Direct optically driven spin-charge dynamics govern the
femtosecond response of ferromagnets
byPhoebe Marie Tengdin
B.A., Mount Holyoke College, 2013
M.S., University of Colorado, 2016

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 Electrical, Computer, and Energy Engineering
2019

This thesis entitled:
Direct optically driven spin-charge dynamics govern the femtosecond response of ferromagnets
written by Phoebe Marie Tengdin
has been approved for the Department of Electrical, Computer, and Energy Engineering

Prof. Margaret Murnane

Prof. Henry 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.

iii
Tengdin, Phoebe Marie (Ph.D., Electrical, Computer, and Energy Engineering)
Direct optically driven spin-charge dynamics govern the femtosecond response of ferromagnets
Thesis directed by Prof. Margaret Murnane

Ferromagnetic materials have strong electron correlations that drive quantum effects andmake the physics that describes them extremely challenging. In particular, the electron, spin, andlattice degrees of freedom can interact in surprising ways when driven out of equilibrium by ultrafast
laser excitation. In this thesis I uncover several previously unexpected connections between theelectronic and spin systems in ferromagnets. Dynamics occur at unexpectedly fast timescales,
driven using femtosecond laser excitation pulses. The tools that I use to observe the exceeding fast
(10s of femtosecond) dynamics are bursts of extreme ultraviolet light resonant with the M-edge oftransition metals and produced via high harmonic generation. We combine time-resolved transverse
magneto-optical Kerr effect and time- and angle-resolved photoemission spectroscopies to showthat the same critical behavior that governs the equilibrium magnetic phase transition in nickel
also governs the ultrafast dynamics within 20 fs of laser excitation. When the electron temperatureis transiently driven above the Curie temperature, we observe an extremely rapid change in thematerial response: the spin system absorbs sufficient energy within the first 20 fs to subsequently
proceed through the phase transition, whereas demagnetization and the collapse of the exchange
splitting occur on much longer, fluence- independent time scales of 176 fs. This observation defines
a new timescale in the field of ultrafast ferromagnetism. The next question is then whether or not a
response at this speed or faster can be directly observed in more complex materials. To investigate
this I perform experiments on the half-metallic heusler compound Co2 MnGe. Here a single infrared
femtosecond laser pulse drives ultrafast transfer of spin polarization from one elemental sublattice
to another within its pulse duration. I simultaneously probe the magnetic response of cobalt andmanganese to make a surprising finding: the magnetization of Co is transiently enhanced, whilethat of Mn rapidly quenches. This marks the first direct manipulation of electron spins via light,

iv
providing a path to spintronic logic devices such as switches and triggers that operate on few
femtosecond or even faster timescales.

Dedication

To my family, both two-legged and four-legged :)vi

Acknowledgements

The first and most important acknowledgment in this thesis can only go to my tireless advisors: Margaret and Henry. These wonderful people have supported my scientific development
over the last three years in ways far beyond the call of duty. Their constant enthusiasm, optimism,
and general good heartedness is what truly drives the research in their large team of incredible
young scientists. The KM Group is more than just a group of researchers working together, it’sa family, and that’s because of the inspiring way that the team is led. There are also a numberof people who played very large roles in my personal story as a graduate student at JILA anddeserve to be mentioned here. On the magnetics and ARPES teams: Dmitriy Zusin, Cong Chen,
Wenjing You, Xun Shi, Christian Gentry, Maithreyi Gopalakrishnan, Adam Blonsky, Drew Morrill. Others in the group contributed in many ways as well: Michael Gerrity, Begoña Abad Mayor,
Nico Hernandez-Charpak, Michael Tanksalvala, TingTing Fan, Jennifer Ellis, Giulia Mancini, Dan
Adams, and many others. From NIST, the always positive and knowledgeable team of Tom Silva,
Hans Nembach, Justin Shaw, and Mark Keller provided me with tons of magnetics support andall of the fabulous samples that were measured here. From Uppsala, an incredible team of theorists: Olle Ericksson, Peter Oppeneer, Yaroslav Krashnik, Erna Delczeg-Czirjak, Danny Thonig,
and Konstantinos Koumpouras. From Kaiserslautern, some very productive collaboration camefrom Martin Aeschlimann, Lukas Hellbrück, and Benjamin Stadtmüller. Finally I need to thankmy family for their constant support and unwavering belief in me, even when things were looking
shaky. Additionally, many of my friends in and around Boulder contributed to this thesis in massive
ways as well. You know who you are (and if you’re reading this, you’re one of them:)!

Contents

Chapter
1 Introduction and Motivation for this Work1

1.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Fundamental Motivation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.3

Application driven Motivation and Results . . . . . . . . . . . . . . . . . . . . . . . .

52 Ferromagnetism in Atoms, Molecules, and Solids

7

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.2

Atom Based Description of Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . .

8

2.2.1

Spin Dependent Atomic Hamiltonian (Pauli Equation) . . . . . . . . . . . . .

9

2.2.2

Exchange Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.3

2.4

Electron Exchange in Molecules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1

Independent Electron Treatment . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2

Heitler-London Treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3.3

Model hamiltonians: Hubbard and Heisenberg models . . . . . . . . . . . . . 19

Band Model of Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.4.1

Shortfall of the atomic picture: predicting the magnetic moment of the transition metal solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5

2.4.2

Bragg scattering and band structure . . . . . . . . . . . . . . . . . . . . . . . 23

2.4.3

The Stoner Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

The Next Frontier: Ultrafast Laser-induced Ferromagnetic Dynamics . . . . . . . . . 28

viii
3 Theory and Method of Experimental Technique
3.1

30

Resonant Processes in Magnetic Materials: Determining the X-ray Absorption Intensity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.2

3.1.1

Transition matrix element for magnetic atoms . . . . . . . . . . . . . . . . . . 32

3.1.2

Transition matrix elements for atoms in solids . . . . . . . . . . . . . . . . . . 34

3.1.3

Intensity Sum Rule for Charge . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.1.4

X-ray Magnetic Circular Dichroism . . . . . . . . . . . . . . . . . . . . . . . . 36

Kerr Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2.1

Transverse magneto-optical effect (TMOKE) . . . . . . . . . . . . . . . . . . 38

3.3High Harmonic Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.4

Details of the Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4 Critical Behavior within 20 fs drives the Out-of-equilibrium Laser-induced Magnetic Phase
Transition in Nickel

44

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.3

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4

Materials and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4.1

Experimental setups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.2

Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4.3

Model of electronic and magnetic heat capacity . . . . . . . . . . . . . . . . . 58

4.4.4

Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.4.5

Details of experimental setups

4.4.6

Method for extracting TMOKE asymmetry dynamics from HHG spectra . . 61

4.4.7

Momentum dependence of TMOKE Measurements . . . . . . . . . . . . . . . 61

4.4.8

Transient electron and magnetic heat capacity . . . . . . . . . . . . . . . . . 63

4.4.9

Effects of electron-phonon coupling and heat diffusion on Eq. 4.1 . . . . . . . 63

. . . . . . . . . . . . . . . . . . . . . . . . . . 60

ix
5 Universal Nature of the Magnetic Phase Transition in Nickel revealed by Correllative Spectroscopies
5.1

68

Supporting information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1

Calculation of absorbed fluence . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.1.2

Derivation of TMOKE Depth Sensitivity Function . . . . . . . . . . . . . . . 79

5.1.3

Comparison of TMOKE fitting between models with and without criticalfluence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.1.4

Fluence dependence of the exchange-splitting dynamics . . . . . . . . . . . . 81

5.1.5

Thermal or non-thermal nature of ultrafast demagnetization? . . . . . . . . . 81

5.1.6

Demagnetization and recovery timescales . . . . . . . . . . . . . . . . . . . . 84

6 Direct Light-induced Spin Transfer between Elemental Sublattices in a Spintronic Heusler
Material via Femtosecond Laser Excitation

87

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

6.2

Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3

Computational Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

7 Concluding Remarks and Future Directions

99

Bibliography

102

Appendix
A Details of the Angle Resolved Photoemission Experiments

111

A.1 Experimental Details of the ARPES Experiments . . . . . . . . . . . . . . . . . . . . 111
A.2 Dynamics of electron temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
A.3 Dynamics of the electron population at 1.6 eV. . . . . . . . . . . . . . . . . . . . . 115

x
A.4 Dynamics of Exchange Splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.5 Temperature dependence of exchange splitting in static ARPES . . . . . . . . . . . . 119
B Details of the Heusler dynamics and corresponding DFT calculations
B.1 Further details of the experimental setup

121. . . . . . . . . . . . . . . . . . . . . . . . 121

B.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
B.3 Change in reflectivity due to optical excitation in both phases . . . . . . . . . . . . . 122
B.4 Element averaged response of the A2 and B2 phases . . . . . . . . . . . . . . . . . . 122
B.5 Dynamics of Co2 MnGe on α-Al2 O3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.6 Method for calculating density of states and magnetic moments from density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
B.7 Probabilities for laser-driven transitions in the B2 and A2 phases . . . . . . . . . . . 126
B.8 Results of the calculations of k-conserving transition probabilities for the L21 phase 128
B.9 Results of atomistic Landau-Lifshitz-Gilbert simulations . . . . . . . . . . . . . . . . 132
B.9.1 Numerical method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

xi

TablesTable
2.1

Number of 3d and 4s electrons in the free transition metal atoms. Source: [1] . . . . 23

4.1

Fitting parameters for the electron and magnetic heat capacity under thermal equilibrium [2] and in the transient state. All units in (J/mol per K]) unless statedotherwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2

Material parameters used in TTM simulation [3, 4].

. . . . . . . . . . . . . . . . . . 65

5.1

Optimum fitting parameters of the TR-TMOKE results in Fig. 5.3 to the model,
consisting of Eqs. 5.1-5.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

B.1 Fitting parameters of the double exponential function to fit M(t)/M(0). . . . . . . . 134
B.2 Site resolved Gilbert damping parameter in the L21 phase. . . . . . . . . . . . . . . 134
B.3 Site resolved Gilbert damping parameter in the A2 phase. . . . . . . . . . . . . . . . 135
B.4 Site resolved Gilbert damping parameter in the B2 phase. . . . . . . . . . . . . . . . 135
B.5 Fitting parameters for fits performed to the functions used in chapter 6. Note that inthis fitting, the numerical values for the A2 and B2 phases are very close (within errorbars) of the experimentally measured values. This corroborates with our analysis
that the LLG equations can capture the later >200 fs time dynamics of the system. 135

Figures

Figure2.1

One electron model in the independent electron treatment: schematic of H2+ atomic
orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2

Stoner Model for ferromagnetic transition metals, and illustration of terms used forthe description of this model. The filled electron states (at zero temperature) below
the Fermi energy EF are shaded. The spin population with greater occupied states
is called the majority band, while the population with less occupied states is called
the minority band. The centers of the two bands are separated by the exchange
energy ∆. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1

Simplified electronic structure for the microscopic picture of X-ray magnetic circulardichroism.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2

Geometrical layout for the transverse magneto-optical Kerr effect. . . . . . . . . . . 39

3.3

Schematic for fiber based generation of extreme ultraviolet light. Figure generated
by the JILA Comunications office. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4

Detailed experimental layout used to capture the element specific ultrafast magnetic
response of various materials. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

xiii
4.1

Schematic of the critical behavior of ultrafast demagnetization in Ni. (A) After
excitation by a femtosecond laser pulse above the critical fluence (Fc ), the transient
electron temperature (Te ) is driven above the Curie temperature (Tc ), inducing highenergy spin excitations within 20 fs, which store the magnetic energy (see text). The
Fermi-Dirac distributions of electrons are also plotted. Demagnetization occurs later,in 176 fs, driven by relaxation of nonequilibrium spins and the likely excitation of
low-energy magnons. Full recovery of the spin system occurs within 500 fs to 76
ps, depending on the laser fluence. (B and C) Experimental setups for time-resolved
ARPES and TMOKE, respectively, using ultrafast high-harmonic sources. IR, infrared. 47

4.2

Magnetization dynamics in Ni. (A) Change of the TMOKE asymmetry and exchange
splitting reduction ∆Eex as a function of time delay for different laser fluences. The
solid lines represent fitting results, from which we extract the three characteristic times for demagnetization (τdemag ), fast recovery (τrecover1 ), and slow recovery(τrecover2 ) (see chapter 5 for more on this). The fit to TMOKE (upper panel) and
ARPES (lower panel and lower fluence) yields the same fluence-independent time
constants. a.u., arbitrary units. (B) Typical TMOKE asymmetry before (td = 360fs) and after (td = 500 fs) excitation with a pump fluence F = 6 mJ/cm2 .(C) Photoelectron spectra of Ni(111) along the Γ - K direction before (td = 500 fs) and after (td= 500 fs) laser excitation, showing the collapse in the exchange splitting Eex after
excitation (blue dashed lines). The dashed-dotted lines represent the momentum
at which photoemission intensities are extracted. The photoemission intensities areplotted in the right panel with Eex extracted from a Voigt function fit to the data
(dashed lines; see appendix A). (D) Constant, fluence-independent demagnetization
time observed for different laser fluences for both ARPES and TMOKE. . . . . . . . 49

xiv
4.3

Ultrafast charge dynamics in Ni. (A) Log plots of the photoemission intensity above
EF for F 6 mJ/cm2 and at different td , integrated from k ≈ 0.85Å−1 to k ≈ 1.3Å−1
in the momentum space. The dashed lines represent the fitting of the photoemission
intensities with the Fermi-Dirac distribution convolved with experimental energy
resolution (see appendix A). Inset: Integrated photoemission intensity as a functionof pump-probe time delay. The yellow dashed box illustrates the integration regionof electron population in (B). (B) Dynamics of the electron temperature and therelative electron population (n/n0 ) within 0.2 eV above EF as a function of td . The
electron population is normalized to the band electron population (n0 ) 0.2 eV below
EF (see appendix A). (C) Comparison of the electron temperature [red dashed line,
same as (B)] and the change of EUV transient reflectivity at a similar pump fluence.
Inset: EUV transient reflectivity measurement. The resonant EUV light (65 eV)
directly probes the charge dynamics around EF induced by the laser pump pulse.
This measurement is averaged over k-space . . . . . . . . . . . . . . . . . . . . . . . 51

xv
4.4

Observation of multiple critical behaviors during ultrafast demagnetization in Ni.
(A) Peak electron temperature extracted 24 fs after excitation as a function of pumpfluence. The open symbols represent the electron temperature extracted at different
k|| using Tr-ARPES. The solid red line is the fit using Eq. 4.1 considering the transient electron and magnetic heat capacity [inset of (B)], whereas the green dashedline considers only the contribution from transient electron heat capacity (see appendix A). The yellow-colored region (∆FS ) is the energy transferred to the spin
system within 20 fs. (B) Change in the exchange splitting at 2 ps as a function ofpump fluence. The red line represents a fit with an error function. The same criticalfluence of Fc ≈ 2.8 mJ/cm2 is observed for the exchange splitting collapse and thepeak electron temperature in (A). The transient electron heat capacity is plotted in
the inset. (C) Peak electron temperature calculated using Eq. 4.1 and (Ce + Cm )Transient [inset of (B)] for the sample temperatures of 300 and 100 K. The red solidline is the same as in (A). (D) Change of exchange splitting at 2 ps as a functionof laser fluence at different sample temperatures. The solid lines represent the error
function fit of the experimental results. The dashed lines align the critical fluences
observed in (C) and (D) for different sample temperatures. . . . . . . . . . . . . . . 52

4.5

Experimental Setup. (A) Experimental setup of Tr-ARPES experiment. (B) Sideband intensity as a function of pump-probe time delay in laser assisted photoemission
measurement for Tr-ARPES (Further details in appendix A). The solid line is thefitting result with the Gaussian function with a FWHM of ≈42 fs. (C) Experimental
setup of Tr-TMOKE experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.6

Momentum dependence of TMOKE measurements. The TMOKE asymmetry dynamics for different crystal orientations. Inset: Illustrations of the relative orientations of the crystal cut relative to the directions of the magnetic field and lightpolarizations in the experimental geometry. . . . . . . . . . . . . . . . . . . . . . . . 62

xvi
4.7

Electron and spin heat capacity. (A) The electron and magnetic heat capacity underthermal equilibrium [2]. The solid line is the fitting result to Eq. 4.4. (B) The transient electron and magnetic heat capacity extracted from our experimental results.
The green line represents the contribution of the electron bath to the total heat capacity. The dashed line is the electron heat capacity under thermal equilibrium forcomparison.

