Electronic and Magnetic Properties of
Impurities Embedded in
Non-Magnetic Finite Hosts
vorgelegt von
Diplom-Physiker
Konstantin Hirsch
aus Berlin
von der Fakult¨
at II - Mathematik und Naturwissenschaften
der Technischen Universit¨
at Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften
- Dr. rer. nat. -
genehmigte Dissertation
Promotionsausschuss:
Vorsitzender: Prof. Dr. Mario D¨
ahne
Berichter/Gutachter: Prof. Dr. Thomas M¨
oller
Berichter/Gutachter: Prof. Dr. Bernd von Issendorff
Tag der wissenschaftlichen Aussprache: 03. Juni 2013
Berlin 2013
D 83
Kurzfassung
Die elektronischen und magnetischen Eigenschaften einzelner ¨
Ubergangsme-
tallatome (Sc, Ti, V, Cr) eingebettet in einen finiten Wirt aus Gold wur-
de sowohl mit Hilfe von R¨
ontgenabsorptionsspektroskopie und zirkularem
R¨
ontgendichroismus als auch mit Hilfe von Dichtefunktionaltheorie am Bei-
spiel von gr¨
oßenselektierten kationischen Clustern in der Gasphase untersucht.
Die elektronische Struktur des Dotieratoms h¨
angt sehr empfindlich von der
Natur des Fremdatoms, von Quanteneffekten im Wirt sowie deren Zusam-
mensspiel ab. Gerade-ungerade Variationen in der Lokalisierung von Elek-
tronen des Dotieratoms findet man in TiAu+
nClustern und andeutungsweise
bereits in ScAu+
nClustern. In VAu+
nClustern hingegen ist die lokale elektroni-
sche Struktur des Dotieratoms unabh¨
angig von der Gr¨
oße des Wirtsclusters.
Dennoch l¨
asst sich schlußfolgern, dass es in allen untersuchten Systemen die
Tendenz gibt eine gerade Anzahl von Elektronen zu delokalisieren, sofern dies
durch die gegebenen Randbedingungen m¨
oglich ist.
Das Anderson impurity model [1] beschreibt die Wechselwirkung eines einzel-
nen magnetischen Dotieratoms mit einem homogenen Elektronengas in einer
ausgedehnten Festk¨
orperprobe. F¨
ur finite Systeme hingegen muss der Einfluß
einer Energiel¨
ucke in der Zustandsdichte des Wirstmaterials ber¨
ucksichtigt
werden. Dies wurde in der vorliegenden Dissertation mit Hilfe eines Hamil-
tonoperators in tight binding N¨
aherung getan. Damit konnte gezeigt werden,
dass f¨
ur finite Systeme grunds¨
atzlich die Beschreibung im Anderson impurity
model zusammenbricht, und insbesondere der ¨
Ubergang von magnetischem
zu unmagnetischem Verhalten falsch vorhergesagt wird. Es konnte weiterhin
gezeigt werden, dass die Energiel¨
ucke zu einer deutlichen Stabilisierung des
Spinmoments des Dotieratoms beitr¨
agt. Die gr¨
oßenabh¨
angige Variation des
Spinmoments in nahezu vollst¨
andig spinpolarisierten CrAu+
nClustern ist den-
noch im Wesentlichen in ¨
Ubereinstimmung mit dem Anderson impurity model
[1], da diese Systeme weit vom ¨
Ubergang von magnetischem zu unmagneti-
schem Verhalten entfernt sind. Es ergibt sich eine Skalierung des Spinmoments
mit der Gr¨
oße der Bandl¨
ucke im Wirtscluster, was durch Austausch des Wirts-
materials von Gold zu Kupfer best¨
atigt werden konnte.
3
Abstract
The electronic and magnetic properties of a single transition metal dopant
(Sc, Ti, V, Cr) embedded in a finite gold host were studied. This was done
by applying x-ray absorption and x-ray circular dichroism spectroscopy as
well as density functional theory to size selected cationic gas phase clusters.
The electronic structure of the impurity sensitively depends on the nature of
the dopant as well as on quantum confinement in the host cluster and their
interplay. An odd-even effect of electron localization in TiAu+
nclusters is fore-
shadowed in ScAu+
nclusters, whereas the local electronic structure in VAu+
n
clusters is almost independent of the gold host size. However, for all investi-
gated systems a strong tendency was found to delocalize an even number of
electrons whenever allowed by the given boundary conditions.
The Anderson impurity model [1] describes the interaction of a single magnetic
impurity with a homogenous free electron gas in a bulk sample. However, in
case of a finite system the influence of an energy gap in the hosts density of
states has to be accounted for. This was done within this thesis by study-
ing the problem using a model Hamiltonian in tight binding approximation.
Thus, it could be shown that the description of finite systems within the An-
derson impurity model breaks down in principle. Especially the magnetic to
non-magnetic transition threshold is predicted incorrectly. In addition it was
shown that the spin polarization of an impurity atom is substantially stabi-
lized by the presence of an energy gap in the host density of states. Still, the
size dependence of the spin magnetic moment of chromium in almost fully
spin polarized CrAu+
nclusters is essentially in agreement with the Anderson
impurity model, since it is far from a magnetic to non-magnetic transition.
Therefore, only a scaling of the spin magnetic moment of the impurity as a
function of the energy gap could be observed, which was confirmed by substi-
tuting the host material from gold to copper.
5
Contents
1 Introduction 9
2 Fundamentals 13
2.1 Spectroscopic Methods . . . . . . . . . . . . . . . . . . . . . . 13
2.1.1 X-ray Absorption Spectroscopy . . . . . . . . . . . . . 13
2.1.2 X-ray Magnetic Circular Dichroism Spectroscopy . . . 15
2.1.3 XMCD Sum Rules . . . . . . . . . . . . . . . . . . . . 16
2.1.4 Ion Yield Spectroscopy and Electronic Relaxation Pro-
cesses ........................... 17
2.2 Correlated and Itinerant Electrons . . . . . . . . . . . . . . . 18
2.3 Density Functional Theory . . . . . . . . . . . . . . . . . . . . 20
2.4 Self-Consistent Calculation of
the On-site Coulomb Repulsion Uin a DFT+UFramework . . 23
2.5 Anderson Impurity Model . . . . . . . . . . . . . . . . . . . . 25
2.6 KondoEffect ........................... 27
3 Experimental Setup 31
3.1 Magnetron Sputtering Cluster Source . . . . . . . . . . . . . . 31
3.2 Radio Frequency Ion Guide and Quadrupole Mass Filter . . . 32
3.3 Radio Frequency Ion Trap and Magnet . . . . . . . . . . . . . 33
3.4 Reflectron Time-of-Flight Mass Spectrometer . . . . . . . . . . 35
3.5 Synchrotron Radiation and Undulator Beam-Lines . . . . . . . 36
4 Experimental and Computational Details 39
4.1 DataAcquisition ......................... 39
4.2 DataAnalysis........................... 42
4.3 Computational Details . . . . . . . . . . . . . . . . . . . . . . 43
5 Electronic Structure of Early 3dTransition Metal Impurities
in Non-Magnetic Gold Clusters 45
7
8Contents
5.1 Local Electronic Structure of Scandium Doped
GoldClusters ........................... 46
5.2 Odd-Even Effects in the Electronic Structure of Titanium Doped
GoldClusters ........................... 49
5.3 Independence of the Local Electronic Structure on Impurity
Coordination: Vanadium Doped Gold Clusters . . . . . . . . . 55
5.4 Local Electronic Structure of Chromium Doped
GoldClusters ........................... 59
5.5 Summary ............................. 61
6 The Anderson Impurity Model in Finite Systems 63
6.1 Testing the Anderson Impurity Model:
A study of a Chromium Impurity in Gold-Clusters . . . . . . . 63
6.2 Modification of the Anderson Impurity Model for Finite Systems 71
6.3 Influence of the Host Material: Chromium Doped Copper Clus-
ters................................. 76
6.3.1 Geometries of Chromium Doped Copper Clusters . . . 76
6.3.2 Electronic and Magnetic Properties of Chromium Doped
CopperClusters...................... 79
6.4 Summary ............................. 84
7 Epilogue 85
7.1 Summary ............................. 85
7.2 Outlook .............................. 86
List of Publications 89
List of Figures 91
List of Tables 92
Bibliography 93
8 Appendix 109
8.1 Coordinates of CrCu+
nClusters ................. 109
Acknowledgements 113
Chapter 1
Introduction
The interaction of localized impurity states with non-magnetic host materials
is a long standing problem in solid state physics. It leads to complex many
body phenomena such as Friedel oscillations [2, 3] or the Kondo effect [4],
which is the at first sight counter-intuitive increase of the resistivity at low
temperatures. The properties of magnetic impurities, such as transition met-
als or rare earths, in non-magnetic hosts have therefore been subject of intense
research for the last 50 years. Especially the 3dtransition metals are a par-
ticularly interesting class of material. Their valence electrons can neither be
classified as fully localized nor as itinerant. This has significant consequences
on their electronic and magnetic properties.
Considerable advances in studying the aforementioned many-body effects have
been made by applying photoemission, x-ray magnetic circular dichroism
(XMCD), and scanning tunneling spectroscopy to adatoms [5–8], clusters on
surfaces [9], or well defined quantum dots [10, 11]. This allows to study and
tune basic parameters such as the on-site Coulomb repulsion, which is the
Coulomb interaction of electrons in the same localized orbital, or the coupling
to the electron bath of the host. Up to now, such techniques have been applied
to atomic scale systems only on surfaces or in bulk metals. In these systems
one can study the interplay between impurity and host material by introduc-
ing different kinds of impurities or by varying the electronic density of the
free electron gas, provided by the host material [5, 6]. In contrast, the study
of single impurities in size-selected clusters allows to characterize the interac-
tion with a finite electron gas, i.e., with a well defined number of electrons
occupying a highly discretized density of states. This approach will be fol-
lowed in this thesis by using doped gold clusters (TMAu+
n, TM = Sc,Ti,V,Cr)
as model systems that combine considerable local magnetic moments carried
by the 3delectrons of the single dopant atom with a finite free electron gas
formed by the host cluster. Isolated, doped gold clusters have been studied in
9
10 Chapter 1. Introduction
great detail by density functional theory calculations [12–21] and experiment:
their electronic and geometrical structure as well as their stability have been
probed by photoelectron [21, 22], infrared photodissociation [23, 24], and laser
photofragmentation [25–29] spectroscopy, as well as electron diffraction [30].
In this work the development of the dopant electronic and magnetic struc-
ture as a function of the host cluster size is studied by x-ray absorption and
XMCD spectroscopy of size-selected gas phase clusters combined with density
functional theory calculations. Applying these techniques paves the way to
addressing elementary problems of solid state physics that call for the use
of methods probing the local electronic and magnetic properties of an impu-
rity rather than the global electronic structure of a system. Such problems
are, e.g., the question how the electronic structure of a finite non-magnetic
host system and a single magnetic impurity are interrelated. Under which
circumstances is the impurity’s magnetic moment quenched and when does
it survive? Can the size of the impurity’s magnetic moment be understood
based on bulk models? Furthermore, many-body effects, like the Kondo effect
can be expected to change upon going from extended systems to systems con-
taining only a few particles. In this sense, the present work can be regarded
as a first step in observing a Kondo effect in finite systems.
This thesis is structured as follows: In chapter 2 next to x-ray absorption
and XMCD spectroscopy, basic theoretical models and methods such as den-
sity functional theory and the Anderson impurity model are introduced. In
chapters 3 and 4 experimental and computational details are given. Chapter
5 contains a discussion of the electronic structure of transition metal doped
gold clusters. It is shown that the electronic structure sensitively depends on
the nature of the dopant and quantum confinement in the host material as
well as their interplay. Following the periodic table of elements, differences
in the dopant-host interaction are discussed considering remarkably different
behavior of ScAu+
n, TiAu+
nand VAu+
nclusters. The experimental and theoreti-
cal findings additionally suggest a common ground in the binding mechanism.
In chapter 6, a yet closer look will be given to the magnetic properties of
chromium doped gold clusters. It will be shown that by shell closure in the
two dimensional free electron gas of the gold host that governs the minimiza-
tion of the impurity-host interaction, a nearly atomic spin magnetic moment
of 5 µBcan survive in CrAu+
2and CrAu+
6. A discussion of the size dependence
of the spin magnetic moment in terms of the Anderson impurity model [1]
reveals deeper insight into the interaction of the impurity atom with a highly
discretized density of states. Therefore, the influence of the discrete nature of
the host density of states was studied in tight binding approximation reveal-
ing a break down of the Anderson impurity model in the description of finite
12 Chapter 1. Introduction
Chapter 2
Fundamentals
In this chapter an introduction into the spectroscopic as well as into basic the-
oretical methods will be given. In the first three subsections the benefits and
limitations of x-ray absorption, x-ray magnetic dichroism and ion yield spec-
troscopy will be discussed. In the following sections theoretical approaches to
treat the electronic structure will be introduced. Focus will be laid on the no-
toriously difficult correlated electron systems, by discussing density functional
theory (DFT) and DFT+Umethods and introducing the Anderson impurity
model.
2.1 Spectroscopic Methods
2.1.1 X-ray Absorption Spectroscopy
An ensemble of atoms exposed to x-ray photons will attenuate the incoming
photon beam according to the Lambert-Beer law
I(x, E) = I0·e−µ(E)·x.(2.1)
Here x is the distance the light traveled in the medium and µ(E) is the absorp-
tion coefficient that depends linearly on the absorption cross section σand
the number of atoms n. At certain energies peaks can be found in the absorp-
tion that have their origin in dipole transitions with finite oscillator strength
between core levels and unoccupied valence states. In x-ray absorption the
energy of these transitions is characteristic since the core level binding energy
strongly depends on the atomic number Z. This allows a local and element
specific investigation even of compound systems. In addition to resonant tran-
sitions, photoemission of core electrons gives rise to the well known absorption
edges. The absorption cross section above the continuum threshold decreases
13
14 Chapter 2. Fundamentals
0.0 0.2 0.4 0.6 0.8 1.0
r
a
d
i
a
l
p
r
o
b
a
b
i
l
i
t
y
2p electrons
3d electrons
radius in Bohr units
Figure 2.1: Radial probability of
one-particle 2pand 3dwave func-
tions shown for atomic number
Z= 23.
monotonically since the overlap of continuum and core level wave functions
drops with increasing energy.
The absorption signal IXAS is determined by the transition probability which
can be expressed by Fermi’s Golden rule:
Wfi =2π
¯h| hφf|T|φii |2·ρ(Ef)·δ(Ef−Ei−¯hω)∝σ∝IXAS.(2.2)
Here φi,f are the many-body initial and final state wave functions, respec-
tively, Tis the operator mediating the transition which in this case is the
dipole operator1and ρ(E) is the density of states. The delta function ensures
the conservation of energy. The absorption process is often described in a sin-
gle particle approximation, since the many body wave functions are usually
unknown. Thus, equation 2.2 simplifies to
IXAS =| hφf|T|φii |2·ρ(E)≈ | hφi¯c|T|φii |2·ρ(E) = | hc|T|i |2·ρ(E) = M2ρ(E).
Here and care the wave-functions of the excited electron and core hole,
respectively. Since the matrix element Mvaries only very slowly with energy
[31, 32], x-ray absorption in principal measures the density of states of the
unoccupied energy levels.
This simple picture reaches its limit for the L-edge absorption of 3dtransition
metals. This is evident from the very complex multiplet structure in their
x-ray absorption spectra [32, 33]. The comparable mean radii of 2pand 3d
electrons, cf. figure 2.1, lead to a strong Coulomb interaction of the core hole
created in the x-ray absorption process and the emitted photoelectron leading
to the breakdown of the single particle approximation.
1In dipole approximation the explicit form of Tis given by T=Pqeq·r(qdirection of
polarization, rposition, eunit vector).
2.1. Spectroscopic Methods 15
2.1.2 X-ray Magnetic Circular Dichroism Spectroscopy
The x-ray magnetic circular dichroism spectroscopy (XMCD) can be used to
study the spin and orbital momentum resolved magnetic properties of a sam-
ple.
XMCD spectroscopy can be described in a two step model invented by St¨ohr
3d-levels
2p3/2-level
2200
0-1
1
1
1
1
7,5%
37,5% 62,5%
2,5% 45%15%15%15%
1/3 2/3
0-1
1/3
2/3 ++
+h
Figure 2.2: Excitation at the L3-edge with a photon of positive helicity. The given
probabilities for the individual transitions result from Clebsch-Gordon coefficients. The
figure is adapted from [34].
[35]. In a first step, a circular polarized photon transfers its angular mo-
mentum ±¯hin an absorption process to the photoelectron. Assuming the
photoelectron stems from a spin-orbit split 2pstate2(2p3/2and 2p1/2) part of
the angular momentum can be transferred to the spin degree of freedom via
spin-orbit coupling. The spin polarization of the photoelectrons created in an
absorption process is opposite for light with left and right helicity, respectively.
Additionally, spin and orbital momentum are coupled parallel (2p3/2) and anti
parallel (2p1/2) at the two absorption edges (L2,3) leading to an inversion of
photoelectron spin polarization for both helicities at the two edges (L2,3). The
transition probabilities are given by the transition matrix elements which are
also displayed in figure 2.2.
In the second step the spin polarized photoelectron beam, created in the
mL=0 mL=1 mL=2mL=-2 mL=-1
mL=0 mL=1mL=-1
3d-levels
2p-level
Figure 2.3: Shown are the accessible mLsubsets in a dipole transition for left (dotted
line) and right (solid line) handed photons. The figure is adapted from [34].
first step, is used to probe the spin polarization of the exchange split valence
2In reality, there is no spin-orbit splitting of the 2plevels in the initial state.
16 Chapter 2. Fundamentals
states resulting in different absorption cross sections for both helicities.
Sensitivity to the orbital moment is given by the dipole selection rule ∆m=
±1 for circular polarized light, as illustrated in figure 2.3. Since different mL
subsets are probed with left and right circular polarized light, an unequal oc-
cupation among these subsets leads to a dichroic absorption signal.
2.1.3 XMCD Sum Rules
Quantitative determination of the expectation value of orbital hLziand spin
hSzimoment along the quantization axis znecessitates the introduction of
XMCD sum rules [36–38] relating them to the measured XMCD asymmetry.
In a dipole transition cln→c−1ln+1 between a core state with orbital momen-
tum cand a valence state with orbital momentum loccupied by nelectrons,
the XMCD sum rules in their general form read[36, 37]
Rj++j−(τ+−τ−)dE
Rj++j−(τ++τ−+τ0)dE =1
2
l(l+ 1) + 2 −c(c+ 1)
l(l+ 1)(4l+n−2) hLzi(2.3)
and
Rj+(τ+−τ−)dE −c+1
cRj−(τ+−τ−)dE
Rj++j−(τ++τ−+τ0)dE =
l(l+ 1) −2−c(c+ 1)
3c(4l+ 2 −n)hSzi+
l(l+ 1)[l(l+ 1) + 2c(c+ 1) + 4] −3(c−1)2(c+ 2)2
6cl(l+ 1)(4l+ 2 −n)hTzi
(2.4)
Here τ+,−,0are absorption spectra taken with helicity parallel and anti-parallel
to the sample magnetization and linear polarized light. Eis the energy. The
core spin split states are denoted j+,−=c±1/2 respectively.
