Diploma - Max Planck Institute for Solid State Research

4.2 Hybridization: localized versus itinerant states 49
justifies the choice of the basis set. Due to this simplification, the matrix representation
can be directly written as:
⎛
⎞
ɛ 1 0 . . . V 11 . . . V Nf 1
. 0 .. . . .. .
. ɛ Nd V 1Nd . . . V Nf N d
V 11 . . . V 1Nd ˜ɛ 1 C 12 . . . C 1Nf
(4.4)
. . C .. . .. ⎜ . .. 21 .
.
.
⎝
. .. . .. ⎟
CNf −1N f
⎠
V Nf 1 . . . V Nf N d
C Nf 1 . . . C Nf N f −1 ˜ɛ Nf
This matrix is k-dependent and so the solution yields a rearrangement of the eigenenergies
for each k-point corresponding to the hybridization matrix elements V ij , which
are chosen proportional to the overlap of the hybridizing states. The number of valence
states N d corresponds to the number of eigenenergie values in the respective energy
range of the band structure calculation (itinerant electron system). In contrast to that,
the number of localized levels N f and their mixing C ij depends on the interaction of
different levels in photoemission, or more precise, on their origin. As mentioned before,
the m J -degeneracy can be lifted by an electro-magnetic field whereat the field strength
is proportional to the splitting (cf. ch. 4.1.4). The different symmetry considerations
shall be illustrated exemplarily by the atomic PE spectra of Yb and Eu.
Ytterbium [68–71] exhibits as well as europium [49, 67, 72, 73] a complex PE multiplet
structure which has been extensively studied. Since Rare-Earth elements are metals,
they act mainly as electron donors in compounds. This charge redistribution with
respect to the simple superposition of free atoms’ charges causes an increasing electric
field due to the charge separation which lifts the m J -degeneracy. This is known as
crystal electric field splitting. In EuRh 2 Si 2 , this effect cannot be resolved because the
intensity of 4f emission at the Fermi level is rather weak. Additionally, the resolution
gets worse with increasing binding energy due to finite lifetime effects of final states in
photoemission. On the contrary, in YbRh 2 Si 2 the 4f 7/2 multiplet component is located
in the range of the Fermi level and exhibits a resolvable splitting into four different
levels [74]. Since these are mixed representations which lift the 4f shell degeneracy,
they are partially coupled to each other and accordingly the model should respect their
symmetry. In this case N f equals four having two representation ( Γ i t6 and Γi t7 ) with
two elements each. To adopt the symmetry relations, each representation is coupled
only once to the VB and a repulsive “force” is added between elements of the same
representation because they are forbidden to be degenerate (Pauli principle). This
has already been presented by Vyalikh et. al. [68] (thereafter called model 1 “coupled
localized levels”). Contrariwise, the coupling between states with different J is negligible
and thus a superposition of the hybridized spectra of each final state configuration with