4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

Two-temperature model. (A) The electron temperature measured using Tr-ARPES
in comparison with the simulation results from TTM for the top 1nm thick layerof Ni with different pump fluence. (B) The transient EUV reflectivity results incomparison with the TTM simulation considering the average over a EUV probing
depth of ≈10 nm. (C) The electron temperature simulated by TTM considering
the situations: 1) thermally isolated, same as Eq. 4.1; 2) only heat diffusion; 3)heat diffusion + e-ph coupling. The dashed line represents the simulation results
convolved with the experimental time resolution. . . . . . . . . . . . . . . . . . . . . 67

5.1(a) Schematic of EUV ARPES and TMOKE measurements on Ni(111). The fluence
profile of the laser excitation below the sample surface separates the magnetization
response into two different regions (i) and (ii), depending on whether the in-situ
fluence is above the critical fluence Fc . Using Tr-ARPES, the probed depth is on theorder of a monolayer, while Tr-TMOKE probes the entire laser-heated depth of ≈10
nm. (b) Schematic of the excitation present in the laser-induced phase transition in
Ni when critical phenomena are taken into consideration [5]. When the laser fluence
exceeds the critical fluence Fc , the electron temperature exceeds Tc and the sample
rapidly undergoes a magnetic phase transition, as evidenced by multiple criticalphenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

xvii
5.2

Change in the exchange splitting (∆Eex ) in Ni measured using TR ARPES, for theabsorbed laser fluence below (0.21 mJ = cm2 , grey) and above (1.7 mJ = cm2 , red)the critical fluence Fc . The solid lines are the fits to Eq. 5.1. Inset: Static ARPES
spectrum plot along the Γ K direction recorded using He Iα photons. . . . . . . . . 71

5.3

Magnetization dynamics in Ni measured using Tr-TMOKE over a full range of laser
fluences. The highest fluence is sufficient to fully suppress the sample magnetization.
The data are offset for clarity. Red curves: Fits to our microscopic model which considers the critical behavior, as well as the depth-average effects in the Tr-TMOKE
measurements. Inset: Fluence-dependent amplitudes of the demagnetization and recovery processes directly extracted from the Tr-TMOKE results. In the Tr-TMOKE
results, the magnetization < M > and the extracted amplitudes < A1 >, < A2 >,and < A3 > are averaged over the entire probed depth. The dashed yellow line
highlights the linear relation of the amplitude < A3 > to the absorbed fluence when
the fluence is above the critical fluence. . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.4(a) Top panel: Schematic magnetization of a ferromagnet as a function of temperature under thermal equilibrium with a single critical point (Tc ). Bottom panel:
Extracted amplitudes of the change of magnetization in a monolayer of Ni as a function of in-situ fluence and heat source. The correspondence of Tc to the two criticalfluences (Fc and Fc ’) is highlighted. (b) The laser-induced magnetization variation in
Ni as a function of time and depth. The black dashed lines represent the contours ofequal magnetization. The white dashed lines separate different regions for the in-situ
fluence relative to the two critical fluences Fc and Fc ’. (c) The relative contributionsof the fast (< A3 >) recovery process directly extracted from the TR-TMOKE results in Fig. 5.3. Inset: Potential scenarios for the coexistence of ferromagnetic andparamagnetic phases in different fluence regions. . . . . . . . . . . . . . . . . . . . . 78

xviii5.5(a) Comparison of Asymmetry computed from depth sensitivity model (red) andmeasured asymmetry (blue). The harmonic peaks used to compute the measured
asymmetry are shown (green) beneath. Note that the experimental asymmetry is
only reliable in the regions sampled by the harmonic comb, and that in these regions
the measurements agree very well with the modelled asymmetry. (b) Calculated
TMOKE sensitivity as a function of photon energy and depth. . . . . . . . . . . . . 80

5.6

Comparison between models with and without including the critical behavior. The
symbols are the Tr-TMOKE data, the red solid line is model results with Fc and theblue dashed line is model results without Fc . . . . . . . . . . . . . . . . . . . . . . . 82

5.7

The change of exchange splitting measured by Tr-ARPES for a range of absorbed
fluence both below and above Fc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.8

Comparison between the experimentally extracted demagnetization at long delay
time and the values expected under the thermal equilibrium. The sample magnetization under the thermal equilibrium is extracted from Ref. [6] and the temperatures
are calculated using the TTM for different laser fluences. . . . . . . . . . . . . . . . . 85

5.9

Demagnetization timescale extracted from a two-exponential function fitting to the
TMOKE data for different pump fluences. The grey line is a guide to eye at the
average value to indicate the timescales are fluence-independent within experimental
uncertainties. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

xix
6.1

(A) Representation of spin dynamics in Co2 MnGe. Before excitation, Mn atoms(orange arrows) have a 3x larger magnetic moment than Co atoms (blue arrows),
which are 2x more abundant in the bcc lattice. The purple arrow represents thenet magnetic moment of the compound. Upon excitation, the Mn moment begins
to decrease and the Co magnetic moment immediately grows by 10%. This occurs
within the duration of the pump pulse. Hundreds of femtoseconds later, the Mn and
Co atomic spins become disordered, and the angular momentum begins to transfer
to the lattice. The net magnetic moment begins to decrease. After 1-2 ps, thespins have reached their maximum quenching. (B) Schematic of the experimental
apparatus. Ultrafast femtosecond infrared pulses are used to excite the sample, while
the magnetization dynamics are tracked with femtosecond EUV pulses recorded ona spectrometer. (C) Density of states for each element in the half metal. Notethat the minority spin channel is gapped, with no available states at the Fermi level
for the minority channel. Critically, this gap is larger for Mn than for Co. After
excitation, the conductions band states are hybridized, as illustrated by the shared
red wavefunction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

xx
6.2

Element resolved ultrafast magnetization dynamics following excitation by femtosecond laser in the (A) half-metallic B2 phase. Note that the Co magnetization increases
as the Mn decreases, followed by a lag of 100 fs between the demagnetization of the
Co and Mn sublattices. Inset: Dynamics of ultrafast spin transfer. Solid lines are
the element specific dynamics after excitation by a 55 fs (full width half maximum)pump pulse. Dashed lines indicate the dynamics after excitation by a 90 fs pumppulse. Note that the location of the peak of the enhancement is shifted in time by 25
fs for the enhancement driven by a 90 fs pump (half the difference between the duration of the two pulses), underlining that this process is a direct optical manipulation.
(B) Element resolved ultrafast magnetization dynamics in the non half-metallic A2
phase. There is no enhancement of the Co magnetization. The subsequent lag indemagnetization dynamics between the Co and Mn sublattices decreases to 57 fs.
(C) Atomic structure of compounds studied. In the B2 phase, the Co atoms haveordered so that they occupy sites at the edges of the bcc structure, while the centers
are randomly interspersed between Mn and Ge. In the A2 phase, the material has
formed the ordered bcc structure, but the location of the atoms within the structure
are random. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

6.3

Density of states for Co2MnGe in the (A) B2 and (B) A2 phases. Note that thehalf-metallic character is only present in the B2 phase.

. . . . . . . . . . . . . . . . 93

xxi
6.4

The probability for exciting a spin up (majority) vs. spin down (minority) electronfrom the valence band in the B2 phase for different pump energies in (A) Mn sites.
Note that for a 1.55 eV pump, the probability is higher for spin up electrons to be
excited from Mn. (B) Probability for excitations in Co sites. In contrast to the
Mn result, the probability is higher for minority electrons to be excited in Co. (C)Illustration of process that leads to direct optical transfer of spin polarization from
Mn to Co. The initial state wavefunction is hybridized and composed of both Mn
and Co d-states, with a larger contribution from the Mn atom. In the final state,
the situation is reversed, and the Co d-states dominate. Hence, when an electron is
optically excited from the initial to the final state wavefunction, this is associatedwith a transfer of spin polarization from Mn to Co.

. . . . . . . . . . . . . . . . . . 95

A.1 Electron temperature fitting. (A) Photoemission spectrum of Ni(111) excited by 16
eV HHG photons at room temperature. The white dash-dotted lines represent the
momentum range for the photoemission intensity in the analysis for the transient
electron temperature. (B) and (C) Photoemission intensity and fitting results at td= -500 fs and td = 24 fs for F ≈ 6 mJ/cm2 . The red solid lines are the fitting results
using Eq. A.1 and the blue dashed lines represent the DOS function used in thefitting. The black dashed-dot lines are the Fermi energy obtained from the fitting.
(D) Photoemission intensity as a function of pump-probe time delay from the raw
experimental data. (E) Same as (D), but the EF shifts due to pump-induced spacecharge effects are corrected.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

xxii
A.2 Electron population dynamics. (A) The relative change of photoemission intensity
as a function pump-probe time delay, taking the intensity before pump excitation(td = -500 fs) as the reference. The yellow dashed boxes represent the two regionsof energy where the changes of electron population are extracted. (B) Normalized
electron population as a function of pump-probe time delay extracted from (a) and(b) regions in (A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
A.3 Analysis of exchange splitting. (A) and (B) Photoemission spectra of Ni(111) before(td = -500 fs) and after (td = 500 fs) pump excitation. The solid black boxes
represent the regions the photoemission intensities are extracted for the analysis on
the exchange splitting. (C)-(D) The photoemission intensities and the Voigt function
fitting (Eq. S2) results for different pump fluences and different pump-probe timedelays. The red lines are the overall fitting results. The Cyan, blue and green curves
represent the extracted Voigt peaks for m=1, 2 (majority band) and 3(minority band).118
A.4 Global fitting of the exchange splitting dynamics. The global fitting results of theexchange splitting dynamics to Eq. 4.2 for a range of pump fluences (F = 1.17
mJ/cm2 to F = 14.12 mJ/cm2 ). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
A.5 Collapse of exchange splitting at the Curie temperature. (A) Static photoemission
spectrum excited by He Iα photons (hν = 21.218 eV) at the room temperature.
Right panel: Energy distribution curve extracted at the momentum where the reddashed line is located. The exchange splitting (Eex ) can be clearly extracted using
fitting procedure described in the previous section. (B) Eex change reduces as the
sample temperature increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

xxiiiB.1 (A) Static asymmetry measurements for the two phases of the material. The magnitude of the magnetic signal in Co is larger in the A2 phase, while that of Mn is larger
in the B2 phase. (B) DFT calculation of the magnetic moment for each element incompound. The increase in magnetic moment for Co and decrease in the magnetic
moment for Mn in the A2 phase (compared to the B2 phase) is consistent with thesevalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
B.2 Change in reflectivity measured for both the A2 and B2 phases. Note that to within
experimental error, the signal is identical for the two samples. This confirms thatthe non-magnetic contribution to the signal is the same, while the magnetic signal
is the cause of the transient enhancement. . . . . . . . . . . . . . . . . . . . . . . . . 123
B.3 Total magnetization of sample plotted with element resolved signal. Total signal constructed from the weighted contributions of the Mn and Co magnetizations measured
with TMOKE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
B.4 Element resolved dynamics of Co2MnGe (B2 phase) on sapphire. . . . . . . . . . . . 125
B.5 Element and orbital specific DOS for (a) B2 and (b) A2 phases. Note that the porbitals are an order of magnitude smaller than the d orbitals, however transitions
take place from p to d or d to p orbitals. . . . . . . . . . . . . . . . . . . . . . . . . . 127
B.6 Transition probabilities for the dipole allowed transitions as a function of photon
energy in the B2 phase of Co2 MnGe.

. . . . . . . . . . . . . . . . . . . . . . . . . . 129

B.7 Transition probabilities for the dipole allowed transitions in the A2 phase. . . . . . . 130
B.8 Transition probabilities for the dipole allowed transitions in the A2 phase. . . . . . . 130
B.9 Transition probabilities for the dipole allowed transitions in the A2 phase. . . . . . . 131
B.10 Element resolved demagnetization in Co2MnGe for the L21, A2 and B2 phase. The
peak temperature of the simulations is set to 1000 K, whereas the relaxation timesof the temperature profile are τ1 = 0.01 ps and τ2 = 1.5 ps.

. . . . . . . . . . . . . . 133

Chapter 1

Introduction and Motivation for this Work

1.1

Introduction
This thesis successfully harnesses a novel quantum light source technology: coherent extreme

ultraviolet light generated via high harmonic generation, to uncover fundamental spin dynamics
underlying magnetism in transition metals and their compounds. Before my thesis work, for the
past 20 years, it was widely accepted by theory and experiment that ultrafast laser light coupled
to the spin bath in materials such as nickel indirectly. The laser energy was believed to be firstabsorbed by the charge (electron) bath, which subsequently led to ultrafast demagnetization of thematerial via spin-flip scattering processes that occurred over timescales of ≈200-500 fs. The lattice(phonon bath) was heated on timescales >1 ps, through electron phonon scattering processes.
My thesis research discovered that this prior understanding is not correct for materials suchas nickel and the Co2 MnGe Heusler alloy — and is likely not to be the correct picture for spin
excitation in many other materials. Rather, in the following I make two surprising new findings,
that show that light can couple to spins in magnetic materials on surprisingly fast timescalesof <10 fs. First, I show that ultrafast laser excitation can create superexcited spin states innickel within <20 fs, when the spin system has absorbed sufficient energy to proceed through amagnetic phase transition. Moreover, there is a beautiful correspondence between the ultrafast
magnetic phase transition and the equilibrium magnetic phase transition — the ultrafast magnetic
phase transition is indeed a true phase transition and occurs when the non-equilibrium electron
temperature exceeds the Curie temperature. Second, I show the first direct optical manipulation

2of spins within a single femtosecond laser pulse, directly transferring spin polarization from oneelement to another in Heusler alloys — on few-femtosecond timescales. These results show how
band structure engineering can individually select for specific desirable behaviors in a quantum
material, and provide a path towards spintronic logic devices such as switches that can operate on
few femtosecond or even faster timescales.
Key to these surprising findings was the use of coherent light at short wavelengths (and thus
high photon energies) as a probe of ultrafast spin dynamics, and multiple advanced spectroscopy
techniques. To probe the bulk magnetic state, I used extreme ultraviolet high harmonics that areresonant with the M-edges of ferromagnetic elements to capture the fastest (femtosecond) spin
dynamics, with element-specific resolution. To probe the new ultrafast critical behavior in nickel,
we correlated time- and angle- resolved photoemission and time-resolved transverse magneto-optical
Kerr effect measurements, both utilizing light from high harmonic generation. The combination ofthese techniques provided powerful new insight into the dynamical processes that happen in thismaterial.
In the next sections of this chapter I briefly outline the background and motivation for thisthesis: the history of ultrafast ferromagnetism and the forces that continue to drive research in thisfield. The motivation is both fundamental: based on the current state of understanding of ferromagnets, and also technology driven: developing a path towards next-generation magnetic storage
devices. Chapter 2 provides a first-order theoretical description of the modern day understandingof ferromagnetism: beginning from an atomic picture of the He atom, proceeding to the simplest
molecular description, and finally to a band picture of magnetic solids. Here I will describe the two
commonly used hamiltonians in the study of magnetism: the Heisenberg and Hubbard hamiltonians, and discuss which specific situations warrant the preferred use of each description. The studyof these models provides an introduction to the difficulty of predicting the behavior of the complicated many-body systems studied in this thesis. In chapter 3 I derive a theoretical background forthe light-matter interaction between extreme ultraviolet or soft x-ray light and magnets. I describe
the experimental apparatus used for most of the work of this thesis, along with how we generate

3
the very special extreme ultraviolet (EUV) light that is central to this work. I show how characterization techniques based on high harmonic generation (HHG) have the unique opportunity toprovide new insights into the physics of these challenging systems.
Next, in chapters 4-6 I describe several of the experiments I performed with this experimental
apparatus and the insights into fundamental magnetism that we uncovered. Chapter 4 introduces
experimental measurements taken on single crystal nickel (111), using both magneto-optical spectroscopy and angle-resolved photoemission. These measurements were the first to observe spectroscopic signatures of an extremely fast and short-lived spin excitation in the ultrafast response to adriving infrared laser pulse. In the next chapter, I demonstrate how knowledge of this ultrafast spin
excitation allows us to map out all of the future time- and fluence- dependent dynamics of the system. Here we also use a careful quantitative comparison between the two experimental techniques
to show that our findings are consistent across both experiments. In chapter 6, I present resultsfrom magneto-optical measurements of the half-metallic Heusler alloy, Co2 MnGe. These studies
show direct experimental evidence for optically driven spin transfer between elemental sublattices
within the time duration of the driving laser pulse. I conclude the thesis with an outlook on somefuture experiments to be continued investigating these topics and others.

1.2

Fundamental Motivation and Results
Magnetization in magnetic materials can be strongly suppressed by ultrafast laser irradiation

on femtosecond timescales[7]. The first experiment to show that magnets this process could occur
was performed using magneto-optics in 1996. This result was surprising at the time because previous
to that experimental discovery, people believed that dynamics in ferromagnets could never proceedon timescales faster than hundreds of picoseconds, because the spin-lattice relaxation times had
previously been estimated to be larger than 30 ps [7]. Following the seminal discovery of Beaupaireet.al., many experiments have been performed on transition metal ferromagnets (Co, Ni, and Fe)
to show that the magnetization is quenched within ≈100 to 500 fs, before subsequently recovering
within tens of picoseconds [8, 9, 10]. However, the charge dynamics in the systems were measured

4
to be even faster: with hot electron distributions being excited in 10s of fs. Although opticaltransitions are spin conserving, electron scattering events do not necessarily conserve spin, andmany of these events must occur in order to reach a hot thermal electron population distribution.
Given this information, and the inability for theories using both atomistic and ab-initio models toagree with experiments, the question instead became: why is the demagnetization process so slow?
This dynamic process has been studied using many experimental techniques, including magnetooptical spectroscopy in first the visible and then extreme ultraviolet regimes [11, 8, 12, 13, 14],photoelectron spectroscopy [15, 16, 9], and x-ray magnetic circular dichroism [10, 17]. In theKM group, magnetic materials were previously studied with extreme ultraviolet (EUV) transverse
magneto-optical Kerr effect (TMOKE) measurements [12], angle- and polarization- resolved EUV
magneto optical measurements [18, 19], and angle resolved photoemission (ARPES) [20], however
this thesis is the first to combine both the magneto-optical and photoemission spectroscopies inorder to understand exactly what each method was measuring, and to show that when driven withthe right fluence, there is an extremely fast spin excitation at 20 fs, and the material actually
undergoes a true magnetic phase transition [5].
This insight was critical because the underlying physical mechanisms that drive ultrafast magnetization dynamics are still under debate. A number of microscopic models based on mechanisms
such as Elliott-Yafet spin-flip scattering [8, 21], dynamic exchange splitting reduction [22, 23, 24],
as well as ultrafast spin-polarized or unpolarized currents [25, 26] have been proposed. In addition, coherent optical excitation [27], spin-orbit coupling [28, 29], and collective magnon excitation[11, 30, 31] are also believed to play an important role in this process. The difficulty in determining
the correct underlying mechanism was due to several issues: first, standard magneto-optic spectroscopies are simply not sensitive to highly nonequilibrium excited magnetic states; they cannot
simultaneously monitor the coupled electron, spin, and lattice degrees of freedom (d.o.f.). Second,
these spectroscopies average over different depths of the material, which can mask the physics of
the ultrafast phase transition.
In this thesis, we experimentally investigate nickel (111) using two advanced spectroscopy

5
techniques based on extreme ultraviolet light from high harmonic generation. These results aredescribed in detail in chapters 4 and 5.

1.3

Application driven Motivation and Results
On the technologically driven side of things, ultralow-power, high-performance nonvolatilememory and logic devices based on magnetic spin (spintronics) are starting to make inroads into conventional computing and represent prime candidates for practical quantum technologies. However,fully exploiting the capabilities of new materials and technologies will require a detailed understanding of the underlying physics, as manifested by nanoscale dynamic magnetization properties.
At present, our understanding of spin interactions is crude and predominantly phenomenological: acomprehensive, self-consistent, microscopic model that rigorously includes the spin, electronic, photonic and phonon-degrees of freedom and their interactions does not yet exist. This understanding
is fundamentally constrained in large part by a limited ability to directly observe magnetism on
all relevant time and length scales. While the fundamental length- and time-scales for magnetic
phenomena are nanometers (exchange length) and femtoseconds (exchange splitting), tools thatenable the exploration of dynamics at these scales have only recently become available.
Band structure engineering can significantly enhance the capabilities of quantum materials
by controlling their magnetic and transport properties [32, 33]. One particular focus is the development of materials with the ability to efficiently generate and sustain pure spin currents foruse in spintronics devices. Half-metallic Heusler compounds are particularly promising candidates
for this application due to their unique band structure [34, 35]: one spin-channel (the majorityband) is metallic in nature, but the other spin-channel (minority) is insulating, with a band-gap
at the Fermi energy. This property is of intense interest for spintronics applications because it
could enable spin-based logic devices such as transistors, diodes, and gates [36]. It also puts uniquerestrictions on the spin excitation pathways that can be driven by a coherent laser pulse in eachelement of the compound, which can lead to exciting new dynamics, such as directly manipulating
the magnetic state of elements in a compound [37, 38]. Although all-optical switching has been

6demonstrated experimentally in both ferrimagnetic and ferromagnetic materials using circularly
and linearly polarized light, the response of the material is not instantaneous, but takes place on
picosecond timescales long after the laser excitation pulse [39], and in some cases also requires the
cumulative effect of many pulses [40, 41].
To date, researchers have investigated several Heusler alloys using ultrafast femtosecond laser
pulses to drive the system into a nonequilibrium state while monitoring the response of the systemwith magneto-optics [34, 35, 38, 42]. These experiments can help to reveal the rate at which the totalmagnetization dynamics evolve in a material and estimate how different microscopic mechanisms
contribute, driving further development of functional materials. However, visible lasers probe
only the net magnetization averaged over all elements in the material. Very recent theoretical
papers exploring laser-excited Heusler compounds have suggested the possibility of light-induced
spin transfer from one element to another on extremely fast (<10 fs) timescales [43, 37]. This couldin theory enable the ultimate goal of ultrafast direct optical manipulation of the magnetic state ofa material, provided these dynamics can be observed.
Ultrafast extreme ultraviolet (EUV) high harmonic pulses make it possible to uncover theelement-specific spin dynamics in multi-component magnetic systems, providing rich new information not accessible using visible light. Recent work explored ultrafast laser-induced spin dynamics
in ferromagnetic alloys and multilayers, where distinct responses such as the existence of a time lag
between the quenching of the magnetization of different elements in an FeNi alloy were observed in
the KM Group in 2012 [13]. In the sixth chapter of this thesis we provide results demonstrating thata single ultrafast laser pulse can directly transfer spin polarized electrons, and thus magnetization,from one magnetic sublattice to another within the time duration of the excitation pulse. Thismeasurement was made possible by the element specific nature of the EUV probe that was used
to map out the dynamics of each element’s magnetic state in time. The possibilities for exploiting
the unique behavior of this material extend to the realm of switches and triggers that can operate
on few femtosecond or even attosecond timescales.

