Ultrafast Dynamics of Magnetic Multilayer Films:
Magneto-Optical Spectroscopy and Resonant Scattering in
the Extreme Ultraviolet and Soft X-ray Spectral Regions
byDmitriy Zusin
B.S., Belarusian State University, 2009
M.S., University of Colorado Boulder, 2014

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Physics
2018This thesis entitled:
Ultrafast Dynamics of Magnetic Multilayer Films: Magneto-Optical Spectroscopy and Resonant
Scattering in the Extreme Ultraviolet and Soft X-ray Spectral Regions
written by Dmitriy Zusin
has been approved for the Department of Physics

Prof. Margaret M. Murnane

Prof. Henry C. Kapteyn

Date

The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

Zusin, Dmitriy (Ph.D., Physics)
Ultrafast Dynamics of Magnetic Multilayer Films: Magneto-Optical Spectroscopy and Resonant
Scattering in the Extreme Ultraviolet and Soft X-ray Spectral Regions
Thesis directed by Prof. Margaret M. Murnane and Prof. Henry C. Kapteyn

Magnetism has been known for millennia. Yet, even today, our understanding of the interplayof various interactions and processes in a material that give rise to its magnetic properties and,in particular, are responsible for their dynamics is quite limited. Such understanding, however, iscritical not only from the standpoint of fundamental research. From an applied perspective, there isa strong motivation to explore the possibility of utilizing the spin degree of freedom in the quest of
improving the efficiency, increasing the speed, and reducing the size of logic devices, which requires
a detailed understanding of magnetic interactions and their dynamics. Part of the reason for a lackof such understanding is related to experimental challenges in capturing materials’ magnetism at
its natural time and length scales. Recent advances in ultrafast extreme ultraviolet (EUV) and soft
X-ray sources have enabled element-specific studies of magnetic materials with nanometer spatial
and femtosecond temporal resolution by use of magneto-optics.
In this thesis, I present temporally and spatially resolved studies of ferromagnetic thin films.
In Chapter 4, I develop a new magneto-optical technique that allows a direct measurement of
the full resonant complex magneto-optical coefficient of a material with tabletop high harmonic(HHG) sources of EUV light. I apply it, along with a conventional magneto-optical technique,to a thin Co film in a pump-probe experiment to study the dynamic response of the film to a
femtosecond laser pulse. By comparing the experimental results with theoretical predictions, aconnection to the microscopic mechanisms of ultrafast demagnetization is made, and it is foundthat the transient magnetization dynamics in Co are mostly dominated by magnon excitations
with possible smaller contributions from other mechanisms. In Chapter 5, I develop an approachfor resonant magnetic scattering (RMS) on a tabletop with an HHG source that does not require

iv
wavelength-selective optics and instead relies on the resonant nature of the scattering process. I
use this approach to study magnetic textures with spatial resolution and, by applying a field to aferromagnetic multilayer film, observe a transition from a disordered network of stripe domains toan ordered lattice of magnetic vortices. In Chapter 6, I present the results of a dynamic soft X-ray
RMS experiment on a disordered domain network performed at the Linear Coherent Light Source(LCLS) at SLAC. By directly applying the experimental data to a carefully simulated domain
pattern, I capture laser-induced transient changes in the domains in real space and observe strong
non-uniformities in the demagnetization across the sample, which, with the help of simulations,are attributed to a combined effect of ultrafast spin-polarized currents and a gradient in the pumpabsorption throughout the thickness of the sample. In Chapter 7, I show preliminary simulations
and provide an outlook towards time-resolved lensless magnetic spectro-microscopy with HHG
sources.

Dedication

To my family.

vi

Acknowledgements

I would like to thank my advisors Margaret Murnane and Henry Kapteyn for giving me theopportunity to join their group at JILA and for their generous support both within and outside ofacademic research. I am grateful to you for this truly remarkable experience. My special thank you
goes to Tom Silva. I have aspired to absorb even a small fraction of your infinite wealth of knowledge
and expertise in magnetism. I would also like to thank our long-time collaborators Justin Shaw,
Hans Nembach, Eric Fullerton, Raj Medapalli, Dominik Legut, Karel Carva and Peter Oppeneer
for putting up with sample requests, teaching techniques, lending equipment and providing theory
support. The beam time at LCLS was a great opportunity, and I want to thank the entire SXRteam and, in particular, Alex Reid, Loı̈c Le Guyader, Ezzio Iacocca, and Hermann Dürr for their
help during and after the experimental run. Thank you to the entire Kapteyn-Murnane groupfor letting me learn from their broad scope of expertise, especially to the magnetics team, which
at various points in time included Ronny Knut, Patrik Grychtol, Emrah Turgut, Adam Blonsky,
Christian Gentry, Phoebe Tengdin, and Drew Morill. It has been a privilege to work with you.
Michael Tanksalvala, thank you for offering your help so selflessly in a last minute push experiment.
A special thank you to Dan Adams for patiently explaining to me the concepts of lasers, optics,
and imaging and for always lifting my spirits. You are one of the smartest people I have ever
met and an undefeated ping pong partner. But most importantly, I want to thank my parents,
my sister, my grandparents, and my entire family, and, of course, Maithreyi Gopalakrishnan andJavier Orjuela-Koop, who are like family. Without you, none of this would have happened, and,frankly, none of this would matter either.

Contents

Chapter
1 Motivation and Scope

1

2 Introduction

6

2.1

Magnetism of metals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.1.1

Origin of the band structure and the Stoner model . . . . . . . . . . . . . . .

7

2.1.2

Localized electrons and the Heisenberg model . . . . . . . . . . . . . . . . . . 13

2.1.3

Magnetic interactions in metals: itinerant and localized electrons . . . . . . . 22

2.1.4

Spin-Orbit Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.1.5

Magnetic domains and textures . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.1.6

Optically-induced ultrafast magnetization dynamics . . . . . . . . . . . . . . 30

2.2

Origin of magneto-optical effects in the EUV and soft X-ray regions . . . . . . . . . 33

2.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3 Ultrafast Sources of Coherent EUV and Soft X-ray Radiation
3.1

3.2

37

High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1

The three-step model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.2

Electron wavefunction dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1.3

Circularly polarized HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.1.4

Phase matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

X-ray free-electron lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii
3.3

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4 Extension of the EUV Magneto-Optical Spectroscopy

54

4.1

Time-resolved EUV T-MOKE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2

Measurement of the full complex magneto-optical permittivity. . . . . . . . . . . . 62

4.2.1

Angle-resolved T-MOKE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.2

Diagonal magneto-optical effect . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.2.3

Magneto-optical multilayer formalism . . . . . . . . . . . . . . . . . . . . . . 68

4.3

Understanding the microscopic picture of ultrafast demagnetization . . . . . . . . . . 77

4.4

Ultrafast demagnetization with angle-resolved T-MOKE . . . . . . . . . . . . . . . . 80

4.5

Ultrafast demagnetization with D-MOE . . . . . . . . . . . . . . . . . . . . . . . . . 87

4.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

5 EUV Magneto-Optics with Spatial Resolution
5.1

5.2

97

Resonant magnetic scattering with linearly and circularly polarized light . . . . . . . 98
5.1.1

Maximization of the scattered intensity . . . . . . . . . . . . . . . . . . . . . 102

5.1.2

Estimation of the RMS efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 103

Resonant magnetic scattering on a tabletop . . . . . . . . . . . . . . . . . . . . . . . 104
5.2.1

Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.2

Beamline efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.3

Application of an external magnetic field . . . . . . . . . . . . . . . . . . . . 114

5.3

Field-dependent scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6 Laser-Induced Spin Dynamics in a Network of Magnetic Domains

123

6.1

Overview of previous work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6.2

Soft X-ray scattering experiment at LCLS . . . . . . . . . . . . . . . . . . . . . . . . 125

6.3

Data processing considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

ix
6.4

Separation of magnetic and electronic scattering . . . . . . . . . . . . . . . . . . . . 131
6.4.1

Mathematical basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.4.2

Implementation of the separation procedure . . . . . . . . . . . . . . . . . . . 137

6.5

Summary of the experimental RMS results

6.6

Reconstruction of real-space transient domain dynamics . . . . . . . . . . . . . . . . 142

6.7

Spin transport in a laser-excited domain network . . . . . . . . . . . . . . . . . . . . 149

6.8

Effects of temperature gradients on domain wall dynamics . . . . . . . . . . . . . . . 153
6.8.1

6.9

. . . . . . . . . . . . . . . . . . . . . . . 139

Electron, lattice, and spin temperatures . . . . . . . . . . . . . . . . . . . . . 155

Propagation of X-rays through a thick sample . . . . . . . . . . . . . . . . . . . . . . 157
6.9.1

Evolution of the magnetic anisotropy . . . . . . . . . . . . . . . . . . . . . . . 161

6.10 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
7 Outlook: Towards Time-Resolved Element-Specific Magnetic Microscopy

164

7.1

Principles of lensless imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

7.2

Ptychography CDI of magnetic textures with linearly polarized light . . . . . . . . . 170
7.2.1

Algorithm constraints for magnetic samples . . . . . . . . . . . . . . . . . . . 171

7.3

Element-specific spectro-microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.4

Further work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

Bibliography

177

Appendix
A Depth Sensitivity of T-MOKE

201

B Rigorous Coupled-Wave Analysis

205

C Material Parameters for the 3-Temperature Model

219

C.1 Electron specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

x
C.2 Lattice specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
C.3 Magnetic specific heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
C.4 Thermal conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
C.5 Electron-lattice-spin coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
D Bad Pixel Map for the Tabletop RMS Experiment

230

xi

TablesTable
4.1

Summary of the analysis of the angle-resolved T-MOKE data . . . . . . . . . . . . . 84

4.2

Summary of the analysis of the angle-resolved T-MOKE data with longitudinal excitations only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3

Summary of the analysis of the angle-resolved T-MOKE data with transverse excitations only . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

C.1 Fitting parameters for electron thermal conductivities . . . . . . . . . . . . . . . . . 226

Figures

Figure1.1Time and length scales in the nano-world . . . . . . . . . . . . . . . . . . . . . . . .

4

2.1

Fermi surfaces of fcc Co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2

Density of states of fcc Co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

Comparison of the experimental M (T ) and the Stoner model prediction for Co . . . 12

2.4

Comparison of the experimental M (T ) and the Heisenberg model prediction for Co . 15

2.5

One-dimensional spin chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.6

Spin wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.7

A 3d effective potential and wavefunction . . . . . . . . . . . . . . . . . . . . . . . . 23

2.8

Exchange in the Stoner and Heisenberg models . . . . . . . . . . . . . . . . . . . . . 25

2.9

Bloch and Néel domain walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.10 Bloch (chiral) and Néel (non-chiral) skyrmions . . . . . . . . . . . . . . . . . . . . . 29
2.11 Timescales of electron, lattice, and spin dynamics after a laser excitation . . . . . . . 31
2.12 3p → 3d transitions in a model ferromagnet . . . . . . . . . . . . . . . . . . . . . . . 33
2.13 MOKE geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1

Three-step model of high harmonic generation . . . . . . . . . . . . . . . . . . . . . . 39

3.2

Solutions of the three-step model for different ionization phases . . . . . . . . . . . . 40

3.3

Ground-state and dynamics of an electron wavefunction . . . . . . . . . . . . . . . . 43

3.4

Experimental HHG spectrum in He . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

xiii
3.5

Dynamics of an electron wavefunction in a single-color circular and a bichromatic
trefoil-shaped field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6

The field of a two-color laser driver for circularly polarized HHG . . . . . . . . . . . 46

3.7

Setup for circularly polarized HHG . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.8

Experimental spectrum of circularly polarized HHG in He . . . . . . . . . . . . . . . 48

3.9

Hollow-core waveguide for phase-matched HHG . . . . . . . . . . . . . . . . . . . . . 50

4.1

An example T-MOKE asymmetry spectrum at the M -edge of Co . . . . . . . . . . . 58

4.2

Time-resolved EUV T-MOKE setup . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.3

Time-resolved EUV T-MOKE data collection flow . . . . . . . . . . . . . . . . . . . 61

4.4

An example ultrafast demagnetization curve . . . . . . . . . . . . . . . . . . . . . . . 61

4.5

Static angle-resolved T-MOKE signal from a Co film . . . . . . . . . . . . . . . . . . 63

4.6

D-MOE geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.7

Spectrometer part of the D-MOE setup . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.8

Static D-MOE asymmetry at the M -edge of Co . . . . . . . . . . . . . . . . . . . . . 69

4.9

Fields at an interface between two media . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.10 Azimuthal and polar angles for the magnetization vector. . . . . . . . . . . . . . . . 71
4.11 Sample structure for angle-resolved T-MOKE . . . . . . . . . . . . . . . . . . . . . . 75
4.12 Difference in diffraction efficiencies in the D-MOE geometry for the two opposite
magnetization directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.13 Im (xy ) at the M -edge of Co with a reduced exchange splitting calculated from firstprinciples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.14 Ultrafast demagnetization in a 10 nm Co film . . . . . . . . . . . . . . . . . . . . . . 80
4.15 Time-resolved data and analysis of angle-resolved T-MOKE . . . . . . . . . . . . . . 81
4.16 Comparison of the experimental and reconstructed D-MOE, T-MOKE, and L-MOKE
signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xiv
4.17 Comparison of the experimental and calculated polarization scans based on xy extracted from the T–MOKE and L-MOE scans. . . . . . . . . . . . . . . . . . . . . . 89
4.18 Static experimental xy of Co and comparison with theory and Kramers-Kronig transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.19 Average demagnetization response of a 5 nm Co film based on the D-MOE data . . 92
4.20 Transient laser-induced changes in xy measured with time-resolved D-MOE . . . . . 93
4.21 Direct comparison between theory and experiment for the static and transient xy
of Co . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.1

Concept of an EUV RMS experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2

Schematic of the EUV HHG light source for tabletop RMS . . . . . . . . . . . . . . 106

5.3

RMS chamber layout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.4Transverse profile of a phase-matched HHG beam . . . . . . . . . . . . . . . . . . . . 110

5.5

HHG spectrum and scattering efficiency in the RMS setup . . . . . . . . . . . . . . . 111

5.6

Toroidal mirror . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.7

Calculated reflectivity of the toroid . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.8

Calculated and measured magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . 115

5.9

Example RMS pattern from an Fe-Gd multilayer alloy . . . . . . . . . . . . . . . . . 117

5.10 Field-dependent RMS patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.11 RMS from ordered stripe domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
5.12 Azimuthally averaged RMS pattern for the Fe-Gd sample . . . . . . . . . . . . . . . 119
5.13 Field dependence of the domain size of the Fe-Gd sample . . . . . . . . . . . . . . . 120
5.14 RMS from a skyrmion lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.15 2D hexagonal close-packed lattice of skyrmions . . . . . . . . . . . . . . . . . . . . . 121
6.1

Experimental setup for TR-RMS at LCLS . . . . . . . . . . . . . . . . . . . . . . . . 126

6.2

Sample structure and MFM image of domains . . . . . . . . . . . . . . . . . . . . . . 127

6.3

Static RMS intensity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

xv
6.4

Mask of bad pixels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.5

Time-resolved soft X-ray scattering data . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.6

Separation of the spin and charge scattering . . . . . . . . . . . . . . . . . . . . . . . 138

6.7

Summary of the transient RMS dynamics . . . . . . . . . . . . . . . . . . . . . . . . 139

6.8

Background subtraction for scattered intensity . . . . . . . . . . . . . . . . . . . . . 140

6.9

Simulated domain pattern and comparison of the simulated and experimental RMS . 143

6.10 Fitted spatial filter in Fourier space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.11 Extracted laser-induced evolution of magnetic domains . . . . . . . . . . . . . . . . . 147
6.12 Laser-induced changes in domains with a Gaussian spatial filter . . . . . . . . . . . . 148
6.13 Schematic of the spin transport model . . . . . . . . . . . . . . . . . . . . . . . . . . 149
6.14 Simulated laser-induced changes in domains . . . . . . . . . . . . . . . . . . . . . . . 152
6.15 Depth gradient of the IR laser excitation . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.16 Temperature-dependent magnetization of the FeCo/Ni multilayer . . . . . . . . . . . 156
6.17 Electron, lattice and spin temperatures of FeCo/Ni . . . . . . . . . . . . . . . . . . . 156
6.18 Calculated depth-dependent magnetization profile and comparison with the XMCD
data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.19 Subdivision of a thick domain sample into layers . . . . . . . . . . . . . . . . . . . . 159
6.20 Experimental temperature dependence of the magnetic anisotropy for the FeCo/Nisample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.21 Calculated time-dependent domain wall width . . . . . . . . . . . . . . . . . . . . . . 162
7.1

Geometry and notations for a diffraction problem . . . . . . . . . . . . . . . . . . . . 167

7.2

CDI algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

7.3

Flow of the ePIE algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

7.4

Simulation of a ptychographic reconstruction of magnetic domains . . . . . . . . . . 175

A.1 An approach for a depth-sensitive T-MOKE calculation . . . . . . . . . . . . . . . . 202
A.2 T-MOKE EUV depth sensitivity function of a 10 nm Co film . . . . . . . . . . . . . 203

xvi
B.1 A periodic scattering structure for RCWA . . . . . . . . . . . . . . . . . . . . . . . . 206
C.1 Electron specific heat of Ni, Co, Cu, and Ta . . . . . . . . . . . . . . . . . . . . . . . 221
C.2 Lattice specific heat of Ni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222
C.3 Magnetic specific heat of Ni and Co . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
C.4 Electron thermal conductivity of Ni. . . . . . . . . . . . . . . . . . . . . . . . . . . 227

C.5 Temperature-dependent electron-phonon coupling . . . . . . . . . . . . . . . . . . . . 228
D.1 Bad pixel map for the tabletop RMS expereiment . . . . . . . . . . . . . . . . . . . . 230

Chapter 1

Motivation and Scope

Throughout its history, the research of magnetism has always been at the frontier of scientific knowledge. The discovery of electromagnetic induction by Michael Faraday in 1831 laid thefoundation for the unification of electricity and magnetism and the subsequent development of theclassical electromagnetic theory by James Maxwell.
In the late 1920’s, the concept of spin, an intrinsic angular momentum of elementary particles,
was introduced in quantum mechanics by George Uhlenbeck, Samuel Goudsmit, and Ralph Kronig,
and the theoretical formalism was later developed by Wolfgang Pauli and Paul Dirac for the nonrelativistic and relativistic cases, respectively. The property of spin gives rise to the intrinsic
magnetic moments of elementary particles, and, therefore, the magnetic moments of atoms, which
they make up, and, in turn, the magnetic moments of solids, which those atoms constitute. Aroundthe same time, in 1928, Werner Heisenberg proposed his model of the spin-dependent exchangeinteraction, which became the first successful quantum mechanical picture of ferromagnetism andexplained the nature of the phenomenological molecular field inside a ferromagnet postulated by
Pierre-Ernest Weiss in 1907.
In the 1930’s, developments in the band theory of ferromagnetism were made by Edmund
Stoner, John Slater, and Neville Mott. However, despite this progress and the success of the density
functional theory in the second half of the 20th century, our understanding of magnetism is stilllacking. Even for the simple itinerant ferromagnets (Fe, Co, and Ni) there is no satisfactory theory
that simultaneously explains their fractional magnetic moments and finite-temperature properties.

2
The dynamical evolution of a ferromagnet arising in response to a strong excitation, e.g. by a laser
pulse, is a similarly poorly understood phenomenon.
Yet such understanding is of paramount importance for reasons that are both fundamental
and applied in nature. From a fundamental standpoint, the understanding of magnetic materials,
perhaps, holds an answer to high temperature superconductivity and, in more general terms, to abetter understanding of strongly correlated quantum systems.
From an applied standpoint, the motivation for a better understanding comes from the technology sector, as it historically has. Specifically, the chip manufacturing industry is at a point where
it has become increasingly difficult to fulfill the projections of Moore’s law. Modern lithographic
techniques are exploring the emerging extreme ultraviolet technology in order to manufacture chips
with smaller features, but this same challenge of miniaturizing features on a chip will have to be
faced again once the benefits of this technology have been exhausted. In addition, such miniaturization is often associated with increased energy dissipation.
A possible solution to these problems would be to exploit the spin degree of freedom in order toencode and transfer information, which is the subject of the field of spintronics [1, 2, 3]. It has beenshown that spintronic devices, such as the non-volatile magnetic random access memory (MRAM),can be very energy-efficient [4, 5], which is highly relevant in the context of wide-spread batterypowered personal devices, and can be readily integrated with existing semiconductor microchips.
Spintronic devices are also remarkably well suited for neuro-inspired computer architectures [6].
Such architectures, as opposed to the classical von Neumann architecture with centralized processing, are very desirable for data mining and deep learning algorithms that are needed to analyze
the vast amount of data that is being generated worldwide at an ever increasing rate. A significant
portion of this data can be of great value. New markets relying on big data analytics are alreadyto emerging, and tailor-engineered magnetic materials can be an underpinning technology for allof this.
Additionally, a problem associated with high rates of data production is its storage. Despite
significant progress in the flash-based data storage technology, the cost and lifetime of solid-state3drives prohibit their use for archival purposes. Hard-disk drives thus remain relevant—although
their use has started to shift from personal devices to cloud storage facilities—and research is still
being done to increase their capacity and efficiency while lowering the cost. For a new generationof spintronic and magnetic devices, a possibility of switching their state with laser pulses—thefastest man-made events in the Universe—is being explored in order to achieve the fastest possible
switching speeds.
As elements on microchips become smaller and faster, and are expected to reach nanometerspatial and femtosecond temporal scales in the future, the applications listed above will ultimately
require a coherent control of the spin degree of freedom in order to overcome the speed limit
imposed by the induction laws of classical electromagnetism. This goal requires a fundamental
understanding of the magnetic interactions in a material at various time and length scales.
A lack of such understanding is caused by the fact that, until recently, there hadn’t been
experimental tools that allowed us to observe magnetism at its relevant time and length scales.
The relevant timescale in magnetism can be estimated from the time-energy uncertainty relation∆E∆t ∼ ~. With the characteristic exchange interaction energy of ∼ 0.5 eV, ∆t ∼ ~/∆E ∼ 1 fs.
The relevant length scales for magnetism range from interatomic distances (∼ 1 Å) for the exchangeinteraction to correlation lengths extending to several tens of nm for magnetostatic interactions[7, 8]. This motivates the need for a tool that can probe magnetism on femtosecond time and,ideally, sub-nanometer length scales.
Figure 1.1 demonstrates the characteristic time and length scales for various systems andprocesses in the nano-world, and the temporal and spatial resolution of the tools currently available
to carry out measurements of such systems. Unsurprisingly, there is no universal tool capable ofproviding both a very high temporal and spatial resolution . As shown in the figure, synchrotrons
have the highest spatial resolution [9], while laser-driven high harmonic sources are unrivaled interms of their temporal resolution [10, 11, 12, 13, 14]. Free-electron lasers (FELs) occupy the middle
ground between synchrotrons and high harmonic sources with regards to their spatial and temporal
resolution [15] and, in addition, provide a very high brightness. There has also been recent progress

4
in time-resolved electron [16] and Lorentz [17] microscopy. For the purposes of studying magnetism,
10 -5
Protein folding

10 -6
10 -7
10 -8

Synchrotrons

10 -9Time (s)10 -10Microchips

Lattice vibrations

10 -11
Nanostructured
materials

10 -12
10 -13FEL

10 -1410 -15

Electron dynamics
Molecular dynamics

10 -16

Spin dynamics

10 -17

HHG

10 -18
10 -19 -13
10

10 -12

10 -11

10 -10

10 -9

10 -8

10 -7

10 -6

10 -5

Length (m)Figure 1.1: Relevant time and length scales of nano-systems and spatial and temporal resolutionof selected tools available to study them.

the tool also needs to be sensitive to the spin degree of freedom. Apart from purely scientificmerits, usability considerations, such as ease of access to the tool, must be taken into account. The
latter makes tabletop high-harmonic sources very attractive for time-resolved EUV magneto-optical
spectroscopy. Additionally, such sources generate short wavelength light, which makes them well
suited for imaging magnetic textures with ∼10 nm features, which is relevant in a wide variety ofmagnetic systems and devices. Current developments in the mid-infrared (mid-IR) laser technology
will enable soft X-ray (SXR) high harmonics, which will further improve the resolution to ∼1 nm.
High harmonic sources thus have a potential of becoming a mainstream instrument for dynamic
magnetic microscopy with high temporal and spatial resolution. This development would make theresources of state-of-the-art large-scale facilities available for more demanding experiments that,for example, require a high-brightness X-ray probe and would ultimately accelerate the progress inthe study of materials and their magnetism.

5This thesis is primarily focused on studying the phenomenon of ultrafast magnetization dynamics in multilayer films with tabletop high harmonic sources by use of both spectroscopic andspatially-resolved resonant scattering approaches. A portion of the thesis is dedicated to a development of high harmonic sources with full polarization control and to a time-resolved resonant magnetic scattering experiment at the LCLS FEL. In the introductory Chapter 2, I discuss the nature ofmagnetism in metals and its interaction with polarized light that leads to various magneto-opticalphenomena, which thus allow us to measure magnetic systems with an optical probe. In Chapter 3,
I discuss the process of high harmonic generation in detail and present a method of generatinghigh harmonics with a two-color laser driver, which makes it possible to fully control their polarization. I also briefly discuss generation of coherent soft X-ray pulses in an FEL. In Chapter 4, thedevelopment of new EUV magneto-optical techniques is described. These techniques enable a complete characterization of a material’s magneto-optical permittivity and, through ab initio density
functional theory calculations, can capture the microscopic mechanisms at play during ultrafast
demagnetization. In Chpater 5, I develop a spatially resolved EUV magneto-optical approach byuse of resonant magnetic scattering (RMS) and study the formation of field-induced magnetic textures. In Chapter 6, I report the experimental results of a dynamic RMS experiment performed
at LCLS. The analysis of the results, along with the relevant simulations, reveals the role of spintransport and non-uniform pump absorption in the laser-induced dynamics of magnetic domains.
Finally, in Chapter 7, I provide an outlook for future work in the field and explore avenues forreal-space time-resolved EUV magnetic microscopy.

Chapter 2

Introduction

In this chapter, I review several fundamental aspects pertaining to the field of materials’
magnetism and its dynamics. I discuss the fundamentals of magnetism in materials and the effectit has on their optical properties, which, in turn, give rise to various magneto-optical phenomena
and thus allow us to observe magnetism with an optical probe.

2.1

Magnetism of metals
Our present-day understanding of magnetism is rooted in two distinctly different models:the Heisenberg model and the Stoner model. The former describes the macroscopic magnetizationas arising from an alignment of microscopic magnetic moments in the Heisenberg exchange field
inside a material and successfully explains finite-temperature phenomena such as the temperaturedependence of the magnetization and magnetic susceptibility. However, this model fails to explain
the magnitude of the microscopic moments. The latter Stoner model considers delocalized electrons
which form energy bands in the periodic potential of the crystal lattice. In this model, the differencein the number of electrons in the bands with opposite spins gives rise to an uncompensated magnetic
moment. The Stoner model, on the other hand, cannot predict the finite-temperature propertiesof magnetic materials. Historically, attempts to develop a unified description of magnetism havetaken either the Stoner or the Heisenberg model as a starting point, but, to date, a first principles
unified theory is still lacking.
In the following, I review the Stoner and Heisenberg models with an emphasis on the 3d tran-

7
sition metals (Fe, Co, and Ni). I also provide a brief overview of the other two aspects of magnetismthat this thesis is concerned with: Magnetic domains and textures, and ultrafast magnetization
dynamics.

2.1.1

Origin of the band structure and the Stoner model

The Stoner model of ferromagnetism in metals was developed in the 1930’s by Mott, Slater,and Stoner in an attempt to explain the broken magnetic moment values of ferromagnetic metals[18, 19, 20, 21, 22]. The magnetic moments of atoms are well known to be multiples of the Bohrmagneton µB : m = (2s + l)µB /~, where s and l are the spin and orbital moments, respectively.
Consider, for example, atoms of the 3d magnetic metals Fe, Co, and Ni. For Fe, the ground stateelectronic configuration is [Ar]3d6 4s2 , which leaves 4 unfilled electron states in the d -shell. According to Hund’s rules, the minimum energy configuration must maximize the total spin quantum
number in that shell, which leads to 4 unpaired electrons and thus to a magnetic moment of Fe
of 4µB . Similarly, for Co and Ni, the corresponding moments are 3µB and 2µB . In principle,
the orbital moment also needs to be included, but in any case, a prediction based on the single
atom picture gives integer magnetic moments. This compares poorly with the measured magnetic
moment values in solids of 2.216µB for Fe, 1.715µB for Co, and 0.616µB for Ni. The Stoner model
was able to successfully predict the correct values of the magnetic moments of metals by taking
into account the fact that in solids atomic levels become broadened and form energy bands. The
formation of the bands can be understood qualitatively as a result of a perturbation of the atomic
Coulomb potential due to bonding interactions. Valence states are less localized and are subject tostronger perturbations than the more localized core-level states, which is why valence energy levels
are broadened stronger than the core levels. The periodicity of the lattice potential determines theexact details of the band structure.
Today, density functional theory (DFT) [23, 24, 25, 26] can be used to carry out precise band
structure calculations. The details of DFT are beyond the scope of this thesis, but I shall briefly
outline its main principles.

8
DFT is concerned with finding the ground state of a many-electron system, which could bea bulk solid, an interface, or a nano-particle, for example. In other words, the goal is to find theminimum energy eigen-solution of the non-relativistic time-independent Schrödinger equation fora system of N electronsĤ |ψ (~r1 , . . . , ~rN )i = E |ψ (~r1 , . . . , ~rN )i(2.1)

where Ĥ is the Hamiltonian of the system, E is its eigenvalue, and |ψ (~r1 , . . . , ~rN )i is the eigenfunction, ~ri are the spatial coordinates of the particles. The Hamiltonian has several terms, each
describing a specific type of interaction
 X
N 
N
N
X
~2 2
1X
Ĥ = T̂ + V̂ + Û =−U (~ri , ~rj )∇i +V (~ri ) +
2mi
2
i

i

(2.2)

i6=j

where T̂ is the kinetic energy term, V̂ is the potential energy of the electron-lattice Coulombinteraction, and Û describes the electron-electron interaction. Because of this interaction, electrons
cannot be considered as independent, and the eigenvalue problem (2.1) becomes virtually intractablewithout applying special techniques.
One particular advantage of DFT is that it simplifies the problem by reformulating it interms of a single-particle in an effective potential. As a result, the number of degrees of freedom
is reduced to four (three spatial coordinates and spin) from 4N . This is done by introducing anelectron density ρ (~r). In the framework of the Hohenberg-Kohn-Sham DFT method [27, 28], theground-state wavefunction |ψ0 (~r1 , . . . , ~rN )i is uniquely determined by the ground-state electron
density. In other words, the wavefunction is a functional of the electron density:|ψ0 i = |ψ [ρ0 ]i .

(2.3)

By extension, any observable, particularly the ground-state energy, is also a functional of ρ
E = E [ρ] = hψ [ρ]| T̂ + V̂ + Û |ψ [ρ]i .

(2.4)

The second Hohenberg-Kohn theorem [27] states that E [ρ] is minimized for the ground-state electron density ρ0 , and it thus follows that the first order variation of the energy with respect to ρ is

9stationary. Under a particle number constraint N =R

d3~rρ(~r) this statement can be expressed as

δ (E [ρ] − µN )
= 0,
δρ

(2.5)

where µ is the chemical potential. Problem (2.5) is solved by introducing an auxiliary system of
non-interacting particles moving in an external effective potential Vˆs . Equation (2.5) then yields
the Kohn-Sham equations:



~2 2
ˆ
−
∇ + Vs |φi (~r)i = i |φi (~r)i ,2mρ (~r) =

N
X

(2.6)
2

hφi (~r)i .

i

The effective potential Vˆs consists of three terms: the periodic lattice potential V̂l (~r), the electronelectron Coulomb repulsion V̂H , known as the Hartree term, and the exchange correlation term
V̂XC [ρ]e2V̂s = V̂l + V̂H + V̂XC = V̂l +4π0Zd3~r 0ρ (~r 0 )+ V̂XC [ρ] .
|~r − ~r 0 |(2.7)

Since V̂s depends on ρ, equations (2.6) must be solved in a self-consistent manner. Typically, theprocedure starts with a guess for ρ (~r), then V̂s is calculated and eqs. (2.6) are solved. The solution
is then used to update ρ (~r). The steps are repeated iteratively until a convergence condition isreached.
In magnetic materials, the total electron density is a sum of the spin-up and spin-down
densities ρ (~r) = ρ↑ (~r) + ρ↓ (~r). The exchange correlation potential in eq. (2.7) then becomes spindependent, i.e., V̂XC = V̂XC [ρ (~r) , σ (~r)], where σ (~r) = ρ↑ (~r) − ρ↓ (~r). The first equation in (2.6)
is split into two—one for each spin direction, and the iterative solution procedure described above
yields a spin-dependent band structure which defines a dispersion relation for each band index n ↑↓ ~in a 3-dimensional k-space ↑↓
k . From this dispersion relation, the total (spin-resolved)n = ndensity of states (DOS) can be obtained by counting available states per unit volume in an energy
interval [;  + d]. Mathematically, this is expressed as an integral over surfaces of constant energy (such as the Fermi surface shown in Fig. 2.1) in k-space [29]
Z
dS~k
1 X
g↑↓ () =,3(2π) n∇~k ↑↓
n

(2.8)

10
where dS~k is a surface element (see Fig. 2.1). The spin-resolved density of states for fcc cobalt is

Figure 2.1: Fermi surfaces of fcc cobalt for spin-up and spin down states in the Brillouin zone. The
surfaces were computed using the Elk code [30]. An area element used in eq. (2.8) is shown on the
spin-down surface. Note that there are several spin-up Fermi surfaces originating from different
bands.

shown in Fig. 2.2. The spin-up energy band is shifted by an amount ∆ex relative to the spin-downband. This shift is called the exchange splitting, and it originates from the exchange-correlation
potential V̂XC in eq. (2.7). On a more fundamental level, it is caused by the Pauli exclusion
principle. Because electrons are fermions, the same state cannot be occupied by more than one
electron, and when a spin is flipped from, say, the spin-up to the spin-down band, it will occupy
a higher energy state. The Pauli exclusion principle thus gives rise to an energy cost associatedwith a single spin flip in the presence of all the other spins. For that reason, the exchange splitting
is called the spin-flip energy. Spin-flip excitations are known as Stoner excitations and are on theorder of ∼1 eV, which is determined by the magnitude of the exchange splitting. The relative
strength of interactions within the Stoner and Heisenberg models is discussed in subsection 2.1.3.
Notice that the number of occupied states for the spin-up band (blue shaded area) is smaller thanthat for the spin-down band (orange shaded area). This is because in equilibrium, the chemical
potentials of particles in both bands must be equal, and so the highest filled energy level is the11same in the spin-up and the spin-down bands. This leads to a greater total number of electrons inthe spin-down band, which is called the majority band. The spin-up band is called the minority
band. The choice of up and down is arbitrary, but it is important that the spins in the two bands
are aligned in opposite directions relative to one another.

Density of states of fcc Cog↑↓ (eV−1 atom−1 )2spin-upspin-down

1
0
1
∆ex

2
−10

−5

0

5

10

ǫ − ǫF (eV)Figure 2.2: Spin-up (minority states) and spin-down (majority states) DOS for fcc Co. Filled states
up to the Fermi level are shaded with color. The majority and minority bands are shifted in energy
with respect to one another by an amount ∆ex called the exchange splitting.

The difference in the number of majority and minority electrons gives rise to an uncompensated magnetic moment~
mtot = γe2Z(g↓ () − g↑ ()) f (, T ) d,

where γe is the electron gyromagnetic ratio, and f (, T ) = e(−µ)/kB T + 1

(2.9)
−1

is the Fermi-Dirac

distribution, where µ is the chemical potential that also depends on the temperature T . Because
the gyromagnetic ratio for the electron is negative, the magnetic moment points in the directionof the minority spins. Equation (2.9) yields fractional magnetic moments for ferromagnetic metals
that agree well with experiment.
Because of its ability to explain fractional magnetic moments, the Stoner model became a12significant breakthrough in the 1930’s. To reiterate, in the Stoner model, valence electron states
form energy bands, one spin-up and one spin-down. The two bands are shifted in energy withrespect to one another, which, in equilibrium, results in a difference in the number of electrons
filling each band. This difference leads to an uncompensated magnetic moment that is, generally
speaking, non-integer, according to eq. (2.9). However, despite the success of this model, it still
has major limitations. In particular, the Stoner model cannot predict the temperature dependenceof the magnetic properties, such as, for instance, the Curie temperature of metals.

M (T )=M (0)

1
0.8
0.6
0.4
experiment
Stoner Model

0.2
0
0

2000

4000

6000

T (K)
Figure 2.3: Comparison of the experimental temperature-dependent magnetization [31] and theStoner model prediction for Co.

The Curie temperature is a temperature at which the macroscopic magnetization of a ferromagnet vanishes. In eq. (2.9), the temperature dependence of the magnetic moment is entirely due
to the temperature dependence of the Fermi-Dirac distribution f (, T ). This leads to an overestimation of the Curie temperature by a very significant amount. A comparison of the experimental
temperature-dependent saturation magnetization of Co and a Stoner model prediction calculatedwith eq. (2.9) is shown in Fig. 2.3. In the figure, the calculated decay of the magnetization withtemperature is underestimated because the density of states from Fig. 2.2 used in eq. (2.9) remains
rigid, i.e., it is temperature-independent. If the exchange splitting was reduced at elevated temperatures, the magnetization would reduce faster. But even if an exchange splitting reduction were to

13
be accounted for, the large magnitude of ∆ex on the order of ∼1 eV, as mentioned above, requires
temperatures of ∼ 104 K in order for the exchange splitting, and thus the magnetic moment, tovanish. This renders the Stoner model unsuited for explaining the temperature-dependent magnetic
properties of materials. This problem is solved in the Heisenberg model.

2.1.2

Localized electrons and the Heisenberg model

Unlike the Stoner model, the Heisenberg model does not start from first principles. Rather,it considers a chain of localized magnetic moments that interact with one another by means of aquantum mechanical exchange. The strength of this exchange is an input parameter for the model.
The Heisenberg model is closely related to the Weiss mean-field model of ferromagnetism, and forthe sake of simplicity, I will first derive the temperature dependence of the magnetization in the
Weiss mean-field approximation and then explain the nature of the mean field in the context of the
Heisenberg exchange.
In 1907, Peter Weiss postulated the existence of a molecular field inside a ferromagnet thataligned all the microscopic magnetic moments in the direction of the field and that was proportional
to the magnetization with a field constant β [32]HW = βM.

(2.10)The mean B-field in the material is BW = µ0 (1 + 1/β) HW , where µ0 is the magnetic permeabilityof vacuum.
~ = B/µ~ 0−M~ . By thisNote that this is based on the SI system definition of the H-field H~ has the units of A/m, the same as the field strength H.
~ An alternative
definition, magnetization M~ = 1/µ0 B~ −M~ , and B~ and M~ both have the units
definition is sometimes encountered where H~ = B/µ~ 0−M~ is used throughout this thesis.
of Tesla. The first definition H~ W . The potential energy of the ith dipole
Consider N identical magnetic dipoles µ~ in a field B~ W is Ei = −~~ W = −µBW cos θi , where θ is the angle between µ~W . A
in the field Bµi · B~ and B

14
single-dipole partition function is then [33]
Z=

X
i




µBW cos θi
exp
.
kB T

(2.11)~ W is along the z-axis, the average magnetic moment µz along this axis can be written
Assuming B
as [33]hicos θiµ cos θi exp µBWkB Thiµz = i P.
µBW cos θi
exp
.
kB T
P

(2.12)i

And the total magnetization of a system of N dipoles is M = N µz . If the system is quantized andthe dipole moments are due to unpaired spin-1/2 electrons, then the angle θi takes only two values:
0 and π, and the magnitude of the magnetic moment is one Bohr magneton µ = µB . In this case,eq. (2.12) can be simplified, and the magnetization becomes
h
i
h
i


exp µkBBBTW − exp − µkBBBTW
µB BW
h
i
h
i
M = µB N= µB N tanh.
kB T
exp µkBBBTW + exp − µkBBBTW(2.13)

This equation, together with eq. (2.10) and the magnetization at 0 K M (0) = µB N , gives anequation for the temperature-dependent magnetization
M (T )= tanhM (0)µB µ0 (β + 1)M (T )
kB T


.

