Diploma - Max Planck Institute for Solid State Research

32 4 EuRh 2 Si 2 – semi-localized electrons
(b) first-principle force and energy minimization [47]
Stretching the z-axis of a supercell (1 × 1 × n, n ∈ N ) and performing force
(relaxation) minimization for each configuration yields different interlayer distances
for Eu-Si and Si-Rh because of their different bond strength. For a z > a z, crit.
either the Eu-Si or Rh-Si bond will break up and two surfaces will emerge (which
corresponds to a slit in the crystal). The resulting interlayer distances as functions of
the z-axis lattice constant a z are depicted in fig. 4.6. Obviously the bond strength of
Eu-Si is smaller than that of Rh-Si, because the distance of the former is increasing
more intensely than that of the latter with growing strain. For a z being larger than
the critial lattice constant a z, crit. ≈ 28 Å, the evolution of two surfaces – either
terminated by Eu or Si – can be observed (cf. fig. 4.6c). Besides, one can note that
the distance of the topmost layer (labelled as surface in fig. 4.6a) is smaller than
the corresponding interlayer distance in the bulk. This relaxation due to bonding
asymmetry sometimes severely influences the available surface features.
It has to be mentioned, that this method presumes a large amount of processing
power, because before each force optimization step a self-consistency calculation has
to be performed and the process can diverge if the energy self-consistency cycle does
not converge sufficiently (usually the first self-consistency cycles in a stretched cell
do not achieve the same convergence criteria in the limits of iteration and deviation
in charge density in comparison to the bulk calculation, but if the density at least
converges linearly, the next force minimization mostly does not diverge). For this
calculations a moderate k-mesh of 8 × 8 × 6 (a compromise between accuracy and
computing time) as well as SPG 99 has been used, because the latter does not
reflect the mirror symmetry of the z-axis compared to SPG 123. All atom positions
were allowed to be varied, therefore the slit position is totally arbitrary (it depends
additionally on the convergence algorithm). Because our processing resources were
limited, the force minimization was stopped after 4 weeks having reached an overall
minimum force F min ≈ 0.1 eV/Å. Although the minimization has not been finished,
the results are representative. Moreover it should be noted, that the Eu 4f–Rh 4d
interaction (because it is rather weak) has been completely neglected due to the
open core approximation.
This is in coincidence wtih experimental observations: upon cleaving along the Eu-
Si plane, the Eu atoms stick either on one or the other side of the cleaved crystal.
Regarding cohesive energies, the formation of smooth surfaces is energetically favoured.
Therefore, Eu atoms are expected to form large islands and hence the cleaved surface
constists of regions terminated either by Eu or Si atoms. Thus one should search an
area with mainly one kind of surface atom to measure plain terminated surfaces. The
corresponding spectral signatures will be discussed below.