Chapter 2

Ferromagnetism in Atoms, Molecules, and Solids

2.1

Introduction
The topic of magnetism is truly so broad that it is far beyond the reach of a doctoral disser-

tation to provide a proper theoretical grounding for the concepts involved. This fact is underlinedby the lack of the existence of a complete ab-initio theory of magnetism. In addition, the topic
of electron-electron correlations and how they impact the traditional understanding of condensed
matter systems is a rapidly developing field that deserves a treatment of its own, but is also relevantto the materials studied here. The theoretical platforms described here either treat the system in
an atomic picture, where the number of particles is extremely limited, or describe electron-electron
correlations as a mean field, neglecting any individual interactions. This is because including allinteractions and all particles is not computationally possible at this time. Instead, by working
with both of these models and switching between them when required, we can have a reasonable
understanding of the magnetic phenomena described in this thesis.
I begin with the electronic and magnetic structure of atoms and simple molecules. I will showthat the fundamental magnetic properties that emerge from a description of two electron systems
such as the He atom or the H2 molecule form the basis for our modern understanding of magnetism.
Even at the molecular stage, the difficulty of treating the interactions between electrons and solving the complete hamiltonian without significant approximations becomes apparent. Indeed, theproblem quickly becomes intractable for a system with more than three particles. Here we resort toa mean-field picture of electron-electron interactions, which allows us to derive the concepts such

8
as the exchange interaction and the strength of the exchange field felt by electrons. I also provide several additional techniques and approximations for understanding the two-electron systems
described, that can help us to understand certain magnetic phenomena. With these techniques, I
explain how to accurately predict magnetic properties such as the magnetic moment of an element.
This section will highlight one of the primary dilemmas in modern day magnetics research:whether to treat the electrons in the system as localized and correlated, or as delocalized anduncorrelated. I discuss which approximate solutions and the corresponding wavefunctions provide
a more accurate description for the two scenarios, and which physical situations give rise to them.
This key property of ferromagnets is also a topic of intense debate within the field of ultrafast
ferromagnetism. The experiments conducted in this thesis on ferromagnetic nickel have the ability
to probe which picture is better suited for this system and will be discussed in more detail inchapters 4 and 5.
The atomic and molecular description is then followed by a band theory of magnetism, firsttreating a single electron within a periodic potential (the crystal lattice), and progressing to theStoner model, which includes two free electrons, and forms a basis for our modern day understandingof magnetism. Much of the discussion in this section follows from the text of [1], and the reader is
encouraged to refer to this source for any further information required.

2.2

Atom Based Description of Magnetism
I will begin from the He atom, the simplest system with electron-electron interactions, andshow that the basic concept of exchange can be derived from this system, and thus arises in avery fundamental way from the interaction of two electrons in a single potential. The exchangeinteraction is the largest force in magnetism, and provides an origin for the observed alignment of
spins in a ferromagnet.

9
2.2.1

Spin Dependent Atomic Hamiltonian (Pauli Equation)

Although the primary scientific results contained in this thesis come from strong effects ofcorrelated spin-charge electron-electron interactions, we must first begin by understanding whatmagnetism looks like without including any strong correlations between electrons and their spins.
I do this here by treating the problem in the context of a single electron hamiltonian that includes
spin. To properly treat the electronic and spin degrees of freedom, we cannot use the Schrödinger
equation, since it does not include spin. Instead, I begin from the Pauli equation:[He + Hs ] ψ (r, t) = Eψ (r, t)(2.1)

with He the term responsible for the electronic interaction at origin R=0, nuclear charge qn =Ze
and electrons having a charge qe =-e, mass me , position ri,j and momentum pi :
He =

N  2
X
p

Ze2
−
2me 4π0 |ri |
i

i=1


+

X
i<j

e2
.
4π0 |rj − ri |(2.2)

The terms in this hamiltonian correspond respectively to the kinetic energy, Coulomb interaction
with the nucleus, and Coulomb interaction between electrons. The last term is responsible for theexchange interaction, as we will discuss later. The second term in the Pauli equation 2.1 corresponds
to the spin energy of the electrons:Hs =e~
S · B∗
me(2.3)

with S the atomic spin and B ∗ the magnetic induction. This term ultimately gives rise to thespin-orbit interaction.
In order to solve the complicated system described by this hamiltonian, I will need to makesome important approximations. I will first address He , which we immediately see has terms thatcome from the electron’s interaction with the nucleus, and it’s interaction with other electrons.
Thus, I would like to re-write this hamiltonian in two parts: first as a spherically symmetricproblem that contains only single electron operators and will eventually lead to solutions in theform of Bohr’s atomic shell model, and the second a perturbative term corresponding to two-electron
operators.

10
Since the single electron Coloumb term is negative, while the two-electron term is positive, I
can write the problem as a central field felt by the electron combined with a ”screening” term thatis due to the electron-electron interaction.
Thus we rewrite Eqn. 2.2 as:
He =X

H 0 (ri ) + H 1

(2.4)

p2i
Ze2
1
e2 X
−
+
2me 4π0 |ri | 4π0|rj − ri |

(2.5)

i

with
H 0 (ri ) =j(j6=i)which is the sum of all one-electron (non-perturbative) terms. Note that the last term is an average
strength of the electron-electron interaction, and thus does not depend on the individual positions
at a given point and time. The second (perturbative) term H 1 can then be written as a difference
between two large terms and is thus ”small” compared to H 0 :


2
X
X
1
1
e


−
H1=
4π0
|rj − ri ||rj − ri |
i<j

(2.6)

i,j(6=i)This formalism, called the central field hamiltonian, decouples individual electrons from each other,positions of each electron are not correlated to the position of any other electron. We can now solve
the three dimensional Schrödinger equation for H 0 , and obtain the single electron eigenfunctions
which are composed of a radial part and spherical harmonics:
ψn,l,m (r) = Rn,l (r)Yl,m (θ, φ)(2.7)

In practice, however, the solutions to this hamiltonian are nontrivial and usually calculated self consistently by starting with an approximate parameterized solution which is then optimized according
to additional criteria. Additionally, these solutions are incomplete, since they do not include spin.
The complete spin-orbitals must be eigenfunctions of the central field Hamiltonian in order for us
to use them as our zero-order function for the full perturbative solution of the Pauli equation (Eqn2.1). I specify the direction of the spin relative to the z-axis, using the one electron spin functions
α=(sz = +1/2) and β = (sz =-1/2). The spin dependence is then characterized as χ(sz ) and the11
final one electron spin orbitals are written asΨ(r, s) = Rn,l (r)Yl,m (θ, ψ)χ(sz ).

(2.8)

Although I will not here show the solutions for these wavefunctions, this treatment of the
problem for the two-electron atom should give the reader some concept for the fundamental forces
involved in the problem of magnetism, and the difficulty of finding solutions even on an atomic
scale.

2.2.2

Exchange Interaction

Next, I will show how this formalism can be used to understand the exchange interaction in
He, the simplest atomic species with two unpaired electrons. The Schrödinger equation for helium
is represented by the Hamiltonian (for Z=2):
H (r1 , r2 ) =

p21
2e2
e2
p2
2e2
−
+
+ 2 −
2me 2me 4π0 |r1 | 4π|r2 | 4π|r2 − r1 |(2.9)

where the first four terms can be grouped together and written as the central field portion,
H 0 (r1 ,r2 ), while the final term is the two electron part He−e (r1 ,r2 ). This method is given by
Sakuri [44]. We know that the final solutions of H 0 (r1 ,r2 ) must be anti-symmetric. I begin fromthe ground state of helium, with both electrons occupying the 1s orbital. Here the spatial wavefunction is symmetric since both electrons are in the same orbital with the quantum numbers nlm=100.
Thus the spin wavefunction must be antisymmetric, with each electrons having the different spinfunctions α and β. The requirements for symmetric and antisymmetric states come from the Pauli
exclusion principle, due to the fact that electrons are fermions, have spin 1/2, and cannot occupy
the same state. The ground state wavefunction is written as:
Ψgs (a, b) = Ψsym (r1 , r2 )χas (s1 , s2 )1
= [Ψ100 (r1 )Ψ100 (r2 ) + Ψ100 (r1 )Ψ100 (r2 )][αβ − βα]2(2.10)(2.11)

with solutions:
H 0 (r1 , r2 )Ψgs (a, b) = E0 Ψgs (a, b)(2.12)

12that give the zeroth order ground state energy E0 , which must be corrected by He−e (r1 , r2 ), thatcan be calculated from perturbation theory:
E1 = hΨgs (a, b)|He−e (r1 , r2 )|Ψgs (a, b)i(2.13)The exact calculation of these energies can be found in Sakuri [44]. The solutions are E0 =−4e2 /(4π0 a0 ) and E1 = +5e2 /4(4π0 a0 ), with a0 with Bohr radius. Notably, E0 and E1 havecomparable sizes, and thus both need to be considered to obtain an energy that compares well tothe experimental one of 79.01 eV. This calculation of the ground state energies can actually be
carried out without the inclusion of spin (it enters in only in the anti-symmetrization of the spinstates).
However, the importance of the anti-symmetrization requirement and inclusion of spin reallycomes into play for the calculation of excited states in helium. This calculation will give us afundamental insight into the concept of exchange.
I begin with one electron in the 1s orbital, nlm=100 and the second electron in an excitedstate that has a general form of nlm. For this second electron, the central field wavefunction caneither be symmetric or antisymmetric. For the antisymmetric case, I have the singlet excited state:1
ΨSes (a, b) = √ [Ψ100 (r1 )Ψnlm (r2 ) + Ψ100 (r2 )Ψnlm (r1 )]χas (s1 , s2 )2

(2.14)

where the spin contribution χas (s1 , s2 ) is given as:1
χas (s1 , s2 ) = √ [αβ − βα]2(2.15)

And in the symmetric case I have the triplet excited state:1
ΨTes (a, b) = √ [Ψ100 (r1 )Ψnlm (r2 ) + Ψ100 (r2 )Ψnlm (r1 )]χsym (s1 , s2 )2

(2.16)

13
and the triplet state spin wavefunction χsym (s1 , s2 ) given as




αα





χsym (s1 , s2 ) = √1 [αβ + βα]

2






ββ
The two solutions 2.14 and 2.16 have degenerate energies: Ees = E100 + Enlm . However, theperturbative contribution to the two states is not identical. For the singlet (antiparallel spinalignment) state I have:S
Ee−e= hΨSes (a, b)|H 1 (r1 , r2 )|ΨSes i = I + J(2.17)

and for the triplet (parallel alignment):TEe−e= hΨTes (a, b)|H 1 (r1 , r2 )|ΨTes i = I − J(2.18)and the energy splitting between the two states is given by:ST
Ee−e− Ee−e= 2J(2.19)To understand this splitting we need to know what the energies I and J correspond to. These canbe evaluated using the orthogonality of the radial, angular and spin wavefunctions. The electronelectron Hamiltonian H (r1 , r2 ) does not depend on spin, and thus the matrix elements of 2.17 and2.18 are determined by the orthogonality of α and β. I have h± 21 | ± 12 i=1 and h± 12 | ∓ 12 i=0 and
can obtain:
Z Z
I=
Z Z
J=

e2
|Ψnlm (r2 )|2 dr1 dr2
4π0 r12(2.20)e2Ψ∗ (r2 )Ψ∗nlm (r1 )dr1 dr2 .
4π0 r12 100(2.21)|Ψ100 (r1 )|2Ψ100 (r1 )Ψnlm (r2 )These expressions have a physical meaning. The term I is the Coulomb integral, and it is theelectrostatic Coulomb repulsion between the electron densities |Ψ100 (r1 )|2 and |Ψnlm (r1 )|2 and has
a positive sign, which is opposite to the negative attractive force of the electron-nuclear Coulomb

14
force. The second quantity J is the exchange integral, and it represents the energy cost associatedto switching two electron’s quantum states with each other. In the singlet state the spatial wavefunction is symmetric and thus electrons will be closer together, giving it an electrostatic repulsionthat is larger than in the triplet state, where the wavefunction is antisymmetric and electrons will
be farther apart. Thus the effect of electrostatic repulsion is greater in the singlet state making it
higher in energy, with energy difference ∆E = 2J. This is the singlet-triplet splitting.
We see now that the exchange interaction arises from two main factors: the Coulomb repulsion
between two electrons, and the requirement of a total antisymmetric wavefunction, due to thePauli exclusion principle. With these two ingredients we can understand the concept of exchangeinteraction, and furthermore we can distinguish between systems for which symmetrization leads toeither parallel or antiparallel spins (ferromagnetism or antiferromagnetism). For He, J is positive,
and the spins point into the same direction, parallel to each other. However for H2 , J is negative,
giving rise to antiparallel alignment of spins.

2.3

Electron Exchange in Molecules
Now that I have shown that the fundamental basis for magnetism can be derived even froman atomic picture, I will take the next step and move to a description of the exchange interaction
in molecules, beginning from the simplest case of a H2 molecule. In this way we can understandhow the chemical bond affects magnetism.

2.3.1

Independent Electron Treatment

I begin with the independent electron (IE) approximation; first considering a one-electron
H2+ model and then adding a second electron. The Hamiltonian for the H2+ molecule is:


p2
e2
1
e2
1
1
H (r) =−+−+
2me 4π0 |r − R1 | 4π0|r − R1 | R1 − R2

(2.22)

here the first two terms are the central field hamiltonian for a single hydrogen atom: Hatom , andthe coordinates R1 and R2 correspond to the positions of the ions in the H2+ atom. Solutions of this15
equation can be obtained from the solutions of the Schrödinger equation for separate H atoms. The
one-electron atomic wavefunctions for the H atoms are given as φ1 (r) and φ2 (r) and from these wecan construct normalized molecular orbital wavefunctions with linear combinations of these single
electron solutions. We first define an overlap integral O:
O = hφ1 |φ2 i(2.23)

And then I construct the linear combination atomic orbitals (LCAO) or tight-binding wavefunctions(bonding and anti-bonding):1
ΨB (r) = p(φ1 (r) + φ2 (r))
2(1 + O)1
ΨAB (r) = p(φ1 (r) − φ2 (r))
2(1 − O)

(2.24)(2.25)

These atomic orbital wavefunctions are illustrated in Fig 2.1. The two wavefunctions have theeigenvalues:
E0 + E121+O
E0 + E12
AB =
1−O
B =

(2.26)(2.27)

With the quantities E0 and E12 the atomic energy and interaction energy, respectively:
E0 = hφ1 |H (r)|φ1 i = hφ2 |H (r)|φ2 i(2.28)

E12 = hφ1 |H (r)|φ2 i = hφ2 |H (r)|φ1 i(2.29)

If the atoms are pushed very close together, the overlap becomes unity, and the bonding function
becomes the ground state function. However, as the distance between atoms becomes large, theinteraction energy and overlap integral decay exponentially to zero, and the two energies become
proportional to each other with opposite sign.

16

Figure 2.1: One electron model in the independent electron treatment: schematic of H2+ atomic
orbitals.

17
2.3.2

Heitler-London Treatment

Now I will consider the two-electron H2 molecule. The Hamiltonian is written as
e2
1
p2
e2
1
p2
+ 2 −
H (r1 , r2 ) = 1 −2me 4π0 |r1 − R1 | 2me 4π0 |r2 − R2 |


2
e
1
1
1
1
+
−
.
−
+
+
4π0
|r1 − R2 | |r2 − R1 | |r1 + r2 | |R1 − R2 |(2.30)

This Hamiltonian contains terms for the kinetic energy of each electron, the individual electronnucleus Coulomb terms, and finally a term for the electron-electron interactions. In order to find asolution, I can first try an independent electron model: drop the electron-electron interaction term:
He−e = -e2 /4π0 |r1 − r2 |. Re-written in this way I have
 2

p21
e
1
1
H (r1 , r2 ) =−+2me
4π0 |r1 − R1 | |r1 − R2 |
 2

p22
e
1
1
+
−
+
2me
4π0 |r2 − R2 | |r2 − R1 |1e2.
+
4π0 |R1 − R2 |(2.31)

Now the solutions of this Hamiltonian can be expressed as one electron LCAO wavefunctions
as outlined in the previous section. However, we are still missing one more piece, the symmetrization
postulate for a two-electron system. I can use the spin wavefunctions described in section 2.2.2 towrite the two electron singlet ground state:1
ΨS (r1 s1 ; r2 s2 ) = √ [ΨB (r1 )ΨB (r2 ) + ΨB (r2 )ΨB (r1 )]χas (s1 , s2 )2(2.32)

And the triplet state solution is created from the singlet spin state and the products of both bonding
and antibonding LCAO wavefunctions:1ΨT (r1 s1 ; r2 s2 ) = √ [ΨB (r1 )ΨAB (r2 ) − ΨB (r2 )ΨAB (r1 )]χsym (s1 , s2 )2(2.33)

Next I insert Eqns. 2.24 and 2.25 into Eqns. 2.32 and 2.33 and renormalize by the double overlap
integral S:
S = hφ1 (r1 )φ2 (r2 )|φ1 (r2 )φ2 (r1 )i(2.34)

18
I can rewrite the singlet state solution as:
1
ΨSIE = √[φ1 (r1 )φ2 (r2 ) + φ2 (r1 )φ1 (r2 )][α1 β2 − β1 α2 ]
2 1+S
1
+ √
[φ1 (r1 )φ1 (r2 ) + φ2 (r1 )φ2 (r2 )][α1 β2 − β1 α2 ]2 1+S

(2.35)and the triplet state as:
1[φ2 (r1 )φ1 (r2 ) − φ1 (r1 )φ2 (r2 )]χsym
ΨTIE = p2(1 − S)(2.36)

Now we need to check that these states behave in a physically reasonable manner when the limits of
the atoms being close together or far apart are investigated. When the atoms are pushed together,
the IE two-electron singlet state 2.35 behaves correctly, since its components are LCAO functions
that reduce to the He+ ion limit. However, in the other limit of large distances between ions, |ΨSIE |does not behave in a reasonable manner, because there exist two ionic states φ1 φ1 and φ2 φ2 which
each have two electrons on the same atom. These have equal probability with the states φ1 φ2 andφ2 φ1 which have a single electron in each atom. When pulled apart, the wavefunction does not
reduce to the limit of two H atoms, but rather a 50/50 mixture of atoms and ions. Additionally,at large distances, the triplet state would actually be the ground state, since its expectation value
does not have the additional contribution from the ionic states.
Both these inconsistencies can be overcome by the Heitler-London approximation:
1
ΨSHL = √[φ1 (r1 )φ2 (r2 ) + φ2 (r1 )φ1 (r2 )][α1 β2 − β1 α2 ]2 1+S

(2.37)

However, we must recognize that this approximation will not be good for tight bonds, since we haveseen that the IE wavefunction does reduce to the correct form for this case. At the same time, itturns out that at the equilibrium molecular distance, the Heitler-London wavefunction is a better
approximation for H2 .
Now I want to find the energies associated with the singlet and triplet Heitler-London wavefunctions, in order to express the singlet-triplet and bonding-antibonding splitting in closed form.
I do this by connecting the total hamiltonian for H :ES =hΨSHL |H (r1 , r2 )|ΨSHL i
C +X= 2E0 +S
S
1+S
hΨHL |ΨHL i(2.38)

19
and
ET =

hΨTHL |H (r1 , r2 )|ΨTHL i
C −X= 2E0 +TT
1−ShΨHL |ΨHL i(2.39)

With S the double overlap function given by 2.34, and E0 is the atomic energy of the H atom:12
E0 = hφ1 (r1 )|Hatom|φ1 (r1 )i = hφ2 (r2 )|Hatom|φ2 (r2 )i(2.40)and C the Coulomb integral,
C = hφ1 (r1 )φ2 (r2 )|H 0 |φ1 (r1 )φ2 (r2 )i(2.41)

X = hφ1 (r1 )φ2 (r2 )|H 0 |φ1 (r2 )φ2 (r1 )i(2.42)and X the exchange integral

Note that the exchange term X links two electron terms where each electron is attached to different
nuclei and thus it falls off exponentially similar to the overlap function, and more rapidly thanthe Coulomb interaction C. This means that magnetism of solids is primarily due to short-range
interactions. Finally, the singlet-triplet splitting is:
E s − E T = 2J = 2

X − SC
1 − S2(2.43)The Heitler-London function ΨSHL is the prototypical quantum mechanical description for correlated
or localized electrons, because of the importance of the singlet-triplet correlation for bonding. At thesame time, the independent electron function ΨSIE is the prototype for independent, delocalized, or
itinerant electrons. In practice, this is accomplished when the field experienced by a single electron
can be well approximated by a spatial average over the positions of all the other electrons in thesystem. This approach is put into practice by the Hartree-Fock method and underlies density
functional theory (DFT).