(2.14)

The argument of the tanh is small near the Curie temperature TC , and, keeping only the first orderterm in its Taylor series expansion, one can obtain an expression for TCTC =µBµ0 (β + 1)M (0).
kB

(2.15)

With an experimental value of the Curie temperature for a given material, eq. (2.14) can be solved
numerically to find M (T )/M (0). A result for Cobalt is shown in Fig. 2.4. An agreement with theexperimental data is much better than that in Fig. 2.3.
Because the Weiss mean-field theory gives a fairly accurate prediction of the temperaturedependent magnetization, it is possible to use expressions (2.15) and (2.10) to estimate the magnitude of the molecular field BW inside a material. From typical values of ∼ 103 K for the Curie

15

M (T )=M (0)

1
0.8
0.6
0.4
experiment
Heisenberg Model

0.2
0
0

500

1000

1500

T (K)
Figure 2.4: Comparison of the experimental temperature-dependent magnetization [31] and theHeisenberg model prediction for Co.

16
temperature of the ferromagnetic 3d metals, such as Fe, Co, and Ni, an estimated molecular fieldBW ∼ 103 T. If this was an actual magnetic field, it would require the presence of charge currentsof such a high density that it would be above the damage threshold of the material itself.
The fact that ferromagnets exist means that the molecular field is not generated by charge
currents. It arises from a purely quantum phenomenon of exchange interaction due correlation
terms in the Hamiltonian describing a particular system, such as, for example, Û in eq. (2.2).
The minimal system necessary to understand the exchange interaction is a system with twoelectrons, e.g., the He atom. Because electrons are fermions, the total wavefunction of a multielectron system must be antisymmetric under an exchange of any pair of electrons, according to
the spin-statistics theorem [34]. The Pauli exclusion principle is a consequence of this theorem. Fora two-electron system, the total wavefunction is a product of the spatial part |ψ (~r1 , ~r2 )i and thespin part |χ (~s1 , ~s2 )i|ψasym (~a, ~b)i = |ψsym,asym (~r1 , ~r2 )i |χasym,sym (~s1 , ~s2 )i .

(2.16)In order for |ψ(~a, ~b)i to be antisymmetric, either |ψ (~r1 , ~r2 )i must be symmetric and |χ (~s1 , ~s2 )i
antisymmetric or vice versa. These two possibilities correspond to the singlet and triplet statesof the two-electron system, respectively. In the singlet state, the spin part is antisymmetric, andthe total spin quantum number is S = 0; hence, the projection on the z-axis MS = 0. In the
triplet state, the spin part is symmetric, and the total spin quantum number S = 1, and the spin
projection quantum number MS = −1; 0; 1. When applied to the He atom, the electron-electroninteraction term in the Hamiltonian of the form Ĥee = e2 /4π0 |~r1 − ~r2 |, treated as a first order
perturbation, gives rise to an energy difference ∆E between the singlet and triplet excited states.
This difference is called the exchange energy. Thus, the exchange interaction is a result of the
Coulomb repulsion between the electrons and a requirement of the Pauli exclusion principle thatthe total wavefunction of the system must be antisymmetric. For a detailed derivation of ∆E see
Refs. [8] and [35].
In the Heisenberg Hamiltonian, which describes a system of interacting spins, the interaction

17
between the spins is parametrized with an exchange coupling Jij
ĤHeis = −NX

Jij ~sˆi · ~sˆj = −2N
X

Jij ~sˆi · ~sˆj ,(2.17)i<ji6=jwhere the exchange integral (or exchange coupling) Jij is similar to that for the He atom:
ZZ
Jij =

ψi (~r1 )ψj (~r2 )e2ψ ∗ (~r2 )ψj∗ (~r1 )d3~r1 d3~r2 .
4π0 |~r1 − ~r2 | i(2.18)

Unlike in the He atom case, this exchange integral cannot be readily evaluated in a solid where theinteractions are much more complex. An additional aspect that makes the evaluation of Jij evenmore difficult for a realistic ferromagnetic system is the fact that the magnetic moments are notspin-1/2 electrons but are composite atomic spins. For those reasons, Jij is normally taken as amodel parameter.
With the Heisenberg Hamiltonian (2.17), it is possible to express the energy of the interactionof the ith spin with all other spins as
Ei = −2~si ·X

Jij ~sj .

(2.19)j~ W , where the magnetic moment µ
In the Weiss model, Ei = −~µi · B~ i = gµB ~si , and g is the
gyromagnetic factor. With this, the Weiss mean field can be identified in eq. (2.19).
X~W = − 2
B
Jij ~sjgµB(2.20)j

From this expression, it becomes immediately clear that the origin of the Weiss molecular field isentirely due to the quantum mechanical exchange interaction. For this reason, it is also called theHeisenberg exchange field. The DFT result discussed in the previous subsection is mapped onto
the Heisenberg model by separating the spin ~si into a constant component h~si i and a fluctuating
component ∆~si = ~si − h~si i and making a substitution ~si → h~si i in eq. (2.20). For a lack of asatisfactory first principles theory, this approach is able to capture the fractional magnetic moment
predicted by the band theory at T = 0 K as well as the finite temperature predictions of the
Heisenberg model. More details on this topic can be found in Refs. [23] and [8].

18
2.1.2.1

Spin waves

The Heisenberg model together with the mean-field theory gives a good prediction of thefinite temperature magnetic properties. However, the Hamiltonian in (2.17) needs to be examinedfurther in order to understand what microscopic excitations lead to the predicted temperaturedependence of the magnetization.
Consider a one-dimensional chain of spin-1/2 moments, such as in Fig. 2.5. It is described by𝑎𝑛−1 𝑛 𝑛+1

Figure 2.5: One-dimensional infinite periodic spin chain. The lattice period is a. The position ofspin n is na

the Hamiltonian (2.17). I will consider a simplified case where only the nearest neighbors interact.
In this case, the exchange integral Jij = J when j = i ± 1 and Jij = 0 otherwise. This simplifies
the expression for the Heisenberg Hamiltonian. It can be simplified even further by using a realtion
between the Pauli vectors and the permutation operator ~sˆi · ~sˆj = 2P̂ij − 1, which yields [36]ĤHeis = −2JXP̂n,n+1 − 1 .

(2.21)n

I label the state of the spin chain with the nth flipped moment as |ni. In this notation, the actionof the permutation operator on the states is as follows:P̂n,n+1 |mi = |mi if m 6= n, n + 1,P̂n,n+1 |ni = |n + 1i,P̂n,n+1 |n + 1i = |ni.

(2.22)

19
With these relations, the matrix elements Hn,m = hn|ĤHeis |mi of the Hamiltonian (2.21) can beevaluated [36]Hn,n = 4J,
Hn,n+1 = Hn,n−1 = −2J,

(2.23)

Hn,m = 0 if |n − m| > 1.
Now, consider a state of the chain which is a superposition of states with a spin flipped at eachpossible site
|ψi =

X

|nihn|ψi =X

Cn |ni.

(2.24)

nn

where Cn is the probability amplitude that the spin on the nth site is flipped. If the spin-flip
excitation propagates down the chain, the probability that on any given site the spin is flipped
evolves with time, and this evolution must obey the time-dependent Schrödinger equationi~d|ψi= ĤHeis |ψi,dt

(2.25)

which, with eqs. (2.23) and (2.24), turns into an infinite system of equations
i~

dCn X
=
Hmn Cm = −2J (Cn−1 + Cn+1 − 2Cn ) .
dtm(2.26)

If an excitation were to propagate unaltered, all the probability amplitudes Cn would have tochange at the same rate. In other words, this means that |ψi is a definite energy state. Thus anansatz solution for Cn is
Cn = α n e

−iEt
~

.

(2.27)

Substituting this expression into eq. (2.26), a linear system of equations for the coefficients αn ,which determine the properties of an excitation, can be obtainedEαn = −2J (αn−1 + αn+1 − 2αn ) .

(2.28)

Since the spin chain under consideration is periodic with a lattice constant a, it is reasonable toexpect that the coefficients αn are connected with one another also in a periodic manner. With a20
periodic ansatz αn = einka , a dispersion relation for an excitation propagating along the chain witha wavenumber k = 2π/λ can be derived [36]
E = −2J e−ika + eika − 2 = 4J (1 − cos ka) .

(2.29)For long wavelengths (small k’s), the cosine in the expression above can be Taylor expanded, and,keeping terms up to second order, the dispersion relation is simplified to
E = 2Jk 2 a2 .

(2.30)

Through the kinetic energy of a particle Ekin = ~2 k 2 /2m, such an excitation corresponds to aparticle with an effective mass [36]mef f =

~2.
4Ja2

(2.31)

These particles are called magnons. A magnon can be interpreted as a propagating excitation
corresponding to a single spin flip distributed across an entire lattice. In the Heisenberg model,
these excitations are responsible for the finite-temperature properties of ferromagnets.
Even though the derivation above was performed for a one-dimensional spin chain, it contains
all of the essential physics, and the result can be generalized to three dimensions. Since an electron
spin is 1/2, a single spin flip reduces the total spin angular momentum of the chain by 1, whichmeans that magnons are integer-spin particles and, therefore, obey the Bose-Einstein statistics.
A magnetization reduction due to magnon excitations can then be calculated simply by counting
magnons. This can be done analytically at low temperatures (long magnon wavelengths) by integrating the Bose-Einstein distribution, with the energy given by (2.30), over k. Since the number ofmagnons is not conserved, the chemical potential is zero, and the Bose-Einstien distribution turns
into the Planck distributionnk =1eE(k)/kB T − 1.

(2.32)

Integrating this expression, with E(k) given by eq. (2.30), over k in three dimensions gives thefamous Bloch law for the temperature dependence of the magnetization [37]
M (T ) = M (0) 1 − (T /Θ)3/2 .

(2.33)

21
This dependence holds at low temperatures, and there are deviations from it near surfaces wheresome magnon states are prohibited.
Because magnons are bozons, a single state can be occupied by an unlimited number ofmagnons. Therefore, when the temperature is raised above 0 K, very long wavelength excitations
would destroy the magnetic order. In order to prevent this from happening, a small energy gap
must exist that prohibits such magnons. Magnetic anisotropy, which I discuss below, provides sucha gap.

Figure 2.6: Spin waves are classical analogs of magnons. The magnetic moments precess aroundthe direction of the mean field in a coherent manner: the phases of the precession for each spin are
correlated.

Sometimes, it is helpful to consider the classical analog of a magnon—a spin wave. I shallprovide a breif description of a spin wave, which is shown in Fig. 2.6. Just like a magnon, a spinwave excitation is distributed along an entire lattice. Each magnetic moment precesses around the
direction of the mean magnetization. As a result, the magnitude of the moments’ projections on
this axis is reduced, which leads to an overall magnetization reduction. The precessional motionof different spins is coherent, i.e., their phases are locked relative to one another. Such collective
dynamics amount to a propagating precession phase. This classical approximation can be usefulwhen DFT results are mapped onto the Heisenberg model. Particularly, there are DFT methods

22that yield correct finite-temperature results by applying a “frozen magnon” scheme, i.e., a spin
spiral in which magnetic moments are tilted in the same way as they are in a classical spin wave
[38]

2.1.3

Magnetic interactions in metals: itinerant and localized electrons

The reason that the Stoner and Heisenberg models result in such drastically different predictions and have limited applicability is that they start from two contrasting initial assumptions. The
Heisenberg model assumes that the electrons are localized in real space, while the Stoner model
assumes that they are localized in reciprocal space [39]. In reality, the applicability of each of thesemodels is determined by the localization of the wavefunction of the electrons responsible for themagnetic properties of a material, which is highly system dependent. In general, it depends on the
shape of the effective atomic potential, which includes a centrifugal term due to the orbital angularmomentum as a correction to the screened Coulomb potential [8]Vef f (r) = −Z ∗ e2l(l + 1)~2+,
4π0 r
2me r2(2.34)

where l is the orbital angular momentum quantum number. and Z ∗ is an effective charge of thenucleus that takes into account the screening by the inner shell electrons. An example effective
potential and a 3d radial wavefunction are shown in Fig. 2.7. With a typical lattice constant ofseveral Bohr radii a0 , the wavefunctions from the neighboring lattice sites will overlap.
In rare-earth metals, such as Gd, the 4f valence shell is only partially filled and thus produces
an uncompensated magnetic moment. The wavefunction of the 4f electrons is highly localized [40],and, therefore, the magnetic moments have a local, atomic character. Rare earth metals are welldescribed by the Heisenberg model.
The situation with the 3d transition metals Fe, Co, and Ni, is more complicated. The spatial
extent of the 3d orbitals is in the intermediate range (although they are localized in some insulators[8]). The orbitals are localized enough to generate a significant local moment and, at the same time,broad enough to provide a sizable overlap with the neighbors. The 3d electrons are often referred

23

3d atomic potential and wavefunction

Ve, (Ry)

0.5

-5

-10

A3d (r)

1

0

0
0

1

2

3

4

5

r=a0
Figure 2.7: An effective potential and an outer shell wavefunction of a 3d element (Z ∗ = 10).
The wavefunction extends to a few Bohr radii, which, in a solid, provides an overlap with theneighboring wavefunctions.

24
to as “itinerant” meaning that they are neither fully localized nor fully independent but rather
jump from atom to atom. The Stoner and Heisenberg models by themselves cannot fully explain
the full range of magnetic properties of the 3d ferromagnets, which necessitates a new theory ofmagnetism that is currently lacking.
There have been attempts to develop a unified theory of magnetism that have chosen either
the Stoner or the Heisenberg model as a starting point. At present, a promising approach, called
the self-consistent renormalization theory, starts with the Stoner model and considers the feedbackof spin fluctuations on the band structure in a self-consistent manner. The details of this theory
are beyond the scope of this thesis and can be found in Ref. [39].
For a lack of a better theory, at the moment, it is customary to map the DFT zero-temperatureground state onto the Heisenberg model, as mentioned above. In doing so, it is important tounderstand the meaning of exchange in both models. In the Heisenberg model, the exchange
energy is the energy required to reverse a local moment composed of unpaired moments inside an
atom in the molecular mean field of all other moments. This energy is typically ∼200 meV. In the
Stoner model, the exchange energy corresponds to a single electron spin reversal in the sea of allother delocalized electrons. Its value is close to the energy difference of the singlet and triplet statesof a two-electron atom and is on the order of ∼1 eV. The difference in the meaning of exchange inthe two models is illustrated in Fig. 2.8.
The large value of the exchange splitting in the Stoner model is the reason why it predicts
such high Curie temperatures, as discussed above. Because the exchange energy in the Heisenberg
model is much lower, the TC values predicted by it are much closer to the experimental ones.
Ultimately, this brings up the question of whether the magnetic moment is completely quenchedabove TC or whether the macroscopic magnetization vanishes because of the disorder, while themicroscopic moments persist. The agreement of the Curie temperature predicted by the Heisenberg
model with the experimental values suggests that the latter is the case. This is important in thecontext of ultrafast demagnetization, which I discuss below, because it limits the number of available
relaxation channels for an excited spin system.

25

Figure 2.8: Exchange in the Stoner and Heisenberg models. In the Heisenberg model (left), theexchange energy corresponds to a reversal of a local magnetic moment in the mean exchange fieldof other moments. In the Stoner model, it corresponds to a spin flip of a single electron in the seaof all other electrons.

26
2.1.4

Spin-Orbit Interaction

The spin-orbit interaction couples spins to the lattice and leads to the magnetocrystalline
anisotropy Ku [41] by breaking the symmetry of the isotropic exchange interaction and thus sets amacroscopic magnetization direction. It was discovered spectroscopically through the fine structureof atomic spectra of alkali metals. Qualitatively, it can be understood as the coupling of the spin of abound electron to its orbital motion. In the frame of reference of the electron, the nucleus effectively~ The field strength depends on the orbital angular momentum
orbits it creating a magnetic field B.
~ of the electron and its distance to the nucleus, which is determined by the n quantum number.
L~ with this field causes a shift in its energy by Eso = −B
~ · S.
~The interaction of the electron spin S~ is proportional to L,~ the spin-orbit Hamiltonian Ĥso can be written as [8]Since B
~ · L,~
Ĥso = ξnl S(2.35)

where ξnl is referred to as the spin-orbit coupling constant which depends on the quantum numbersn and l and has the units of energy. It determines the strength of the spin-orbit interaction, which israther weak in the 3d ferromagnets—on the order of ∼ 10 − 100 meV and leads to a relatively small
orbital magnetic moment [8]. Despite its low magnitude, the spin-orbit coupling plays an importantrole in setting a macroscopic magnetization direction. Because the crystal lattice has a preferred
direction, due to the anisotropy of the bonding between atoms, it breaks the isotropic symmetry~ [42], which, in turn, sets the orientation of S
~ since, in the minimum energy configuration, S~
of L~ The energy difference between the parallel and perpendicular alignment ofmust be parallel to L.
~ and L~ is equal to the energy required to rotate the spins from their preferred easy-axis direction
S
to one orthogonal to it. This energy is referred to as the magnetocrystalline anisotropy energy.
Despite a direct connection with the spin-orbit interaction, the magnetocrystalline anisotropy
is still very difficult to estimate accurately because of the complex band structure calculations
involved, which precludes a clear physical insight into its origins [43]. Nonetheless, it is understoodthat the magnetocrystalline anisotropy can be enhanced at interfaces due to missing chemical bonds[44, 45] and used to engineer magnetic films with a magnetization direction perpendicular to their

27
surface. The latter is critical for the creation of nanoscale magnetic domains and other magnetic
textures, which is why in order to obtain them, stacks of ultrathin magnetic layers, with a highnumber of layer repeats to increase the number of interfaces, are typically grown.

2.1.5

Magnetic domains and textures

If a magnetic film were uniformly magnetized in the easy direction determined by its magnetocrystalline anisotropy, it would produce a stray magnetic field. In order to minimize this field,
the magnetization breaks into domains of opposite directions. Additionally, the shape anisotropy
Ks also contributes to the total internal energy of the film and thus affects the domain formation[46]. In nanometer-thin samples with the easy axis parallel to the surface of the film, the strayfield is very small because it exits the film through the side surface whose area is negligible compared to the front and back surfaces of the film, and, therefore, the film typically stays uniformly
magnetized. In samples with perpendicular magnetic anisotropy, the stray magnetic field is large,
and, in the absence of an external field, magnetic domains are formed. The domain size and shapedepend strongly on the magnetic interactions inside the sample, which can be controlled throughthe multilayer structure of the film. Typically, the thinner the film the smaller the domains.
The transition region between two oppositely magnetized domains in which the magnetization
rotates from one direction to the other is called the domain wall. There are two distinct types ofdomain wall, the Néel wall, where the magnetic moments rotate around the axis parallel to theplane of the wall, and the Bloch wall, where the moments rotate around the axis perpendicular
to the plane of the wall. The two wall types are shown in Fig. 2.9. The type, size and energy of
domain walls are determined by the energy of the exchange interaction and magnetic anisotropy,
which compete with one another. For example, if the anisotropy is large, an alignment of the
moments parallel to the easy axis is favored. In order to minimize the total internal energy of thefilm, the volume with any other orientation of the magnetic moments would have to be minimized.
For that reason, the width of the domain walls would have to be reduced. To the contrary, if theexchange energy is high, changing the orientation of a magnetic moment relative to other moments

28
becomes energetically unfavorable. Hence, in order to minimize the energy cost, the wall width
increases [8].

Bloch Wall

Néel

Wall

Figure 2.9: Bloch (left) and Néel (right) domain walls. The magnetization in the Bloch wall rotatesaround the axis perpendicular to the plane of the wall, while in the Néel wall, it rotates around theaxis parallel to the wall plane.

Similarly to the formation of magnetic domains, various competing interactions lead to othermagnetic textures. A particularly interesting example is magnetic vortices referred to as skyrmions[47, 48, 49], which arise due to the µ~ ×~µ-type Dzyaloshinsky-Moriya interaction [50, 51] or competingdomain wall energy and long-range dipolar interactions [52, 53, 54]. Skyrmions are said to be
topologically stable in a continuous field approximation, which breaks down on the scale of a discrete
atomic lattice. This topological stability makes them particularly attractive as information carriersin energy-efficient logic devices [55, 56]. A skyrmion can be thought of as a magnetic domain
collapsed into a point and surrounded by a domain wall. Depending on the wall type, skyrmionscan either be chiral or non-chiral. The former originate from Bloch-type walls and the latter fromthe Néel-type, as shown in Fig. 2.10.
Apart from topologically protected skyrmions, there is a variety of other textures such as
magnetic bubbles, spin spirals, etc. Because magnetic textures span a broad range of correlation
lengths from sub-nm to tens of nm and even µm, studying their static and dynamic behavior
could provide critical insight into the microscopic origins of magnetism. I report such studies in
Chapters 5 and 6.

29

Chiral

Non-Chiral

Figure 2.10: Bloch (chiral) and Néel (non-chiral) skyrmions.

30
2.1.6

Optically-induced ultrafast magnetization dynamics

The phenomenon of ultrafast magnetization dynamics driven by a femtosecond laser pulse wasdiscovered in Ni in 1996 [57]. In this pioneering work, a magneto-optical polarization rotation of theprobe pulse reflected from a Ni film was observed at various delay times after an excitation of thefilm by a 60 fs laser pulse. A sharp drop in the magneto-optical signal followed by a slower recovery
was observed. An example curve measured in Co is shown in Fig. 4.4 in Chapter 4. This work
opened up a new field in the magnetism research with the goal of understanding the microscopic
mechanisms behind ultrafast magnetization dynamics. Such an understanding is needed not only
from the perspective of fundamental research but also from an applied standpoint as it could
lead to a new generation of spintronic devices that can be operated on the natural timescales ofmagnetism through a coherent control of their properties with light. Additionally, it could also
have implications for high-temperature superconductivity as it is closely related to magnetism.
Since the original discovery, various types of laser induced magnetization dynamics have beenfound. They include type I and type II demagnetization dynamics and the phenomenon of all-opticalswitching. Type I demagnetization is observed in the 3d ferromagnets where a recovery phasehappens immediately after an initial rapid loss of magnetization [58]. Type II demagnetization,
where the rate of magnetization loss slows down significantly with time, is observed in rare-earth
metals, their alloys, and other systems[59, 60, 61]. A magnetization reversal referred to as the
all-optical switching (AOS) by a single as well as multiple laser pulses has also been demonstrated[62, 63, 64]. AOS suggests that light could directly couple to the spin system.
The difficulty in solving the puzzle of ultrafast demagnetization has to do with the complexity of the interactions involved. The coupled electron, phonon, and spin systems are strongly
out of equilibrium, particularly on femtosecond timescales (see Fig. 2.11). There is also evidence
suggesting that the phonon system could be out of equilibrium even on picosecond timescales [65].
This presents a great challenge for both theory and experiment. The theoretical challenge is indescribing the transient dynamics of a many-body strongly coupled quantum system from first prin-

31
ciples. The experimental challenge is to unambiguously capture the dynamics of each of the threeinteracting systems, i.e., the electrons, lattice, and spins. Typically, a single technique is used forthe purpose of measuring the response of a particular system to a laser excitation. For example, thelattice response can be accessed using EUV nanometrology [66], which measures the reflectivity anddiffraction efficiency of patterned films, or resonant X-ray diffraction. Electron dynamics can bemeasured using photoemission or X-ray absorption spectroscopy or EUV and X-ray reflectometry,
and the response of the spin system can be captured with the help of magneto-optical spectroscopies
in the visible, EUV and X-ray spectral ranges as well as spin polarized photoemission. However,each experimental method measures a certain observable, which might not necessarily represent apure electron, phonon, or spin system. For example, visible magneto-optical measurements havebeen shown to be affected by non-magnetic contributions to the signal [67, 68]. For that reason,studying the sample using multiple techniques can be advantageous because it can help disentangle the magnetic and non-magnetic contributions, as, for example, was done in Ref. [69], which
reported a laser-induced ultrafast phase transition in Ni.
Although ultrafast demagnetization is a difficult subject, and, to date, no universal picture ofthis phenomenon exists, the general underlying microscopic mechanisms, illustrated in Fig. 2.11, canbe understood, at least qualitatively, based on the conservation of energy and angular momentum,
which is key for magnetism. On timescales of a few femtoseconds to tens of femtoseconds, theExcitationof electrons(and spins)

10 fs

Demagnetization via
Electron scattering local mechanisms
and thermalization and spin currents

100 fs1 ps

Electron-phonon
coupling andthermalization Thermal transport

10 ps

100 psFigure 2.11: Timescales of electron, lattice, and spin dynamics after a laser excitation.

32
electron system absorbs the energy from the laser pulse. If the laser pulse is circularly polarized, it
carries angular momentum, which upon absorption is transferred to the material and can eventually
result in a magnetization reversal via the AOS process discussed above. If the pulse is linearlypolarized, its net angular momentum is zero, and the energy absorbed by electrons is transferredto the spin and lattice systems on timescales that range from tens of femtoseconds to tens ofpicoseconds. Simultaneously, the three systems thermalize by means of various electron, magnon,and phonon scattering processes. Spin excitations lead to a decrease in spin polarization. Because
angular momentum must be conserved and the net angular momentum transferred to the material
by the pulse is zero, a reduction in spin polarization generates a net angular momentum anti-parallelto the direction of the majority spins. This means that an equal and opposite angular momentum
must be absorbed somehow.
One possibility is to transfer this excess angular momentum to the lattice via the spin-orbit
coupling, which is greatly enhanced for highly excited electrons and can thus happen on subpicosecond timescales [8]. This process would generate a phonon and a magnon with opposite
angular momenta, and magnons would reduce the net magnetization of the sample. Since thespecific heat of the spin system is much smaller than that of the lattice (see Appendix C), therise in the lattice temperature accompanying the exchange of angular momenta would be relatively
small, compared to the rise in the spin temperature.
Another possibility is to drive the excited majority spins out of the pumped volume. The
magnetization would then be reduced locally and enhanced elsewhere. In the case of magnetic
domains or two oppositely magnetized films, such spin-polarized currents would reduce the magnetization everywhere due to a direct exchange of angular momentum between either the domains
or the films.
Alongside all these processes, thermal diffusion also takes place. In metals, thermal transport
is mediated by electrons and can thus be rather fast, while lattice heating is limited by the strengthof electron-phonon coupling.
Laser-driven ultrafast magnetization dynamics is one of the main subjects of this thesis and33
is considered in both uniformly magnetized films as well as in the presence of a network of magnetic
domains in Chapters 4 and 6, respectively.

2.2

Origin of magneto-optical effects in the EUV and soft X-ray regions
The magnetic state affects the transition probabilities between different energy levels and cantherefore be probed optically. I start by discussing the effect of X-ray magnetic circular dichroism(XMCD) which results in different absorption of left and right circularly polarized light. XMCD
was first predicted theoretically in 1975 at the M -edge of Ni[70] and measured experimentally in
1987 at the K-edge of Fe [71]. It can be understood with the help of the two-step model of XMCD
[72]. In the first step of the model, electrons are excited from the core 3p-states to 3d states bycircularly polarized photons, as seen in Fig. 2.12, which shows XMCD at the M -edge of a model
ferromagnet. A careful analysis of the transitions with appropriate dipole selection rules, namely

left

right

Figure 2.12: 3p → 3d transitions in a model ferromagnet.

∆l = ±1, ∆s = 0, ∆ml = ±1, and ∆ms = 0, yields an imbalance in the transition intensities forspin-up and spin-down electrons (see Refs. [8] and [72] for details). In the absence of the exchange
splitting of the d-bands ∆ex , however, this imbalance does not manifest itself as XMCD. The latter
requires a second step, in which the exchange-split d-bands act as a spin detector. In Fig. 2.12,

34
unoccupied states above the Fermi level exist only for the minority spin-up states, which meansthat only spin-up electrons photoexcited from the 3p states can be absorbed. Combined with anasymmetry in transition intensities between spin-up and spin-down electrons for circularly polarized
incident light, this results in an absorption asymmetry between the two circular polarizations.
The dipole matrix elements of the transitions between initial i and final f states for linear
and right and left circular polarizations are [72]
Dlinear = hi| ẑ |f i ,Dright = hi| x̂ + iŷ |f i ,(2.36)

Dlef t = hi| x̂ − iŷ |f i .
First principles calculations of the transition elements are difficult yet necessary for the computationof the optical properties of materials by use of the linear response theory [73], which allows one tocalculate the rank 2 optical conductivity tensor σ̂ [74, 75] related to the dielectric tensor ˆ asˆ = Î +

4πi
σ̂,
ω

(2.37)

where Î is the identity matrix.
The dielectric tensor, in general, has non-zero off-diagonal components xy , referred to as themagneto-optical dielectric permittivity, which give rise not just to XMCD, but to other magnetooptical effects as well. Because the transition matrix elements given by eq. (2.36) are complex, xy is
also complex and therefore affects both the amplitude and the phase of the transmitted EUV light.
The latter, in the case of linearly polarized light incident normal to the surface of an out-of-planemagnetized film, leads to a magneto-optical rotation of the polarization, known as the Faraday
effect. More details on the magneto-optical effects in a transmission geometry can be found in
Chapter 5.
In reflection geometries, xy affects the Fresnel reflection coefficients and leads to magnetooptical changes in the amplitude, phase and polarization of the reflected light, known as themagneto-optical Kerr effect (MOKE). Based on the symmetry of the dielectric tensor, there arethree special cases when the magnetization is oriented in three distinct directions with respect to35
the material’s surface and the plane of incidence, as shown in Fig. 2.13, which are called longitudinal(L), polar (P), and transverse (T) MOKE.

Figure 2.13: P- (blue), L- (green), and T-MOKE (red) geometries. The geometries are definedwith respect to the plane of incidence of the probe and the surface of the sample. In P-MOKE, themagnetization mz is orthogonal to the surface and parallel to the plane of incidence; in L-MOKE,
my is parallel to both; in T-MOKE, mx is parallel to the surface and orthogonal to the plane ofincidence.

An analysis of the Maxwell boundary conditions at an interface between vacuum and a semiinfinite magnetized layer yields magneto-optical Kerr rotation θM OKE and ellipticity EM OKE for
the incident s- and p-polarizations for L- and P-MOKE, which, combined in a single complex
quantity, are written as [73]xy
xx (xx − 1)

θL−M OKE + iEL−M OKE ≈ √(p)(p)
θL−M OKE + iEL−M OKE ≈ √xyxx (xx − 1)



(s)
(s)
θP −M OKE + iEP −M OKE ≈ √xyxx (xx − 1)



(p)
(p)
θP −M OKE + iEP −M OKE ≈ √xyxx (xx − 1)



(s)

(s)

cos φi tan φtcos (φi − φt )

cos φi tan φtcos (φi + φt )

cos φicos (φi − φt )

cos φicos (φi + φt ),,
(2.38),,

where φi is the angle of incidence and φt is the angle of refraction, and xx is the diagonal elementof the dielectric tensor. L-MOKE and P-MOKE geometries require a polarization analysis of themagneto-optical reflections, which is difficult in the EUV spectral range due to a lack of efficientpolarizers. Unlike L- and P-MOKE, T-MOKE shows no change in the polarization, however, for ppolarization, there is a magneto-optical change in amplitude, which makes this geometry well suited36
for EUV magneto-optical spectroscopy. More details on EUV T-MOKE as well as an introductionof a new magneto-optical spectroscopic technique can be found in Chapter 4.

2.3

Conclusions
In this chapter, the Stoner and Heisenberg models of magnetism have been discussed. In theformer model, electrons are delocalized whereas in the latter they are treated as local. The Stonermodel explains the fractional magnetic moments of metals, and the Heisenberg model correctlypredicts their thermodynamic properties. Spin fluctuations in the two models play an important rolein the magnetization dynamics of materials following an ultrafast laser excitation. I also discussed
the origin of magnetic domains and other textures, which may provide additional microscopic
relaxation channels for laser-driven magnetization dynamics.
The splitting of the spin-up and spin-down bands in a ferromagnet leads to a polarization- andmagnetization-dependent absorption of light, which underlies magneto-optical effects. The latter
effects allow us to access the magnetic state of a material with light, and the magnetic contrast isenhanced at the elemental absorption edges.

Chapter 3

Ultrafast Sources of Coherent EUV and Soft X-ray Radiation

In this chapter, I discuss sources of extreme ultraviolet (EUV) and soft X-ray radiation, which
are capable of producing temporally and spatially coherent femtosecond pulses and can thus be
used in experiments studying ultrafast dynamics of materials and devices at the nanoscale with ahigh temporal and spatial resolution. My main focus is on tabletop laser-driven sources based onhigh harmonic generation (HHG), as they are the primary tool used in the experiments reported
in this thesis. I also briefly discuss the working principles of a free-electron laser (FEL), which wasused in the experiment reported in Chapter 6.
Historically, the first X-rays were produced in vacuum discharge tubes via bremsstrahlung
and X-ray fluorescence. Synchrotron radiation was discovered in 1947 [76], and conversions ofsynchrotrons from particle accelerators into light sources started in the 1950’s. In synchrotrons,
radiation is produced by accelerating charged particles (most often electrons) moving at nearly
the speed of light, and the spectrum ranges from infrared wavelengths to hard X-rays. Due to its
small wavelengths, synchrotron light has been used to study many material and molecular systems,
but it lacks the temporal resolution necessary to study such systems at their natural timescales(see Fig. 1.1). Today, laser-driven HHG sources and FELs are capable of providing both a highspatial and temporal resolution and can be used for element-specific ultrafast studies of material
and molecular systems. HHG sources are particularly attractive as they are suited for laboratoryscale experiments and provide ease of access and require less maintenance and operation costs thanlarge scale experimental facilities.

38

3.1

High Harmonic Generation
The first observation of HHG was reported in 1987 [77, 78], and a broad HHG spectrum

generated with IR lasers with roughly similar harmonic intensities throughout the spectrum up to
an HHG cutoff energy was reported shortly thereafter [79, 80]. The development of the theory of
HHG started around the same time.

3.1.1

The three-step model

Historically, the first model of HHG was a semiclassical three-step model [81, 82, 83], shownin Fig. 3.1. Within this model, in a single-atom picture, an electron starts in the ground state
and, when the atomic potential is distorted by the electric field of an incident laser pulse, it cantunnel out of the atom. Given the typical ionization energies of noble gases of ∼15-20 eV, the tunnel
ionization requires laser intensities > 1014 W/cm2 , which can be easily achieved using chirped pulse
amplification (CPA) [84, 85] of femtosecond laser pulses. The ionization rate is known as the ADK
rate after Ammosov, Delone, and Krainov who performed the original fully quantum calculations[86]. In the second step, the electron is accelerated in the laser field. Finally, for a range of phasesof the laser field, the electron trajectory is closed, and the electron recombines with the parent ion
emitting the energy it gained from the laser field in the form of a short-wavelength burst of light.
An electron motion in the field can be described classically by Newton’s equations of motion.
For a linearly polarized laser driver, the problem becomes one-dimensional. Consider an electron(charge −e, and mass me ) in a harmonic field E0 cos ωt, where ω is the angular frequency of the
field. An equation of motion then reads
me ẍ + eE0 cos ωt = 0.

(3.1)

Integrating this equation with the initial condition that, at the time of the ionization ti , of zero

39

Figure 3.1: Three-step model of HHG. (a) Original and perturbed atomic potential along with thedriving laser potential. (b)-(d) Steps of the three-step model. (b) The atomic potential is disortedby the laser field allowing tunnel ionization (the electron wavepacket is shown in red). (c) The
ionized electron is accelerated in the field gaining energy, (d) which is released in the form of a highenergy photon (purple) upon recombination with the parent ion.

40
position x and velocity ẋ of the electron yields
eE0ẋ = −(sin ωt − sin ωti ) ,
me ω


1
eE0
(t − ti ) sin ωti + (cos ωt − cos ωti ) .
x=
me ω
ω

(3.2)

Solutions for different ionization phases ωti are plotted in Fig. 3.2. Note that in order for the

Figure 3.2: Solutions of the three-step model for different ionization phases.

electron to recombine with the parent ion, the solution must cross zero, which is only true when
the initial phase 0 ≤ ωti ≤ π/2, within a single cylce of the laser driver.
The ponderomotive energy Up is defined as the time-averaged kinetic energy of the ionizedelectron, and from eq. (3.2), it is found thatUp =e2 E02.
4me ω 2

(3.3)

Eq. (3.2) can be solved numerically to find ti that gives the maximum possible kinetic energy
Tmax of an electron upon recombination. The solution yields Tmax ≈ 3.17Up . The maximum
possible HHG photon energy hνmax is given by Tmax and the atomic ionization energy Ip , which is41
the potential energy of the ionized electron relative to its ground statehνmax = Ip + 3.17Up .

(3.4)

hνmax is also called the cutoff energy. It scales linearly with Ip , which, of the noble gases, is thehighest for He (24.59 eV), and is also linearly dependent on the laser intensity IL and the squareof the laser wavelength λ2 . Therefore, in order to maximize the HHG photon energy, one mustuse a gas with a high Ip , a laser driver with a high peak intensity IL , which can be obtained bycompressing the pulse temporally, and a long wavelength λ.

3.1.2

Electron wavefunction dynamics

The three-step model is included in the more general quantum picture of HHG [87, 88]. A
rigorous quantum model must take into account the selection rules in a multi-electron atom. An
approximation of a single active electron in an effective field created by the nucleus and the restof the electrons in an atom allows one to simplify the full Schrödinger equation to a single particle
problem [87], for which the time-dependent Schrödinger equation (TDSE) reads


~2 2
∂
−
∇ + V̂ (~r, t) |ψ (~r, t)i = i~ |ψ (~r, t)i .
2me
∂t

(3.5)

The time-dependent potential V̂ (~r, t) includes a static atomic potential V̂a (~r) and a time-dependent
laser potential V̂L (~r, t)V̂ (~r, t) = V̂a (~r) + V̂L (~r, t) = −e2~√− eE(t)· ~r,
4π0 r2 + α2

(3.6)

~
where the laser field E(t)in the case of a linearly polarized laser driver is~~ 0 (t) cos ωt,
E(t)
=E

(3.7)

~ 0 (t) is the envelope of the pulse.
where EAn initial condition for |ψ (~r)i is the ground state wavefunction of an unperturbed atom.
2~
Therefore, the minimum energy eigenvector of the time-independent Hamiltonian Ĥ = − 2m∇2 +eV̂a (~r) must be found prior to solving the time-dependent problem. This can be done by use of a

42
finite-difference numerical scheme [89]. TDSE (3.5) can then be solved. The solution can formally
be written as


 
i
~2 2
|ψ (~r, t)i = exp − −∇ + V̂ (~r, t) t |ψ (~r, t = 0)i .
(3.8)
~
2me

i 
h
~2
2 + V̂ (~With a known evolution operator U ˆ(t) = exp − ~i − 2m∇r,t)t , eq. (3.8) can be usede
to find the laser-driven dynamics of the electron wavepacket. The difficulty in evaluating Û is2~
that the kinetic and potential energy operators T̂ = − 2m
∇2 and V̂ (~r, t) do not commute and,e

6 eT̂ eV̂ . However, on small time steps ∆t, the Trotter approximation can be
therefore, eT̂ +V̂ =1/∆t 1/∆tapplied lim∆t→0 e(T̂ +V̂ )∆t= eT̂ ∆t eV̂ ∆t[90], and thus the evolution operator becomesÛ (∆t) ≈ eT̂ ∆t eV̂ ∆t .

(3.9)

The spectral theorem [91] states that for a diagonal operator Ô there is a space, on which therepresentation of Ô is a multiplication operator. For the potential energy operator V̂ this is real
space. Also, for a diagonal operator, eÔ = diag(eOi ), where Oi are the eigenvalues of Ô. Hence,
the action of eV̂ ∆t on |ψi in real space is equivalent to multiplying |ψ (~r)i by eV (~r)∆t . The kinetic
2 2energy operator T̂ = ~2mke is diagonal in momentum space, which is related to real space via Fourier~transform, and, therefore, the multiplication by eT (k)∆t is performed in momentum space. With
this, the time propagation of |ψ (~r, t)i can be written as

n i
o
2
− i~k
∆t
V
(~
r
,t)∆t
−1
−
|ψ (~r, t + ∆t)i = F
e 2me F e ~|ψ (~r, t)i.

(3.10)In the numerical implementation of eq. (3.10), the multiplication in real space was split in two
steps before and after the multiplication in Fourier space in order to improve the stability of themethod, and an apodizing mask was applied at each time step to simulate an open boundary of afinite computation window.
Simulated dynamics of an electron wavefunction in a hydrogen-like atom are shown in Fig. 3.3
along with the ground-state wavefunction. The highly localized ground-state wavefunction is spreadaround by the laser field and forms time dependent ripples. The spreading amounts to the tunnel
ionization discussed above, and the ripples lead to a time-dependent dipole moment which produces
dipole radiation.

