Diploma - Max Planck Institute for Solid State Research

4.2 Hybridization: localized versus itinerant states 45
culation the localized 4f electrons are treated as core orbitals, we introduce additional,
higher-lying 5f-orbitals to the basis set of the Eu sphere, because they only differ in the
radial wave function in comparison to the 4f orbitals. Bonds 5 of 4f-like electrons with
the itinerant VB should be apparent by additional 5f weight inside the Eu sphere after
the self-consistency process. Doubtless, this weight depends particularly on the chosen
ratio of the atomic spheres and therefore the result can only be regarded in a qualitative
way. Thus different compounds are only comparable if their spherical ratios are similar.
But since the ratios are used to adopt the calculated band structure to experiment, it
is a firm task.
The weight’s distribution of the Eu 5f is depicted in fig. 4.16a for Si and Eu terminated
surfaces considering surface and subsurface emission. The respective Eu spheres from
which the 5f weight has been taken are highlighted in the insets. The surface originated
states S1 (Si) / S3 (Eu) are shifted to higher binding energies for both configurations,
but the bulk projected bands are similar to that of the FPLO calculations. Consequently
it is not possible to use the coefficients for the surface band structure directly which
could be related with the states shown in the PE spectrum. A comparison between the
Rh 4d characters of FPLO and the distribution of the 5f weight in LMTO is used as a
starting point for the modelling of the coupling strength assuming that a large portion
of 5f weight initially stems from Rh 4d which in fact is supported by the WF analysis.
To rate the hybridization you thus use largly the distribution of the 5f weight of LMTO
and shift the corresponding surface bands / states according to the results obtained by
FPLO. This will be discussed seperately for both configurations:
(a) The 5f weight distribution for the Si terminated surface is dominated by two
triangularly-shaped areas A1 and A2 (see fig. 4.16a).
These are, if we compare
them to the FPLO calculation, based on the Rh 4d yz surface and bulk component
for silicon terminated surfaces (cf. fig. 4.16b). Since the number of bands in the
projected band structure of a slab calculation depends on the size of the slab, an
analytical approximation (given below) is used. For A1 a superposition of parabola
and for A2 a linear combination of cosine is applied. To emphasize the origin as
projected band structure, calculations with variable number of bands are made (cf.
fig. 4.19a-c). Moreover, the surface state S1 is shifted according to the FPLO calculation
(mainly Rh 4d xy ), so the nodal point appears 0.2 eV above the Fermi level.
Additionally, the weight of the projected bulk band structure below the surface
state S1 is taken into account (cf. Rh 4d xy ), because the 5f weight of LMTO in
this region is shared between several eigenenergy values for one k-point. The resulting
distribution of the coupling parameter V ij (k) in Γ-X direction is presented
in fig. 4.17a.
used dispersions (k ‖ in [π/a x ] and ɛ(k ‖ ) in [eV]):
5 bonds illustrate overlapping wave functions of different sites, which are signatures of “charge shifts”
inside solids compared to spherical symmetric atomic configurations