2.3.3

Model hamiltonians: Hubbard and Heisenberg models

Due to the difficulties of establishing a first-principles theory as illustrated above, anothercommon technique is to utilize model hamiltonians to understand the experimental properties of20
magnetic systems. The most well-known of these models are the Heisenberg and Hubbard models.
In the Heisenberg model, the spin is explicitly included, and has been used extensively to model
the behavior of ferromagnets in 1, 2, and 3 dimensions. The Hubbard model, in contrast, has no
explicit inclusion of the spin, but has particular importance in describing the splitting electronic
states in correlated materials. This model provides the transition between an independent electron
model and a model that includes correlated electrons. We will discuss these models in the context
of the H2 molecule, where the two models can be solved exactly.

2.3.3.1

Heisenberg HamiltonianThis model requires the existence of two states; the singlet and triplet states, in order to couple
the two spins with correct degeneracy and relative energy. These energies are given by Eqns. 2.32
and 2.33, and there are three terms responsible for the energies of these states. Therefore I write
the effective hamiltonian in terms of these energies:
Hef f = 2H0 + Hcoul + Hexch(2.44)

E0 is the expectation value of the atomic central field hamiltonian H0 , Coulomb energy C is the
expectation value of Hcoul and ±X are the singlet and triplet expectation values of Hexch . To
explicitly include spin in this model, I use the model hamiltonian,
Hexch = As1 · s2

(2.45)

where A is a constant, and the spins s1 = ±1/2 and s2 =±1/2 couple to the total spin as S =s1 +s2 , with S 2 = (s1 )2 + (s2 )2 + 2s1 · s2 and the combination results in a triplet S=1 or singlet
S=0 state. Using the operator expectation values, it can be shown that the triplet states has anenergy of s1 · s2 = 1/4, while the singlet has energy s1 · s2 = −3/4.
Extending the two-electron Heisenberg hamiltonian from two spins to a many-electron systemis done:Hef f = −

N
X
i6=j

Jij si · sj = −2

N
X
i<j

Jij si · sj

(2.46)

21
with an exchange integral defined similar to the atomic case,
Z Z
Jij =

Ψi (r1 )Ψj (r2 )e2Ψ∗ (r2 )Ψ∗j (r1 )dr1 dr2 .
4π0 r12 i(2.47)

There are several key points I need to summarize about the Heisenberg model. First, thecoupling energy is positive for ferromagnetic coupling and negative for antiferromagnetic, withdifferent ground states (singlet or triplet), respectively. The coupling of individual spins, located on
the same atom, is called the intra-atomic exchange, while coupling of atomic moments on different
atoms is the inter-atomic exchange. Finally, when reduced to a single dimension, the Heisenberg
Hamiltonian becomes the famous Ising model,
NX

Jij (sz )i (sz )j(2.48)

i<jwhich is often used to describe magnetic interactions in the first order.

2.3.3.2

Hubbard HamiltonianIn the Hubbard model, the magnetic ground state is found via a balance of two competing
energies. Spin enters into the model only through the Pauli exclusion principle. The two balanced
forces are the hopping energy and the Coulomb energy. The hopping energy is the motion of electrons of the same spin between different atoms, and favors delocalized, band-like behavior. On theother hand, the Coulomb energy is experienced by electrons of different spins, and keeps electrons
apart and confined to different atoms. This force drives the formation of localized moments.
For the H2 atom, the Hubbard hamiltonian is written as:
Hef f = 2H0 + HHub

(2.49)(c†1σ c2σ + c†2σ c1σ ) + U (n1↑ n1↓ + n2↑ n2↓ )(2.50)

with HHub :HHub = −tXσ=↓,↑where the c†iσ and ciσ are the creation and annihilation operators for electrons with spin σ on atomi, and thus the first term describes the motion of the electron hopping from site to site. The

22
operator niσ = c†iσ ciσ is the number operator that counts the occupation on atom i for a given spin
σ. For H2 we have: hσ|niσ |σi=1 and hσ|niσ |σ 0 i = hσ 0 |niσ |σi = 0. Thus this term describes theinteraction of electrons with opposite spin on the same atom.
One notable detail about the Hubbard Hamiltonian is that the hopping operator only moves
spins between atomic sites, but cannot flip them. In contrast to the Heisenberg Hamiltonian, theHamiltonian actually acts only on the spatial part of the wavefunctions via coordinate changes but
does not affect the spins. However, the concept of hopping quickly transfers to a band model of asolid, and in fact a version of the Hubbard model called the Bose-Hubbard model is commonly used
as a first-order quantum mechanical description of solids. The discussion of band theory brings us
to the next step in our description of magnetism, which is the inclusion of the crystal lattice.

2.4

Band Model of Ferromagnetism
Here we discuss the effects of the bonds between atoms in solids, in the specific context ofthe 3d transition metals.

2.4.1

Shortfall of the atomic picture: predicting the magnetic moment of thetransition metal solids

In the atomic picture, the spin and orbital magnetic moments of atoms are multiples of the
Bohr magneton according to m = (2s+l)µB /. Thus an initial guess for the magnetic moments forthe transition metals would be even multiples of µB . The magnetic moment could be determinedby the number of unpaired electrons in each atom’s shell. The number of 3d and 4s electrons in
the transition metal atoms are given in Table 2.1.
For example, in Fe, there is a possibility of 12 electrons occupying the 4s and 3d orbitals,
and only 8 electrons occupying these positions, giving rise to 4 unfilled ”holes”, for which situation
Hund’s rule would predict a magnetic spin moment of µs =4µB for Fe, and similarly 3µB for Co
and 2µB for Ni.
However, if the spontaneous magnetization M of Fe, Co and Ni is measured and divided by

N3d
N4s
N3d+4s

K
0
1
1

Ca
0
2
2

Sc
1
2
3

Ti
2
2
4

V
3
2
5

Cr
5
1
6

Mn
5
2
7

Fe
6
2
8

Co
7
2
9

Ni
8
2
10

Cu
10
1
11

23

Zn
10
2
12

Table 2.1: Number of 3d and 4s electrons in the free transition metal atoms. Source: [1]the number of atoms, the resulting values are mexp = 2.215 µB for Fe, 1.715 µB for Co, and 0.616µB for Ni. The sizes of these values immediately illustrates that the atomic picture cannot provide
an accurate description of the magnetism of solids. In the next section, we will take a different
approach, accounting for the itinerant nature of the electrons, and treating the effects of bondingof the atoms in a metal lattice.

2.4.2

Bragg scattering and band structure

Before I introduce a model that includes magnetism in the context of a crystalline lattice, I
will first introduce the language of band structure and the effect of periodic boundary conditions
on the solutions of Schrödinger’s equation for electrons in a periodic potential such as a crystalline
lattice of ions. I begin with Bragg reflection of the electron from the lattice.
The most important consequence of Bragg reflection in this discussion is that it leads to thepresence of an energy gap. In the 1-D case, the Bragg equation becomes:
2dsinθ = nλ(2.51)
k = nπ/a
The first reflection occurs at k = ±π/a and the first energy gap is there as well. Other energy gaps
occur for the other positive and negative integral values of n.
Reflection at k = ±π/a arises because the wave reflected from the (p ±1)st atom interferesconstructively with the original wave at the pth atom. The region in k space between ±π/a is thefirst Brillouin zone.
At k = ±π/a the stationary state wave functions are not traveling waves as for the free particle
model, but the solutions at these particular k values are made up equally of waves traveling to the24right and the left: standing waves. For k = ±π/a I have two independent standing wave solutions:Ψ1 ≈ sin(πx/a) ≈ (eiπx/a − e−iπx/a )(2.52)
iπx/a

Ψ2 ≈ cos(πx/a) ≈ (e−iπx/a+e)

These solutions are made up of equal parts waves traveling to the right and left. Moreover, thesesolutions correspond to different values of the potential energy of the particle, owing to the positional
shift between the two wavefunctions. Ψ1 distributes charge preferentially midway between ion cores,
while Ψ2 distributes charge preferentially on the ion cores. On calculating average values of thepotential energy over the three charge distributions, we expect the potential energy of Ψ1 to begreater than that of a plane wave while that of Ψ2 to be less than that of a plane wave.
If the potential energies of the states Ψ1 and Ψ2 differ by an amount ∆E, we have an energy
gap of width ∆E. The physical reason for the gap is this: If wave functions at values of large k (farremoved from the Brillouin zone boundaries ±π/a) may be represented by plane waves eikx , thenin forming a solution of the wave equation as the boundaries are approached and Bragg reflection
becomes imminent the wave gradually becomes an admixture of the wave ei[k−(2π/a)]x . Finally at
the zone boundary k = ±π/a and the solution is eiπx/a ± e−iπx/a .

2.4.3

The Stoner Model

The problem of predicting the value of the magnetic moment for transition metals was solved
by developing a band theory for magnetic systems, which was done in 1935 by Mott, Slater, andStoner [45, 46, 47, 48, 49]. This model is called the Stoner model.
In this model, the bonding interaction in the 3d electrons causes a smearing of the energies
into a band. The periodicity condition of the lattice leads to the characteristic periodic variations
and as seen in section 2.4.2, their band width increases with the inverse lattice constant. For an
intuitive description of the Stoner model, I consider the average finite energy width of the valenceband states and approximate the density of states as a simple semicircle as shown in Fig. 2.2.

25

EnergyEF
exchangeenergymajority ofelectrons# of states/eVfilled statesminority ofelectrons

Figure 2.2: Stoner Model for ferromagnetic transition metals, and illustration of terms used for thedescription of this model. The filled electron states (at zero temperature) below the Fermi energy
EF are shaded. The spin population with greater occupied states is called the majority band, while
the population with less occupied states is called the minority band. The centers of the two bands
are separated by the exchange energy ∆.

26
Here I introduce the terminology of majority and minority charge carriers. The electrons
with spin states corresponding to the larger electron population are the ”majority band” while
those with the smaller population are the ”minority band”. This terminology will be extensively
used in Section 6, since the distinction between minority and majority carriers becomes especially
important for the special class of ferromagnets called half-metals studied in this project.
Next I proceed to derive an expression for the magnetic moment that is relevant for bothelectron and hole pictures. From the spin resolved density of states we can compute the number ofspin states in the majority and minority bands by energy integration:
Nemaj = Ne↓ =Z EF

D↓ (E)dE(2.53)− inf

From electromagnetic theory, the atomic magnetic moment is given by m = −2µB s with shaving the units of [~]. The absolute value of m can be found via the expectation of the electron
spin operator < sz > according to the expression |m| = −2µB hsz i /~. Using the convention thatthe spin up states are labeled |↑i = |+i with eigenvalue sz = +~/2 and the spin down states |↓i =|−i with eigenvalue sz = −~/2, we have for sz :1
1
1
1
hsz i = h+ |sz | + i Nemin + h− |sz | − i Nemaj
2
2
2
2
~ min
~
= (Ne − Nemaj ) = (Ne↑ − Ne↓ )22(2.54)(2.55)

and we have the maximum number of electrons in the d shell: (2s + 1)(2l + 1) = 10. We also have
Ne + Nh = 10. Thus the number of holes in the minority band Nhmin = Nh↑ and for the majority
band Nhmaj = Nh↓ is Ne↓ + Nh↓ = 5 and Ne↑ + Nh↑ = 5 and thus,
hsz i =~
~ ↑(Ne − Ne↓ ) = − (Nh↑ − Nhmaj ).
22(2.56)

And finally we can conclude that the magnetic moment |m| is given by the difference in the numberof electrons or holes in the majority and minority bands, according to|m| = µB (Nemaj − Nemin ) = µB (Nhmin − Nhmaj ).

(2.57)

27
When finally including these numbers, we find that in the band picture, the number ofmajority and minority states is a non-integer, and thus the magnetic moments are non-integer
multiples of the Bohr magneton as experimentally measured. This simple relation between electron
and hole moments is exact only if the band states are well defined in their energy spread about the
Fermi level. In reality, there is hybridization between the d states and the s-p states such that even
10 eV above the Fermi level, there is some small d character in the s-p bands. Thus the numberof d holes and the magnetic d moment is slightly reduced relative to the values expected from thefilled electron states [50].
Since I now have a value for the magnetic moment, the exchange energy can also be written as
an ”exchange field”, an effective magnetic field that splits the levels and can be used to describe theoriginal splitting as ∆ = 2mḢex . In addition, it exerts a torque T = m × Hex on the electron spins.
This gives a surprisingly giant value for the effective exchange field of 103 T. The size however gives
a quantitative understanding for the giant strength of the magnetic interaction in 3d ferromagnets.
It is important to keep in mind that this exchange field does not act on the orbital moment or the
nuclear magnetic moment, and thus the effect of the exchange interaction on the spin system isseparate from the orbital system. The two systems can interact, but only via the (much weaker)
spin-orbit interaction.
Although the Stoner model does a good job of predicting the magnetic moments of the transition metals by accounting for the bonds between atomic sites, there are some major shortfalls
with the model. In particular, when attempting to calculate the Curie temperature for the transition metals, one comes up with a number several thousands of Kelvin higher than experimentallymeasured. This is because the Stoner model fails to account for interactions between electron spins
directly. It is basically an extension of the Hubbard model to band theory rather than the Heisenberg model. In practice, a combination of the Stoner and Heisenberg models must be used in orderto understand macroscopic behavior of ferromagnets.

28

2.5

The Next Frontier: Ultrafast Laser-induced Ferromagnetic Dynamics
As a ferromagnetic metal approaches the Curie temperature, it reaches a critical point where

the magnetic properties of the material change dramatically. Under equilibrium conditions, thisferromagnetic-to-paramagnetic phase transition is associated with critical phenomena, characterized by a vanishing of the spontaneous magnetization as well as a divergence of the magnetic heat
capacity and susceptibility [1]. A faster route to change the magnetization is to use femtosecond
laser irradiation: Since the first experimental observation of ultrafast laser-induced demagnetization[7], femtomagnetism has been a subject of intense experimental and theoretical studies. Although
one might expect critical phenomena to play an important role in laser-induced de-magnetizationof ferromagnetic metals, to date, there has been no clear evidence of this.
When a ferromagnetic metal is heated with a femtosecond laser pulse, the energy is directly
coupled into the electron bath, creating an out-of-equilibrium electron distribution. This electronenergy distribution quickly thermalizes (within tens of femtoseconds) to a hot Fermi-Dirac energy
distribution. In most past work, the laser-induced demagnetization process is described as a sequence of events where the energy of the hot electron bath transfers first to the spin and later tothe lattice degrees of freedom. This cascade of energy relaxation processes is used to explain why
ultrafast demagnetization occurs over a range of time scales from 100 to 500 fs [7, 21]. These multiple time scales observed in past experiments obscured any contributions from critical phenomena.
Although there is still no consensus on the important microscopic mechanisms that drive ultrafast
demagnetization, or their relevant time scales, a number of microscopic models have been proposed.
These are based on spin-flip scattering [21, 51, 24, 27, 11, 23] that transfers spin angular momenta
during the demagnetization process, as well as laser-induced polarized [52, 25, 53] or unpolarized[54, 26] spin currents that can also lead to ultrafast demagnetization.
We note that the ability to manipulate the magnetic state on femtosecond timescales is important both scientifically and technologically. Although ferromagnetic metals are some of thesimplest materials that exhibit strong interactions between the electron, spin, and lattice degrees

29of freedom, there is yet no comprehensive theory that describes their non-equilibrium behavior.
Past work concluded that many different timescales were associated with laser-induced magnetic
dynamics and that these depended on the pump fluence [28, 55] and sample geometry [31, 18].
This made it challenging to develop complete theories and compare with experiments. In contrast, by showing the essential contribution of critical behavior associated with a magnetic phasetransition, we reveal that only a few characteristic timescales are needed to fully explain ultrafast
demagnetization in Ni.

Chapter 3

Theory and Method of Experimental TechniqueIn the previous chapter we provided a microscopic description of ferromagnetism and the
motivation for how certain properties of ferromagnets arise. In this section, we will proceed totreat the interaction of light with these properties and explain how light can be used as a sensitive
probe for magnetism.
Although visible light interacts with magnetism in materials in a non-resonant fashion, theresonant interaction of high energy light with the core level electrons in a ferromagnet provides
a much stronger and more detailed source of information about the material. Traditionally, the
2p core level shells have been probed using light from synchrotrons and free electron lasers. The
physics that describes these resonances is almost identical to the physics that occurs when lightresonant at the 3p level shells is used. In this thesis, we generate light resonant with the 3p level
electrons via the process of high harmonic generation. This technique has the advantage that thetemporal duration of the probe is very short: < 10 fs and carries a broad range of energies that canbe simultaneously resonant with multiple elements and chemical states in a compound, and evengive broad information across a resonance.
The resonance associated with the 3p core level electrons in transition metals is commonlyreferred to as the M-edge, while the 2p level resonance is called the L-edge. In this section, we will
explain how the resonance at the L-edge can be used to directly measure the magnetic moment ofa material. This description also applies to light resonant at the M-edges. Additionally, we will
focus on circularly polarized light, because it provides an intuitive and convenient basis for our

31
discussion. However, all of the experiments performed in this thesis were in fact carried out using
linearly polarized light. This is possible because the transition affects not only the amplitude ofthe light but also the phase, and linear polarization can be written as a combination of right andleft circularly polarized light. For certain geometries, the microscopic physics that we will discuss
here leads directly to a complex dielectric permittivity tensor that interacts with linearly polarized
light in a useful fashion.
Erskine and Stern first proposed the use of X-rays resonant with the core level to valenceband transitions to measure magneto-optical properties in the transition metals [56]. The first
X-ray magneto-optical effects were then discovered in 1986 (linear effect) [57] and in 1987 magnetic
circular dichroism was demonstrated at the K-edge in iron [58].

