Ultrafast spin dynamics of a ferrimagnet revealed by femtosecond
soft X-ray and XUV radiationUltrafast spin dynamics of a
ferrimagnet revealed by femtosecond soft X-ray and XUV
radiation
vorgelegt von M. Sc.
an der Fakultät II - Mathematik und Naturwissenschaften der
Technischen Universität Berlin
zur Erlangung des akademischen Grades
Doktor der Naturwissenschaften - Dr. rer. nat. -
genehmigte Dissertation
Vorsitzender: Prof. Dr. Mario Dähne Gutachter: Prof. Dr. Stefan
Eisebitt Gutachter: Prof. Dr. Jan Lüning
Tag der wissenschaftlichen Aussprache: 13.08.2020
Berlin 2020
Die vorliegende Arbeit handelt von zeitaufgelösten experimentellen
Untersuchungen ul- traschneller Spinphänomene in ferrimagnetischen
metallischen Legierungen aus GdFeCo. Neuere Studien solcher
ferrimagnetischen Systeme haben gezeigt, dass eine optische
Anregung mit Femtosekundenpulsen nicht nur in der Lage ist, ein
solches System auf einer Zeitskala von weniger als einer
Picosekunde zu entmagnetisieren, sondern auch seine Magnetisierung
dauerhaft umzuschalten, ohne dass dafür ein anderer externer Impuls
wie ein Magnetfeld erforderlich wäre. Obwohl mehrere Theorien
aufgestellt wurden, um die ultraschnelle Abnahme oder sogar
Umkehrung der magnetischen Ordnung zu erklären, sind die
tatsächlichen Mechanismen hinter diesen Phänomenen noch immer
unklar. Eine der Schlüsselfragen ist z.B., wie der Spin- und
Bahndrehimpuls der Elektronen, welche zusammen das magnetische
Moment hervorrufen, auf einer so kurzen Zeitskala nach Anregung aus
dem elektronischen System oder in dieses hinein übertragen werden.
Eine weitere Frage betrifft die Art und Weise, auf die ein
Lichtpuls während der Anregung mit dem magnetischen Medium
interagiert. Dabei kann es sich im Allgemeinen entweder um einen
thermischen Prozess handeln, der sich auf eine ultraschnelle
optisch induzierte Erwärmung des elektronischen Systems stützt,
oder um einen nichtthermischen, optoma- gnetischen Mechanismus wie
den inversen Faraday-Effekt (IFE), der bei Anregung mit zirkular
polarisiertem Licht eine helizitätsabhängige Magnetisierung oder
einen effektiven optomagnetischen Feldpuls induziert. Die
tatsächlichen Einflüsse der letztgenannten nicht- thermischen
Mechanismen auf einer Femtosekundenzeitskala sind in stark
absorbierenden Materialien wie in Metallen, wo thermische Effekte
normalerweise dominieren, jedoch stark umstritten. Die im Rahmen
dieser Arbeit präsentierten Ergebnisse sollen Licht in die beiden
zuvor genannten Fragen bringen.
Um die Drehimpulsübertragung während der ultraschnellen optisch
induzierten Entma- gnetisierung einer ferrimagnetischen
GdFeCo-Legierung zu untersuchen, verwenden wir zeit- und
elementaufgelöste Messungen des zirkularen magnetischen
Röntgendichroismus (XMCD) mittels weicher Röntgenpulse, die an der
FemtoSpeX fs-Slicingquelle am Syn- chrotron BESSY II erzeugt
werden. Die Anwendung magnetooptischer Summenregeln auf die
fs-XMCD-Daten ermöglicht es, die Dynamik von Spin- und Bahnmomenten
der Fe- und Gd-Atome individuell zu verfolgen. Anhand unserer
experimentellen Daten lässt sich auf eine vollständige Übertragung
sowohl des Spin- als auch des Bahndrehimpulses von Fe und Gd an das
Atomgitter schließen, die während weniger hundert Femtosekunden
nach Anregung stattfindet. Im Rahmen unserer experimentellen
Zeitauflösung von ≈ 130 fs
i
gibt es dabei keinen Hinweis auf einen interatomaren
Drehimpulsaustausch zwischen Fe und Gd oder eine Ansammlung des
Drehimpulses im Bahnmoment. Letzteres kann somit als Engpass für
eine Drehimpulstransfer an das Gitter ausgeschlossen werden.
Um den Einfluss eines nichtthermischen, optomagnetischen
Mechanismus wie des IFE in metallischen Ferrimagneten wie GdFeCo zu
untersuchen, verwenden wir einen neuartigen Ansatz, um eine
helizitätsabhängige ultraschnelle Entmagnetisierungsdynamik durch
resonante Anregung von Elektronen aus den inneren 3p-Zuständen von
Fe zu induzieren (resonant zur Fe M3,2-Absorptionskante). Dafür
nutzen wir intensive Femtosekunden- pulse im XUV-Spektralbereich
mit sowohl linearer als auch zirkularer Polarisation, die am
Freie-Elektronen-Laser FERMI erzeugt werden, um den Prozess der
ultraschnellen Entmagnetisierung als Funktion des
Polarisationszustands und der Photonenenergie des
XUV-Anregungspulses zu untersuchen. Die Motivation hinter diesem
Ansatz ist die starke Spin-Bahn-Kopplung der inneren elektronischen
Zustände. Da die Spin-Bahn-Kopplung der vermittelnde Mechanismus
hinter jedem magnetooptischen oder optomagnetischen Ef- fekt ist,
könnte dieser Ansatz Zugang zu einem Bereich bieten, in dem der
nichtthermische IFE viel stärker ist im Vergleich zu sichtbaren
Wellenlängen. Bislang liegen jedoch keine experimentellen Daten
oder Theorien zur Existenz eines IFE im XUV-Spektralbereich vor.
Unsere Messungen zeigen in der Tat einen starken dynamischen,
helizitätsabhän- gigen Effekt, der sowohl für resonante als auch
nichtresonante Anregung nahe der Fe M3,2-Absorptionskante existiert
und der nicht durch dichroitische Absorption aufgrund des ebenfalls
vorhandenen XMCD-Effekts erklärt werden kann. Unsere Ergebnisse
ent- sprechen vielmehr den erwarteten Eigenschaften des IFE und
deuten daher stark auf die erste Beobachtung eines IFE im bisher
unerforschten XUV-Spektralbereich hin.
ii
Abstract
In this thesis, we will present time-resolved experimental
investigations of ultrafast spin phenomena in ferrimagnetic
metallic alloys of GdFeCo. Recent studies of such ferrimagnetic
systems have revealed that a femtosecond optical excitation is not
only able to demagnetize such a system on a subpicosecond
timescale, but also to permanently reverse its magnetization
without any other external stimulus like a magnetic field. Although
several theoretical approaches were made to explain the ultrafast
loss or even reversal of the magnetic order, the actual mechanisms
behind such spin phenomena still remain unclear. One of the key
issues is, e.g., how the spin and orbital angular momentum of the
electrons, which gives rise to the magnetic moment, is transferred
out of, or into the electronic system on such a short timescale
after excitation. Another issue concerns the way a light pulse can
interact with the magnetic medium during the excitation process. In
general, this can either be a thermal mechanism relying on
ultrafast light-induced heating of the electronic system or a
non-thermal, opto-magnetic mechanism like the Inverse Faraday
Effect (IFE) generating a helicity-dependent induced magnetization
or an effective opto-magnetic field pulse upon excitation with
circularly polarized light. However, the actual role of the latter
non-thermal mechanisms on a femtosecond timescale in highly
absorbing materials like metals, where thermal effects usually
dominate, is highly debated. The scope of this thesis is to shed
light upon both of the aforementioned issues.
To reveal the path of angular momentum flow during ultrafast
laser-induced demagne- tization of a ferrimagnetic GdFeCo alloy, we
employ time- and element-resolved soft x-ray magnetic circular
dichroism (XMCD) measurements at the FemtoSpeX fs-slicing facility
of the BESSY II synchrotron light source. A magneto-optical sum
rules analysis of the fs-XMCD data allows us to disentangle and
monitor the dynamics of both spin and orbital moments individually,
at Fe and Gd sites. Our experimental data enable us to conclude on
a full transfer of angular momentum from both spin and orbital
moments of Fe and Gd to the atomic lattice during the first
hundreds of femtoseconds after excitation. Within our experimental
time resolution of ≈ 130 fs, there is also no indication for an
interatomic exchange of angular momentum between Fe and Gd, as well
as for an accumulation in the orbital moment. Thus, the latter can
be ruled out as a bottleneck for the angular momentum transfer to
the lattice. In order to reveal the influence of a non-thermal,
opto-magnetic mechanism like the
IFE in metallic ferrimagnets like GdFeCo, we use a novel approach
of inducing helicity-
iii
dependent ultrafast demagnetization dynamics by resonantly exciting
electrons from the 3p core levels of Fe (corresponding to the Fe
M3,2 resonance). Therefore we employ highly intense, femtosecond
XUV pulses with both linear and circular polarization generated at
the free electron laser FERMI to study the ultrafast
demagnetization process as a function of polarization state and
photon energy of the XUV excitation pulse. The motivation behind
this approach is the strong spin-orbit coupling of the core levels.
While the spin- orbit coupling is the mediating mechanism behind
any magneto-optical or opto-magnetic effect, this could provide
access to a regime where the non-thermal IFE is much stronger
compared to the visible wavelength regime. However, no experimental
data or theory is available so far on the existence of an IFE in
the XUV spectral range. Our results reveal a strong dynamic
helicity-dependent effect present for both off- and on-resonant
excitation around the Fe M3,2 resonance that can not be explained
by dichroic absorption due to the XMCD effect, which is present as
well. Our findings rather resemble the expected IFE fingerprints
and strongly indicate that we have made the very first observation
of an IFE in the so far unexplored XUV spectral range.
iv
Acknowledgments
First of all I would like to thank Stefan Eisebitt, who gave me the
opportunity to join his division at the Max Born Institute and to
become a PhD student in such a capable, collaborative and friendly
team. Besides profiting from his experience and scientific advice,
I was given great freedom to realize and work on my research
projects and to extend my qualification by attending international
conferences and summer schools, where I had the chance to present
my work and connect with scientists from all over the world.