43
According to the Larmor formula, the total power radiated by an accelerating charge is [92]µ0 q 2 |ẍ|2µ0 |p̈|2
P (t) =
=
,
6πc
6πc

(3.11)

where µ0 is the permeability of free space and q is the electric charge, and the dipole momentp is defined as p = qx (for a dipole in the x-direction). With a known time dependence of the

Figure 3.3: (a) Ground-state and (b) laser-driven dynamics of the electron wavefunction in ahydrogen-like atom. The ripples in the wavefunction in (b) give rise to dipole radiation.

wavefunction, the dipole moment can be computed asp~(t) = e hψ (~r, t)| ~r |ψ (~r, t)i .

(3.12)The power spectrum of HHG is proportional to F {P (t)}, where the time-dependent radiated poweris given by eq. (3.11). The dipole selection rules allow only odd harmonics in the spectrum, whichis confirmed experimentally in Fig. 3.4.

3.1.3

Circularly polarized HHG

Intuitively, it would seem that in order to produce circularly polarized harmonics, the HHG
process would have to be driven by a circularly polarized laser, for which the field in the (x, y)plane is given by~ circ (t) = E0 (t)(x̂ cos ωt + ŷ sin ωt),E

(3.13)

44

Intensity (counts)

2

#10 4

1.5
1
0.5
0
25

30

35

40

45

Harmomnic Order
Figure 3.4: Experimental spectrum of linearly polarized HHG in He. The fundamental wavelengthis 800 nm. Only odd harmonic orders are present.

45
where x̂ and ŷ are unit vectors in the x and y directions, respectively. However, solving eq. (3.5)~ circ given by eq. (3.13) yields no dipole oscillations, as seen in Fig. 3.5(a). Instead thewith E
wavefunction is driven away from the ion and the recombination never occurs.

Figure 3.5: Dynamics of an electron wavefunction in (a) a single-color circular and (b) a bichromatic
trefoil-shaped field. With a circularly polarized laser driver, the wavefunction is driven away fromthe ion and never recombines. The dipole oscillations responsible for HHG are missing. With abichromatic ω + 2ω trefoil-shaped driver, the dipole oscillations appear in three directions, which
effectively leads to circularly polarized HHG.

It would seem that circular polarization could only be achieved by conversion from linear [93].
However, such an approach is very lossy and can only be implemented in a rather narrow spectral
range. A direct production of circularly polarized HHG is possible, but it requires a specially shaped
laser field that would not only drive the wavefunction in both x- and y-directions, but would also
allow its re-collision with the parent ion. Such an approach was first proposed in 1995 [94, 95]. In
it, two co-propagating circularly polarized pulses rotating in opposite directions make up the laser
driver. The two pulses have different wavelengths: one is at the fundamental wavelength and theother is at its second harmonic. When added together, the two pulses produce an electric field thatis given by the following expression~ ω+2ω (t) = E0 (t)(x̂(cos ωt + cos 2ωt) + ŷ(sin ωt − sin 2ωt)).
E(3.14)

46~ ω+2ω (t) travels on a trefoil-like trajectory over a single cycle of the fundamental,
The field vector E
as shown in Fig. 3.6. When an electron is driven by such a field, it can recombine with the parent+=

Figure 3.6: The field of a two-color laser driver for circularly polarized HHG. A combination of
two circularly polarized co-propagating fields at the fundamental wavelength (red) and its second
harmonic (blue) that rotate in opposite directions results in a trefoil-like field pattern.

ion producing dipole oscillations (see Fig. 3.5(b)) and thus making HHG possible. A measurementof the ellipticity of HHG produced using the method described above is reported in Ref. [96] andthe helicity was measured in Ref. [97].
Notably, upon closer inspection of Fig. 3.5(b), it can be seen that the ripples in the wavefunction occur predominantly in three distinct directions. This threefold symmetry is imposed by thesymmetry of the bichromatic driver and results in three linearly polarized EUV bursts per one laser
cycle of the fundamental that are delayed with respect to one another, and the plane of polarizationof each burst is rotated by 120◦ relative to the other bursts. Together the three EUV bursts resultin right and left circularly polarized HHG [98]. The existence of the sequential linearly polarized
bursts was confirmed experimentally by analyzing quantum pathways of photoemitted electrons
from a metallic surface irradiated by circularly polarized harmonics [99].
A setup for circularly polarized HHG is shown in Fig. 3.7. A pulse train from a Ti:Sapphirelaser amplifier at a central wavelength of ∼790 nm travels through a BBO crystal to generate pulses
at 395 nm. The two beams are split into separate arms by a dichroic mirror, and a delay stage

47
is introduced in one arm to overlap the red and blue pulses in time; a set of waveplates in eacharm controls the polarization of the respective beam; another dichroic mirror recombines the two
beams in space, and the combined beam is focused into a hollow-core waveguide filled with He
where the HHG process takes place. The light emitted from the waveguide is measured with an
EUV spectrometer.

EUV Spectrometer

BBO
CCD

DM1

grating
DM2

WaveguideTM

Al filters

L2
λ400/4λ400/2L1

λ800/2λ800/4delay stage

Figure 3.7: Setup for circularly polarized HHG. A delay stage is used for a temporal overlap of the
red and blue pulse trains. DM1 and DM2 are dichroic mirrors that split and recombine the two
beams, respectively. The lenses L1 and L2 focus each beam into the waveguide, and the toroidal
mirror TM re-images the output of the waveguide onto the CCD.

The conservation of energy and angular momentum determines which circularly polarizedharmonic orders are allowed in the HHG spectrum. The harmonic order q is defined relative to thefundamental frequency ω. First, for n absorbed photons of frequency ω and m absorbed photons
of frequency 2ω, the conservation of energy givesqω = nω + m2ω.

(3.15)

Second, because the harmonics are either right or left circularly polarized, the conservation of thespin angular momentum requires thatn − m = ±1.

(3.16)

48
From eqs. (3.15) and (3.16), it follows that the allowed harmonic orders are
q = 3m ± 1.

(3.17)

Therefore, circularly polarized harmonics driven by a bichromatic driver defined by eq. (3.14) are
expected to appear in pairs, which is confirmed by an experimental spectrum in Fig. 3.8, in which
every third harmonic is suppressed.

Intensity (counts)

2

#10 4

1.5
1
0.5
0
20

25

30

35

Harmomnic Order
Figure 3.8: Experimental spectrum of circularly polarized HHG in He. Every third harmonic issuppressed, in agreement with the selection rules.

The helicity of each harmonic in the spectrum can be derived from eq. (3.16) and (3.17). For
instance, if n − m = 1, the helicity of the corresponding harmonic order q = 3m − 1 is determinedby the helicity of the field at the fundamental frequency ω, while for n − m = −1, the harmonicorder q = 3m + 1 has the same helicity as the second harmonic 2ω.
Circularly polarized HHG has been demonstrated in non-collinear geometries, where harmonics of different helicities could be separated spatially [100], and in the soft X-ray region with abichromatic driver composed of optical fields at 1.3 µm and 800 nm [101].
Both the semiclassical and quantum models of HHG discussed above deal with radiation bya single atom. In reality, the HHG process takes place in a finite volume of gas that is pumped by49
a laser, and the total HHG flux is built up from the light emitted by multiple atomic sources. If
radiation from these sources can be added in-phase, a significant improvement in HHG flux can beobtained. Such coherent addition is known as phase matching.

3.1.4

Phase matching

The concept of phase matching originated in nonlinear optics [102] in the context of parametric processes, e.g. four-wave mixing, second harmonic generation, sum and difference frequency
generation, etc. In order for such processes to be efficient, the correct phase relationships between
the interacting waves have to be maintained. Similarly, in HHG, the phase of EUV light emitted at
different points along the propagation direction of the laser driver needs to be the same in order for
the HHG signal to add constructively. The requirement that the phase between the laser driver andthe EUV light remains the same along the propagation direction means that their phase velocities
defined as vp = ω/k have to be equal, i.e.,
ωf
ωq
=
,
kf
kq

(3.18)

where ωf and kf are the fundamental angular frequency and wavenumber, and ωq and kq are theangular frequency and wavenumber of the harmonic of order q. Assuming that ωq = qωf , eq. (3.18)
becomes∆k = qkf − kq = 0,

(3.19)

where ∆k is called the phase mismatch. In the limit of perfect phase matching, when ∆k = 0, theEUV photon flux grows quadratically with the interaction length, greatly exceeding the non-phasematched flux, and reaches saturation when the absorption by the medium is taken into account(see Ref. [103] for details).
The difficulty in achieving phase matching for HHG has to do with the dispersion of therefractive index of the neutral gas and plasma. At EUV wavelengths, the refractive index is close
to 1, and their phase velocity is nearly equal to the speed of light c = 3 · 108 m/s, while for the IR
laser driver that is not the case. A solution to this problem was proposed in Ref. [104]. Instead of

50
a gas jet, the HHG process took place in a gas filled hollow-core waveguide (see Fig. 3.9), whichallowed the balancing of the dispersion of neutral atoms and plasma by a mode dispersion due tothe confinement of light in the waveguide.
IR pumpGas out

Gas inEUV

Figure 3.9: Hollow-core waveguide for phase-matched HHG. The laser driver is focused into the
waveguide such as to match its lowest order mode (EH11 )The expression for the phase mismatch in a hollow-core waveguide for a harmonic order q is
given by [105, 106] 2u11 λf2π− P (1 − η) (∆δ + n2 IL ) + P ηNa re λf ,∆k ≈ q4πa2
λf(3.20)

where λf is the fundamental wavelength of the driving laser, u11 = 2.405 is the first zero of thezeroth order Bessel function of the first kind J0 , a is the radius of the waveguide, P is the gaspressure, η is the ionization fraction of the gas, ∆δ = Re(n(λf ) − n(qλf )) is the difference in thereal parts of the indices of refraction at the fundamental wavelength and its q th harmonic, n2 is
the nonlinear refractive index, which is small and is often neglected [105], IL is the driving laser
intensity, Na is the number density of the gas per unit pressure, and re is the classical electron
radius. The first term in parentheses represents the waveguide dispersion, the second term is dueto the dispersion of neutral atoms and the third is due to plasma dispersion. The parameters thatcan be fairly easily controlled in an experiment are the waveguide radius a, the gas pressure P and
the ionization fraction η, which depends on the intensity of the laser and can be computed fromthe ADK tunnel ionization rate discussed above.
With ∆k = 0, eq. (3.20) can be solved to find the phase-matching pressureP ≈u211 λ2f4πa2 (2π∆δ − ηNa re λ2f ).

(3.21)At the critical ionization level ηc ≈ 2π∆δ/Na re λ2f , the denominator vanishes and P → ∞, whichmeans that phase matching above the critical ionization level is impossible. In He, the critical

51
ionization is ∼0.5%, and the corresponding laser fluence, based on the ADK rate calculations
mentioned above and assuming a 150 µm diameter waveguide and a 25 fs pulse with the central
wavelength of 800 nm, is ∼1.5 mJ/cm2 [107, 108]; the maximum possible harmonic order is 91[108].
The concept of phase matching is applicable for linearly as well as circularly polarized HHG,
albeit with helicity-dependent considerations [109], and bright phase-matched circularly polarizedharmonics were demonstrated in Ref. [97].

3.2

X-ray free-electron lasers
Another source of ultrafast coherent EUV and X-ray pulses is a free-electron laser (FEL).

Although the idea of an FEL originated in the 1970’s [110], X-ray FELs are fairly recent [111, 15].
In an FEL, a coherent X-ray pulse is produced when a relativistic electron bunch passes througha periodic series of alternating magnets, known as the undulator [112]. The power radiated by anaccelerating electron is given by the relativistic generalization of the Larmor formula (3.11)
P =iq 2 γ 6 h ~˙ 2
~˙ 2 ,(β) − (β~ × β)6π0 cwhere β~ is the ratio of the charge velocity to the speed of light, and γ = 1/

(3.22)
p1 − β 2 is the relativistic

Lorentz factor. At relativistic speeds, γ > 1 and the γ 6 factor in eq. (3.22) leads to a significant
increase in radiated power, which makes FELs so bright.
The key components of an X-ray electron laser are an electron injector that sends a shortelectron bunch into an accelerator section, in which it is accelerated to GeV energies, and an
undulator where the X-ray light is emitted. The discussion of FELs below is limited to the Linear
Coherent Light Source (LCLS) FEL at SLAC since the experiment reported in Chapter 6 was
performed there.
At LCLS, electrons are emitted from the copper photocathode of an RF electron gun by a
UV laser operating at 253 nm (third harmonic of an amplified Ti:Sapphire laser) [113]. The RF
gun boosts the energy of photoelectrons to 135 MeV and sends them into two sequential bunch52compressors, which accelerate and shorten the duration of electron bunches. The stochastic natureof the photoesmission process at the cathode results in fluctuations of the X-ray pulse brightness
and introduces an uncertainty in the pulse arrival time thus limiting the temporal resolution of
the FEL, which can be improved by measuring the relative arrival time of the pulses [114]. Afterthe compressors, an electron bunch is injected into the tunnel of a linear accelerator re-purposedfrom particle physics experiments. After the accelerator section, high energy electrons enter theundulator where the generation of an X-ray pulse takes place as the electrons are forced to move on a
curved trajectory in an alternating magnetic field. As they propagate along the undulator, electrons
and X-ray photons couple to one another. Such coupling results in self-amplified spontaneousemission (SASE) from the spontaneous emission arising from the initial electron beam shot noise.
Again, due to its stochastic nature, SASE gives rise to fluctuations in the beam brightness and leads
to a rather low temporal coherence [115]. To improve the latter, SASE can be used to self-seed
the FEL [115]. Because X-rays are amplified through interactions with relativistic electrons, thelatter can be viewed as a laser gain medium, and X-ray FELs can thus be considered single-passlaser amplifiers. Gain saturation in a SASE FEL is achieved by use of very long undulators or gain
media [111].
In the soft X-ray and EUV regions, external seeding methods, which include seeding with
HHG, have been proposed to reduce fluctuations [116, 117, 118] and a fully seeded operation has
been implemented at the FERMI FEL in Trieste [119, 120].
FELs can provide coherent ultrashort X-ray pulses of very high brightness. However, due totheir complex design and high operational costs, there are only a few FEL user facilities available
worldwide. Currently, they can deliver an X-ray beam to only one experimental end station ata time. For these reasons, their accessibility is severely limited. Due to their compactness andrelatively low cost, HHG sources are much more accessible and can be used in a broad range ofmaterials science, biology and chemistry experiments that are not as demanding with regards to thephoton flux but require a high temporal resolution, broad spectrum, or good coherence propertiesof the probe.

53

3.3

Conclusions
In this chapter, I reviewed HHG and FEL sources of EUV and soft X-ray light suited forstudying material and molecular systems with high temporal and spatial resolution as well as
element specificity. Both the classical three-step model of HHG in a single atom and a full quantum picture based on a solution of the time-dependent Schrodinger equation for an electron in a
combined atomic and laser potential were presented. Quantum simulations show that circularly
polarized HHG cannot be achieved with a cicularly polarized single-color laser driver and a complex
field pattern is required to generate circular polarization directly. Such a field, called a bichromatic
driver, can be obtained by mixing two laser pulses of different colors. Conservation laws constrain
the allowed harmonic orders: even and every third orders are suppressed for linearly and circularly
polarized HHG, respectively. This is confirmed by the experimental HHG spectra for the two polarizations. Neutral gas and plasma dispersions in an active HHG region can be compensated for byconfining light in a hollow-core waveguide, and the HHG process can thus be phase-matched, which
results in a significant increase in the intensity of an HHG-based tabletop source of EUV light withfull polarization control. FELs produce very bright X-ray pulses by use of accelerated relativistic
electron bunches but due to their complexity and cost allow very limited access to users. HHG andFEL sources can compliment one another in their respective applications in order to realize the fullpotential of ultrafast, coherent X-rays in the study of material, molecular, and biological systems
and nanoscale devices.

Chapter 4

Extension of the EUV Magneto-Optical Spectroscopy

In this chapter, I extend the traditional magneto-optical techniques and present two approaches that are capable of measuring the full resonant EUV complex magneto-optical permittivity xy , which depends on the microscopic state of a magnetic material, in a reflection geometry. In
the first approach, xy can be extrated from the magneto-optical reflectivity spectra measured at
multiple angles of incidence in the transverse magneto-optical Kerr effect (T-MOKE) geometry. In
the second, more efficient approach, the angle of incidence is kept fixed, and a series of magnetooptical spectra is taken by scanning the polarization direction of a linearly polarized EUV probe.
I show that xy can be unambiguously determined from this extensive data set.
The two techniques were applied in a pump-probe experiment on thin Co films in order totrace the evolution of xy during the course of laser-induced ultrafast demagnetization. With thehelp of ab initio DFT simulations (carried out at Uppsala University), a connection was made to
the microscopic processes, i.e., ultrafast longitudinal and transverse excitations and their role in the
demagnetization process on femtosecond and picosecond timescales. The magnetization response to
an ultrashort IR laser excitation was found to be dominated by transverse excitations, or magnons,with a possible smaller contribution from a transient reduction of the exchange splitting. Thisconclusion is supported independently by the angle- and polarization-resolved techniques.

55

4.1

Time-resolved EUV T-MOKE
I start by describing T-MOKE in the EUV spectral region, which has an advantage, comparedto the visible one, that stems from its resonant nature because magneto-optical reflection spectra
taken at the resonant absorption edges (particularly the M -edges of the ferromagnetic 3d metals Fe,
Co, and Ni in the range of ∼ 50-70 eV) can provide access to the microscopic picture of magnetism
and its dynamics.
The three MOKE geometries P-, L-, and T-MOKE are discussed in Chapter 2. The former
two require an analysis of the polarization state of the reflected light and are thus not very well suitedfor the EUV photon energies, due to a lack of efficient polarization analyzers. In the transverse
geometry, however, the polarization state of the reflected light remains unchanged, and the contrast
is purely an amplitude one. This can be shown analytically by considering the dielectric tensor(relative to the permittivity of free space 0 ) for the transverse magnetization direction along thex-axis (see Fig. 2.13) in an isotropic material [121]




0
0 
xx




ˆ =  0
,

−iQ
xx
xx




0 iQxx
xx(4.1)

where Q is the magneto-optical constant, and the off-diagonal element of ˆ is defined as xy =−iQxx .
Consider the Maxwell curl equations together with the constitutive relations (in a medium
with a magnetic permeability µ = 1, relative to the permeability of free space µ0 , and in the absenceof current sources)~~ ×E~ = − ∂B ,
∇
∂t
~
~ ×H
~ = ∂D ,∇∂t~ = 0 ˆE,~D
~ = 1 B.
~
H
µ0

(4.2)

56
√
By taking a curl of the first equation in (4.2) and using c = 1/ 0 µ0 , the following form of
the wave equation can be derived


2~
~ ·E
~ = − ˆ ∂ E .
~ +∇~ ∇− ∇2 Ec2 ∂t2

(4.3)

~ = E
~ 0 ei(ω~n·~r/c−ωt) in eq. (4.3), where the refractive index
By inserting a plane wave solution E~n is defined along the direction of the wavevector ~k as ~n = ~kc/ω, an eigenvalue equation can beobtained [73]
~ 0 = 0,|~n|2 − ˆ + ~n ⊗ ~n · E(4.4)

where ⊗ denotes an outer product of two column vectors. The eigenvalues are found from
det |~n|2 − ˆ + ~n ⊗ ~n = 0,(4.5)

~ 0 are found from eq. (4.4).
and the corresponding eigenvectors E
For the dielectric tensor in the T-MOKE geometry, defined by eq. (4.1), the eiegenvectors
correspond to the s- and p-polarizations, and the respective eigenvalues of the refractive index are
n2s = xx ,n2p = xx +

2xy
.
xx

(4.6)

Therefore, in the T-MOKE geometry, only the p-polarization is affected by magneto-optical changes
to the refractive index. Taking eqs (4.6) into account, the Fresnel reflections in the transverse
geometry at an interface of a semi-infinite magnetic layer are [121]rpp =ns cos φi − cos φt
2ns cos φi sin φt+ iQ,
ns cos φi + cos φt(ns cos φi + cos φt )2
cos φi − ns cos φt
rss =,cos φi + ns cos φt(4.7)

where φi and φt are the incident and refracted angles, and rpp and rss are the reflections for the pand s-polarizations, respectively. Generally speaking, the polarization can rotate from s to p andvice versa, upon reflection from a surface magnetized in an arbitrary direction. For that reason,
it is convenient to add reflection coefficients rsp and rps , which vanish in the transverse geometry[121].

57
Typically, in an experiment, a reflected intensity is measured Ir = I0 |rpp |2 , where I0 is theincident intensity, and thus the information about the phase of the complex reflection coefficient
(0)

(1)

cos φi −cos φt
2ns cos φi sin φt
rpp is lost. For brevity, I define the parameters rpp = nnss cos
φi +cos φt and rpp = i (n cos φ +cos φ )2 ,sit

and rewrite the p-polarization reflection coefficient as(0)(1)
rpp = rpp
+ Qrpp.

(4.8)

To a first order in Q, the reflected intensity is



(0) 2
(1) (0)Ir = I0 |rpp
| + 2Re Qrpp
rpp
.

(4.9)

For a source with a fluctuating I0 , it can be difficult to distinguish the magnetic contribution to
the total reflected intensity. For that reason, a differential measurement is usually performed by(+)

magnetically saturating the sample in a positive direction, measuring the reflected intensity Ir ,and then reversing the magnetization direction, upon which the sign of Q in eq. (4.9) is also(−)

reversed, and measuring the reflected intensity Ir . From the two measurements, a quantity called
the magneto-optical asymmetry is constructed
A=

(+)

− Ir

(+)

+ Ir

Ir
Ir

(−)

(−)

.

(4.10)

Using eqs. (4.7) and (4.9) and Snell’s law sin φt = sin φi /ns , an explicit expression for A can bederivedA = 2Re(1) (0)
Qrpp rpp(0)|rpp |2

!

= 2Re (0)
rpp

iQ sin 2φi2n2s cos2 φi − 1 + sinn2φi
s




(0) ∗


.

(4.11)

rpp



(0)
(0) ∗
With an approximation rpp / rpp≈ 1, which holds relatively well for materials’ refractive indicesin the EUV spectral range, and given that xy = −iQn2s , the asymmetry can be expressed as (up toan overall sign, which depends on an arbitrary definition of the positive and negative magnetization)A ≈ 2Rexy sin 2φi
42
ns cos φi − n2s + sin2 φi
.

(4.12)

It is important that A ∝ Q, which allows one to access the magnetic state by performing a magnetooptical differential measurement. An advantage of T-MOKE measured with an HHG source is58that the HHG spectrum is so broad that it can access the absorption edges of multiple elements
simultaneously and thus probe laser-induced magnetization dynamics of alloys and multilayer films
with elemental specificity. The asymmetry is maximized at the Brewster angle, which, for a typicalmetal, is near 45◦ . An example magneto-optical asymmetry spectrum taken with an HHG sourceat the M -edge of Co is shown in Fig. 4.1. The figure clearly shows a difference between the
spectra measured at the opposite magnetization directions (blue). The difference results in an
EUV spectrum of the magneto-optical asymmetry shown in orange.

Figure 4.1: An example T-MOKE asymmetry spectrum (orange) at the M -edge of Co. The HHG
spectra reflected from the sample when it was positively (solid) and negatively (dashed) magnetized
are shown in blue.

The spectra in Fig. 4.1 were measured with a pump-probe T-MOKE setup shown in Fig. 4.2.
In the setup, an IR laser pulse train is delivered by a KMLabs Dragon Ti:Sapphire laser amplifier
producing 20 fs pulses at a ∼795 nm central wavelength with an energy of 2 mJ/pulse at a repetition
rate of 4 kHz. This pulse train is split into two beams: 10% of the original beam goes into the pumparm, which has a delay stage in it to control the arrival time of pump pulses on the sample, while
90% is focused into a hollow-core waveguide where a Ne gas flows under a pressure of ∼630 Torr.
In the waveguide, the IR laser drives a phase-matched HHG process, which produces broadband
EUV light whose spectrum is shown in Fig. 4.1. A toroidal mirror with an imaging distance of59
60 cm re-images the output of the waveguide onto a CCD camera (Andor Newton 920), which isoperated in a full vertical binning mode, to increase the readout rate. Before reaching the CCD,
an EUV beam impinges on a magnetic sample that has grooves etched throughout its thickness
with a line spacing of 500 lines/mm in order to achieve spectral resolution and thus simultaneously
acts as a spectrometer diffraction grating. The magnetic field at the sample is controlled by anelectromagnet. Thin film Al filters are placed right after the waveguide and right in front of the
CCD camera to block any residual IR light coming from the waveguide and reflected from thesample, respectively. The pump and the probe beams are combined on a mirror with an apertureat the center. The EUV beam is transmitted through the aperture while the pump beam is reflectedfrom the mirror as close to the aperture as possible to minimize the angle between the two beams
and thus improve the time resolution. The angle of the goniometer arm, that includes both thesample and the CCD, can be adjusted, which allows a measurement of the T-MOKE response atmultiple angles of incidence. The importance of such a measurement is discussed below.
The diagram in Fig. 4.3 below shows the data collection flow of the T-MOKE setup. The
synchronization of the instruments and the data acquisition was implemented in LabView.
Fig. 4.4 shows an example ultrafast demagnetization and recovery trace in pure cobalt
measured with the setup in Fig. 4.2 and the data collection procedure in Fig. 4.3. While thedemagnetization trace in Fig. 4.4 represents an average response of the material’s magnetization toan ultrafast laser pulse, it, nonetheless, is of limited utility because the T-MOKE asymmetry definedby eq. (4.12) and measured at a fixed angle does not allow the separation of the real and imaginaryparts of xy . Knowledge of the evolution of the full resonant complex xy is important because
it would allow one, through first principles calculations, to make a connection to the microscopic
mechanisms that drive the magnetization response in Fig. 4.4. Specifically, Ref. [70] makes aprediction that longitudinal Stoner excitations will have a different effect on the magneto-optical
spectrum than transverse magnon excitations (see Chapter 2 for details). Namely, longitudinal
excitations change the spectral shape of xy , while transverse excitations simply lead to a uniform
reduction of its magnitude across the entire spectrum. It is, therefore, important to have a technique

60

λ/2

Figure 4.2: Time-resolved EUV T-MOKE setup. Laser-driven HHG in the waveguide produces a
train of EUV pulses that are inherently synchronized with the IR laser pulses in the pump arm,
which has a delay stage in it to control the delay time between the pump and the probe at the
sample. The toroidal mirror re-images the output of the waveguide onto the CCD camera, whichcollects the EUV light diffracted by the grating sample. The projection-field electromagnet applies
a magnetic field to the sample. The CCD and the sample can be tilted simultaneously, and thus
the T-MOKE response can be measured at various angles of incidence.

61
Collect statistics
Set
magnet𝐼+
Expose
CCD

Move to a new pumpprobe delay

ReversemagnetNo

Asymmetry𝐼−

Expose
CCD

Done?

Yes

Done?

Yes

Outputdata

No

Figure 4.3: Time-resolved EUV T-MOKE data collection flow.

Figure 4.4: An example ultrafast demagnetization curve measured from the T-MOKE signal at the
M -shell absorption edge in pure Co.

62
that would allow a direct extraction of xy .

4.2

Measurement of the full complex magneto-optical permittivity
Such an extraction can be preformed in two ways—by adding an angle resolution to T-MOKEor by setting the magnetization direction at an angle to its transverse orientation and performing ascan of the polarization angle of a linearly polarized EUV probe. I outline both approaches below.

4.2.1

Angle-resolved T-MOKE

I start with the angle-resolved T-MOKE approach as it is a straightforward extension of
the conventional T-MOKE given by expression (4.12) [122]. I define an angle-dependent factor
2φip(φi ) = n4 cos2 φsin−n2 +sin2 φ and rewrite eq. (4.12) as
s

i

s

i

A = p(φi )xy + p∗ (φi )∗xy .

(4.13)For two different angles of incidence, eq. (4.13) yields a linear system with two equations and two
unknowns xy and ∗xy , which can be solved uniquely to find both the real and imaginary partsof xy . The factor p(φi ) depends on the refractive index, which might change when the sample
is pumped by a femtosecond IR laser pulse. However, it has been shown that the non-magnetic
contribution to the EUV T-MOKE asymmetry from ultrafast changes in the refractive index is
two orders of magnitude smaller than the magnetic one and can thus be neglected [123, 124]. In
practice, due to experimental uncertainties and noise in the data, more than two angles of incidence
are required in order to solve for xy . An angle-resolved T-MOKE measurement can be carried out
with the setup shown in Fig. 4.2 by titlting the sample and, correspondingly, the CCD camera. A
static angle-resolved T-MOKE asymmetry measured at the M -edge of Co is shown in Fig. 4.5

4.2.2

Diagonal magneto-optical effect

I now discuss the second approach that allows a direct measurement of xy [125]. In thisapproach, the magnetization is set ”diagonally”, i.e. it has both a transverse and a longitudinal

63

Figure 4.5: Static angle-resolved T-MOKE signal from a Co film: (a) Raw, (b) Interpolated

64
component, as shown in Fig. 4.6, and, for that reason, the effect is called a diagonal magneto-opticaleffect or D-MOE.

Figure 4.6: D-MOE geometry. The magnetization is set in-between the longitudinal and transverse
orientations and the polarization angle is rotated. An example multilayer structure of a Co sample
is also shown. θ is the polarization angle with repsect to s-polarization.

I start the derivation of the D-MOE asymmetry at the interface of a semi-infinite magnetic~ i in the basis of s- andlayer by expanding the amplitude of an incident linearly polarized field E
p-polarizations (Es , Ep )


 

E
cos
θ
s


~i = 
E
 =
 E0 ,
Epsin θ(4.14)

where θ is a polarization angle relative to s-polarization, as shown in Fig. 4.6, and E0 is themagnitude of the incident field, which I set to 1 in the derivation below without loss of generality.
It is convenient to assemble the Fresnel reflection coefficients similar to the ones in eqs. (4.7) intoa 2 × 2 Fresnel reflection matrix r̂. The relationship between the incident and reflected fields thenbecomes




cos
θ

~ r = r̂E
~ i = r̂ 
E

,
sin θ

(4.15)

where r̂ depends on the magnetization m~ [121, 126]. For an in-plane magnetization direction, to a65
first order in magneto-optical Voigt parameter Q = ixy /xx , r̂ can be written as

 

(0)
(1)
rsp my Q rss rsp   rss
r̂(m)
~ =
'
,
(1)
(0)
(1)
rps rpp−rsp my Q rpp + rpp mx Q

(4.16)

where the superscripts (0) and (1) denote non-magnetic and magnetic first order contributions to(0)

the reflection coefficients, respectively. Note that rss ≡ rss . With eq. (4.10) and expressions (4.15)
and (4.16), the magneto-optical asymmetry for a sample magnetized in-plane with magnetization
components mx and my along the x- and y-axis, respectively, can be derived (again, keeping only
terms up to first order in Q)




(0)
(1)
(0)∗ (1)my sin 2θ < (rss − rpp )∗ rsp Q − mx (1 − cos 2θ) < rpp rpp Q
A=

.

(0)

|rpp |2 sin2 θ + |rss |2 cos2 θ(4.17)For transverse (mx = 1, my = 0) and longitudinal (mx = 0, my = 1) configurations, the familiar
expressions for the asymmetries AT and AL are recovered [127, 128](0)∗ (1)(cos 2θ − 1) < rpp rpp Q,
AT := A(mx = 1; my = 0) = (0)|rpp |2 sin2 θ + |rss |2 cos2 θ


(0)
(1)
sin 2θ < (rss − rpp )∗ rsp Q
AL := A(mx = 0; my = 1) =.
(0)
|rpp |2 sin2 θ + |rss |2 cos2 θ(4.18)(4.19)

Note that the T-MOKE asymmetry AT maximizes when θ = π/2, i.e. for p-polarization, andAL = 0 for θ = π/2 and θ = 0, which means that a mixed polarization is required to measure an
asymmetry in the longitudinal configuration.
√
In the diagonal geometry, |mx | = my = 1/ 2, and the D-MOE asymmetry is11
AD := A mx = − √ ; my = √= FD (θ)Q + FD∗ (θ)Q∗ ,2
2(4.20)

where the complex factor FD (θ) has been defined as
(0)

(1)

(0)∗ (1)sin 2θ(rss − rpp )∗ rsp + (1 − cos 2θ) rpp rpp

FD (θ) =.
√  (0)2 2 |rpp |2 sin2 θ + |rss |2 cos2 θ(4.21)

Note that eq. (4.20) would hold for any other in-plane orientation of m,~ as long as both the x and√y magnetization components are non-zero. Here a symmetric configuration |mx | = my = 1/ 2 ischosen for simplicity and without loss of generality.

66
I now show that eqs. (4.20) and (4.21) can be used to uniquely determine Q, and thus xyfrom measurements taken at multiple polarization angles, while the transverse and longitudinalgeometries are not suited for that purpose. I rewrite AT and AL in a form similar to eq. (4.20)
(0)∗ (1)(0) (1)∗ ∗AT = FT (θ)Q + FT∗ (θ)Q∗ ≡ fT (θ) rpp
rpp Q + rpp
rpp Q ,

(4.22)(0) ∗ (1)(0) (1)∗ ∗AL = FL (θ)Q + FL∗ (θ)Q∗ ≡ fL (θ) (rss − rpp) rsp Q + (rss − rpp)rsp Q ,(4.23)

where the parameters fT (θ) and fL (θ)—as well as FT (θ) and FL (θ) implicitly—are defined asfT (θ) ≡fL (θ) ≡FT (θ)(0)∗ (1)rpp rpp(cos 2θ − 1)=
2

FL (θ)(0)(1)(rss − rpp )∗ rsp



(0)|rpp |2 sin2 θ + |rss |2 cos2 θ,(4.24)

sin 2θ

=2



(0)|rpp |2 sin2 θ + |rss |2 cos2 θ.

(4.25)

A system of two linearly independent equations is required in order to find a unique solution for Q
and Q∗ . Such a system can be obtained by measuring the magnetic asymmetry spectrum at two
different polarization angles θ1 and θ2 and, in the most general form, can be written as  

ax bx   Q  Ax (θ1 )
  = 
,
cx dx
Q∗
Ax (θ2 )(4.26)

where the subscript x = T, L or D. From eqs. (4.22), (4.23), and (4.20), for the transverse,
longitudinal, and diagonal geometries, the system matrix Λx has the following explicit expressions 
(0)∗ (1)(0) (1)∗
aT bT  fT (θ1 )rpp rpp fT (θ1 )rpp rpp ΛT := (4.27)
=,(0)∗ (1)(0) (1)∗
cT dT
fT (θ2 )rpp rpp fT (θ2 )rpp rpp (0) ∗ (1)
aL bL  fL (θ1 )(rss − rpp ) rsp
ΛL := 
=
(0)
(1)
cL dL
fL (θ2 )(rss − rpp )∗ rsp aD bD  FD (θ1 )ΛD := =cD dD
FD (θ2 )(0) (1)∗fL (θ1 )(rss − rpp )rsp 
(0)

(1)∗

fL (θ2 )(rss − rpp )rsp∗
FD (θ1 )
.
∗
FD (θ2 ),(4.28)(4.29)The determinants of the system matrices are
det ΛT = fT (θ1 )fT (θ2 )

(0)
rpp

2

(1)
rpp

2

−

(0)
rpp

2

(1)
rpp

2


≡ 0,

(4.30)

67
det ΛL = fL (θ1 )fL (θ2 )(0)
rss − rpp

2

(1)
rsp

2

(0)
− rss − rpp

2

(1)
rsp

2


≡ 0,

det ΛD = FD (θ1 )FD∗ (θ2 ) − FD∗ (θ1 )FD (θ2 ) 6= 0.

(4.31)(4.32)The only non-trivial determinant (when cos (θ1 ) sin (θ2 ) 6= sin (θ1 ) cos (θ2 ), which is satisfied when
θ1,2 6= 0 and θ1 6= θ2 ) is the one that corresponds to the diagonal geometry, which means that thisgeometry can be used for a unique determination of Q and xy by scanning the polarization angle,
while the transverse and longitudinal geometries are not suited for that purpose.
The experimental setup in Fig. 4.2 needs to be adapted for D-MOE measurements. The
rotation of the polarization of the EUV probe is easily achieved by simply rotating the polarizationof the IR driving laser with a half-wave plate placed before the waveguide in which the HHG
process takes place. The direction of the linear polarization of HHG pulses follows that of thelaser driver [88, 82]. Therefore, no modification to the HHG source part of the setup is required,
other than replacing a manual rotation stage for the half-wave plate with a motorized one, sothat the polarization scans could be integrated into an automated data acquisition procedure in
Fig. 4.3. In order to apply a magnetic field in a diagonal direction, the sample mount needs tobe redesigned because with the mount in Fig. 4.2 the field can only be applied in the transverse
direction. The spectrometer arm of the setup adapted for D-MOE measurements is shown in
Fig. 4.7. Because scanning the angle of incidence is no longer required, the spectrometer can bepositioned horizontally, which has an added benefit of increased stability. Moreover, the diffraction
grating can now be separated from the sample. Not only does this design simplify the sample
making process by eliminating the need to pattern a grating on top of the film, but it also improves
the diffraction efficiency as the grating can now be blazed and positioned in a conical mount [129].
Additionally, the separation of the grating from the sample removes any ambiguity in the sample’s
response as it can be attributed to the sample itself without the need to consider the effects of thegrating deposited on the sample. Below, I do consider the polarization sensitivity of the grating
response, nonetheless, in order to confirm that it does not affect the D-MOE signal. A different
Ti:Sapphire amplifier, a KMLabs Wyvern, was used in the D-MOE experiment. The laser was

68
operated at a 5 kHz repetition rate, with a pulse duration of ∼40 fs, and a pulse energy of 2 mJ.

Figure 4.7: Spectrometer part of the D-MOE setup. A projection-field electromagnet is placed
behind the sample. The diffraction grating is separated from the sample and is oriented in a conicalmount, with the grating lines parallel to the plane of incidence for higher diffraction efficiency.

A static polarization-resolved D-MOE asymmetry is shown in Fig. 4.8, as measured at theharmonic peaks. Note that overall, the amount of data in Fig. 4.8 is greater than that in Fig. 4.5
because it is easier to take finer steps in the polarization angle than in the angle of incidence. In
addition, when the angle of incidence is changed the amount of the absorbed IR pump light alsochanges slightly. That is not the case for time-resolved D-MOE since it keeps the angle of incidence
constant.

4.2.3

Magneto-optical multilayer formalism

For thick samples, expressions (4.12) and (4.20) could be applied directly to the data in
Figs. 4.5 and 4.8, respectively, to extract xy . However, the data in the figures were taken withmultilayer thin films, which can significantly modify the Fresnel reflections. A formalism describing
the propagation of light in magnetic multilayers was developed in Ref. [130], and I outline it below.

69

Figure 4.8: Static polarization-resolved D-MOE asymmetry at the M -edge of Co measured at theharmonic peaks.

70
Within this formalism, each layer in a multilayer stack is represented by a set of matrices:
the boundary matrix L that describes the reflection from and transmission through the boundaryof two layers and the propagation matrix D̄ that describes the propagation of light through a given
layer. The entire multilayer stack then is represented by a combination of those matrices for eachlayer.
The medium boundary matrix L is derived from the Maxwell boundary conditions. Consider
an interface between two semi-infinite media, as in Fig. 4.9. In each half-space, there are forward(incident) and backward (reflected) propagating fields marked by the superscripts (i) and (r),respectively. The medium above the boundary is denoted with an index 1, and the one below withan index 2. The medium boundary matrix L is defined as a matrix that relates two 4-element

(𝑖)

𝐸1𝑝

(𝑖)
𝐸1
(𝑖)

𝐸1𝑠

(𝑟)

𝐸1

𝑥

Θ

𝑧=0
𝑦

(𝑟)

(𝑖)

𝐸2

𝐸2
𝑧

Figure 4.9: Fields at an interface between two media. The x- and y-axes are in the plane of theinterface, and the z axis is normal to it. Adapted from [130].

vectors
F = LP,(4.33)

where vectors F and P contain the x- and y-components of the E and H fields and the incident

71
and reflected s- and p-components of the E-field, respectively, all in a given medium Ex 
   Ey 

F =
 ,
Hx 
 
 
Hy





(i)
 Es  (i)  Ep 

P =
 (r)  .
Es 




(r)
Ep

(4.34)At the boundary between media 1 and 2, F1 = F2 , and from eq. (4.33), matrix L for a medium with

𝑥
𝛾

𝑦

𝜙
𝑀
𝑧
Figure 4.10: Azimuthal and polar angles for the magnetization vector. For a yz plane of incidence,the polar geometry corresponds to φ = 0; transverse φ = π/2, γ = 0; longitudinal φ = π/2,
γ = pi/2, and diagonal φ = π/2, 0 < γ < π/2.

a refractive index N and a magnetization direction determined by the polar angle φ and azimuthal
angle γ (see Fig. 4.10) can be obtained (details can be found in Ref. [130])




1
0
1
0




 i αyα 2 αz Q (αy gi − 2Sφ,γ ) αz + iαy Cφ,γ Q − 2i αyz Q (αy gr − 2Sφ,γ ) −αz + iαy Cφ,γ Q . (4.35)
L=

i
i


N
g
Q
−N
N
g
Q
−N
i
r


2
2


iN
iN
N αz
−N αz
− 2αz gr Q
2αz gi Q

72
In the expression above, the following definitions have been introduced
Sφ,γ = sin φ sin γ,
αy = sin Θ,
gi = αz cos φ + αy Sφ,γ ,Cφ,γ = sin φ cos γ,
αz = cos Θ,(4.36)

gr = −αz cos φ + αy Sφ,γ ,where Θ is the angle of incidence.
The propagation matrix D̄ relates the fields at the top of a layer with the fields at a depth z
in the layer and is defined by
P (0) = D̄(z)P (z).

