Diploma - Max Planck Institute for Solid State Research

48 4 EuRh 2 Si 2 – semi-localized electrons
4.2.3 The hybridization model
To this end, a simple hybridization model 6 (already demonstrated in [67, 68]) will
be applied. Therewith band gaps and hybrid states can be assessed treating the hybridization
as a small pertubation to the pre-calculated atomic final state PE spectrum.
Furthermore the transfer of spectral weight to the Fermi level can be estimated. The
corresponding Hamiltonian
H =
N∑
ɛ i d † i d ∑N f
i + ˜ɛ i f † i f i
i
} {{ }
(1)
i
} {{ }
(2)
N,N
∑ f
+
i,j
V ij
(f † j d i + d † i f j
)
} {{ }
(3)
N∑
f ,N f
+
i≠j
C ij
(f † j f i + f † i f j
)
} {{ }
(4)
(4.3)
consists of two basic count terms, one for itinerant states (1) and one for localized
states (2), and a hopping term (3) describing the probability for mixing 7 . In addition,
there is a fourth term (4) for symmetry relations between the localized states. Regarding
them as possibly energy degenerate states and using fermionic operators, contradicts
itself and does not allow to determine any quantitative measure for occupation. Strictly
speaking, knowing that the final states of PE are many-body states, the usage of a
fermionic operator for each of them is not justified. Nevertheless knowing this peculiarity
and neglecting further many-body effects, you gain qualitative corrections to the
eigenenergies as long as the coupling strength V ij is small enough. The k-dependence
of H is solely introduced by the parameters C ij , V ij . Therefore, the Hamiltonian can
be evaluated in the following one-particle pseudo basis set
| d j , k 〉 = d † jk
| 0 〉 . . . itinerant valence state
| f j , k 〉 = f † jk
| 0 〉 . . . localized state
whereat | 0 〉 denotes the free vacuum state. Since many-body effects are already covered
by LDA (generally: exchange-correlation functional in DFT) and corresponding interactions
are included in Gerken’s et. al. [49] calculation of the Eu PE spectrum as well,
6 In general, the task would have been to solve the Anderson model [63]. It involves two sorts of
electrons: itinerant and localized ones. Furthermore, it includes interaction between them (hybridization)
and adds auxiliary Coulomb repulsion between the localized electrons if they occupy
the same orbital at one site, the “correlation” depends on the degeneracy of the orbital. Since the
Eu impurities are translational invariant, one should include this periodicity, which is not possible
without severe approximations [64–66]. Therefore the extracted hybridization model has been used.
A motivation on how to renormalize the periodic Anderson model to obtain a hybridization model
is given in [44].
7 This model is not limited to a coupling between localized and itinerant states. In principle, there is
a term proportional to the number operator for each species and one describing the mixing of them,
whereat the dispersion of the states can be arbitrary, because ɛ/˜ɛ as well as V ij are k-dependent.
Nevertheless, there exists no intrinsic mixing between different k-dependent variables (in contrast
to the Anderson model).