Special thanks go to my supervisor Ilie Radu, who introduced me to
the field of ultrafast spin dynamics and whose extensive knowledge
and experience gave me the opportunity to learn a lot about the
underlying physics and scientific methods. During the last three
and a half years of my PhD, he not only actively supported and
encouraged me during our numerous beam times, at conferences and in
the lab, but was also a very patient and committed supervisor, who
always had an open ear for questions and discussions.
Of course the latter was also the case for many other members of
our team. In particular I would like to thank Clemens von Korff
Schmising, Daniel Schick, Dieter Engel, Christian Strüber, Bastian
Pfau and Michael Schneider for always being ready to help in case I
needed scientific or experimental advice and feedback on my talks,
work and latest results. I would also like to thank Dieter Engel
and Sascha Petz for spending so much time in the sputtering lab to
prepare and optimize samples for me, as well as Marc Zieglarski for
his software development and support in all Labview-related
questions.
Thanks also go to my fellow PhD students Felix Willems, David
Weder, Piet Hessing and the later joined Kelvin Yao and Martin
Borchert, with whom I have shared the office for the last years.
Discussing and solving scientific questions together, helping us
out at beam times and in the lab as well as talking about the
latest news and share prices made my PhD time much more productive
and motivating.
I would also like to thank all my colleagues and the involved
researchers mentioned in the Preface of this thesis, who also
contributed greatly to the completion of this work, either by their
assistance in preparing and conducting the experiments, or by
providing the theory and contributing to the interpretation of the
experimental data. Without their support and expertise, the
experiments and results presented in this thesis could not have
been achieved. Last but not least I would like to thank Ilie Radu,
Felix Willems, Karsten Holldack,
Flavio Capotondi and Peter Oppeneer for proofreading various parts
of my thesis, which helped me to increase the quality of the
manuscript.
v
Contents
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . i Abstract . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
List of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . ix List of Figures . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . xi List of Tables . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xii Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . xiii
1 Introduction 1
2 Fundamentals and experimental techniques 5 2.1 Magnetic
properties of rare earth and 3d transition metal systems . . . .
5
2.1.1 Magnetic interactions . . . . . . . . . . . . . . . . . . . .
. . . . . 6 2.1.2 Magnetic order in RE and 3d-TM elements . . . . .
. . . . . . . . 8
2.2 Ultrafast laser-induced magnetization dynamics in RE and 3d-TM
systems 12 2.2.1 Mechanisms behind ultrafast magnetization dynamics
. . . . . . . 12 2.2.2 Local spin-flip scattering processes . . . .
. . . . . . . . . . . . . . 13 2.2.3 Non-local spin transport
processes . . . . . . . . . . . . . . . . . . 15 2.2.4 All-optical
magnetization switching in RE and 3d-TM systems . . 17
2.3 Magneto-optical effects (probing magnetization with light) . .
. . . . . . . 18 2.3.1 Magneto-optical Faraday and Kerr effect
(MOFE/MOKE) . . . . . 19 2.3.2 X-ray/XUV magnetic circular
dichroism (XMCD) . . . . . . . . . 21 2.3.3 Magneto-optical sum
rules analysis . . . . . . . . . . . . . . . . . . 22
2.4 Opto-magnetic effects (inducing magnetization by light) . . . .
. . . . . . 25 2.4.1 Inverse Faraday Effect (IFE) . . . . . . . . .
. . . . . . . . . . . . 26
2.5 Soft x-ray and XUV radiation sources . . . . . . . . . . . . .
. . . . . . . 29 2.5.1 FemtoSpeX fs-slicing facility at BESSY II .
. . . . . . . . . . . . . 29 2.5.2 Free electron laser FERMI . . .
. . . . . . . . . . . . . . . . . . . . 33 2.5.3 Static
spectroscopy setup at BESSY II . . . . . . . . . . . . . . . .
35
3 Angular momentum flow during ultrafast demagnetization of GdFeCo
37 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 37 3.2 Results of static soft x-ray
spectroscopy . . . . . . . . . . . . . . . . . . . 38
vii
Contents
3.3 Time-resolved optical pump – x-ray probe experiment . . . . . .
. . . . . 40 3.3.1 Experimental setup . . . . . . . . . . . . . . .
. . . . . . . . . . . . 40 3.3.2 Data acquisition and data
treatment . . . . . . . . . . . . . . . . . 41 3.3.3 Applying sum
rules to the time-resolved data . . . . . . . . . . . . 43
3.4 Time-resolved demagnetization of GdFeCo . . . . . . . . . . . .
. . . . . . 46 3.4.1 Time-resolved XMCD measurements . . . . . . .
. . . . . . . . . . 46 3.4.2 Spin and orbital moments dynamics
during demagnetization . . . 49
3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 53
4 X-ray driven ultrafast demagnetization of GdFeCo 55 4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 55 4.2 Results of static XUV spectroscopy . . . . .
. . . . . . . . . . . . . . . . . 58
4.2.1 Absorbed fluence and penetration depth . . . . . . . . . . .
. . . . 61 4.3 Time-resolved XUV pump – optical probe experiment .
. . . . . . . . . . 63
4.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . . .
. . . . . . 63 4.3.2 Data sorting and data treatment . . . . . . .
. . . . . . . . . . . . 67
4.4 Results of the time-resolved measurements . . . . . . . . . . .
. . . . . . . 67 4.4.1 Pump-probe delay scans . . . . . . . . . . .
. . . . . . . . . . . . . 68 4.4.2 Fluence-dependence . . . . . . .
. . . . . . . . . . . . . . . . . . . 71 4.4.3 Comparison to the
XMCD . . . . . . . . . . . . . . . . . . . . . . . 75
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 78 4.6 Summary . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . 81
5 Conclusions and outlook 83
Bibliography 87
A Appendix 101 A.1 Magnetic hysteresis loops . . . . . . . . . . .
. . . . . . . . . . . . . . . . 101 A.2 Faraday vs. Kerr probing .
. . . . . . . . . . . . . . . . . . . . . . . . . . 101 A.3
Pump-probe delay scans . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 102
viii
2TM two-temperature model 2D two-dimensional 3D three-dimensional
3TM three-temperature model AO-HDS all-optical
helicity-dependent
switching APD avalanche photo diode bw. bandwidth DFT density
functional theory FEL free electron laser fs femtosecond FWHM full
width at half maximum FZP Fresnel zone plate GMD gas monitor
detector HHG high harmonic generation HGHG high-gain harmonic
generation IFE Inverse Faraday Effect lin.hor. linear horizontal
MAE magnetic anisotropy energy M3TM microscopic
three-temperature
model MCD magnetic circular dichroism MOFE magneto-optical Faraday
effect MOKE magneto-optical Kerr effect nm nanometer ns nanosecond
OISTR optical inter-site spin transfer PGM plane grating
monochromator ps picosecond RE rare earth
RKKY Ruderman-Kittel-Kasuya-Yosida RZP reflection zone plate SASE
self-amplified spontaneous emission SHG second harmonic generation
SOC spin-orbit coupling THG third harmonic generation TM transition
metal VIS visible XAS x-ray absorption spectroscopy XMCD x-ray
magnetic circular dichroism XUV extreme ultra violet ZPM zone plate
monochromator
ix
List of Figures
2.1 Temperature-dependence of the sublattice magnetizations in GdFe
. . . . 10 2.2 Illustration of experiments employing the
magneto-optical Faraday effect . 20 2.3 Illustration of the XMCD
effect at the L edges of a 3d ferromagnet . . . . 23 2.4 Schematic
illustration of the quantities needed to apply the sum rules . . 25
2.5 Schematic illustration and mechanism of the Inverse Faraday
Effect . . . . 27 2.6 Static ab-initio calculations of the IFE in
GdFe2 . . . . . . . . . . . . . . 28 2.7 Schematic top view on the
FemtoSpeX fs-slicing facility at BESSY II . . . 30 2.8 Femtoslicing
technique as implemented at BESSY II . . . . . . . . . . . . 32 2.9
Schematic illustration of the external seeding at FERMI . . . . . .
. . . . 34
3.1 Static soft x-ray absorption (XAS) and XMCD spectra of GdFeCo .
. . . 39 3.2 Schematic illustration of the pump-probe experiment at
FemtoSpeX . . . 40 3.3 Time-resolved pumped and unpumped XMCD at Fe
L3,2 . . . . . . . . . . 42 3.4 Comparison of the XMCD spectra
measured at FemtoSpeX and PM3 . . 44 3.5 Time-resolved XMCD at Fe
L3,2 and Gd M5,4 . . . . . . . . . . . . . . . . 47 3.6
Two-temperature model (2TM) simulation of GdFeCo . . . . . . . . .
. . 48 3.7 Time-resolved XMCD spectra of GdFeCo measured at
FemtoSpeX . . . . 50 3.8 Time-resolved evolution of spin and
orbital moment at Fe and Gd sites . . 51
4.1 Static XUV absorption (XAS) and XMCD spectra measured at Fe
M3,2 . 60 4.2 Static XUV absorption (XAS) and XMCD spectra measured
at Gd N5,4 . 61 4.3 Selecting the XUV photon energies used for
excitation . . . . . . . . . . . 62 4.4 Estimated XUV excitation
and VIS depth profiles in the sample . . . . . 63 4.5 Schematic
illustration of the pump-probe experiment at FERMI . . . . . 64 4.6
Analysis of the shot-resolved FEL pulse energies from the I0 GMD .
. . . 67 4.7 Time-resolved normalized Faraday rotation . . . . . .
. . . . . . . . . . . 69 4.8 Demagnetization amplitudes upon σ−, σ+
and lin. polarized excitation . . 72 4.9 Fluence-dependence of the
difference σ = σ− − σ+ . . . . . . . . . . . . 73 4.10
Demagnetization for lin. polarized XUV excitation vs. absorbed
fluence . 74 4.11 Estimation of asymmetry needed for the observed
hel.-dependent effect . . 76 4.12 Comparison of the XMCD to the
estimated asymmetries . . . . . . . . . . 77 4.13 Static ab-initio
calculations of the IFE in the XUV spectral range . . . . .