(4.37)

This matrix takes into account phase shifts, absorption, and polarization rotation of the lightpropagating through a given layer. For a layer of thickness d, the medium propagation matrix canbe written explicitly as



U
U
δ
0
0
i






−U
δ
U
0
0


i

,
D̄ = 

−1
−1
 0
0
U
−U δr 

00 U −1 δr
U −1
where the elements U , δi , and δr are defined as follows
2π
U = exp −i N dαz ,λπQδi = N d gi ,λαzπQδr = N d gr ,λαz(4.38)(4.39)

where λ is the free-space wavelength of the light propagating in the layer. Here, it is important tonote that a convention where a forward propagating wave is given by E = E0 exp i(N ~k0 · ~z − ωt)is used. It requires a positive imaginary part of the refractive index, i.e. N = 1 − δ + iβ. It also
determines the sign of the Kramers-Kronig transform, which I describe below.
The action of an entire multilayer stack on an initial 4-element vector Pi containing forward
and backward propagating waves (zero for an incident wave) for s- and p-polarizations is described

73
by a single 4 × 4 matrix M defined as
Pi = M P f ,(4.40)

where Pf is the final 4-element vector, and M is equal to a product of the medium propagation
and boundary matrices and their inverses for all the layers [130]M = L−1iY

Lm D̄m L−1
m Lf .

(4.41)m

M can be written in a block-diagonal form asG H 
M =
,I J(4.42)

where G, H, I, J are 2 × 2 matrices. The Fresnel transmission and reflection matrices (as in eq.
(4.16)) can then be identified as [130]tsst̂ = 
tps
rssr̂ = 
rps


tsp 
−1
=G ,
tpp

rsp 
−1
 = IG .
rpp(4.43)

With the expressions above, the following procedure can be used to compute magneto-opticalreflections and transmissions for a magnetic multilayer stack(1) Set the angle of incidence and the polarization state of an incident beam(2) Set the magnetization directions for all layers(3) For each layer, determine the refractive index N and the magneto-optical Voigt parameter
Q at the wavelength of interest(4) For each layer, compute the medium boundary and propagation matrices according toeqs. (4.35) and (4.38), respectively(5) Compute the total matrix M for an entire stack according to (4.41)

74(6) Compute the Fresnel transmission and reflection matrices t̂ and r̂ using eqs. (4.42) and(4.43)~ t = t̂E
~ i and E~ r = r̂E
~ i , respectively(7) Compute the transmitted and reflected fields as E
Sometimes, it can be advantageous to use a slightly different procedure in the interest ofcomputational expediency. One example is when a large parameter space of wavelengths andincidence angles needs to be swept. In that case, the code would be easier to parallelize if reflections and transmissions at each boundary were calculated explicitly from the boundary condition
Li−1 Pi−1 = Li Pi . For the most general case of arbitrary magnetization directions in the adjacentlayers, the expressions are too unwieldy to provide here, but such a calculation was, nonetheless,performed in Mathematica. With known boundary reflections and transmissions, a wave can bepropagated through a multilayer stack until a convergence condition (e.g. an error tolerance) isreached. An added benefit of such an approach is that it allows a calculation of the fields throughout an entire sample. This is useful when, for example, the pump absorption profile needs to be
calculated, or an estimation of the depth sensitivity of an EUV probe needs to be performed. I
discuss the latter estimation in Appendix A.

4.2.3.1

Effect of optical elements on the magneto-optical signal

With the multilayer formalism, the extraction of xy from the angle-resolved and polarizationresolved data sets is performed by a fitting procedure, i.e., xy is adjusted in order to minimize aroot mean square (rms) error between the experimental data and the calculated angle-resolved
T-MOKE or polarization-resolved D-MOE signals.
In the case of the angle-resolved T-MOKE, a 500 lines/mm diffraction grating was lithographically patterned on the sample (see Fig. 4.11). Because the diffraction grating is patterned directlyon the sample, it might reduce the T-MOKE asymmetry because the Si substrate is non-magnetic
and has a higher reflectivity. Upon diffraction from the grating, it is mixed with the light reflectedfrom the magnetic part and, therefore, needs to be taken into account. This is done by calculating

75
the Fresnel reflections for both the grooves and ridges of the grating using the multilayer approach.
This gives a spatially varying distribution of the reflection coefficient r(y). The reflected field at
the detector is equal to a Fourier transform of a product of the incident probe and r(y)Ẽ(ky ) ∝ F {r(y)Ei (y)} ,(4.44)

where Ei (y) is a Gaussian illuminating approximately 50 grating lines. The calculated intensity istaken at the first order diffraction peak position k1 , for a given wavelength2

Ir = Ẽ(ky = k1 ) .

(4.45)

Si3N4 3 nm
Co 10 nm

1 mm

1 mm

Ta 3 nm
SiO2 150 nm/ Si substrate

Figure 4.11: Sample structure for angle-resolved T-MOKE. A 10 nm Co film is grown on a Si wafer
with a 3 nm Ta seed layer and a 3 nm Si3 N4 cap. The sample was lithographically patterned toform a 500 lines/mm grating.

For the D-MOE experiment, the effect of the toroidal mirror on the polarization of theprobe and the effect of the spectrometer grating on the D-MOE asymmetry must be examined.
I will first consider the effect of the toroidal mirror. Because the measurement is polarization
sensitive, one might expect that, upon reflection from the toroid, the linear polarization of theprobe becomes elliptical. I confirm that the ellipticity of the reflected EUV probe is very small.
The polarization ellipse is uniquely described by a set of four parameters known as the Stokes
parameters [131]. The Stokes parameter S3 is related to the ellipticity, and S3 = 0 for linear
polarization and S3 = ±1 for circular. By calculating multilayer reflections at a 6◦ grazing incidence
angle for different polarization angles ranging from s- to p-polarization from a toroid made out of

76
Pyrex glass and coated with a 100 nm of B4 C, I find that S3 ≤ 0.11. This means that approximately
95% of the reflected field maintains the original incident polarization, while only ∼5% is rotatedperpendicularly to that direction. In other words, the ratio of the minor and major semi-axes of thereflected elliptical polarization is 0.055. This ellipticity is rather small and thus can be neglected
for the purposes of the D-MOE experiment.
A diffraction grating could also introduce artifacts in the D-MOE signal. The reason forthis is a possible difference in diffraction efficiencies for the light reflected from the positively
and negatively magnetized sample. This is because the D-MOE configuration in Fig. 4.6 is not
symmetric, which leads to a difference in the polarization states of the probe reflected from thesample when its magnetized in the +m~ and −m~ directions. Because the spectrometer grating is asawtooth grating and is mounted in a conical configuration, it breaks the symmetry further, which
could result in different diffraction efficiencies for different polarizations. The calculation in Fig.
4.12 was carried out for different polarization angles of EUV light incident on a thin Co film with amultilayer composition of Si/SiO2 (150)/Ta(3)/Co(10)/Si3 N4 (3), where all layer thicknesses are in

Polarization angle (degrees)

nm. First, I calculate the magneto-optical reflections from this sample for opposite magnetization
2

50

1
0

100

-1
-2

150
-3

50
60
Photon energy (eV)

70

×10-3

Figure 4.12: Difference in diffraction efficiencies in the D-MOE geometry for the two opposite
magnetization directions. Complex magneto-optical reflections from a Co multilayer were calculated
and used as inputs for an RCWA model (see Appendix B) in order to compute diffraction efficiencies
from a blazed grating mounted in a conical configuration. The difference is small and does notexceed 0.35%.

77
directions in the D-MOE geometry at a 52◦ incidence using the multilayer approach describedabove. Next, I use the calculated complex reflected fields as inputs for a diffraction efficiency
calculation. The latter is performed using the rigorous coupled-wave analysis (RCWA) formalism
described in Appendix B. The sawtooth grating used in the calculation was made of Zerodur glasswith a 30 nm B4 C coating. The periodicity of the grating was 2 µm, and the blaze angle was 4.7◦ . It
was mounted in a conical configuration at a 5◦ grazing incidence, and the grating vector was turnedby 2◦ with respect to the normal to the plane of incidence, to allow for any possible misalignment.
The difference in the calculated diffraction efficiencies in Fig. 4.12 does not exceed 0.35%, and,
therefore, the effect of the spectrometer grating on the D-MOE asymmetry is negligible.

4.3

Understanding the microscopic picture of ultrafast demagnetization
Despite the active interest in the phenomenon of ultrafast demagnetization over the past two

decades and a large volume of experimental data [57, 132, 133, 123, 134, 135, 136] the consensus
as to the microscopic picture of laser-driven magnetization dynamics is still lacking in the scientificcommunity [137, 138, 139, 140, 141]. There are several reasons for this. First, as discussed in
Chapter 2, our understanding of ferromagnetism in transition metals is incomplete due to theitinerant nature of 3d electrons in those metals, i.e. they exhibit both localized and delocalized
behavior. Second, measurements are often performed on one material, e.g. Ni, and the conclusions
are sometimes extrapolated to other materials, such as Co and Fe. Even though they are alltransition metals, they are quite different from one another and such an extrapolation is not always
correct. Third, measurements often employ different techniques, such as visible MOKE, EUV
T-MOKE, spin-resolved photoemission, or XMCD. These techniques have inherent limitations,and, in general, they measure different aspects of the phenomenon. For example, photoemission
is sensitive to the first few monolayers near the surface of the sample, and the measurement isperformed along a specific direction in the Brillouin zone. EUV T-MOKE, on the other hand,is capable of measuring polycrystalline materials, which gives an average signal across the entire
Brillouin zone, and the T-MOKE response is a convolution of the magnetization profile throughout

78
the thickness of the sample with the T-MOKE depth sensitivity function (see Appendix A), which
can extend up to a few tens of nm into the sample. Due to a non-uniform absorption of the IRpump light, the conditions at the surface and in the bulk of the film can be quite different, andcare must be taken while comparing the results of photoemission experiments with those of visible
or EUV T-MOKE [142, 143]. For example, such a comparison was made for a single crystal bulk
(400 nm) Ni film, and the difference in the T-MOKE and photoemission data was attributed to adepth dependence of the demagnetization process [144, 145]; additionally, in these references, anultrafast phase transition from the ferromagnetic to the paramagnetic state was reported.
The following processes have been reported in the literature as potential microscopic driversof ultrafast demagnetization, in the absence of a phase transition:• Ultrafast magnon generation, or transverse spin excitations [146, 147, 148, 149, 150, 151].
• Longitudinal spin-flips driven by the spin-orbit interaction in the valence band [152] throughthe Elliott-Yafet electron-phonon scattering process [153]. In the context of the microscopic
three-temperature model, these lead to a reduction of the exchange splitting ∆ex (see Fig.
2.2) [58].
• Superdiffusive spin currents [154, 155, 156].
The former two items are known as the local demagnetization mechanisms because they lead to a
local reduction of the magnetization, while the superdiffusive spin currents are non-local because
they involve transport of the spin polarization from one part of the film to another. Because the films
studied here are grown on insulating substrates and are rather thin, the non-local demagnetization
channel is blocked, and only the local excitations are considered. Longitudinal spin fluctuations
exist within the framework of the Stoner model of ferromagnetism, and transverse fluctuations are
based on the Heisenberg model. More details on the two models and the types of spin excitations
within them can be found in Chapter 2. It should be noted that very short-wavelength magnons
overlap with the Stoner continuum of spin-flip excitations, and they have very short lifetimes

79[157, 158], on the order of femtoseconds for Co, and, therefore, such short wavelength magnons cancontribute to ultrafast magnetization dynamics.
As mentioned above, transverse and longitudinal spin fluctuations affect the magneto-opticalpermittivity xy differently: transverse excitations lead to a uniform reduction of xy across all
photon energies, while longitudinal excitations change its spectral shape [70]. This was confirmedby first principles DFT calculations of the DOS for hcp Co. The calculations were carried out
at Uppsala University, Sweden, and Charles University, Czech Republic, using the WIEN2k DFT
code [159]. The dielectric tensor was computed from the linear response theory [73]. More details regarding the calculations can be found in Ref. [122] and its Supplementary Information.
These calculations confirmed that if the magnetization was tilted by an angle of up to 22◦ , theamplitude of xy would decrease uniformly without a change in its spectral shape. On the other
hand, if a decrease in magnetization was due to, at least in part, a reduction in the exchange
splitting, the spectral shape of xy would be affected, as shown in Fig. 4.13. This provides aIm(0xy ) for a reduced z-projection of the moment from 1.63 7B to 0.97 7B
0.1

0% exchange, 100% magnons
22% exchange, 78% magnons
100% exchange, 0% magnons

Im(0xy )

0.05

0

-0.05

-0.1

-0.15
53

54

55

56

57

58

59

60

61

62

63

Energy (eV)Figure 4.13: Im (xy ) at the M -edge of Co with a reduced magnetization calculated from firstprinciples for different values of exchange splitting ∆ex . Note the changes in the spectral shape as
the exchange splitting is reduced.

means of distinguishing between longitudinal and transverse excitations in a material following afemtosecond optical pump, i.e., tracing the transient evolution of the measured full complex resonant magneto-optical permittivity xy . Such tracing was carried out using the angle-resolved EUV

80
T-MOKE and polarization-resolved D-MOE techniques discussed above. In the following, I present
the experimental results obtained using the two methods and the respective conclusions.

4.4

Ultrafast demagnetization with angle-resolved T-MOKE
First, with a 10 nm Co thin film whose composition is shown in Fig. 4.11, a T-MOKE

demagnetization curve was measured (see Fig. 4.14) at the M -edge of Co at a 45◦ incidence using
the setup in Fig. 4.2. Because the experimental procedure for angle-resolved T-MOKE involves

5% demag.

12% demag.

Figure 4.14: Ultrafast demagnetization in a 10 nm Co film. The maximum demagnetization is 12%(at 700 fs) and the demagnetization at 3 ps is 3%. The absorbed pump fluence is estimated at
1.3 mJ/cm2 .

a manual adjustment of the angle of incidence, which is rather cumbersome and time consuming,
only two time points on the curve in Fig. 4.14 were chosen for an angle-resolved scan (from 37.5◦
to 52.5◦ ): at the time of maximum demagnetization at 700 fs, and at 3 ps, when the system hasreached thermal equilibrium and is in the recovery phase. Transient changes to the static angleresolved T-MOKE scan (Fig. 4.15(a)) at these times are shown in Fig. 4.15(b) and (c). The static
data and the dynamic changes to it can now be analyzed to determine the relative contributionsof the longitudinal and transverse spin fluctuations. Note that a dissipation channel through spinpolarized currents is blocked because the ferromagnetic films are grown on insulating substrates.

81

Figure 4.15: Time-resolved data and analysis of angle-resolved T-MOKE. The top row is the experimental data, the second row is the simulations that correspond to the best fit to the experimental
data, and the bottom row is the rms analysis. The columns (from left to right) correspond to
the times of < 0 ps, 0.7 ps, and 3 ps. (a) Static angle-resolved T-MOKE asymmetry. (b), (c)Differential changes to the static asymmetry at 0.7 and 3 ps, respectively. (d), (e), (f) are the static
angle-resolved T-MOKE and the differential changes to it at 0.7 ps and 3 ps, respectively, calculated
based on the theoretical values of xy . (g), (h), (i) are quadratic surface fits to the rms residue
calculated by fitting the experimental data with calculated T-MOKE signals based on different
theoretical values of xy calculated at various electron temperatures and exchange splittings.

82
The analysis involves fitting the data with simulated angle-resolved T-MOKE signals thatare based on xy ’s calculated for different electron temperatures and exchange splittings in a selfconsistent manner (see Ref. [122] for details on the calculation of xy ). This is preferable to a
direct extraction of xy from the experimental data because a robust extraction would require a lot
more points in the incidence angle than were actually measured. In the first step of the analysis,
a fit to the static data in Fig. 4.15(a) is found. This is done by applying a Gaussian broadening
and an energy shift to the ab initio calculated xy at a zero electron temperature Te = 0 K andexchange splitting ∆ex = 1.672 eV, which corresponds to the ground-state magnetic moment of Co
of 1.63 µB . These transformations are necessary in order to include the effects that the theoretical
calculations do not take into account, such as a finite lifetime of electrons that are optically excitedfrom the 3p core states to the conduction band of Co, which requires a Gaussian broadening, anenergy offset of the theoretical 3p levels relative to the experiment (requires an energy shift), and an
overestimation of the magnitude of xy as well as magnon excitations present at 300 K (requires anapplication of a scaling factor of < 1). Additionally, a small offset of < 1◦ in the incidence angle aswell as a 2% s-polarized component of the probe were included in order to allow for a misalignmentof the EUV probe with respect to a purely horizontal direction and its depolarization on opticalsurfaces (e.g. the toroid). An additional broadening by a Gaussian kernel with a FWHM of 1.28 eV
was applied to the simulated angle-resolved T-MOKE asymmetry in order to take into account thefact that an EUV probe illuminated only ∼50 grating lines, which lowered the spectral resolution.
The optimal energy shift δE, the width of the Gaussian broadening kernel w, and the scaling factor
G are found by minimizing a mean square difference between the calculated asymmetry and thePexperimental one in Fig. 4.15(a) Θ,E [Aexp (Θ, E) − Acalc (Θ, E, x y (w, G, δE))]2 = min, where
the sum is taken over all measured incidence angles Θ and harmonic energies E(see Fig. 4.5(a)).
The optimal parameters w, δE, and G are then applied to the rest of xy ’s calculated for all otherelectron temperatures Te and exchange splittings ∆ex .
In the next step, a fit to the time-resolved experimental data in Fig. 4.15(b) and (c) is
performed. This is done by minimizing an rms residue r between the experimental transient change

83
in asymmetry ∆Aexp (both at 0.7 ps and 3 ps) and one calculated for a given theoretical xy (Te , ∆ex )at a specific Te and ∆ex . The only fitting parameter that minimizes r is the magnon contribution
to the total demagnetization ∆mt = ∆µt /µ, where µ = 1.63 µB is the ground state magnetic
moment of Co, and ∆µt is a reduction of its projection on the axis of average magnetization due
to transverse excitations. The effect of transverse excitations is included by multiplying xy by afactor (1 − ∆mt ). The rms residue mentioned above is expressed as
s
i2
1 Xh
(Θ,E)
(Θ,E)
r(∆t, Te , ∆ex , ∆mt ) =∆Aexp (∆t) − ∆Acalc ((1 − ∆mt )xy (Te , ∆ex )) ,N

(4.46)

Θ,E

where ∆t is a delay time between the pump and the probe, which is equal to either 0.7 ps or 3 ps,
and N is the total number of experimental data points in the energy-angle parameter space (seeFig. 4.5). In addition to the transverse contribution to demagnetization ∆mt , the fitting procedureyields an rms residue, calculated according to eq. (4.46), as a function of the electron temperature
Te and the exchange splitting ∆ex (or rather its reduction δ∆ex (t) = ∆ex (t = 0) − ∆ex (t)).
The smaller the rms residue defined by eq. (4.46) the better the fit. For a theoretical xy
calculated for certain Te and ∆ex , r(∆t, Te , ∆ex , ∆mt ) would be minimized, and thus the optimal(min)values Te(min)and ∆exof the electron temperature and the exchange splitting can be determined.

Because xy ’s were calculated on a discrete grid of temperatures and exchange splittings, the rms
residues calculated on the same grid through the fitting procedure described above are fitted witha parabolic surface given byf (Te , ∆ex ) = A + B(min)

where A, B, C, D, Te





2

2
(min)
(min)
+
D
,
∆ex − ∆(min)
+
C
T
−
T
T
−
T
e
e
ex
e
e

(min)

, ∆ex

(4.47)

are fitting parameters. The parabolic fitting function (4.47) isjustified because this is the lowest order approximation to a function around its minimum. The
parabolic fits to the rms residues calculated at 0, 0.7, and 3 ps are shown in Fig. 4.15(g), (h),and (i), respectively. The minimum of the parabolic surface moves from the upper left corner,corresponding to the ground state with zero exchange splitting reduction Te = 300 K, towards thecenter at 0.7 ps, indicating an increase in Te and a reduction in ∆ex , and then at 3 ps, as the system
recovers, back to the upper left corner.

84
(min)

According to eq. (2.9), with a reduced exchange splitting ∆ex, the magnetic moment

per atom also becomes reduced by an amount ∆µl , and a longitudinal contribution to the total
demagnetization ∆ml = ∆µl /µ can be found. In addition, at an elevated electron temperature, thethermal filling of the spin-up and spin-down bands changes, and the magnetic moment is reduced
by an additional amount ∆me (Te ) = 1 − µ(Te )/µ(Te = 0). The temperature dependence of µ(Te ) isshown in Fig. 2.3 in orange. Thus the total demagnetization ∆m is a sum of the transverse ∆mt ,longitudinal ∆ml , and thermal smearing ∆me contributions∆m = ∆mt + ∆ml + ∆me .
(min)∆ml and ∆me are determined from ∆ex(min)and Te(4.48), respectively, which are found fromfitting a parabolic surface (4.47) to the rms residues calculated using eq. (4.46). The procedure ofcalculating the rms residues also yields a dependence of ∆mt on ∆ex and Te , and, once the optimalvalues for the exchange splitting and electron temperature at the minimum of the parabolic surface
are found, the corresponding ∆mt can also be determined.
The results of the analysis are summarized in Table 4.1 According to this table, at 0.7 ps,
Table 4.1: Summary of the analysis of the angle-resolved T-MOKE data.

Te (K)
Measured ∆m
Fitted ∆m
Fitted net moment (µB /atom)∆ml∆µl (µB /atom)∆mt∆µt (µB /atom)∆me∆µe (µB /atom)< 0 ps
300
0
0
1.62
0
0
0.008 [38]
0.013
0
0

0.7 ps1018
0.12 ± 0.01
0.115 ± 0.014
1.44 ± 0.02
0.032 ± 0.013
0.05 ± 0.02
0.073 ± 0.006
0.12 ± 0.01
0.01
0.016

3 ps
352
0.05 ± 0.01
0.049 ± 0.012
1.54 ± 0.02
0.021 ± 0.011
0.03 ± 0.02
0.028 ± 0.004
0.05 ± 0.01
00approximately 2/3 of the total 12% magnetization reduction comes from transverse excitations∆mt , about 10% of the signal is due to thermal repopulation of the bands ∆me , and the rest

85
is from longitudinal excitations ∆ml . At 3 ps, longitudinal and transverse excitations contribute
approximately 40% and 60% to the total demagnetization, respectively.
In order to confirm that both types of excitations drive the demagnetization process simultaneously, the above analysis was repeated for two additional cases where, in addition to the thermal
repopulation of the bands ∆me , only a single type of excitation was allowed (either transverse or
longitudinal). The results are summarized in Tables 4.2 and 4.3
Table 4.2: Summary of the analysis of the angle-resolved T-MOKE data with longitudinal excitations only.

Te (K)
Measured ∆m
Fitted ∆m
Fitted net moment (µB /atom)∆ml∆µl (µB /atom)∆mt∆µt (µB /atom)∆me∆µe (µB /atom)< 0 ps
300
0
0
1.62
0
0
0.008
0.013
0
0

0.7 ps3225
0.12 ± 0.01
0.082 ± 0.025
1.49 ± 0.04
0.032 ± 0.025
0.052 ± 0.041
0.008
0.013
0.05
0.08

3 ps
796
0.05 ± 0.01
0.034 ± 0.009
1.56 ± 0.04
0.029 ± 0.009
0.047 ± 0.014
0.008
0.013
0.005
0

In both Tables 4.2 and 4.3, the predicted net demagnetization ∆m at 0.7 ps as well as 3 ps
is below the experimentally measured values. It should be noted that in Table 4.3, only the lower
limit of ∆m is estimated because xy ’s calculated for higher electron temperatures were required tominimize the rms residue. In addition, both analyses predict very high electron temperatures. For
the experimental value of absorbed fluence of F = 1.3 mJ/cm2 , the upper and lower temperature
limits can be estimated by solving for T from
Z T
F/d =
300

C(τ )dτ,(4.49)

K

where d is the thickness of the film, and C(T ) is the temperature dependent specific heat. Here, a
uniform distribution of the absorbed fluence throughout the sample thickness is assumed. Because
only an average temperature needs to be estimated, such an approximation is acceptable. The

86
Table 4.3: Summary of the analysis of the angle-resolved T-MOKE data with transverse excitations
only.

Te (K)
Measured ∆m
Fitted ∆m
Fitted net moment (µB /atom)∆ml∆µl (µB /atom)∆mt∆µt (µB /atom)∆me∆µe (µB /atom)< 0 ps
300
0
0
1.62
0
0
0.008
0.013
0
0

0.7 ps3500
0.12 ± 0.01
> 0.087< 1.4800> 0.037> 0.06
0.05
0.016

3 ps
1136
0.05 ± 0.01
0.033 ± 0.008
1.57 ± 0.012
000.025 ± 0.008
0.041 ± 0.012
0.008
0.012

upper electron temperature limit corresponding to the 0.7 ps delay time is found when all of theabsorbed energy goes into the electron system, which means that C = Ce . With the electron heat
capacity of Co (see Appendix C), eq. (4.49) yields an upper limit of 2295 K for Te . Similarly, alower limit of 620 K, which corresponds to 3 ps, is found when the total specific heat includes all
three energy baths: electrons, lattice, and spins, and so the total heat capacity C = Ce + Cl + Cs
(see Appendix C). The temperatures in Tables 4.2 and 4.3 at 0.7 ps exceed the upper limit. Even
though they are within the limits at 3 ps, the values of ∆m disagree with the experimental ones.
This supports the conclusion that both longitudinal and transverse excitations, rather than a single
type of excitation, contribute to the total demagnetization at both delay times.
These results must interpreted with caution because of the rather low temperature sensitivityof the method, which leads to a large uncertainty of the temperatures derived from the fitting
procedure. This can be seen from Figs. 4.15(h) and (i), where the iso-contours are stretched in
the horizontal direction, indicating a shallow minimum along Te . For that reason, the analysis
was repeated with the coefficients C and D in eq. (4.47) set to zero, which yields a temperaturedependence of both ∆ml and ∆mt . The analysis confirms that regardless of the temperature, bothtypes of excitations are required in order to explain the net amount of demagnetization observed

87
in the experiment, albeit with different ratios of longitudinal to transverse contributions dependingon the temperature.

4.5

Ultrafast demagnetization with D-MOE
Unlike angle-resolved T-MOKE, D-MOE is a very efficient technique, which allows one tocollect a sufficient amount of data relatively quickly, thus making possible a direct measurement of
the full resonant complex xy .
First, I validate D-MOE by extracting xy of Co from a static polarization scan (shown in
Fig. 4.8) performed on a Co multilayer with a composition of Si/SiO2(150)/Ta(3)/Co(10)/Si3N4(3)(all layer thicknesses are in nm) shown in Fig. 4.6. The extraction is performed by fitting the DMOE data in Fig. 4.8 with a multilayer model of the sample based on the multilayer formalism
introduced above. Material refractive indices used in the model were taken from Ref. [160]. xy
is found, independently at different photon energies, from the best fit of the simulated D-MOE
signal to the experimental data. In order to validate the technique, the extracted xy is thenused to successfully reproduce the polarization-resolved asymmetries measured in the transverseand longitudinal geometries. A comparison of the experimental D-MOE, T-MOKE, and L-MOE
signals with the calculated ones based on the xy extracted from D-MOE is shown in Fig. 4.16. The
abbreviation L-MOE introduced here stands for longitudinal magneto-optical effect, which is given
by eq. (4.19). It should not be confused with L-MOKE, as the latter requires a polarization analysis
whereas the former simply results in a magneto-optical asymmetry. The agreement between theexperimental data and the reconstructions based on the extracted xy in Fig. 4.16 demonstrates
the validity of the D-MOE technique.
An additional test was also performed where the same extraction procedure was applied
to the T-MOKE and L-MOE data. The results are summarized in Fig. 4.17. Importantly, thereconstructions are successful only for the geometries from which xy was extracted, but not forthe other ones. This confirms the conclusion of eqs. (4.31), (4.30), and (4.32), which states that
only the diagonal geometry can provide a unique xy by use of polarization scanning, while the88θ (degrees)

D-MOE(experiment)
50

T-MOKE(experiment)

a)

0.05

b)

0.2

50

0.1

100

c)

0.05

0

100

-0.05

100

-0.1

150

-0.15

50

60

50

d)

0

150

70

50

D-MOE calculation(using ǫxy from D-MOE)θ (degrees)

50

L-MOE
(experiment)

60

50

150

70

e)

-0.05

50

T-MOKE calculation(using ǫxy from D-MOE)
0.05

0

60

70

L-MOE calculation(using ǫxy from D-MOE)
0.2

50

0.1

100

f)

0.05

0

100

-0.05

100

-0.1

150

-0.15

50
60
70
Energy (eV)

0

150
50
60
70
Energy (eV)

0

150

-0.05

50
60
70
Energy (eV)Figure 4.16: Comparison of the experimental (a)-(c) and reconstructed (d)-(f) D-MOE, T-MOKE,
and L-MOE polarization angle-resolved spectra. The experimental data points are taken at thediscrete HHG peaks of the probe. θ = 90◦ corresponds to p-polarization; θ = 0◦ corresponds tos-polarization. The reconstructions are based on the xy (ω) extracted from (a).

89

100
150

L-MOE experiment

50

0.2

100

0.1

150

0

θ (degrees)

0.05
0
-0.05
-0.1
-0.15

50

θ (degrees)

T-MOKE experiment

50

0.05

100

0

150

-0.05

50
60
70
Energy (eV)D-MOE calculation
using ǫxy from T-MOKE

T-MOKE calculation
using ǫxy from T-MOKE

L-MOE calculation
using ǫxy from T-MOKE

0.05
0
-0.05
-0.1
-0.15

50
100
150

50

0.2

100

0.1

150

0

θ (degrees)

50
60
70
Energy (eV)θ (degrees)

50
60
70
Energy (eV)

50

0.05

100

0

150

-0.05

50
60
70
Energy (eV)

50
60
70
Energy (eV)D-MOE calculation
using ǫxy from L-MOE

T-MOKE calculation
using ǫxy from L-MOE

L-MOE calculation
using ǫxy from L-MOE

0.05
0
-0.05
-0.1
-0.15

50
100
150
50
60
70
Energy (eV)

50

0.2

100

0.1

150

0

50
60
70
Energy (eV)θ (degrees)

50
60
70
Energy (eV)θ (degrees)c)θ (degrees)b)θ (degrees)a)θ (degrees)

D-MOE experiment

50

0.05

100

0

150

-0.05

50
60
70
Energy (eV)Figure 4.17: Comparison of the experimental and calculated polarization scans based on xy extracted from the T–MOKE and L-MOE scans. (a) experimental signals, (b) reconstructions basedon the xy extracted from T-MOKE, (c) reconstructions based on the xy extracted from L-MOE.
The reconstructions can successfully reproduce only the signals in their respective experimental
geometries, which the extraction of xy was based on, but not the other two.

90
transverse and longitudinal geometries cannot.
Another verification of the D-MOE method is done by checking the consistency of the extracted xy with the Kramers-Kronig relations [161]. Because xy arises as a material’s response toan electromagnetic wave [73, 161], which is causal, it must be analytic [161, 162]. As such, it mustsatisfy the Kramers-Kronig relations, which are written as [73]
Z ∞
2
ω 02
Re(xy ) =dω 0 02
P
Im(xy ),πωω − ω20Z ∞ω0
2dω 0 02Re(xy ),Im(xy ) = − Pπω − ω20(4.50)

where P indicates the principal value of the integral. Note that the sign of the second expression
depends on the sign convention for the absorptive part of the refractive index n. Expressions(4.50) are defined for the positive absorptive part of n. An extracted experimental xy for a 5 nm
Co film, which was also used for dynamic measurements, is shown in Fig. 4.18, along with the
Kramers-Kronig transform of an interpolated Im(xy ) and a comparison with a theoretical xy .
The experimental xy satisfies the Kramers-Kronig relations and agrees well with the theoreticalFigure 4.18: Static experimental xy of Co measured on a Si/SiO2(150)/Co(5)/GeO2(3) multilayer.
The data satisfy the Kramers-Kronig relations and compare well with the theoretical values [122].

calculations of Ref. [122].
An experimental uncertainty in the measured xy shown as the errorbars in Fig. 4.18 wasestimated based on the measured standard deviation of the intensity of the EUV probe. Uncertain-

91
ties in the reflected intensities were propagated into the measured asymmetry, for each polarizationaccording to
s
∆A =

∂A
∆R+
∂R+

2


+

∂A
∆R−
∂R−

2

2
=

q
2 ∆R2 + R2 ∆R2R−++−(R+ + R− )2,(4.51)

where ∆R+ and ∆R− are the standard deviations of the reflected intensities R+ and R− for the
positive and negative magnetization directions, respectively. ∆R+ and ∆R− were measured ateach HHG peak independently. In principle, ∆A could be determined from the experimental data,
however, only HHG spectra averaged over 100 acquisitions were saved, which streamlined the dataacquisition process and reduced data storage requirements. Therefore, in order to estimate an
uncertainty in xy , a series of Gaussian-distributed D-MOE polarization scans was simulated withan average D-MOE asymmetry over the series equal to the experimentally measured value anda standard devition ∆A calculated according to eq. (4.51). For each element of the series, an
extraction of xy was performed, which yielded a series of xy . ∆xy was calculated as a standard
deviation of that series.
In the next step, time-resolved D-MOE polarization scans were taken. The polarization
angle was scanned from 30◦ to 150◦ in steps of 6.7◦ , and the delay times were scanned in the range
from -0.5 ps to 2 ps with 25 fs time steps. It should be noted that such fine time steps, which
effectively amount to measuring xy as a continuous function of time, were made possible due tothe efficiency of the D-MOE method. Other methods that are, in principle, capable of directly
measuring xy have certain challenges in the EUV and X-ray spectral regions associated with them,which precludes the use of such methods for dynamic measurements with fine time steps. In thecase of angle-resolved T-MOKE [122], XMCD in transmission [163, 8] and reflection [164], or polar
MOKE [165, 166], one has to vary the angle of incidence. A polarization analysis is required in thecase of longitudinal MOKE [167, 165, 166] and Faraday and Voigt rotation [168, 169, 170]. The
data were taken on a 5 nm Co film grown directly on an insulating substrate with a composition of
Si/SiO2 (150 nm)/Co(5 nm)/GeO2 (3 nm). Such a sample structure was used in order to exclude any
possible non-local microscopic demagnetization mechanisms, e.g. superdiffusive spin currents, and92
thus isolate the contributions of local processes, e.g. magnons, spin-flips, and exchange splitting
reduction, to the total demagnetization. Average demagnetization curves measured from raw D-

Normalized Magnetization (a.u.)

MOE data and based on extracted xy (t) are shown in Fig. 4.19. Both curves in the figure agree
from ǫxy
a fit for demagnetization from ǫxyfrom raw D-MOE dataa fit for demagnetization from raw D-MOE data

1
0.9
0.8
0.7
0.6
-500

0

500
1000
Time delay (fs)

1500

2000

Figure 4.19: Average magnetization response of a 5 nm Co film on an insulator based on raw
D-MOE data as well as extracted time-dependent xy (t).

well. The signal measured from the raw D-MOE data was averaged over multiple harmonic peaks
and all of the polarization angles in the scanned range. To improve the signal-to-noise ratio, athreshold of 0.12 was applied to the D-MOE asymmetry before time-zero. For the blue curve, anaverage response of Im(xy (E, t)) was taken in the energy range from 55 to 63 eV. Based on doubleexponential fits [171] of the two experimental data sets, the demagnetization τD and recovery τRtimes were τD = 224 ± 53 fs and τD = 242 ± 58 fs and τR = 2302 ± 623 fs and τR = 2417 ± 686 forthe blue and orange curves in Fig. 4.19, respectively.
Using the method described above, values of xy were extracted at each delay time between
an IR laser pump pulse and an EUV probe. A time dependent change ∆xy (E, t) = xy (E, t) −xy (E, t < 0) is shown in Fig. 4.20. The data in the figure show a transient reduction and recoveryof xy . This direct time-resolved measurement of an evolving xy allows an analysis of the relative
contributions of the local transverse and longitudinal excitations to the total demagnetization,according to Ref. [70], at various time steps. The non-local mechanisms, i.e. spin-polarized
currents are not allowed in the sample structure considered here.

93

Figure 4.20: Transient changes in xy measured with time-resolved D-MOE: (a) Re(∆xy (E, t)) =Re(xy (E, t) − xy (E, t < 0)), (b) Im(∆xy (E, t)) = Im(xy (E, t) − xy (E, t < 0)). Also shown is acomparison of the (c) real and (d) imaginary parts of xy at the time of maximum demagnetizationt = 450 fs with the unperturbed state at t < 0.

94With the angle-angle resolved T-MOKE technique described above, such an analysis was
performed by comparing experimental and theoretical angle-resolved T-MOKE spectra. The DMOE technique allows a direct comparison of the experimental and theoretical values of xy , whichis shown in Fig. 4.21. The comparison is done between the imaginary parts. In panel (a) of the

Figure 4.21: Direct comparison between theory and experiment for the static and transient xy ofCo. (a) The ground-state case t ≤ 0. The red and yellow theoretical curves do not include any
magnon excitation. A reduction in the magnetic moment from 1.63 µB (red) to 1.42 µB (yellow) is
entirely due to a reduced exchange splitting. For the ground state, the theoretical xy corresponding
to an unperturbed exchange splitting (red) is in better agreement with the experiment (blue). (b)t = 450 fs. The theoretical xy ’s are calculated for varying relative contributions (see the legend)of the longitudinal and transverse excitations to the total demagnetization of 42 %.

figure, the ground state is considered. An ab initio calculated xy corresponding to a magnetic
moment of 1.63 µB per atom (red) is a better fit to the experimental data (blue) than the 1.42 µB
per atom (yellow) obtained with a reduced exchange splitting. At t = 450 fs, in panel (b), several
ab initio calculations are compared with the experimental result. For each of those cases, therelative contributions to the total demagnetization of 42 % from transverse excitations and exchange
splitting reduction are varied. For the red curve, it was assumed that the entire 42 % reduction inmagnetization is driven by magnon generation; whereas for the purple curve it is purely due to adecrease in the exchange splitting. The yellow curve is an intermediate case, for which about 3/4of the demagnetization comes from magnons and the rest is from the exchange splitting reduction.

95
Because the highest signal-to-noise ratio is at the peak of Im(xy ), which is around 60.5 eV, acomparison between theory and experiment is more reliable at that energy. The red and yellow
curves are within the experimental uncertainty indicated by the errorbars in the figure, while thepurple curve is outside the error bars. This helps estimate an upper limit on the relative contribution
from the exchange splitting reduction to the total demagnetization at ∼25 %. Thus a reductionof xy is predominantly uniform across the spectrum, and at least 75 % of the demagnetization in
Co is driven by magnons, or transverse excitations, which compares well with the result obtainedwith the angle-resolved T-MOKE reported above as well as the recent time- and spin-resolved
photoemission measurements [172].
The spectral resolution of D-MOE is limited by the spectrum of the HHG source, which is acomb of harmonic peaks separated by approximately 3.1 eV. More information about the transient
evolution of xy can be obtained using a source with a continuous spectrum. Such an HHG spectrum
can be achieved by generating isolated attosecond pulses[173, 174, 175].

