Diploma - Max Planck Institute for Solid State Research

4.2 Hybridization: localized versus itinerant states 47
Figure 4.17: coupling strength depending
on the surface termination
(a) hybridization strength for the
Si terminated surface as a superposition
of surface and projected bulk
contributions
(b) weight distribution of the hybridization
for Eu terminated surface
which is mainly based on surface
contributions
white S1 : ɛ(k ‖ ) = −5.8 · k ‖ + 0.1
white S1 projected: ɛ(k ‖ ) = −5.8 · k ‖ − 0.5 · i/30, i ∈ [0, 29]
white A1 : ɛ(k ‖ ) = 23 · (k ‖ − 0.25) 2 − 0.75 + i/50, i ∈ [0, 49]
white A2 : ɛ(k ‖ ) = 0.03 · cos(30 · k ‖ ) − 0.6 · i/26 − 0.15, i ∈ [0, 25]
(b) Since the surface and subsurface emission are originated in different slab layers for
Eu termination, one has to evaluate both positions. The two distributions depicted
in fig. 4.16a are rather similar, thus in both cases S2 and S3 are dominating
(see fig. 4.16a). Both are shifted in comparison to the Rh 4d xy and Rh 4d yz surface
contribution, so their nodal points remain below the Fermi level. Due to the similarity
the 5f weights were not chosen differently for surface and subsurface emission.
The corresponding distribution is depicted in fig. 4.17 b.
used dispersions (k ‖ in [π/a x ] and ɛ(k ‖ ) in [eV]):
whiteS3 : ɛ(k ‖ ) = −5.8 · k ‖ − 0.3
whiteS2 part1 : ɛ(k ‖ ) = −30 · k‖ 2 ⎧
− 0.6
⎪⎨
−5.8 · k ‖ − 0.1 ‖k ‖ ‖ < 0.19
whiteS2 part2 : ɛ(k ‖ ) = −5.8 · k ‖ + 2.5 · (k ‖ − 0.19) 2 − 0.2 0.19 < ‖k ‖ ‖ < 0.38
⎪⎩
−1.6 ‖k ‖ ‖ > 0.38
In the subsurface layer, additional weight of projected band structure can be seen,
which seem to have minor influence (in comparison to the measured spectrum, see
fig. 4.8). At a first glance it is not obvious, that in case of Eu termination no
effect of the projected band structure is observed. It may be argued, that since
the mean free scattering path of electrons is in the same range as the depth of
the subsurface layer for Eu termination, the influence of the bulk band structure is
much weaker in comparison to the Si termination. Perhaps it can be evaluated at
higher photon energies increasing the mean free path if the instrumental resolution
admits to resolve the final state multiplet structure.
In the following, the motivated coefficients V ik (k) will be used to introduce an interaction
between the two electronic subsystems. Due to the analytical expressions, not
all details have been taken into account.