80
A.1 Magnetic hysteresis loops of the studied GdFeCo samples . . . .
. . . . . 103
x
List of Figures
A.2 Fluence diagrams (Faraday vs. Kerr probing) at 51.0 eV . . . .
. . . . . . 104 A.3 Fluence diagrams (Faraday vs. Kerr probing) at
54.1 eV . . . . . . . . . . 105 A.4 Fluence diagrams (Faraday vs.
Kerr probing) at 56.1 eV . . . . . . . . . . 106 A.5 Fluence
diagrams (Faraday vs. Kerr probing) at 64.0 eV . . . . . . . . . .
107 A.6 Demagnetization vs. lin. pol. absorbed XUV fluence (Faraday
vs. Kerr) . 108 A.7 Pump-probe delay scans obtained from Faraday
data (1/7) . . . . . . . . 109 A.8 Pump-probe delay scans obtained
from Faraday data (2/7) . . . . . . . . 110 A.9 Pump-probe delay
scans obtained from Faraday data (3/7) . . . . . . . . 111 A.10
Pump-probe delay scans obtained from Faraday data (4/7) . . . . . .
. . 112 A.11 Pump-probe delay scans obtained from Faraday data
(5/7) . . . . . . . . 113 A.12 Pump-probe delay scans obtained from
Faraday data (6/7) . . . . . . . . 114 A.13 Pump-probe delay scans
obtained from Faraday data (7/7) . . . . . . . . 115 A.14
Pump-probe delay scans obtained from Kerr data (1/7) . . . . . . .
. . . 116 A.15 Pump-probe delay scans obtained from Kerr data (2/7)
. . . . . . . . . . 117 A.16 Pump-probe delay scans obtained from
Kerr data (3/7) . . . . . . . . . . 118 A.17 Pump-probe delay scans
obtained from Kerr data (4/7) . . . . . . . . . . 119 A.18
Pump-probe delay scans obtained from Kerr data (5/7) . . . . . . .
. . . 120 A.19 Pump-probe delay scans obtained from Kerr data (6/7)
. . . . . . . . . . 121 A.20 Pump-probe delay scans obtained from
Kerr data (7/7) . . . . . . . . . . 122
xi
List of Tables
4.1 Spin-orbit coupling constants of Fe, Co and Ni at 3d, 3p and 2p
states . . 57 4.2 XUV photon energies and wavelengths used for
excitation . . . . . . . . . 65 4.3 Demagnetization amplitudes upon
σ+, σ− and lin. polarized excitation . . 70
A.1 Demagnetization time constants (Faraday vs. Kerr probing) . . .
. . . . . 108
xii
Preface
Some parts of this thesis have already been published as an article
in a peer reviewed journal. This concerns the experiment and
results presented in Chapter 3. For this reason, several text
passages and figures from this publication, including the
supplemental material that was published alongside the article,
also appear in this thesis in an identical or slightly modified
version. The corresponding paragraphs and figure captions are
marked by the following symbols:
† M. Hennecke, I. Radu, R. Abrudan, T. Kachel, K. Holldack, R.
Mitzner, A. Tsukamoto, and S. Eisebitt. “Angular Momentum Flow
During Ultrafast De- magnetization of a Ferrimagnet”. In: Phys.
Rev. Lett. 122, 157202 (15 Apr. 2019) (cited as Ref. [1] in the
bibliography)
‡ Supplemental Material belonging to Ref. [1].
Adapted with permission and copyright (2019) by the American
Physical Society.
The author of this thesis (M. Hennecke) is also the principal
author of the aforementioned article and supplemental material. The
experiment was primarily conceived by I. Radu and M. Hennecke. It
was conducted by M. Hennecke and I. Radu with the support of R.
Abrudan, T. Kachel, K. Holldack and R. Mitzner. The sample was
prepared by A. Tsukamoto. The data treatment and evaluation was
done by M. Hennecke, with I. Radu and S. Eisebitt contributing to
the interpretation of the experimental data. The experimental
results presented in Chapter 4 have not yet been published in
an
article. The corresponding experiment was primarily conceived and
conducted by M. Hennecke and I. Radu. Several co-workers
contributed to the successful conduction of the experiment: C. von
Korff Schmising, K. Yao (Berlin, Germany), E. Jal, B. Vodungbo, V.
Chardonnet (Paris, France), K. Légaré (Varennes, Canada), F.
Capotondi, D. Naumenko and E. Pedersoli (Trieste, Italy). The
sample was prepared by D. Engel (Berlin, Germany). The data
treatment and evaluation was done by M. Hennecke. The ab-initio
calculations of the Inverse Faraday Effect were carried out and
contributed by L. Salemi, M. Berritta and P. M. Oppeneer (Uppsala,
Sweden).
xiii
xiv
1 Introduction
The fundamental interactions between light and matter provide the
foundation for a variety of versatile techniques for non-invasive
investigations of static and dynamic properties of matter. Commonly
employed experimental techniques involve spectroscopic measurements
as well as scattering and imaging techniques in order to gain
microscopic insights into electronic, structural and magnetic
properties of a material. The use of ultrashort light pulses with
durations down to the femto- or even attosecond regime makes it
possible to obtain information on the elementary processes
(electron-electron, electron-magnon, phonon-magnon scattering etc.)
and interactions (spin-orbit, exchange etc.) in solids on their
intrinsic/characteristic time and length scales. In this thesis, we
will present such time-resolved studies on the interaction of light
and magnetic moments. In magnetically ordered materials, it is well
established that such an interaction allows to probe the
magnetization of the system via so called magneto-optical effects,
which influence either the polarization state or intensity of a
light wave while it is transmitted through or reflected by the
magnetic material. A rather novel approach, however, is to pursue
the opposite direction and use femtosecond light pulses to
influence the magnetization and to optically excite ultrafast spin
phenomena.
Over the last years, the use of such ultrashort light pulses to
manipulate and control the magnetic order in a material on
fundamentally limiting time and length scales has become an
important quest in modern magnetism [2–4]. In a pioneering work in
1996, it was discovered that ferromagnetic nickel can be
demagnetized on a subpicosecond timescale solely due to an optical
excitation with a femtosecond laser pulse [5]. At that time, the
impact of a laser excitation on the magnetic moments was understood
as a sudden heating of the atomic lattice due to absorption of the
laser pulse, followed by the formation of a new equilibrium
magnetization state via spin-lattice relaxation, a process which
typically happens on timescales of hundreds of picoseconds [6, 7].
Thus, the intriguing observation of such an ultrafast
demagnetization process approaching the characteristic timescales
of fundamental spin-orbit and exchange interactions (≈ 0.01–1 ps
[2]) was extremely surprising and became the foundation for a new
research area in condensed matter, the “Ultrafast Magnetism” or
“Femtomagnetism” field. A few years later, it was shown that it is
not only possible to demagnetize, but also fully and
deterministically switch the magnetization of a ferrimagnet (i.e.,
rotate the direction of the magnetic moments by 180 ) by means of
single femtosecond laser pulses without applying an external
magnetic field [8]. This kind of all-optical magnetization
switching was first observed
1
1 Introduction
in ferrimagnetic GdFeCo systems using circularly polarized
optical/visible laser pulses. The outcome of the process, i.e., the
final magnetization direction, was thereby found to depend on the
helicity of the light used for excitation, which is why the process
was called all-optical helicity-dependent switching (AO-HDS). These
findings were not only of large interest for fundamental science,
but also for possible applications, as the optically induced
demagnetization and magnetization reversal processes were several
orders of magnitude faster than the typical times needed to
demagnetize a material or reverse its magnetization by externally
applied magnetic fields commonly available in a laboratory (≈ 1 ns
[2]). As such, a process that allows such an ultrafast and
deterministic control over magnetization could potentially be a
large step forward in the further development of, e.g., spintronic
devices and magnetic data storage technologies, which require
manipulating the magnetization state of a system as fast and
energy-efficient as possible [9].
The action of a femtosecond laser pulse on a magnetic sample, i.e.,
how such a pulse can interact with the magnetic moments and
ultimately cause an optically induced demagnetization and even
magnetization switching, could thereby be related to both thermal
and non-thermal effects [2]: The first type relies on the
absorption of the fs laser pulse leading to an ultrafast heating of
the electronic system close to or above the Curie temperature,
triggering secondary processes which quench the local magnetic
moment. The second type is usually related to opto-magnetic effects
like the Inverse Faraday Effect (IFE) [10–12], which does not
involve the absorption of photons but describes a coherent
interaction of the angular momentum of the circularly polarized
light with the electron spins, inducing a helicity-dependent
magnetization or effective magnetic field pulse in the material.
Similar to other magneto-optical effects, such an interaction
between light and electron spins is mediated by the spin-orbit
coupling (SOC), suggesting a large induced magnetization in case of
the involved electronic states possessing a strong SOC [13].
Relying on a coherent and dissipationless light-matter interaction,
the IFE could thereby provide a path to ultrafast manipulation of
magnetization (that avoids heating up the magnetic material) on
very short timescales predicted to be fundamentally limited only by
the pulse length of the excitation [14–16]. From a technological
point of view, this would be of particular interest, as the
repetition rate of the all-optical switching process is highly
limited by the necessity of the system to cool down between
subsequent laser shots [17]. Thus, a non-thermal control of the
magnetic order could provide a way of achieving a significantly
faster magnetization reversal and much higher switching
rates.