The magnetic dipole term hTzivanishes in our case due to angle averaging of
the randomly oriented sample [39]. Thus the total magnetic moment µtotal is
given by the sum of spin µSand orbital µLmagnetic moment
µtotal =µS+µL= 2nh
µB
¯hhSzi+nh
µB
¯hhLzi,
µBbeing the Bohr magneton, ¯h=h/2πthe reduced Planck constant and nh
the number of vacancies in the 3dstate.
The limitations of the applicability of XMCD sum rules are discussed in detail
2.1. Spectroscopic Methods 17
1.0
0.8
0.6
0.4
0.2
0.0
908070605040302010
atomic number Z
Auger yield
Fluorescence yield
3d transition metals
yield
Figure 2.4: Comparison of flu-
orescence and Auger yield at the
L2,3edges as a function of the
atomic number Ztaken from [46].
The Auger process is the dominant
relaxation channel for the 3dtran-
sition metals.
in a series of papers [40–44]. Nonetheless, some of the assumptions involved
in the derivation of the XMCD sum rules will be given here.
If the orbital angular momentum cof the core level differs from zero, transi-
tions into valence states with orbital momentum c±1 are allowed in a dipole
transition. As can be seen from equations 2.3 and 2.4 the XMCD sum rules
differ for the two transitions. Moreover, these transitions cannot be separated
in an x-ray absorption experiment. However, since 2p→4stransitions con-
tribute only up to 5 % to the L2,3-edges absorption of 3dmetals they can be
neglected. This means that despite this first limitation, XMCD sum rules are
still applicable.
In the derivation of the XMCD sum rules the initial and final states were
restricted to be pure configurations. This cannot be expected at least for the
highly excited final state. However, Thole et al. showed that the sum rules
still hold in case of mixed configurations [36].
The spin XMCD sum rule implies the separation of j+and j−states. In case
of L-edge absorption of the 3dtransition metals this is only satisfied if the
2pspin orbit splitting is larger than the 3dbandwidth. Obviously, the spin
XMCD sum rule is inapplicable for the early and middle 3dtransition metals
because of the strong overlap of 2p3/2,1/2→3dtransitions. The break-down
of the spin XMCD sum rule has already been shown for atomic manganese
deposited on potassium [45].
2.1.4 Ion Yield Spectroscopy and Electronic Relaxation
Processes
After x-ray absorption the cluster is highly excited and can in principle re-
lax via radiative and non-radiative channels, namely fluorescence and Auger,
Coster-Kronig or Super-Coster-Kronig processes. The fluorescence probabil-
18 Chapter 2. Fundamentals
ity is given by the dipole matrix element hc|r|vi ∝ E3≈Z6(|cicore state,
|vivalence state), while the probability of the aforementioned non-radiative
processes depend on the wave-function overlap. For a given pair of states
the overlap does not vary strongly with the atomic number Z. Hence, which
process is dominant strongly depends on the atomic number Zand the states
involved. The fluorescence and Auger yield at the L2,3-edges are shown in
figure 2.4 indicating a negligible contribution of fluorescence to the total yield
in case of L2,3-edge absorption of 3dtransition metals. Therefore electron
or ion yield spectroscopy is the method of choice for the investigation of 3d
transition metal L-edge absorption. Upon deposition of 500 −900 eV in the
absorption process the system relaxes via Auger cascades leaving the cluster
highly charged. If a critical charge at a given cluster size is exceeded, Coulomb
repulsion will lead to fragmentation [47]. The yield of photofragments created
in the absorption process is proportional to the cross section [48] and will here
be used as the x-ray absorption signal.
2.2 Correlated and Itinerant Electrons
Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb
Th Pa U Np Pu Am Cm Bk Cf Es Em Md No
Sc Ti V Cr Mn Fe Co Ni Cu
Y Zr Nb Mo Tc Ru Rh Pd Ag
Lu Hf Ta WRe Os Ir Pt Au
Figure 2.5: In the Kmetko-Smith diagram the periodic table is arrange by the localization
of the valence orbitals. Since itinerant and correlated behavior is closely related to local-
ization, magnetic metals with atomic-like valence orbitals can be found in the upper right
(blue background color) whereas non-magnetic metals can be found in the lower left (red
background color) corner of the diagram. The diagonal of the diagram is the cross-over
region where itinerant and correlated behavior can be found simultaneously. The diagram
is adapted from [49].
In the late 1920s and early 1930s Bloch [50] and Wilson [51] among others
developed an one-electron theory for bulk materials. Within this theory non-
interacting electrons move in the periodic charge background of the crystal as
2.2. Correlated and Itinerant Electrons 19
1
0.5
0
1
0.5
0
DENSITY OF STATES
1
0.5
0
EF
ENERGY
+
EF
U0
2
W
EF
U0
2
1
0.5
0
-
U0/W=0.5
U0=0
U0/W=1.2
U0/W=2
U0
(a)
(b)
(c)
(d)
Figure 2.6: Mott transition: Density of
states as a function of U0/W adapted from
[56]. (a) Non interacting Fermions form-
ing a band of width W(b) Weakly interact-
ing Fermions: A quasi-particle peak forms at
EF. The system can be well described within
Fermi liquid theory. (c) Weakly correlated
electronic systems: Coexistence of Hubbard
bands and quasi-particle peak. (d) Strongly
interacting electrons: Spectral density of the
quasi-particle peak is completely transferred
to the lower and upper Hubbard bands sepa-
rated by the on-site Coulomb interaction U0.
plane waves and occupy energy bands up to the Fermi energy, cf. figure 2.6 (a).
Depending on the the location of the Fermi energy the material is a metal or
an insulator. Electrons which are well described within this theory are called
itinerant. However, it was quickly realized that this simple theory fails in
describing transition metal compounds and rare earths. Hubbard [52, 53] and
Mott [54] showed that the one-electron theory will fail if charge fluctuations
are suppressed by large values of the on-site Coulomb interaction U0,cf. sec-
tion 2.4, as compared to the one-electron band width W. Since the mobility of
electrons moving through the lattice is hindered by the Coulomb interaction
with electrons occupying localized orbitals, these electrons are called corre-
lated. If U0W, the on-site Coulomb repulsion splits the valence band and
result in a Mott-Hubbard insulator as was pointed out by Mott and Peierls in
1937 [55]. In between these two extremes two other regimes exist as depicted
in figure 2.6 (b) and (c). If correlation is still small the interacting system
can be described as an ensemble of non-interacting quasi-particles exhibiting
some collective excitations. In case of comparable U0and W, Hubbard bands
and quasi-particle peak coexist making the theoretical description very chal-
lenging.
The itinerant and correlated behavior of electrons is closely related to the
degree of localization of the valence electrons. The valence electrons can be
considered localized if their mean radius is small compared to inter-atomic
distances in the material, thus the Coulomb interaction among electrons in
20 Chapter 2. Fundamentals
these orbitals is large. On the other hand, the valence orbitals overlap and
form bands if their radius is comparable to equilibrium bond distances in the
system. Then the electrons exhibit itinerant character.
By arranging the periodic table by localization of the valence orbitals, the
Kmetko-Smith diagram [57] can be obtained which is shown in figure 2.5. In
the upper right magnetic metals with atomic-like valence shells can be found
whereas in the lower left corner non-magnetic metals are located. The cross-
over regime is the diagonal of the diagram where valence electrons exhibit
itinerant as well as correlated behavior, cf. figure 2.6. Correlated materials
exhibit a wealth of interesting phenomena such as high temperature supercon-
ductivity [58], giant magneto resistance [59, 60], large effective electron mass
(heavy fermions) [61] and many more [62]. Doubtless, the diagram shown in
figure 2.5 is oversimplified and does not reflect the whole complexity of the
physics in bulk materials. Yet, the point I want to emphasize is that the sys-
tems which will be investigated throughout this thesis consist of elements very
close to the diagonal of the Kmetko-Smith diagram and thus are expected to
exhibit itinerant as well as localized behavior.
2.3 Density Functional Theory
In quantum physics a system is solely described by its wave function Ψ, which
can in principle be calculated by solving the Schr¨odinger equation. In Born-
Oppenheimer approximation the Schr¨odinger equation for a non-relativistic
Coulombic system consisting of Nelectrons at positions riand with spin χi
and Mionic cores is given by 3:
T+V+˜
VΨ (r1,r2,...,rN, χ1, χ2, . . . , χN)
=EΨ (r1,r2,...,rN, χ1, χ2, . . . , χN).(2.5)
Here Tis the N-particle kinetic energy
T=
N
X
i=1
−¯h2
2m∇2
i.(2.6)
The Coulomb interaction of Mionic cores at positions Rkpossessing a nuclear
charge Zkand Nelectrons is described by the external potential
V=
N,M
X
i,k=1
Zke2
|ri−Rk|.(2.7)
3Here I use the convention: 1
4π0= 1
2.3. Density Functional Theory 21
The most intriguing contribution to the total energy is the electron-electron
interaction
˜
V=
N
X
i,i0=1 i6=i0
e2
|ri−ri0|.(2.8)
Here arises the main challenge in atomic and condensed matter physics. The
many-body wave-function Ψ depends on 3Nspatial and Nspin coordinates.
Since ˜
Vcouples the spatial coordinates they cannot be treated separately.
Therefore, solving the Schr¨odinger equation for a many-electron system is not
only very difficult and computationally demanding, but also exceeds storage
space available today even for small systems.
Tackling the many-body problem without directly involving the many-body
wave-function Ψ is at the heart of density functional theory (DFT). Only a
short introduction to DFT will be given here, detailed information can be
found in excellent review articles [63, 64] or monographs [65–68].
In 1964 Hohenberg and Kohn proved that there is a bijective mapping of the
ground state many-body wave-function Ψ and electron density n(r) [69]. Con-
sequently, the expectation value of every observable and therefore in particular
the ground state energy can be expressed as a functional of the density n(r):
E[n] = hΨ [n]|H|Ψ [n]i=ZV(r)n(r)d3r+F[n].(2.9)
By construction the functional F[n] = T[n] + ˜
V[n] is universal, i.e., not ex-
plicitly depending on the external potential V. The Hohenberg-Kohn theorem
proves that the density n(r) can be used as the basic variable reducing the
many-body problem tremendously. The functional 2.9 exhibits a minimum
for the ground state density under the constraint Rn(r)d3r=N:
E0[n] = minn(r)E[n].(2.10)
Thus the variational principle can be stated as:
δE[n]−µZn(r)d3r−N= 0,(2.11)
where the Lagrangian multiplier µtakes care of the constraint of constant
particle number N.
In practical calculation the functional E[n] will be expressed as
E[n] = Ts[n] + ZV(r)n(r)d3r+EHartree[n] + Exc[n],(2.12)
as proposed by Kohn and Sham [70]. The functional Ts[n] is the kinetic en-
ergy functional of non-interacting electrons and EHartree the classical Coulomb
22 Chapter 2. Fundamentals
interaction of two charge distributions:
EHartree[n] = e2
2Z Z n(r)nr0
|r−r0|d3rd3r0.(2.13)
The functional Exc[n] collects all the dynamical and static exchange and cor-
relation effects and can by definition be expressed as:
Exc[n] = F[n]−Ts[n]−EHartree[n].(2.14)
Application of the variational principal yields
δTs[n]
δn (r)+VKS =µ, (2.15)
VKS being an effective potential, which is called Kohn-Sham potential
VKS =V(r) + e2
2Znr0
|r−r0|d3r0+δExc[n]
δn(r).(2.16)
Equation 2.15 describes an auxiliary system of non-interacting particles mov-
ing in an effective potential VKS. The Schr¨odinger equation
−¯h2
2m∇2+VKS(r)!φi=iφi(2.17)
of this auxiliary system yields single particle wave-functions φithat reproduce
the density n(r) of the original system
n(r) =
occ
X
i
|φi|2.(2.18)
Equations 2.16-2.18 are known as the Kohn-Sham equations and have to be
solved self-consistently. Solving the Kohn-Sham equations would yield the cor-
rect ground state density, were it not for the exchange-correlation functional
Exc [n], which is in general not known exactly, but has to be approximated.
Several approximations were introduced for Exc[n], the most common are the
local density approximation (LDA) and the generalized gradient approxima-
tion (GGA). These approximations yield accurate descriptions of a wealth of
systems. However, they fail in describing even very fundamental properties of
strongly correlated electronic systems. Notorious examples are the transition
metal oxides FeO an CoO for which they fail to reproduce the antiferromag-
netic ground states, as well as the band gaps predicted for MnO and NiO [73–
2.4. Self-Consistent Calculation of
the On-site Coulomb Repulsion Uin a DFT+UFramework 23
NN+1 N+2N-1
number of electrons
total energy
exact
GGA / LDA
Figure 2.7: Schematic illustration of the to-
tal energy as a function of occupation num-
bers. The exact functional exhibits deriva-
tive discontinuities at integer electron num-
bers [71]. The GGA and LDA approximations
underestimate the total energy for fractional
occupations. This can partly be compensated
by introducing an additional Hubbard term to
the XC functional [72].
75]. Many problems that standard approximations to Exc[n], such as LDA or
GGA, have, can be overcome by correcting for the well known, spurious self-
interaction error that plagues most commonly employed xc-functionals. How-
ever, most self-interaction corrected methods [76] are computationally very
demanding. Alternatively, semiempirical modifications of LDA and GGA,
like DFT+Umethod can be employed. This will be described in the next
section.
2.4 Self-Consistent Calculation of the On-site
Coulomb Repulsion Uin a DFT+UFrame-
work
The introduction of an Hubbard like energy U, which is the Coulomb repul-
sion among localized electrons, to the exchange-correlation functional [77–79]
constitutes a semi-empirical method to compensate for the self interaction er-
ror inherent in the highly localized 3dand 4forbitals of transition metals, cf.
figure 2.7. In general it is implemented in the following way:
EU=U
2X
l,χ
Tr [nl,χ(1 −nl,χ)] .(2.19)
The electrostatic Coulomb energy Uis the energy necessary to add an addi-
tional electron with opposite spin to a localized orbital. The matrix nl,χ gives
the occupation of the localized orbitals at site lwith spin χ. The energy U
24 Chapter 2. Fundamentals
5.26
5.25
5.24
5.23
5.22
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
bare response
screened response
linear fit to bare response
linear fit to screened response
occupation n
rigid shift a
Figure 2.8: Linear response
function for the screened and bare
system. The rigid potential shifts
∆V=PlαlPlinduce a change
in occupation n. The differ-
ence of the slopes gives the onsite
Coulomb repulsion U.
depends on the system and is normally fitted to experimental data, strongly
limiting the predictive power of such a theory. Hence, it is highly desirable to
calculate the energy Ufrom first principles. This is non-trivial, since the direct
Coulomb interaction is partially screened by delocalized electrons. Therefore
simply calculating the respective Coulomb integral will fail.
Several approaches are proposed in the literature [72, 77, 80–85], here I stick to
the linear response method introduced by Cococcioni et al. [72, 84, 85]. The
approach starts by calculating the total energy as a function of the occupation
of the localized orbitals
E[{ql}] = minn(r),αl(E[n(r) + X
l
αl(nl−ql)]).(2.20)
The constraints on the site occupations nlare taken care of by the Lagrangian
multipliers αl. The curvature of the above function contains in principle the
fully screened on-site Coulomb repulsion. However, additionally change of
the occupation induces re-hybridization effects that are even present in non-
interacting system. These effects yield non-vanishing contributions to the
curvature of the energy EKS [{ql}] and therefore have to be subtracted. Thus
the on-site Coulomb repulsion Uis given by
U=∂2E[{ql}]
∂q2
l
−∂2EKS [{ql}]
∂q2
l
.(2.21)
Equation 2.21 can be recasted [72] applying a Legendre transform as
U=∂αKS
l
∂ql
−∂αl
∂ql
=β−1
0−β−1,(2.22)
2.5. Anderson Impurity Model 25
where β0and βare the density response functions:
β0
lm =∂2EKS
∂αl∂αm
=∂nl
∂αKS
m
(2.23)
βlm =∂2E
∂αl∂αm
=∂nl
∂αm
(2.24)
The αlinduce a rigid potential shift ∆V=PlαlPl(Plare generalized projec-
tors onto localized orbitals) at the localized orbitals, leading to a change of oc-
cupation. An example calculation, performed within the quantum espresso
5.0 code [86], is shown in figure 2.8. The difference of both slopes gives the
on-site Coulomb repulsion Uself-consistently.
2.5 Anderson Impurity Model
The Anderson impurity model [1, 61] was introduced to describe the interac-
tion of a single magnetic impurity embedded in a non magnetic host material
based on a simple model Hamiltonian given in second quantization as
H=Hk+Hd+HU+Hkd.(2.25)
The energy of the free electron gas of the non magnetic host is given by
Hk=X
k,χ
kc†
k,χck,χ =X
k,χ
knk,χ,(2.26)
where ck,χ and c†
k,χ are the annihilation and creation operators. The energy
of the unperturbed localized levels of the magnetic impurity is given by
Hd=E c†
dcd=E nd,(2.27)
while the unscreened on-site Coulomb repulsion U0among localized electrons
in the same orbital is treated as
HU=U0nd,↑nd,↓.(2.28)
The last term in equation 2.25 describes the interaction of the localized states
with the free electron gas
Hkd=X
k,χ
Vdk c†
k,χcd,χ +c†
d,χck,χ,(2.29)
by introducing a hybridization matrix Vdk.
A schematic illustration of the Anderson model is shown in figure 2.9 as-
26 Chapter 2. Fundamentals
EFermi
Γ
2
E
E+U0n+
E+U0n-
E+U0
effective
on site
Coulomb
repulsion
Ueff
Spin DOS Spin DOS
energy
on-site
Coulomb
repulsion
U0
Figure 2.9: Schematic illustra-
tion of the Anderson impurity
model for the magnetic case and
maximal spin polarization U0Γ
[1]. Two electrons in the same lo-
calized state are separated by the
energy U0. Hybridization with the
parabolic density of states of the
non-magnetic host causes a broad-
ening 2Γ of the localized states and
reduction of U0to an effective value
Ueff . The system becomes non-
magnetic if Ueff ≈Γ.
suming the simple case of a single non-degenerate impurity state. If in a first
step hybridization of the localized orbitals with host electron gas states is
neglected, adding an electron to a localized level φd(r) results in a state at
energy E+U0,U0being the on-site Coulomb repulsion among the localized
electrons. The sample becomes magnetic if U0pushes the state E+U0above
the Fermi level EF.
However, in real systems interaction will be present resulting in virtual bound
states at energies E+U0·n−and E+U0·n+,cf. figure 2.9. Here, n±are
the occupation numbers of the single impurity majority and minority spin
levels. Hybridization of the localized orbitals with the host electronic states
yield a considerable broadening 2Γ of the localized state. This in turn can
lead to a reduction of the spin polarization (n+−n−), since a certain amount
of the Lorentzian shaped density of states ρ(Ed) of the localized spin up or-
bital is smeared above and in case of the spin down orbital below the Fermi
energy. This reduces the energetic separation Ueff of the virtual states to
Ueff =U0(n+−n−).