4.6

Conclusions
In this chapter, I presented two new magneto-optical techniques, an angle-resolved T-MOKE

and a polarization-resolved diagonal magneto-optical effect, which extend the capabilities of the
traditional transverse, longitudinal, and polar magneto-optical geometries and can be used to unambiguously determine the full resonant complex magneto-optical permittivity of materials xy .
The permittivity depends on the microscopic state of a given material, i.e., on the number of longitudinal spin fluctuations, which lead to a reduction of the exchange splitting, and the number oftransverse spin fluctuations, or magnons. A connection between the measured xy and the relative
amounts of longitudinal and transverse excitations is made via first principles DFT calculations,which confirm that magnons result in a uniform reduction of xy across an entire spectrum, while adecrease in exchange splitting changes the spectral shape of xy , in agreement with previous work.
The two techniques were used to study laser-induced magnetization dynamics on sub-picosecond
timescales of thin Co films, namely, the role of transverse and longitudinal excitations in the ul-

96
trafast demagnetization process driven by a near-infrared laser pulse. By tracing the dynamicalchanges in xy and comparing them with theory, it was found that the demagnetization responseof Co is dominated by magnons, with a possible smaller contribution from an exchange splitting
reduction. The findings obtained independently with both the angle-resolved T-MOKE and thepolarization-resolved D-MOE methods are in agreement.
The character of the microscopic demagnetization mechanisms discussed in this chapter islikely not universal for all 3d ferromagnets, and could be unique for Co because of its rather large
exchange splitting compared to the exchange interaction energy. This could make the microscopic
demagnetization channel by means of magnon generation energetically more favorable. Recent work
comparing photoemission and T-MOKE measurements suggests that the situation is different in
Ni, where the values of the exchange splitting and exchange interaction energies are much closer. A
similar comparison of photoemission and EUV magneto-optical measurements for Co and Fe would
be very useful in the future. In addition, an EUV source with a continuous HHG spectrum will beable to provide much more magneto-optical information than a source with a rather sparse HHG
comb.

Chapter 5

EUV Magneto-Optics with Spatial Resolution

This chapter is dedicated to a coherent resonant magnetic scattering (RMS) experiment witha tabletop EUV HHG light source. Unlike spectroscopic approaches, scattering experiments provide
important information about spatial variations of the magnetization and are capable of capturingnanoscale magnetic features.
Magneto-optical phenomena, such as the Kerr and Faraday effects, have been known since
the nineteenth century, and a quantum theory of the interaction of electron magnetic moments withpolarized photons was developed in 1929 [176]. The interaction of X-rays with magnetic materials
had not been discovered until the 1970’s, when a theoretical paper predicted relativistic corrections
to the Compton and Bragg scattering cross-sections due to magnetic moments [177]. The firstobservation of X-ray magnetic diffraction from an antiferromagnetic NiO crystal was made in 1972
with a cathode tube used as a source of incoherent X-rays [178], and later measurements on ferroand ferrimagnetic compounds were reported along with detailed relativistic quantum calculations[179]. The development of bright X-ray sources, such as synchrotrons, turned magnetic scattering
into a reliable measurement technique by compensating for a low scattering cross-section with abright photon flux and thus boosting the signal [180]. An enhancement of magnetic scattering atelemental absorption edges was observed in 1988 [181].
Tabletop EUV light sources have an advantage over synchrotron facilities in that they bringcoherent RMS experiments to a laboratory setting and thus provide more flexibility and cost efficiency. Additionally, with such sources, RMS could be performed in a time-resolved manner with

98
femtosecond temporal resolution [182]. The first demonstration of coherent RMS with a tabletop
source at the M2,3 absorption edge of Co was reported in Ref. [183], and with a bright phasematched high harmonic source at the M2,3 absorption edge of Fe in Ref. [184].
Given that the magnetic scattering efficiency is very weak, the design of the experimentreported here makes use of the resonant nature of RMS and employs efficient grazing incidence
broadband EUV focusing optics, instead of multilayer coatings as in Refs. [183] and [184], in orderto maximize the photon flux at the sample.
In this chapter, I develop a theoretical description of RMS with linearly and circularly polarized light and show that the contrast mechanism for the scattering with a linearly polarized probe isentirely due to Faraday rotation. I also derive an optimal sample thickness that maximizes the RMS
signal. I design an RMS experiment on a tabletop and apply it to study field-dependent skyrmion
formation in a sample with perpendicular magnetic anisotropy. I report an observation of a transition from a disordered domain phase to an ordered hexagonal lattice of skyrmions accompanied
by an increase in the size of magnetic features.

5.1

Resonant magnetic scattering with linearly and circularly polarized light
As mentioned in Chapter 2, the interaction of light with a magnetized material can be

described using the material’s dielectric tensor ˆ. The off-diagonal elements of the tensor dependon the magnetization direction. For an arbitrary direction, all three off-diagonal elements xy , xz ,and yz are, in general, non-zero, and an electromagnetic wave transmitted through or reflectedfrom such a material will have magneto-optical changes to its amplitude, phase and polarization.
However, in most cases, the magnetization direction is either strictly parallel or perpendicular to thesurface of the sample, which introduces additional symmetries and helps simplify the problem. In
particular, a significant simplification is achieved in the case of magnetic films with perpendicularmagnetic anisotropy. In these films, the magnetization predominantly points into or out of theplane of the film, and the perpendicular magnetic anisotropy leads to a variety of magnetic textures
ranging from networks of nanoscale domains to magnetic bubbles and vortices, known as skyrmions,

99
depending on the composition of the film.
For an out-of plane magnetization, the dielectric tensor becomes xx xy 0 ˆ = −xy yy 0  ,




0
0 zz

(5.1)

where z is the axis orthogonal to the surface of the sample. Note that yx = −xy [185, 186].
With the ˆ defined by eq. (5.1), from eq. (4.4), it can be shown that for left (+) and right (−)
circularly polarized incident fields, the refractive indices in a material uniformly magnetized in an
out-of-plane direction satisfy [73]n2± = xx ± ixy cos φ,(5.2)

where φ is the angle of refraction. For normal incidence, φ = 0, and by taking a square root ofeq. (5.2), and Taylor expanding it up to a first order in xy , the expression for the refractive index
can be simplified ton± =where n0 =

√

√


xx

xy
1±i
xx


= n0 ± δns ,

(5.3)

√
xx is the non-magnetic part of the refractive index, and δns = ixy / xx is its

magnetic variation. For convenience, I write δns in terms of its real (∆δ) and imaginary (∆β)
parts
δns = −∆δ + i∆β.

(5.4)

Equation (5.3) can be viewed as a difference between the refractive indices for left and rightcircularly polarized fields for a fixed out-of-plane magnetization direction, or as a difference inrefractive indices upon magnetization reversal for a field of fixed helicity.
I now consider the transmission of a plane wave with a wavenumber k through a magnetizedfilm of thickness d. I define the basis vectors for the linear s and p and circular (+) and (−)

100
polarizations in terms of the x and y field components as
 
 
1
0
~s = 
~p = 
E , E
 ,
0
1
 
 
1
1
~ + = √1 
~ − = √1 E , E
 .
2 i
2 −i
Starting with a simpler case of circular polarization, the transmitted field is  1 ikn d
1 1 ikn d −ik∆δd −k∆βd~ (t) = √1 Ee,
 e + = √  e 0 e
+
2 i
2 i
 
 
1  ikn d1  1  ikn d ik∆δd k∆βd
~ (t) = √1 Ee,
 e − = √  e 0 e
−
2 −i
2 −i

(5.5)

(5.6)

for the positive and negative magnetization directions, respectively. For a magnetic domain sample,~ (t) and E~ (t) correspond to the fields transmitted through the positively and negatively magnetizedE+−~ (t) and E~ (t) , there is a difference in both phase and amplitude. The difference
domains. Between E+−
in phase is 2k∆δd and the difference in amplitude is approximately 2k∆βd. The polarization stateremains circular and unchanged. The latter is especially important for holographic imaging ofmagnetic domains [187, 188, 189] because, in order for the beams transmitted through the domains
and the reference hole to interfere, they must have identical polarizations. The disadvantage isthat the non-magnetic signal, such as scattering from inhomogeneities in the film, for example, willinterfere with the magnetic one making it impossible to separate the magnetic and the unwanted
non-magnetic scattering.
~ s as the incident field. It canWith linear polarization, the situation is different. I choose E
be written as a superposition of left and right circular fields
   1  1 
~ s = √1 E~+ + E~− = 1 E  +   .
2
2
i
−i

(5.7)

For the positive magnetization direction, the transmitted field is~ (t,+) = √1 E
~ + ein+ kd + E~ − ein− kd = √1 ein− kd E~− .
~ + e−2∆βkd e−2i∆δkd + E
E
2
2

(5.8)

101
A similar expression can be derived for the transmitted field when the magnetization direction isreversed~ (t,−) = √1 E
~ + ein− kd + E~ − ein+ kd = √1 ein− kd E~+ + E~ − e−2∆βkd e−2i∆δkd .
E
2
2

(5.9)

I now expand eqs. (5.8) and (5.9) in the basis of the linear s- and p-polarizations using~ ± = √1 E
~ s ± iE~p .
E
2(5.10)hi~ (t,+) = 1 ein− kd E
~ s + iE
~ p e−2∆βkd e−2i∆δkd + E~ s − iE
~p ,
E
2
i
h
 

~ (t,−) = 1 ein− kd E
~p + E~ s − iE
~ p e−2∆βkd e−2i∆δkd .
~ s + iEE
2(5.11)

Eqs. (5.8) and (5.9) then become~ s and E~ p , these expressions yield
Grouping the terms with E~ (t,±) = 1 ein− kd
E
2

h

 i


~p .
~ s ∓ i 1 − e−2∆βkd e−2i∆δkd E
1 + e−2∆βkd e−2i∆δkd E(5.12)

According to eq.(5.12), for the case of a linearly polarized incident field, the light transmitted
through a material magnetized out-of-plane becomes elliptically polarized and the axes of theellipse are rotated. Such rotation is known as the Faraday effect; the amount of rotation is thesame for both the positive and negative magnetization directions, but the rotation directions areopposite. When decomposed into the s- and p-polarization components, the transmitted fields
for the two magnetization directions have identical s-polarization components (for the incident
s-polarized field), and the p-polarization components have equal magnitudes but opposite signs,
which effectively amounts to a π phase shift between them since −1 = eiπ . For a network of up anddown magnetic domains, this phase shift maps out the transverse magnetization profile and causes
the incident light to scatter. An important consequence of eq. (5.12) is that the light scatteredfrom magnetic feastures is polarized orthogonally to the incident linearly polarized field. Since any
non-magnetic scattering preserves the polarization state, the photons scattered magnetically andnon-magnetically will have orthogonal polarizations and thus will not interfere. This makes theseparation of the magnetic and non-magnetic scattering signals possible by magnetically saturating

102
the sample and subtracting the scattered intensity from the signal measured with an unsaturated
sample. On the other hand, any holographic imaging techniques that rely on an interference of thescattered light with a reference beam are not possible with linear polarization.

5.1.1

Maximization of the scattered intensity

From eq. (5.12), the magnetically scattered fraction of the incident flux is2i in− kd e
1 − e−2∆βkd e−2i∆δkd
=
2

2
1
,
= ein0 kd e∆βkd ei∆δkd − e−∆βkd e−i∆δkd
4

ρRM S =

(5.13)

where the non-magnetic part of the refractive index is defined as n0 = 1 − δ + iβ. The goal now is
to find the optimal sample thickness d that maximizes the RMS signal ρRM S . By Taylor expanding
the exponents of the factor in parentheses in powers of ∆δ and ∆β and keeping the terms only up
to a first order in ∆δ and ∆β, I obtain
1
ρRM S ≈ e−2βkd |(2∆βkd + 2i∆δkd)|2 .
4(5.14)

After expanding the square eq. (5.14) is simplified toρRM S = k 2 d2 e−2βkd ∆β 2 + ∆δ 2 .

(5.15)For the optimal sample thickness, ∂ρRM S /∂d = 0. Taking a derivative of eq. (5.15) with respect
to d yields
∂ρRM S= −2βe−2βkd k 3 d2 ∆β 2 + ∆δ 2 + 2k 2 de−2βkd ∆β 2 + ∆δ 2 = 0,∂d(5.16)

which leads to an equation for d− βkd2 + d = 0.

(5.17)The non-trivial solution of eq. (5.17) is
d=

1
.
βk

(5.18)

Given that βk = 1/2a, where a is an absorption length, according to eq. (5.18), the optimal
thickness of the sample that maximizes the total magnetically scattered intensity is equal to twice

103
the absorption length of a given material at the wavelength of the X-ray or EUV probe. This is a
result of the fact that the Faraday rotation, which leads to the magnetic contrast, increases linearlywith d, while the overall transmittance of light by the sample decreases exponentially with d.
An important consequence of eq. (5.18) for EUV and soft X-ray scattering experiments has
to do with the typical absorption lengths at the M and L absorption edges of metals. Absorption
lengths at the M -edges are on the order of ∼ 20 nm while at the L-edges they are ∼ 10 times
greater. This means that the optimal sample thicknesses of ∼ 40 nm for EUV experiments agree wellwith the typical thicknesses of several tens of nm of magnetic multilayer films with perpendicularmagnetic anisotropy. Soft X-ray experiments, on the other hand, operate far away from the optimal
conditions. This can be partially compensated for by a high photon flux at synchrotrons or freeelectron lasers. In addition, sample surface quality typically improves as its thickness is reduced,
and, in that respect, the situation is more favorable for HHG sources.

5.1.2

Estimation of the RMS efficiency

With the result of eq. (5.18), the maximum scattering efficiency can be computed from eq.
(5.15) as(max)

ρRM S =


1 −2
2
2
e
∆β
+
∆δ
.
β2

(5.19)

With the typical magnitudes of β, ∆β, and ∆δ of 0.1, 10−2 , and 10−2 , respectively, at the M -edgesof Fe, Co, and Ni [168], the highest possible fraction of magnetically scattered flux relative tothe incident intensity can be estimated at ∼ 10−3 . In reality, it is even lower because films withperpendicular magnetic anisotropy are usually composed of multiple layers, which might effectively
reduce ∆β and ∆δ. In addition, given that the diffraction efficiency depends on the transverse
magnetization profile of a particular sample, the actual ρRM S will decrease even further. Thus
RMS experiments are very demanding with respect to the photon flux, and great care must be
taken to generate a high number of photons at the resonance energy of the sample of interest as
well as to preserve it as the probe beam is transported down the beamline.

104

5.2

Resonant magnetic scattering on a tabletop
In order to generate a high photon flux at EUV wavelengths, laser driven phase-matchedhigh harmonics were used [104]. The phase matching process is critical for the HHG brightness
and is described in more detail in Chapter 3. In order to preserve the photon flux, broadband
grazing incidence optics rather than wavelength-selective multilayer EUV mirrors were used. Thisapproach relies on the resonant nature of the EUV magnetic scattering: in eq. (5.15), ∆β and ∆δ
depend on the photon energy and are greatly enhanced at the resonant energy of the absorption
edge for a given material. Therefore, on resonance, ρRM S is much greater than off resonance, andonly the photons at the resonant photon energy will scatter. A caveat with this approach is thatit puts constraints on the sample composition, i.e., the sample cannot contain multiple elements
whose absorption edges lie within the bandwidth of the EUV source because this will create an
overlay of multiple scaled versions of the scattering pattern, each corresponding to the absorption
edge of a given element within the sample, and will ultimately lead to decoherence effects. For
example, if the photon energies of the EUV beam range from ∼30 to 72 eV (absorption edge ofAl filters), and the sample contains approximately equal amounts of Fe and Co whose M -shellabsorption edges are at ∼53 eV and ∼59 eV, respectively, the scattered intensity at 53 eV will
overlap with that at 59 eV, and the total scattered intensity will be smeared because of this. Notethat for uniform transverse distributions of the Fe and Co atomic species, the scattering pattern isdetermined purely by the magnetic texture of the sample, and will have the same profile at bothphoton energies, but because higher energy photons scatter at smaller angles, it will appear smallerat the Co edge than at the Fe. However, if only one element with an absorption edge within the
bandwidth of the EUV source is present in the sample, which was the case with the Fe-Gd magnetic
film studied here, this concern is alleviated, and broadband optics are safe to use.
Fig. 5.1 illustrates the concept of an EUV RMS experiment. Instead of multilayer EUV
mirrors, more efficient broadband optics are used, which increased the photon flux incident on thesample. A toroidal mirror focuses the beam, and the flat mirror directs it through the sample.

105
In addition, it allows small adjustments of the beam pointing. A permanent ring magnet can be
moved along the beam, and thus magnetic fields of up to 270 mT can be applied to the sample. A
beam block placed in front of the CCD which collects diffraction patterns prevents the undiffracted
light from saturating it. This is necessary because a direct transmitted beam carries no magnetic
information and is several orders of magnitude more intense than the magnetic scatter. I outline

Beam block

Sample
CCD
MagnetFlat mirror
Toroidal mirror

Figure 5.1: The concept of an EUV RMS experiment. Broadband grazing incidence optics are
more efficient than wavelength-selective multilayers. A permanent ring magnet applies a field tothe sample. The CCD placed behind the sample is used to collect the scattered intensity.

the design considerations in more detail below.

5.2.1

Experimental Design

The design of the tabletop RMS setup is concerned with optimizing the brightness and thebeam profile of the EUV source as well as the general layout and efficiency of the optics in the
scattering chamber itself.
I start by discussing the design of the EUV source. A schematic of the source is shown in Fig.
5.2. EUV light is obtained by driving the process of high harmonic generation with an infrared

106
amplified Ti:Sapphire laser at the central wavelength of 795 nm. The repetition rate of the laser
was 5 kHz, and the pulse energy was 1.9 mJ. The pulse duration was intentionally stretched inorder to minimize the B-integral [190]B(~r, t) =

2π
λ

Z

n2 |E(~r, t)|2 dz,(5.20)

where n2 is the second-order nonlinear component of the refractive index. The quantity in eq. (5.20)
corresponds to a nonlinear phase accumulation during the propagation of a laser pulse, which caninterrupt the phase-matched HHG process and thus cause instabilities of the EUV pulses. B(~r, t)
can be minimized by reducing the magnitude of the electric field |E(~r, t)|, which can be achievedby stretching the laser pulse.
FromuTi:Sapphire
5kHzIuPv9umJApulse
AMP

WP
CMP
AM2
CM2

MP
4Wufs
LensIufu=u5Wucm
Touscatteringchamber

BSP
Capillary
Cartridge
BS2

CCDP

Beam
pointulock

In-vacuum Alufilters
iris

CCD2
M2

Figure 5.2: Schematic of the EUV source. Amplified 40 fs pulses from a Ti:Sapphire laser (KMLabs
Wyvern) at the central wavelength of 795 nm are converted into bright EUV pulses in a hollow-core
glass waveguide filled with He. The pointing of the infrared beam into the fiber is stabilized. Any
residual infrared light emitted from the fiber is blocked by an in-vacuum aperture and a series of
Al filters.

HHG, however, requires a high-intensity laser driver, and for that reason, laser pulses need tobe compressed before they enter the gas filled capillary in Fig. 5.2. The compression is performed
with a pair of chirped mirrors CM1 and CM2, which compensate for the quadratic spectral phase
introduced into the pulse for the purpose of increasing its duration and thus compress the pulse

107back to its nearly transform-limited length of ∼40 fs. Because this pulse compression is performed
relatively close to the HHG region, it accomplishes the goal of minimizing the B-integral by optimizing the long pulse vs. short pulse propagation distance while delivering short pulses to thegas-filled capillary.
A CaF2 lens with a focal length of 50 cm is used to focus the beam into the hollow-corewaveguide. The material of the lens has a low group velocity dispersion, which allows one tominimize the stretching of laser pulses as they travel through the lens. The laser spot size at thefocus of the lens is matched with the inner diameter of the fiber of 150 µm in order to optimize
the coupling of the laser light into the lowest order mode of the fiber. The fiber entrance is placed
close to the focus of the lens.
The beam splitter immediately after the lens (BS1) in Fig. 5.2 reflects 99% of the IR beam
into the fiber, and 1% goes into the beam stabilization system composed of a 50/50 beam splitter
BS2, a mirror M2, two CCD cameras CCD1 and CCD2, control software, and two actuated mirrors
AM1 and AM2. The purpose of the beam stabilization system is to lock the position and angle ofthe IR laser beam at the entrance of the fiber as they may change due to thermal drifts in the laser
amplifier and, therefore, can cause a reduction in the coupling efficiency into the fiber, which, inturn, may lead to instabilities and decreased efficiency of the HHG process and result in a low total
photon flux. The cameras track the centroids of non-focused (CCD1) and focused (CCD2) beams
and the control software makes adjustments to the actuated mirrors AM1 and AM2 based on the
feedback from the cameras. The two actuated mirrors are placed before the pair of chirped mirrors
in order to keep the incidence angle of the laser beam on CM1 and CM2 the same. Changes in
the incidence angle on the chirped mirrors can result in a fluctuating pulse duration and adversely
affect the HHG process. More details on the beam stabilization setup can be found in Ref. [107].
The HHG process itself takes place in a KMLabs XUUSTM hollow waveguide cartridge. The
waveguide is filled with He gas at ∼900 Torr. Because the end section of the waveguide is very
short, an efficient pumping scheme must be used in order to evacuate any residual gas from thebeamline to minimize re-absorption of high harmonics. Two Agilent TriScroll 300 roughing pumps

108
and a Pfieffer HiPace 300 turbo pump were used for that purpose. A pressure of 10−4 Torr inthe beamline was reached even at the highest gas pressures in the waveguide. To maximize theamount of IR light coupled into the waveguide, a sapphire Brewster window was used in front of
the waveguide at the entrance into the vacuum section of the beamline. The Brewster window
was oriented such that the HHG linear polarization would be in the s-polarization direction withrespect to the toroidal mirror in Fig. 5.1 in order to maximize its reflectivity. A half-wave plate
WP in Fig. 5.2 was used to match the polarization of the laser to the orientation of the Brewster
window and maximize the transmission through the window. After the waveguide, a 500 nm anda 200 nm Al filters were used to reject any residual IR light emitted from the waveguide. Because
the divergence of the IR beam is higher than that for the EUV, an aperture was placed in front of
the filters to select a small central portion of the IR beam and transmit the full EUV beam. Suchan approach greatly reduces the total amount of the IR power incident on the Al filters and thus
lowers the risk of their failure.
I now proceed to discussing the layout of the scattering chamber shown in Fig. 5.3. Threebeam paths are implemented in the chamber. The main one is path 1 shown in purple in the figure.
In this path, the EUV beam from the waveguide impinges on a toroidal mirror, which focuses it
through the sample. The toroid is oriented at a ∼10◦ grazing incidence, and, in order to redirect
the beam onto the sample, a flat mirror M2 oriented at the same angle is used. The beam isthen transmitted through a sample with a magnetic texture, and the diffracted light is collectedwith an imaging CCD (Andor DO-436). The beam block in front of the imaging CCD blocks the
un-diffracted transmitted beam as it contains no magnetic information, and, due to its very highintensity compared to the scattered light, can saturate the camera and thus make the observation ofmagnetic scattering impossible. The beam block is mounted on translation stages and its position
can be adjust to match the position of the un-diffracted peak on the CCD. The sample can betranslated in three directions. Translating it along the x and y directions perpendicular to thenormal to the sample helps align it with the incident beam, while translation along the beamadjusts the distance between the imaging CCD and the sample and thus, for a fixed scattering

109
angle, controls the size of the measured diffraction pattern. In the experiment reported here, thesample was placed at a distance of 36 mm away from the CCD. The sample mount was intentionallymade rather big, so that it would prevent any light scattered in the chamber from illuminating theimaging CCD. A ring magnet placed in front of the sample applies a magnetic field while letting
the EUV beam to go through. More details on the magnet mount are given below.
In path 2, shown in yellow in Fig. 5.3, a 45◦ gold coated mirror M1 can be inserted in the
beam before the toroid to reflect it onto the beam diagnostic CCD (Andor Newton 920). It helps
ensure that the HHG process in the waveguide cartridge is phase-matched and the EUV mode isgood with a symmetric Gaussian profile and low divergence as shown in Fig. 5.4. This is a criticalstep because the EUV photon flux depends on how well phase-matched the HHG process is, andthe focusing of the beam depends on its divergence and profile. A beam with a high divergence
and/or an asymmetric profile cannot be focused into a small spot by the toroid. Typically, a goodmode indicates that the phase-matching is also good, and, therefore, the photon flux is high.
In path 3, shown in Fig. 5.3 in blue, mirror M2 can be moved out of the beam in order todirect it into a spectrometer that consists of a 500 lines/mm gold coated diffraction grating anda CCD (Andor DO-420 BN). The imaging CCD and the spectrometer CCD are both mounted at
the focus of the toroid, which, in the case of the spectrometer, improves the spectral resolution bymaking the spot sizes of the diffracted beamlets small in the plane of the CCD. The spectrometerallows a small tuning of the HHG spectrum to match a harmonic peak with the resonant absorption
peak of the sample. An example spectrum is shown in Fig. 5.5(a) together with the maximum
estimated scattering efficiency for Fe calculated according to eq. (5.19) with the optical constants(max)

for Fe taken from Ref. [168]. As can be seen from the figure, the spectral overlap of ρRM S and the
HHG spectrum could be improved, but even with the existing overlap, the scattering predominantly
occurs at a single harmonic energy, as evidenced by Fig. 5.5(b), which shows a normalized productof the two curves in (a).
Another important consideration in the design of the setup is the EUV spot size on thesample. It can be estimated by assuming that the EUV beam is a perfect Gaussian (M 2 = 1)

110

SampleholderToroidalEmirror

M11magnetM2

3

Imaging
CCD

Spectrometer
CCD

Beam
diagnosticECCD

2

Diff.
grating

beamEblock

FromEEUV
sourceFigure 5.3: RMS chamber layout. Three beam paths are implemented in the chamber: In path 1(purple), the EUV beam is transmitted through the sample, and the diffraction pattern is collectedwith the Imaging CCD; in path 2 (yellow), the beam is directed onto the Beam diagnostic CCD,
which measures its spatial profile; in path 3 (blue), the beam is directed into the spectrometer.

Figure 5.4: Transverse profile of a phase-matched HHG beam. The vertical streaks are from thescratches on the gold mirror.

111

(max)

Figure 5.5: (a) An example HHG spectrum and its overlap with the scattering efficiency ρRM S . (b)(max)
The product of the spectrum and ρRM S demonstrates that the scattering occurs predominantly ata single harmonic energy.

112
and propagating it through the beamline to the sample. The beam radius at the output of the
waveguide can be calculated from the divergence in Fig. 5.4 as [191]
w0 =

λ
,
πθ

(5.21)

where θ is the divergence half-angle. With θ ≈ 0.5 mrad and λ ≈ 23 nm at the M -edge of Fe,
w0 ≈ 15 µm, and thus the spot size 2w0 is 30 µm. A toroid has two focal lengths—sagittal andtangential—that correspond to the sagittal and tangential radii of curvature of its surface r and R,
respectively, shown in Fig. 5.6. The two focal lengths depend on the angle of incidence α and arerαR

Figure 5.6: A sketch of a toroidal mirror. r and R are the sagittal and tangential radii of curvature,
respectively. α is the angle of incidence with respect to the surface normal.

the same when
r

r
.
R

(5.22)

1√
rR.
2(5.23)

cos α =For α defined by eq. (5.22), the focal length isf=

For the toroid used in this experiment, r = 90.5 mm and R = 3220 mm, which, according toeqs. (5.22) and (5.23), yields α = 80.35◦ , which is equivalent to a 9.65◦ grazing angle, and f = 27 cm.
The distance between the toroid and the EUV source is 160 cm, and with a known focal lengthand the size of the EUV beam at the output of the waveguide, the beam size at the sample position
can be calculated with the ABCD-matrix method [191]. At the sample position 29 cm away fromthe toroid (and 3.6 cm from the imaging CCD), an estimated beam diameter is ∼150 µm. This isa rather large beam, compared to typical illumination spot sizes of a few µm in EUV microscopy;
however, for scattering experiments this can be beneficial because an average over a bigger area is113
taken. Additionally, if a magnetic texture consists of repeated features, such as worm-like domains
or a skyrmion lattice, the scattering peak is sharper for a larger illumination area. This is analogous
to the resolution of a diffraction grating: the resolution improves and the diffraction peaks become
sharper if a larger area of the grating is illuminated.

5.2.2

Beamline efficiency

As mentioned above, a high EUV photon flux from the laser driven HHG source in Fig. 5.2
needs to be preserved in the beamline. Typical reflectivities of multilayer wavelength-selective
EUV mirrors do not exceed 50%, and approximately 3/4 of the photon flux would be lost on a pairof such mirrors. For that reason, broadband grazing incidence optics were used in the chamber.
Reflectivities at grazing incidence are normally quite high. Fig. 5.7 confirms that this is indeed thecase for the toroidal mirror and flat mirror M2 (see Fig. 5.3). Both optics are made from Pyrex
glass and coated with 30 nm of B4 C. The reflectivities in the figure are calculated for a grazing
angle of 10◦ with the multilayer approach described in Chapter 4. For a toroid-flat mirror pairFigure 5.7: Reflectivity of a toroidal mirror made from Pyrex glass and coated with 30 nm of B4 C
at a 10◦ grazing incidence.

114
in path 1 in Fig. 5.3, the total reflectivity is approximately 0.71 for s-polarization and 0.64 forp-polarization, according to the data in Fig. 5.7. In the case of an s-polarized EUV probe, there isalmost a threefold improvement compared to multilayer EUV mirrors.
Two alternative designs are possible to reject the residual IR light emitted from the fiber.
The first design includes a pair of rejector mirrors, typically made from Ru or another material,that are very efficient at absorbing the IR light and reflecting the EUV. After the Ru rejectors,
a thin (200 nm) Al film is placed to filter out any IR light that either was not absorbed or was
scattered. This design is robust in that it ensures that no damage to the camera by direct laser lightwill be caused should the Al film break. However, the rejectors are most efficient when the incident
light is p-polarized. For example, for a pair of Ru rejectors at an 8◦ grazing incidence, the totalreflectivity is 0.36 at the Fe absorption edge, and if the 0.72 transmissivity of the 200 nm Al filter is
included, the efficiency of the Ru rejectors with an Al filter becomes 0.26. Alternatively, a series of
Al filters can be used. A combination of a 500 nm and a 200 nm Al film transmits virtually no IR
light, and its transmissivity at the Fe M -edge is 0.32, which is higher than the reflectivity of a pairof Ru rejectors, and, therefore, this design is preferable. Additionally, at normal incidence, thesefilters are not sensitive to the polarization direction. The disadvantage is that the Al filters could
fail under very high thermal load from the IR laser pulses thus putting the imaging camera andthe sample at risk. To mitigate this risk, an aperture is used in front of the filters (see Fig. 5.2).
It blocks most of the IR beam, whose divergence is high, while transmitting a slowly diverging
EUV beam. The total efficiency of the beamline includes the transmissivity of the Al filters andthe reflectivities of the toroid and the flat mirror and is equal to 0.26 at the Fe absorption edge
meaning that approximately 26% of the EUV light generated in the waveguide is delivered to thesample.

5.2.3

Application of an external magnetic field

An external magnetic field can either be applied to the sample with an electromagnet ora permanent magnet. For an in-vacuum experiment, the latter is preferable because it does not115
require water lines in the vacuum chamber to cool the magnet. Additionally, stronger fields can beachieved with a permanent magnet, and the field at the sample can be adjusted by changing the
distance to the magnet. In Fig. 5.8, the measured on-axis (a) and the calculated (b) fields of asystem of two permanent ring magnets are shown. Two magnets, rather than a single magnet are
required for the mounting. The maximum on-axis field magnitude is ∼ 280 mT, which should be
sufficient to induce the skyrmion phase in the Fe-Gd sample, according to Ref. [192]. In addition,
the field near the axis at z > 8 mm is relatively uniform and thus the experiment should not bevery sensitive to a small misalignment of the magnet relative to the sample and the EUV beam.

Figure 5.8: Magnetic field of a system of two permanent ring magnets (shown in dark blue) (a)measured on-axis and (b) calculated.

The calculation in Fig. 5.8(b) was carried out using a magnetostatic scalar potential approach[92]. According to this approach, the magnetic potential, in the absence of boundary surfaces, isgiven byΦM (~r) = −Z ~0 ~ 0∇ · M (~r ) 3 0
d ~r ,|~r − ~r0 |(5.24)~ (~r) is a spatially dependent magnetization, and the magnetic field in free space can thenwhere Mbe computed using~ = −µ0 ∇Φ~ M.
B(5.25)The magnet mount is placed on a translation stage, and the distance between the sample

116
and the magnet, and thus the field at the sample, can be adjusted.

5.3

Field-dependent scattering
The magnetic field dependence of resonant magnetic scattering from an Fe-Gd film wasstudied. The nominal structure of the sample was [Gd (0.41 nm)/Fe (0.34)nm]×80. However, due
to an interdiffusion of the Fe and Gd layers, the interfaces disappeared, and the entire film could beconsidered an amorphous alloy. The sample was prepared at the Center for Memory and Recording
Research at the University of California San Diego, and more details about it can be found in Refs.
[192] and [53]. The magnetic film was grown on a 50 nm Si3 N4 membrane, and 5 nm Ta films wereused as a seed and cap layers.
Since the scattering experiment is performed at the Fe M -edge, the total thickness of the Felayers needs to be optimized in order to improve the magnetic contrast, according to eq. (5.18).
With an attenuation length of Fe equal to ∼15 nm at 54 eV, the optimal thickness is 30 nm, which
is very close to the sample’s total thickness of the Fe layers of 27.2 nm.
In the sample studied here, skyrmions are formed due to a competition between long-range
magnetic dipolar interactions and the energy of domain walls [54, 52, 193]. Such skyrmions arealso called dipole-stabilized skyrmions to emphasize the fact that, although they are topologicallysimilar to the more common skyrmions stabilized by the chiral Dzyaloshinskii-Moriya interaction[194, 195, 196], the mechanism that leads to their formation is different from the latter. The
sample was designed to form a hexagonal lattice of dipole-stabilized skyrmions in a magnetic field.
Here, I study a transition from a disordered stripe-domain phase to an ordered skyrmion phase bymeasuring RMS patterns at various applied magnetic fields.
An example RMS pattern is shown in Fig. 5.9. The pattern was taken at zero applied fieldand shows two intensity lobes around q ≈ 0.03 nm−1 , where q is the momentum-transfer vector
defined as q = 4π sin(θ/2)/λ (with λ = 23.16 nm) and θ are the wavelength and scattering angle,
respectively. This is a signature of a disordered nanoscale worm-like domain network commonly
seen in thin films with perpendicular magnetic anisotropy. At q ≈ 0.06 nm−1 , additional lobes

117
can be seen. They correspond to second-order scattering and indicate that there is an asymmetry
between up and down domains. To understand this, consider a Fourier series expansion of a squarewave. If the wave is perfectly symmetric, only odd harmonics will be present in the expansion 1 .
If the duty cycle is not equal to 50%, or the shape of the wave deviates from a perfect square wave,
e.g. becomes a sawtooth wave, even harmonics will appear in the expansion. In the case considered
here, those are manifested as two additional intensity lobes.

Figure 5.9: Example RMS pattern from an Fe-Gd multilayer alloy at zero field. Notice a weaksecond order scattering ring at q ≈ 0.06 nm−1 .

Fig. 5.10 shows RMS patterns taken as the field was increased from 0 to 180 mT. The
intensity in all of the images is normalized to 1. The patterns in the figure change from large lobes
at low fields to more concentrated intensity distributions in q-space at higher fields. This indicatesa transition from a disordered network of worm-like domains to a more ordered phase of stripe
domains. For a perfectly ordered stripe domain phase, the scattering pattern has two concentrated
peaks at ~q0 and −~q0 , as shown in Fig. 5.11. The magnitude of q0 corresponds to the periodicity ofthe domain pattern, and its direction is orthogonal to the direction of the stripes. At B > 100 mT,
the RMS patterns look like an overlay of an RMS pattern from perfectly ordered stripe-domains
and that from a more disordered domain distribution. This is likely due to a co-existence of an1

http://mathworld.wolfram.com/FourierSeriesSquareWave.html

118

Figure 5.10: Field-dependent RMS patterns. The film transitions from a disordered worm-likedomain phase at low fields to a more ordered mixed phase of stripe domains at high fields.

Figure 5.11: RMS from ordered stripe domains in Fe-Gd (taken at 54 eV, B = 0 mT).

119ordered and disordered domain phases within the illuminated area.
The domain periodicity at a given field can calculated from the RMS patterns. For small
scattering angles, 2 sin(θ/2) ≈ sin θ = mλ/p, where p is the periodicity, and m is a diffraction order.
In Fig. 5.10, m = 1. The domain size d = p/2, and, given the definition of the scattering vector q,
an expression for the domain size is
d=

π
.
q

(5.26)In order to obtain an accurate estimate of the position of diffraction peaks in q-space, the diffraction
patterns in Fig. 5.10 are averaged in the azimuthal direction. The averaging must exclude areasof the detector with pixels that do not carry any scattering information because they were either
obscured by the beam block (see the white shadow in Fig. 5.9) or exposed to scattered light (seethe left edge of Fig. 5.9). Such a map of bad pixels is shown in Fig. D.1 in Appendix D. A detailed
description of the averaging procedure is given in Chapter 6. An example azimuthally averaged
intensity distribution for B = 25 mT in Fig. 5.10 is shown in Fig. 5.12. In the figure, the peakFigure 5.12: Azimuthally averaged RMS pattern for the Fe-Gd sample at B = 25 mT.

is located at q = 0.0315 nm−1 , which corresponds to a domain size d = 99.7 nm, according toeq. (5.26). By performing the averaging procedure for each field value in Fig. 5.10, the dependence

120of the diffraction peak position, and thus of the domain size, on the field can be obtained. The
results are shown in Fig. 5.13. As the field increases, the scattering peak in (a) shifts towards lower
q-values, and the domain size in (b) increases, accordingly.

Figure 5.13: The dependence of the (a) angular RMS spectrum and (b) domain size of the Fe-Gd
sample on the applied magnetic field.

Upon further increase of the magnetic field to 220mT, a hexagonal lattice of dipole-stabilized
skyrmions formed in the film, which is reflected in the hexagonal shape of the scattering patternshown in Fig. 5.14, in agreement with Ref. [192]. The lattice constant, as determined from thefigure, is a = 228 nm. Assuming that the lattice is two-dimensional and close-packed, as illustrated
in Fig. 5.15, the radius of the skyrmions is rs = 124 nm. This result is a first demonstration ofresonant magnetic scattering from a skyrmion lattice on a tabletop.

5.4

Conclusions
This chapter presented the design of a resonant magnetic scattering experiment on a tabpletop

and its implementation for studying field-dependent scattering from a Fe-Gd magnetic film, as well
as some general theoretical considerations pertaining to the scattering process with linearly andcircularly polarized probes.
I showed analytically that the contrast mechanism of resonant magnetic scattering withlinearly polarized light arises from the Faraday rotation, which is mapped onto the magnetic texture,

121

Figure 5.14: RMS from a hexagonal skyrmion lattice.

rsaFigure 5.15: Two-dimensional hexagonal close-packed lattice of skyrmions. a is the lattice constantrs = a/2 is the skyrmion radius.

122
and derived the optimal sample thickness, equal to twice the attenuation length, that maximizes
the scattered flux.
A high photon flux at the M -shell absorption edge of Fe was achieved with a phase-matchedlaser-driven HHG process. The resonant nature of magnetic scattering in the EUV spectral region
allowed an improvement of the beamline efficiency through the use of efficient broadband grazing
incidence optics instead of multilayer mirrors. Even though the spectrum of the EUV probe incident
on the sample is broad, only the photons whose energy is on resonance with the absorption edgeof a given magnetic material are scattered.
The tabletop RMS setup was employed to study the field dependence of magnetic texturesin a Fe-Gd multilayer alloy with perpendicular magnetic anisotropy. At low fields, the sample
exhibited disordered domain networks. As the field increased, the domain size also increased, andthe domains themselves became more ordered. At B = 220 mT, an RMS pattern with a hexagonal
symmetry was observed. It was attributed to a formation of a two-dimensional hexagonal closepacked lattice of dipole-stabilized skyrmions. This constituted the first observation of magnetic
skyrmions with a tabletop EUV source.