While the IFE was originally discovered in transparent paramagnetic
materials and found to be strong in nearly non-absorbing systems
which exhibit a large spin-orbit coupling like garnets [14] and
orthoferrites [17, 18], it was also proposed to explain the
helicity-dependent sign reversal of the magnetization during the
switching phenomena observed in metallic ferrimagnets like GdFeCo
[8]. However, in case of garnets and orthoferrites, the photon
energy of the visible light which was used to trigger the IFE is
typically below the band gap of these systems, inhibiting
electronic excitation and
2
heating due to the absorption of photons. Therefore, very high
fluences can be used to generate a strong IFE-induced magnetization
in such systems without heat-induced effects quenching the magnetic
order. In contrast, the electronic and phononic heating is much
more significant in highly absorbing materials like metals, leading
to strong thermal effects which overlay any IFE-induced
magnetization, making the observation of a non-thermal effect like
the IFE in metallic systems particularly difficult [19]. Because of
this reason, no experimental reports on time-resolved studies are
available so far, by which one can systematically distinguish
between the influence of thermal (ultrafast heating) vs.
non-thermal (IFE) effects in metals on the femtosecond timescale of
ultrafast demagnetization. Furthermore, it was later shown that
also a purely thermal excitation with linearly polarized light can
lead to all-optical switching in ferrimagnets, relying solely on
the intrinsic ferrimagnetic properties like, e.g.,
antiferromagnetically coupled sublattices and the existence of a
magnetization compensation temperature [20–24]. Thus, the influence
of a non-thermal process like the IFE on the observed
demagnetization and switching phenomena remained unclear so
far.
The aforementioned effects are related to the excitation stage of
the magnetization dynamics while the laser pulse is interacting
with the material. Another very important issue in ultrafast
magnetism concerns the microscopic mechanisms leading to a loss or
reversal of the magnetic order, which has to involve an angular
momentum transfer on a subpicosecond timescale [4, 25]. As the
magnetic moment is fundamentally connected to the angular momentum
of the electrons, which is a conserved quantity contained in both
their spin and orbital moments, any change in magnetization
requires a transfer of angular momentum out of, or into the
electronic system. This transfer can either be a local angular
momentum transfer from electron spins to another reservoir (e.g.,
orbital moment, atomic lattice or electrons of other atoms) [26–29]
or a selective transport of spin-polarized electrons [30–32].
However, despite the existence of various theoretical models trying
to explain the angular momentum transfer, the actual process is
still highly debated and the experimental studies available so far
do not provide definitive answers on the path of ultrafast angular
momentum transfer during demagnetization and switching events
[4].
The scope of this thesis is to shed light upon both of the
aforementioned issues, i.e., the path of angular momentum transfer
after femtosecond optical excitation, as well as the influence of
an opto-magnetic effect like the IFE on timescales of ultrafast
demagnetization in a metallic ferrimagnet. Therefore we will employ
different pump-probe techniques which allow us to monitor the
transient behavior of the magnetic moments in the studied sample
systems of ferrimagnetic GdFeCo after excitation. For probing the
magnetization state, we will utilize magneto-optical effects like
the Faraday and Kerr effect (MOFE, MOKE) [33, 34] in the visible
light regime as well as the x-ray magnetic circular dichroism
(XMCD) [35] for element-selective studies. An introduction to the
intrinsic magnetic properties of the studied ferrimagnetic GdFeCo
alloys and to the ultrafast magnetization
3
1 Introduction
phenomena that can be observed in such systems will be given in
Chapter 2, which also explains the magneto-optical and
opto-magnetic effects employed for probing and excitation.
Furthermore, in this chapter the femtosecond soft x-ray and XUV
radiation sources at which the experiments presented in this thesis
were carried out are described.
Chapter 3 will present element-selective and time-resolved XMCD
studies at the L3,2
and M5,4 absorption edges of Fe and Gd, respectively, in a
ferrimagnetic GdFeCo alloy. Employing a pump-probe technique allows
us to monitor the transient changes of the element-specific
magnetic moments of Fe and Gd after excitation with laser pulses of
800 nm wavelength and linear polarization. A magneto-optical sum
rules analysis [36, 37] of the time-resolved XMCD data is employed
to disentangle the individual contributions of spin and orbital
moments at each atomic site during the ultrafast demagnetization
process; an analysis which will enable us to draw conclusions on
the path of angular momentum transfer in such systems. The
experiments presented in this chapter were carried out at the
FemtoSpeX fs-slicing facility at the BESSY II synchrotron light
source, providing 100 fs short, circularly polarized soft x-ray
pulses needed for time-resolved and element-specific XMCD studies
[38]. In Chapter 4, the influence of an opto-magnetic effect like
the IFE in ferrimagnetic
GdFeCo is studied by using a novel approach of inducing
helicity-dependent ultrafast demagnetization dynamics by resonantly
exciting electrons from the 3p core level states of Fe,
corresponding to the Fe M3,2 resonance. This approach employs
highly intense, 90 fs XUV pulses with both linear and circular
polarization for excitation, which were generated at the free
electron laser FERMI [39]. In contrast to previous studies on the
IFE excited by visible light, the resonant XUV excitation could
provide access to a regime where the opto-magnetic effect
potentially gets larger due to the much stronger spin-orbit
coupling of the core levels, allowing the influence of a
non-thermal IFE to be distinguished from the thermally induced
demagnetization. To quantify the magnitude of a helicity-dependent
effect on the magnetization, the ultrafast demagnetization process
is studied as a function of polarization state and photon energy of
the XUV excitation pulse. The wavelength- and helicity-dependent
dynamics are probed by light pulses in the visible wavelength
regime, utilizing the magneto-optical Faraday and Kerr effects. As
no theoretical and experimental studies are available so far
regarding the existence of an IFE in the XUV spectral range, the
experiment presented in this thesis is the first study aiming at
the observation of an IFE in this wavelength regime.
4
2 Fundamentals and experimental techniques
This chapter serves as an introduction to the fundamentals and
experimental techniques employed in the time-resolved studies
presented in the subsequent parts of this thesis. As all
experiments were carried out on ferrimagnetic GdFeCo alloys, the
first two sections will give an overview over their intrinsic
magnetic properties and the ultrafast magnetization dynamics that
can be observed in such systems. Afterwards, the physical
principles behind the magneto-optical and opto-magnetic effects
used for probing and inducing magnetization by light will be
explained. Finally, the instrumentation used for generating
femtosecond soft x-ray and XUV pulses at large-scale facilities
will be described.
2.1 Magnetic properties of rare earth and 3d transition metal
systems
The magnetic samples studied in this work are ferrimagnetic alloys
of GdFeCo, consisting of the rare-earth (RE) element Gd and the 3d
transition metals (TMs) Fe and Co. In their elemental form, both
types of materials (RE and 3d-TM) undergo ferromagnetic ordering,
but due to different origin. While the RE element Gd is a classical
Heisenberg-ferromagnet due to its well-localized 4f magnetic
moments (binding energy ≈ 8 eV below the Fermi level [40]), Fe and
Co belong to the group of itinerant band-ferromagnets described by
the Stoner model. In an alloy, they couple antiferromagnetically to
each other and exhibit ferrimagnetic properties like a
magnetization compensation temperature. Furthermore, thin films of
GdFeCo can possess perpendicular magnetic anisotropy, leading to an
out- of-plane orientation of the magnetic moments. Such properties
can thereby be tuned by the composition and stoichiometry of the
layer [41, 42]. In the first section of this chapter, we will thus
briefly explain the underlying interactions leading to ferro- or
antiferromagnetic ordering and magnetic anisotropy. In the second
section, the individual magnetic properties and interactions in RE
and 3d-TM elements and the studied GdFeCo alloys will be described.
As a thorough description of the underlying physics and theories
can be found in Ref. [40, 43, 44], this chapter will focus more on
a qualitative description of the physical principles which are
referred to in the later chapters. Unless mentioned otherwise, all
formulas and values are taken from Ref. [40].
5
2.1.1 Magnetic interactions
Exchange interaction
The mechanism responsible for the long-range magnetic ordering in
solids is the micro- scopic exchange interaction. In general, there
exist different types of exchange mechanisms, which can be divided
into direct and indirect interactions. While the first describe a
direct coupling between the electronic spins of two neighboring
atoms due to a spatial overlap of their electronic wave functions,
the latter describe indirect mechanisms involv- ing an intermediary
site (e.g., double-exchange, superexchange, Dzyaloshinskii–Moriya
interaction) or between strongly localized states mediated by
conduction band electrons (RKKY interaction) [40]. Regarding the
materials studied in this thesis, i.e., ferrimagnetic GdFeCo
alloys, the only relevant mechanisms are the direct exchange
interaction (see below) and an RKKY-type coupling present in the RE
element Gd (see Chapter 2.1.2). The direct exchange is thereby
based on an interplay of Coulomb repulsion forces between the
electrons and the Pauli exclusion principle which states that the
total electronic wave function has to be antisymmetric, i.e.,
electrons with the same spin state have to be in different orbitals
[40]. When the valence orbitals of adjacent atoms overlap, the
electrons try to minimize their ground state energy which can in a
simple Hubbard-like picture be seen as an interplay of Coulomb
forces (repulsion of electrons in the same orbital) and their
orbital degrees of freedom (inter-atomic hopping). In the classical
Heisenberg model of fully localized spins, the direct exchange
interaction can be described by an effective Hamiltonian, coupling
the electronic spins Si,j of neighboring atoms (i,j):
Heff = − ∑ i,j
JijSi · Sj , (2.1)
where Jij is the exchange coupling constant. Depending on the
electronic occupation of the overlapping orbitals and the bonding
between different ligands, this can lead to the electrons either
favoring a parallel (Jij > 0) or antiparallel (Jij < 0) spin
alignment in the ground state, which gives rise to ferro- and
antiferromagnetism. The exchange energy is thereby defined as the
energetic difference between the states with different spin
alignment.