3.1

Resonant Processes in Magnetic Materials: Determining the X-ray
Absorption Intensity
In this thesis, two different resonant processes are used to monitor fundamental changes

in magnetic materials after ultrafast laser excitation. The first is to directly monitor the charge
response using the change in absorption of resonant extreme ultraviolet light, and the second is tomonitor the change in magnetic moment using the magneto-optical Kerr effect. In this section, wetheoretically derive the origin for each of these effects, starting with the sum rule for charges andhow the absorption intensity for a transition from a core to valence band depends directly on thenumber of holes in the valence band. This is the process responsible for changes in reflectivity that
result from (non-magnetic) changes in the electron population occupation in a metallic sample.
Next, using this information it is a straightforward extension to see how circularly polarized lightis sensitive to the number of spin up and spin down electron states available in the valence band.
This information can then be used to monitor the magnetic moment of the material and is thefundamental basis for the magneto-optical effects that are exploited in the measurements taken
here.

32
3.1.1

Transition matrix element for magnetic atoms

We begin from the most basic quantum mechanical expression available to describe polarization dependent X-ray absorption and resonant scattering. We want to calculate the intensity of astrong resonance such as the 2p3/2 ,2p1/2 → 3d transition.
The transition matrix elements can be written (by quantizing the electromagnetic field) in
the general form:

M = hb|pp ·  eikk ·r |ai(3.1)

with p the electron momentum vector,  the unit photon polarization vector, and k the photon
wave vector. We can simplify this expression by taking the dipole approximation, which is welljustified in both the soft X-ray (L-edge) and extreme ultraviolet (M-edge) regimes. This is simply
the assumption that the size of the absorbing atomic shell is small relative to the X-ray wavelength,
so that the electric field that drives the electronic transition is constant over the atomic volume.
For the L-edge, in the photon energy range ~ω ≤ 1,000 eV, with a wavelength λ > 1.2 nm andtransitions from the 2p core shell of radius |r| ≈0.01 nm so that |r| ≈0.01 nm < λ/2π ≈0.2 nm.
Given that the wavelength of EUV is an order of magnitude larger, while the 3p shell is only
marginally larger in spatial extent to the 2p shell, it is reasonable to use the dipole approximationfor our discussion here. In this approximation we have:
M = hb|pp ·  (1 + ik · r + . . . )|ai ≈ hb|pp ·  |ai = ime ω hb|rr ·  |ai(3.2)

with me the electron rest mass, and ω = ωb − ωa the photon frequency for the transition from state|ai to state |bi. The X-ray absorption cross section can be written
σabs = 4π 2e2
~ω| hb| · r |ai |2 δ[~ω − (Eb − Ea )]ρ(Eb )4π0 ~c(3.3)

with the density of final states per unit energy ρ(Eb ) depending on the normalization of the electronic wavefunctions |ai and |bi. When these functions have been volume normalized to unity, the33following expression can be obtained:
Ires = A| hb| · r |ai |2(3.4)

with the proportionality A given as
A = 4π 2e2~ω = 4π 2 αf ~ω4π0 ~c(3.5)

and the dimensionless fine structure constant αf
αf =

e2
1
=
.
4π0 ~c137.04(3.6)

The resonance intensity Ires has dimensions of [length2 × energy].
Now the calculation of the transition matrix element depends on the initial and final state
wavefunctions |ai and |bi which, in the one electron picture, consist of the core level and valence
electron wavefunctions, respectively.
In a one-electron picture an ”initial” state wavefunction for a core shell n with angularmomentum c is given by|ai = |Rn,c (r); c, mc , s, ms i

(3.7)

with Rn,c (r) the radial component of the core shell with the principle quantum numbers n andorbital quantum number c, |s = 1/2, ms i the spin state of the electron. The final state can also be
written as|bi = |Rn0 ,l (r); l, ml , s, m0s i(3.8)

where Rn0 ,l is the radial component of the valence state of shell n’ with angular momentum l. These
final states are determined by the empty states in the l subshell. Thus, we need to calculate thetransition matrix element
hb|Pαq |ai = hRn0 l (r); l, ml , s, m0s |Pαq |Rn,c (r); c, mc , s, ms i

(3.9)

with the direction and polarization dependent dipole operator Pαq written in the general form:
Pαq /r =X
p=0,±1

(1)

(1)

(1)

eqα,p Cp(1) = eqα,1 C1 + eqα,0 C0 + eqα,−1 C−1(3.10)

34(1)

are the Racah operators for spherical harmonics, and the coefficients aqα,p may bePimaginary with p |eqα,p |2 =1. For a transition from a core shell with angular momentum c to an
where Cq

unfilled valence shell with angular momentum l we then have0hb|Pαq |ai = δ(ms , ms ) hRn0 , l|r|Rn,c (r)iX

eqα,p hl, ml |Cp(1) |c, mc i .

(3.11)

mc ,ml ,pSeveral things become immediately apparent from this description. First, the dipole operator
does not act on spin, and thus only spin conserving transitions are allowed. Next, the polarization
dependence is entirely contained in the angular part of the wavefunction. And finally, the radial
part determines the angle integrated transition strength. It is also the part of the wavefunction
responsible for the elemental specificity of X-ray absorption spectroscopy. Even though the final
valence state has a relatively large radial wavefunction, the radial extent of the transition matrix
element between the 2p and 3d states is relatively localized. This localization leads to a fundamental
difference between x-ray absorption spectroscopy and optical spectroscopy.
In general, optical transitions occur only between filled and empty valence states, and thus
are not localized on any specific atom, but instead the intensity of the transition is determined by
the group theoretical symmetry of the valence state of the crystal or molecule.
(1)

The angular part of the transition matrix element has the form hl, ml |Cq |c, mc i, and thenonzero matrix elements here dictate the dipole selection rules.

3.1.2

Transition matrix elements for atoms in solids

The next step is to continue the calculation of the transition matrix elements for a realsystem, namely that of atoms in a solid, as measured experimentally in this thesis. To do this, weneed to use realistic initial and final state wavefunctions. Fortunately, the wavefunctions of bonded
atoms can be constructed from linear combinations of the atomic wavefunction.
Within this approximation, called the tight binding model, we can write the wavefunctions

35
for the electrons as:= |Rn,L (r)i+LX

|Ψi (kk , r)i = |Rn,L (r)|φi (kk )i(3.12)

ai,M (kk ) |LM χ+ i + bi,M (kk ) |LM χ− i

(3.13)

M =−L

with energy Ei (k), the X-ray absorption transition intensity can be written,
X

Iαq = A| hΨi (kk , r)|Pαq |Φjm (r)i |2(3.14)

X(3.15)

k ,m,jEi >EF ;i,k

X

= AR2

| hφi (kk )|eqα,p Cp(1) |cmχj i |2p=0±1

k ,m,jEi >EF ;i,kand by using the fact that the dipole operator does not act on spin we can write the transition
intensity
Iαq = AR2

X|

X

ai,M (kk )eqα,p hLM |Cp(1) |cmi |2

k ,m p,M
E1 >EF ;ik+|X(3.16)

bi , M (kk )eqα,p hLM |Cp(10) |cmi |2 .

p,M

The important thing to notice about this expression is that the matrix elements involve only atomic
orbitals. The band states contain wavevector dependent expansion coefficients, however this canbe done analytically for the case that one averages over all polarization states (or, equivalently, all
x-ray incidence angles). For the next section, we will discuss transition matrix element intensities
and intensity sum rules for charge and spin in the context of polarization averaged (averaging bothleft and right circularly polarized), and normal incidence measurements. However, it is important
to note that in a real measurement, the angle of incidence and polarization of the x-ray plan an
important role in the results of the measurement. This will be the case for all the experimental
work detailed in the later parts of this thesis.

3.1.3

Intensity Sum Rule for Charge

The sum rule for charge states that the transition intensity is proportional to the total numberof empty d states Nh when a sum is taken over the 2p3/2 and 2p1/2 contributions. By separating

36
the diagonal and cross terms in Eqn. 3.16 we have:X

Iαq = AR2X[|ai,M (kk )|2 + |bi,M (kk )|2 ]|eqα,p |2 | hLM |Cp(1) |cmi |2(3.17)

k ,m p,M
Ei >EF ;i,k

+AR2

X

X

(1)

∗

eqα,p (eqα,p0 )∗ hLM |Cp(1) |cmi hLM 0 |Cp0 |cmi

(3.18)

×[ai,M (kk )(ai,M 0 (kk ))∗ + bi,M (kk )(bi,M 0 (kk ))∗ ](3.19)

k ,m p6=p0 ,M 6=M 0
Ei >EF ;i,kfrom here we can perform an orientation average by summing over the three orthogonal polarization
states q = 0,±1 or over crystal dimensions α = x,y,z. The cross term then vanishes and we are left
with the polarization averaged transition intensity:1
hIi = AR2
3

X(|ai,M (kk )|2 + |bi,M (kk )|2 )X| hLM |Cp(1) |cmi |2

p,m

k ,M
Ei >EF ;i,kX

|eqα,p |2(3.20)

qorα

where the first sum is the number of holes in the material, the second sum is characterized by
L/2L+1, and the final sum in the term is equal to 1. Finally, then we have the transition intensity
for a core to valence nc → nc0 can be written as
hIi = AR2L
Nh
3(2L + 1)(3.21)

with A = 4π 2 ~ωα where α is the fine structure constant and R is the radial nc → n0 L matrix
element.
This relation is called the intensity sum rule for charge and allows us to directly relate theintensity of an electronic transition to the number of valence holes in the electronic ground state.
This is the underlying principle behind the transient reflectivity measurements performed in chapter
4 of this thesis. We can also use this principle to derive X-ray magnetic circular dichroism, which
is how one can monitor transient changes in the magnetic state as described in the next section.

3.1.4

X-ray Magnetic Circular Dichroism

It is now a reasonable extension to think that if we could make the absorption of photons
spin dependent, we could measure an intensity difference corresponding to the difference between

37
spin up and spin down electrons, which is directly proportional to the magnetic moment, as shownin chapter 2.
For simplicity, we will assume a one-electron picture that corresponds to the Stoner splitting
as derived in the previous chapter. For the XMCD effect to have a maximum, the magnetization
direction M of the sample and the photon spin (angular momentum L ph ) should be collinear. We
then measure the difference in absorption of the p → d transition for photons with positive andnegative angular momentum (q = ±1, Lph points in ±kk direction). Experimentally, it is more
typical to fix the X-ray photon spin direction and instead switch the magnetization direction.
This is in fact what we will now derive. The expression for the difference in intensity for thep → d transition with negative/positive photon spin (q ± 1),∆I = I ↑↓ − I ↑↑ = I − − I + .

(3.22)In this expression, I ↑↑ corresponds to the intensity with the spin-orbit coupling is positiveand is directly added to the exchange energy, and I ↑↓ corresponds to the case when the spin-orbit
coupling is negative and subtracted from the exchange energy. This is experimentally the casewhen the magnetization and photon angular momentum are parallel and anti parallel, respectively.
The quantum mechanical origin of the XMCD effect can be explained using the Stoner model
for ferromagnetism as outlined in chapter 2. The available density of hole states is larger for theminority band, which is important for the probability of excitation of spin-polarized electrons.
When the sample is magnetized in the ”up” direction, the spin down states are filled, but the spinup states are only partially filled. When an electron is excited by a circularly polarized photon, theangular momentum of the photon must be conserved, and is transferred to the electron. Part of
the angular momentum is then transferred to the spin via the spin-orbit coupling and the excited
photoelectrons are spin polarized. Since the 2p3/2 (L3 ) and 2p1/2 (L2 ) levels have opposite spinorbit coupling (l + s and l - s), the spin polarization will be opposite for the two edges. Then, theexchange splitting in the valence shell with unequal spin up and spin down available hole states

38
acts as a detector for the spin of the excited photoelectrons. Figure 3.1 illustrates this process.

3.2

Kerr Effect
The microscopic origin for XMCD is identical to that of resonant magneto optical Kerr effect.

This is because the origin of the dielectric permittivity tensor that interacts with any incident light isthe same as that outlined above. From here, we will treat the interaction of light with the dielectric
tensor rather than directly from the transition matrix elements, since it provides a convenient way
for computing magneto-optical effects that depend on inccidence angle and polarization of the lightas described above.
When linearly polarized light reflects from a magnetized solid it is modified in two ways: firstthe direction of polarization is rotated over an angle ΘK , and secondly the phase of the light is
modified so that it becomes elliptically polarized. When magnetization is applied in a transverse
geometry (see Fig. 3.2), the amplitude of the light is modified as well. It is this effect that we will
employ for the majority of the work in this thesis due to the fact that changes in intensity are much
simpler to monitor in the EUV light regime than changes in the polarization of the light.

3.2.1

Transverse magneto-optical effect (TMOKE)

We now consider the interaction of magnetic media magnetized in the transverse direction
with linearly polarized light as shown in figure 3.2
In the s- and p- polarization basis (Es ,Ep ), as shown in figure 3.2, we write the incident~ i , i.e. the time- and space- independent part of the (plane wave) radiation aselectric field E

 

E
cos(θ)−→  s  Ei = 
(3.23)
= E0
Epsin(θ)
with E0 the amplitude of the incoming electric field vector. In the following, we set E0 to 1. Θ isthe angle of the linearly polarized radiation relative to the s-polarization direction. The reflected39

spin down e-spin up e-absorption
spectrumexchange
interaction

spin-orbit
coupling

Figure 3.1: Simplified electronic structure for the microscopic picture of X-ray magnetic circulardichroism.

s-pol

Figure 3.2: Geometrical layout for the transverse magneto-optical Kerr effect.

40−→
field Er can be related to the incident field through the Fresnel reflection matrix r̂ as


−
→
−
→
 cos(θ) Er = r̂Ei = r̂ .
sin(θ)(3.24)

Now this matrix depends on the magnetization direction of the solid and can be written whenm
~ lies in the (x,y) plane as,






(0)
rss

(1)
rsp my Q rss rsp  

r̂(m)
~ =
=

(1)
(0)
(1)
rps rpp−rsp my Q rpp + rpp mx Q

(3.25)

with the magneto-optical Voigt constant Q=ixy /xx and the subscripts (0) and (1) indicate thecoefficients in terms independent and linear in Q, respectively. Note that for the special case of themagnetization lying entirely in the x direction, the dielectric tensor will cause only the p-polarized
light to be directly affected by the magnetization. In this work and many others, it is common todefine an experimentally measurable quantity, the magnetic asymmetry:A=

R+ − R−
R+ + R−(3.26)

where R+ is the reflected intensity when the sample is fully magnetized in one direction (arbitrary),
and R− the reflected intensity when the sample is fully magnetized in the other direction. When themagnetization lies only along the x-direction (mx =1, my =0, mz =0), then the asymmetry depends
on the angle of incidence of the light and the material’s Voigt constant as:(0∗) (1)A=

mx (1 − cos 2θ)Re(rpp rpp Q)(0)|rpp |2 sin2 θ + |rss |2 cos2 θ.

(3.27)

Here it becomes clear that the asymmetry depends linearly on the material’s Voigt constant,or magnetic moment.

3.3

High Harmonic Generation
To study the non-equilibrium properties of transition metals as they pass from the ferro-magnetic to paramagnetic transition, the ideal probe to use would be one that drives transitions
originating from the core levels, which are nearly unaffected by the change in occupancy of the41states at the fermi level that occurs with the sharp increase in temperature of the electrons. In
this thesis, we use a process called high harmonic generation to up-convert femtosecond pulses oflight at 800 nm to wavelengths as short as 13 nm. This process was first demonstrated in 1987
[59, 60], and has since been extensively developed [61, 62] as a method for generating coherent,
ultrashort bursts of light in the extreme ultraviolet and soft x-ray regions of the spectrum. In
this thesis we focus on generating light which is resonant at the 3p edges of the transition metals
from approximately 45 eV to 72 eV in energy. Our method, which is now relatively standard forgenerating EUV light at these energies, is both robust and stable.
Pulses of light from a regenerative Ti:Sapphire amplifier are focused into a hollow core waveguide filled with He gas. The pulses have a time duration of 40 fs FWHM, a pulse energy of 1.6
mJ and the temporal characteristics are preserved by propagating the beam from the amplifier tothe waveguide in a slightly chirped out form of approximately 100 fs. Then in the final step before
focusing the beam into the waveguide, chirped mirrors perform the final compression down to 40
fs. The probe beam is focused with a 50 cm lens into a hollow core fiber with a diameter of 150µm. The hollow core fiber is filled with He gas at a pressure of 1 Torr, where high harmonics ofthe fundamental beam are generated.
Inside the waveguide, the light is converted from 1.5 eV up to energies of 100 eV or greater.
The energies generated are odd harmonics of the driving laser, due to quantum mechanical selection
rules relevant for the process. A simplified version of the high harmonic process called the 3-stepmodel [63]. The driving electric field from the laser pulse is strong enough to warp the Coulomb
potential of the atom, allowing electrons to escape from their orbital positions. Next, the electrons
are accelerated in the electric field, changing direction with the field, until finally they recombinewith their parent ion, and emit a photon with an exceedingly high energy.
After the high harmonic generation process, we want to remove the gas used in the process
as quickly as possible, since the X-rays are quickly re-absorbed by any kind of atmosphere inside
the experimental chamber. To do this, we employ a differential pumping geometry, where gas is
pumped into a hollow core waveguide through small laser-cut openings on the top, and pumped

42
out on the exit side of the waveguide. This process is explained in grater detail in ref. [64] andillustrated in figure 3.3.

3.4

Details of the Experimental Setup
We then focus the light which is diverging after exiting the fiber using a toroidal mirror withfocal length of 30 cm. Because the EUV light comes at a 6 degree angle into this mirror, theefficiency can be as high as 80 percent. This is important because we want to preserve all of the
EUV light that we can from this point forward in the experiment. Next, we need to remove theresidual 1.5 eV driving laser light (which is still very intense). To do this, we using an aluminum
foil filter of 500 nm thickness. This also removes EUV light with energies greater than 72 eV (thealuminum absorption edge), however that light is not needed for any of the experiments performedin this thesis. The EUV light with energies less than 72eV can pass through the aluminum filter
with an efficiency ≈ 50 percent.
Next, the harmonic beam is rejoined by the p-polarized pump beam, and the two are incident
on the sample at an angle of 48 degrees measured with respect to the normal. The timing of thepump beam with respect to the probe is controlled by a mechanical delay stage. The powerin the pump arm is controlled by a polarizer and mechanical half waveplate combination (forfluence dependent studies). The FWHM of the pump pulse for both the TMOKE and reflectivity
measurements at the location of the sample was determined to be 55 fs by in-situ frequency resolved
optical gating measurements. After reflecting off the sample, the HHG probe reflects off a grating(500 lines/mm) placed in the conical mount configuration and passes through two more aluminum
filters of thickness 200 nm before being measured by an Andor Newton X-ray CCD detector.
This layout is illustrated in figure 3.4.