Whether the system is magnetic or non-magnetic sensitively depends on the
ratio of U0and Γ. A transition between these two states is predicted by the
Anderson model at a ratio of U0and Γ which is equal π, if we assume a sym-
metrical arrangement of the impurity levels around the Fermi energy [1]. In
figure 2.10, the occupation numbers of the majority and minority impurity
2.6. Kondo Effect 27
1.0
0.8
0.6
0.4
0.2
0.0
1.00.80.60.40.20.0
n+
n-
U0/Γ = 5π
U0/Γ = 0.5π
U0/Γ = 1.5π
Figure 2.10: Occupation num-
bers for majority and minority im-
purity spin levels. Self consistent
solutions can be found at the inter-
sections of the curves. Spin polar-
ization can only be found for val-
ues U0/Γ> π, whereas the system
is non magnetic for U0/Γ=0.5π.
spin levels is plotted versus each other for three different values of U0/Γ. Sta-
ble solutions can be found at the intersections. As can be seen in the figure
only for the cases where U0/Γ is larger than πspin polarized solutions can be
found, whereas for the value of U0/Γ=0.5πthe system is non-magnetic.
To further clarify the nature of the virtual bound states, the situation has to
be explained in the framework of a scattering process [2, 3, 87]. The unfilled
dand forbitals of transition metals and rare earths exhibit a large centrifu-
gal barrier and are therefore trapped or localized at the impurity site. Still
the barrier is not attractive enough to form a real bound state. If the impu-
rity states lie within the conduction band of the host material, free electrons
with momentum kcan resonantly scatter at the impurity potential and form
a sharp resonance. Thus, the free conduction electrons can tunnel through
the centrifugal barrier to the impurity state, reside there for a long but finite
time and in the end tunnel back into the conduction sea of electrons with a
momentum k0. Since the electron stays only a finite time at the impurity site,
the impurity state is called a virtual bound state.
2.6 Kondo Effect
Since the Anderson impurity model is closely related to the Kondo model and
furthermore the present study is a first step towards the investigation of a
28 Chapter 2. Fundamentals
k, ↑k′,↑
k′′ ,↓
↓ ↓↑
S+S−
Figure 2.11: Second
order scattering process
leading to the Kondo ef-
fect: A conduction elec-
tron in state k,↑scatters
at the magnetic impurity
with spin ↓under spin flip
transition into the inter-
mediate state k00 ,↓and
impurity spin ↑. A second
scattering process leads
to the final state k0,↑and
impurity spin ↓.
Kondo effect in finite systems, the Kondo effect will be briefly discussed here.
For a detailed description I refer the reader to the original work and excellent
reviews on the subject [4, 61, 88].
The Kondo effect describes transport anomalies, like an increase in resistivity
at low temperatures found in non-magnetic metals sparsely doped with mag-
netic impurities, first observed in 1934 by de Haas et al. in FeAu [89]. Hence,
there has to be an additional contribution to the scattering of conduction
electrons beyond electron-phonon and defect scattering, which increases the
resistivity at low temperatures.
In case of small hybridization strength Vdk and large U0,i.e. the local moment
regime, the Anderson model can be transformed by the use of the Schrieffer-
Wolff transformation [90] to the sd-model Hamiltonian [61], which has the
form
Hsd =X
k,k0
Jk,k0S+c†
k,↓ck0,↑+S−c†
k,↑ck0,↓+Szc†
k,↑ck0,↑−c†
k,↓ck0,↓.(2.30)
The spin ladder operators are defined as S±=Sx±iSyand Jk,k0is the ex-
change coupling.
Allowing only low energy excitations k,k0≈kFin the system, i.e. par-
ticularly forbid excitations into the upper local moment state, the exchange
coupling can be expressed as follows [90]:
JkF,kF=J0=|VdkF|2U0
Ed(Ed+U0)<0.
Since J0is negative, conduction electron and local impurity spin couple anti-
ferromagnetically. The formation of this singlet state was first described by
Kondo [4]. This formation of a singlet state in the low energy regime can be
understood as a higher order scattering processes as depicted in the Feynman
2.6. Kondo Effect 29
diagram in figure 2.11. The scattering process involves a spin flip of the im-
purity and conduction electron. An incoming electron k,↑is scattered under
spin flip into an intermediate state k00 ,↓. Another spin flip scattering leads to
the final state of electron k0,↑.
Perturbation theory in third order of the exchange coupling J0of the local-
ized spin and the spin of itinerant electrons treating this spin flip scattering
process [4], yields a contribution to the resistivity of the form ln(T/TK). The
temperature TKis the Kondo temperature which defines the energy scale for
the singlet formation. This leads to the anomalous resistivity increase at low
temperatures as observed experimentally.
It should be mentioned that the transition to the Kondo regime as a function
of temperature is not a phase transition but a continuous cross over, since the
cloud of conduction electrons screening the local impurity spin grows as the
temperature decreases.
30 Chapter 2. Fundamentals
Chapter 3
Experimental Setup
The following chapter will give an overview of the experimental techniques ap-
plied. The data presented throughout this thesis was taken using two different
setups. Since a detailed description of the setup used to obtain linear x-ray
absorption spectra, which was designed by me, can be found in the literature
[48, 91], focus will be brought to the second setup enabling us to perform
XMCD spectroscopy on mass selected clusters. This setup, a modification
of the first one, was designed by Andreas Langenberg [92] and is shown in
figure 3.1. Clusters were produced in a magnetron gas aggregation source and
transmitted with an adjacent radio frequency ion guide to a quadrupole mass
filter. Transfer of the mass selected cluster beam into a radio frequency ion
trap situated in a high field magnet is accomplished by high transmission ion
optics. The low target density of a gas phase sample and the low absorption
cross sections for inner shell excitations are overcompensated by using an ion
trap and intense synchrotron radiation. Detection of the ion yield, resulting
from inner shell excitation, is achieved by time-of-flight mass spectrometry.
All inner shell spectra were taken at the synchrotron radiation facility BESSY
II, currently operated by the Helmholtz association. A short introduction to
synchrotron radiation is given in section 3.5.
3.1 Magnetron Sputtering Cluster Source
Clusters were produced in a standard magnetron gas aggregation source devel-
oped in the Haberland/von Issendorff group. Production of binary clusters is
achieved by co-sputtering of two metal targets by introducing 4 mm drillings
on a reference circle of 27 mm radius in the upper target, which is normally
the host material. The size of the drillings depend on the ratio of the sputter
yields of both metals and the desired amount of doping. High purity argon
31
32 Chapter 3. Experimental Setup
cluster source
ion guide
mass filter
x-ray beam
solenoid magnet
ion guide
ion trap
reflectron time-of-flight
mass spectrometer
Figure 3.1: Schematic view of the setup for XMCD spectroscopy on free mass selected clus-
ters [34]. Clusters were produced in a magnetron gas aggregation source and subsequently
mass selected in a quadrupole mass filter. The mass selected cluster beam is transferred by
several ion optics into a quadrupole ion trap cooled to liquid He temperature and situated in
a 5T solenoid superconducting magnet. Synchrotron radiation is coupled in co-axially to the
trap axis. Ion yield spectroscopy is performed using a reflectron time-of-flight spectrometer.
is used for sputtering, the plasma discharge is stabilized by a magnetic field.
Clusters grow by aggregation in a helium atmosphere. The aggregation vol-
ume is cooled to liquid nitrogen temperature. Since cooling conditions are
very mild in the aggregation process, preferentially ground state structures
are formed.
By adjusting the argon and helium flow, the aggregation length and the pres-
sure as well as the sputtering power, the size distribution of the clusters can
be shifted in a wide range, from monomers up to clusters containing several
hundreds of atoms. For a very detailed description of the source I refer the
reader to references [48, 93].
3.2 Radio Frequency Ion Guide and
Quadrupole Mass Filter
Adjacent to the cluster source a radio frequency ion guide is mounted, trans-
mitting the cluster beam to a quadrupole mass filter, through a differential
pumping stage. Mass selection is then performed in a commercial quadrupole
3.3. Radio Frequency Ion Trap and Magnet 33
Figure 3.2: Magnetization as a
function of temperature and exter-
nal magnetic field shown for a par-
ticle with J=S= 5/2. Magne-
tization follows the Brillouin func-
tion. At a typical temperature of
about 15 K and a maximum mag-
netic field of 5 T magnetization is
about 45 %.
mass filter 1featuring high transmission at a resolution of up to m/∆m=
1200. Typical currents of the mass selected cluster beam are 10 −100 pA.
The divergent mass selected cluster beam is focused by an ion lens. Addi-
tionally, an electric field transverse to the cluster propagation direction can
be applied to correct the ion trajectories in the stray field of the high field
magnet. Subsequently the cluster beam is bended by a 90 ◦static quadrupole
deflector. The cluster beam is transmitted to the ion trap situated in the
homogeneous part of the magnetic field by a quadrupole ion guide. By colli-
sions with residual gas the cluster beam is compressed to the ion guide axis
before entering the magnetic field, which forces the ions to precess. Therefore
a maximum overlap of cluster ion and synchrotron radiation beam is ensured.
Details can be found in the master thesis of M. Niemeyer [34] and reference
[39].
3.3 Radio Frequency Ion Trap and Magnet
The low cross section of the inner shell absorption and a dilute gas phase
sample necessitate the accumulation of target density in an ion trap in order
to perform XAS and XMCD spectroscopy [48]. Since the basic principles of
the operating mode of the ion trap are discussed thoroughly in the literature
[34, 48, 91, 94] a detailed description will not be given here. Instead technical
details affecting the magnetization of the sample will be brought into focus.
As an example the magnetization of particles with a magnetic moment of
5µBas a function of the external magnetic field and the temperature is shown
in figure 3.2. In this example a magnetization of about 45 % is obtained in a
1Extrel QMS equipped with 9.5 mm electrodes and operated at a frequency of 880 kHz.
A mass range of 10 amu
/eup to 4000 amu
/eis accessible.
34 Chapter 3. Experimental Setup
ion trap
Figure 3.3: Variation of
the magnetic field at 5 T
along the magnets sym-
metry axis [95]. The
trap is aligned so that the
magnetic field variation is
less than 1 %, as shown
by the solid line (mag-
netic field on axis) and
the dotted line (magnetic
field 10 mm off axis).
5 T magnetic field and at a typical operating temperature of 15 K. To achieve
magnetization of the sample, the trap is placed in the homogeneous part of
the magnetic field of a solenoid superconducting magnet and cooling of the
sample is accomplished by using a helium buffer gas coupled to a liquid he-
lium reservoir. The variation of the magnetic field along the magnet axis is
shown in figure 3.3. As can be seen the trap is not placed in the center of the
magnet, but in an area where the magnetic field varies less than 1%. The trap
is moved as close as possible to the focus of the x-ray beam, which is only
limited by the presence of a vacuum chamber hosting the static deflector and
several supply feedthroughs for the experiment, cf. figure 3.1.
The design of the liquid helium reservoir is shown in figure 3.4. The radio fre-
quency (rf) and direct current (dc) electrodes are coupled to a double walled
tube containing the liquid helium. This tube is shielded against infrared radi-
ation by a second double walled tube cooled to liquid nitrogen temperature,
which is thermally isolated by ceramic spacers against the inner liquid he-
lium tube and the vacuum chamber. As already mentioned cooling of the gas
phase sample itself is achieved by a helium buffer gas. The temperature was
controlled using two temperature sensors2mounted on the helium reservoir
and the dc electrodes, respectively. However, the temperature reading of the
probes can in principle deviate from the true cluster temperature due to rf
heating and imperfect coupling to the helium reservoir. For that reason the
probes were calibrated by taking magnetization curves of large iron clusters,
which was shown in detail in [34, 39]. To obtain temperatures Tas low as
T= 10 −20 K a partial pressure of about 10−4−10−3mbar is required [93,
2Cernox sensor, model: CX-1050-SD-HT-1.4L-SMOD-4-QT36-4
3.4. Reflectron Time-of-Flight Mass Spectrometer 35
dc electrodes
mounting
100mm outer
LN shield
LHe shield
rf electrodes
90mm
tube
Figure 3.4: Sectional
view of the ion trap [34].
The rf electrodes con-
fine the clusters radially.
The dc electrodes cre-
ate a electric field push-
ing the clusters towards
the exit aperture. Cool-
ing is achieved by a he-
lium buffer gas coupled to
the liquid helium reser-
voir, which is shielded
against infrared radiation
by a liquid nitrogen cool-
ing shield.
96] inside the ion trap.
During x-ray absorption measurements a considerable number of helium atoms
is ionized, although the cross section of direct photoionization is rather small
at the transition metal L-edge photon energies. This is overcompensated by
the high partial pressure present in the trap. The helium ions, that are trapped
very efficiently by the magnetic field, suppress the cluster density and further-
more spark discharges in the trap’s electric fields. Therefore, an additional
dipole rf field is applied to the trap’s electrodes tuned to the cyclotron reso-
nance ωcyclotron =e·B/2mof helium at the given magnetic field strength. In
this way helium ions can efficiently be removed from the trap.
3.4 Reflectron Time-of-Flight Mass
Spectrometer
Ion yield spectra were recorded using a modified reflectron time-of-flight mass
spectrometer. In a pulsed electric field particles of different masses gain the
same amount of energy. The mono-energetic particles spatially disperse in a
field free region according to their mass-to-charge ratio. Hence, by measuring
the time of flight, ion yield spectra can be obtained.
The initial space and velocity distribution of the particles was corrected by
using two acceleration stages and a reflector. This standard reflectron time-of-
flight mass spectrometer [97–99] was modified by three additional einzel-lenses
that adjust the ion trajectories which are disturbed by the stray field of the
high field magnet.
36 Chapter 3. Experimental Setup
e-
e-
g
λ
Figure 3.5: Sketch of an undulator and
electron bunch trajectories taken from
[34]. Light emitted by the electrons
within the undulator can interfere con-
structively enhancing the synchrotron ra-
diation intensity. By tuning the gap g the
energy of the radiation can be varied. By
tuning the shift in a split undulator cir-
cular and elliptical radiation can be gen-
erated.
3.5 Synchrotron Radiation and
Undulator Beam-Lines
Only a very short description of synchrotron radiation will be given here since
excellent reviews are available in the literature [100–102]. All x-ray absorption
experiments presented throughout this thesis were recorded at the third gen-
eration synchrotron radiation facility BESSY II. Synchrotron radiation arises
from acceleration of charged relativistic particles (here electrons).
Electrons with an energy of 1.7 GeV circulate in a storage ring, which con-
sists of straight segments and deflecting dipole magnets. Electron trajecto-
ries were corrected by sextupole and octupole magnets and energy losses are
compensated in micro wave resonators. Whenever passing a dipole magnet
synchrotron radiation is emitted according to the Maxwell equations. By
Lorentzian transformation from the reference system of the electron to the
laboratory frame of reference the Hertzian dipole characteristic of the radia-
tion is modified in such a way that radiation is mainly emitted in the direction
of electron motion. Higher radiation intensities were generated by using un-
U49-2-PGM UE52-SGM
energy range 85-1600 eV 90-1500 eV
resolving power 25000 (85-500 eV) >4000
15000 (500-1500 eV)
flux 1013 photons
/s(85-500 eV) >1012 photons
/s
1012 photons
/s(500-1500 eV)
polarization horizontal variable
divergence 2 mrad ×2 mrad 6 mrad ×1 mrad
Table 3.1: Technical specification of the undulator beam lines U49-2-PGM [103] and
UE52-SGM.
3.5. Synchrotron Radiation and Undulator Beam-Lines 37
dulators inserted in the straight segments of the storage ring. Undulators are
periodical magnetic structures with a variable gap, cf. figure 3.5. Linear po-
larized radiation emitted by electrons on a sinusoidal trajectory can interfere
constructively enhancing the intensity by a factor of N(Nnumber of magnet
pairs in the undulator). When introducing a shift in a split undulator, as
shown in the lower panel of figure 3.5, electrons on spiral trajectories emit
elliptical or circular polarized light depending on the undulator shift. The
degree of polarization is well defined and about 90 %.
The undulator radiation is coupled to a beam line where the spectral purity
of the radiation is further improved by plain or spherical grating monochro-
mators. The resolving power and transmitted intensity strongly depends on
the beam line. All experiments were carried out at U49-2-PGM [103] and
UE52-SGM beam lines, for technical details see table 3.1 and reference [104].
38 Chapter 3. Experimental Setup
Chapter 4
Experimental and
Computational Details
In this chapter details on the data acquisition and analysis as well as some
preparatory calculations are presented. Purpose of the first section is to get a
detailed overview of the preparation procedure before taking x-ray absorption
or x-ray magnetic circular dichroism spectra and the data acquisition cycle.
Details of the data analysis can be found in the second section of this chapter,
focusing on background subtraction and normalization procedures. In the last
section some preparatory calculations are presented, which aim to establish
parameters for further calculations.
4.1 Data Acquisition
Before recording x-ray absorption spectra three preparatory steps were always
performed. Firstly, the cluster ion yield in the mass region of interest is op-
timized, whereas all ion optics are operated in transmission mode. A typical
mass spectrum is shown in panel (a) of figure 4.1. Subsequently, the mass se-
lection is switched on and transmission is optimized for the cluster of interest.
Typically, the ion yield drops by a factor of two in selection mode, as can be
seen from panel (b) of figure 4.1.
The second step - accumulating target density - is achieved by operating the
trap in storage mode by tuning the trap parameters to efficiently store the
selected cluster. In the last preparative step, the x-ray beam is coupled in and
the trap parameters are adjusted in such a way that the ion yield of fragments
induced by x-ray absorption is maximal, which can be seen in panel (d) of
figure 4.1. Hereby the trapping efficiency of the parent cluster is eventually
reduced.
39
40 Chapter 4. Experimental and Computational Details
2
4
6
8
104
2
400350300250200150100500
2
4
6
8
104
2
4000
2000
4000
2000
transmission
transmission
and selection
trapped
trapped
and x-ray beam
(a)
(b)
(c)
(d)
ion yield in arb. units
mass in amu
Figure 4.1: Mass spec-
tra taken in prepara-
tion for recording x-
ray absorption spectra of
CrCu+
4. Panel (a) shows
a mass spectrum opti-
mized for the mass re-
gion of interest. Panel
(b) shows the very same
mass range but with
the quadrupole mass fil-
ter in selecting mode.
The ion yield of CrCu+
4
is reduced by about
50 %. Target density
is accumulated by stor-
ing the selected cluster,
cf. panel (c). Panel
(d) shows a mass spec-
trum with additional x-
ray beam induced frag-
ments.
X-ray absorption spectra were then obtained by recording the ion yield of
photo-induced fragments, cf. panel (d) of figure 4.1, as a function of the pho-
ton energy. A schematic view of the data acquisition is given in figure 4.2.
While recording the ion yield, several quantities were simultaneously collected
such as synchrotron and beam-line parameters as well as photon flux measured
using a GaAsP diode, cf. figure 4.2. When setting a new photon energy the
x-ray beam is blocked with a beam shutter.
In order to be able to subtract a background, x-ray absorption spectra are
recorded from well below the resonances up to about 20 eV above resonant
excitations. In most of the cases, the energy resolution was chosen to be 250-
625 meV at the resonances to resolve multiplet structures. At every energy
point mass spectra were averaged for 8-12 s at frequencies of 100-500 Hz. After-
wards, a x-ray absorption spectrum with opposite photon helicity is recorded.
Taking at least an additional pair of x-ray absorption spectra ensures repro-
ducibility of the measurement. Depending on the signal to noise ratio, up to
six pairs of x-ray absorption spectra were taken.