Spin-orbit coupling
The spin-orbit coupling (SOC) is a relativistic microscopic
interaction that couples the spin and orbital parts of the
electronic wave functions. It can be understood as an interaction
between the spin of an electron and its orbital motion within the
electrostatic Coulomb potential Φ(r) = Ze/4πε0r of the positively
charged nucleus (q = +Ze). The SOC thereby describes the coupling
between spin (s) and orbital (l) angular momentum vectors to a
total angular momentum j = l + s. In a semi-classical picture, it
can be understood by treating the orbital motion around the nucleus
as a current loop with
6
2.1 Magnetic properties of rare earth and 3d transition metal
systems
radius r, generating a magnetic field that acts on the spin moment
of the electron. In a quantum-mechanical description, the SOC
couples electronic states with spin and orbital quantum numbers s
and l to new eigenstates with total quantum numbers j = l ± s and
lifts the degeneracy of the corresponding energy levels. For a
single electron (j = l± 1/2), the SOC energy can be described by
the following Hamiltonian, scaling with the radial gradient ∇Φ =
(r/r) dΦ(r)/dr of the nuclear Coulomb potential:
Hso = − e~2
8πε0m2 ec
2r3 (2.2)
where the expectation value ζnl = ξnl(r) is the so called SOC
constant, corresponding to the energetic splitting of the two j
states in an orbital with quantum numbers n,l, which also scales
with the atomic number and thus gets larger for heavy elements with
large Z values. The total magnetic moment mj of the electron is
thereby given by both spin and orbital angular momentum:
mj = µB ~
(gels + l), (2.3)
where gel ≈ 2 is the electronic g-factor. For light atoms (Z ≤ 30)
consisting of multiple electrons i, the same formalism can be
applied to the total spin and orbital moments S =
∑ si and L =
∑ li of the atom (L-S or Russel-Saunders coupling,
respectively).
The SOC gives rise to many different types of effects as it
mediates the interaction of electronic spins with the atomic
lattice and the angular momentum of light. Exemplary consequences
are the magneto-crystalline anisotropy and the existence of
magneto-optical and opto-magnetic effects (see Chapters 2.3 and
2.4).
Magnetic anisotropy
The magnetic anisotropy (MA) describes the tendency of the total
magnetic moment to align along a certain crystallographic axis of a
magnetically ordered system, which is then called the “easy axis”.
A correlated quantity is the magnetic anisotropy energy (MAE),
which is the energy that has to be spent in order to tilt the
magnetic moment from the easy axis to the perpendicularly oriented
hard axis. In magnetic materials, there is usually a competition
between shape, magneto-crystalline and surface/interface anisotropy
[40, 45]. The latter becomes especially relevant in magnetic thin
films and multilayer structures consisting of them, where surface
or interface effects can dominate over the 3D bulk properties of a
magnetic crystal. Depending on the strength of each term, the
magnetic layer can exhibit an overall in-plane or out-of-plane
magnetic anisotropy. The shape anisotropy is an externally
imprinted property given by the macroscopic
shape of the magnetic material and describes the urge of the system
to lower its ground state energy by reducing the magnetic stray
field. This can easily be understood in terms of a long rod magnet
which always wants to magnetize along its longitudinal axis because
a perpendicular alignment would lead to much higher stray fields
and is thus the
7
2 Fundamentals and experimental techniques
energetically less favorable configuration. In thin films, the
shape anisotropy therefore tends to align the magnetic moments
parallel with respect to the surface.
The magneto-crystalline anisotropy, on the other hand, is of
microscopic and quantum- mechanical origin. Given its complexity,
only a qualitative description based on the Bruno model will be
presented here [46, 47]. Due to the crystal field potential of the
neighboring atoms in an atomic lattice, certain directions of the
orbital motion of the electrons can be effectively suppressed. In
conjunction with the spin-orbit coupling, which aligns the spins
parallel or antiparallel with respect to the orbital moment, a
preferential direction of the total magnetic moment can arise. The
magneto-crystalline anisotropy thereby scales with the spin-orbit
coupling strength and its direction depends on the individual
composition and orbitals of the involved atoms in the
lattice.
The surface or interface anisotropy arises due to the symmetry
breaking of the crystal- lographic structure at the surface of a
magnetic layer or at the interface between different layers. While
in the plane of a magnetic thin film, the environment of each atom
and thus the overlap of the orbitals of neighboring atoms is
similar to a 3D bulk crystal, it is very different in out-of-plane
direction at a surface or interface, where there are either no
neighboring atoms on one side or they are of different elements
with differing electronic structures. Thus, the surface or
interface anisotropy can prefer an orientation of the magnetic
moments which is different compared to the bulk material. With
decreasing layer thicknesses, the surface/interface contributions
get larger compared to the bulk term. Well established systems
where the surface/interface anisotropy dominates are Co/Pt or Co/Pd
multilayers that forms perpendicular magnetic anisotropy when the
Co layers deceed a certain thickness, typically on the order of ≈
1.2 nm or less [45].
2.1.2 Magnetic order in RE and 3d-TM elements
Heisenberg magnetism and intra-atomic 4f5d-exchange in RE
elements
In case of the rare-earth elements (e.g., Gd, Dy, Tb, Ho), the
magnetic moment is mainly given by the partially filled and
strongly localized 4f electrons, while the valence electrons in the
outer more band-like 5d, 6s and 6p states only yield a minor
contribution [40]. Therefore, the magnetic exchange interaction can
be treated in the classical Heisenberg- model by the effective
Hamiltonian shown in Eq. 2.1. All RE elements are paramagnetic at
room temperature and above, and show either antiferromagnetic or
ferromagnetic ordering below a certain temperature. Gd (Z = 64) is
a special case since it is the only RE element that has a
ferromagnetic phase up to TC ≈ 289 K (close to room temperature)
and an electronic configuration of [Xe]4f75d16s2, i.e., a
half-filled 4f shell. Following Hund’s rules, the ground state is
thus given by the spin and orbital quantum numbers S = 7/2 and L =
0, thus its local 4f magnetic moment is exactlymJ = 7µB/atom and
arises completely due to spin angular momentum, while the orbital
moment is zero. An additional yet small contribution arises from
the itinerant magnetism of the 5d valence band (see below),
8
2.1 Magnetic properties of rare earth and 3d transition metal
systems
which for Gd is on the order of ≈ 0.6µB, thus less than 10 % of the
total magnetic moment. An important consequence of the strong
localization of the 4f orbitals is their vanishing overlap with
neighboring atoms. Thus, the direct exchange interaction between
the 4f electrons is very small, leading to strong, atomic-like
local moments but cannot explain the long-range ferromagnetic
ordering. The reason why Gd still enters a ferromagnetically
ordered phase is an indirect exchange mechanism based on a
RKKY-type coupling [40, 48]. A very strong intra-atomic direct
exchange between the spatially overlapping 4f and 5d orbitals
(J4f5d ≈ 100–130 meV [4, 49]) leads to a spin polarization of the
delocalized 5d electrons. The latter then mediates an inter-atomic
exchange between the neighboring atoms. The strong 4f-5d coupling
also plays an important role in studies of ultrafast laser-induced
demagnetization and switching phenomena as light in the visible
wavelength regime can only excite electrons within the 5d band and
not directly from the tightly bound 4f states which carry the
largest part of the magnetic moment. However, it was shown
experimentally in Gd and other RE elements that an optically
excited quenching of the 5d magnetic moment leads to a respective
change in the 4f magnetic moment, matching both relative
demagnetization magnitudes and times [50, 51]. The intra-atomic
coupling was thus shown to be quasi-instantaneous, on timescales
much faster than the experimental time resolution and predicted to
be as fast as given by the uncertainty relation (~/J4f5d ≤ 10
fs).
Itinerant (delocalized) ferromagnetism in 3d-TMs
In case of the 3d transition metals Fe, Co and Ni, the more than
half-filled 3d shell leads to the formation of a strong
ferromagnetic coupling (TC ≈ 600–1400 K) due to a spin polarization
of the partially delocalized valence electrons of the 3d band [40].
The partial delocalization thereby leads to a strong overlap and
exchange interaction with the 3d orbitals of the neighboring atoms.
For that reason, Fe, Co and Ni are called itinerant or delocalized
band-ferromagnets. As the Heisenberg model of localized spins
cannot describe the itinerant magnetism anymore, the Stoner model
of metallic ferromagnetism has been introduced. In this model, the
ferromagnetic ordering leads to a spin polarization of the 3d
valence band, shifting the density of states for spin up and down
electrons with respect to the Fermi level. Thus, the relative
occupation of spin up vs. down states is different, leading to the
formation of a majority and minority spin band. The magnitude of
the magnetic moment m and the Stoner splitting energy is thereby
given by the difference between the occupation number of electrons
in the majority (Nmaj
e ) and minority (Nmin e )
spin band, which in case of the band model can have non-integer
values:
|m| = µB(Nmaj e −Nmin
e ), = 2mHex, (2.4)
where Hex is an effective field describing the exchange
interaction. In case of Fe, the resulting total magnetic moment is
on the order ≈ 2.2µB/atom (bcc Fe), given mainly by its spin moment
with an orbital contribution of only ≈ 0.1µB.
9
Gd
Fe
TC0
|M|
TTcomp
0
Figure 2.1: Illustration of the typical temperature-dependence of
Gd and Fe sublattice magneti- zations (red and blue curves) in
ferrimagnetic GdFe leading to the formation of a magnetization
compensation temperature Tcomp. For temperatures below Tcomp, the
magnetic moment of the Gd sublattice (red arrows) dominates and
determines the direction of the net magnetization (green arrows).
Exactly at Tcomp, both sublattice magnetizations are equal and
cancel each other out (zero net magnetization). Above Tcomp, the
magnetic moment of the Fe sublattice becomes dominant, thus the net
magnetization reverses its sign. When the Curie temperature TC is
approached, the magnetic order disappears and both sublattices
enter a paramagnetic phase.
Coupling between RE elements and 3d-TMs
Alloys of RE elements and 3d-TMs can be treated as two magnetic
sublattices of the corresponding elements exhibiting either ferro-
or antiferromagnetic coupling between their total magnetic moments.