Chapter 6

Laser-Induced Spin Dynamics in a Network of Magnetic Domains

In this chapter, I report the results of an RMS experiment performed at LCLS. The goal of theexperiment was to study optically-induced dynamics of nanoscale magnetic domains and domain
walls in a ferromagnetic multilayer film by use of resonant soft X-ray (SXR) magnetic scattering.
In the experiment, photons at high scattering angles were captured, which allowed access to finespatial features such as domain walls. I develop a method of reconstructing dynamic changes tomagnetic domains in real space by applying the experimental time-resolved scattering data to asimulated domain pattern whose Fourier spectral intensity agrees well with the static experimental
scattering intensity. The reconstructed changes in the domain pattern reveal feature-dependent
non-uniform demagnetization. I develop a phenomenological spin transport model and apply it
to the same simulated domain pattern and find that the model predictions agree well with theexperimental reconstructions. I, therefore, attribute the observed spatially resolved dynamics toa transient exchange of angular momentum between magnetic domains of opposite magnetization.
I also find that, on short timescales, domain walls remain sharp and demagnetize less than thedomains themselves, but their demagnetization and broadening proceed even on long timescales. I
perform thermal transport simulations and correlate this behavior with the heat diffusion throughthe crystal lattice throughout the depth of the sample, which affects the total magnetic anisotropyof the film and thus the width of the domain walls.

124

6.1

Overview of previous work
RMS is a very powerful technique because it allows one to access spatial variations of magnetic

properties with light. This is especially important in time-resolved studies since additional nonlocal demagnetization channels, such as superdiffusive spin currents [154], can arise due to magnetic
inhomogeneities, e.g. in a network of magnetic domains, and contribute to the demagnetization
process. Because of this and the nanometer magnetic correlation lengths involved, laser-induced
evolution of a magnetic texture can be substantially different from that of a uniformly magnetizedfilm. To date, most studies of ultrafast demagnetization have either adopted a spatially averaged
approach [197, 172, 144] or used uniformly magnetized samples [127, 57, 122, 133, 125, 198, 199].
They have been able to uncover the role of Stoner excitations and ultrafast magnon generation in
ultrafast demagnetization [122, 125], as described in Chapter 4, and also reported an observationof an ultrafast phase transition from the ferromagnetic to the paramagnetic state [144, 172]. In
the absence of spatial resolution, however, observation of lengthscale-dependent ultrafast magnetic
phenomena would be impossible without specifically tailoring the design of the sample, as wasdone in Refs. [200, 201, 156, 202], where the authors observed ultrafast spin transport by carefully
designing the magnetic multilayer stacks and measuring a spectroscopic time-resolved MOKE response in the EUV or visible spectral range. An ability to perform measurements with both a hightemporal and spatial resolution is critical not only for gaining insight into the fundamental aspectsof materials’ magnetism but also from the practical standpoint when, for instance, fast switchingof a nanoscale magnetic device needs to be observed.
I reviewed the tools for studying materials systems and their respective temporal and spatial
resolution in Fig. 1.1 in Chapter 1. FELs and laser-driven tabletop HHG sources are best suited forthe task of capturing magnetization dynamics with a high temporal and spatial resolution by meansof time-resolved resonant magnetic scattering. Such experiments have been performed to study
ultrafast spatially resolved magnetization dynamics in real space in patterned [203] and disorded[204] systems by use of X-ray holography at an FEL. Time-resolved experiments in reciprocal space

125
have been carried out using both FEL [205, 206, 207] and HHG [182] sources. Refs. [206, 207]
report on magnetization dynamics in granular media, which are quite different from the domain
sample studied here. Relevant results concerning the evolution of a pattern of disordered magnetic
domains are reported in Refs. [182, 204, 205], where evidence of ultrafast spin-polarized transport
between domains is observed. While Ref. [182] does not make any claims regarding the dynamics
of domain walls, Refs. [205, 204] state that the the domain walls are softened in a laser-excited
sample. This softening is attributed to the effect of spin currents. However, in all of the references
above, the spatial resolution was limited by the relatively low scattering angles, which included
only the first order scattering ring. Therefore, the interpretation of the results is based on an
extrapolation of the experimental signals to higher scattering angles.
In the experiment reported here, in addition to the first order ring, third and fifth orders were
measured, which allowed an extraction of the dynamics of very small magnetic features includingdomain walls. In contrast with Refs. [204, 205], I observe that domain walls demagnetize less andremain sharp on short timescales, and only start to broaden at longer times. In this chapter, I
discuss possible sources of this disagreement with the previous results in the literature.

6.2

Soft X-ray scattering experiment at LCLS
The experiment was performed at the SXR beamline at LCLS, and the setup is shown in

Fig. 6.1. Soft X-ray pulses with a 60 fs pulse length were produced by the FEL at a repetition rate of
120 Hz. The energy of the photons in the pulse was set to 852.7 eV to match the L3 absorption edgeof nickel. The linear polarization of the soft X-ray pulses was converted to circular in the Deltaundulator [208]. The X-rays were scattered by a domain sample whose multilayer composition
was Si3 N4 (50)/ Ta(3)/ Cu(5)/ (Co90Fe10(0.2)/ Ni(0.6))x50/ CoFe(0.2)/ Cu(3)/ Ta2 O5 (3), where
the layer thicknesses in parentheses are in nm. Such a sample structure generated perpendicularmagnetic anisotropy (PMA), which lead to a formation of worm-like magnetic domains with aperiodicity of 160 nm. The multilayer structure of the sample and its magnetic force microscopy
image are shown in Fig. 6.2. As discussed in Chapter 5, the magnetic contrast mechanism with

126
circularly polarized X-rays is due to the X-ray Magnetic Circular Dichroism (XMCD), which ismanifested as a dichroism in the X-ray absorption of light with a well defined helicity by domains
with opposite magnetization directions. XMCD thus maps the domain pattern to a difference in
X-ray absorption. In the far field, the X-rays scattered by the domain sample produce a series ofconcentric rings corresponding to odd orders of diffraction. This scattering pattern was recordedwith a pnCCD detector placed 275.3 mm away from the sample. The detector consisted of four
512×512 pixel independent panels with adjustable positions, and the pixel size was 75 µm. Such an
experimental geometry allowed detection of X-rays scattered at angles of up to ∼ 8◦ . The pnCCD

XMCD CCD

Electromagnet

Al filter

Domain Sample

pnCCD

IR Pump
X-ray Probe

Figure 6.1: Experimental setup for time-resolved soft X-ray resonant magnetic scattering. The
pnCCD is used to detect scattered X-rays at different delay times between an IR pump and an X-ray
probe, and the detector behind the pnCCD is used for XMCD measurements. The electromagnet
allows one to control an external magnetic field at the sample.

had an opening at the center to allow the unscattered X-rays propagate through. These X-rays
were detected with an Andor Newton CCD camera placed behind the pnCCD. An Al filter in frontof the Andor CCD suppressed the IR light from the pump laser. An electromagnet could apply
up to 0.6 T of external magnetic field perpendicularly to the surface of the sample. In addition to
zero-field RMS measurements, both the scattering (with the pnCCD) and XMCD (with the Andor

127

Sample structure
Ta2O5 (3 nm)
Cu (5 nm)

x50{

Co90Fe10 (0.2 nm)

Ni (0.6 nm)
Co90Fe10 (0.2 nm)

Cu (5 nm)Ta (3 nm)
Si3N4 (100 nm)

Figure 6.2: Sample structure and MFM image of domains. The domains are disordered, and theirperiodicity is ∼160 nm.

128CCD) signals were taken when the sample was magnetically saturated. Those data were used toremove any non-magnetic contributions from the zero-field scattering data. I describe the detailsof this removal procedure in Section 6.4.
An X-ray probe pulse arrived at the sample with a time delay relative to an IR pump pulse
from an amplified Ti:Sapphire laser at 795 nm. The pump pulses were 60 fs long, and the average
incident pump fluence was ∼23 mJ/cm2 . The pump laser was time-locked to the FEL. However
the FEL had an intrinsic jitter associated with an uncertainty in the emission time, energy, andintensity of individual electron bunches, which was manifested as fluctuations in the arrival time,energy, and brightness of X-ray pulses. The delay time between the IR pump and the X-ray probe
was scanned in the range from −3 ps (to probe an unperturbed sample at negative delays) to 20
ps. Scattering patterns at each delay time were taken in a signle-shot manner. Because of thejitter in the arrival time of the probe, the position of the delay stage in the pump arm does notcorrespond to a fixed pump-probe delay from pulse to pulse. However, a correlation method used
to time stamp the X-ray pulses at LCLS allows one to correct for such jitter. If both the position of
the delay stage and the arrival time of the probe pulse are known, then the total delay between thepump and the probe can be calculated. For this reason, the delay stage is moved continuously inorder to improve the statistics. In the next step, all of the collected scattering patterns are sorted
according to their delay, and the pattern at a requested delay is computed as an average of all
the patterns recorded within a specified time window (400 fs for this experiment) centered at thatdelay. Such a binning procedure, while lowering the temporal resolution, leads to a significantly
improved signal-to-noise ratio.

6.3

Data processing considerations
With the experimental setup in Fig.

6.1, two-dimensional RMS intensity distributions

Sq2 (qx , qy ) shown in Fig. 6.3(a) were measured. Because of the azimuthal symmetry of Sq2 (qx , qy ),129
the dimensionality of the data set can be reduced by computing an azimuthal average of Sq2 (qx , qy )1
hSq2 i(q) =2π

Z2π

Sq2 (q cos φ, q sin φ)dφ,

(6.1)

0

where the scattering vector q is defined as q = 4π sin(θ/2)/λ (λ is the incident wavelength and θ isthe scattering angle). An azimuthally averaged RMS intensity distribution is shown in Fig. 6.3(b).
Prior to computing hSq2 i with eq. (6.1), it is important to ensure that several aspects are properly-0.6

RMS Intensity Sq2 (qx , qy ) (a.u.)(a)Azimuthally Averaged RMS Intensity
10 0

100

(b)

-0.4
-2

10-4

0
0.2

10

-6

hSq2 i (a.u.)qy (nm−1 )

10

-0.2

10 -2

10 -4

0.4
0.6
-0.5

0
qx (nm−1 )

0.5

10-8

10 -6

0

0.2
0.4
q (nm−1 )0.6

Figure 6.3: RMS intensity distribution from an unperturbed domain sample at t = 0: (a) as
measured by the pnCCD, (b) averaged in the azimuthal direction. In (b), the first and third orderscattering peaks are clearly visible at 0.04 and 0.12 nm−1 , respectively. A small fifth order peak isdiscernable at 0.2 nm−1 .

accounted for.
First, the panels of the pnCCD must be aligned correctly. The spacing in pixels between
the panels seen as white lines cutting through the data in Fig. 6.3(a) represents the actual physical spacing between the panels. If this distance-to-pixel calibration is incorrect, the azimuthally
averaged signal will be distorted.
Second, the center of the scattering pattern must be determined accurately. This is important
for the correct computation of hSq2 i, especially at low q-values. For a continuous detector this canbe done simply by finding the centroid of the image. For the pnCCD detector consisting of four
separate panels, the centroid does not give the true position of the center of the scattering pattern
because of the missing pixels in between the panels. I, therefore, threshold the data and fit a circle

130
to the pixels whose values are above the specified threshold. A function that performs the fitting
can be found online.1
Third, the missing pixels in between the pnCCD panels (as well as pixels at the edges ofthe pnCCD that are saturated by the pump) must be properly accounted for. Eq. (6.1) assumes
that on a circle of radius q, there are 2πq pixels. However, in the case considered here, data on
some pixels might be missing, so the total number of pixels with data is less than 2πq. This is
illustrated with the pixel mask in Fig. 6.4, where the pixels with missing or bad data are shownin black, and a circle of constant q is shown in blue. The intersection of the circle with the black
pixels should not contribute to the total average. Assuming that the scatter pattern is azimuthally
symmetric the missing data can be accounted for by multiplying hSq2 i(q) defined by eq. (6.1) by
R
−1
2π
a correction factor fc (q) = 0 m(q, φ)dφ
, where m(q, φ) is a mask of pixels with bad data
shown in Fig. 6.4, such that m(q, φ) = 1 for good pixels (white) and m(q, φ) = 0 otherwise (black).

1
0.8

q
φ

0.6
0.4
0.2
0

Figure 6.4: Mask of bad pixels m and a circle of constant q. m(q, φ) = 1 for good pixels (white)and m(q, φ) = 0 otherwise (black).

With the procedure described above, an azimuthally averaged spectrum is obtained (seeFig. 6.3(b)). The procedure is repeated for each delay time between the pump and the probe andyields a time-dependent azimuthally averaged dataset hIq i(t) shown in Fig. 6.5 as a density plot of
the time-resolved data normalized by the static signal hIq i(0). Here, I have changed the notation1

https://blogs.mathworks.com/pick/2008/03/14/fitting-a-circle-easily/

131
for the scattered intensity from Sq2 to Iq to emphasize the fact that the measured scattering is
not necessarily purely magnetic (which will be denoted by Sq2 further in the text) but also has acharge component Cq2 contributing to the total scattered intensity Iq . In Section 6.4, I show thatthe oscillations seen in Fig. 6.5 around q ≈ 0.3 nm−1 come from the charge scattering and develop
a method of isolating the purely magnetic component of the scattered intensity Sq2 .

Total Scattered Intensity Iq (t)=Iq (0)1

q (nm!1 )

0.5
0.8

0.4
0.3

0.6

0.2
0.4

0.1
0

10
t (ps)

20

Figure 6.5: Time-resolved azimuthally averaged scattered intensity normalized by the static signal
hIq i(t)/hIq i(0). Note the oscillations at q ≈ 0.3 nm−1 .

6.4

Separation of magnetic and electronic scattering

6.4.1

Mathematical basis

In the following, I derive the intensity of the probe beam at the pnCCD after it is scattered by a sample with some real-space distribution of the magnetization (or spin) s(~r) and chargec(~r). I employ the refractive index formalism to describe X-ray interactions with matter, whichis equivalent to the scattering factor formalism often used for that purpose [160]. The derivation
takes into account contributions from both the charge distribution c(~r), as well as two types of spin
distribution—one that is uncorrelated (s(~r)) and one that is correlated (αc(~r)) with the charge,
where α is a scalar that represents the magnitude of the charge-correlated spin distribution. Because the sample is probed with circularly polarized X-rays, the magnetic and charge scattering

132
components have the same polarization and can interfere in the detector plane. Therefore, caremust be taken while separating the two scattering components.

6.4.1.1

General caseA wave transmitted through the sample is called the exit surface wave (ESW). If the electricfield of the incident wave is E0 , the exit surface wave is
E = E0 exp [ikd (n0 + δns s(~r) + δnc c(~r) + αδns s(~r))] ,(6.2)

where d is the sample thickness, δnc is the magnitude of the refractive index variation due to charge,
δns is the magnetic refractive index variation, i.e. it is defined by the difference in the refractive
index between the positive (+) and negative (−) magnetization directions δns = 21 (n+ − n− ),k = 2π/λ. s(~r) is determined by, for example, the magnetic domain topography of a given sample and is equal to the spatial dependence of the out-of-plane component of the magnetizations(~r) = Mz (~r). c(~r) is determined by spatial variations of the charge distribution that arise due
to, e.g., inhomogeneities in the chemical composition and/or sample roughness. In the most general case, there is a charge-correlated spin distribution αc(~r), which depends on how strongly themagnetization is sensitive to local charge variations c(~r). In the case of granular media [206, 207],
such correlation is indeed very strong. In the case considered here, it is expected to be negligible
because the sample is relatively flat and homogeneous. Nonetheless, I shall keep the αc(~r) term inorder to develop a rigorous approach. Note that c(~r), s(~r), and α can all be time-dependent, andthe time dependencies of these terms are not necessarily the same.
The spot size of the X-ray probe is on the order of 100 µm in the experiment, which is muchbigger than the characteristic lengths of the domain sample. Hence, I assume that the incident
beam is a plane wave, and, in eq. (6.2), E0 = 1. Additionally, I divide out the factor exp [ikdn0 ]because it does not contribute to the scattered intensity. For brevity, I define Cr = ikdδnc c(~r) andSr = ikdδns s(~r). With these definitions and the assumptions made above, eq. (6.2) simplifies toδns
E = exp Sr + Cr + α
Cr .
δnc

(6.3)

133
I Taylor expand this expression up to second order
δns
1
1
1
E = 1 + Cr + S r + α
Cr + Cr2 + Sr2 + α2
δnc
2
2
2



δns
δnc

2

Cr2 + αδns 2
δnsCr + αCr Sr + Cr Sr . (6.4)δnc
δnc

The complex electric field of the scattered wave at the detector is obtained by Fourier transforming
expression (6.4)



2 !
δns
1
δns
2 δns
Eq = δ(~q) + 1 + α
Cq + Sq +1 + 2α+αF{Cr2 }+
δnc
2
δnc
δnc


1
δns
2
+ F{Sr } + 1 + α
F{Cr Sr },
2
δnc

(6.5)

where I defined Cq = F{Cr } and Sq = F{Sr }. This expression can be simplified by introducingsa = 1 + α δnand canceling the Dirac-Delta δ(~q), as it is only non-zero at ~q = 0, i.e. for theδnc
unscattered light, which I am not considering here. Eq. (6.5) now simplifies to
1
1
Eq = aCq + Sq + a2 F{Cr2 } + F{Sr2 } + aF{Cr Sr },
2
2

(6.6)

The scattered intensity is Iq = |Eq |2 . Using eq. (6.6) and neglecting the third order terms,
an expression for the scattered intensity can be derived
Iq = a2 Cq2 + a3 Cq F{Cr2 } cos ξaCq − ξa2 F {Cr2 } ++ 2 a2 Cq F{Cr Sr } cos ξaCq − ξaF {Cr Sr } + aCq F{Sr2 } cos ξaCq − ξF {Sr2 } +

+ 2 |aCq Sq | cos ξaCq − ξSq +2

a F{Cr2 }Sq

(6.7)



cos ξa2 F {Cr2 } − ξSq +
+ 2 |aF{Cr Sr }Sq | cos ξSq − ξaF {Cr Sr } + F{Sr2 }Sq cos ξSq − ξF {Sr2 } + Sq2 ,where ξx are the phases of the respective terms indicated by the subscript x. The expression
above is very unwieldy, but it is simplified upon azimuthal integration because, for disordered anduncorrelated distributions s(~r) and c(~r), many of the cosines average out to zero. The phases in
the following pairs are random and uncorrelated ξaCq and ξSq , ξa2 F {Cr2 } and ξSq , ξF {Sr2 } and ξaCq .

Therefore cos ξaCq − ξF {Sr2 } = 0, cos ξaCq − ξSq = 0, and cos ξa2 F {Cr2 } − ξSq = 0, where
R 2π
hXi = 1/2π 0 X(φ)dφ is an azimuthal average.
The terms with the factors a3 Cq F{Cr2 } and F{Sr2 }Sq are vanishingly small and, as such,can be eliminated. The reason for this is that Cq and F{Cr2 }, and Sq and F{Sr2 } have a different

134
q-dependence. To illustrate this, consider a one-dimensional example f (x) = eiωx . The Fouriertransforms of f (x) and f 2 (x) are F{f (x)} = δ(ω) and F{f 2 (x)} = F{ei2ωx } = δ(2ω). The
product of the two Dirac-Deltas is zero δ(ω)δ(2ω) = 0. Similarly, if the Fourier transform of Sr
is a ring of a certain diameter, then the Fourier transform of Sr2 is a ring of twice the diameter,
and the product of the two rings is negligible if the widths of the rings are smaller than theirradii, which is the case for the data set being considered. The same argument applies to Cr . With
these approximations and the fact the, from the convolution theorem, F{Cr Sr } = Cq ⊗ Sq , where
the symbol ⊗ denotes the convolution operation, the final expression for the azimuthally averaged
scattering intensity isEδns 2 Dδns 2
|Cq |2 + 2 1 + α|Cq | |Cq ⊗ Sq | cos ξaCq − ξaCq ⊗Sq +
δnc
δnc
D
E

δns
+2 1+α|Cq ⊗ Sq | |Sq | cos ξSq − ξaCq ⊗Sq + |Sq |2 .
δnc

hIq i = 1 + α(6.8)

This expression is valid in the general case and can be simplified for the special cases of zero andnear-saturation applied magnetic field B.

6.4.1.2

Zero applied field

Without an external magnetic field, Cr and Sr are random an uncorrelated, and, thereforecos ξaCq − ξaCq ⊗Sq = cos ξSq − ξaCq ⊗Sq = 0. The measured azimuthally averaged intensity
then becomeshIq (B = 0)i = 1 + α

E DEδns 2 D
|Cq |2 + |Sq |2 .
δnc

(6.9)

If α = 0, the signal is simply the sum of the magnetic and charge scattering intensities.

6.4.1.3

Near-saturation applied fieldI have assumed above that the magnetic part has two components—a non-uniform charges
correlated ±α δnr) = ±1, so that Sr = ±idkδns when theδnc Cr and a uniform charge-uncorrelated s(~

sample is magnetically saturated by an external field. The Fourier transform of Sr , in this case, isa delta-function Sq = F{Sr } = ±idkδns δ(~q). Therefore, Sq (~q 6= 0) = 0, and Sq ⊗ Cq = ±idkδns Cq .

135
Note that I am not assuming the existence of a non-uniform and charge-uncorrelated magnetization
component because, in the experiment, only scattering into q-vectors corresponding to the charge
scattering was observed with a saturating applied field without any additional time-dependent
scattering into any other q-vectors. It would be an improbable coincidence if a charge-uncorrelated
magnetic component would scatter X-rays into the same cone of angles as the Cr . The assumption
employed here can, therefore, be considered reasonable.
Under these conditions, only the first two terms in eq. (6.8) remain and the expression for
hIq i is simplified tohIq (B 6= 0)i = 1 + αEEδns 2 Dδns 2 D
|Cq |2 + 2 |dkδns | 1 + α|Cq |2 cos ξaCq − ξ±idkδns aCq .
δnc
δnc(6.10)In order to compute the scattered intensity using this expression, the argument of the cosine has
to be evaluated for positive and negative applied fields. This can be done using the fact that thephase of a product of two complex numbers is equal to the sum of their individual phases. The
following expressions thus hold
ξaCq = ξa + ξCq ,(6.11)
ξ±idkδns aCq = ξa + ξCq + ξ±idkδns .
The phase of the complex number ±idkδns for positive and negative fields, respectively, is simply
Im(δns )πξ+idkδns = arctan+ ,Re(δns )2(6.12)
Im(δns )πξ−idkδns = arctan+ + π.
Re(δns )2
Using eqs. (6.10), (6.11), and (6.12), I derive the final expression for the scattered intensity undera saturating applied magnetic fieldhIq (B 6= 0)i = (1 ± 2dkIm(δns )) 1 + αEδns 2 D
|Cq |2 .
δnc(6.13)

Note that Cq and α can be time-dependent. Additionally, in the saturation-field case, I allow
δns to be time-dependent in order to take into account the possibility of ultrafast demagnetizationof a uniformly magnetized sample via local demagnetization mechanisms. Because of the possible

136
additional demagnetization channels in the zero-field case, the time dependence of δns in that caseis not necessarily the same. Finally, in the limit of α = 0 the total scattering intensity is simply
equal to the charge scattering intensity modulated by the XMCD absorption.

6.4.1.4

Separation of the spin and charge contributions

I now use eqs. (6.9) and (6.13) to separate the magnetic and charge scattering intensities.
D
E
First, I use eq. (6.13) to solve for |Cq |2 . To streamline the derivation, I introduce a2s
substitution f = α δnδnc . With this substitution, I expand the term |1 ± f | and rewrite the intensities

for the positive and negative fields
DE|Cq |2 ,DE
2|Cq |2 .
hIq (B < 0)i = (1 − 2dkIm(δns )) 1 + |f |2 + 2Re(f )hIq (B > 0)i = (1 + 2dkIm(δns )) 1 + |f |2 + 2Re(f )2(6.14)

I define the sum and difference intensities as and1(hIq (B > 0)i + hIq (B < 0)i) ,2
1∆ = (hIq (B > 0)i − hIq (B < 0)i) .
4
Σ=Using eqs. (6.14), ∆ and Σ can be written explicitly asED∆ = dkIm(δns ) 1 + |f |2 + Re(f ) |Cq |2 ,ED
Σ = 1 + |f |2 + 4dkIm(δns )Re(f ) |Cq |2 .

(6.15)

Assuming that f is small (because α must be small in this experiment) and δns is small, andneglecting second order terms, these expressions are simplified to


 D
E
δns
∆ ≈ dkIm(δns ) + Re α
|Cq |2 ,δncDEΣ ≈ |Cq |2 .

(6.16)

The charge-uncorrelated (time-dependent) term Sxmcd = dkIm(δns (t)) can be found fromthe measured XMCD signal since d and k are known, δns (t = 0) can be found from the CXRO
database [209], and the time dependence of δns (t) is given by the time dependence of the XMCDδnsintensity ∆Ixmcd (t). Thus the charge-correlated (time-dependent) term Re α δnc can be foundfrom ∆.

137
In order to finalize this derivation and separate the charge and magnetic contributions to thesscattering intensity, I Taylor-expand eq. (6.9) in powers of Re α δnδnc up to a first orderhIq (B = 0)i ≈

 D


E D
E
δns
1 + Re α|Cq |2 + |Sq |2 .
δnc

(6.17)

D
E
Using eqs. (6.16) and (6.17) and the definition Sxmcd = dkIm(δns (t)), I now solve for |Sq |2and thus completely separate the spin and charge contributions to the scattering intensity
E
1|Cq |2 = Σ = (hIq (B > 0)i + hIq (B < 0)i) ,2(6.18)
DE∆
2|Sq | = hIq (B = 0)i − 1 + − Sxmcd Σ.
ΣDE
2
Notice that in the limit of α = 0, ∆
−S
=
0,
and
|S
|
simply becomes the difference between
q
xmcd
Σ
D

the total measured intensity and the average of intensities measured at the opposite values of an
applied saturating field.
With eq. (6.18), the separation procedure can be summarized as follows(1) Measure the scattering intensity hIq (B = 0)i at zero field.
(2) Apply a saturating magnetic field to the sample and measure the scattering intensities
hIq (B > 0)i and hIq (B < 0)i for the opposite directions of the field. Simultaneously, takean XMCD signal ∆Ixmcd (t) with the XMCD detector in Fig. 6.1.
(3) From ∆Ixmcd (t) compute Sxmcd
DE
DE(4) Using eqs. (6.18), compute |Cq |2 and |Sq |2 .

6.4.2

Implementation of the separation procedure

The result of the application of the separation procedure derived above to the experimental
dataset in Fig. 6.5 is shown in Fig. 6.6(c). The oscillations seen in the total scattered intensity
in Fig. 6.6(a) are present in the sum intensity Σ, shown in Fig. 6.6(b), which confirms that their
origin is non-magnetic. They appear due to laser-induced acoustic strain waves that originate
at charge inhomogeneities and travel through the film. As can be seen from Fig. 6.6(c), the

138

(a)

(b)

(c)

Figure 6.6: Separation of the spin and charge scattering: (a) Total measured scattering intensity hIq (B = 0, t)i / hIq (B = 0, t = 0)i, (b) Charge scattering Σ(t)/Σ(t = 0), (c) Spin scattering
Sq2 (t) / Sq2 (t = 0) .

139
oscillations that appear in the original signal in Fig. 6.6(a) are no longer present in the magnetic
scattering intensity in Fig. 6.6(c) computed according to eq. (6.18), which confirms the validity ofthe separation procedure. The XMCD signal used to separate out the spin contribution is shownin Fig. 6.7. While the XMCD trace is noisy its effect on the final result is small because themagnitude of the charge-correlated spin distribution represented by α is small.

6.5

Summary of the experimental RMS results
Using the purely magnetic scattering signal in Fig. 6.6(c), I can now carry out its time-

resolved analysis, which is summarized in Fig. 6.7.

Sq2 (t)/Sq2 (t = 0)

ΔM (q) = 1 −
(a)

0.25

q (nm−1 )1st Order Peak Intensity Sq2  (a.u.)

0.5
0.4

0.2

0.3

0.15

0.2

1

(b)
0.05

q (nm−1 )

0.3

0.8

0.045
0.04

0.6

0.035

0.4

0.1

0.03
0.1

0.05

0.2

0.025

0

0

5

10

15

20

0

20
×10-3

3

0.13

q (nm−1 )Magnetization (a.u.)15

0.14 (d)1st order
3rd order
5th order
XMCD

1

10

3rd Order Peak Intensity Sq2  (a.u.)

RMS and XMCD Magnetization Dynamics
(c)

5

0.8

2.5

0.12

2

0.11

1.5
1

0.1

0.6

0.5

0.09
-5

0

5

10

Δt (ps)

15

20

0

5

10

Δt (ps)15

20

Figure 6.7: Summary of the transient RMS dynamics in the CoFe/Ni multilayer sample: (a) qdependent relative demagnetization; (c) demagnetization as determined from the first, third, andfifth order peaks as well as the XMCD signal; dynamics of the (b) first and (d) third order scatteringpeaks. The solid black lines represent the center positions of the peaks.

First, I compute the magnetization dynamics in reciprocal space as a function of momentum
transfer q and time t according to∆M (q) = 1 −q

Sq2 (t)/ Sq2 (t = 0).

(6.19)

140
The result is shown in Fig. 6.7(a). Surprisingly, the magnetization behaves differently at differentq’s, which suggests some interesting feature-dependent spatial dynamics of magnetic domains. In
order to analyze the first, third, and fifth order scattering rings separately, they need to be isolated.
Because the magnitude of the first order peak is ∼2 orders of magnitude greater than that of
the third order (see Fig. 6.3(b)), the first order can be considered isolated without any further
processing. The fifth order peak is too small—only a few detector counts above the noise level—
and a reliable background subtraction for this peak has been impossible. The third order peakis sitting on an exponential background, which is manifested as a straight line with a constantnegative slope on the log scale in Fig. 6.3(b). As such, the background under the third order peakcan be fitted with a decaying exponential, at each time step. An example of such fitting for t = 0
ps is shown in Fig.6.8.

6#10 -3
hSq2 iBackground Fit

hSq2 i (a.u.)

4
2
0

-2
0.05

0.1

0.15 0.2
q (nm!1 )0.25

Figure 6.8: Background subtraction for scattered intensity at t = 0 ps in order to isolate the thirdorder peak (blue signal above the orange background).

For the isolated scattering peaks, I track changes in their position and magnitude withtime. Square roots of the peaks’ magnitude correspond to changes in magnetization M (q, t) =q
Sq2 (t)/ Sq2 (t = 0), according to eq. (6.19). They are shown in Fig. 6.7(c). Note that the

141first order demagnetization trace agrees well with the XMCD signal. The third and fifth orderdemagnetization traces match well with the first order at short times, but start to decay after ∼7 ps. The discrepancy between the fifth order and the rest of the data stems from the fact that the
fifth order could not be properly isolated. For that reason, I did not perform a peak shift analysis
for the fifth order.
I track the peak positions of the first and third order scattering peaks by separately fitting
each peak with a split Pearson type VII distribution [205]


−m1
−m2
(q − q0 )2(q − q0 )2
hIq i = I0 1 +H(q0 − q) + I0 1 +H(q − q0 ),α12 m1
α22 m2(6.20)

where I0 is the peak amplitude, α1,2 and m1,2 are the parameters of the distribution that accountfor its skewness and sharpness, q0 is the position of the peak maximum, which I define as the centerof the peak, and H is the Heaviside step function. At each time delay, the fitting of the peaks witheq. (6.20) is performed, which yields q0 (t) for the first and third order peaks shown with a black line
in Fig. 6.7(b) and (d), respectively. Both peaks shift rapidly towards lower q’s by approximately
7% within the first 1.6 ps, after which their positions remain almost constant. A similar peak shiftobserved only in the first order was interpreted as evidence of a softening of domain walls causedby superdiffusive spin currents in Ref. [205]. Here, I show that the additional data provided by thethird and fifth order scattering rings are inconsistent with such a scenario, and, to the contrary,domain walls remain sharp.
I summarize the main features of the time-resolved data set in Fig. 6.7 as follows:• Demagnetization traces extracted from the first, third and fifth orders agree on short times.
• Third and fifth order magnetization starts to decay relative to the first order after ∼7 ps.
• First order demagnetization agrees well with the XMCD trace.
• First and third order scattering peaks shift rapidly towards lower q’s within the first 1.6 ps.

142

6.6

Reconstruction of real-space transient domain dynamics
In order to gain insight into spatially resolved magnetization dynamics, it would be very

advantageous to have the capability of time resolved real-space imaging of magnetic domains. Coherent diffractive imaging techniques are capable of reconstructing an object from its diffraction
intensity distribution by use of phase retrieval methods, which rely on specialized computer algorithms [210, 211, 212, 213, 214]. However, such phase retrieval methods require that the datasatisfy certain constraints. One of such constraints is oversampling [215], which requires that thediffraction amplitude is sampled at a spatial frequency exceeding the Nyquist frequency. This
oversampling ratio is expressed as [107]
ρ=

λz
,
pD

(6.21)

where λ is the wavelength of the probe, z is the distance between the sample and the detector, Dis the lateral size of the object, and p is the pixel size of the detector. In order for phase retrieval
algorithms to work, ρ must be greater than 2. In the experiment reported here, ρ ≈ 0.1, whichmakes the reconstruction of the real-space domain pattern impossible.
Therefore, in order to observe spatially resolved transient changes to the magnetization profile, particularly to the out-of plane magnetization component Mz (x, y), a different approach has
to be adopted. I develop such an approach below. It involves application of the measured RMS
data to a simulated domain pattern, for which the phase is known and does not need to be retrieved computationally, and thus enables the computation of transient laser-induced changes inthe simulated pattern. A caveat with this approach is that the simulated domain pattern has torepresent the actual domain pattern of the sample to a high degree of accuracy. The arrangementof the domains is not so important because the system is disordered, and the experimental data
were averaged over multiple domain configurations due to shot-to-shot variability of the domain
pattern caused by a re-nucleation of the domains after an optical pump pulse. In principle theMFM image in Fig. 6.2 could be used for the purpose of reconstruction. However, the quality
and the resolution of the image are not sufficient. Instead, I use a domain pattern that was sim-

143
ulated using the MuMax software [216] with the first and second order magnetic anisotropy equal
to the experimental values of 343 kJ/m3 and -143 kJ/m3 , respectively. The simulation grid size
was 1024×1024×8 cells, and the dimensions of each cell were 5×5×5 nm. The resulting pattern
and the comparison of its scattered intensity Sq2 with the experimental one are shown in Fig.
6.9 The agreement between the simulated and the experimental RMS intensities is reasonable: theFigure 6.9: (a) Part of the simulated domain pattern and (b) comparison of the simulated andexperimental scattering intensities for the first (left) and third (right) orders.

magnitudes and positions of the peaks agree well, while the experimental peaks are broader than
the simulated ones. The cause for the latter is the fact that the probed area in the experiment was∼ 100 × 100 µm, while the simulation window was only ∼ 5 × 5 µm. For perfectly regular domains,
increasing the width of the window would lead to a narrowing of the scattering peaks. However,
because in the FeCo/Ni sample domains are stochastic, broader areas show more domain variability
and thus broaden the scattering peaks. Simulating a larger area is computationally prohibitive,
and for that reason, given that the agreement between the RMS intensities is satisfactory, I use thesimulated domain pattern in Fig. 6.9(a). Caution must be exercised in order not to over-interpretthe results of the reconstruction procedure due to the discrepancies between the simulated andexperimental scattering intensities mentioned above.
The base assumption underlying the method of reconstructing laser-induced changes in magnetic domains is that, at any given time t, the profile Mz (x, y, t) can be represented by a convolution

144of the original magnetization profile Mz (x, y, t = 0) with a spatial filtering kernel G(x, y, t)
Mz (x, y, t) = G(x, y, t) ⊗ Mz (x, y, t = 0).

(6.22)

This assumption is valid if there are no new spectral components in the Fourier spectrum of Mz attime t compared to t = 0 ps. Fig. 6.6 clearly demonstrates that this is indeed the case: none of thevalues in the figure exceed unity, and thus a Fourier spectrum at any time is just a modified original
Fourier spectrum. Note that eq. (6.22) does not require a linear response of the magnetization
to an optical pump in the time domain. Since the RMS intensity is related to Mz via a Fouriertransform as Sq2 = |F{Mz }|2 , from the convolution theorem and eq. (6.22), it follows that the RMS
intensity at time t can be expressed as a product of the unperturbed RMS intensity and a squared
Fourier transform of G(x, y, t)
Sq2 (t) = gq2 (t) · Sq2 (t = 0),(6.23)

where gq (t) = F{G(x, y, t)}. Eq. (6.23) can be solved to find the spatial filter in reciprocal space|gq (t)| =q

Sq2 (t)/Sq2 (t = 0).

(6.24)

Because of the azimuthal symmetry, eq. (6.24) holds for the azimuthal averages as well. Thus
|gq (t)| is simply equal to the time-resolved RMS data set normalized by the unperturbed RMS
intensity, and the quantity shown in Fig. 6.6(c) is equal to |gq |2 (t). In practice, I obtain gq (t)by fitting the square root of the normalized RMS data with a smoothing spline and extrapolating
to higher q-vectors with a constant value equal to the average demagnetization at a given time
step. This is illustrated in Fig. 6.10, where gq (t = 1.6 ps) is shown at the time of both maximum
demagnetization and maximum shift in the peak positions. gq is then rotated around q = 0 nm−1
to produce a rotationally symmetric two-dimensional Fourier filter.
In the most general case, gq is complex and thus its phase would be lost in eq. (6.24).
However, here, gq is purely real. This is because the spatial filtering kernel G(x, y, t) is even andreal, and, therefore, its Fourier transform gq is also real. The even parity of G(x, y, t) follows fromits azimuthal symmetry which requires that G(x, y, t) = G(−x, −y, t). The azimuthal symmetry, in145
turn, is based on an assumption that laser-induced changes in the magnetization profile are driven
by an isotropic process because, for a disordered domain network, there should not be any preferred
direction. The real-valuedness of G(x, y, t) follows from the real-valuedness of the magnetization
profile Mz (x, y, t). Hence, as a Fourier transform of a real and even function, gq is real. It is also
positive. If gq became negative for certain times the XMCD signal in Fig. 6.7(c) would switch sign.
Such a reversal of the magnetization was not observed. If gq crossed zero at certain q’s, gq2 would
be equal to zero at those q’s. However, gq2 is always positive, according to Fig. 6.6(c). Therefore,
gq is real and positive, which means that its phase is zero, and that gq = |gq |.

g(t = 1:6 ps)

1
0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

!1

q (nm )
Figure 6.10: Spatial filter gq (t = 1.6 ps) (red) is fitted to the data (black) with a smoothing spline.
A constant value is extrapolated to high q’s.

With the simulated magnetization profile Mz (x, y, t = 0) and the experimentally measured
spatial filter gq (t), an excited domain pattern at time t can now be found from eqs. (6.22) and(6.23) asMz (x, y, t) = F −1 {gq (t)F {Mz (x, y, t = 0)}} .

(6.25)

At each time t, I apply eq. (6.25) and thus follow the evolution of an excited domain patternin real-space. Fig. 6.11 shows the original simulated unperturbed domain pattern (panel (a)) and146
snapshots of the relative changes to this pattern at the time of maximum demagnetization (andmaximum RMS peak position shifts) of 1.6 ps (panel(b)), and at the longest time step of 20 ps(panel (d)). Fig. 6.11(c) shows the lineouts of the domain profiles along the red line in Fig. 6.11(a)at 0 ps, 1.6 ps and 20 ps.
The reconstructed domain pattern exhibits several distinct features(1) The demagnetization process is spatially non-uniform: smaller domains and domain ends
demagnetize stronger(2) This feature-dependent asymmetric quenching causes a perceived motion of domain walls
by shifting the magnetization zero-crossings. In the two-dimensional images (b) and (d) in
Fig. 6.11, it is manifested as sharp color transitions from red to blue at the domain walls.
In Fig. 6.11(c), the zero-crossing of the orange and yellow curves at around −0.27 µm is
shifted by 2.6 nm to the right relative to the unperturbed blue curve.
(3) The domain walls themselves demagnetize less than the centers of domains.
(4) At long times, the magnetization profile becomes smoother as evidenced by the recovery of
the magnetization at the domain centers of the yellow curve in Fig. 6.11(c).
To finalize the discussion of the reconstruction of laser-induced changes, I repeat the reconstruction procedure with a Gaussian spatial filter based only on the first order scattering ring inorder to see the effect of the data at higher q-vectors on the reconstruction and directly compare
the results with Ref. [205]. Fig. 6.12 shows both, a Gaussian fit for the filter at 1.6 ps (panel (a))
and changes to the original domain pattern that were reconstructed using this filter.
A comparison of Fig. 6.12(b) and Fig. 6.11(b) reveals that if the high q data are removed, thedemagnetization process remains spatially non-uniform, but the domain walls are quenched muchstronger than the domain centers. This is in agreement with Ref. [205], but contradicts the resultof this experiment. Thus, discarding scattering data at high q’s does not allow one to capture thepersistence of domain walls and their motion in the course of ultrafast demagnetization.