42 Chapter 4. Experimental and Computational Details
620610600590580570560550
residual
xas of CrCu4+
linear fit to
background
CrCu4+
subtracted
background
photon energy in eV
ion yield in arb. units
Figure 4.3: The background of a x-ray
absorption spectrum (solid red line) is fit-
ted below and above the absorption edge
as a linear function (dashed blue line) with
the same slope. The residuum of the fit is
shown in the upper panel, the straight line
indicates the region which was not included
in the linear fit. The x-ray absorption spec-
trum without background is shown as dot-
ted green line.
4.2 Data Analysis
As mentioned in the previous section, x-ray absorption spectra were obtained
as the photon energy dependent ion yield of photo induced fragments. The
dominant photo fragments are Au+
1in case of doped gold and Cr+
1and Cu+
1in
case of chromium doped copper clusters, respectively. In the x-ray absorption
spectra of small clusters the ion yield of doubly charged photo fragments be-
comes dominant. All ion yield spectra were normalized to the incident photon
flux obtained by measuring the photo current of a GaAsP diode with known
quantum efficiency.
The background, which mainly originates from direct photo ionization of
weaker bound electrons 1, is fitted as a linear function below and above the res-
onances with the constraint of identical slopes and is subsequently subtracted
from the x-ray absorption spectrum, cf. figure 4.3. The direct photoionization
of the x-ray absorption spectra was normalized to unity. The XMCD asymme-
try as well as the sum spectra shown in figure 4.4 were subsequently obtained
as the average of all possible combinations of x-ray absorption spectra taken
with positive and negative helicity. For comparison all XMCD spectra were
normalized to the number of absorbers and 3d-holes. These numbers can be
inferred from the integral of the x-ray sum spectra after subtracting contribu-
tions from direct photoionization and resonant excitations into higher orbitals
1Direct photo ionization at chromium and copper M-edges and gold Nand Oedges
[105].
4.3. Computational Details 43
normalized xmcd asymmetry
in arb. units
x-ray absorption in arb. units
565 570 575 580 585 590 595 560 570 580 590 600 610
photon energy in eV photon energy in eV
linear xas
step function
integral
of
linear xas
- step function
Figure 4.4: Data processing exemplary shown for CrAu+
2. In the left panel linear XAS,
step function, to account for transitions other than 2p→3d, and integral of linear XAS
(previously the step function was removed) is shown. In the right panel the normalized
XMCD asymmetry is shown. The error is obtained as the standard error of the mean value.
allowed by dipole selection rules. The corresponding step functions were di-
rectly placed at resonant excitation energies of 2p3/2,1/2→3dtransitions,
which can be seen in figure 4.4 and was shown to be reasonable procedure
[32]. The intensity ratio of the step functions was set to 2/3 according to the
degeneracy of the 2p3/2,1/2levels. A normalized XMCD spectrum is shown in
the right panel of figure 4.4.
4.3 Computational Details
Most of the calculations were carried out using the quantum espresso 5.0
code [86]. In quantum espresso 5.0 the calculations are based on plane
waves exploiting the periodicity of a bulk material. In practice this is done
by periodically repeating a super cell. Since properties of an isolated cluster
should be calculated here, the cell has to be large enough so that the periodic
images of the cluster do not interact with each other, but as small as possible
to limit the computational cost. This is even more challenging because of the
long-range Coulomb interaction of cationic clusters. Therefore, prior to the
calculations the convergence with respect to cell size and kinetic energy wave
function cutoff was tested. In all calculations only one k-point (the Γ point)
was used for the k-point sampling because no dispersion is expected for the
finite systems investigated throughout this thesis.
44 Chapter 4. Experimental and Computational Details
-700
-695
-690
-685
-680
6055504540353025201510
-699.85
-699.80
-699.75
-699.70
-699.65
-699.60
-699.55
60555045403530252015
total energy in Ry
total energy in Ry
wave function cut-off energy in Ry cell dimension in a. u.
Figure 4.5: Convergence test for CrAu+
n: Total energy shown as a function of cell size
and wave function kinetic energy cut-off. Optimal values used for subsequent calculations
are indicated by an arrow.
The total energy as a function of the cell size for CrAu+
6is shown in the right
panel of figure 4.5. The arrow indicates the cell size used in all following cal-
culations, since here the total energy starts to oscillate, which is typical for
the energy convergence in plane wave codes. In the left panel of figure 4.5 the
total energy as a function of the kinetic energy wave function cut-off is shown.
Again the arrow indicates the value used for all following calculations. This
value of 50 Ry for the kinetic energy wave function cut-off was chosen since
the change in total energy dropped below 50 meV, which is the accuracy of
the calculations presented here.
The search for ground state structures in case of CrCu+
nclusters was done
using the turbomole 6.2 program suite. Here a basis set of triple-ζquality
(def2-TZVP) was used in all calculations. This is known to yield good quan-
titative results for 3dtransition metals [106].
All calculations were carried out by employing the PBE approximation [107]
to the exchange-correlation functional.
Chapter 5
Electronic Structure of Early 3d
Transition Metal Impurities in
Non-Magnetic Gold Clusters
In this chapter the electronic structure of doped gold clusters will be studied
by systematically exchanging the dopant atom and varying the cluster size.
This was done in order to vary the number of 4sand local 3delectrons of the
impurity as well as the number of delocalized electrons in the host material.
The local electronic structure and details in the binding strongly depend on
the nature of the dopant atom. For example, starting from the early transition
metal scandium a slight odd-even effect in the electronic structure is evident.
This effect becomes very pronounced in titanium doped gold clusters. In con-
trast, the electronic structure of the impurity is almost independent of the
number of gold atoms in vanadium doped gold clusters. In chromium doped
gold clusters the impurities electronic structure is dominated by shell closure
in the gold host.
Linear x-ray absorption spectra presented throughout this chapter are taken
with an energy resolution of 125 meV, which enables us to resolve the multi-
plet structure present in the x-ray absorption of all investigated clusters. The
multiplet structure is very characteristic for the symmetry of the environment,
the occupation of the 3dstates as well as their hybridization due to bonding.
This sensitivity makes x-ray absorption spectroscopy a finger print method
[31, 108]. Although the energy resolution in principle allows us to resolve
individual transitions, the high density of states at the Fermi level and the
multiplet structure causes overlap of the life time broadened lines [109].
In addition density functional theory calculations were performed for all clus-
ters investigated throughout this chapter. Established ground state structures
[12] were re-optimized and the total density of states (DOS) as well as pro-
45
46
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
jected density of states (PDOS), obtained from projecting the density of states
onto atomic orbitals, were calculated. These are used to analyze the electronic
structure and to interpret the experimental x-ray absorption data.
We follow the periodic table and start with a discussion of scandium doped
gold clusters.
5.1 Local Electronic Structure of Scandium
Doped Gold Clusters
In figure 5.1 the linear x-ray absorption spectra of mass selected ScAu+
nclus-
ters in a size range of n= 1 −6 are presented. Additionally, the spectrum
of atomic Sc+is shown [32]. The x-ray absorption spectra of ScAu+
1and
Sc+feature almost identical spectral shapes, which indicates a very similar
local electronic structure. The ground state electronic configuration of Sc+is
[Ar]3d14s1, hence it can be expected to find one 3d-electron localized at the
scandium site in ScAu+
1and that bonding is mediated by the 4selectron.
Localization of the 3d-orbital can indeed be observed from a comparison of
the total and the scandium d-projected density of states, cf. figure 5.2. Here
the state close to the Fermi energy (highlighted by the box) has almost pure
scandium d-character, which points to a complete localization of this state at
the scandium site.
In case of ScAu+
2the spectral signature becomes very narrow and exhibits
only two main lines comparable to the x-ray absorption spectrum of atomic
calcium cation [32]. This hints at a local 3d0configuration of the scandium
atom in ScAu+
2, which is confirmed by the scandium d-projected density of
states shown in figure 5.2 where no localized 3d-electron can be found. Fur-
thermore the 3d-spin polarization obtained from a L¨owdin population analysis
[110, 111] is 0 µB,cf. figure 5.3, which is in perfect agreement with the as-
sumption of a local 3d0configuration due to delocalization.
The described observations for ScAu+
1,2can be generalized to larger clusters.
Indeed, odd numbered ScAu+
nclusters always feature pronounced multiplet
signatures in their x-ray absorption spectra while even numbered clusters have
more symmetric and narrow spectral features. Although the odd-even effect
is less pronounced for larger ScAu+
nclusters it clearly survives up to n= 6
as can be seen for example from the difference of 3dspin up and down occu-
pation shown in figure 5.3. The 3d-electron in odd numbered ScAu+
nclusters
(n > 1) hybridizes more strongly with the gold electronic states as compared
to ScAu+
1, since the coordination of the scandium impurity increases linearly
with the cluster size.
5.1. Local Electronic Structure of Scandium Doped
Gold Clusters 47
420416412408404400396392
photon5energy5in5eV
x-ray5absorption5in5arb.5units
ScAu1
+
ScAu2
+
ScAu3
+
ScAu4
+
ScAu5
+
ScAu6
+
Sc+
L3L2
Figure 5.1: Linear x-ray absorption spectra of scandium doped gold clusters at the L2,3
edges of scandium. The spectral signature changes drastically as a function of the cluster
size. There is some resemblance of the x-ray absorption spectrum of ScAu+
1and Sc+indi-
cating a localized 3d-electron in ScAu+
1. Additionally shown are the ground state structures,
taken from [12] and re-optimized.
48
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
ScAu1
+
ScAu2
+
ScAu3
+
ScAu4
+
ScAu5
+
ScAu6
+
spinFresolvedFdensityFofFstatesFinF1/eV
energyFE-EFermiFinFeV
d-projected
ScFPDOS
totalFDOS
100
50
0
-50
-6 -5 -4 -3 -2 -1 0 1
100
50
0
-50
-100
-100
-50
0
50
100
-50
0
50
-100
-50
0
50
100
100
50
0
-50
-100
Figure 5.2: Spin re-
solved total and scan-
dium d-projected den-
sity of states shown for
ScAu+
n,n= 1 −6. Pos-
itive and negative val-
ues of the density of
states represent spin-up
and spin-down states,
respectively.
Since the mean radius of the 3d-orbitals in atomic scandium, which is about
1.6˚
A, is quite large compared to an inter-atomic separation of about 2.5˚
A
in ScAu+
nclusters, one could expect the participation of the 3d-electron in
bonding for all cluster sizes. Furthermore 3dand 4sstates in scandium may
benefit from sd-hybridization since they are energetically almost degenerate
[112]. From these observations a strong hybridization with the gold electronic
states might be expected independent of the cluster size. However, the 3d-
electron remains localized at least to some extend in odd numbered cluster.
Hence, it can be deduced that it is energetically favorable to delocalize an even
number of electrons. This will be discussed in more detail in the following sec-
tion, since the odd-even effect in the electronic structure is more pronounced
5.2. Odd-Even Effects in the Electronic Structure of Titanium
Doped Gold Clusters 49
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
654321
number of 3d electrons
host cluster size
total
spin up
spin down
difference
Figure 5.3: Population
analysis of the scandium
3d-orbitals in ScAu+
n
clusters using a L¨owdin
population analysis.
The total 3d-occupancy
exhibits a slight odd-
even effect, while it
is pronounced for the
difference in spin-up and
spin-down occupation.
in TiAu+
nclusters, due to the more compact 3d-orbitals and opening of the
3d-4sgap along the 3dtransition metal series [112].
5.2 Odd-Even Effects in the Electronic Struc-
ture of Titanium Doped Gold Clusters
In figure 5.4 linear x-ray absorption spectra of titanium-doped gold clusters
at the L2,3edges of titanium are presented in a size range of n= 1 −9. At
first glance there is a strong odd-even effect visible in the x-ray absorption
of TiAu+
nclusters. The planar, odd numbered TiAu+
nclusters (up to n= 5)
exhibit almost identical spectral signatures. The x-ray absorption signal of
these clusters nicely resembles the x-ray absorption spectrum of an isolated
titanium cation ([Ar]3d24s1), cf. the Hartree-Fock calculation shown in figure
5.4. This finding suggests that both 3delectrons in odd numbered TiAu+
n
clusters remain localized at the titanium site to a large extend and do not
participate in bonding.
Even numbered TiAu+
nclusters have very similar spectral shapes among each
other as well, although a slightly increasing broadening of the spectral features
with increasing cluster size can be observed.
However, in contrast to odd numbered TiAu+
nclusters, no similarity to the
x-ray absorption spectrum of atomic titanium can be found. As already men-
tioned, x-ray absorption locally probes the unoccupied density of states, there-
fore changes in the symmetry or the unoccupied density of states due to bond-
ing or fractional electron transfer to or away from 3dstates would result in a
deviation from the atomic spectral shape. Indeed, a small odd-even variation
50
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
472468464460456452448
TiAu1
+
TiAu2
+
TiAu3
+
TiAu4
+
TiAu5
+
TiAu6
+
TiAu7
+
TiAu8
+
TiAu9
+
Ti+xHFxcalc
photonxenergyxinxeV
x-rayxabsorptionxinxarb.xunits
L3L2
Figure 5.4: Linear x-ray absorption spectra of titanium doped gold clusters at the L2,3
edges of titanium. The spectral signature of odd numbered TiAu+
nin a size range of
n= 1 −5 show a strong resemblance with an atomic Hartree-Fock calculation of the XAS
of Ti+. Furthermore, a second set of similar spectral shapes are found for even numbered
TiAu+
nclusters in a size range of n= 2 −8. Additionally shown are the ground state
structures, taken from [12].
5.2. Odd-Even Effects in the Electronic Structure of Titanium
Doped Gold Clusters 51
2.5
2.0
1.5
1.0
0.5
0.0
987654321
totale3deoccupation
(thisework)
(spineupe-espinedown)e(Torreseeteal.)
(spineupe-espinedown)e(thisework)
spinedowne(thisework)
spineupe(thisework)
3deoccupation
hosteclusteresize
Figure 5.5: Total and
spin resolved 3doccu-
pation derived from a
L¨owdin (this work) and
Mulliken (Torres et al.
[12]) population analysis
of TiAu+
nclusters. A
small odd-even effect in
the total 3doccupation
is evident.
of the 3doccupancy exists, as can be seen from the total and spin resolved 3d
occupation presented in figure 5.5, which is derived from a L¨owdin population
analysis [110, 111]. Still, both odd and even numbered TiAu+
nclusters have
about two 3delectrons, with a variation in 3doccupancy smaller than 10 %.
Such a small effect might not result in the different spectral signatures for odd
and even numbered TiAu+
nclusters.
To shed some more light on the odd-even aspect the density of states, shown
in figure 5.6, will be analyzed. Again, the occupied density of states is very
similar among odd and even numbered TiAu+
nclusters, respectively. In figure
5.6 the total density of states and the titanium d-projected density of states
is displayed for the whole series of TiAu+
nclusters, n= 1−9. From a compar-
ison of the DOS and the PDOS the amount of hybridization of the titanium
3dstates can be deduced. There is a severe difference between odd and even
numbered TiAu+
nclusters, as can be seen from the figure. In case of even
numbered TiAu+
nclusters only one non-hybridized 3dstate remains, whereas
for odd numbered TiAu+
nclusters there are two almost non-hybridized 3d
states present. In contrast to these well localized states near the Fermi level
(highlighted by the boxes), in even numbered TiAu+
nclusters the second 3d
electron strongly hybridizes with the sd-states of gold well below the Fermi
level, cf. figure 5.6.
The finding of two atomic-like 3delectrons in odd numbered TiAu+
nclusters,
which do not participate in bonding, is in very good agreement with the exper-
imental x-ray absorption spectra. Since the titanium atom in odd numbered
TiAu+
nhas an almost undisturbed atomic-like 3d2electron configuration, these
clusters nicely resemble the atomic x-ray absorption spectrum of titanium, cf.
figure 5.4.
52
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
spin6resolved6density6of6states6in61/eV
energy6E-EFermi6in6eV
TiAu1
+
TiAu2
+
TiAu3
+
TiAu4
+
TiAu5
+
TiAu6
+
TiAu7
+
TiAu8
+
TiAu9
+
total6DOS
d-projected
Ti6PDOS
100
50
0
-50
-6 -5 -4 -3 -2 -1 0 1
-40
0
40
80
40
0
-40
100
50
0
-50
100
50
0
-50
-100
-50
0
50
100
-100
-50
0
50
100
-100
-50
0
50
100
-100
-50
0
50
Figure 5.6: The total density of states (black dotted line) and the 3d-projected titanium
density of states (red solid line) of TiAu+
nare shown. While odd numbered TiAu+
nclusters
exhibit two states at the Fermi level of almost pure 3dcharacter, even numbered TiAu+
n
clusters exhibit only one of these states. In case of even numbered TiAu+
nclusters the
second 3delectron is strongly hybridized with the gold sd states. Positive and negative
values of the density of states represent spin-up and spin-down states, respectively.
5.2. Odd-Even Effects in the Electronic Structure of Titanium
Doped Gold Clusters 53
454.0
453.8
453.6
453.4
453.2
453.0
452.8
452.6
987654321
absorption onset in eV
host cluster size
Figure 5.7: The absorption on-
set of TiAu+
nclusters as func-
tion of size suggests strong 3d-
electron localization for odd num-
bered TiAu+
nclusters compared to
even numbered ones.
The stronger localization of the 3delectrons in odd numbered TiAu+
nclusters
is also supported by the size dependent x-ray absorption onset shown in fig-
ure 5.7. Here, odd numbered TiAu+
nclusters with two localized 3delectrons
exhibit an absorption onset at lower excitation energy than even numbered
TiAu+
nclusters with one localized and one hybridized and therefore delocal-
ized 3delectron state. As was shown for pure transition metal clusters, a shift
of the absorption onset towards lower excitation energy indicates a stronger
localization of the 3delectrons [113, 114].
Additionally, the difference in occupation of titanium 3dspin-up and spin-
down states is shown in figure 5.5, obtained from a L¨owdin [110, 111] (this
work) and Mulliken [115] population analysis (Torres et al. [12]). The dif-
ference of about 0.2 electrons between the L¨owdin and Mulliken population
analysis is not surprising, since both methods are strongly basis set dependent
[116]. The size dependent trend, however, is, apart from the case of TiAu+
7,
that will be discussed below, similar. It shows that odd numbered TiAu+
n
clusters exhibit larger spin polarizations than even numbered clusters. This
finding indicates that only localized, but not itinerant electrons contribute to
the spin polarization. One can compare this behavior with what one would
expect from a very simple model of molecular bonding, namely the Heitler-
London model [117]. In this model, participation in bonding results in low
spin states which enhance the stability of the molecule by effective screening
of the Coulomb interaction among the ionic cores. In odd numbered TiAu+
n
clusters where two localized 3delectrons are present the local spin magnetic
moment is about 2 µB, whereas in even numbered TiAu+
nclusters the delo-
calized electron participating in bonding does not contribute to the local spin
polarization, hence even numbered TiAu+
nclusters exhibit a local spin mag-
netic moment of about 1 µB. However, there is a discrepancy in the local spin
moment of TiAu+
7obtained in this work and the work of Torres et al. [12].