As described in Ref. [52], the coupling of the spin system is
thereby determined by the density of states around the Fermi level
and the hybridization of the 5d and 3d valence bands of the RE and
TM, respectively. Due to the strong exchange shift of the
ferromagnetically polarized 3d band, the 5d band of the RE element
energetically overlaps mainly with the minority spin band of the
TM, which results in a much stronger hybridization of 5d electrons
with the 3d minority spin electrons compared to their majority spin
counterparts. As a result, the 5d electrons will always polarize
according to the minority spin of the TM and thus show an
antiferromagnetic spin polarization. Due to the strong intra-atomic
exchange with the 4f electrons, the latter will polarize
accordingly. The type of coupling between the total magnetic
moments is then determined by the polarity of the spin-orbit
coupling in the RE element:
In alloys with light RE elements (less than half-filled 4f shell),
the spin-orbit coupling of the RE is negative (L−S) and its orbital
moment exceeds the spin moment. Thus, the total magnetic moments of
RE and TM exhibit a ferromagnetic coupling. In alloys with heavy RE
elements (half- or further filled 4f shell), the spin-orbit
coupling of the RE is positive (L+S). Thus, the total magnetic
moments will couple antiferromagnetically. Due to their different
amount of magnetic moment per atom, an antiferromagnetic coupling
between the sublattices usually does not lead to a compensation of
their magnetic moments, which means the alloy is a ferrimagnet and
possesses a non-zero net magnetization.
10
2.1 Magnetic properties of rare earth and 3d transition metal
systems
The exchange between RE element and 3d-TM further leads to a common
Curie temperature TC, which was theoretically derived and
experimentally confirmed by the authors of Ref. [41] and shown to
depend on the RE concentration in the alloy. However, due to the
different exchange interactions in the RE and 3d-TM sublattices,
their magnetic order and thus magnetization scales differently with
the temperature. In particular, the power law describing the M(T )
behavior up to the Curie temperature TC where the magnetic order
disappears is not the same for both sublattices [41]. As
illustrated in Fig. 2.1 for the case of ferrimagnetic GdFe, this
can lead to the existence of a magnetization compensation
temperature Tcomp, where both sublattice magnetizations are equal
and the net magnetization vanishes. Below and above Tcomp, either
the Gd or Fe sublattice dominates and determines the direction of
the net magnetization. Thus, static heating across Tcomp leads to a
reversal of the net magnetization. If the heating is done under an
applied magnetic field, the net magnetization will always align
along the field direction, so the sublattice magnetizations will
reverse. As shown in the next chapter, the ferrimagnetic
compensation point was found to play a significant role with
respect to ultrafast magnetization dynamics, when excitation by a
laser pulse leads to ultrafast heating across the compensation
point.
Ferrimagnetic GdxFeCo1-x alloys
The amorphous GdFeCo alloys studied in this thesis consist of the
RE sublattice Gd coupled ferrimagnetically to a 3d-TM sublattice of
Fe and Co. As the latter two exhibit a strong ferromagnetic
exchange coupling to each other and show the same temperature
dependent behavior up to TC, they can be treated as a single
sublattice of FeCo [41]. The total magnetic moment of the Gd
sublattice is thereby provided mostly by the spin moment of the 4f
electrons (≈ 7µB/atom) due to the half-filled 4f shell which has
zero orbital angular momentum. In case of FeCo, the total magnetic
moment is provided by both the spin and orbital angular momentum of
the 3d electrons, although its main contribution (≈ 2µB/atom) is
given by the spin moment, as the orbital moment is much smaller and
quenched by the crystal field potential of the lattice. The
critical temperatures (TC, Tcomp) and the magnetic anisotropy of
thin GdFeCo films can be tuned by varying the composition and
thickness of the magnetic layer [45]. Systematic studies on the Gd
concentration influencing the Curie and compensation temperatures
of 30 nm thin Gdx(FeCo)1−x layers can be found in Ref. [41],
leading to a common Curie temperature of TC ≈ 550 K and a
ferrimagnetic compensation point of Tcomp ≈ 60–350 K for Gd
contents of x = 23.4–29%. Respective studies on the magnetic
anisotropy, which is also temperature dependent and leads to the
layer magnetizing either in- or out-of-plane, were carried out in
Ref. [42]. The latter showed out-of-plane magnetic anisotropy at
room temperature in the low Gd contents regime of x = 20–34 %
(close to an existing compensation point) and also for higher Gd
concentrations between 52–59 %, where the Gd magnetic moments
dominate over the whole temperature range.
11
2.2 Ultrafast laser-induced magnetization dynamics in RE and 3d-TM
systems
†Since it was discovered in 1996 that ferromagnetic Ni can be
demagnetized on a sub- picosecond timescale by a femtosecond laser
pulse excitation [5], the investigation of ultrafast magnetization
dynamics has become an intense field of research.† At that time,
this was a very surprising result, as the laser-induced
demagnetization was so far un- derstood in terms of a sudden
lattice temperature increase due to the laser absorption, followed
by a spin-lattice relaxation until a new equilibrium magnetization
is reached; a process which usually happens on timescales of ≈ 100
ps in case of Fe and Gd [6, 7]. This led to the question, how the
spin moment of an electron can be reduced so fast and which
microscopic mechanisms allow a transfer of angular momentum out of
the spin system on a femtosecond timescale. More recent studies on
ferrimagnetic rare earth and 3d transition metal systems like,
e.g., GdFe alloys have shown that it is even possible to
permanently switch magnetization (i.e., rotation/reversal of the
magnetization vector by 180 ) on ultrashort timescales by a single
optical laser pulse without any other external stimulus [8, 22,
23]. This type of all-optical magnetization switching was first
discovered in ferrimagnetic GdFeCo alloys and could be related to
the intrinsic properties of the ferrimagnetically coupled RE and
3d-TM elements described in the previous section, and how they
react to a femtosecond laser excitation. Therefore, revealing the
microscopic processes leading to the distinct dynamics observed in
RE-TM alloys and multilayers gained large amount of interest in the
last years and is the motivation behind the choice of ferrimagnetic
GdFeCo for the studies presented in this thesis. In the following,
we will thus describe the so far proposed mechanisms which allow an
optical laser pulse to influence the magnetization state of a
system on femtosecond timescales. At the end of this section, we
will also give an overview over the all-optical magnetization
switching dynamics that could be observed in RE-TM systems.
2.2.1 Mechanisms behind ultrafast magnetization dynamics
As the total magnetic moment is given by the spin and orbital
moments of the electrons, any change in magnetization is
fundamentally connected to an ultrafast transfer of angular
momentum out of or into these moments; a process which has to be
induced by the femtosecond laser excitation. It is thereby
important to distinguish between different types of mechanisms:
Local processes describe an actual change of the local magnetic
moment, which due
to the conservation of angular momentum is connected to its
transfer between different reservoirs like the spin and orbital
moments, electrons of different atoms or the atomic lattice. Such
processes are the underlying mechanisms of most ultrafast
magnetization phenomena, including the laser-induced switching
observed in ferrimagnetic RE-TM
12
2.2 Ultrafast laser-induced magnetization dynamics in RE and 3d-TM
systems
systems [2]. Typical processes leading to such a local dissipation
of angular momentum are Elliot-Yafet type spin-flip scattering
events (see Chapter 2.2.2). Non-local processes, on the other hand,
describe a non-local change of magnetic moment
where the laser-excitation of the electronic system leads to the
generation of spin-polarized currents which transport the magnetic
moment out of the probed volume or to another atomic site. Typical
mechanisms leading to such a non-local transport of magnetic moment
are super-diffusive spin transport [30, 31] and the recently
discovered optical inter-site spin transfer [32, 53, 54] (see
Chapter 2.2.3). The ultrafast demagnetization phenomena studied in
the Chapters 3 and 4 are local
processes of both thermal and non-thermal origin. While
opto-magnetic effects leading to a non-thermal change of
magnetization will be discussed in more detail in Chapter 2.4, the
following sections will give an overview over the different local
spin-flip scattering and non- local transport mechanisms that were
proposed to explain the ultrafast demagnetization process after
thermal excitation.
2.2.2 Local spin-flip scattering processes
Local effects describe a change of magnetization due to a local
dissipation of angular momentum following the excitation of
electrons via the absorption of a femtosecond laser pulse. As
optical dipole transitions are forbidden to directly change the
spin state of an electron due to the dipole selection rules, the
ultrafast demagnetization can be understood in terms of the excited
electrons undergoing spin-flip scattering events which ultimately
lead to a loss of magnetic order. The underlying mechanisms behind
such scattering events which are accompanied by a certain spin-flip
probability include Elliott-Yafet type electron-phonon and
electron-impurity/defects scattering [55, 56] as well as scattering
of an excited electron with other electrons or quasi-particles like
magnons. This can be understood as follows: In presence of
spin-orbit coupling, the spin itself is not a good quantum number
anymore, thus an electron does not have a pure spin eigenstate
|Ψ↑,↓, but a mixed spin state with majority (a↑,↓) and minority
(b↑,↓) spin components [4]:
|Ψ↑ = a↑|↑+ b↑|↓
|Ψ↓ = a↓|↓+ b↓|↑ (2.5)
Such mixed spin states, the so called “spin hotspots”, were
predicted to appear as an avoided crossing between spin up and down
bands in the band structure of a ferromagnet and have been indeed
observed in photoemission studies (see, e.g., Ref. [57]). The spin-
mixing was shown to open up a channel for transitions into states
with a changed dominant (majority) spin component, allowing a
scattering event to effectively flip the spin of an electron. The
spin-flip probability thereby depends on the minority spin
coefficient, corresponding to the strength of spin mixing.
13
Electron-phonon scattering
The first microscopic model explaining the ultrafast
demagnetization on the basis of an Elliott-Yafet type mechanism
included phonon-mediated spin-flip scattering events [26]. The
theory is based on a Hamiltonian describing the interactions
between the three subsystems of electrons, spins and phonons, and
is the basis for the microscopic three- temperature model (M3TM) as
explained later. Despite the classical electron-electron and
electron-phonon equilibration terms, a further term is added which
accounts for the Elliott-Yafet spin-flip probability during
electron-phonon scattering events, leading to a transfer of angular
momentum from the electrons to the lattice. It was shown that this
model allows to obtain demagnetization times which are much faster
than the electron-phonon thermalization processes. However,
ab-initio calculations carried out by the authors of Ref. [58–60]
showed that, although there is a significant contribution from
phonon-mediated spin-flips, the induced demagnetization rate is too
small in order to explain the observed dynamics, thus further
mechanisms have to be taken into account.