147

Figure 6.11: Laser-induced evolution of magnetic domains: (a) original domain pattern; changes inthe domain pattern at (b) t = 1.6 ps (the time of maximum demagnetization and maximum peakshifts) and (d) 20 ps after the pump pulse; (c) lineouts of domain profiles at various times along
the red line in (a).

148

Figure 6.12: Laser-induced changes in domains with a Gaussian spatial filter according to Ref.
[205]: (a) Gaussian fit to the filter; (b) Changes in the domains reconstructed with the filter in (a).

149

6.7

Spin transport in a laser-excited domain network
To explain the features observed in the spatially-resolved magnetization dynamics, namely,

the non-uniformity of the demagnetization process, the smaller quenching of domain walls and theirmotion, I develop an effective model that describes two interacting spin populations: one is a cold
, i.e., below the Fermi level, and stationary ground-state population Scold (x, y), and the other isan excited spin population Shot (x, y), which is hot, i.e., above the Fermi level, and mobile. The
model is illustrated in Fig. 6.13. The two populations interact with one another because the cold

Up-domain

Down-domain

Shot
Shot

Shot

Fermi
Level

Scold
DOS

DOS

Figure 6.13: Schematic of the spin transport model.

electrons are promoted above the Fermi level by the laser pump, and the hot electrons cool down
and eventually decay. Additionally, the hot electrons can move to reduce a magnetization gradient.
Mathematically, the dynamics of the two spin populations are described by the following coupledequations∂Scold (x, y, t)= −γ(t)Scold (x, y, t) + β(t)Shot (x, y, t),∂t∂Shot (x, y, t)~ D(x, y, t)∇S
~ hot (x, y, t) ,= γ(t)Scold (x, y, t) − β(t)Shot (x, y, t) + ∇∂tMz (x, y, t) = Scold (x, y, t) + Shot (x, y, t),(6.26)

150
where γ(t) is the rate of generation of hot electrons, β(t) is the rate of their relaxation, the diffusion
coefficient D(x, y, t) determines the transport properties of hot electrons, and Mz (x, y, t) is the totaltransverse profiles of the z-component of the magnetization.
The parameters γ, β and D are phenomenological, but they can have a microscopic origin.
For example, β can be viewed as an ensemble-averaged decay rate of electrons above the Fermi
level
β = RE

1

F,(6.27)−∞ τ ()g()f (, T )dwhere τ () is the lifetime of excited electrons [217], and g() and f () are the density of states andthe non-equilibrium electron distribution function, respectively.
The diffusion coefficient D(x, y, t) is an effective parameter, which, similarly to eq. (6.27), is
also an ensemble average of the hot spin population. The underlying processes of spin transport
might not necessarily be diffusive, and might evolve from ballistic transport at early times after
the laser pump to diffusive transport at longer times, according to Ref. [155], which refers to suchtransient evolution as the superdiffusion of spins. Note that this is different from the standard
definition of superdiffusion, where the particle displacement ∆x ∝ tp>1 [218]. However, since theformer nomenclature has become a standard convention in the field of ultrafast magnetism, I shallrefer to the transient evolution from ballistic to diffusive transport as superdiffusive spin transport.
In the spin transport model described by eqs. (6.26), the superdiffusion can effectively be describedthrough the time dependence of D.
The spin transport model presented here requires that the polarization of spin currents ispreserved over hundreds of nm. Although such lengths exceed the mean free path of hot electrons,
efficient spin-polarized currents mediated by hot electrons through up to 300 nm thick Cu films
have been observed in ultrafast experiments [219, 220].
The results of applying eqs. (6.26) to the domain pattern in Fig. 6.11(a) are shown in Fig.
6.14. The parameter values used in the simulations were β = 2.5 ps−1 , γ was time-dependentγ(t) = (1 − exp(−t/τ ))102 ps−1 , where τ = 100 fs, and the diffusion coefficient D in Fig. 6.14(a)151
was higher for the domains than for the domain walls D(x, y) = 2 · 104 (0.15 + 0.85Mz2 (x, y, t =0)) nm2 /ps whereas in Fig. 6.14(b) it was constant D = 104 nm2 /ps. Note that the values of theparameters were chosen with the goal of reproducing the extracted changes in the domain pattern in
Fig. 6.11 and are derived neither from first principles nor from a rigorous fitting procedure, so theiruniqueness cannot be guaranteed. Nonetheless, the simulations do capture the main features ofthe experimentally extracted dynamic domain patterns and clearly demonstrate that the observed
laser induced changes in the magnetization profile are driven by spin transport.
Fig. 6.14(a) simulated with a varying diffusion coefficient D agrees well with the extractedchanges in the domain pattern in Fig. 6.11(b), while the result of the simulation with a constant
D in Fig. 6.14(b) is in good agreement with Fig. 6.12 obtained by neglecting scattering data
at high q-vectors. This leads to a conclusion that the smaller quenching of the magnetization at
the domain walls is caused by a dependence of the spin transport properties on the local magnetization vector, which reduces the spin-polarized flux at the domain boundaries. The lineouts
in Fig. 6.14(c) illustrate this point further. The orange curve, which corresponds to the simulation with a magnetization-dependent D, exhibits sharper domain walls, albeit not as sharp as inFig. 6.11(c), and a clear shift of the domain wall at −0.27 µm to the right, in good agreement withthe experimental domain reconstructions. For a constant D, the broadening of the domain walls is
too strong.
A comparison in Fig. 6.14(d) of the azimuthally averaged scattering intensities corresponding
to the simulations with a magnetization-dependent and constant D with the scattering intensity ofthe original domain pattern reveals that, in both cases, there is a shift of the first order peak (leftpanel) towards lower q’s, in agreement with the experimental data in Fig. 6.7(b). However, thethird order peak (right panel) disappears when D is assumed to be constant. For the magnetizationdependent D, the third order peak persists, again, in agreement with the experimental data, albeit
with a somewhat decreased magnitude, and, like in the experiment, it is shifted relative to the blue
curve at 0 ps. A small reduction of the simulated third order peak suggests that the dependence ofthe diffusion coefficient on the magnetization might be more complex than what is assumed here.

152

Figure 6.14: Laser-indeced changes to the ground-state of a domain pattern (Fig. 6.11(a)) simulatedusing eqs. (6.26) with (a) a magnetization dependent diffusion coefficient D = 2 · 104 (0.15 +0.85Mz2 (x, y, t = 0)) nm2 /ps and (b) constant diffusion coefficient D = 104 nm2 /ps. The whitelines in (a) and (b) indicate the paths along which (c) the lineouts of the magnetization profile aretaken. (d) is a comparison of the hSq2 i Fourier spectra of the original domain pattern and calculated
for the cases (a) and (b).

153
Nonetheless, the figure provides clear proof of the dependence of the spin transport properties on
the magnetization.
In summary, the simulations in Fig. 6.14 demonstrate that ultrafast spin transport gives riseto the observed non-uniformity of the demagnetization in Fig. 6.11. Because the spin-polarized
flux is proportional to the size of a domain boundary ∼ d2 , where d is the characteristic domain
size, and the total number of spins within a domain is proportional to its volume ∼ d3 , the numberof spins leaving the domain per unit time relative to the number of spins contained within thedomain scales as 1/d. Therefore, the smaller the domain the more it demagnetizes, which is why in
Fig. 6.11, the magnetization reduction is stronger for smaller domains and domain ends. Such an
asymmetry in the demagnetization leads to shifts in zero-crossings of the magnetization, which areperceived as domain wall motions. The persistence of domain walls requires a reduced transportof spins at the walls compared to the domain centers. This could either be an intrinsic propertyof ultrafast spin transport or it could be related to the behavior of the magnetic anisotropy. The
latter is responsible for the slow decay of the magnetization measured at the third order peak in
Fig. 6.7, as I show below.

6.8

Effects of temperature gradients on domain wall dynamics
The thickness of the CoFe/Ni magnetic film in Fig. 6.2 was 40 nm. Given that the absorption

length of the pump at 800 nm in Ni is ∼13 nm, one could expect a significant gradient of laser
excitation throughout the film. Indeed, this is the case for the sample considered here, as can beseen from Fig. 6.15. The heat source in the figure was computed as
S = −dFabs (z)/dz,
where the absorbed fluence Fabs (z) of the infrared laser pump was calculated using the full multilayer formalism described in Chapter 4. In this section, I explore how this gradient in the pumpabsorption affects the magnetization dynamics.
I start with the three-temperature model (3TM) proposed in the 1996 pioneering work on ul-

Ta2O5

154

Multilayer
Cu

Ta
Cu

Si3N4

Figure 6.15: Depth gradient of the IR laser excitation in the CoFe/Ni multilayer in Fig. 6.2 for an
incident pump fluence of 26.7 mJ/cm2 .

155
trafast magnetization dynamics [57]. The model consists of three coupled thermodynamic equations
that describe the interacting electron, lattice, and spin energy baths∂Te~ · κe (Te , Tl )∇T~ e + Gel (Te )(Tl − Te ) + Gel (Ts − Te ) + S(z, t),
=∇
∂t


∂Tl
~ · κl (Te , Tl )∇T~ l + Gel (Te )(Te − Tl ) + Gls (Ts − Tl ),=∇Cl (Tl )∂t∂Ts
Cs (Ts )= Ges (Te − Ts ) + Gls (Tl − Ts ),∂t
Ce (Te )(6.28)

where indices e, l, and s correspond to the electron, lattice, and spin systems; Tx is the temperatureof the respective system, Cx is the specific heat, κx is the thermal conductivity, and Gij are thecoupling parameters between the electron and lattice, electron and spin, and lattice and spin energy
baths. The source term in first equation in (6.28) describes the direct absorption of the pump pulse
by the electron system. The details regarding the material parameters used in the 3TM simulations
can be found in Appendix C.
The 3TM is fitted to the experimental XMCD signal in Fig. 6.7(c) by converting the spintemperature Ts to the magnetization using the temperature dependence of the magnetization of
the sample measured with a vibrating-sample magnetometer (VSM) (see Fig. 6.16).

6.8.1

Electron, lattice, and spin temperatures

The electron, lattice, and spin temperatures for the best fit of the 3TM to the experimental XMCD data are shown in Fig. 6.17, and the corresponding calculated depth profile of themagnetization, as well as its depth average, are shown in Fig. 6.18
While the equilibrium in the electron temperature is established quickly throughout the
thickness of the sample (see Fig. 6.17(a)) due to a high electron thermal conductivity κe , thegradients in the lattice and spin temperatures persist out to longer times. The thermal conductivityof the latter two is negligibly small (see Appendix C) and thus the equilibriation of the temperature
distribution of the lattice and spin systems throughout the sample is mediated by electrons. For
the case of the lattice, for example, at early times, the hot electrons at the top portion of thesample strongly couple their energy to phonons and also diffuse downward into the cooler bottom

156

M (T )=M (T = 0 K)

1
0.8
0.6
0.4
0.2
0
0

500

1000

T (K)
Figure 6.16: Temperature-dependent magnetization of the FeCo/Ni multilayer. The experimental
data are represented by black circles, and the blue curve is a fit with M (T ) = 1 − (T /TC )l (TC =982.2 K and l = 3.166).

Figure 6.17: Depth-dependent (a)-(c) and depth-averaged (d)-(f) temperatures of the (a), (d)electron, (b), (e) lattice, and (c), (f) spin baths. The total averages as well as separate ones for theNi and Co layers are shown in (d)-(f).

157portion of the film where they again transfer some of their energy to the lattice. Even if theelectrons and the lattice are in thermal equilibrium locally, any gradient in the lattice temperature
will manifest itself as a gradient in the electron temperature through the electron-lattice coupling
parameter Gel . The cycle of coupling-diffusion-coupling will thus continue until a full thermal
equilibrium is reached. For the spins, the situation is similar. These temperature gradients resultin a magnetization gradient shown in Fig. 6.18
Such a non-uniform simulated demagnetization suggests that, for a domain pattern, thedynamics could be different at different depths. It is important to understand that the scattering
data provide depth-averaged information, as I show below.

6.9

Propagation of X-rays through a thick sample
Because the wavelength of the X-ray probe at 1.45 nm is much smaller than the thicknessof the magnetic multilayer film of 40 nm, strictly speaking, the propagation of X-rays throughsuch a film should be treated as a three-dimensional problem, and a gradient in the magnetization
can be expected to affect the overall scattering intensity. In general terms, the propagation ofan electromagnetic wave through a non-uniform medium with a spatially varying refractive index
n(x, y, z) is described by the Helmholtz equation [221]∇2 E + k02 n2 (x, y, z)E = 0,

(6.29)

where E is the electric field envelope, which excludes the fast oscillating part eiωt , and k0 is thewavenumber in vacuum. For an arbitrary refractive index n(x, y, z), this equation can be solved
numerically using the split-step method [222] to obtain an ESW out of a thick sample. The splitstep method involves subdividing the sample into thin layers, as shown in Fig. 6.19; each layer
modifies the phase and the amplitude of the propagating electric field, according to the refractive
index in that layer n(x, y, zi ), where i is the index of the layer. Propagation in between the layers
is computed by applying a free-space spatial filter in Fourier space [222]. However, for a weakly
scattering magnetic domain sample, the method can be simplified. Because the probability of a

158

Figure 6.18: Calculated (a) depth-dependent and (b) depth-averaged time-dependent magnetization. The average calculated magnetization agrees well with the experimental XMCD trace.

159

Figure 6.19: Subdivision of a thick domain sample into layers. The transverse magnetization profile
varies with depth.

160
scattering event is small compared to non-magnetic scattering, the likelihood that a photon canscatter twice as it travels through the sample is negligible. As such, the ESW can be written as asum of the waves coming from individual layers
ESW =

N
X

s(x, y, zi ),(6.30)iwhere N is the total number of layers, and s is a spatially-dependent magnetic profile as definedin eq. (6.2). It should be noted that the phase and amplitude of each s(x, y, zi ) is determined
only by the transverse variations of the magnetization in layer i. The total accumulated phase andattenuation for each layer do not depend on the position of the layer zi and are determined by
the total thickness of the film and its composition. In the detector plane, the field Sq is a Fouriertransform of the ESW
Sq (qx , qy ) = F(NX)
s(x, y, zi )

=

i

N
X

F {s(x, y, zi )} =i

NX

Sq(i) ,

(6.31)iand is equal to the sum of Fourier transforms of the scattered waves coming from individual layers.
Assuming that the transverse magnetization profile is the same for all zi at t = 0 ps, eq. (6.23) canbe applied to each layer
Sq(i) (t) = gq(i) (t)∆Sq (t = 0),(6.32)

where ∆Sq (t = 0) corresponds to the field in the detector plane scattered from a single layer at
t = 0 ps. From eqs. (6.32), (6.31), and (6.24), an expression for the net measured spatial filter canbe derivedv
u22
u PN (i)u i gq (t)∆Sq (t = 0)u.
gqΣ (t) = t
PNi ∆Sq (t = 0)

(6.33)

The factors |∆Sq (t = 0)|2 in the numerator and denominator cancel one another, and eq. (6.33) issimplified toNgqΣ (t) =1 X (i)gq (t).
N(6.34)i

Therefore, for a thick sample, the measured spatial filter in reciprocal space gqΣ (t), i.e., a root of the
RMS intensity at time t normalized by the static RMS intensity, is equal to the depth-average of

161
the spatial filters for the layers at different depths zi . Since the scattering amplitude is proportional
to the magnitude of the magnetization component Mz at a particular depth, with a magnetization
gradient, layers that are demagnetized stronger would contribute less to the detected RMS intensity.

6.9.1

Evolution of the magnetic anisotropy

I now consider the effect of the magnetic anisotropy Ku . Its temperature dependence measured with a VSM is shown in Fig. 6.20. Assuming that the anisotropy energy depends on the

5

#10 5

Ku (J/m3 )

4
3
2
1
0
200

400

600

800

T (K)
Figure 6.20: Experimental temperature dependence of the magnetic anisotropy Ku for the FeCo/Nisample.

lattice temperature, the time-dependent depth profile of Ku can be found by use of the calculated
lattice temperature in Fig. 6.17. The anisotropy energy determines the domain wall width [8]
r
d=π

A
,
Ku

(6.35)

where A is the exchange stiffness. According to Ref. [147], A(T ) ∝ M 2 (T ). Hence, the timedependent domain wall width z-profile can be found from eq. (6.35) using the lattice temperaturein Fig. 6.17(b), the temperature dependence of Ku in Fig. 6.20, and the transient magnetization
profile in Fig. 6.18. A proper calculation of the depth-averaged domain wall width needs to take

162
into account account the magnetization gradient because, as mentioned above, film layers that aredemagnetized stronger contribute less to the scattering signal. For that reason, the RMS data
should be expected to be more sensitive to the colder bottom layers of the sample. I, therefore,
define the average domain wall width asRL
hdi (t) =0

M (z, t)d(z, t)dz
RL
0 M (z, t)dz

(6.36)and plot it from 8 to 20 ps in Fig. 6.21. The result in Fig. 6.21 is rather qualitative and has to be

hd(t)=d(t = 0 ps )i

1.205

1.2

1.195

1.19
10

15

20

t (ps)
Figure 6.21: Calculated time-dependent domain wall width changes relative to the static case.

interpreted with caution because the true temperature dependence of the exchange stiffness for the
sample considered here is unknown. In addition, it is not entirely clear whether the magnetizationand temperature gradients can be converted directly into a domain wall width gradient using eq.
(6.35) which is applicable in thermal equilibrium (I only used the data at times greater than 8 ps
for that reason). Nonetheless, even with such a qualitative approach, there is evidence of a slow
increase of the domain wall width with time. This suggests that the observed slow decrease of themagnitude of the third order in Fig. 6.7(c) could be caused by an increase in the domain wall width
related to the thermal diffusion throughout the thickness of the sample.

163
Further investigation of this phenomenon is needed, and additional insight can be gained
from both rigorous micromagnetic simulations as well as time-resolved experiments, which employ
techniques, such as X-ray vector nanotomography, capable of imaging magnetic structures in three
dimensions [223].

6.10

Conclusions
In this chapter, experimental results on time-resolved resonant magnetic scattering of soft

X-rays from a magnetic domain sample were presented. The scattering data were collected athigh q-vectors, which, in contrast with previous work, allowed an extraction of the magnetization
dynamics of small spatial features, such as domain walls. Spatially-dependent demagnetization
can be represented as a convolution of the static domain pattern with a time dependent spatial
filter, and such filter can be extracted from the scattering data. By applying this filter to a simulated domain pattern, spatially-resolved transient changes in the transverse magnetization profile
were reconstructed. The reconstructions showed that the demagnetization process was spatially
non-uniform with stronger quenching of smaller domains and domain ends. In addition, the demagnetization of domain walls was found to be surprisingly weak, and their broadening occurred
only on long timescales. The spatial non-uniformity of the demagnetization and the persistence ofdomain walls was explained with a phenomenological spin transport model that included a reduced
transport at the walls, thus confirming that the laser induced demagnetization process in a domain
network is primarily driven by spin-polarized currents which facilitate a direct exchange of angularmomentum between domains of opposite magnetization. The slow broadening of domain walls onlong timescales was confirmed by energy transport simulations in combination with simulationsof the propagation of X-rays through a thick sample. This broadening was attributed to a slow
reduction of the temperature and magnetization gradients—initially caused by a non-uniform absorption of the laser pump—through thermal diffusion. In the future, further insight can be gained
from experiments that can capture magnetic structures in three dimensions as well as rigorous
simulations.

Chapter 7

Outlook: Towards Time-Resolved Element-Specific Magnetic Microscopy

Magnetic materials exhibit a rich variety of spin textures ranging from macroscopic domains
to nanoscale topologically protected magnetic vortices known as skyrmions. Those textures playa very important role because they correspond to the minimum-energy or meta-stable states ofa particular material system and, as discussed in Chapter 6, can provide additional dissipation
channels, e.g. via spin currents, after the system has been excited by a laser pulse. Thus magnetism
and its dynamics are inherently spatially dependent. Most spectroscopic studies overlook thisspatial dependence by either magnetically saturating the sample in a specific direction and thus
erasing the features or by effectively averaging across them. Yet, an ability to observe magnetism
not only with a high temporal but also with a high spatial resolution is critical for its fundamental
understanding.
There are multiple techniques that can provide a spatial resolution for studying magnetic
materials. With resonant magnetic scattering discussed in Chapters 5 and 6, the spatial information is accessible in reciprocal space. Real-space imaging techniques include Kerr and Faraday microscopy [224, 225], scanning Hall probe [226, 227] and SQUID [228, 229] microscopy,
Lorentz microscopy [230], scanning transmission X-ray microscopy [231], photoemission electron
microscopy [232, 233, 234], magnetic force microscopy [235, 236], holographic methods [188, 237,
238, 239, 203, 204], and coherent diffractive imaging (CDI) techniques also known as lensless imaging [187, 210, 212, 214, 213, 240, 241, 242, 243]. Detailed reviews of various imaging techniques
can be found in Refs. [244] and [230].

165
An ideal technique for magnetic microscopy would have to have both a high spatial andtemporal resolution. These requirements necessitate the use of a small wavelength probe with ashort pulse duration. HHG driven EUV and soft X-ray coherent light sources are particularly suitedfor this purpose. With such sources, magnetic microscopy can be implemented on a tabletop ina laboratory setting. EUV microscopes that rely on re-imaging with Fresnel zone plates [245] areinefficient and suffer from aberrations. Their resolution is limited by the width of the outer ring.
Similarly, the resolution of holographic approaches is limited by the size of reference holes, and,due to the small size of the latter, such approaches are also lossy. This is particularly important
for weakly scattering objects such as magnetic textures. Lensless imaging techniques are very
attractive since they require imaging geometries that are very similar to those used for resonant
magnetic scattering and thus can be easily adapted to existing scattering experiments (see Chapters
5 and 6). In addition, their resolution is truly diffraction limited meaning that for a wavelength ofλ ∼ 20 nm near the M-edges of the 3d ferromagnets the theoretical resolution limit is ∼ 10 nm.
CDI has been used for imaging magnetic textures with both synchrotron [246, 247] and HHG
sources [187]. A remarkable example is a reconstruction of a three-dimensional magnetization map
in a microscopic rod using X-ray vector nanotomography [223]. FELs can also be used for timeresolved magnetic CDI and can provide a high spatial resolution. However, HHG sources havea better temporal resolution and an added advantage of easy access, relative compactness, andcomparatively low operation costs. Thus developing an approach for time-resolved CDI magnetic
microscopy with an HHG source would be beneficial.
As mentioned above, static CDI has been demonstrated with an HHG source in Ref. [187].
However, this work utilized circularly polarized harmonics, for which the information about themagnetization magnitude and direction is mixed in the reconstructed phase and amplitude, andthe method relied on a holographic reconstruction thus making the sample design quite complicated.
Below, I demonstrate the feasibility of lensless magnetic imaging with linearly polarized HHG, forwhich the experimental setup is simpler, and show that the reconstructed amplitude represents themagnitude of the out-of-plane magnetization component, and the phase represents its direction.

166
The method does not rely on holography and, therefore, no special considerations are required indesigning the sample. At the end of this chapter, I also provide an outlook for future work which
includes studying antiferromagnets and phase transition materials.

7.1

Principles of lensless imaging
A basic lensless imaging problem is shown in Fig. 5.1. A coherent EUV probe interacts withthe object and the scattered electric field forms a diffracted intensity distribution in the CCD plane.
Because the detector only measures the intensity, the information about the phase is lost. If thephase were known, however, the electric field from the detector plane could be propagated backto the object, and, with a known probe, the phase and amplitude of the object could be found.
Hence, the lensless imaging problem becomes a phase retrieval problem, which can be solved bya number of iterative algorithms. CDI has a lot in common with the phase retrieval problem incrystallography, although the two fields have developed largely independently [248]. Historically,
the first optical phase retrieval algorithm was proposed in 1971 and is known as the error reduction
algorithm [249]. In this algorithm, the phase is found by iteratively propagating the field between
the two planes and setting the amplitude of the field to be equal to the measured amplitudes in both
the detector and the object planes at each iteration. The constraint that the solution must satisfy
the measured intensities in both planes allows the algorithm to converge. Generally speaking, thesolution does not have to be constrained in terms of the measured intensities only; any other set of
constraints suited for the problem will also work. The scalar propagation between the object andthe detector planes is described by the Helmholtz equation [221]∇2 E(~r) + k 2 E(~r) = 0,

(7.1)

where k = 2π/λ. Assuming that z is the propagation direction, a Fourier transform of (7.1) in thetransverse direction reads− |~q⊥ |2 + ∂z2 + k 2 Ẽ(~q⊥ ) = 0,

(7.2)

167
where ~q⊥ = k∆~r⊥ /z denotes the spatial frequencies in the transverse plane, and Ẽ(~q⊥ ) is the~ ⊥ , where R~ ⊥ is theFourier transform of E(~r). In the definition of ~q⊥ , I introduced ∆~r⊥ = ~r⊥ − R
transverse coordinate in the object plane (see Fig. 7.1 for notations). The solution to eq. (7.2) is
𝑌

𝑦

𝑋

𝑥
𝑟⊥

𝑅⊥

𝑧

𝑧=0
Object plane

Detector plane

Figure 7.1: Geometry and notations for a diffraction problem.

q

2
2
Ẽ(~q⊥ , z) = Ẽ(~q⊥ , z = 0) exp i k − |~q⊥ | z .

(7.3)

In real-space, the field at the detector is then found as an inverse Fourier transformE(~r⊥ , z) = F

−1


q

n
o
2
−1
2
Ẽ(~q⊥ , z) = FẼ(~q⊥ , z = 0) exp i k − |~q⊥ | z
.

(7.4)

In the paraxial approximation, the diffraction angle is small, and the transverse components of the
wavevector are much smaller than the longitudinal component, i.e., |~q⊥ |  k. The square root in
the exponent can then be Taylor expanded up to a second order, which yields
!#)
(
"
2
|~
q
|
⊥
.
E(~r⊥ , z) = F −1 Ẽ(~q⊥ , z = 0) exp ikz 1 −
2k 2

(7.5)

Eq. (7.5) can be written in terms of a convolution of the field at z = 0 with a Fresnel propagation
kernel [250, 222]E(~r⊥ , z) = E(~r⊥ , z = 0) ⊗ h(~r),

(7.6)

168
where, using the definition of ~q⊥ above and eq. (7.5), h(~r) is defined as
ik 2

h(~r) = eikz e 2z r⊥ .

(7.7)

Note that for simplicity, a factor 1/iλz present in the Fresnel-Huygens integral is not included in
the definition of h(~r). For a more rigorous derivation see Refs. [250] and [222]. The convolution,
up to a phase factor, can be written explicitly as
ZZ
E(~r⊥ , z) =~ ⊥ , z = 0)ei
E(R

~ )2
k(~
r⊥ −R
⊥
2z

~ ⊥,
d2 R

(7.8)

where the integration is carried out over the entire spatial extent of the object, and the eikz phasefactor, again, has been omitted. Expanding the square under the exponent in eq. (7.8) yields
kr 2

i 2z⊥

E(~r⊥ , z) = e

ZZ~ ⊥ , z = 0)e−i
E(R

~
k~
r⊥ ·R
⊥
z

kR2

⊥
~ ⊥.
ei 2z d2 R

(7.9)

If z  a2 /λ, which is known as the Fraunhofer approximation, where a is the characteristic size of2kR⊥

an object, it holds that ei 2z ≈ 1, and eq. (7.9) simply becomesno~ ⊥ , z = 0) ,E(~r⊥ , z) ∝ F E(R(7.10)

where the phase factor has been omitted. Eq. (7.10) has an important consequence, i.e., it tells
us that the fields at the object and the detector are related via a Fourier transform. This is true
if the condition z  a2 /λ is satisfied. For typical parameters for EUV imaging of λ ≈ 20 nm anda ≈ 10 µm, z  5 mm. In the scattering experiment discussed in Chapter 5, the distance between
the sample and the detector was 3.6 cm, which satisfies the Fraunhofer approximation.
Now that the connection between the fields in the two planes has been established, a scheme
for a CDI algorithm can be developed (see Fig. 7.2). In this scheme, the algorithm starts with aguess for the phase of the Fourier transform of the object and then interates between Fourier andobject spaces. At each iteration, a set of constraints is applied. In Fourier space, the constraint isthat the modulus of the Fourier transform of the object G(~q) must be equal to a square root of themeasured diffracted intensity. In real space, various constraints are possible. They include [107]169
the isolation constraint, the non-negativity constraint, and the overlap constraint for ptychography.
Diffraction from magnetic samples leads to different constraints, which are discussed below.
The phase retrieval procedure in Fig. 7.2 guarantees an almost unique solution (up to atranslation, reflection, and phase conjugation) in two dimensions as long as the diffraction patternis oversampled in the Fourier domain [215]. It is a consequence of the Nyquist-Shannon theorem,
which states that a continuous signal can be recovered from a discrete dataset if the data aresampled at a rate above a certain frequency, known as the Nyquist frequency [251]. The data is

Real-Space Constraints
𝑔𝑛+1 𝑟 = 𝐶 𝑔𝑛 𝑟
Start with a guess forthe phase of 𝐺 𝑞

Updated Object
𝑔𝑛 𝑟 = ℱ−1

At the detector
𝐺𝑛+1 𝑞 = ℱ 𝑔𝑛+1 𝑟

𝐺𝑛 𝑞

Converged?

𝑛 →𝑛+1Modulus Constraint
𝐺𝑛+1 𝑞𝐺𝑛+1 𝑞 =𝐼𝑚𝑒𝑎𝑠 𝑞𝐺𝑛+1 𝑞Output:𝑔𝑛+1 𝑟Figure 7.2: CDI algorithm. n is the current iteration number, g(~r) is the complex object, G(~q) is its
Fourier transform, Imeas is the measured diffraction intensity, and Ĉ represents a set of real-spaceconstraints.

said to be oversampled if the oversampling ratio ρ ≥ρ=√2 [215]. The ratio is given by

λz
,
pD

(7.11)

where λ is the wavelength of the probe, z is the distance between the object and the detector, Dis the size of the illuminated area of the object, and p is the detector pixel size.
The convergence criterion for the algorithm in Fig. 7.2 is the difference between the modulus√of the Fourier transform of the reconstructed object and the measured diffraction amplitude Imeas ,170
for which the metric is the sum squared error χ2n =P q~Imeas (~q) − |Gn (~q)|2

2. The iterations are

stopped when a specified error tolerance is reached.
Apart from the update condition in Fig. 7.2, there are multiple other ways to update theobject at each iteration, which give rise to faster converging CDI algorithms such as the hybrid
input-output (HIO) [210] and relaxed averaged alternating reflections (RAAR) [252]. These algorithms rely on a single exposure and are thus well suited for single-shot dynamic imaging ofmagnetic textures that are non-repeatable, such as magnetic domains, which can nucleate differently each time after being excited by a laser pulse. For static and stroboscopic imaging, wherea magnetic texture can be repeated reliably at each time step, a ptychographic phase retrieval,
which is a scanning technique that relies on multiple exposures, is preferable due to its superior
performance and reliability [242, 253, 254]. In addition, ptychography can reconstruct not only
the object, but also the profile of the probe [242]. Repeatable dynamic magnetic textures include
magnon waveguides and spintronic logic elements. If magnetic domains are weakly excited, they
can also relax to the original configuration and could thus be repeatable. In the following, I briefly
outline an approach to ptychography and derive constraints for magnetic samples that are applicable for most CDI algorithms. Using these constraints, I simulate a ptychographic reconstructionof nanoscale magnetic domains.

7.2

Ptychography CDI of magnetic textures with linearly polarized light
Because ptychography is a scanning technique, it can be used for imaging extended objects,

which relaxes the requirements on the sample fabrication because microscopic apertures no longer
need to be made. An extend ptychographical algorithm (ePIE) [242] does not require an accurate
model of the illumination, and would, therefore, be most useful in dealing with actual data, forwhich the exact illumination is, in general, unknown. In ptychography, a complex object O(~r) is~ s ), where the illumination position R~ s is scanned acrossilluminated by a complex probe P (~r − R
the object (the subscript s enumerates the illumination position), and the main constraint in real
~ s . An exit surface
space (Ĉ in Fig. 7.2) is an overlap of the illuminated areas at various positions R

171
wave g(~r) is simply a product of P and O~ s ).
g(~r) = O(~r)P (~r − R(7.12)One iteration of ePIE involves a loop over all illumination positions, which are addressed in arandom order to avoid artifacts associated with the periodicity in the reconstructed object andprobe. At each position, the object and the probe are updated separately. The object is updated
by dividing out the probe from the ESW and vice versa. The update condition for the object is~ s (n))Pn∗ (~r − R

On+1 (~r) = On (~r) + α~ s (n))Pn (~r − R2
gn0 (~r) − gn (~r) ,(7.13)

max

where n is the current iteration number, α is an algorithm feedback parameter (typically, α = 1),
gn (~r) is the current guess for the ESW, and gn0 (~r) is an updated guess, calculated according to
gn0 (~r) = F −1q

F{gn (~r)}Is(n) (~q),|F{gn (~r)}|(7.14)~ s . An update for thewhere Is(n) is the diffracted intensity measured at the illumination position R
probe is written similarly to eq. (7.13)
Pn+1 (~r) = Pn (~r) + β~ s (n))On∗ (~r + R~ s (n))On (~r + R

2
gn0 (~r) − gn (~r) .

(7.15)

max

Again, the feedback parameter β is typically set to 1. The overlap constraint is implicitly containedin eqs. (7.13), (7.14), and (7.15). With these update conditions, the flowchart in Fig. 7.2 is modified,and, for the ePIE algorithm, it is shown in Fig 7.3. Next, I discuss real-space constraints that are
specific for magnetic imaging and are applicable for ePIE as well as for other CDI algorithms.

7.2.1

Algorithm constraints for magnetic samples

In order to derive the constraints, the interaction of linearly polarized light with a magnetic
sample must be considered. A detailed analysis of such interaction was conducted in Chapter 5,
and it was found that, at normal incidence, the polarization of the transmitted magnetically scattered light is orthogonal to the polarization of the incident probe beam, while the unscattered light

172

ESW

Current guesses
𝑃𝑛 𝑟 , 𝑂𝑛 𝑟

Update current guesses,
𝑛 →𝑛+1

𝑔𝑛 𝑟 = 𝑃𝑛 𝑟 − 𝑅𝑠 𝑛𝑔𝑛′ 𝑟

= ℱ −1 𝐺𝑠 𝑛

⋅ 𝑂𝑛 𝑟𝐺𝑛 𝑞 = ℱ 𝑔𝑛 𝑟𝑞Modulus constraint
𝐺𝑛 𝑞
𝐺𝑠 𝑛 𝑞 = 𝐼𝑠 𝑛 𝑞
𝐺𝑛 𝑞

Figure 7.3: Flow of the ePIE algorithm. At n = 1, ePIE uses supplied guesses for the object O(~r)~ s(n) must be iterated over before the algorithm
and the probe P (~r). All illumination positions R
proceeds to the n + 1 iteration. For magnetic imaging with linear polarization, an additional
real-space constraint must be added.

173
maintains the original polarization but contains no magnetic information (see eq. (5.12)). The
magnetic contrast thus arises due to the Faraday rotation (in combination with XMCD), and, inthe first approximation, the amount of rotation is proportional locally to an out-of-plane magnetization component mz . A larger rotation results in a larger orthogonal, i.e., magnetically scattered,polarization component, and the amplitude of the ESW thus maps the magnitude of mz (x, y). The
direction of the rotation depends on the local sign of mz . For the two opposite rotations, the magnetically scattered polarization components have opposite signs, which amounts to a π phase shift.
This phase shift, therefore, maps the sign of mz (x, y). These two statements are mathematically
expressed as
ESW (x, y) = A0 (x, y) |mz (x, y)| em (x,y)i π2 1+ |mz (x,y)|z,(7.16)

where A0 (x, y) is the incident probe. Note that the expression above includes only the magneticallyscattered polarization component, which is orthogonal to the incident polarization. In eq. (7.16),
the phase only takes the values of 0 and π, which, makes the object purely real. Therefore, aconstraint that is specific for magnetic samples probed with linearly polarized light is simply thatthe object is realO(x, y) = O∗ (x, y).

(7.17)

The fact that the object can be negative seemingly contradicts the non-negativity constraint commonly used in CDI. However, it does not mean that the absorption becomes negative. Rather,
it indicates that, due to magnetism, locally, the absorption can be less than its mean value. In
addition, according to eq. (7.16), the magnetically scattered ESW has no undiffracted (DC) component, which is a consequence of mz (x, y) changing sign across the sample. This means that theDC peak has no magnetic information in it and can be disregarded, which is convenient because inan actual experiment, the DC peak has to be blocked due to its much higher intensity comparedto the RMS intensity (see the discussion in Chapter 5), in order to avoid saturating the detector.
A simulation of a ptychographic reconstruction of magnetic domains was performed usingePIE with the constraint given by eq. (7.17). The results shown in Fig. 7.4 demonstrate a174
successful reconstruction of the original domain pattern. The parameters of the simulation wherechosen to be close to the experimental parameters: the energy of the EUV probe was 53 eV, thedomain size was ∼80 nm. With the probe spot size of approximately 1.5 µm, a field of view of∼2.5 µm was achieved by scanning the probe spot across the sample. The resolution of the spatial
grid in the simulations was ∼20 nm/pixel. Note that the sign of the reconstructed mz is reversed
compared to the original domain texture. This is a manifestation of the ambiguity of the sign of thephase in the solution of the phase retrieval problem. The simulation demonstrates the feasibilityof magnetic imaging with a linearly polarized tabletop HHG source.

7.3

Element-specific spectro-microscopy
Using a technique known as ptychographic information multiplexing (PIM) [255, 256, 257],

it is possible to reconstruct multiple objects corresponding to different modes of illumination that
add incoherently at the detector. The modes can include illumination with different polarizations,spatial profiles, or wavelengths [258, 259, 260]. The latter is particularly important as it canbe used for visualization of magnetic textures at the M -shell absorption edges of multiple 3d
ferromagnets simultaneously, which paves a way for lensless magnetic microscopy with an elemental
contrast. A repeatable magnetic texture, e.g. a patterned nanoscale magnetic device, can be imaged
stroboscopically in a pump-probe experiment using PIM, and thus spatially-resolved dynamics
of multiple elements can be measured with femtosecond time resolution. Finally, the dynamics
of the deterministic all-optical magnetization reversal [62] can be studied stroboscopically using
ptychographic imaging.

7.4

Further work
In addition to magnetic microscopy, there are multiple other promising areas of interest.

Ferromagnets represent a rather narrow class of strongly correlated materials. Another class is
antiferromagnets, which play an important role in high-TC superconductivity [261]. An antiferromagnetic spin alignment typically occurs in transition metal compounds such as FeMn and NiO175

Figure 7.4: Simulation of a ptychographic reconstruction of magnetic domains. (a) and (b) are theoriginal probe beam profile and domain texture, respectively; (c) and (d) are the corresponding
reconstructions; (e) is the evolution of the squared error with the number of algorithm iterations.

176
and can be studied magneto-optically using X-ray magnetic linear dichroism (XMLD) [73, 128],
which has been used in combination with photoelectron microscopy to image antiferromagnetic domains [262, 263, 264]. Due to its small magnitude, XMLD requires an improvement in the photon
flux and stability of HHG sources, but, in principle, can be readily implemented in a pump-probe
experiment to study laser-induced spin dynamics in antiferromagnets.
Studying magnetization dynamics in these and other complex magnetic systems, such as halfmetals (Heusler alloys) [265, 266] and materials with phase transitions at cryogenic temperatures,
e.g. magnetite [267, 268, 269], with EUV resonant magneto-optical spectroscopy can improve theunderstanding of magnetism in the context of strongly correlated quantum phenomena and bring
us closer to the ultimate goal of its coherent control.