54
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
The deviation from the odd-even pattern in TiAu+
7is probably caused by a
geometrical 2D →3D transition, increasing the mean coordination of the ti-
tanium atom and therefore maximizing the titanium-gold bonding. This in
turn results in a reduction of the local spin moment as found in our population
analysis. This hypothesis is additionally supported by the size dependent ab-
sorption onset shown in figure 5.5. Although the absorption onset in TiAu+
7
is reduced compared to the neighboring cluster sizes, the absolute value is
comparable to even numbered TiAu+
nclusters and significantly enhanced with
respect to the value for small odd numbered TiAu+
nclusters. Therefore a sim-
ilar delocalization of the 3delectrons as in even numbered TiAu+
nclusters can
be expected, which is consistent with our DFT calculations. This is strong
evidence that the structure of TiAu+
7presented here, is indeed the the struc-
ture present in our experiment.
The origin of the strong odd-even effect in TiAu+
nclusters is still unclear. It
might have its origin in electron pairing effects in the free electron gas formed
in the host cluster, as observed in pure gold clusters [118–120].
To further elucidate this question, we neglect the 3delectrons of titanium for
now. There are then n+ 1 delocalized electrons in the TiAu+
nsystem, nelec-
trons stem from the 6s-orbitals of gold while one electron is contributed by the
4s-orbital of titanium. Hence, there is an odd (even) number of delocalized
electrons present in case of even (odd) numbered TiAu+
nclusters. It seems
energetically favorable for even numbered TiAu+
nclusters to delocalize one 3d
electron, since the 3d24s1→3d14s2promotion energy of about 3 eV [121] is
overcompensated by the gain from kinetic energy reduction and reduction of
intra-atomic Coulomb interaction by delocalization. This is very different for
odd numbered TiAu+
nclusters: Here a delocalized titanium 3delectron would
have to populate a free electron gas state higher in energy, since the jellium
states are already doubly occupied.
Thus, titanium in TiAu+
nclusters serves as an electron donor providing exactly
the right number of electrons to stabilize the free electron gas bond in the gold
host cluster, without a strong overall disturbance of the gold cluster geometry.
All the ground state structures of TiAu+
n(n= 1 −9) clusters posses identical
symmetry as the ground state structures of neutral gold clusters [122, 123],
except for TiAu+
7and TiAu+
9.
Moreover, there is a second effect beneath the strong odd-even effect. From
the density of states presented in figure 5.6 it can be seen that the free electron
gas states (mainly all states which do not exhibit titanium 3d-character) are
more strongly bound in case TiAu+
1and TiAu+
5as compared to the other clus-
ter sizes. This may originate from shell closure in a two dimensional potential
well in these to systems with two and six delocalized electrons, respectively.
In summary, all these results suggest that one always finds an even number
5.3. Independence of the Local Electronic Structure on Impurity
Coordination: Vanadium Doped Gold Clusters 55
of delocalized electrons in TiAu+
nclusters.
5.3 Independence of the Local Electronic
Structure on Impurity Coordination:
Vanadium Doped Gold Clusters
In figure 5.8 the linear x-ray absorption spectra of vanadium doped gold clus-
ters together with their ground state geometries [12] are presented in the size
range of n= 1 −7.
Apart from VAu+
2which will be discussed later, clusters of sizes n= 1 −5 ex-
hibit almost identical spectral shapes. Because of large life-time broadening
due to additional electronic relaxation channels (Coster-Kronig and Super-
Coster-Kronig transitions) the L2-edge is not as well resolved as the L3-edge.
The similarity of the mentioned spectral shapes is therefore especially strik-
ing at the L3-edge. The similar spectral shapes indicate that these clusters
have the same local electronic structure. Therefore the bonding mechanism
of vanadium to the gold host has to be the same or at least very similar for
clusters with 1 and 3-5 gold atoms. At first glance this is somehow surprising,
since here the coordination of the vanadium impurity increases linearly with
the cluster size. Therefore it could be expected that the interaction strength
increases with the cluster size.
An atomic Hartree-Fock calculation shows, that the x-ray absorption spectra
of the small VAu+
nclusters do not resemble the electronic structure of the
vanadium cation with its 3d44s0ground state configuration. Since the vana-
dium cation offers no 4s-electron for bonding, in contrast to scandium and
titanium, at least some of the vanadium 3d-states are expected to hybridize
with the host materials electronic states. However, even more likely is the
promotion of a 3delectron into the far more extended 4sorbital which then
forms a molecular orbital with the host electronic states. This can be assumed
to be energetically favorable since the 3d44s0→3d34s1promotion energy is
with 320 meV [124] rather small. However, since the x-ray absorption spectra
do not resemble a pure local 3d34s1electron configuration and the local en-
vironment is very different in VAu+
nclusters n= 1 −7, vanadium 3dorbitals
have to be involved in the bonding, at least partly.
The similar x-ray absorption spectra for all VAu+
nclusters, n= 1 −5, suggest
that the number of 3d-electrons is almost identical. That this is indeed the
case can not be easily seen from the density of states, cf. figure 5.10, but
from the L¨owdin population analysis, which is shown in figure 5.9. The total
number of 3d-electrons is about 3.5 for all investigated cluster sizes, except
56
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
500 505 510 515 520 525 530 535
x-ray absorption in arb. units
photon energy in eV
VAu1+
VAu2
+
VAu3
+
VAu4
+
VAu5
+
VAu6
+
VAu7
+
L3L2
Figure 5.8: Linear x-ray absorption spectra of vanadium doped gold clusters at the L2,3
edges of vanadium. The spectral signature remains almost unchanged as a function of the
cluster size, although dramatic structural changes are present, as can be seen from the
ground state structures (taken from [12]).
5.3. Independence of the Local Electronic Structure on Impurity
Coordination: Vanadium Doped Gold Clusters 57
4
3
2
1
0
7654321
(spin up - spin down)
spin up
spin down
total
host cluster size
3d occupation
Figure 5.9: Total and
spin resolved 3doccupa-
tion of VAu+
nclusters in
a size range of n= 1 −
7. The occupation num-
bers are obtained from a
L¨owdin population anal-
ysis. The total number
of 3delectrons varies only
very weakly with the clus-
ter size pointing to al-
most identical local elec-
tronic structure of the
vanadium atom in VAu+
n
clusters. Only the VAu+
2
cluster exhibit a slightly
enhanced 3doccupation.
for the VAu+
2which exhibits a slightly enhanced number of 3d-electrons. This
in turn explains the deviation of the spectral shape for the VAu+
2cluster.
However, vanadium-doped gold clusters are qualitatively different from ScAu+
n
and TiAu+
nclusters since they are missing the pronounced odd-even effect in
their electronic structure. In order to feature an even number of delocalized
electrons in even numbered VAu+
nclusters, the vanadium atom would have
to delocalize a second 3d-electron. Again this involves an energy cost which
can be estimated by the promotion energy. Since it is larger than 5 eV [124]
delocalization of a second 3delectron seems to be improper. Therefore lo-
calization of three 3d-electrons is stabilized and results in considerable spin
magnetic moments of about 3 µBat the vanadium site, cf. figure 5.9.
Contrary to the rather simple size dependence of the electronic structure for
the smaller clusters, the sudden change in the spectral shape from cluster size
five to six is somehow puzzling. Neither changes in the local 3d-occupation
nor a 2D→3D transition, the factors that were responsible for the changes in
the spectral shapes of scandium and titanium doped gold clusters, emerge in
VAu+
nclusters.
A possible reason might be that the structure depicted in figure 5.8 is not the
one experimentally present. However, it is unlikely that the clusters produced
in the cluster source are not the the ground state species, because of the very
mild cooling conditions present in the source, which was shown to produce
preferably the geometrical ground state structures [125]. As the potential
energy surface of metal clusters is very complex even for small cluster sizes,
it seems more likely that the theoretically predicted structure [12] does not
represent the ground state geometry of VAu+
6.
58
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
spin7resolved7density7of7states7in71/eV
energy7E-EFermi7in7eV
VAu1
+
VAu2
+
VAu3
+
VAu4
+
VAu5
+
VAu6
+
VAu7
+
total7DOS
d-projected7V7PDOS
-100
-50
0
50
100
-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 0.0 1.0
40
20
0
-20
-40
80
40
0
-40
100
50
0
-50
-100
-50
0
50
100
100
50
0
-50
-100
100
50
0
-50
-100
Figure 5.10: Density of states of VAu+
nclusters, n= 1 −7. Positive and negative values
of the density of states represent spin-up and spin-down states, respectively.
5.4. Local Electronic Structure of Chromium Doped
Gold Clusters 59
5.4 Local Electronic Structure of Chromium
Doped Gold Clusters
In figure 5.11 linear x-ray absorption spectra of chromium doped gold clusters
at the L2,3edges of chromium are presented in a size range of n= 1 −8.
They were obtained as described in section 4.2. As already discussed, x-
ray absorption can be used as a fingerprint method. As can be seen from
figure 5.11, the spectral signature changes drastically with the host cluster
size, indicating changes in occupation of the 3dstates or the symmetry of the
environment. The x-ray absorption spectra of three clusters attract particular
attention: CrAu+
n,n= 2,6,8. Here a strong resemblance with the spectra of
atomic Cr+can be found [32], also evident from a comparison to an atomic
Hartree-Fock calculation of Cr+[126, 127], cf. dotted lines in figure 5.11
(direct photoionization was not taken into account in the calculation). Hence,
in these cases the 3d-electrons of chromium seem to be mainly undisturbed
by the gold host matrix or, put differently, do not participate in bonding.
Additionally the spectral shapes, cf. figure 5.11, indicate that the 3delectrons
are strongly localized in CrAu+
n,n= 2,6,8. As already pointed out, increasing
localization of 3delectrons yield a shift of the absorption onset towards lower
photon energy as was shown by us [114] for pure transition metal clusters.
Therefore the shift of the absorption onset towards lower energy compared to
the neighboring cluster sizes, as plotted in figure 5.12, is consistent with the
assumption of strong localization of the 3delectrons in CrAu+
2,6,8.
The interaction of the host with the chromium impurity in these particular
systems is strongly suppressed by shell closure in the free electron gas of the
gold host cluster. Shell closure is found for two and six electrons in a two
dimensional and eight electrons in a three dimensional potential well. Indeed,
the gold hosts in CrAu+
2,6,8feature appropriate shapes, i.e. two dimensional
structures in CrAu+
2, CrAu+
6and a three dimensional structure in CrAu+
8, as
can be seen from the structures depicted in figure 5.11. Therefore, in CrAu+
2,6,8
the chromium impurity is less strongly bound to the host as compared to the
other cluster sizes, as will be discussed in detail in the next chapter.
However, there is also a striking similarity among the x-ray absorption spectra
of CrAu+
1,5,7,cf. figure 5.11. Actually, this again can be understood in shell
closure in a free electron gas, assuming chromium delocalizes one electron. In
order to do so, chromium has to delocalize one 3delectron, due to its cationic
ground state configuration 3d54s0missing a 4selectron. This in turn leads to
the deviation of the spectral signature from a local 3d5configuration. Since
the 3d54s0→3d44s1promotion energy is about 1.5 eV [121] it can be assumed
that it is feasible to delocalize that one 3delectron. This is especially true if
60
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
600595590585580575570565560
CrAu1+
CrAu2
+
CrAu3
+
CrAu4
+
CrAu5
+
CrAu6
+
CrAu7
+
CrAu8
+
photon.energy.in.eV
x-ray.absorption.in.arb..units
L3L2
Figure 5.11: Linear x-ray absorption spectra of chromium doped gold clusters at the L2,3
edges of chromium. The spectral signature changes drastically as a function of the cluster
size. Still, strong resemblance with the atomic XAS can be seen for CrAu+
2,6by comparison
to an atomic Hartree-Fock calculation of Cr+(dotted line). Additionally shown are the
ground state structures, taken from [12].
5.5. Summary 61
571.6
571.4
571.2
571.0
87654321
host cluster size n
absorption onset in eV
Figure 5.12: Size dependent ab-
sorption onset in the x-ray absorp-
tion of CrAu+
n. Strong localization
of the 3delectrons in CrAu+
2,6,8is
reflected in a shift towards lower
energy compared to the neighbor-
ing cluster sizes.
the energy gain by delocalizing one electron is large, which can be expected
in these cases, since it results in shell closure in the free electron gas with two,
six and eight electrons, respectively.
The x-ray absorption spectra of CrAu+
nn= 3,4 are almost identical and differ
from those of the other CrAu+
nclusters.
5.5 Summary
Albeit the very different size dependence in the x-ray absorption spectra of
TMAu+
n(TM=Sc,Ti,V,Cr), a common ground in the bonding mechanism of
the impurity to the host can be deduced. Whenever possible the TMAu+
n
systems tend to delocalize an even number of electrons at least partly mediat-
ing the bond. This however depends on an interplay of the host size and the
impurity nature. While the gold host cluster contributes nelectrons, the num-
ber of electrons contributed by the impurity depend not only on its cationic
ground state configuration, e.g., the presence of a 4selectron, but also on the
3dk4sl→3dk−14sl+1 promotion energy. Hybridization with the gold sd-states
is favored by the spatially extended 4sorbitals rather than the compact 3d
orbitals. Therefore, in order to delocalize an electron the promotion energy
must be rather small.
These particular boundary conditions make the size dependence in the elec-
tronic structure of transition metal (Sc,Ti,V,Cr) doped gold clusters so rich
and lead to the different size dependencies of the spectral shapes.
62
Chapter 5. Electronic Structure of Early 3dTransition Metal
Impurities in Non-Magnetic Gold Clusters
Chapter 6
The Anderson Impurity Model
in Finite Systems
In this chapter the Anderson impurity model will be put to test in finite sys-
tems. The influence of a discrete density of states of the host material on the
impurity spin magnetic moment will be studied in the model system CrAu+
n.
Starting from the XMCD spectra as well as DFT calculations, the size de-
pendence of the local spin moment of CrAu+
nclusters will be discussed: It
turns out that the size dependence of the impurity magnetic moment essen-
tially is in agreement with the Anderson impurity model. In order to obtain
a more quantitative insight into the influence of the discretized density of
states, represented by the host clusters highest-occupied–lowest-unoccupied
molecular orbital (HOMO-LUMO) gap, the problem is modeled in a simple
tight binding approximation indicating strong stabilization of the impurity
spin magnetic moment in presence of an energy gap. The dominance of the
HOMO-LUMO gap in stabilizing the impurities spin magnetic moment will
be illustrated by exchanging the host material from gold to copper.
6.1 Testing the Anderson Impurity Model: A
study of a Chromium Impurity in Gold-
Clusters
In section 5.4 of the previous chapter the electronic structure of CrAu+
nwas al-
ready discussed. It turned out that the chromium impurity in CrAu+
nn= 2,6
features an undisturbed atomic-like 3d5electronic configuration, while partici-
pation of the 3d-electrons in bonding can be deduced from the spectral shapes
of the x-ray absorption for the other cluster sizes.
63
64 Chapter 6. The Anderson Impurity Model in Finite Systems
600595590585580575570565560
CrAu1+
CrAu2
+
CrAu3
+
CrAu4
+
CrAu5
+
CrAu6
+
photonxenergyxinxeV
normalizedxxmcdxsignalxinxarb.xunits
Cr+xHFxcalculation
Figure 6.1: Normalized XMCD spectra of CrAu+
n,n= 1 −6. All spectra were taken
with an energy resolution of 625 meV. Additionally, a calculated XMCD spectrum of Cr+
is shown (dotted line). Structures of CrAu+
ndepicted are taken from [12].
6.1. Testing the Anderson Impurity Model:
A study of a Chromium Impurity in Gold-Clusters 65
cluster average nearest neighbor number of nearest neighbors
Cr-Au bond distance in ˚
A
CrAu+
12.60 ˚
A 1
CrAu+
22.72 ˚
A 2
CrAu+
32.70 ˚
A 3
CrAu+
42.71 ˚
A 4
CrAu+
52.71 ˚
A 5
CrAu+
62.88 ˚
A 5
Table 6.1: Average nearest neighbor chromium-gold bond distance and number of nearest
neighbors. Large distances or small number of nearest neighbors can be found for CrAu+
2,6.
Solely from these observations, a high spin magnetic moment can be expected
in CrAu+
2,6whereas a reduction of the spin magnetic moment should be found
in the other cases, since participation of the 3delectrons in bonding should
reduce the spin polarization.
However, the magnetic properties can be probed directly by performing x-ray
magnetic circular dichroism spectroscopy. The XMCD spectra of CrAu+
nin
a size range of n= 1 −6 are shown in figure 6.1. Again, there is a strong
resemblance of the XMCD spectra of the calculated atomic Cr+on the one
hand and CrAu+
2and to a lesser extent for CrAu+
6on the other hand. Hence,
an atomic spin moment of about 5 µBfor CrAu+
2,6can directly be deduced
from this comparison without using XMCD sum rules [36–38]. The applica-
tion of XMCD sum rules that allow to determine spin magnetic moments is
not possible, since 2p-spin-orbit and multiplet splitting have the same order of
magnitude in the case of the early 3dtransition metal chromium, cf. section
2.1.3. The aforementioned atomic spin magnetic moment in CrAu+
2,6is also
supported by DFT studies: In agreement with previous theoretical results [12]
local chromium spin magnetic moments of 4.85 µBfor CrAu+
2and 4.75 µBfor
CrAu+
6were found, as can be seen from panel (a) of figure 6.2.
The situation is different for the experimental spectra of CrAu+
1,3−5, where
no agreement with atomic Hartree-Fock calculations can be found. This sug-
gests that the chromium 3delectrons participate in bonding, which leads to
a reduction, but not to a quenching of the local spin moment. Non-vanishing
orbital magnetic moments in CrAu+
1,3−5, which are indicated by the negative
sign of the XMCD spectra at the L2edges, further underline a deviation from
the local 3d5configuration of chromium in CrAu+
1,3−5.
In a next step the validity of the Anderson impurity model [1] for CrAu+
n
clusters will be tested. Therefore, a brief recapitulation of the size-dependence
of the chromium-gold bond mechanism will be given here, because the inter-
66 Chapter 6. The Anderson Impurity Model in Finite Systems
μSrinrμB
EintrinreVfU764Jz8rΓr
HOMO2LUMO
gaprinreV
faz
fbz
fcz
fdz
localrchromiumrspin
momentrf3dz
Cr2Aurinteractionr
energy
Anderson
criterion
HOMO2LUMO
gaprofrAu
host
hostrclusterrsize
populationranalysis
AIM
26
25
24
23
21
98765431
597
495
497
395
397
.1
.7
8
6
4
197
.95
.97
795
Figure 6.2: (a) Cal-
culated local chromium
spin moments µSof
CrAu+
nfrom popula-
tion analysis (black
triangles) and spin
moments deduced from
the Anderson criterion
(red bullets) presented
in panel (c), (b) Cr-Au
interaction energy Eint,
(c) Anderson criterion
(U0+ 4J)/Γ, and (d)
energy gap ∆EHL of the
gold host.
action strength of the impurity and host-material states is a crucial ingredient
in the Anderson impurity model, which changes with cluster size.