Electron-electron scattering
The contribution of an electron-electron interaction was proposed
in Ref. [28] by describing an inelastic Coulomb scattering process
involving the optically excited majority and minority electrons of
a band-ferromagnet. In presence of spin-orbit coupling, interband
scattering events change the spin-mixture and lead to a
redistribution of electrons from majority to minority bands, which
consequently reduces the magnetization. It was shown that
simulations carried out using this model lead to a good agreement
with the ultrafast demagnetization effects observed in
ferromagnetic Co and Ni.
In an extended model developed by the authors of Ref. [61], the
influence of electron- electron vs. Elliott-Yafet type
electron-phonon scattering was studied. It was proposed that a
combination of both mechanisms needs to be considered in order to
fully describe the ultrafast dynamics in Ni, as they act
differently on demagnetization and remag- netization timescales,
while the phonon-mediated scattering also leads to an increased
demagnetization efficiency.
Electron-magnon scattering
Another proposed mechanism is electron-magnon scattering. In this
model, excited electrons undergo electron-electron interactions
leading to the excitation of collective spin modes and thus
emission of a magnon [27]. This is mediated by the spin-orbit
coupling, which allows the corresponding transfer of angular
momentum out of the electronic spin system into the orbital moment
and eventually to the lattice that acts as a final sink of angular
momentum. However, experiments have shown that there is no
indication for an increase of orbital angular momentum on the so
far experimentally accessible timescales (see, e.g., Ref. [25, 62]
and also Chapter 3). Therefore, a second and much faster
process
14
2.2 Ultrafast laser-induced magnetization dynamics in RE and 3d-TM
systems
was assumed, leading to a rapid quench of the orbital moment by the
crystal field, which efficiently transfers the angular momentum to
the lattice. While the latter would be in line with another recent
theory, proposing a dissipation of angular momentum to the lattice
within just ≈ 1 fs [29], calculations carried out by the authors of
Ref. [63] have shown that the spin-flips caused by electron-magnon
scattering do not lead to sufficient demagnetization rates and only
play a minor role. Instead, a combined mechanism was proposed,
taking also electron-phonon scattering into account [60, 63].
Three-temperature model (3TM)
The three-temperature model (3TM) was originally proposed in 1996
[5] and is a phe- nomenological model describing the
demagnetization process by assigning temperatures to the three
subsystems of electrons, spins and lattice. The excitation by the
laser is regarded as an ultrafast increase of the electronic
temperature, which subsequently ther- malizes with the other baths
coupled via rate equations. The 3TM thereby allows only a
phenomenological description of the process by fitting the transfer
rates between the heat baths to the experimentally obtained data.
As such, no insights into the microscopic mechanisms or angular
momentum transfer can be provided. A simplified version of the 3TM
is the two-temperature model (2TM), which treats only the
electronic and phononic bath independently and is used in Chapter 3
to estimate the electron-lattice thermalization time.
The model was later extended to the microscopic three-temperature
model (M3TM) by including Elliott-Yafet type phonon-mediated spin
flips, considering also angular momen- tum conservation by allowing
its transfer from the electrons to the lattice [64]. Although the
M3TM can be used to successfully reproduce experimental data, it
usually fails to predict the actual dynamics [4]. Reasons are that,
as mentioned before, phonon-mediated spin flip scattering was found
to be not sufficient to fully explain the demagnetization process,
and that it assumes a full internal equilibrium of each heat bath
by assigning it a certain temperature. It was shown, however, that
the largest spin-flip rate arises from nonequilibrium electrons
before they are thermalized [58].
2.2.3 Non-local spin transport processes
In contrast to the local effects described in the previous section,
non-local effects do not rely on a process changing the spin state
of an electron but on a transport mechanism leading to a reduced
net magnetic moment due to laser-excited spin-polarized currents
which transport mainly the majority spin electrons out of the
probed volume or atomic site [31, 32]. Such effects were thus
proposed to explain ultrafast demagnetization and switching
phenomena without requiring the existence of a local (on-site)
angular momentum dissipation channel and are briefly introduced in
the following.
15
Super-diffusive spin transport
Super-diffusive spin transport describes a non-local change of
magnetic moment in conductive materials due to a diffusive motion
of mostly majority spin electrons out of the probed volume [31]. It
was experimentally observed for the first time in ultrafast
demagnetization studies of antiferromagnetically vs.
ferromagnetically coupled Co/Pt multilayers which were separated by
a spacer layer of either insulating NiO or metallic Ru [30]. It
could be shown that using a conducting spacer layer leads to a
faster and more efficient demagnetization of the antiferromagnetic
phase of the Co/Pt multilayer, which was explained by a
spin-polarized current leading to a transport of majority spin
electrons from Co to Pt and vice versa. A semi-classical model
describing such spin transport processes in a 3d band-ferromagnet
was proposed by the authors of Ref. [31]. It relies on a
spin-polarized diffusion of electrons due to different electronic
scattering probabilities (which are treated as spin-conserving in
this model) and thus mobility of the excited majority vs. minority
spin electrons. The larger mean free path and higher velocity of
the majority spin electrons thereby lead to a super-diffusive
motion of majority carriers out of the probed volume, reducing the
net magnetic moment in this region.
A specific type of super-diffusive spin transport has also been
observed in amorphous alloys of ferrimagnetic GdFeCo, which can
occur due to the amorphous state and thus not necessarily
homogeneous elemental distribution in the sample. This ultrafast
spin transport was shown to happen laterally (i.e., in the sample
plane) between nanometer scale regions with different Gd and Fe
concentrations (e.g., from Gd-rich towards Fe- rich regions) [65].
However, such non-local spin transport or its observation is
usually suppressed when the magnetic layer is grown on an
insulating substrate and probed over its whole thickness and a
sufficiently large area, preventing spin-polarized currents between
different layers and suppressing the observation of both
longitudinal (i.e., along the sample normal) and lateral spin
diffusion which averages out due to the macroscopic probing volume.
This was the case in the experiments presented in Chapters 3 and
4.
Optical inter-site spin transfer (OISTR)
Another recently discovered mechanism is optical inter-site spin
transfer (OISTR), which describes an optically induced
spin-selective charge flow between different sublattices (A,B) of
multi-component magnetic systems [32, 53, 54]. Although it is a
short ranged interaction involving a charge transfer between
neighboring atoms, it is referred to as a non-local process in this
overview as the electronic spin state is not changed during this
process and it does not generate local (on-site) spin flips.
Instead, the process relies on an optical spin-conserving
excitation of electrons from occupied states of an atom in
sublattice A to unoccupied states of an adjacent atom in sublattice
B. Being an electronically coherent effect, it is expected to act
only during the presence of the laser excitation pulse and to occur
on timescales shorter than the intrinsic electronic lifetimes
16
2.2 Ultrafast laser-induced magnetization dynamics in RE and 3d-TM
systems
(given by inelastic electron-electron scattering times). As it was
shown exemplary for antiferromagnetically coupled Mn atoms by the
authors of Ref. [32], the OISTR process can lead to a direct
excitation of majority spin electrons from Mn site A to empty
minority spin states of Mn site B, leading to an extremely fast
demagnetization on timescales corresponding to the pulse duration
of the laser. Very recent theoretical and experimental studies have
shown that the OISTR process is also present in ferromagnetically
coupled CoPt alloys, playing a decisive role in the ultrafast
demagnetization dynamics observed in such systems [54]. However,
the presence of OISTR highly depends on the density of states
around the Fermi level and so far it was neither predicted nor
observed to happen in ferrimagnetic RE-TM systems.
2.2.4 All-optical magnetization switching in RE and 3d-TM
systems
Laser-induced all-optical magnetization switching was first
observed in amorphous ferri- magnetic GdFeCo alloys [8], employing
circularly polarized light pulses to permanently switch the
magnetization. The direction of switching, i.e., the final
magnetization state, was thereby found to be dependent on the
helicity of the light, with no external magnetic field required.
This type of switching was initially explained by a combination of
both thermal and non-thermal mechanisms: In a first step, the laser
pulse excites the electronic temperature up to slightly below TC,
leading to an ultrafast demagnetization. In a second step, the
circularly polarized light acts as a helicity-dependent effective
magnetic field due to the Inverse Faraday Effect (see also Chapter
2.4.1), which is antiparallel or parallel to the small remaining
magnetization and thus determines the direction of the subsequent
relaxation. A different and purely thermal mechanism was proposed
to be based on ultrafast
heating of the system across its ferrimagnetic compensation point
(Tcomp) in presence of an external magnetic field [20, 21].
Time-resolved XMCD studies showed that this type of thermally
induced switching is mediated by a transient ferromagneticlike
state, where the magnetic moment of the usually
antiferromagnetically coupled Gd and FeCo sublattices is aligned
parallel [22]. It was later shown that the external field is not
required and that a thermal excitation alone is sufficient to
switch the magnetization forth and back (so called “toggle
switching”) [23]. The magnetization reversal could be related to
the different demagnetization times of the two sublattices,
depending on their amount of magnetic moment per atom [22, 24]. A
multi-sublattice model was used to give a phenomenological
explanation for the switching process in terms of two thermal
regimes [66]: Directly after excitation, the electrons are heated
far beyond the Curie temperature (T TC), leading to a
temperature-dominated regime with negligible exchange coupling
between the sublattices. Consequently, they will demagnetize
independently on their intrinsically different timescales, with Gd
still demagnetizing when the Fe sublattice magnetization approaches
zero. In the mean time, the electronic temperature thermalizes with
its
17
2 Fundamentals and experimental techniques
environment, approaching a second regime of T < TC, where the
angular momentum exchange between the two sublattices dominates.
Any further spin-flip in Gd will then oppositely flip an Fe spin
due to their antiferromagnetic coupling. Subsequently, the
magnetization of the FeCo and later the Gd sublattices reverse
sign. It was shown in Ref. [67], that the vicinity to the
compensation temperature is thereby crucial for an efficient
magnetization reversal process, as it leads to an increased domain
wall mobility that favors the relaxation to a state with reversed
magnetization.
