute for Solid State Research

4.2 Hybridization: localized versus itinerant states 51
the VB yields a more appropriate model [67] (denoted by model 2 “superposition of
localized levels”).
To illustrate this distinction, both models are presented in fig. 4.18. Exemplarily, two
localized levels differing by a factor of two in spectral intensity and a hole-like parabolic
VB with its apex located 0.2 eV above the Fermi level are taken. On the left-hand side
in (a) as well as in (b) the initial configuration is depicted followed by the rearranged
eigenenergy distribution. A sketch of both components in model 2 the sum of which
represents the spectrum, clarifies the degeneracy of the spectral eigenvalues in regions
of the VB separated from the localized levels. The main disparity between both is the
occurance of a band gap in the case of coupled localized levels, which is not present in
case of superposition. Furthermore, due to the constant symmetry parameter C ij also
the localized states in model 1 will be pushed apart if no valence band approaches. A
feasible correction would be to make it proportional to the distance between the valence
band and the second localized level or solving the model self-consistently re-adjusting
the input parameters of the localized levels so the solution fits in the limit of ɛ i → ∞
to the original PE spectrum.
In the following part, the motivated model 2 for europium (hence C ij (k) = 0 and
N f = 1) will be evaluated for Si and Eu terminated surfaces. As input for the subsurface
PE levels the spectrum calculated by Gerken et. al shifted by 0.15 eV is used. In the
case of a Eu terminated surface, the surface 4f emission is chosen proportional to the
subsurface emission adopted in intensity (6x) and energy position (shifted by 1 eV to
higher binding energies) to the experiment. The distribution of V ij (k), the evolution
of which already has been sketched, is given in fig. 4.17. To verify the coupling to the
projected band structure for the Si terminated surface, different numbers of bands have
been taken to simulate the transition from a single band to a continuously-projected
band structure. On the one hand, this effect displaces spectral weight to the edges of the
region where the VB is located and on the other hand distributes the residual weight in
the coupled area yielding an almost homogeneous intensity distribution. In fig. 4.19, this
process is indicated for the triangularly shaped area A1 next to the linear dispersive state
at Γ. The final spectra for Si and Eu terminated surfaces in Γ−X direction are depicted
in fig. 4.20. The main spectral features are well-reproduced for both terminations.
Besides, the measured spectrum for Si termination evidences additional interaction with
the projected band structure especially in the region marked by 2. Furthermore, there
has to be a some part of the band structure which causes the accumulation of spectral
weight at the tip of the state S1 (1) not evidenced in the calculated spectrum. Because
only f emission is taken into account in the simulation, the weak VB like contribution (3)
is missing. For the Eu terminated surface, the shift of surface 4f emission with respect
to the subsurface emission has been estimated imprecisely, as well as its width. The
latter is probably related to a different multiplet structure, because the potential is no
longer rotationally symmetric as it can be regarded in bulk as a first approximation.