43

Figure 3.3: Schematic for fiber based generation of extreme ultraviolet light. Figure generated by
the JILA Comunications office.

Delay Stage
80/20 BS

Half Waveplate+ Polarizer + Lens

800nm, 1.6 mJ
Toroid
HHG
Lens Waveguide

Electromagnet
SampleGra�ngAl Filters
CCD

Figure 3.4: Detailed experimental layout used to capture the element specific ultrafast magnetic
response of various materials.

Chapter 4

Critical Behavior within 20 fs drives the Out-of-equilibrium Laser-induced
Magnetic Phase Transition in Nickel

In this chapter we describe experiments carried out on single crystal nickel (111). Although
this material was in fact the first one where ultrafast laser-induced demagnetization was observed[7], many things about that process are still poorly understood. In this study, we compare theresults of time resolved transverse magneto optical Kerr effect spectroscopy and angle resolved
photoemission spectroscopy both using extreme ultraviolet light from high harmonic generation.
By comparing the results of these two experimental techniques, we are able to gain fundamental new
insights about the laser-induced demagnetization process and show that the same critical behavior
that dominates the static temperature-driven magnetic phase transition also governs the ultrafast
phase transition in nickel.
It has long been known that ferromagnets undergo a phase transition from ferromagnetic toparamagnetic at the Curie temperature, associated with critical phenomena such as a divergence inthe heat capacity. A ferromagnet can also be transiently demagnetized by heating it with an ultrafast laser pulse. However, to date, the connection between out-of-equilibrium and equilibrium phasetransitions, or how fast the out-of-equilibrium phase transitions can proceed, was not known. By
combining time- and angle-resolved photoemission with time-resolved transverse magneto-optical
Kerr spectroscopies, we show that the same critical behavior also governs the ultrafast magnetic
phase transition in nickel. This is evidenced by several observations. First, we observe a divergenceof the transient heat capacity of the electron spin system preceding material demagnetization. Sec-45
ond, when the electron temperature is transiently driven above the Curie temperature, we observe
an extremely rapid change in the material response: The spin system absorbs sufficient energy
within the first 20 fs to subsequently proceed through the phase transition, whereas demagnetization and the collapse of the exchange splitting occur on much longer, fluence- independent timescales of 176 fs. Third, we find that the transient electron temperature alone dictates the magnetic
response. Our results are important because they connect the out-of-equilibrium material behavior
to the strongly coupled equilibrium behavior and uncover a new timescale in the process of ultrafast
demagnetization.

4.1

Introduction
Here we present clear evidence that critical behavior on a new ultrafast 20 fs time scale

governs laser induced demagnetization in nickel. We show this by correlating time- and angleresolved photoelectron spectroscopy (Tr-ARPES) with Tr-TMOKE spectroscopy, as well as with
EUV transient reflectivity, all using tabletop high-harmonic sources [52]. Through fluence andtemperature dependent studies, we make several surprising observations. First, we observe criticalbehavior as the spin system undergoes a transient magnetic phase transition: As the laser fluence
approaches a critical fluence Fc of ≈2.8 mJ/cm2 , corresponding to a hot electron temperatureapproaching the Curie temperature (631 K), significantly more laser energy is required to increase
the peak electron temperature, indicating a divergence in the heat capacity of the spin system.
Second and very surprisingly, the spin system absorbs sufficient energy within the first 20 fs to
subsequently proceed through the phase transition. This defines a new time scale in the processof ultrafast demagnetization. Demagnetization (measured using TMOKE) and the collapse of theexchange splitting (measured using ARPES) both occur on much longer and similar timescales of
176 fs, independent of the laser fluence. Third, the recovery dynamics of the exchange splitting also
exhibit a critical behavior, changing from a full recovery within 500 fs when pumped at fluences
below Fc to a much longer 76-ps recovery above Fc . These critical phenomena indicate that thetransient electron temperature alone dictates the magnetic response.

46
We note that our findings are in contrast to past work assuming that the electron bath alone
absorbs the laser energy, which is then slowly transferred to the spin system throughout the demagnetization process [21, 27, 23, 14]. Instead, our results imply that the ultrafast demagnetization ofa ferromagnetic metal is driven by a highly nonequilibrium process that takes place within the first
20 fs, with sufficient energy transfer from the optical excitation to the spin system to subsequently
proceed through the transient magnetic phase transition. This ultrafast, highly nonequilibrium
process can likely be driven by superdiffusive spin currents [25, 53] or spin mixing via spin-orbit
coupling [28, 30, 65]. Then, demagnetization of the sample occurs on a longer time scale of 176 fs,
likely mediated by processes such as low-energy magnon generation.

4.2

Results
The observed critical behavior in laser-induced ultrafast demagnetization in nickel is shown

schematically in Fig. 4.1A. Our time-resolved ARPES and TMOKE measurements were both
performed using 780-nm laser-driven EUV high-harmonic generation (HHG) probe beams, as shownin Fig. 4.1 (B and C) (see appendix A). For all measurements, the sample used is a 400-nm Ni(111)single-crystalline film grown on an α − Al2 O3 (0001) substrate [66]. The Ni film thickness is muchgreater than both the pumping (13 nm) and probing depth (1 nm for ARPES; 10 nm for TMOKE)
to avoid any influence of the substrate, capping layer, or multilayer structure [53, 54, 26].
The characteristic dynamics of Ni demagnetization measured by TMOKE, as well as of theexchange splitting (Eex ) measured by ARPES, are plotted in Fig.4.2A as a function of pump-probe
time delay (td ). The TMOKE asymmetry As at 360 fs before the pump pulse arrives and 500 fs
after the pump pulse arrives is plotted in Fig. 4.2B, where a clear reduction of the asymmetry
can be observed after laser excitation. Tr-TMOKE is a momentum-averaged measurement, asshown in section 4.4.7. For ARPES, the photoelectron spectra along the Γ - K direction at room
temperature are shown in Fig. 4.2C. Before the pump pulse excites the sample (td = 500 fs),
the exchange splitting (Eex ) between the majority and minority bands of Ni can be observed at
momentum k|| ≈ 1.05Å1, where the d band crosses the Fermi energy (EF )[9, 67]. The exchange

47

Figure 4.1: Schematic of the critical behavior of ultrafast demagnetization in Ni. (A) After excitation by a femtosecond laser pulse above the critical fluence (Fc ), the transient electron temperature(Te ) is driven above the Curie temperature (Tc ), inducing high-energy spin excitations within 20
fs, which store the magnetic energy (see text). The Fermi-Dirac distributions of electrons are alsoplotted. Demagnetization occurs later, in 176 fs, driven by relaxation of nonequilibrium spins andthe likely excitation of low-energy magnons. Full recovery of the spin system occurs within 500 fs
to 76 ps, depending on the laser fluence. (B and C) Experimental setups for time-resolved ARPES
and TMOKE, respectively, using ultrafast high-harmonic sources. IR, infrared.

48
splitting Eex reduces after laser excitation, as indicated by the spectrum taken 500 fs after thepump pulse and the extracted photoemission intensity. The values of Eex are obtained by fitting
the extracted spectra with multiple Voigt functions (see Fig. 4.2C and appendix A).
The dynamics of the exchange-splitting change (∆Eex )and TMOKE asymmetry (As ) can begenerally fit by an exponential decay and bi- exponential recovery function (see section 4.4). We
find that the dynamics of ∆Eex can be described by a set of time constants, with a dramatic change
in recovery times above a critical laser fluence. For all pump laser fluences (below or above thecritical fluence), the collapse of the exchange splitting exhibits a constant and fluence-independent
time scale of 176 ± 27 fs. For fluences below a critical laser fluence of Fc ≈ 2.8mJ/cm2 , the magnetic
response exhibits a fast recovery (τrecover1 = 537 ± 173 fs). For fluences above Fc , the magnetic
response exhibits the same fast recovery as well as a slower recovery (τrecover2 = 76 ± 15 ps; seeappendix A). These characteristic time constants are obtained using a global fitting scheme. We
note that the minima of dynamics shown in Fig. 4.2A depend on the ratio of demagnetization andfast recovery amplitudes. Although they appear at different time delays for different fluences, theextracted time constants (and characteristic time scales of the processes) are the same.
To correlate the dynamics of the exchange-splitting collapse ∆Eex (probed by ARPES) withthe laser-induced demagnetization of the sample, we analyze TMOKE measurements taken atsimilar pump fluences. We find that the time scale of demagnetization measured using TMOKE is
the same as the exchange-splitting collapse measured using ARPES and that both are independentof the pump laser fluence (Fig. 4.2D). This is consistent with the conclusion that the collapse of
Eex obtained in Tr-ARPES directly represents the quenching of the magnetization in the material,
although the dynamics are obtained at a specific momentum.
In time-resolved ARPES, the dynamic electron temperature can be directly extracted by analyzing the photoemission intensity distribution above EF . In Fig. 4.3A, we plot the photoemission
intensity from above EF at different time delays, for a fluence of 6 mJ/cm2 . Before the pump pulse
arrives (td = 500 fs), the photoemission intensity corresponds to a Fermi-Dirac distribution across
EF at room temperature. The electron temperature reaches its maximum value at 24 fs after the

49

Figure 4.2: Magnetization dynamics in Ni. (A) Change of the TMOKE asymmetry and exchange
splitting reduction ∆Eex as a function of time delay for different laser fluences. The solid lines
represent fitting results, from which we extract the three characteristic times for demagnetization(τdemag ), fast recovery (τrecover1 ), and slow recovery (τrecover2 ) (see chapter 5 for more on this). The
fit to TMOKE (upper panel) and ARPES (lower panel and lower fluence) yields the same fluenceindependent time constants. a.u., arbitrary units. (B) Typical TMOKE asymmetry before (td =360 fs) and after (td = 500 fs) excitation with a pump fluence F = 6 mJ/cm2 .(C) Photoelectron
spectra of Ni(111) along the Γ - K direction before (td = 500 fs) and after (td = 500 fs) laser
excitation, showing the collapse in the exchange splitting Eex after excitation (blue dashed lines).
The dashed-dotted lines represent the momentum at which photoemission intensities are extracted.
The photoemission intensities are plotted in the right panel with Eex extracted from a Voigt function
fit to the data (dashed lines; see appendix A). (D) Constant, fluence-independent demagnetization
time observed for different laser fluences for both ARPES and TMOKE.

50
peak of the pump pulse and then rapidly decreases due to cooling to the lattice. By 2 ps after
excitation, the electron temperature is close to room temperature, as evidenced by the fact thatthe slope of photoemission intensity as a function of energy is very close to that obtained in the
ground state (td = 500 fs) (Fig. 4.3A). The electron temperature can be reliably extracted byfitting the photoemission intensity with the Fermi-Dirac function convolved with the experimental
energy resolution (see the appendix A). The time evolution of the electron temperature after laser
excitation is plotted in Fig. 4.3 (B and C). These results are further corroborated by the dynamics
of electron population at 0.2 eV (Fig. 4.3B) and the transient EUV reflectivity measurements
performed at similar laser fluences (Fig. 4.3C). These measurements probe the charge dynamics
around EF averaged over the entire Brillouin zone, by exciting the 3p core level electrons of Ni toelectronic states around EF (see inset of Fig. 4.3C). The agreement between the transient electron
temperatures extracted from both ARPES and EUV reflectivity measurements suggests that themeasured electron dynamics across EF are uniform throughout k-space. The short electron thermalization time in Ni is not surprising if we consider that the lifetime of photoexcited electrons at
1.6 eV above EF is extremely short (1 fs) [68], indicating very strong electron-electron interactions(see appendix A) [20].
In Fig. 4.4A, we plot the maximum electron temperature extracted around 20 fs, which isthe characteristic time scale for the rise of the electron temperature after excitation, as a functionof the laser pump fluence. Here, we observe the first critical behavior: The increase of the electron
temperature is strongly suppressed around the critical fluence Fc ≈ 2.8 mJ/cm2 , indicating that asignificant amount of energy is transferred into the spin system within 20 fs, preventing a further
increase in electron temperature. The critical behavior of the peak electron temperature (Fig.
4.4A) can be explained by the divergence of the heat capacity of the strongly coupled electron andspin systems in Ni.
To a first-order approximation, the maximum temperature that the electrons can reach (Tmax )51

Figure 4.3: Ultrafast charge dynamics in Ni. (A) Log plots of the photoemission intensity above EF
for F 6 mJ/cm2 and at different td , integrated from k ≈ 0.85Å−1 to k ≈ 1.3Å−1 in the momentum
space. The dashed lines represent the fitting of the photoemission intensities with the Fermi-Dirac
distribution convolved with experimental energy resolution (see appendix A). Inset: Integrated
photoemission intensity as a function of pump-probe time delay. The yellow dashed box illustrates
the integration region of electron population in (B). (B) Dynamics of the electron temperature andthe relative electron population (n/n0 ) within 0.2 eV above EF as a function of td . The electron
population is normalized to the band electron population (n0 ) 0.2 eV below EF (see appendix A).
(C) Comparison of the electron temperature [red dashed line, same as (B)] and the change of EUV
transient reflectivity at a similar pump fluence. Inset: EUV transient reflectivity measurement.
The resonant EUV light (65 eV) directly probes the charge dynamics around EF induced by thelaser pump pulse. This measurement is averaged over k-space

52

Figure 4.4: Observation of multiple critical behaviors during ultrafast demagnetization in Ni. (A)Peak electron temperature extracted 24 fs after excitation as a function of pump fluence. The
open symbols represent the electron temperature extracted at different k|| using Tr-ARPES. The
solid red line is the fit using Eq. 4.1 considering the transient electron and magnetic heat capacity[inset of (B)], whereas the green dashed line considers only the contribution from transient electron
heat capacity (see appendix A). The yellow-colored region (∆FS ) is the energy transferred to thespin system within 20 fs. (B) Change in the exchange splitting at 2 ps as a function of pumpfluence. The red line represents a fit with an error function. The same critical fluence of Fc ≈2.8 mJ/cm2 is observed for the exchange splitting collapse and the peak electron temperature in(A). The transient electron heat capacity is plotted in the inset. (C) Peak electron temperature
calculated using Eq. 4.1 and (Ce + Cm ) Transient [inset of (B)] for the sample temperatures of
300 and 100 K. The red solid line is the same as in (A). (D) Change of exchange splitting at 2 ps
as a function of laser fluence at different sample temperatures. The solid lines represent the error
function fit of the experimental results. The dashed lines align the critical fluences observed in (C)and (D) for different sample temperatures.

53
at the sample surface with a given pump fluence (F) can be calculated as:
F (1 − R)=δZ TB[Ce (T ) + Cm (T )]dT

(4.1)

Temax

where R is the reflectivity, δ is the penetration depth, TB is the ground state sample temperature,
and Ce and Cm are the heat capacities of the electron and spin systems of Ni, respectively. It is wellknown that the magnetic heat capacity Cm diverges when the temperature approaches the Curie
temperature to fulfill the energy required for the ferromagnetic-to-paramagnetic phase transitions
under thermal equilibrium conditions [1, 2]. Here, we model Ce (T ) + Cm (T ) using a power-law
function and fit the measured peak electron temperature as a function of pump fluence to Eq. 4.1(see appendix A). The fitting results are shown as the solid line in Fig. 4.4A, which essentially
captures the critical behavior of the peak electron temperature observed in our experiments. The
parameters for the optimum fitting are listed in table 4.1, whereas the corresponding transientheat capacity is plotted in the inset of Fig. 4.4B. We find that the divergence in the heat capacity
around the critical temperature (Tc ) can quantitatively explain the critical behavior observed in
the electron temperature. Tc obtained for the ultrafast transient phase transition is very close to
the Curie temperature (see table 4.1), which further corroborates that the critical phenomena weobserved are related to the intrinsic magnetic properties of Ni. The peak electron temperatureswith the contribution of electron heat capacity alone are plotted as the dashed line in Fig. 4.4A
(see section 4.4.3 and appendix A). We estimate that there is an energy of ∆Es = ∆Fs (1−R)≈ 105
δ
meV per unit cell transferred to the spin bath within 20 fs. This energy is enough to go throughthe magnetic phase transition because it is even higher than the energy required under thermal
equilibrium conditions (68 meV per unit cell) [2].
As noted above, the slow recovery of the magnetization is only observed when the pumpfluence is higher than the same critical fluence Fc ≈ 2.8mJ/cm2 , which represents the second criticalbehavior. This can be illustrated if we plot the exchange-splitting collapse ∆Eex at td = 2 ps as afunction of pump laser fluence, as shown in Fig. 4.4B. We note that the observed critical fluence
here is consistent with that shown in Fig. 4.4A for the electron temperature, corroborating that

Eq.
TransientA
0.775±0.05
0.775

A’
2.46±0.05
2.46

β-0.0718±0.01-0.0718B
13.5±1.812.6±3.4

B’
37.0±5.542.0±3.7TC634±1
646±12854γ3.03±0.1
7.69±3.7

Table 4.1: Fitting parameters for the electron and magnetic heat capacity under thermal equilibrium[2] and in the transient state. All units in (J/mol per K]) unless stated otherwise.

the observed critical behavior of the transient electron temperature is related to the laser- induced
magnetic phase transition in the spin-electron system. Considering that the electron and lattice
temperatures are both less than the Curie temperature of Ni at td = 2 ps (see below), our results
indicate that a transient paramagnetic-like state exists when the pump fluence is above Fc , which
recovers through a path distinctly different from the fast recovery for lower pump laser fluences.
We note that the different time scales observed in many magnetic spectroscopy experiments to date[21, 14] can be explained by depth averaging, where parts of the material near the surface undergo
a transient magnetic phase transition with slow recovery dynamics, whereas layers deeper within
the material exhibit faster recovery dynamics because there the effective fluence is lower. This was
confirmed by comparing an extensive set of fluence-dependent time-resolved ARPES and TMOKE
data that show that the same critical behavior can explain the entire data set [69].
To further investigate the driving mechanisms for the observed critical phenomena during
ultrafast laser-induced demagnetization, we repeated the same measurements for a sample temperature of 100 K using Tr-ARPES. Figure 4.4D plots the change of exchange splitting at a timedelay of 2 ps as a function of pump fluence for sample temperatures of 300 and 100 K. The peakelectron temperatures reached for those same sample temperatures are calculated using Eq. 4.1
and are shown in Fig. 4.4C. For a sample temperature of 100 K, the required critical fluenceincreases by ∆FT =0.58mJ/cm2 compared to room temperature (Fig. 4.4D). As shown in Fig.
4.4C, this offset can be precisely captured by Eq. 4.1 using the same transient heat capacity (inset
of Fig. 4.4B), thus further validating the extracted value of this transient heat capacity. This
result strongly suggests the following criterion for critical behavior in ultrafast demagnetization of
Ni: whether or not the transient electron temperature exceeds the Curie temperature. Hence, in55
our work, the critical fluence corresponds to the fluence required to drive the transient electron
temperature above the Curie temperature. We believe that the fact that the critical temperatureobserved in ultrafast demagnetization is very close to the Curie temperature under thermal equilibrium cannot be a simple coincidence but underscores the importance of the connection between
the nonequilibrium physics and its equilibrium counterpart. Similar ideas were recently exploredby studies on the ultrafast spin-density-wave transition in Cr [70]. However, in that case, the spindensity-wave phase transition occurs immediately when the transient electron temperature reaches
the critical point. In contrast, in our work, critical spin excitations occur within 20 fs, and theferromagnetic-to-paramagnetic phase transition then happens on a longer time scale that is characteristic of the material. We believe that these mechanisms are broadly applicable to many different
materials that undergo phase transitions at a critical temperature, for example, superconductors
and charge-density-wave materials.