While all-optical magnetization switching was first discovered in
GdFeCo, such phenom- ena can also be observed in other
ferrimagnetic RE-TM alloys and multilayers like, e.g., TbFeCo, DyCo
and HoFeCo, exhibiting similar properties regarding
antiferromagnetically coupled sublattices and compensation
temperatures [68, 69]. Although several theoretical approaches and
phenomenological models tried to give
explanations for the microscopic mechanism behind the ultrafast
demagnetization and magnetization switching phenomena [66, 70, 71],
the actual path of angular momentum transfer mediating both
processes remains elusive so far [4]. X-ray scattering experiments
on thin films of GdFeCo alloy have revealed that there is also a
non-local contribution due to an angular momentum transfer mediated
by lateral spin currents on the nanometer length scale, which
further complicates the microscopic picture of all-optical
switching processes [65]. Furthermore, the influence of a
non-thermal effect like the Inverse Faraday Effect on the
magnetization dynamics is unclear, as the authors of Ref. [72] were
able to explain the helicity-dependent all-optical switching
process originally observed in Ref. [8] by a purely thermal effect,
relying on magnetic circular dichroism leading to a
helicity-dependent threshold fluence. Thus, further studies are
required to shed light upon the path of angular momentum transfer
during ultrafast demagnetization and switching events, as well as
to disentangle thermal from non-thermal contributions. Such studies
will be presented in the subsequent chapters, investigating the
angular momentum flow in GdFeCo during ultrafast demagnetization
(see Chapter 3) as well as non-thermal contributions due to an
opto-magnetic effect (see Chapter 4), respectively.
2.3 Magneto-optical effects (probing magnetization with
light)
One possibility to probe the magnetization of a system is its
interaction with light via magneto-optical effects that influence
either the polarization state or the magnitude (i.e., intensity) of
a light wave while it is being transmitted through or reflected by
a magnetic medium. In this thesis, both light from the visible (≈
380–780 nm [73]) and soft x-ray spectral range (≈ 250 eV up so
several keV [74]) was used to perform time-resolved measurements of
magnetization on a femtosecond timescale. Chapter 4 will employ the
magneto-optical Faraday and Kerr effect to probe the magnetization
of a ferrimagnetic GdFeCo system using linearly polarized visible
light pulses. While an advantage of using visible light for probing
is the flexibility and feasibility of the
18
2.3 Magneto-optical effects (probing magnetization with
light)
detection scheme as well as the high stability of femtosecond
optical laser systems, it lacks element-selectivity by probing only
the valence bands of the FeCo and Gd sublattices due to its low
photon energy. Therefore, Chapter 3 makes use of circularly
polarized soft x-ray pulses to probe element-specific magnetic
moments of Fe and Gd by doing core level spectroscopy, utilizing
the x-ray magnetic circular dichroism (XMCD). The latter technique
also allows to extract information about the elemental spin and
orbital angular momentum of the electrons by applying
magneto-optical sum rules [36, 37]. While the XMCD is mainly used
as a probing technique, it can also affect the magnetization when
employing highly intense and circularly polarized x-ray pulses;
such short and intense pulses could potentially trigger fast
helicity-dependent magnetization dynamics via the dichroic (i.e.,
x-ray helicity dependent) absorption of the magnetic sample. One
such example is described in Chapter 4 where resonant circularly
polarized XUV excitation is studied and thus the XMCD in the XUV
spectral range has to be taken into account.
In this section, we will briefly introduce both types of
magneto-optical effects employed for static characterization and
time-resolved experiments. As the underlying physics of these
effects are well known and understood, we will focus mainly on the
aspects which are important regarding their application in our
experiments and the ferrimagnetic GdFeCo alloys under
investigation.
2.3.1 Magneto-optical Faraday and Kerr effect (MOFE/MOKE)
The magneto-optical Faraday and Kerr effects (MOFE/MOKE),
originally discovered in 1845 [33] and 1877 [34], describe the
influence of magnetization on the polarization of linearly
polarized light when it is transmitted through (Faraday) or
reflected by (Kerr) a magnetized medium, leading to a rotation and
ellipticity of the polarization plane of the light wave scaling
linearly with the magnetization via magneto-optical constants.
Qualitatively, both magneto-optical effects can be described in the
following way [4]:
Due to linearly polarized light being a superposition of left- and
right-circularly polarized waves of equal amplitudes, the rotation
of the polarization can be understood in terms of a circular
birefringence caused by the magnetization of the system, resulting
in different refractive indices for left- and right-circularly
polarized light. This leads to different phase velocities of the
circularly polarized waves while traveling through the magnetic
medium and thus an accumulated phase shift between both components.
As a result, the polarization plane of the exit wave is rotated
with respect to the incident wave by the so called Faraday or Kerr
angle, while the magnitude and direction of rotation depends
linearly on the magnetization vector. In addition, an ellipticity
can be introduced due to different absorption of the left- and
right-circularly polarized components in the magnetized medium. The
underlying microscopic mechanism is of quantum mechanical origin,
due to the interaction of the two helicities of circularly
polarized light with the spin-orbit coupled and exchange split
states [75, 76].
19
VIS
D+
D-
Θ
F
L
B
magnetic layer
Wollaston prism
Figure 2.2: Illustration of an experiment employing the
magneto-optical Faraday effect as a contrast mechanism to probe the
magnetization state of a sample. When a linearly polarized, visible
light pulse (VIS) is transmitted through the magnetic layer, the
polarization plane of the pulse gets rotated by the Faraday angle
ΘF, depending on the direction of the applied magnetic field B and
the thickness L of the sample. A Wollaston prism splits the
transmitted wave into two linearly polarized components with
orthogonal polarization axes. Their difference and sum intensities
are then measured using a balanced detector consisting of two
photodiodes D+,−.
A quantitative description of the Faraday and Kerr effect can be
derived macroscopically via Maxwell’s and Fresnel equations, based
on the magnetization inducing antisymmetric off-diagonal components
in the dielectric tensor. This allows to derive expressions for the
complex Faraday and Kerr angles depending on the geometry between
the magnetization vector M and the propagation direction k of the
light [77]. In case of the Faraday effect in polar geometry (M k
along the z direction), this leads to the following complex
relation:
ΘF + iεF ≈ iπL λv
, (2.6)
where εxx, εxy are the diagonal and off-diagonal components of the
dielectric tensor, λv is the wavelength of the light in vacuum and
L the distance the light travels inside of the material, leading to
a rotation ΘF and ellipticity εF of the polarization. While the
latter definition is typically used for the magneto-optical Faraday
effect in ferromagnetic materials, a more general definition of the
Faraday effect in transparent materials subjected to an external
magnetic field B expresses the rotation via a material-specific
Verdet constant VF [78]:
ΘF + iεF = VFLB ≈ iπL λv
εBxy√ εxx
, (2.7)
where εBxy is now the perturbation of the off-diagonal term in the
dielectric tensor due to the magnetic field B. In time-resolved
experiments, the Faraday and Kerr effect can be used to probe
the
magnetization state of a magnetic layer by using linearly polarized
visible light pulses
20
2.3 Magneto-optical effects (probing magnetization with
light)
generated by a femtosecond laser system. Fig. 2.2 shows a schematic
illustration of an experiment probing the Faraday rotation of a
light pulse after transmission through a sample by employing a
polarization-sensitive detection scheme, consisting of a Wollaston
prism and a balanced photodetection. The prism separates the
transmitted wave into two linearly polarized components with
orthogonal polarization axes. Their difference and sum intensities
are then measured using a balanced detector consisting of two
photodiodes. Any change of magnetization and thus rotation of the
polarization plane can then be detected by a change of the
difference signal. Normalizing to the sum provides a quantity which
is proportional to the Faraday rotation and thus magnetization of
the magnetic layer, averaged over the magnetic moments of all
constituent elements.
2.3.2 X-ray/XUV magnetic circular dichroism (XMCD)
The magnetic circular dichroism in the (soft) x-ray or XUV spectral
range (XMCD) describes a helicity- and magnetization-dependent
absorption of circularly polarized x-rays or XUV radiation while
being transmitted through a magnetic medium. If the magnetic moment
is oriented (anti-)parallel to the k vector of the light wave, this
leads to a different absorption cross section of left- (σ−) and
right-circularly (σ+) polarized light. The existence of the XMCD
effect was theoretically predicted in 1975 [79] and experimentally
demonstrated for the first time in 1987 by hard x-ray spectroscopy
at the K absorption edge of iron [35]. In contrast to the
magneto-optical Faraday and Kerr effect in the visible light regime
as presented in the previous section, doing core-level spectroscopy
provides much larger dichroic effects and high element-selectivity
due to resonant excitation of well separated absorption edges of
the different elements [74]. Because a full description of the XMCD
effect can be found in Ref. [40, 80], the following paragraph will
only give a brief summary.
As the absorption of a photon requires the excitation of an
electron from an occupied state below the Fermi energy to an empty
state above, the absorption probability of the photon depends both
on the density of available states above the Fermi level and the
possible dipole transitions between initial and final states
allowed by the selection rules. As the latter forbid spin-flips
during optical transitions, the secondary spin quantum number ms of
an electron cannot be changed when it is promoted from initial to
final states, i.e., ms = 0. Additionally, the magnetic quantum
number ml of the orbital has to change depending on the helicity,
i.e., ml = ±1 for left- or right-circularly polarized radiation,
respectively. Due to the spin-orbit coupling of initial and final
states, this leads to an imbalance in the number of allowed
transitions from spin up or down states comparing left- and
right-circularly polarized excitation. Thus, one helicity compared
to the other possesses a larger probability of exciting spin up vs.
spin down electrons and vice-versa, resulting in a
helicity-dependent, spin-polarized excitation. In a non-magnetic or
fully demagnetized medium, where the number of available states
above the Fermi
21
2 Fundamentals and experimental techniques
level is equal for both spin up and down electrons, this does not