The gold subunits of CrAu+
2,6, depicted in figures 5.11 and 6.1, are struc-
turally close to pure Au2and Au6[119], i.e., they remain nearly undistorted
when adding the chromium impurity. In CrAu+
3−5, in contrast, the number
of Cr-Au bonds is maximized, and the gold host is strained and deformed
as compared to its isolated, relaxed counterpart. The enhanced stability of
Au2,6stems from shell closure for two and six delocalized 6selectrons [119],
which, in spite of strong spd hybridization [120], form a free electron gas con-
fined in a two dimensional potential well [26]. Therefore, Au2,6are known to
feature large second differences in binding energy and HOMO-LUMO gaps
∆EHL of about 2 eV [119]. Consequently, the chromium cation can be ex-
pected to interact more weakly in CrAu+
2,6in comparison to the other cluster
sizes. That this is indeed the case can be inferred from the chromium-gold
interaction energy Eint, depicted in figure 6.2 (b). Here, Eint is calculated as
6.1. Testing the Anderson Impurity Model:
A study of a Chromium Impurity in Gold-Clusters 67
Eint =ECrAu+
n−E(Cr+)−E(Aun). The positive charge is mainly lo-
cated at the chromium site, which can be deduced from a population analysis.
This further supports the notion of electronic shell closure in the gold host
of CrAu+
2,6. To obtain Eint and to extract the contribution of the chromium
interaction with the gold cluster from the total energy, E(Aun) is calculated
in the same geometric configuration of Aunas in CrAu+
n. As expected, the
weakest impurity-host interactions of 2.5 eV and 3.3 eV are found for CrAu+
2
and CrAu+
6, respectively.
In contrast to electronic shell closure, the coordination of the impurity atom
only has a minor effect, cf. table 6.1. It does lead to a slight decrease of
the spin magnetic moment with increasing coordination number for n= 2 to
n= 5 (two-dimensional), and n= 6 to n= 9 (three-dimensional) clusters.
Nonetheless, chromium in CrAu+
6exhibits a larger spin magnetic moment than
in CrAu+
3, although it is higher coordinated in the former, as can be seen in
figures 6.1 and 6.2 (a) and from table 6.1.
We now turn to analyzing the results in terms of the Anderson impurity model
[1]. In its original formulation, cf. section 2.5, it describes the interaction of
a single orbital magnetic impurity embedded in a free electron gas. Within
this model, the size of the magnetic moment of the impurity atom sensitively
depends on the interplay of the on-site Coulomb repulsion (direct Coulomb
interaction of two electrons in the same localized orbital) and the width 2Γ of
the localized state. The width is determined by the amount of hybridization
with the free electron gas states of the host and by the total density of states
at the Fermi level [1]. In the absence of interaction with the free electron gas,
the impurity states Eand E+U0are separated by the bare Coulomb inter-
action U0that preserves the local magnetic moment if U0pushes the state
E+U0above the Fermi level. In the case of interaction, virtual states are
formed at energies E+U0·n−and E+U0·n+, where n±are the occupation
numbers of the impurity atom’s majority and minority state. The separation
of the virtual levels is reduced to an effective value Ueff =U0(n+−n−) by
hybridization of the localized impurity states with free electron gas states,
which also leads to an increasing width 2Γ of the virtual states [1]. Here,
(n+−n−) is the magnetic moment of the localized state, which depends on
the ratio of U0and Γ.
A generalization of the Anderson impurity model for a five-fold degenerate
impurity state, as for 3d-elements, predicts a transition from a magnetic to a
non-magnetic impurity state for (U0+4J)/Γ≤π, where Jis the intra-atomic
exchange [128].
Therefore, quantitative insight into the impurity-host interaction can be
acquired from analyzing the Anderson criterion (U0+ 4J)/Γ. To this end,
68 Chapter 6. The Anderson Impurity Model in Finite Systems
size 1 2 3 4 5 6 7 8 9
U0in eV 1.68 4.00 3.15 2.91 2.92 3.80 3.26 3.43 3.46
Γ in eV 0.56 0.66 0.84 0.82 0.94 0.85 0.99 1.18 1.45
Table 6.2: Calculated on-site Coulomb repulsion U0and width 2Γ obtained as the weighted
standard deviation of d-projected density of states of the chromium impurity.
Ueff was calculated in a self consistent scheme for the multi-orbital systems
CrAu+
n, which then takes the form Ueff = (U0−J) (n+−n−) [72, 85]. The
effective on-site Coulomb repulsion Ueff is derived from the slope of the lin-
ear response of the occupation number of the 3dimpurity states to a rigid
potential shift introduced at the impurity site, for details see section 2.4 or
[72]. Since Ueff is obtained from ab initio calculations, the Coulomb inter-
action among the impurity 3delectrons, and all screening and hybridization
effects, are intrinsically accounted for. Hence, U0can be determined from
U0=Ueff/(n+−n−) + Jand is listed in table 6.2. The atomic exchange
interaction of chromium JCr = 1 eV [127] is weakly screened, therefore Jis
expected to only slightly depend on the cluster size and to be of the order of
0.5-1 eV [129]. The half width Γ, cf. table 6.2, of the localized impurity states
is obtained from the weighted standard deviation of the d-projected density
of states of the impurity, which is shown in figure 6.3. The resulting values
for the Anderson criterion (U0+ 4J)/Γ, using a mean value for the exchange
J±∆J= (0.75 ±0.25) eV, are shown in figure 6.2 (c).
As can be seen the Anderson criterion for a magnetic impurity state is well
satisfied throughout the whole size range, which is in perfect agreement with
the non-vanishing XMCD signal for CrAu+
npresented in figure 6.1. In par-
ticular, a large value of the (U0+ 4J)/Γ is found for CrAu+
2,6, where, as was
already shown, the interaction of the impurity with the host is reduced in com-
parison to the other cluster sizes. Although U0+ 4Jexhibits a pronounced
size dependence, the size-dependence of the Anderson criterion is dominated
by the width of the virtual bound state, which can be related to the host
clusters HOMO-LUMO gap ∆EHL,i.e. the gap of Aunin the geometry of
CrAu+
n,cf. figure 6.2. Since correlation effects are expected to be weak in the
free-electron gas states of the gold host, ∆EHL should be a good representa-
tion of the energy gap. In a very simple picture, the impurity state resides
in the center of the host cluster’s energy gap, and amount of hybridization,
and therefore the width 2Γ, depends on the size of ∆EHL. For example in
CrAu+
2and CrAu+
5, the experimentally investigated clusters with the largest
and smallest magnetic moment, Γ increases from 0.66 eV to 0.94 eV whereas
the energy gap ∆EHL is reduced from 1.95 eV to 0.48 eV.
From the values (U0+ 4J)/Γ it is possible to calculate the impurity’s spin
6.1. Testing the Anderson Impurity Model:
A study of a Chromium Impurity in Gold-Clusters 69
magnetic moment within the Anderson impurity model by graphically solving
n±=1
πarctan U0+ 4J
Γ(n∓−0.5)+ 0.5 (6.1)
[1, 128]. As can be seen in panel (a) of figure 6.2, the size dependence of the
spin magnetic moment obtained from population analysis, which is in accor-
dance with the experimental data, shows a good qualitative and satisfying
quantitative agreement with the spin magnetic moments deduced from the
Anderson criterion. The deviation can mainly be attributed to the rough es-
timations of U0,Jand Γ. Still, the magnetic moments observed in CrAu+
n
clusters are essentially in agreement with the Anderson impurity model. The
non-linear relation in equation 6.1 also explains the small changes in the
spin magnetic moments below 10 % for values of Anderson criterion π,
although interaction energy Eint and the energy gap ∆EHL, strongly influenc-
ing (U0+ 4J)/Γ, vary by more than a factor of two.
However, the good agreement of the experimental data with the Anderson
impurity model is somehow surprising, since the discrete nature of the host
density of states was expected to have a substantial influence on the spin mag-
netic moments. Presumably the good agreement is a result of the large values
(U0+ 4J)/Γ resulting in almost full spin polarization. Therefore the presence
of a gap only marginally influences the impurity spin magnetic moment. In
contrast, the size of the impurity spin may benefit strongly from a gap in the
host density of states in systems featuring small values (U0+ 4J)/Γ≈π. To
get a more profound insight into the influence of a gap in the host density of
states, the problem will be treated theoretically in the next section.
In summary, the experimentally observed size dependence of the XMCD spec-
tra of mass selected chromium doped gold clusters are essentially well de-
scribed within the Anderson impurity model. The size dependent variation
of the spin magnetic moment can be linked to the hybridization strength of
impurity and host density of states governed by the HOMO-LUMO gap of
the host gold cluster. Electronic shell closure, resulting in large energy gaps
∆EHL in the free-electron gas, reduces the interaction with the impurity and
therefore results in spin magnetic moments of 5 µBfor CrAu+
2,6. This effect is
a result of quantum confinement and unique to finite systems.
70 Chapter 6. The Anderson Impurity Model in Finite Systems
-50
0
50
100
-6 -5 -4 -3 -2 -1 0 1
-50
0
50
100
-50
0
50
100
-50
0
50
100
-50
0
50
100
-50
0
50
100
-100
-50
0
50
100
-50
0
50
100
50
0
-50
CrAu1
+
CrAu2
+
CrAu3
+
CrAu4
+
CrAu5
+
CrAu6
+
CrAu7
+
CrAu8
+
CrAu9
+
spinVresolvedVdensityVofVstatesVinV1/eV
energyVE-EFermiVinVeV
d-projectedVCrVPDOS
totalVDOS
Figure 6.3: Total density of states (dotted black lines) and density of states projected onto
chromium d-orbitals (red solid lines). Resemblance of XAS and XMCD spectra of CrAu+
2,6
originates in similar electronic structure obvious from the d-projected DOS. Positive and
negative values of the density of states represent spin-up and spin-down states, respectively.
6.2. Modification of the Anderson Impurity Model for Finite
Systems 71
1.0
0.8
0.6
0.4
0.2
0.0
1.00.80.60.40.20.0
analyticalrAIM
tightrbinding
Hamiltonian
n+
n-
1050-5-10
densityrofrstatesrinrarb.runits
energyrinreV
Figure 6.4: Comparison of the
self-consistent solutions for the oc-
cupation numbers n±of spin-up
and -down states, obtained ana-
lytically from the standard Ander-
son impurity model and the tight
binding Hamiltonian, equation 6.2.
Both models yield almost identi-
cal results. For details see the
text. Inset: Density of states re-
sulting from diagonalization of the
tight binding Hamiltonian. The
Lorentzian fit nicely matches the
density of states.
6.2 Modification of the Anderson Impurity
Model for Finite Systems
The Anderson impurity model in finite systems, also known as the Anderson
box, has already been discussed intensively in the literature [130–139]. Still,
none of these theoretical studies addressed the stabilization of the impurity
spin magnetic moment by the discretized host density of states.
Here, this problem will be tackled by modeling the system using a tight binding
Hamiltonian [140]. To this end, the following Hamiltonian will be diagonal-
ized:
HTB =
Eda· · · a
a Ek,10 0
.
.
. 0 ...0
a0 0 Ek,n
(6.2)
Here the localized impurity state Edis coupled to a finite number of contin-
uum states Ek,n with a coupling strength a. The separate diagonalization for
majority and minority spin states E±
dyields the spin resolved eigenvalues ±
i
and eigenvectors φ±
i. The robustness of the model is tested by comparing the
self-consistent solution of the occupation of the two spin states
n±=1
πarctan U0·(n∓−0.5)
Γ!+ 0.5 (6.3)
72 Chapter 6. The Anderson Impurity Model in Finite Systems
within the standard Anderson impurity model [1] with the solution obtained
from the tight binding Hamiltonian 6.2 by approximating the continuous band
of the host by dense but discrete levels. The occupation number of the ma-
jority and minority spin states within the tight binding approximation was
obtained by summation over the spin resolved density of states ρ±(E) as
follows1:
ρ±(E) = X
iφ±
i
2δE−±
i(6.4)
n±=ZEF
−∞ ρ±(E)dE (6.5)
The results obtained by using a ratio of U0/Γ = 4.1 in both cases, are almost
identical as can be seen from figure 6.4. Additionally, the density of states
obtained from the tight binding Hamiltonian has a Lorentzian shape as can
be seen from the inset in figure 6.4. This is also what the standard Anderson
model predicts. Hence, in the continuous band limit the model Hamiltonian
6.2 describes the system quite well and is equivalent to the analytical solution
of the Anderson impurity model.
To study the influence of an energy gap in the host band, parameters U0and
Γ were chosen such that the spin polarization of the impurity states vanish
when interacting with the continuum states, i.e. U0/Γ< π in the Ander-
son impurity model. The relative energetic positions of impurity states and
free electron gas states of the separated systems are sketched in the inset of
panel (a) of figure 6.5. The self-consistent solution of spin-up and spin-down
states shown in panel (c) of figure 6.5 indicates the vanishing impurity spin
polarization, since the curves only cross at n+=n−= 0.5. The absent spin
polarization can also be seen from the degeneracy of the Lorentzian shaped
spin-up and spin-down states formed as expected within the Anderson impu-
rity model, cf. section 2.5.
However by introducing a small gap of 0.1 eV to the host density of states
at the Fermi energy of the isolated host, cf. inset in panel (b) of figure 6.5,
the majority as well as the minority spin density of states of the combined
host-impurity system get poles at the Fermi energy, cf. panel (b) of figure
6.5. This finally leads to a non-vanishing impurity spin polarization as shown
in panel (d), due to transfer of weight induced by the poles from minority
spin density of states above the Fermi energy to the majority spin density
of states below the Fermi energy. In addition, the curves for n±which form
horizontal and vertical regions which suppress the quenching of the impurity
spin moment. These curves no longer form inverse tangent functions as in the
1δ(E) is the delta function.
6.2. Modification of the Anderson Impurity Model for Finite
Systems 73
energy)in)eV energy)in)eV
density)of)states)in)arb.)units
density)of)states)in)arb.)units
without)gap 0.1)eV)gap)at
EFermi=0
-2 -1 01 2
-2 -1 01 2
energy)in)eV energy)in)eV
host
impurity
host impurity
n+
n-
n+
n-
spin)majority
spin)minority
spin)majority
spin)minority
(a) (b)
(c) (d)
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
1.0
0.8
0.6
0.4
0.2
0.0
1.00.80.60.40.20.0
1.0
0.8
0.6
0.4
0.2
0.0
1.00.80.60.40.20.0
-3 -2 -1 0 1 2 3
-3.0 -2.0 -1.0 0.0 1.0 2.0 3.0
Figure 6.5: Upper panels: DOS obtained using the tight binding model Hamiltonian,
equation 6.2, for an impurity interacting with a dense discrete host DOS without a gap
(a) and with a gap (b). Parameters aand U0were kept constant. Lower panels (c) and
(d) show the resulting self-consistent solution for the occupation numbers for spin-up and
-down states. The impurity magnetization is restored when introducing a small gap in the
host density of states, panel (d). Parameters used here: a= 0.04 eV, U0= 1 eV, 2000 states
within a bandwidth of 20 eV.
standard Anderson impurity model.
Thus, the presence of a gap in the host density of states stabilizes the impu-
rity’s magnetic moment.
To obtain a more quantitative insight into the influence of the energy gap,
the spin polarization as a function of the hosts energy gap is plotted in figure
6.6 keeping the coupling parameter aas well as the on-site Coulomb repulsion
U0constant. As can be seen, the spin polarization adopts finite values as the
74 Chapter 6. The Anderson Impurity Model in Finite Systems
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
2.01.81.61.41.21.00.80.60.40.20.0
energy gap in eV
spin polarization
Figure 6.6: Spin polarization
of the impurity as a function of
the host’s energy gap, keeping
the coupling parameter a= 0.08
and the on-site Coulomb repul-
sion U0= 2 eV constant and us-
ing a host density of states of
50 states/eV.
hosts energy gap opens and increases rapidly. Interestingly, the spin polariza-
tion can be approximated to scale almost linearly with the host energy gap
for gaps larger than ≈0.5 eV, cf. shaded area in figure 6.6.
In order to get a deeper insight into the influence of the hosts energy gap the
spin polarization within the standard and modified Anderson impurity model
will be studied as a function of the coupling strength a. This is shown in
panel (a) of figure 6.7. As can be seen from the figure, only in case of very
small coupling a < 0.03 eV the spin polarization is almost identical with and
without the gap, whereas for larger athere is a profound deviation. Most
strikingly for coupling parameters a > 0.04 eV, where the spin polarization is
lost in absence of an energy gap in the host materials density of states, but
survives in the other case. This seems to be a quite robust effect, since even
for a coupling strength athree times as large as necessary to quench the spin
magnetic moment in the continuous band case, the spin is still finite in the
presence of a gap.
The vanishing of the spin polarization can clearly be attributed to the strongly
increasing width Γ of the virtual bound state as function of increasing coupling
strength a, as depicted in figure 6.7 (b). The width was calculated as:
Γ = U0(n−−0.5)
tan (π(n−−0.5)),
following equation 6.3 and assuming charge neutrality n++n−= 1. In pres-
ence of a gap the Anderson description with a Lorentzian shaped virtual bound
state breaks down as shown above, therefore a width Γ cannot be given in
this case. Furthermore, the Anderson criterion marking the magnetic to non-
magnetic transition is not valid anymore. As can be seen from figure 6.7 (c)
the Anderson criterion calculated for the continuous band limit drops below
πfor values a > 0.04 eV. Still, at these large coupling strengths which can
6.2. Modification of the Anderson Impurity Model for Finite
Systems 75
2.0
1.5
1.0
0.5
0.0
0.120.100.080.060.040.02
1.0
0.8
0.6
0.4
0.2
0.0
0.120.100.080.060.040.02
coupling(parameter(a(in(eV coupling(parameter(a(in(eVcoupling(parameter(a(in(eV
width(Γ(in(eV
Anderson(criterion
spin(polarization
AIM(without(gap
AIM(with(gap AIM(without(gap
AIM(without(gap
(a) (b) (c)
20
15
10
5
0
0.120.100.080.060.040.02
Figure 6.7: Comparison of the Anderson model and the modified version incorporating
an energy gap in the host density of states (U0= 1 eV and 0.5 eV energy gap). Panel
(a): Spin polarization as a function of coupling strength a. Only for very small coupling
parameters similar spin polarizations can be found. The spin magnetic moment is quenched
for a > 0.04 eV in the continuous band case, whereas it survives in presence of a gap. Panel
(b): Width of the virtual bound state. In absence of a gap the width increases strongly with
aresulting in vanishing spin polarization for a > 0.04 eV. Panel (c): Anderson criterion
drops below πfor coupling strengths a > 0.04 eV marking the magnetic-to-nonmagnetic
transition.
be related to values U0/Γ< π in the continuous limit, the spin polarization
survives in presence of a gap.
It can be expected that the influence of the gap is more pronounced for sys-
tems which are very close to the transition from magnetic to non-magnetic
state in the continuous limit. Systems already exhibiting a large spin polar-
ization in the bulk limit should be less affected by the presence of a gap in
the host density of states. Still, the size of the spin magnetic moment of the
impurity scales with the energy gap in the host density of states for a given
coupling strength a,cf. figure 6.6. This is in good agreement with the ex-
perimental findings presented in the previous section 6.1, where a correlation
of the magnitude of the chromium impurity spin magnetic moment and the
HOMO-LUMO gap of the gold host was revealed.
Summing up, in the present section the influence of a gap on the impurity’s
spin magnetic moment was studied. It was shown that the presence of a gap
stabilizes the impurity magnetic moment. This result is expected to hold even
when the actual discrete nature of the density of states is taken into account.