4.3

Discussion
Our results also shed light on the microscopic time evolution of electron spins during ultrafast

demagnetization in Ni. The presence of critical behaviors in the electron temperature and heat
capacity shows that, within 20 fs after excitation, the spin system in Ni has already absorbed
sufficient energy to go through the ferromagnetic-to-paramagnetic phase transition, without yet
exhibiting significant demagnetization. This time scale is much shorter than the time scale ofdemagnetization (TMOKE) and collapse of the exchange splitting (ARPES) (176 fs). The verylarge difference between these two time scales indicates the importance of a highly nonequilibriumtransient state before the sample reaches maximum demagnetization. The initial transferred energy
must be stored in the spin system in the form of high-energy spin excitations, leaving the spin system
in a nonequilibrium state. Then, this energy likely decays into low-energy magnons over time,leading to demagnetization of the sample in 176 fs. In this picture, the time scale and critical fluenceof the demagnetization is intrinsic to the material itself and is determined by the Curie temperature,heat capacity, and exchange energy. This is strongly supported by the fact that the time constant56
of demagnetization in Ni (176 fs) is independent of pump laser fluence, as shown by both the TrARPES and Tr-TMOKE measurements (Fig. 4.2D). It has been suggested both experimentallyand theoretically that the emission of low-energy magnons on a time scale oftens of femtoseconds
could have significant contributions to ultrafast demagnetization in ferromagnetic materials [11,30, 26, 31, 18]. Finally, we note that our results cannot be explained by the models based on spinflip scattering, because, in those models, the demagnetization of the material occurring in several
hundred femtoseconds is driven by the gradual transfer of the energy from hot electrons to electron
spins via spin-orbit and electron-phonon interactions [21, 24, 23, 14]. Moreover, these mechanisms
should give rise to a fluence-dependent demagnetization and recovery times.
Now, an important question remains what is the mechanism that could lead to high-energy
spin excitations during the first 20 fs? On this time scale, we believe that superdiffusive spin currents
could be an important candidate, considering the significant difference between the lifetimes of the
majority and minority electrons at 0.2 eV above EF [25, 53], as well as the extremely short time scaleof spin transport (1fs over 1nm distance). However, it has also been shown both experimentally[10, 71] and theoretically [28, 65, 29] that the spin mixing via spin-orbit coupling can happen in avery short time, which might also contribute.
Furthermore, another interesting finding is that, as evidenced by the critical behavior of∆Eex in Fig. 4.4B, because the spin system recovers to the ground-state magnetization, twodifferent recovery regimes exist, depending on the pump fluence. We note that this is the firsttime that such a critical behavior in the ultrafast decay and recovery time scales has been observedin ferromagnetic metals such as Ni. As noted above, the demagnetization and exchange-splitting
collapse times are fluence-independent at 176 fs. However, when the pump fluence is lower than Fc ,we find that the magnetization of the sample undergoes a fast recovery to the ferromagnetic phasewith a time constant of 500 fs (Fig. 4.2A). This fast recovery might be explained by damping of
magnons (relaxation of the precessing magnetic moment into its equilibrium configuration) underhthe exchange field (Hex ). The typical damping time is given by τdamp = gµB µ0|H
using theex |αLandau-Lifshiz-Gilbert equation [72, 8], where µB is the Bohr magneton, g ≈ 2 is the Landé factor,

57
and α is the damping factor. Considering that α = 0.065 [73, 74]and | Hex | = 939 T for Ni [75], we
find tdamp ≈ 580 fs, which is in quantitative agreement with τrecover1 observed in our experiments.
On the other hand, when the pump fluence is above Fc , the sample evolves into a transient magnetic
state with low magnetization and Hex is melted (see Fig. 4.1A). As a result, the magnetizationmust recover by coupling to phonons and the lattice, which occurs over much longer times.
In conclusion, by investigating ultrafast laser-induced demagnetization in Ni using fluenceand temperature-dependent ARPES, TMOKE, and EUV transient reflectivity measurements, weunambiguously show that critical phenomena govern the ultrafast demagnetization response. We
find that although demagnetization and the collapse of the exchange splitting occur on the same
fluence-independent time scale of 176 fs, sufficient energy for the transient magnetic phase transition
has been transferred to the spin system already within 20 fs. Our results suggest the existence ofa high-energy nonequilibrium transient magnetic state and that the transient electron temperaturealone is responsible for crossing the spin electronic phase transition.

4.4

Materials and Methods

4.4.1

Experimental setups

In the ARPES experiment, photoelectrons from the Ni(111) surface are mapped using 16 eV
HHG probe pulses at near-normal incidence (5°) and collected by a momentum-resolved hemispherical analyzer. The output of a Ti:Sapphire laser amplifier system (KM Labs Dragon) is frequency
doubled to 390 nm and used to generate high harmonics with 160 meV energy resolution and wellseparated (by 6 eV) harmonic orders that allow us to use the fifth harmonic (16 eV) directly [76].
For the TMOKE and EUV transient reflectivity experiments, the harmonics are driven directlywith the 780-nm laser. Light near HHG orders 43 and 45, at photon energies of 65 and 68 eV, isused to probe the 3p absorption edge of Ni. For this purpose, the HHG probe beam is reflectedfrom the sample at an angle of 48° from normal, spectrally dispersed by a grating, and recordedwith a charge-coupled device (CCD) camera [52, 53, 18]. The sample is fully magnetized in-plane58
by an electromagnet placed in the transverse geometry. These experimental setups are illustrated
in figure 4.5.

4.4.2

Data analysis

In TMOKE, the sample magnetization is characterized by the change in reflected EUV intensity at the 3p absorption edge for opposite orientations of the initial in-plane magnetization vector.
The TMOKE asymmetry (As ) is then calculated fromAs =I+ − I−
I+ + I−(4.2)

where I+ and I are the reflected EUV intensities for two magnetization directions [52, 13, 12].
Details of data analysis of ARPES experiments are presented in detail in appendix A. The dynamics
of TMOKE asymmetry (As ) and change of the exchange splitting (∆Eex ) can both be fit to




Y0
Y (td ) =
(4.3)



Y0 + ae−td τdemag − be−td τrecover1 − (a − b)e−td τrecover2
where Y represents either ∆Eex or As , Y0 is the ground-state signal, and a and b are the amplitudesof the collapse and the first fast recovery of the signal, respectively. τdemag , τrecover1 ,and τrecover2
are the time constants of the collapse, fast recovery, and slow recovery of the signal, respectively.

4.4.3

Model of electronic and magnetic heat capacity

The electronic and magnetic heat capacity Ce (T ) + Cm (T ) can be modeled with the following
power-law function as it passes through the critical temperature (Tc ) [77]Ce (t) + Cm (t) =A −β|t | + B + γt
β

(4.4)

c
where t = T −T
is the reduced temperature for T > Tc . We used the same function with primed
Tcparameters (A’, B’) for T < Tc . The parameters β, γ, and Tc are kept identical on both sides of thephase transition. The electronic and magnetic heat capacities under thermal equilibrium conditions[2] can be fit to Eq. 4.4, allowing us to extract parameters A, A’, and β, which determine the critical

59

Figure 4.5: Experimental Setup. (A) Experimental setup of Tr-ARPES experiment. (B) Side band
intensity as a function of pump-probe time delay in laser assisted photoemission measurement for
Tr-ARPES (Further details in appendix A). The solid line is the fitting result with the Gaussianfunction with a FWHM of ≈42 fs. (C) Experimental setup of Tr-TMOKE experiment.

60
behavior of the heat capacities around Tc . The contribution of the electron heat capacity alone canbe estimated under the free-electron gas approximation Ce = γ’T (see appendix A) and gives riseto the modeled peak electron temperature using Eq. 4.1, as shown by the dashed line in Fig. 4.4A.
We also noted that in Eq. 4.1, we neglected electron-phonon coupling and heat diffusion; however,these effects have negligible influence on our results in the first 20 fs, as shown in appendix A.

4.4.4

Statistical analysis

In both Tr-ARPES and Tr-TMOKE experiments, the data were collected with multiple pumpprobe delay cycles to achieve a sufficient signal-to-noise ratio. The exchange splitting dynamics wereanalyzed by fitting the extracted photoemission spectra with multiple Voigt functions and definedas the energy difference between peaks of electrons with majority and minority spins (see appendix
A for more details). The error bars in Figs. 4.2A and 4.4 (B and D) were due to uncertainties in thefitting procedure. The transient electron temperature was obtained by fitting the photoemission
intensity above EF with Fermi-Dirac distribution, and the error bars in Figs. 4.3B and 4.4A were
due to uncertainties in the fitting procedure, as well as the energy resolution (see appendix A).
The errors on the pump fluence were due to fluctuations of pump power and laser spot size. The
error bars of Tr-TMOKE data in Fig. 4.2 (A and D) were defined by the standard deviation ofexperimental data before pump excitation.

4.4.5

Details of experimental setups

The detailed experimental setups of Tr-ARPES and Tr-TMOKE is shown in figure 4.5. Further experimental details of the ARPES measurement are given in appendix A. The details of the
Tr-TMOKE experiment are given in chapter 3.
In both experiments, the spot sizes of pump are measured using the knife-edge method in-situ
at the sample location. The pump fluence in this work is defined as F = πD4Q
, where Q thex Dyis the pump-pulse energy and Dx/y is the FWHM of the pump spot along two orthogonal axes.
In order to compare the fluence-dependent results between two experiments, the difference in the61reflectivity at different incident angles has to be taken into consideration. According to the opticalconstants of Ni [78], the reflectivity R at normal incidence is ≈0.74, which changes to ≈0.64 for thep-polarized pump at an incident angle of 48 degrees measured with respect to the normal. Thisvalue is also confirmed by reflectivity measurement on site. The fluence reported in the main text
for Tr-TMOKE results are scaled considering this difference to preserve the same absorbed fluenceas that in Tr-ARPES measurements. In both experiments, the time resolution is estimated to be≈10 fs, considering the fastest changes that can be observed from the data.

4.4.6

Method for extracting TMOKE asymmetry dynamics from HHG spectra

In the Tr-TMOKE experiment, to extract the asymmetry dynamics at the nickel edge, we use
a spectrometer as shown in Fig. 4.5C. We take an average of the signal over an energy range of thespectrometer approximately equal to 1.2 eV for each harmonic, although we note that this widthis also convolved with the spot size of the HHG beam after reflection off the grating. We select theharmonics corresponding to the 43rd and 45th harmonic of the fundamental laser (corresponding
to energies of 65 and 68 eV) in the data analysis procedure. Finally, we normalize the data before
the pump pulse arrives by taking the average value of the static asymmetry. We use the standard
deviation in values for static asymmetry measured before the sample is pumped to determine theerror bars for the TMOKE measurement.

4.4.7

Momentum dependence of TMOKE Measurements

In order to investigate the momentum dependence of the Tr-TMOKE measurements, we took
several different time-resolved asymmetry measurements with different crystal lattice orientations
relative to the applied field shown as M, with an incident pump fluence of 2.8 mJ/cm2 . These
measurements are shown in Fig. 4.6. These results clearly show the momentum-averaged natureof the Tr-TMOKE measurement.

62

Figure 4.6: Momentum dependence of TMOKE measurements. The TMOKE asymmetry dynamics
for different crystal orientations. Inset: Illustrations of the relative orientations of the crystal cut
relative to the directions of the magnetic field and light polarizations in the experimental geometry.

63
4.4.8

Transient electron and magnetic heat capacity

The electron and magnetic heat capacity is modeled using Eq. 4.4 with six parameters(A, A’, B, B’, β, γ and Tc ). In order to reduce the number of free fitting parameters, we first
determine the values of A, A’ and β by fitting the electron and magnetic heat capacity underthermal equilibrium [2] to Eq. 4.4. The fitting results are plotted in Fig. 4.7A. By doing so,
we assume the order parameters for the critical behavior of heat capacities are preserved in laserinduced phase transition. At the same time, the fact that the total heat capacity at T=0 is zero0

places an additional confinement on the parameters: Aβ + B’γ=0. As a result, we have in total
three free fitting parameters when we fit the experimentally measured Te (Fig. 4.4) to Eqs. 4.3 and4.4. The critical behavior of the nickel heat capacity represents the energy transfer into the spinbath when the system crosses the ferromagnetic-to-paramagnetic phase transition. In transient(Ce + Cm ), the contributions of the electron heat capacity can be estimated under the free-electrongas approximation: Ce = γ 0 T , as shown in Fig. 4.7B. We note that when T < Tc , γ’ is close to thevalue for thermal equilibrium (25). However, we find γ’ is about 3 times larger than its thermal
equilibrium counterpart when T > Tc , the reason of which requires further investigations. Using thetransient Ce and Eq.4.3 in the main text, we can estimate the peak electron temperature without
spin contributions, which is shown as the pink dashed line in Fig. 4.4A. The offset in fluence of thiscurve to the experimentally measured peak electron temperature represents the amount of pumpenergy (∆FS ) transferred to the electron spin bath in ≈20 fs.

4.4.9

Effects of electron-phonon coupling and heat diffusion on Eq. 4.1

In Eq. 4.1, we ignored the heat diffusion and electron-phonon coupling that could reduce
the electron temperature at the surface measured using Tr-ARPES method. These two effects
can be explicitly considered using two-temperature model (TTM) [3]. The time (t)- and space

64

Figure 4.7: Electron and spin heat capacity. (A) The electron and magnetic heat capacity underthermal equilibrium [2]. The solid line is the fitting result to Eq. 4.4. (B) The transient electron
and magnetic heat capacity extracted from our experimental results. The green line represents thecontribution of the electron bath to the total heat capacity. The dashed line is the electron heat
capacity under thermal equilibrium for comparison.

τe (fs)10

κe0 (J m−1 s−1 K−1 )91

Cl ( 106 J m−3 s−1 K−1 )
4.1

65δ (nm)
13

tp (fs)28

Table 4.2: Material parameters used in TTM simulation [3, 4].

(z)-dependent evolution of the electron (Te ) and lattice temperatures (Tl ) is described by
Cem −

∂Te
∂
= − Qe − G(Te − Tl ) + S(z, t)
∂t
∂z
∂Qe
∂
τe
+ Qe = κe Te
∂t
∂z
∂Tl
Cl
= G(Te − Tl )
∂t

(4.5)
(4.6)
(4.7)

where Qe is the heat flux of electrons, G the electron-phonon (e-ph) coupling constant, τe theelectron energy relaxation time, κe = κe0 TTel the electron thermal conductivity, Cl the lattice heat
capacity and S(z,t) the optical excitation. In Eq. 4.7, the heat diffusion through lattice is neglected,
because the lattice thermal conductivity (κl ) is much smaller than κe . The values of τ3 , κe0 and Cl
of Ni can be determined by literature [4] as well as material parameters, and are listed in table 4.2.
We note that τe varies from ≈20 fs to 1 fs as the electron energy rises from 0.2 eV to 1.5
eV above EF [68]. This value is shorter than the pump pulse duration (≈28 fs). As a result, thedynamics of electron and lattice temperature have very weak dependence on the choice of τe . For
convenience, we use τe = 10 fs in all the simulations. In Eq. 4.7, the pump laser pulse is modeledasr
S(z, t) =" 2 #z
4 log 2 (1 − R)Ft
× exp − − 4 log 2
π
tp δ
δ
tp

(4.8)

where F is the pump fluence, R the reflectivity, δ the IR-light penetration depth and tp the pumppulse FWHM width. The values of δ and tp are also listed in Table 4.2. The heat capacity Cem
in Eq. 4.7 is defined as the total heat capacity of the electron and spin systems (Ce + Cm ). In
the simulation, we use the temperature-dependent transient (Ce + Cm ) as plotted in the inset
of Fig. 4.4B. This choice is validated by the fact that the electron and spin systems are foundto be strongly coupled in short time after pump excitation, evidenced by the critical behavior of
the electron temperature at ≈24 fs as shown in Fig. 4.4A. The coupled differential equations in

66
Eqs. 4.7 and 4.8 are numerically solved using MacCormack method. In Fig. 4.8A, the simulation
results of Te averaged over 1 nm under the sample surface are plotted in direct comparison with theelectron temperature measured using Tr-ARPES in our experiments. We find very good agreement
can be achieved for a range of pump fluence when a proper temperature dependent e-ph coupling
constant G is implemented. The value we used here is a factor of 3 stronger than values reported
in Ref. [79]. Very interestingly, we find the TTM simulations results show that there exists a kinkin the recovery dynamics of the electron temperature, whenever the electron temperature reduces
to the critical temperature. This reproduces our experimental data very well, which in the time
domain strongly supports that the transient electron and magnetic heat capacity influences thelaser-induced dynamics. On the other hand, in order to account for the dynamics of the EUV
reflectivity, the electron temperature averaged across the EUV probing depth (≈10 nm) has to betaken into consideration, as shown in Fig. 4.8B.
The calculations using TTM also allow us to investigate the contributions from heat diffusionand electron-phonon coupling to the model in Eq. 4.1. In Fig. 4.8C, we plot the Te obtained from
TTM simulation for 1) thermally isolated, same as Eq. 4.1, by setting κe = 0 and G = 0, 2) heatdiffusion only, by setting G=0 and κe 6= 0 and 3) heat diffusion and e-ph coupling with G 6= 0
and κe 6= 0. From the results, we can see that the surface electron temperature could be reducedby less than 100 K within 20 fs through heat diffusion when the sample is pumped by a relatively
high fluence (5.8 mJ/cm2 ). This value is expected to be much smaller at lower pump fluences.
In contract, the e-ph coupling has negligible effects on electron temperature in 20 fs, because thecharacteristic time of e-ph coupling is on a time scale of ≈100 fs. At the same time, we can also
investigate whether the convolution with the time-resolution of our measurement could affect theelectron temperature measurements (dashed line in Fig. 4.8C). We find the time resolution of ourexperiment is sufficient to precisely capture the electron temperature in very short time scales.

67

Figure 4.8: Two-temperature model. (A) The electron temperature measured using Tr-ARPES incomparison with the simulation results from TTM for the top 1nm thick layer of Ni with different
pump fluence. (B) The transient EUV reflectivity results in comparison with the TTM simulationconsidering the average over a EUV probing depth of ≈10 nm. (C) The electron temperature
simulated by TTM considering the situations: 1) thermally isolated, same as Eq. 4.1; 2) only
heat diffusion; 3) heat diffusion + e-ph coupling. The dashed line represents the simulation results
convolved with the experimental time resolution.