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Appendix A

Depth Sensitivity of T-MOKE

To date, depth resolution has been achieved by simultaneously measuring Kerr ellipticity andKerr rotation, which is known as the complex Kerr effect and has been used to capture the depth
profiles of the magnetization in thin films, multilayers, and nano-strucures [270, 271, 272, 273, 274]
as well as to measure transient changes to the depth profile in the course of ultrafast demagnetizationand thus uncover the role of ultrafast spin transport [275, 202, 276]. The depth sensitivity of TMOKE, which is a pure intensity measurement, was also used in simulations in order to determine
the role of laser-induced spin currents in Co [277] as well as to understand the depth-resolved
critical behavior in Ni [144]. Below, I present a calculation of the depth sensitivity of T-MOKE.
Assuming that an infinitesimally thin layer of thickness dz at a depth z contributes an amountξ(z)dz to the total asymmetry A, the latter can be expressed as
Z dA=ξ(z)dz,(A.1)

0where d is the film thickness and ξ(z) is called the depth sensitivity function.
In deriving the depth sensitivity function I follow the approach of Ref. [278]. The approach isbased on subdividing a magnetic film into thin layers of thickness dz and setting the magnetizationof all but one layer at depth z to zero, as shown in Fig. A.1. At each position z of the layer,
the T-MOKE asymmetry dA is calculated using the multilayer formalism described in Chapter
4, and the depth sensitivity is then determined as ξ(z) = ∂A∂z z . Such an approach is based on
the assumption that the responses of individual layers are independent of one another. Generally

202
𝑧

𝑚=0
𝑚=1

𝑑𝑧

𝑚=0

Figure A.1: An approach for a depth-sensitive T-MOKE calculation. The depth sensitivity function
is found by calculating the T-MOKE asymmetry at each position of the magnetized sublayer.

speaking, this is not necessarily true, and small corrections need to be made to ξ(z) calculated
in the manner described above. I implement the corrections by repeating the calculation for the
structure in Fig. A.1, with the only difference that a non-magnetized thin layer is now translated
along the depth of a magnetized material. The resulting calculated quantity at each layer position
is A0 −ξ(z)dz, where A0 is the net T-MOKE asymmetry of the film. The corrected depth sensitivity
is found as an average of the depth sensitivities calculated using the two methods described above.
An example of ξ(z) calculated at a 52◦ incidence for a 10 nm Co film with a multilayer compositionof SiO2 (150 nm)/Co (10 nm)/GeO2 (3 nm) and grown on a Si substrate is shown in Fig. A.2.
Interestingly, the depth sensitivity switches sign a ∼5 nm due to a phase accumulation in theprobe wave propagating through the film, which could mean that the T-MOKE asymmetry canbe optimized by adjusting the thicknesses of the sample’s layers, and, particularly, making themagnetic film somewhat thinner.
It is sometimes assumed that T-MOKE is only sensitive to interfaces, where there is a discontinuity in the optical properties of two media, which gives rise to a reflected wave, and thus any
changes of the magnetization in the bulk of the film cannot be detected. This assumption is incorrect because any variations of the magnetization within the magnetic film are mapped onto spatial
inhomogeneities of its optical properties, which reflect light and, therefore, affect the T-MOKE
signal. Mathematically, this is expressed as
Z d
A [m(z)] =m(z)ξ(z)dz.
0

(A.2)

203

Figure A.2: T-MOKE EUV depth sensitivity function of a 10 nm Co film with a multilayer composition of SiO2 (150 nm)/Co (10 nm)/GeO2 (3 nm) at a 52◦ incidence. The magneto-optical Voigt
parameter Q was taken from Ref. [168].

204
Hence, T-MOKE should, in principle, be able to capture a non-uniform depth profile of the magnetization m(z). The idea is similar to that of X-ray reflectometry, where element composition profiles
are deduced from angle-dependent reflectivity curves [279]. In order to perform such a measurement, a series of configurations has to be found, for example, by taking T-MOKE measurements
at different angles of incidence, for which the corresponding functions ξ(z) are sufficiently different
from one another and sample different depths of the film. In that case, a system of equations canbe obtained based on eq. (A.2), and problem (A.2) can thus be inverted. For EUV T-MOKE,
however, sweeping the energy, polarization, magnetization direction, and angle of incidence parameter spaces did not result in a well-conditioned system of equations that could yield a uniquesolution for m(z). The knowledge of the depth sensitivity function is, nonetheless, useful because
it allows an optimization of the sample design in order to maximize the T-MOKE asymmetry, andcan also be used in simulations, e.g. for understanding the response of a sample to an ultrafast
laser excitation.

Appendix B

Rigorous Coupled-Wave Analysis

In this appendix, I provide a brief overview of the Rigorous Coupled-Wave Analysis (RCWA)
method. RCWA was proposed by M.G. Moharam and T.K. Gaylord in the early 1980’s [280] as
an efficient semi-analytical method to treat the problem of diffraction of light by a general planar
grating. Subsequent developments of the method, such as the Enhanced Transmittance Matrix
approach (ETM), improved the stability and convergence of RCWA [281, 282]. A comprehensive
and thorough review of RCWA as well as additional references can be found in Ref. [283]. I follow[283] in deriving the RCWA below.
RCWA solves Maxwell’s equations without any approximations (hence the name “rigorous”)
in the Fourier space. The propagation in the longitudinal z-direction is treated analytically, while
the solution for the transverse components of the electric and magnetic fields in the (x, y) plane is
obtained numerically. The method treats any scattering object, such as, for example, a diffraction
grating, as a three-dimensional distribution of the dielectric constant r (x, y, z) and requires aperiodicity of r in the transverse directions, such as shown in Fig. B.1. RCWA finds the eigenmodesof the fields in a unit cell of r and stitches them with the fields outside of the scattering object in a
way that satisfies Maxwell’s boundary conditions. If r is non-uniform in z, it can be subdivided intolayers that are approximately uniform in z, and the stitching of the eigenmodes will have to be done
between the subdividing layers as well. This procedure is known as the staircase approximation.
With it, a structure of any complexity in z can be treated, but because each layer requires its own
eigenmode calculation, very complex structures quickly become computationally intensive. Because

206

𝑦𝑥 Region I

𝑘𝑖𝑛𝑐

𝜃

𝑧

𝜙

Unit cell𝜖𝑟 (𝑥, 𝑦, 𝑧)
Region II
Figure B.1: RCWA geometry. The dielectric constant of the scattering structure r (x, y, z) must be
periodic. A unit cell is shown in red. The wave vector of an incident wave is ~kinc . Region I above
the scattering structure contains the incident and reflected waves. Region II below the structure
contains the transmitted waves.

207
RCWA solves Maxwell’s equations separately for each Fourier component, or spatial harmonic, of
the field. Problems that can be represented well by a small number of spatial harmonics are best
suited for this method. As a consequence, this puts a constraint on the transverse contrast in r : itneeds to be relatively smooth without sharp edges. The sharper the features in r , the more spatial
harmonics are required to adequately describe the problem and the higher the computational cost.
I start with Maxwell’s curl equations, which in a source-free material are written as~~ = k0 µr H̃,
∇×E
(B.1)
~ = k  E,~
∇ × H̃
0 r~ is defined as H̃
~ = −ipµ / H,~ and k0 is the wavenumber
where the normalized magnetic field H̃0 0of the incident wave. Using the Levi-Civita symbol eijk and Einstein’s summation convention(summation over repeated indices), these equations can be re-written in a compact form
eijk∂
Ek = k0 µr H̃i ,
∂xj

(B.2)

∂
H̃k = k0 r Ei ,
eijk
∂xj

where the indices i, j, k run through x, y, z each. A Fourier series expansion in the (x, y) plane of
the fields in each z-layer labeled with an index j yields~ j (x, y, z) =E
~ (x, y, z) =
H̃
j

m=+∞
X n=+∞
X
m=−∞ n=−∞m=+∞
X n=+∞X
~j,m,n (z) exp −i k x x + k y y ,
S
m,n
m,n
(B.3)


x
y
~ j,m,n (z) exp −i km,nU
x + km,ny .

m=−∞ n=−∞In (B.3), the wavevector components of the (m, n)th spatial harmonic (Fourier component) are


(x)
(x)
~ 1 + nG
~ 2 · x̂,km,n= kinc − mG


(y)
(y)
~ 1 + nG
~ 2 · ŷ,km,n= kinc − mG
r





(z)

kj,m,n =(j)(x)

k02 r − km,n2(y)− km,n(B.4)
2.

Note that in these equations, depending on the sign of the expression under the square root, thez-component of the wavevector is either purely real or purely imaginary. The latter represents

208
evanescent waves. In order to complete the Fourier series expansion of Maxwell’s equations, thedielectric constant and the magnetic permeability need to be expanded. Their inverses are expanded~ m,n as G
~ m,n = mG
~ 1 + nG
~ 2 and ~r as ~r = x · x̂ + y · ŷ, (i)as well, for better convergence. Defining Gr(i)and µr and their inverses can be expressed asm=+∞
X n=+∞
X

(j)
r =
µ(j)
r =




(j)
r

m=−∞ n=−∞m=+∞
X n=+∞
X

−1

µ(j)
r



~ m,n · ~r ,aj,m,n exp iG

−1



~ m,n · ~r ,cj,m,n exp iG

m=−∞ n=−∞m=+∞
X n=+∞
X(B.5)


~ m,n · ~r ,bj,m,n exp iG

=

=



m=−∞ n=−∞m=+∞
X n=+∞X~ m,n · ~r .
dj,m,n exp iG

m=−∞ n=−∞The Fourier coefficients aj,m,n , bj,m,n , cj,m,n , and dj,m,n are given by1
φj,m,n =AZZ~ m,n · ~r d2~r,
fj (x, y) exp −iG

(B.6)

A

 −1
(j)
(j)
(j)
where φj,m,n stands for aj,m,n , bj,m,n , cj,m,n , or dj,m,n , and fj (x, y) stands for r , r
, µr , or(j) −1µr, respectively. A is the area of the unit cell in the (x, y) plane.
Substituting eqs. (B.3) and (B.6) into eq. (B.2), one obtains a system of Maxwell’s equations
in reciprocal space for the (m, n)th spatial harmonic (see Ref. [283] for details)(y)dSj,m,n (z)

(z)

(y)
−ikm,nSj,m,n (z) −(x)= k0 µxj ⊗ Uj (z),

dz
(x)

(z)

(x)
ikm,n
Sj,m,n (z) +dSj,m,n (z)
dz

(y)

(y)

= k0 µyj ⊗ Uj (z),

(x)

(z)

(x)
(y)
−ikm,nSj,m,n (z) + ikm,nSj,m,n (z) = k0 µzj ⊗ Uj (z),
(y)

(z)

(y)
−ikm,nUj,m,n (z) −dUj,m,n (z)
dz

(B.7)

(x)

= k0 xj ⊗ Sj (z),

(x)

(z)

(x)
ikm,n
Uj,m,n (z) +(y)dUj,m,n (z)
dz
(x)

(y)

= k0 yj ⊗ Sj (z),
(z)

(y)
(x)
−ikm,nUj,m,n (z) + ikm,nUj,m,n (z) = k0 zj ⊗ Sj (z),where the symbol ⊗ denotes the convolution operation. Since for an arbitrary system, in general,
one needs an infinite number of spatial harmonics (terms in the Fourier expansion), the system209(B.7) is also infinite. However, in an actual implementation of RCWA on a computer, the numberof harmonics is finite, and, as mentioned above, the fewer harmonics are needed for an accurate
description of the scattering object the better. With a finite number of spatial harmonics, the systemof equations can be re-written in a matrix form (with the introduction of a non-dimensionalized~
wavevector k̃ = ~k/k0 and longitudinal coordinate z̃ = k0 z)
dsy= [[µr ]]ux ,dz̃
dsx= [[µr ]]uy ,iK̃x sz +dz̃

−iK̃y sz −−iK̃x sy + iK̃y sx = [[µr ]]uz ,
(B.8)

duy
−iK̃y uz −= [[r ]]sx ,dz̃dux
iK̃x uz += [[r ]]sy ,dz̃
−iK̃x uy + iK̃y ux = [[r ]]sz ,where the matrix K̃ and vectors u and s are defined as follows


k̃
(1,
1)
0
.
.
.
0
 q





k̃q (1, 2) . . .
0
 0


,
K̃q = 

..
..
..
 ...

.
.
.




0
...
0 k̃q (M, N ) Uq (1, 1)  Uq (1, 2) ,uq = 

.
..






Uq (M, N )



(B.9)



 Sq (1, 1)  Sq (1, 2) ,
sq = 

.
..






Sq (M, N )(B.10)

where index q represents the subscripts x, y, or z. Note that [[r ]] and [[µr ]] are square matrix
operators that perform the convolution operation mentioned above. They are not dielectric constantand magnetic permeability matrices defined on a real-space grid. [[r ]] and [[µr ]] can be found bycomputing Fourier coefficients of r (x, y) and µr (x, y) and re-arranging them along the matrix
diagonals. The number of coefficients is equal to the total number of spatial harmonics used to210
describe a given problem. More details on the re-arrangement of the Fourier coefficients can befound in Ref. [283]. For a one-dimensional problem, the convolution matrices [[r ]] and [[µr ]] become
matrices with constant left-to-right diagonals known as Toeplitz matrices [284]. This is relevant forthe case of a one-dimensional sawtooth-shaped diffraction grating which I consider in this thesis.
In the next step, the longitudinal field components are eliminated from the system of equations (B.8). This can be done by using the third and sixth equation of the system, which yieldsuz = [[µr ]]−1 −iK̃x sy + iK̃y sx ,−1
sz = [[r ]]−iK̃x uy + iK̃y ux .

(B.11)

Substituting these expressions into (B.8) leaves only four differential equations for the transverse
components of the fields
dux= K̃x [[µr ]]−1 K̃y sx + [[r ]] − K̃x [[µr ]]−1 K̃x sy ,
dz̃


duy
= K̃y [[µr ]]−1 K̃y − [[r ]] sx − K̃y [[µr ]]−1 K̃x sy ,
dz̃


dsx
= K̃x [[r ]]−1 K̃y ux + [[µr ]] − K̃x [[r ]]−1 K̃x uy ,
dz̃


dsy
= K̃y [[r ]]−1 K̃y − [[r ]] ux − K̃y [[r ]]−1 K̃x uy .
dz̃These equations can be re-written in a block-matrix form  d ux sx 
  = Q ,
dz̃
uy
sy
 
 
d sx ux 
  = P ,
dz̃
sy
uy
where the matrices Q and P are defined as


−1
−1
[[r ]] − K̃x [[µr ]] K̃x  K̃x [[µr ]] K̃y
Q=,K̃y [[µr ]]−1 K̃y − [[r ]]−K̃y [[µr ]]−1 K̃x


−1
−1
[[µr ]] − K̃x [[r ]] K̃x  K̃x [[r ]] K̃y
P=.
K̃y [[r ]]−1 K̃y − [[µr ]]−K̃y [[r ]]−1 K̃x(B.12)(B.13)(B.14)

211
From eqs. (B.13), matrix wave equations can be derived. They have an identical form forboth the electric and magnetic fields. For this reason, I only write down the wave equation for thetransverse Fourier components of the electric field
 
 
s
d2 sx 
2  x−Ω
 = 0,d2 z̃
sysy(B.15)

where Ω2 = PQ. The general solution of eq. (B.15) is expressed in terms of forward (+) andbackward (−) propagating waves
 
sx 
−Ωz̃ (+)
s (0) + eΩz̃ s(−) (0).
 =e
sy

(B.16)In this equation, the terms s(±) (0) are the initial transverse Fourier amplitudes of the forward
and backward propagating waves. An exponent of a square diagonalizable matrix exp(A) =W exp(λ)W−1 [285], where λ is a diagonal matrix of the eigenvalues of A, and W is a matrix of theeigenvectors of A in which each column is an eigenvector corresponding to its respective eigenvalueon the diagonal of λ. The solution (B.16) can thus be expressed in terms of the eigenvectors W
and eigenvalues λ2 matrices of Ω2 as
 
sx 
−λz̃
W−1 s(+) (0) + Weλz̃ W−1 s(−) (0) = We−λz̃ c(+) + Weλz̃ c(−) ,  = Wesy

(B.17)

where I defined the coefficients c(±) = W−1 s(±) (0).
A similar procedure can be performed for the transverse Fourier components (spatial harmonics) of the magnetic field. It can be shown by substituting expression (B.17) into the firstequation in (B.13) that the relation between the eigenvector matrix V for the magnetic field andthe eigenvector matrix W for the electric field is V = QWλ−1 [283], and by combining the electricand magnetic fields into a single vector, an entire solution can be expressed as  sx  



 
 
(+)
0  c  sy   W W exp(−λz̃)
=
ψ=



.
 
(−)
ux 
−V
V
0
exp(λz̃)
c
 
 
uy

(B.18)

212
In order to calculate the fields reflected from and transmitted through a periodic structure,
one needs to evaluate the mode coefficients c(±) outside of the structure. This can be done by applying Maxwell’s boundary conditions for the s and u transverse field components. At an interface,~ and H~ fields are continuous in the absence of surface charges
the tangential components of the E
and currents. Therefore, at the boundary between the layers i and i + 1, it follows that ψ i = ψ i+1 .
This condition can be written as




 


˜i
(+)
(+)
−λ
d
i
0 ci   Wi+1 Wi+1  10
 Wi Wi  e
 ci+1 


,

=


˜
(−)
(−)
−Vi Vi0
1 ci−Vi+1 Vi+1
0 e−λi+1 di+1
ci+1

(B.19)

where d˜i is the thickness of the ith layer normalized by k0 . Note that in deriving (B.19) from (B.18)
Pi−1 ˜I have introduced a substitution z̃i → z̃i − j=1
dj , which leaves only the negative exponentialP
th and (i − 1)st˜
exp(−λi d˜i ) when z̃i = i−1
j=1 dj , which is the coordinate of the boundary of the iP
th and (i + 1)st˜
layers, or when z̃i = d˜i + i−1
j=1 dj , which is the coordinate of the boundary of the ilayers. The expression above can be re-written in a more compact form by introducing




(+)
 Wi Wi ci 
Fi = 
 , Xi = exp(−λi d˜i ), Ci = .
(−)−Vi Vici
With these substitutions, the boundary condition (B.19) simplifies toXi 0
Xi+1 0Fi 
 Ci = Fi+1  Ci+1 .
0 1
01(B.20)(B.21)

Special care must be taken when considering boundary conditions at the first and last interfaces of the scattering structure as these are responsible for coupling the incident light into thestructure correctly as well as giving the correct reflections and transmissions of the structure as awhole. I label the region that contains the incident and reflected waves as Region I and the region
that contains the transmitted waves as Region II. In Fig. B.1, Region I is above the blue structure,
and Region II is below it. The boundary condition (B.21) at the first and last interfaces, which are
the interfaces between Region I and the first layer of the structure and Region II and the last layer

213of the structure, respectively, can be written asX0 f irst(last)ψ I (II) = Ff irst(last)  Cf irst(last) .
0
1(B.22)In Region I, ψ I is a sum of the forward propagating incident wave and a backward propagatingreflected wave. In Region II, ψ II is a forward propagating transmitted wave. If the incident electric
field


~ inc = P~ exp −i~kinc · ~r ,E

(B.23)

where P~ is the polarization vector, it can be shown by substitution into eq. (B.2) that the corresponding transverse components of the incident magnetic field are

 


(z)
(y)
~
i exp −ikinc · ~r kinc py − kinc pz H̃x .
  =µI k 0(x)(z)
kinc pz − kinc pxH̃y(B.24)

inc

The same relationship holds for the Fourier components of the fields. The transverse Fouriercomponents of the incident electric field are
 


s
p
δ
 x
 x m,n=0   =,sypy δ m,n=0(B.25)

incwhere I introduced a vector δ m,n=0 that has the same number of elements as the number of spatial
harmonics in a given problem, but all elements of the vector are zero except for the element at
m = n = 0. This is a representation of an incident plane wave as a 0th order spatial harmonic.
From eqs. (B.25) and (B.24), the Fourier components of the incident magnetic field are

 


(z)
(y)
i  k̃inc py − k̃inc pz δ m,n=0 
ux 

.
  =

µI
(x)
(z)
k̃inc pz − k̃inc px δ m,n=0
uy(B.26)

inc

Denoting the reflected Fourier components of the electric field in Region I as rx and ry , thetotal electric field in that region as a sum of the incident and reflected waves is
 

  
sx 
px δ m,n=0  rx   =
 +  .
sy
py δ m,n=0
ryI(B.27)

214
The associated reflected magnetic field in Region I can be found with the help of the first equationin (B.13). The reader is referred to [283] for more details. Just like the electric field, the magnetic
field in Region I is a sum of the incident and reflected waves
 




K̃x K̃y(z)(y)
i  k̃inc py − k̃inc pz δ m,n=0 i  − K̃I,z
ux 
  =

+


µI
µI K̃2I,z +K̃2x
(x)
(z)
uy
k̃inc pz − k̃inc px δ m,n=0
K̃
I

−

I,z

K̃2I,z +K̃2y
K̃I,zK̃x K̃yK̃I,z  rx  .
ry

(B.28)

A similar derivation can be performed for the fields transmitted through the scattering structure.
With the transmitted transverse Fourier components of the electric field tx and ty , the magnetic
field in Region II is K̃x K̃yi  K̃II,z
ux 
  =

K̃2 +K̃2xµII
uy− II,z
K̃
II

 
t
K̃II,z   x 
 .
K̃ K̃
− K̃x yty

K̃2II,z +K̃2yI,z(B.29)

II,zHaving determined the fields, and hence vector ψ in Regions I and II, it is now possible to write
explicitly the boundary conditions (eq. (B.22)) at the interfaces between the scattering structure
and the surrounding media. In order to write them in a more compact form, I first introduce some
auxiliary matrices and vectors −iµI 1


i 
 0
A=

µI  K̃x K̃y − K̃
I,z K̃2 +K̃2

0




−iµI 1 .
K̃2I,z +K̃2y 
− K̃

I,z


K̃ K̃

x

I,z

y

x

K̃I,z

K̃I,z





 −iµII 1


0
i 

B=

µII  K̃x K̃y
 K̃
II,z

 K̃2 +K̃2
II,z
− K̃ xI,z(B.30)

px δ m,n=0







py δ m,n=0,ψ inc =   i k̃ (z) py − k̃ (y) pz δ m,n=0  µI inc

inc
 


(x)
(z)
i
k̃
p
−
k̃
p
δ
z
x
m,n=0
inc
inc
µI

0




−iµII 1 .
K̃2II,z +K̃2y K̃II,z K̃ K̃− K̃x y(B.31)

II,z

 
rx 
 
 
 ry 

R=
 ,
rx 
 
 
ry

 
tx 
 
 
 ty 

T=
 .
tx 
 
 
ty

(B.32)

215
With these definitions, the first boundary condition between Region I and the scattering structure
is



1
0


ψ inc + AR = F1 
 C1 ,
0 X1(B.33)and the condition between the structure and Region II for a structure with N interfaces is
XN 0
BT = FN (B.34)
 CN .
0 1
These boundary conditions (eqs. (B.33), (B.21), and (B.34)) can be solved for an entire structure
in order to find the total reflection and transmission as well as diffraction efficiencies for various
spatial harmonic orders and, if required, electric and magnetic fields inside the structure. There
are several ways of solving this problem.
The solution method that I describe below is called the Enhanced Transmittance Matrix(ETM) approach. It was first proposed in Ref. [282] in 1995 and is a stable implementation of
RCWA. The stability is ensured by analytically avoiding growing exponential terms of the form
˜
X−1
i = exp(λi di ) which could lead to ill-conditioned matrices when eqs. (B.21) are solved.
The unknowns that need to be solved for are the reflection and transmission vectors R and
T in eqs. (B.33) and (B.34). ETM starts with the last interface. The mode coefficients CN are
found from (B.34)
 



−1
−1
XN 0  aN 
XN 0 −1
CN =    T.
 FN BT = 
0
1
0
1
bN
In the equation above, the following definition has been introduced  aN −1  = FN B.
bN(B.35)(B.36)In order to eliminate the growing exponential term X−1
N in eq. (B.35), an auxiliary transmittancevector TN is introduced, and the unknown vector T is expressed in terms of TN as
T = a−1
N XN TN .

(B.37)

216
With the auxiliary vector TN eq. (B.35) becomes


1


CN = 
 TN .
−1bN aN XN

(B.38)

By going backwards through the layers using eq. (B.21) and introducing auxiliary variables ai , bi ,and Ti for each layer, one can arrive at expressions similar to eqs. (B.36), (B.37), and (B.38)


1


−1
Ci = 
 Ti , Ti+1 = ai Xi Ti ,−1bi ai Xi
(B.39)
 



0 
1
 ai 
1
−1
  = Fi Fi+1 .
bi
0 Xi+1bi+1 a−1X
i+1 i+1
With C1 determined by (B.39), the boundary condition (B.33) at the first interface (between the
structure and Region I) is modified to



1

1 0  
ψ inc + AR = F1 
 T1 .
−1
0 X1
b1 a1 X1
From here, R and T1 can be found
 

−1
R
=
ψ inc ,
 
−AB̃
T1


where

(B.40)

1B̃ = F1 .
−1
X1 b1 a1 X1(B.41)

If only reflected spatial harmonics need to be solved for, the method can stop here. To find thetotal transmission through the scattering structure, work forward through the layers using−1−1−1
T = a−1
N XN aN −1 XN −1 . . . a2 X2 a1 X1 T1 .

(B.42)

Now that the electric and magnetic fields are known everywhere in space, the reflection andtransmission diffraction efficiencies, defined as fractions of the incident power, or the magnitude of the Poynting vector, scattered into a particular reflected or transmitted spatial harmonic
(z)

(z)

Dm,n = Pm,n /Pinc , can be found according to [283]
#
"


 2
K̃I,z
−1
2
2
,
DR = |Rx | + |Ry | + K̃I,z K̃x Rx + K̃y Ry
Re
(z)
k̃inc
"
#


 2
µI K̃II,z−122
DT = |Tx | + |Ty | + K̃II,z K̃x Tx + K̃y Ty
Re.
µII k̃ (z)inc(B.43)

217
Below, I provide a step-by-step procedure to implement RCWA using the ETM approach:(1) Define the parameters of the incident wave: direction, wavelength and polarization.
(2) Define the parameters of the scattering structure: r and µr for each layer of the structure
as well as in the transmission and reflection regions, period in the x and y directions, layer
thicknesses of the structure, the number of spatial harmonics in the x and y directions.
(3) Given the parameters defined in the previous step, construct the scattering structure on ahigh-resolution spatial grid.
(4) For each layer of the structure, compute the convolution matrices [[r ]] and [[µr ]].
(5) Compute matrices K̃x , K̃x and K̃y , as defined by (B.9), and, in the reflection and transmission regions, matrix K̃z .
(6) Compute matrices A and B as defined by eqs. (B.30) and (B.31), respectively.
(7) Work backwards through the layers starting from the last one and, for each layer, solve
an eigenvalue problem for matrix Ω2 = PQ, where matrices P and Q are defined by eq.
(B.14). With the computed eigenvector and eigenvalue matrices, obtain matrices F and X
defined by eq. (B.20), and with those, solve for the matrices ai according to the second
equation in (B.39). In order to calculate aN for the last layer (first iteration), use eq.
(B.36).
(8) When the first boundary is reached, solve for the reflected spatial harmonics R and theauxiliary transmission T1 using eq. (B.41).
(9) Propagate forward through the layers to determine the transmitted spatial harmonics T
from eq. (B.42).
(10) Compute the reflected and transmitted diffraction efficiencies DR and DT according to(B.43). The total reflection and transmission of the structure can be found by summing

218
the diffraction efficiencies over all spatial harmonics. This can be used to ensure the conservation of energy, which is a very powerful criterion that determines whether the outputof the algorithm is meaningful. The sum of the total reflection and transmission is equal
to 1 when there is no loss or gain in the structure, and is less or greater than 1 when there
is loss or gain, respectively.
In addition to the reflected and transmitted waves, fields within the scattering structure can also
be calculated. This is useful for gaining insight into the coupling of incident light to a device, but it
requires keeping track of additional parameters as the algorithm proceeds, which, in turn, requires
more memory. For the case considered in this thesis of an EUV beam reflected from a sawtooth
diffraction grating, such a calculation was not necessary, and for that reason I do not discuss it
here. More details on the calculation of the fields within the structure can be found in Ref. [283].
Because the ETM approach requires that multiple matrix variables be stored in memory foreach layer, it can very quickly become computationally intensive. Since the size of the matrices isdetermined by the number of spatial harmonics needed to describe the problem, and the number ofmatrices depends on the number of layers, the memory requirements of the algorithm grow rapidly
as the complexity of the scattering structure increases, i.e., if more layers and/or spatial harmonics
are needed for its adequate description. Nonetheless, to date, ETM is one of the fastest and moststable implementations of RCWA.

Appendix C

Material Parameters for the 3-Temperature ModelIn this appendix, I provide the material parameters used in the simulation of the electron,
lattice, and spin temperatures in the CoFe/Ni multilayer film in Chapter 6. The model requires
the following inputs for each material that are, in general, temperature dependent(1) Electron specific Ce (Te )(2) Lattice specific heat Cl (Tl )(3) Magnetic specific heat Cs (Ts )(4) Electron thermal conductivity κe (Te , Tl )(5) Lattice thermal conductivity κl (Te , Tl )(6) Electron-lattice coupling Gel (Te )(7) Electron-spin coupling Ges (Te )(8) Lattice-spin coupling Gls(9) Material refractive indices to calculate the source term 1
The layer structure of the sample of interest is Si3 N4 (50)/ Ta(3)/ Cu(5)/ (Co0.9 Fe0.1 (0.2)/ Ni(0.6))x50/
CoFe(0.2)/ Cu(3)/ Ta2 O5 (3), where the layer thicknesses in parentheses are in nm. Because thefraction of Fe in the CoFe layer is only 0.1, I assume that the material properties of that layer are
1

Available at https://refractiveindex.info/

220
adequately described by those of pure Co. The sample thus includes six different materials, and foreach one of those materials, nine properties need to be specified, according to the list above. This
amounts to a total of 36 material properties, many of them being temperature dependent.

C.1

Electron specific heatSpecific heat is defined as a derivative of the total internal energy E with respect to temper-

ature C(T ) = dE
dT . The total internal energy of a statistical system isZ∞
E(T ) =

g()f (, T )d,

(C.1)

0

where g() is the density of states of the system, and f (, T ) is its distribution function. Electrons
are described by the Fermi-Dirac distribution for which1f (, T ) =e−µ(T )
kT

,

(C.2)

+1

where µ(T ) is a chemical potential. With eqs. (C.1) and (C.2), the specific heat of the electron
thermal bath can be foundZ∞
Ce (T ) =

−µ

e kT

g()( − µ(T ))  −µ2kT + 1
e
0



1 ∂µ
−µ
+
2
kT
kT ∂T


d.

(C.3)

The temperature dependence of the chemical potential on T , which is required for the calculationof the specific heat according to the equation above, is found from the conservation of the total
electron number densityZ∞N=g()f (, T )d.

(C.4)

0

The formalism described above is applicable in metals where electrons are not strongly bound totheir respective nuclei and are free to move in the periodic potential of the lattice. The electron
heat capacities for Ni, Co, Cu, and Ta (all with the fcc crystal structure) are shown in Fig. C.1.
Note that eq. (C.3) requires the knowledge of the electron density of states g(). For the metals
shown in the figure, g() was calculated with the ELK package [30]. All the other materials in themultilayer structure are insulators, and, because electrons in those materials are strongly bound,

221

! J "

3

#10 6

Ce

m3 K

2

fcc Nifcc Cofcc Cufcc Ta

1

0
0

2000

4000

6000

Temperature (K)Figure C.1: Electron specific heat of Ni, Co, Cu, and Ta.

222
their heat capacity is negligible. As can be seen from Fig. C.1, the dependence of the specific on
the temperature is non-linear, and thus the Sommerfeld model in which Ce = γe T [29] does notprovide an accurate estimate of the electron specific heat, particularly at high temperatures.

C.2

Lattice specific heat
The lattice (or phonon) specific heat is given by the Debye model [29].
Cl = 9N kB

TTD3 TZD /T0

x4 e xdx(ex − 1)2

(C.5)

where N is the number of atoms per unit volume, TD is the Debye temperature, and kB is theBoltzmann constant. Because the functional form of Cl (T ) for different materials is the same,according to eq. (C.5), I only show the lattice specific heat for Ni in Fig. C.2 as an example. Notice

#10 6

! J "

3.8

Cl

m3 K

3.7

3.6

3.5
0

2000

4000

6000

Temperature (K)Figure C.2: Lattice specific heat of Ni.

that at high temperature the electron specific heat in Fig. C.1 is of the same order of magnitude
as that of the lattice. The Debye temperatures can be found online. 22

http://www.knowledgedoor.com/2/elements_handbook/debye_temperature.html

223

C.3

Magnetic specific heat
The only magnetic materials in the sample being considered here are Ni and the Co0.9 Fe0.1

alloy. At low temperatures, low energy magnons are responsible for magnetic excitations and thus
act as a thermal reservoir. Thus, the total internal energy of the magnetic degree of freedom
should be described by expression (C.1). With f (, T ) equal to the Bose-Einstein distribution formagnons and a known magnon density of states, which can be calculated using DFT packages, themagnetic part of the specific heat of a material can in principle be calculated. However, because
at higher temperatures Stoner excitations start to play an important role, this approach is only
valid at low temperatures. Particularly, near the Curie temperature, special considerations must
be taken into account [286, 287, 288]. The situation is complicated by a lack of a first principles
theory of magnetism. A good approximation to the specific heat of the spin degree of freedom canbe obtained from the Weiss mean-field theory [289]Cs =µA Na kB ρ
x2 tanh x,Atanh x cosh2 x − x(C.6)

where µA is the magnetic moment per atom measured in Bohr magnetons, A is the atomic weight,
Na is the Avogadro number, ρ is the mass density, andx=

Ms /M (0K),
T /Tc

where the ratio Ms /M (0K) is defined implicitly through the Weiss law [289]Ms= tanhM (0K)



Ms /M (0K)
Ts /Tc.

In practice, however, a more accurate estimate of the magnetic specific heat can be obtainedby subtracting the electron and lattice contributions from the total specific heat or by simply using
experimentally measured values if those are available. Since the three-temperature model is phenomenological, such an approach can be considered reasonable, and should produce more accurate
results than with purely theoretical values of the magnetic specific heat. For Ni, experimental
values for the total specific heat are available in Ref. [286], and for Co in Ref. [290]. The magnetic

224
specific heat of Ni and Co used in this thesis is shown in Fig. C.3. In simulations, the magnetic
heat capacity is often neglected [58]; however, Fig. C.3 clearly shows that near the Curie temperature, the magnetic specific heat is actually on the same order of magnitude as the lattice, and thus
cannot be neglected. In fact, it can play an important role in the transport of energy in a magnetic
sample [291, 292, 293].

4

#10 6
fcc Nifcc Co

m3 K

! J "

3

Cs

2
1
0

0

500

1000

1500

2000

Temperature (K)Figure C.3: Magnetic specific heat of Ni and Co.

It is important to understand that the values in Figs. C.1-C.3 are for bulk materials. In
multilayer samples they might be altered due to confinement effects and interfaces. This can beespecially significant in magnetic multilayers where sub-nanometer layers can interact and thus
modify the magnon spectrum responsible for the specific heat of the spin energy bath.
Finally, as a practical note, in the calculations presented in this thesis, values of Cs above theCurie temperature were set to a small positive number rather than zero in order to avoid numericalinstabilities in cases where the laser pump fluence is sufficiently large to drive the material above
its Curie temperature.

225

C.4

Thermal conductivity
Thermal transport may play a very important role in laser-induced material dynamics, espe-

cially in an optically thick sample, i.e., when there is a strong gradient in the absorbed laser pumpfluence throughout the sample. Specifically, for the FeCo/Ni multilayer considered in Chapter 6,
there is a strong gradient in the absorption of the pump light, which affects the magnetization dynamics. In order to accurately describe such dynamics, the dependence of the thermal conductivityof materials on the electron and phonon temperatures needs to be taken into account.
First, I consider the case for metals. There, thermal transport is mediated by electrons
in the conduction band. Qualitatively, the temperature dependence of thermal conductivity κecan be understood by considering energy excitations of such electrons. In simple terms, the more
excited an electron is the farther it is from its parent nucleus and the less it feels the atomic
potential. Hot electrons should thus move more freely than cold electrons, and κe should increase
with electron temperature. On the other hand, electron-phonon scattering limits the mean freepath of electrons. The hotter the lattice, the more phonon states are occupied and, for a given
electron-phonon scattering rate, the net probability of a scattering event increases. Therefore, κeshould decrease with the lattice temperature. This behavior is not universal and depends on theelectron band structure and the phonon spectrum for a given material. More details can be foundin Refs. [294, 295]. According to Ref. [294], a good approximation for the temperature-dependent
electron thermal conductivity is
κe (θe , θl ) = C(θe2 + 0.16)5/4 (θe2 + 0.44)θ
,(θe2 + 0.092)1/2 (θe2 + bθl )(C.7)

where θe = kB Te /F , θl = kB Tl /F , and C and b are fitting parameters. The latter two can befound by fitting expression (C.7) to the experimental data in Ref. [296]. The fitting parameters
are provided in Table C.1, and the result for Ni is shown in Fig. C.4 as an example. The lattice
thermal conductivity for metals is, in general, negligible and I thus set κl = 0 for the metals in thetable. The spin thermal conductivity for magnetic materials is also assumed to be zero because
I am not considering spin-dependent thermal transport (not to be confused with the transport of226
Table C.1: Fitting parameters for the electron thermal conductivities of Ni, Co, Cu, and Ta andtheir Fermi energies.
Material
Ni
Co
Cu
Ta

C (W/m·K)
0.94
2.49
79.06
3070

b
0.0017
0.005
0.06
18

F (eV)
13.97
7.97
7.00
10.79

the spin), and, as such, κs should be absorbed in κe .
For the insulators in the multilayer stack, namely, Si3 N4 and Ta2 O5 , κe = 0 because electrons
are bound and do not contribute to the heat conduction which is mediated by phonons. Since thethermal conductivity of Si3 N4 and Ta2 O5 is much smaller than that of the metallic layers in thesample, the coupling of the thermal energy into the Si3 N4 membrane and the Ta2 O5 cap is very
slow, and the temperature dependence of κl can be neglected. For Si3 N4 , κl = 3 W/m·K, according
to [297], and for Ta2 O5 , κl = 1 W/m·K, according to [298].

C.5

Electron-lattice-spin coupling
The strength of the electron-lattice coupling Gel depends on the electron phonon-scattering

rate. In most simulations, it is assumed to be constant and is used as a model fitting parameter. However, the electron-phonon interaction depends on the electrons’ energy and momentum
and should thus depend on the electron temperature. Ref. [299] provides an expression for thetemperature-dependent electron-phonon coupling factor
π~kB λ ω 2Gel (T ) =g(F )Z ∞∂fg () −
d,
∂
−∞
2

(C.8)

where λ is an electron-phonon mass enhancement parameter [300] and ω 2 is the second momentof the phonon spectrum by McMillan [301]. The factor λ ω 2 can be approximated by λ(kB TD )2[299]. In the simulations presented in this thesis, I used λ = 0.26 for Ni [302], λ = 0.41 for Co[303], λ = 0.13 for Cu [304], and λ = 0.88 for Ta [305]. The resulting electron-phonon temperaturedependent coupling parameters for the four metals are shown in Fig. C.5. In the figure, the227

5e of Ni

5000

!W "
mK

800

Te (K)

4000

600

3000

400

2000

200

1000
500

1000

1500

2000

Tl (K)
Figure C.4: Electron thermal conductivity of Ni.

228
temperature of the lattice is assumed to be 300 K. There is no universal trend for the curves. For
Ni, the coupling becomes weaker with temperature, while for Cu and Ta it increases. For Co, thebehavior is non-monotonic. An optical excitation can increase the electron temperature by almost
2 orders magnitude from the room temperature. The coupling parameter may change significantly
over such a broad range of temperatures and thus should not be approximated by a constant value.

4#10 18fcc Nifcc Cofcc Cufcc Ta

m3 K

! W "

3

Gel

2
1
0

0

2000

4000

6000

Temperature (K)Figure C.5: Electron-phonon coupling of Ni, Co, Cu, and Ta.

For the insulating layers Si3 N4 and Ta2 O5 , the concept of electron-phonon coupling is notapplicable, but because the three-temperature model requires those inputs, I set this parameter toa small number ∼ 3 × 108 W/m3 K.
The electron-spin Ges and the lattice-spin Gls coupling parameters are not known and arevaried in order to fit the experimental data. Because Ges and Gls are not independent with respect
to the fitting procedure, one of them needs to be specified explicitly. I set Gls to 6×1016 W/m3 K for
both Ni and Co. This is in agreement with an approximate value of 1.6 × 1016 W/m3 K for Fe given
in Ref. [306]. With Ges = 3 × 1017 W/m3 K for both Ni and Co, a good fit of the magnetization
predicted by the three-temperature model to the experimental data is found.

229
As a final note for this Appendix, there is a useful online resource for the electron-latticecoupling parameter, electron specific heat, chemical potential, and density of states of multiple
materials. 33

http://www.faculty.virginia.edu/CompMat/electron-phonon-coupling/

230

Appendix D

Bad Pixel Map for the Tabletop RMS Experiment

A map of bad pixels is shown in Fig. D.1. These pixels do not carry any information because
they have either been obscured by the beam block or exposed to unblocked scattered IR light. The
analysis of the RMS data in Chapter 5 takes the map in Fig. D.1 into account.

Figure D.1: Bad pixel map for the tabletop RMS expereiment. The areas with bad pixels are shownin black