76 Chapter 6. The Anderson Impurity Model in Finite Systems
6.3 Influence of the Host Material: Chromium
Doped Copper Clusters
In this section experimental and theoretical results on chromium doped cop-
per clusters will be discussed. Investigating this system will not enlarge our
”stamp collection“, but helps to study the influence of varying hybridization
on the magnetic moment. The use of copper as the host material is supposed
to increase the hybridization of the chromium states and the host clusters free
electron gas states. Enhanced hybridization is to be expected since the nearest
neighbor bond distance in bulk copper is 2.56 ˚
A [141] which is a contraction
of about 10 % compared to bulk gold exhibiting a nearest neighbor bond dis-
tance of 2.88 ˚
A [141]. Therefore a reduction of the spin magnetic moment can
be anticipated.
Before studying the electronic and magnetic properties of CrCu+
nclusters, the
next section is devoted to the ground state geometries of chromium doped
copper clusters.
6.3.1 Geometries of Chromium Doped Copper Clusters
Since no structures are reported in the literature so far ground state structures
for CrCu+
nn= 1 −6 clusters have to be found for further detailed analysis.
All calculations shown in this section were carried out in a DFT framework
employing the PBE approximation [107] to the exchange-correlation func-
tional as implemented in the program package turbomole 6.2 [142, 143].
The triple-ζbasis set with two polarization functions (def2-TZVPP) was used
[106], which is known to yield quantitative robust results for chromium and
copper [106]. Additionally, the resolution of the identity approximation [144]
was used to further reduce the computational cost. The spin ground state was
found by additionally introducing a thermal smearing. Here the occupation
is calculated using a Fermi distribution function in every self-consistent cycle
starting at a temperature of 300 K and cooling the system to 10 K in order to
yield integer occupation numbers.
Different approaches were applied to generate candidate structures. In the
small size regime up to a cluster size of n= 3 all possible arrangements of the
atoms were used as candidate structures. In case of larger clusters three differ-
ent methods were used: First, cluster geometries were derived from cationic
copper clusters [122, 145, 146] by substitution of one copper atom by the
dopant atom. Second, the gold host material in small doped gold clusters
taken from literature [12] was replaced by copper. Third, a molecular dynam-
ics simulation was performed, where the cluster freely moves at a constant
6.3. Influence of the Host Material: Chromium Doped Copper
Clusters 77
temperature of 1200 K for about 10 ps. The computational effort in the molec-
ular dynamics simulation was further reduced by using the smaller basis set
def2-SVP [106]. Every 200 fs a structure snap-shot was taken and a geometry
optimization was performed.
In addition, the dopant atom was placed at all non-equivalent positions in the
candidate cluster which was thereupon re-optimized.
The ground state structures and the energetically next two isomers are shown
in figure 6.8 together with their total spin moments. The coordinates can be
found in the appendix section 8.1. As can be seen from the figure there is an
early transition from two to three dimensional structures at n= 3 →n= 4
different from chromium doped gold clusters, where the transition takes place
at cluster sizes n= 6−7. Small gold clusters favor two dimensional structures
because of an energy gain due to the reduction of the kinetic energy of rela-
tivistic electrons [147]. This is less important in copper and therefore three
dimensional compact structures are preferred. It is obvious from figure 6.8
that the dopant atom tends to be coordinated as highly as possible. Still, the
total spin magnetic moments are high: 4 µBand 5 µBfor clusters with odd
and even numbered host atoms, respectively.
The ground state structures presented here will be further analyzed in the
next section.
78 Chapter 6. The Anderson Impurity Model in Finite Systems
4 μB
5 μB
4 μB6 μB
+124 meV
5 μB5 μB
+0.1 meV +230 meV 5 μB
4 μB6 μB
+291 meV +340 meV 4 μB
5 μB5 μB
+228 meV 5 μB
+258 meV
5 μB5 μB
+1727 meV+1192 meV
CrCu1
+
CrCu2
+
CrCu3
+
CrCu4
+
CrCu5
+
CrCu6
+
Figure 6.8: Left column: Ground state geometries of CrCu+
nclusters in a size range
n= 1 −6. Chromium is shown in blue whereas copper is colored orange. Additionally
shown are the two energetically lowest lying isomers. The energy is given relative to the
ground state structure. The total spin moment of the clusters is also given.
6.3. Influence of the Host Material: Chromium Doped Copper
Clusters 79
600595590585580575570565560
XAS in arb. units
photon energy in eV
CrCu1
+
CrCu2
+
CrCu4
+
CrCu5
+
CrCu6
+
CrCu3
+
Figure 6.9: Linear x-ray absorp-
tion spectra at the L2,3edges of
chromium of CrCu+
nin a size range
n= 1 −6. The energy resolu-
tion is quite low (625-1500 meV,
see text for details). Additionally
shown is a Hartree-Fock calculation
of atomic Cr+XAS (dotted line).
6.3.2 Electronic and Magnetic Properties of Chromium
Doped Copper Clusters
Linear x-ray absorption spectra of chromium doped copper clusters in a size
range of n= 1 −6 are presented in figure 6.9. The energy resolution is rather
low compared to the x-ray absorption spectra presented in the previous sec-
tions. The energy resolution was set to 625 meV for CrCu+
2, 1250 meV for
CrCu+
1,3−5and 1500 meV for CrCu+
6. Unfortunately, it was necessary to use
such a low energy resolution, since the beam line was in bad condition at
that particular beam time, delivering about three orders of magnitude less
photons than in principle possible. Therefore the signal to noise ratio in the
x-ray absorption of CrCu+
nis not improved compared to CrAu+
n, although the
background absorption by the host material is strongly suppressed in CrCu+
n
by the relative energetic position of the chromium L-edges and the host ma-
terial absorption edges in contrast to CrAu+
n.
Multiplet structures could only be resolved in case of CrCu+
2. A comparison
to the calculated x-ray absorption spectrum of atomic Cr+,cf. dotted line
in figure 6.9, reveals a local [Ar]3d5configuration of the chromium in CrCu+
2,
80 Chapter 6. The Anderson Impurity Model in Finite Systems
600595590585580575570565560
CrCu1
+
CrCu2
+
CrCu4
+
CrCu5
+
CrCu6
+
CrCu3
+
normalized XMCD signal in arb. units
photon energy in eV
Figure 6.10: Normalized
XMCD spectra at the
L2,3-edges of chromium of
CrCu+
nin a size range of
n= 1 −6. A strong size
dependence of the XMCD
asymmetry is present.
Additionally shown is a
Hartree-Fock calculation of
atomic Cr+XMCD (dotted
line).
similar to the case of CrAu+
2. A comparison to atomic Hartree-Fock calcula-
tions is not meaningful for the other cluster sizes because of the low energy
resolution applied in recording the x-ray absorption spectra. Still, narrow line
widths can be found in the x-ray absorption of CrCu+
1,2,4, whereas a strong
broadening is present in the x-ray absorption spectra of of CrCu+
3,5,6. This
in turn suggests a stronger hybridization of the chromium 3d-states with the
host clusters electronic states in CrCu+
3,5,6than in CrCu+
1,2,4. Hence, reduced
spin magnetic moments should be found for CrCu+
3,5,6. Because larger hy-
bridization leads to a smaller separation of the virtual states formed in the
system and to a transfer of electrons from majority to minority spin states,
cf. section 6.1, this again results in a reduced spin magnetic moment.
In order to directly study the magnetic properties of chromium doped copper
clusters XMCD spectroscopy was performed. The normalized XMCD spectra
taken at a constant temperature of (20 ±5) K are shown in figure 6.10. As
6.3. Influence of the Host Material: Chromium Doped Copper
Clusters 81
65A79
host)cluster)size
intensity)in)
arb6)units
XMCD)L7)peak)intensity
XMCD)L7)area
HU8gAJO2Γ)
in)eV ms)in)μB
ImaxHxmcdO)
in)arb6)units
host)cluster)size
HOMO1LUMO
gap)in)eV
HOMO1LUMO
gap)of
copper
host
local)chromium)spin
moment)H7dO
8656
8659
8688
65A79
568
A69
A68
A67
A66
5A
59
58
8
6
566
569
868
86A
Anderson)criterion
HaO
HbO
HcO
HdO
Figure 6.11: Panel (a):
Maximum intensity at the
L3-edge of the XMCD
asymmetry as a function
of cluster size. In the
inset a comparison of the
maximum intensity and the
area A(see text for details)
is given, showing the same
size dependent trend. Panel
(b): Local chromium spin
moments obtained from
DFT+Ucalculations. Panel
(c): (U0+ 4J)/Γ is larger
than πfor all cluster sizes
and fulfill therefore the
Anderson criterion for a
magnetic impurity state
[1]. Panel (d): The host
clusters HOMO-LUMO gap
exhibiting the same size
dependence as the impurity
spin magnetic moment.
in linear x-ray absorption a resemblance of the calculated XMCD signal of
atomic Cr+and the XMCD spectrum of CrCu+
2exists, indicating a local spin
magnetic moment of about 5 µB. Furthermore, a strong size dependence of
the magnitude of the XMCD asymmetry can be found, cf. figure 6.10. The
spin magnetic moment mSis proportional to mS∝A−2·B2,Abeing the
area at the L3and Bthe area at the L2edge of the asymmetry signal. Since
the area Bis negligibly small in case of chromium doped copper clusters, the
spin magnetic moment is mainly given by the area A. Here we can even use
the maximum intensity at the L3-edge of the XMCD asymmetry, since the
area of the XMCD asymmetry at the L3edge is very narrow and because of
the low energy resolution used the peak intensity mainly covers the whole area
A. A comparison of the area Aand the peak intensity at the L3-edge of the
XMCD signal is given in the inset of figure 6.11 exhibiting almost the same
size dependent trend. As already expected from the line shapes in linear x-ray
absorption, a large XMCD asymmetry signal, which is, in this particular case,
proportional to the local chromium spin magnetic moment, can be found in
2This is basically the XMCD spin sum rule, cf. section 2.1.3.
82 Chapter 6. The Anderson Impurity Model in Finite Systems
cluster average nearest neighbor number of nearest neighbors
Cr-Cu bond distance in ˚
A
CrCu+
12.74 ˚
A 1
CrCu+
22.58 ˚
A 2
CrCu+
32.57 ˚
A 3
CrCu+
42.64 ˚
A 4
CrCu+
52.65 ˚
A 5
CrCu+
62.63 ˚
A 6
Table 6.3: Average nearest neighbor chromium-copper bond distance and number of near-
est neighbors in CrCu+
n.
case of CrCu+
2,4.
A reasonable agreement in the size dependence of the calculated local chromium
spin moments (within a DFT framework), obtained from projecting the total
density of states onto chromium d-states, shown in panel (b) of figure 6.11,
with the measured XMCD asymmetry, displayed in panel (a) of the same fig-
ure, can be found. In order to study the influence of the host material, the
size dependence of the spin magnetic moment of CrCu+
nand CrAu+
nwill be
compared. By analyzing figures 6.2 and 6.11 it is immediately clear that the
size dependent trend of the spin magnetic moment of CrCu+
nis very different
from CrAu+
n. The only agreement can be found for clusters exhibiting the
same overall geometry, namely CrCu+
2and CrCu+
3.
The spin magnetic moments as well as the Anderson criterion are almost
identical for CrCu+
3(mS= 4.67 µB, (U0+ 4J)/Γ) = 10.7) and CrAu+
3(mS=
4.59 µB, (U0+ 4J)/Γ) = 8.2), cf. figures 6.2 and 6.11. The small deviation of
the spin magnetic moments in CrCu+
3compared to CrAu+
3is, at first glance,
remarkable, since the average bond distance in CrCu+
3is contracted by about
5 % compared to CrAu+
3,cf. tables 6.1 and 6.3. The contraction of the bond
distance entails a larger overlap of the impurity and host valence electron
wave-functions and should scale exponentially. Therefore a larger hybridiza-
tion would be expected in case of the chromium doped copper clusters. The
same contraction of the Cr-Cu bond distances can be found in CrCu+
2, again
not altering the electronic and magnetic properties of the chromium atom in
the CrCu+
2cluster compared to CrAu+
2. This of course becomes manifest in
their density of states which are compared in figure 6.12. As can be seen
the chromium 3d-states are equally undistorted in both systems. The main
difference is the relative energetic position of the chromium 3d-states and the
sd-states of the host, which are basically changing positions.
As can be seen from figure 6.11, the impurity spin magnetic moment of CrCu+
n
6.3. Influence of the Host Material: Chromium Doped Copper
Clusters 83
80
40
0
-40
-5 -4 -3 -2 -1 0 1 2
-80
-40
0
40
CrCu2+
CrAu2+
d-projectedyCryPDOS
totalyDOS
spinyresolvedydensityyofystatesyiny1/eV
energyyE-EFermiyinyeV
Figure 6.12: Comparison
of the density of states of
CrCu+
2and CrAu+
2. The
chromium 3d-states exhibit
a similar undistorted struc-
ture. The energetic posi-
tion of chromium 3d-states
and host sd-states are in-
terexchanged from CrCu+
2
to CrAu+
2.
scales with the HOMO-LUMO gap in the host density of states. This was al-
ready shown in section 6.1 for CrAu+
nclusters and in general for systems
exhibiting a gap in section 6.2.
Hence, increasing the hybridization induced by geometrical changes, for exam-
ple bond distance contraction, is of minor importance for the size dependence
of the impurity’s spin magnetic moment, at least in the parameter regime of
(U0+ 4J)/Γ we are investigating here. The size dependence is rather dom-
inated by electronic shell closure in the host material, determining the size
of the HOMO-LUMO gap. For that reason one finds almost identical spin
magnetic moments in CrCu+
2and CrAu+
2. The two 4selectrons of the copper
host form a two dimensional electron gas [148, 149]. This results in a large
HOMO-LUMO gap, comparable to the case of CrAu+
2, yielding a reduced
width of the virtual bound state and therefore an almost identical value of
(U0+ 4J)/Γ, although the Cr-Cu bond distance is contracted by about 5 %
compared to the Cr-Au bond distances.
This observation is in agreement with the line widths found in x-ray absorp-
tion and XMCD spectra as discussed above. The reduced (enhanced) spectral
width in CrCu+
2,4(CrCu+
3,5,6) originate from the sizable (small) HOMO-LUMO
gaps in the host material density of states, cf. panel (d) of figure 6.11 (in agree-
ment with [150, 151]).
The magnetic moments in CrCu+
nclusters in the investigated size regime
clearly exceed 4 µB, which is considerably larger than magnetic moments of
3µB[152–154] predicted for chromium embedded in a bulk copper sample.
The reduction of the moment in bulk copper can have multiple reasons. The
most obvious is the reduction of spin due to stronger hybridization caused by
the absence of a gap in bulk copper. Additionally, the coordination number
84 Chapter 6. The Anderson Impurity Model in Finite Systems
of CrCu+
nclusters linearly increases with the cluster size and has a maximum
value of 6 in CrCu+
6. However, assuming a chromium impurity in fcc bulk cop-
per to favor a substitutional site, its coordination number would be as high as
12. This increase in coordination in comparison to the small clusters probably
results in a stronger hybridization of the chromium 3dstates with the copper
4sderived band. This would in turn lead to a reduced chromium impurity
spin polarization. Also the cationic nature of the clusters may enhance the
spin moment. The excess charge in the bulk material could conceivably lead
to a reduction of the spin polarization by placing the extra electron in the
3d-states of the impurity, which would reduce the spin by 1 µB.
6.4 Summary
In this chapter the applicability of the Anderson impurity model to finite
systems was studied experimentally and theoretically. The experimental data
taken for chromium doped gold clusters are essentially in agreement with the
standard Anderson impurity model, since the Anderson criterion is rather
large in these systems. Therefore CrAu+
nclusters are far above the magnetic
to non-magnetic transition threshold, so the discrete nature of the host density
of states has only minor consequences. Still, it could be shown that the size of
the HOMO-LUMO gap in the host influences the magnitude of the impurity
spin magnetic moment. This was further studied using a tight binding model
Hamiltonian. It could be shown that not only the description within the
Anderson impurity model breaks down, but also that in a certain parameter
regime the size of the gap in the host density of states scales almost linearly
with the impurity spin magnetic moment, which is again in agreement with the
experimental findings. Moreover, significant stabilization of the spin magnetic
moment by the presence of an energy gap in the host density of states for values
of the Anderson criterion close to the magnetic to non-magnetic transition
was demonstrated. By substituting the host material from gold to copper the
importance of shell closure effects determining the size of the HOMO-LUMO
gap for the host-impurity interaction could be revealed.
Chapter 7
Epilogue
7.1 Summary
In this thesis the interaction of a single magnetic impurity embedded in a
finite non-magnetic host, exhibiting a highly discretized density of states,
was studied experimentally as well as theoretically. Gold clusters doped with
early 3d-transition metals (scandium through chromium) served as model sys-
tems. By performing the experiment in the gas phase not only any support
or impurity-impurity interaction was eliminated which allows to study the in-
trinsic properties of the system but a discretized density of states of the host
material is introduced very natural by quantum confinement.
It was shown that the local electronic structure of the impurity depends on a
complex interplay of the nature of the dopant element as well as on cluster size
dependent quantum confinement in the host rather than on simple impurity
coordination. However, a strong tendency of all investigated systems to delo-
calize an even number of electrons, when allowed by the boundary conditions,
was found. This was inferred from x-ray absorption spectroscopy combined
with density functional theory calculations.
In case of chromium doped gold clusters also the magnetic properties were
probed by applying XMCD spectroscopy. The results were discussed within
the Anderson impurity model [1]. Introducing an impurity with sufficiently
localized valence orbitals into the non-magnetic host leads to the formation of
virtual bound states. Two parameters and their relative sizes are essential in
the Anderson impurity model for determining if the impurity retains its spin
degree of freedom. These are for one the on-site Coulomb repulsion U0, which
is the energy necessary to add an electron to a localized orbital and secondly
the width Γ of the virtual bound state caused by hybridization of the localized
impurity orbitals and the electronic states of the itinerant electrons of the host
85
86 Chapter 7. Epilogue
material. In case of a single impurity orbital, the system undergoes a mag-
netic to non-magnetic transition if U0/Γ = π. Since the Anderson impurity
model was developed for bulk materials, severe deviations could be expected
in finite systems.
Surprisingly, it was found that the Anderson impurity model is essentially in
agreement with the experimental data. This agreement can be attributed to
the overall large values of U0/Γπin case of CrAu+
n, which leads to almost
full spin polarization. Therefore, the highly discretized density of states in the
host has only a minor influence on the impurity spin, namely scaling it as a
function of the hosts energy gap. This was confirmed by exchanging the host
material from gold to copper.
In order to get a more profound insight into the influence of the gap, the prob-
lem was studied within tight binding approximation. It turned out that the
presence of a gap in the host density of states can substantially stabilize the
impurity’s magnetic moment. This is especially true for systems approaching
the magnetic to non-magnetic transition, i.e. the parameter regime U0/Γ≈π.
In this parameter regime, the description within the standard Anderson im-
purity model breaks down and magnetic states can also be found for values
U0/Γ< π.
7.2 Outlook
Although the treatment of finite systems within tight binding approximation
already yields significant insight, the model should be extended to multiple
orbitals in order to get a more quantitative description of the impurity-host
interaction. To test this model in a quantitative manner, XMCD spectroscopy
should be applied to FeAu+
n, CoAu+
nand NiAu+
nfor which the XMCD sum
rules hold.
Furthermore, the size dependence of the spin of the magnetic impurity of Fe,
Co, Ni in Auncan be expected to be different from that of Cr in CrAu+
n
clusters. That is because the on-site Coulomb repulsion U0varies along the
3d-transition metal series, as does the hybridization of host and impurity
electronic states. The dopant-host interaction in CrAu+
nclusters is reduced
by shell closure in the gold host as well as the highly stable half filled valence
shell of Cr+(6Sstate). Valence states of Fe, Co, Ni are less stable, as can
be inferred from their lower 3d−4spromotion energy [155, 156]. It can be
assumed that this brings the system closer to the magnetic to non-magnetic
transition. In the regime close to the transition, the influence of the highly
structured density of states in the host material should be more pronounced
7.2. Outlook 87
than in the almost fully spin polarized case of CrAu+
nclusters. This should
lead to a strong size dependence of the magnetic moments, potentially with
complete quenching of the moments for certain sizes. Hence, deviations from
the bulk Anderson impurity model are supposed to be more distinct in FeAu+
n,
CoAu+
nand NiAu+
n.
The presented data is a first step towards the experimental investigation of a
Kondo effect in a finite system which is expected to differ considerably from its
bulk counterpart. This is because the scattering processes in the Kondo effect
involve states near the Fermi level in the energy range of kBTK. Therefore, it
is natural to assume that the Kondo effect should be strongly altered in finite
systems where the mean level spacing ∆ becomes comparable to the energy
scale kBTbulk
K.
Indeed, several theoretical publications point out the implications of the mean
level spacing ∆. For instance, a partitioning of the Kondo resonance is pre-
dicted, which is different for odd and even numbers of electrons in the host
material [138]. Also non-vanishing values of the susceptibility χat low tem-
peratures are found [130].
In principle the latter effect should be accessible by temperature dependent
magnetization curves obtained from XMCD spectroscopy. However, even
without extracting the exact values of the susceptibility χthe formation of
a Kondo state should be observable by the deviation of a temperature de-
pendent Brillouin-like magnetization curve. The spin polarization of the host
material is even more easily to access. Above the Kondo temperature TKno
spin polarization is present in the host electron gas assuming an even number
of electrons. Below TK, upon formation of the low spin state, the itinerant
electrons get spin polarized. This can possibly be observed if the impurity is
underscreened, i.e. the number of itinerant electrons in the host is not suf-
ficient to fully screen the localized impurity spin, because this in turn yields
a finite alignment in an external magnetic field necessary to detect a dichroic
signal. The unambiguous signature of the Kondo effect would then be the
appearance of an antiferromagnetic coupling of impurity and host electrons
spins below the Kondo temperature TK. From a spectroscopic point of view
copper would be a perfect host material due to its nicely structured x-ray
absorption signal at the L2,3-edges [32], which in turn allows to probe the spin
polarization of the 4s-derived free electron gas states.
The investigation of a Kondo effect in a finite system has thus come into
reach. It would open up new paths for experimental many-body physics and
promises new physical insight into an old problem of condensed matter physics:
the scattering of electrons at a magnetic impurity.
88 Chapter 7. Epilogue
List of publications
1. K. Hirsch, V. Zamudio-Bayer, A. Langenberg, M. Niemeyer, B. Lang-
behn, T. M¨oller, A. Terasaki, B. v. Issendorff, and J. T. Lau, Magnetic
Moments of Chromium-Doped Gold Clusters: The Anderson Impurity
Model in Finite Systems, in preparation.
2. V. Zamudio-Bayer, L. Leppert, K. Hirsch, A. Langenberg, J. Rittmann,
M. Kossick, M. Vogel, R. Richter, A. Terasaki, T. M¨oller, B. v. Is-
sendorff, S. K¨ummel, and J. T. Lau, Coordination-driven magnetic-to-
nonmagnetic transition in manganese doped silicon clusters, submitted
to Phys. Rev. Lett.
3. K. Hirsch, V. Zamudio-Bayer, J. Rittmann, A. Langenberg, M. Vogel,
B. v. Issendorff, and J. T. Lau, Initial- and final-state effects on screen-
ing and branching ratio in 2p x-ray absorption of size-selected free 3d
transition metal clusters, Phys. Rev. B 86, 165402 (2012).
4. K. Hirsch, V. Zamudio-Bayer, F. Ameseder, A. Langenberg, J. Rittmann,
M. Vogel, T. M¨oller, B. v. Issendorff, and J. T. Lau, 2p x-ray absorp-
tion of free transition-metal cations across the 3d transition elements:
Calcium through copper, Phys. Rev. A 85, 062501 (2012).
5. M. Niemeyer, K. Hirsch, V. Zamudio-Bayer, A. Langenberg, M. Vogel,
M. Kossick, C. Ebrecht, K. Egashira, A. Terasaki, T. M¨oller, B. von Is-
sendorff, and J. T. Lau, Spin Coupling and Orbital Angular Momentum
Quenching in Free Iron Clusters, Phys. Rev. Lett. 108, 057201 (2012).
89
6. M. Vogel, C. Kasigkeit, K. Hirsch, A. Langenberg, J. Rittmann, V.
Zamudio-Bayer, A. Kulesza, R. Mitri, T. M¨oller, B. v. Issendorff, and
J. T. Lau, 2p core-level binding energies of size-selected free silicon clus-
ters: Chemical shifts and cluster structure, Phys. Rev. B 85, 195454
(2012).
7. J. T. Lau, M. Vogel, A. Langenberg, K. Hirsch, J. Rittmann, V.
Zamudio-Bayer, T. M¨oller, and B. von Issendorff, Communication: High-
est occupied molecular orbital-lowest unoccupied molecular orbital gaps
of doped silicon clusters from core level spectroscopy, J. Chem. Phys.
134, 041102 (2011).
8. J. T. Lau, K. Hirsch, P. Klar, A. Langenberg, F. Lofink, R. Richter,
J. Rittmann, M. Vogel, V. Zamudio-Bayer, T. M¨oller, and B. von Is-
sendorff, X-ray spectroscopy reveals high symmetry and electronic shell
structure of transition-metal-doped silicon clusters, Phys. Rev. A 79,
053201 (2009).
9. J. T. Lau, K. Hirsch, A. Langenberg, J. Probst, R. Richter, J. Rittmann,
M. Vogel, V. Zamudio-Bayer, T. M¨oller, and B. von Issendorff, Local-
ized high spin states in transition-metal dimers: X-ray absorption spec-
troscopy study, Phys. Rev. B 79, 241102 (2009).
10. K. Hirsch, J. T. Lau, P. Klar, A. Langenberg, J. Probst, J. Rittmann,
M. Vogel, V. Zamudio-Bayer, T. M¨oller, and B. von Issendorff, X-ray
spectroscopy on size-selected clusters in an ion trap: from the molecular
limit to bulk properties, J. Phys. B: At. Mol. Opt. Phys. 42, 154029
(2009).
11. J. T. Lau, J. Rittmann, V. Zamudio-Bayer, M. Vogel, K. Hirsch, P.
Klar, F. Lofink, T. M¨oller, and B. von Issendorff, Size Dependence of
L2,3Branching Ratio and 2p Core-Hole Screening in X-Ray Absorption
of Metal Clusters, Phys. Rev. Lett. 101, 153401 (2008).
List of Figures
2.1 Radial probability of 2pand 3dwave functions for atomic num-
ber Z=23. ............................ 14
2.2 Transition probabilities for L3-edge excitation with circular po-
larizedlight............................. 15
2.3 Accessible mLsubsets in a dipole transition at the L2,3-edges. 15
2.4 Comparison of fluorescence and Auger yield at the L2,3edges
as a function of the atomic number Z. ............. 17
2.5 Kmetko-Smith diagram. . . . . . . . . . . . . . . . . . . . . . 18
2.6 Mott transition as a function of U/W .............. 19
2.7 Schematic illustration of the total energy as a function of oc-
cupationnumbers. ........................ 23
2.8 Linear response function for the screened and bare system to
obtain on-site Coulomb repulsion U0. .............. 24
2.9 Schematic illustration of the Anderson impurity model. . . . . 26
2.10 Self consistent solution within the Anderson impurity model for
the occupation numbers n±of spin-up and spin-down state. . 27
2.11 Feynman diagram of the Kondo scattering process. . . . . . . 28
3.1 Schematic view of the experimental setup for XMCD spec-
troscopy............................... 32
3.2 Magnetization as a function of temperature and external mag-
netic field shown for a particle with J=S= 5/2. ....... 33
3.3 Variation of the magnetic field of the solenoid magnet along the
magnets symmetry axis. . . . . . . . . . . . . . . . . . . . . . 34
3.4 Sectional view of the ion trap. . . . . . . . . . . . . . . . . . . 35
3.5 Sketch of an undulator and electron bunch trajectories. . . . . 36
4.1 Preparation of measurement: Mass spectra. . . . . . . . . . . 40
4.2 Schematic of the data acquisition cycle. . . . . . . . . . . . . . 41
4.3 Background subtraction in x-ray absorption spectra. . . . . . . 42
4.4 Data processing of XMCD asymmetries. . . . . . . . . . . . . 43
4.5 Convergence test for the density functional theory calculations. 44
91
5.1 Linear XAS of ScAu+
nn= 1 −6 at the scandium L2,3edges. . 47
5.2 DOS and PDOS of ScAu+
nclusters. ............... 48
5.3 Population analysis of the scandium 3d-orbitals in ScAu+
nclusters. 49
5.4 Linear XAS of TiAu+
nn= 1 −9 at the titanium L2,3edges. . . 50
5.5 L¨owdin population analysis of TiAu+
nclusters. . . . . . . . . . 51
5.6 DOS and PDOS of TiAu+
nclusters, n= 1 −9. ......... 52
5.7 X-ray absorption onset of TiAu+
nclusters. . . . . . . . . . . . 53
5.8 Linear XAS of VAu+
nn= 1 −7 at the vanadium L2,3edges. . . 56
5.9 Total and spin resolved 3doccupation of VAu+
nclusters in a
size range of n= 1 −7....................... 57
5.10 Density of states of VAu+
nclusters, n= 1 −7. ......... 58
5.11 Linear XAS of CrAu+
nn= 1 −8 at the chromium L2,3edges. . 60
5.12 Size dependent absorption onset in the x-ray absorption of
CrAu+
nclusters........................... 61
6.1 Normalized XMCD spectra of CrAu+
n,n= 1 −6. ....... 64
6.2 Local magnetic moment, chromium-gold interaction energy, An-
derson criterion and gold HOMO-LUMO gap for CrAu+
nclusters. 66
6.3 Total and d-projected density of states of CrAu+
nclusters, n=
1−9. ............................... 70
6.4 Comparison of the self-consistent solutions for the occupation
numbers from the analytical Anderson impurity model and in
tight binding approximation. . . . . . . . . . . . . . . . . . . . 71
6.5 DOS of a finite system obtained from Hamiltonian in tight bind
approximation. .......................... 73
6.6 Comparison of the Anderson impurity model for continuous and
discrete host density of states. . . . . . . . . . . . . . . . . . . 74
6.7 Comparison of spin polarization for the Anderson impurity
model with and without gap. . . . . . . . . . . . . . . . . . . . 75
6.8 Geometries of CrCu+
nclusters in a size range n= 1 −6. . . . . 78
6.9 Linear x-ray absorption spectra at the L2,3edges of chromium
of CrCu+
nin a size range n= 1 −6................ 79
6.10 Normalized XMCD spectra at the L2,3-edges of chromium of
CrCu+
nin a size range of n= 1 −6................ 80
6.11 Maximum intensity at the L3-edge of the XMCD asymmetry,
local chromium spin magnetic moment, Anderson criterion and
HOMO-LUMO gap of the copper host of CrCu+
nclusters. . . . 81
6.12 Comparison of the density of states of CrCu+
2and CrAu+
2. . . 83
List of Tables
3.1 Technical specification of the undulator beam lines U49-2-PGM
andUE52-SGM. ......................... 36
6.1 Average nearest neighbor chromium-gold bond distance and
number of nearest neighbors in CrAu+
nclusters. . . . . . . . . 65
6.2 Calculated on-site Coulomb repulsion U0and width 2Γ for CrAu+
n
clusters. .............................. 68
6.3 Average nearest neighbor chromium-copper bond distance and
number of nearest neighbors in CrCu+
n.............. 82
93
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Chapter 8
Appendix
8.1 Coordinates of CrCu+
nClusters
All distances are given in ˚
A.
CrCu+
1x y z
ground state
Cu -2.8096432 2.1443083 0.0000000
Cr -0.0724060 2.1773405 0.0000000
109
CrCu+
2x y z
ground state
Cu -2.7820222 2.0691877 0.0000000
Cu -1.7583713 -0.0300738 0.0000000
Cr -0.2368152 2.0116150 0.0000000
CrCu+
2x y z
first isomer
Cu -0.0000000 -0.0000000 0.0557006
Cu 0.0000000 0.0000000 2.3941831
Cr 0.0000000 0.0000000 -2.4498837
CrCu+
2x y z
second isomer
Cu 0.0000000 0.0000000 -2.5504625
Cu 0.0000000 0.0000000 2.5504626
Cr 0.0000000 0.0000000 0.0000000
CrCu+
3x y z
ground state
Cu -2.1505923 -0.3483559 1.2545071
Cu -3.5330069 0.5047884 -0.4706835
Cu -2.8398571 1.8686943 -2.2797694
Cr -1.1131110 1.1857723 -0.5157236
CrCu+
3x y z
first isomer
Cu -3.8090847 1.3503527 -0.0364732
Cu -2.7780393 0.7544706 -2.0562519
Cu -5.1325497 0.3423142 -1.7752698
Cr -1.2350227 1.8593313 -0.2323240
CrCu+
4x y z
ground state
Cu -4.1048699 1.7694397 0.0140550
Cu -1.8233374 0.9714893 -0.4527150
Cu -5.8467743 0.9137523 -1.5031206
Cu -3.8247137 -0.3771064 -0.9450439
Cr -3.4683194 1.6351437 -2.4223941
CrCu+
4x y z
first isomer
Cu -3.4783818 0.3372498 0.0000000
Cu -1.7865323 -1.5293825 -0.0000000
Cu 0.5394157 -1.1959252 0.0000000
Cu -2.9157225 2.6200534 -0.0000000
Cr -1.0131866 0.8742243 0.0000000
CrCu+
4x y z
second isomer
Cu -3.7749336 2.2958538 -0.0000007
Cu -0.9186915 -1.5478900 0.0000006
Cu -1.4550412 2.6911512 0.0000008
Cu -2.2178529 0.4366716 0.0000000
Cr 0.3526315 0.6234522 -0.0000008
CrCu+
5x y z
ground state
Cu -5.3025571 -2.6623380 0.5859749
Cu -4.3078724 -0.5817841 -0.1887269
Cu -1.8702858 -0.6260682 -0.1921873
Cu -0.9491436 -2.7407853 0.5799354
Cu -3.1284857 -2.7701099 -0.4867023
Cr -3.1075401 -1.7666213 1.7702456
CrCu+
5x y z
first isomer
Cu -5.2401760 -2.6827573 0.7113284
Cu -4.4019011 -0.6094780 -0.1708348
Cu -1.0526985 -2.8318225 0.7355800
Cu -3.1188604 -2.7245492 -0.5611099
Cu -3.0989335 -1.8133781 1.5979678
Cr -1.7533153 -0.4857217 -0.2443920
CrCu+
5x y z
second isomer
Cu 1.5167037 0.6420737 -0.1712256
Cu 0.7337849 -1.5356580 0.1936203
Cu -2.4023426 1.0673920 1.0762432
Cu -0.1806753 0.2853206 1.6103346
Cu 1.0121314 -0.7627699 -2.1441690
Cr -0.9553931 0.1766146 -0.7765314
CrCu+
6x y z
ground state
Cu 1.2680685 1.7445335 -0.0458994
Cu 0.0000000 0.0004092 -1.1595719
Cu 2.0510249 -0.6666058 -0.0463570
Cu -2.0510249 -0.6666058 -0.0463570
Cu -1.2680685 1.7445335 -0.0458994
Cu -0.0000000 -2.1565600 -0.0454908
Cr -0.0000000 0.0003053 1.3895754
CrCu+
6x y z
first isomer
Cu 1.2961846 1.6998437 0.0000000
Cu 0.0045176 -0.0054914 -1.1897664
Cu 0.0045176 -0.0054914 1.1897664
Cu 2.0501430 -0.7076098 0.0000000
Cu -2.0171049 -0.7071098 0.0000000
Cu 0.0388649 -2.1680662 0.0000000
Cr -1.3771227 1.8939348 0.0000000
CrCu+
6x y z
second isomer
Cu -2.0054171 -5.5562396 -2.6403927
Cu -4.0808899 -4.8003748 -1.6800297
Cu -3.7190511 -3.0086048 0.0779657
Cu -1.3857893 -2.4886761 0.3709471
Cu -1.9027572 -4.7156239 -0.3859186
Cu -5.1192088 -2.6331658 -1.8449734
Cr -2.4285410 -2.9255677 -2.1013761
Acknowledgments
The presented thesis results from research activities at Technische Universit¨at
Berlin and Helmholtz Zentrum Berlin, where I had the pleasure to work with
many people who supported me during the last years and whom I would like
to thank here.
First of all I thank my supervisor Thomas M¨oller who not only gave me the
opportunity to join his group, but also the freedom to choose a research topic
not belonging to his main focus of research. I will remember his friendly, un-
complicated attitude and the possibility to drop by into his office whenever
finding myself in any kind of trouble.
I am deeply indebted to Tobias Lau who has drawn me into cluster physics.
He impressed me with his stamina in long beam-times, especially in the early
stages of the experiment, and his everlasting enthusiasm. Scientifically, I
benefited from countless discussions and extensive lab experience with him.
Moreover, working with him always was a lot of fun.
I am grateful to Bernd von Issendorff, not only for reviewing this thesis.
He constantly supported us with his enormous experimental experience and
played a brilliant devil’s advocate, thereby giving me insights into his deep
understanding of physics.
I was very lucky to work together with two good friends: Jochen Rittmann
and Vicente Zamudio-Bayer with whom I shared an office during the begin-
ning of the PhD project at the TUB. We also spend uncounted hours together
in the lab and at the beam-line. It really was a great time!
Coffee breaks at Chez Marcel were essential to overcome the fatigue in the
afternoon, thanks for letting me disturb you in your office Marcel Pagels and
Falk Reinhardt. I thank also the the last active member of the legendary PG
272, Max B¨ugler.
I like to thank Andreas Langenberg, the most patient guy I know, for design-
ing the XMCD setup and sharing loads of funny night shifts.
Of course I want to thank all the other great people who were associated to the
group at some time: Marlene Vogel, J¨org Wittich, Markus Niemeyer, J¨urgen
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Probst, and Arkadiusz Lawicki.
I am very thankful for the constant support by my family. Especially, I thank
my brother Alexander Hirsch and his wife Mae Voon for proof-reading parts
of this thesis.
For delivering numerous excellent parts I like to acknowledge the workshop.
I thank Akira Terasaki for providing the superconducting solenoid essential
to perform the XMCD studies.
Calculations were carried out on the FOR 1282 computing cluster.
I thank my beloved girlfriend Linn Leppert for countless things. She encour-
aged me, gave me lot of advice concerning the DFT calculations, cheered me
up, and furthermore proof-read this thesis over and over again. I thank you!