Chapter 5

Universal Nature of the Magnetic Phase Transition in Nickel revealed by
Correllative Spectroscopies

In this chapter, we further investigate the ultrafast magnetic phase transition in Ni by quantitatively comparing measurements taken with time-resolved transverse- magneto- optical Kerr effect(Tr-TMOKE) and time- and angle- resolved photoemission (Tr-ARPES) spectroscopies based onhigh harmonic generation. Using the critical behavior and the timescales of demagnetization andrecovery processes observed from Tr-ARPES, and by taking the depth- dependent signal contributions in Tr-TMOKE into account, we show that critical phenomena are also key for the correctinterpretation and a full understanding of ultrafast optical or x-ray magnetic spectroscopies. With
this knowledge, we can now fully explain the Tr-TMOKE response of Ni over the full range oflaser fluences, using only three universal timescales to describe the demagnetization and recovery
dynamics in distinct physical regions. Although the spin system absorbs all the energy required toproceed through a magnetic phase transition within 20 fs, the spectroscopic signatures of demagnetization take 176 fs to develop. Moreover, these timescales are fluence independent. In contrast,
the speed of remagnetization dynamics depends on whether the applied laser fluence is below or
above the critical fluence. Our data show that the demagnetization amplitudes scale linearly withpump fluence. Finally, we observe a competition between the fast and slow recovery channels with
distinct timescales, suggesting a potential coexistence of ferromagnetic and paramagnetic phases
during the phase transition.
A schematic of the physics studied in the experiment is shown in Fig. 5.1. The sample

69

Figure 5.1: (a) Schematic of EUV ARPES and TMOKE measurements on Ni(111). The fluence
profile of the laser excitation below the sample surface separates the magnetization response into
two different regions (i) and (ii), depending on whether the in-situ fluence is above the criticalfluence Fc . Using Tr-ARPES, the probed depth is on the order of a monolayer, while Tr-TMOKE
probes the entire laser-heated depth of ≈10 nm. (b) Schematic of the excitation present in the laserinduced phase transition in Ni when critical phenomena are taken into consideration [5]. When thelaser fluence exceeds the critical fluence Fc , the electron temperature exceeds Tc and the sample
rapidly undergoes a magnetic phase transition, as evidenced by multiple critical phenomena.

70
used in our experiments was a 400 nm Ni(111) single-crystal film. We intentionally chose a thickfilm sample to minimize nonlocal effects due to interfaces or poor substrate thermal conduction[80, 53] and also verified that the observed dynamics were not dependent on the orientation of
the sample (see Fig. [?]). In both the Tr-TMOKE and TR-ARPES experiments, the sample was
excited by 45 fs pulses from a Ti:sapphire laser amplifier system at a wavelength of 800 nm. In
the Tr-TMOKE measurements, the subsequent change of the sample magnetization was probed
by extreme ultraviolet (EUV) pulses produced by high harmonic generation (HHG). The sample
magnetization can be quantitatively determined by recording the asymmetry of the reflected HHG
spectrum at the 3p edge of Ni [12, 13, 18]. In the Tr-ARPES measurements, the magnetization
dynamics was probed by monitoring the magnitude of the exchange splitting at different time delays[15, 9, 5].
In order to determine if Tr-TMOKE and Tr-ARPES give spectroscopic signatures that areconsistent with the same microscopic physics and interactions, we measured the de- and remagnetization dynamics in Ni excited by a wide range of fluences, with the highest fluence sufficient tofully suppress the Tr-TMOKE asymmetry (i.e., demagnetize the sample). The pump penetration
depth in Ni is δL ≈ 13 nm [78], which is comparable to the probing depth of the EUV light usedin the Tr-TMOKE experiments (10 nm). In contrast, the probing depth of photoelectrons is close
to a monolayer for the photon energy (16 eV) used in the TR-ARPES experiments [81], which suggests that the Tr-ARPES signal can probe the elementary magnetization dynamics in an individual
surface layer of the sample. In Fig. 5.1(b), we conceptually summarize the electron, spin, andmagnetization dynamics after laser excitation, with the critical behavior taken into consideration[5].

Y (td ) =1 (td < 0)

(5.1)




1 + a1 (z)e−td τdemag − a2 (z)e−td τrecover1 − a3 (z)e−td τrecover2 (td >)In Fig. 5.2, we plot the change of the exchange splitting (∆Eex ) at the transverse momentum
k ≈ 1.05 Å1 along the Γ - K direction of Ni (inset of Fig. 5.2) observed in the TR-ARPES

71

Figure 5.2: Change in the exchange splitting (∆Eex ) in Ni measured using TR ARPES, for theabsorbed laser fluence below (0.21 mJ = cm2 , grey) and above (1.7 mJ = cm2 , red) the criticalfluence Fc . The solid lines are the fits to Eq. 5.1. Inset: Static ARPES spectrum plot along the ΓK direction recorded using He Iα photons.

72
measurements [80]. Due to <1 nm probing depth, Tr-ARPES probes the elementary magnetization
dynamics in a monolayer of the material, which can be well described by an exponential decay andbiexponential recovery function as shown in Fig. 5.2:
Here we obtain three time constants that correspond to the following physical processes: thecollapse of the exchange splitting τdemag = 176 ± 27 fs, a fast recovery time τrecover1 = 537 ±173 fs, and a slow recovery time. τrecover2 = 76 ± 15 ps [5]. In Eq. 5.1, a1 , a2 , and a3 are theamplitudes. These amplitudes are independent since the magnetization will processes, with a1 = a2
+ a3 . Note that only two of the recover fully at long times. Their values depend on the strength ofthe laser fluence and, hence, are depth dependent due to the profile of the optical pump below thesample surface [Fig. 5.1(a)]. From the ARPES results, we map the dynamics in monolayers of thematerialwe can now test whether this understanding can fully explain the TR- TMOKE results.
The magnetization dynamics in the same sample excited by fs laser irradiation were alsomeasured using TR TMOKE. In the inset of Fig. 5.3, we present the bulk-averaged < A2 >, and< A3 >) as a function of pump fluences, by fitting averaged amplitudes of de- and remagnetization(< A1 >, < A2 >, < A3 >) as a function of pump fluences, by fitting the Tr-TMOKE resultspresented in Fig. 5.3 with the same exponential decay and bi-exponential recovery function. Herethe amplitudes represent the change of the sample magnetization normalized to the magnetizationof the ground state. Even from the raw data traces, the presence of three timescales is very apparent:a fast and universal demagnetization time followed by fast and slow recoveries depending on the
fluence. From these results, we find that the slow-recovery process (< A3 >) only turns on when
the absorbed laser fluence (Fabs ) is above the critical fluence (Fc ≈ 0.59 mJ/cm2 ), which highlights
the importance of the critical behavior to the interpretation of the Tr-TMOKE results (Note thatin [30] we quoted the incident fluence of 2.8 mJ/cm2 , which is consistent with an absorbed fluenceof 0.59 mJ/m2 within error bars). Moreover, a linear response of the slow-recovery amplitude canbe clearly observed, as highlighted in the inset of Fig. 5.3.
Under the assumption of linear absorption, in both experiments the in-situ laser fluence F
decays exponentially with the depth z i.e. F(z) = F0 ez/δL , where F0 is the fluence at the surface.

73

Figure 5.3: Magnetization dynamics in Ni measured using Tr-TMOKE over a full range of laser
fluences. The highest fluence is sufficient to fully suppress the sample magnetization. The data areoffset for clarity. Red curves: Fits to our microscopic model which considers the critical behavior,as well as the depth-average effects in the Tr-TMOKE measurements. Inset: Fluence-dependent
amplitudes of the demagnetization and recovery processes directly extracted from the Tr-TMOKE
results. In the Tr-TMOKE results, the magnetization < M > and the extracted amplitudes < A1 >,< A2 >, and < A3 > are averaged over the entire probed depth. The dashed yellow line highlights
the linear relation of the amplitude < A3 > to the absorbed fluence when the fluence is above thecritical fluence.

74
To take into account the true absorption at different depths, the heat source q can be calculated by
q(z)=F(z)/δL . When F0 > Fc , the Tr-TMOKE signal arises from different regions, each exhibiting
different recovery dynamics depending on whether the laser excitation is above or below the criticalfluence (Fig. 5.1a). In Region (i) where the in-situ fluence is always lower than the critical fluence,and the sample re-magnetizes through the fast recovery channel. In contrast, in Region (ii), the insitu fluence is above Fc , and re-magnetization occurs through both slow and fast recovery channels.
Here, we further assume that the change of magnetization is a linear function of the in-situ fluence,
which is strongly supported by our experimental results (inset of Fig. 5.3) and previous work [82].
Given this linear relation, we havea1 (z) = min[b1 F (z), 1]and
a3 (z) =0 [F (z) < Fc ]

(5.2)

(5.3)




min [b3 [F (z) − Fc ]] ; [F (z) ≥ Fc ]where b1 and b3 are the proportionality constants. The Tr-TMOKE signals can be modeled
as the bulk-averaged magnetization < M >, given by the integral of the unit magnetization m(td ,z) over the probed depth z:
R∞< M > (td ) =0m(td , z)W (z)dz
R inf
0 W (z)dz(5.4)

Here, W(z) is the depth sensitivity function of TMOKE [83], which is explicitly calculated for Ni.
Using the model described above, we now fit the TR-TMOKE results for the different fluences
shown in Fig. 5.3 to Eqs. 5.1 through 5.4, taking b1 , b3 , and Fc as the fitting parameters. We
use the characteristic times obtained from the TR-ARPES measurements as the time constants in
Eq. 5.1. As shown in Fig. 5.4, there is excellent agreement between the model (solid lines) andexperimental data (symbols) over the full range of pump fluences, even though the limited numberof fitting parameters places a strong constraint on our fitting. We note that the extracted value of
Fc is in good agreement with values obtained from the Tr-ARPES experiments [5], which further

b1 (cm2 /mJ)
0.65/pm0.01

b3 (cm2 /mJ)
0.31/pm0.01

Fc (mJ/cm2 )0.59/pm0.05

75

Table 5.1: Optimum fitting parameters of the TR-TMOKE results in Fig. 5.3 to the model,
consisting of Eqs. 5.1-5.4.

validates our model. From these results, we find that the apparent presence of a fluence-dependent
remagnetization time is a direct result of the bulk-averaged signal in TR-TMOKE: the surface
layers of the material undergo a phase transition and exhibit slow recovery dynamics, while layers
deeper within the material do not undergo a magnetic phase transition and as a result, exhibit
only fast recovery dynamics. We note that similar fluence-dependent remagnetization times havebeen often observed in previous Tr-TMOKE experiments on ferromagnets these were interpreted asa frustration-induced slowdown of the spin dynamics [82] or were regarded as important evidence
supporting the Elliott-Yafet spin-phonon interaction as the relevant microscopic mechanism [14, 21].
In contrast, our model provides an alternative interpretation validated over the full demagnetization
parameter space: there indeed exists a transient magnetic phase transition in Ni when the excitation
laser fluence is higher than a critical value, which can completely explain the observed TR-TMOKE
data. The optimum values of fitting parameters are listed in Table 5.1.
From our model which correlates the TR-TMOKE and TR-ARPES results, we can extract
the time- and depth- dependent magnetization dynamics in Ni. In Fig. 5.4(a), we plot the amplitudes of the exponential functions in Eq. 5.1 for a monolayer Ni as a function of the heat source.
A complete temporal and spatial profile of the laser- induced ultrafast demagnetization in Ni isplotted in Fig. 5.4(b). Physically, the characteristic fast and slow recovery timescales (τrecover1
and τrecover2 ) indicate the existence of two distinct physical mechanisms. The fast remagnetization
timescale (τrecover1 ) can be explained by the damping of magnons under the strong exchange fieldin Ni [82], which yields a damping time of 580 fs, in quantitative agreement with the observed fast
recovery timescale (τrecover1 , within experimental error) [5]. On the other hand, from molecularfield theory, the exchange field is dissolved when the sample crosses the critical point and enters theparamagnetic state. In this case, we can expect the damping time to approach infinity and cooling76of the spin system can only be achieved via other mechanisms, e.g., coupling to the lattice andthermal transport. The latter is consistent with the appearance of the slow remagnetization process (τrecover2 ), when the fluence is above the critical fluence. As a result, the distinct timescales in
our ultrafast measurement provide a way to probe the exchange field present on microscopic scales.
Our results, hence, suggest the competition and coexistence of paramagnetic (slow recovery) andpartially suppressed ferromagnetic (fast recovery) phases during the ultrafast demagnetization process, as well as the variation of their relative contributions as a function of pump fluence [Fig.
5.4(c)]. Indeed, it has been shown by simulations based on atomic level classical spin Hamiltonianthat the recovery from a highly disordered magnetic state involves the growth of many small magnetically ordered and disordered regions, with a size comparable to the magnetic correlation length[82]. Very interestingly, the fluence for which the fast- remagnetization contribution completely
disappears [Fc ’ in Fig. 5.4(a)] coincides with the fluence that drives the lattice temperature above
the Curie temperature. This is consistent with the thermodynamic limit. We note, however, that
we cannot simply conclude that the variation of sample magnetization is only determined by thetemperature of electron-lattice system. One obvious evidence is that the magnetization at long delay times (a3) increases linearly as a function of the laser fluence (and, hence, of the temperature),
as shown in Fig. 5.4(a)this cannot be explained by the typical nonlinear relationship between thesample magnetization and temperature under thermal equilibrium conditions. This result suggeststhat the spin system is far from thermal equilibrium on time-scales of picoseconds, a finding whichis consistent with previous theory [82]. By separating the different degrees of freedom in the time
domain, our results suggest that the single critical point under thermal equilibrium is expanded
into a critical region for the nonequilibrium magnetic phase transition in Ni (Fig. 5.8(a)), spanning
critical fluences that first drive the electron temperature above the Curie temperature (Fc ) andthen the lattice to the Curie temperature (Fc ’).
Finally, another interesting conclusion we can make from our work is how to achieve very fast
manipulation of spins, which has been an important goal ever since the first observation of ultrafast
demagnetization [7]. The fundamental speed of the demagnetization process we study is limited by

77
the slow recovery dynamics, which typically occur on picosecond-to-nanosecond timescales [7, 15,
11, 16, 8, 12, 13, 9, 14, 10, 18]. From our data, one way to achieve faster all-optical spin control on
sub-ps timescales is to apply a laser fluence lower than Fc to take advantage of a faster recovery
timescalealthough in this case, the maximum demagnetization is <50 in Ni, as shown in Fig. 5.4(a).
Another alternative would be to use a nanostructured magnetic material, with adjustable magnetic
interactions and more optimal thermal transport.
In conclusion, we show that by correlating Tr-ARPES and Tr-TMOKE measurements on Ni,
we obtain new insights into the laser-induced magnetic phase transition. All results consistently
reveal a critical behavior associated with a true magnetic phase transition and universal timescales for spin excitation, demagnetization, and recovery. Moreover, the linear response and twocompeting channels observed in the recovery process suggest the possible presence of coexisting
phases in the material.

5.1

Supporting information
Here we provide an additional set of information required in order to asses the validity of ourclaims in the previous section.

5.1.1

Calculation of absorbed fluence

In order to quantitatively compare the results obtained from Tr-ARPES and Tr-TMOKE
experiments, which possess different geometries, we carefully calculated the absorbed fluence at thesurface of Ni, taking into account different incidence angles and polarizations of the laser pump.
In both measurements, absorbed fluence is defined as Fabs = 2E(1−R)πwx wy , where E is the pump pulse
energy, R the reflectivity and wx(y) the Gaussian beam radius along the x(y) axis of the laser spot
at 1/e2 of the maximum intensity. Hence, the fluence is defined as the peak fluence of the laser
pump. The reflectivity R is 0.74 in ARPES (s-polarized, normal incidence) and 0.64 in TMOKE(p-polarized, 48 degrees incidence angle). The values of reflectivity were confirmed by the opticalconstants of Ni [82], as well as the in-situ calibration measurements.

78

Figure 5.4: (a) Top panel: Schematic magnetization of a ferromagnet as a function of temperature
under thermal equilibrium with a single critical point (Tc ). Bottom panel: Extracted amplitudes of
the change of magnetization in a monolayer of Ni as a function of in-situ fluence and heat source.
The correspondence of Tc to the two critical fluences (Fc and Fc ’) is highlighted. (b) The laserinduced magnetization variation in Ni as a function of time and depth. The black dashed lines
represent the contours of equal magnetization. The white dashed lines separate different regions
for the in-situ fluence relative to the two critical fluences Fc and Fc ’. (c) The relative contributionsof the fast (< A3 >) recovery process directly extracted from the TR-TMOKE results in Fig. 5.3.
Inset: Potential scenarios for the coexistence of ferromagnetic and paramagnetic phases in different
fluence regions.

79
In order to take into account the true absorption of a monolayer sample, we used the heatsource in Fig. 5.4a. The heat source is given by Q = δFL and δL is the penetration depth of thelaser pump.

5.1.2

Derivation of TMOKE Depth Sensitivity Function

We followed the technique outlined in Refs. [73] to construct the depth sensitivity function
for EUV TMOKE in a thick nickel film. In this method, a thin layer is selected and assumed to bemagnetic, while the rest of the film is treated as non-magnetic, but with otherwise identical opticalconstants. This thin film is then varied in its location throughout the film, while the contributions
are summed. In this way, the contribution to the asymmetry is broken into a set of orthogonalbasis functions, corresponding to discrete locations in the sample.
Finally the depth sensitivity function is computed, which accounts for the difference in phasebetween parts of the EUV light that penetrate to different depths in the material. To derive thisresult, several approximations are made. The first is that the optical constants used for both themagnetic and non-magnetic parts of the sample are taken to be their equilibrium values. Thisis a fairly good approximation to within a few percent, as evidenced by the very small change
in reflectivity observed in our previous work [5]. The optical constants used for this calculationin the EUV regime are given by Ref. [78], a static measurement. We note that even more exact
values for these optical constants could be determined by using the dynamic measurement techniquedescribed in Ref. [19]. Second, we assume that this numerical model offers a good approximation
for the continuous magnetism of the real material. To verify this, we checked the predicted static
asymmetry values calculated by integrating the weighting function across the material to predict
the asymmetry across the M edge, and comparing this result to the asymmetry measured at specificphoton energies measured with the harmonic comb. We show the results of this comparison in Fig
5.5(a). Given the agreement of the integrated weighting function with the static measurementof asymmetry, we believe the model is robust for the static case. Since the depth sensitivity
function is derived from the assumption that the magnetization of each individual layer does not80

Figure 5.5: (a) Comparison of Asymmetry computed from depth sensitivity model (red) and measured asymmetry (blue). The harmonic peaks used to compute the measured asymmetry are shown(green) beneath. Note that the experimental asymmetry is only reliable in the regions sampled by
the harmonic comb, and that in these regions the measurements agree very well with the modelledasymmetry. (b) Calculated TMOKE sensitivity as a function of photon energy and depth.

depend on the magnetization of the other layers in the sample, it can be used for the simulation ofasymmetry for an arbitrary magnetization profile (such as the one present in the sample after laserinduced demagnetization). This means that the weighting function holds constant, to within our
experimental error, throughout the dynamics of the demagnetization and re-magnetization process.
The calculated TMOKE sensitivity as a function of both photon energy and depth is plotted inFig. 5.5(b).

5.1.3Comparison of TMOKE fitting between models with and without criticalfluence

In Fig. 5.3, we plot the comparison between the measured magnetization dynamics and ourmodel, which takes the critical fluence Fc into account. The critical fluence can also be revealedby the threshold behavior of hA3 i in the inset of Fig. 5.3, which highlights the importance ofthe critical phenomena to the interpretation of Tr-TMOKE results. Here, we further confirm
the existence of the critical fluence by fitting the TMOKE data at different fluences without thecritical behavior in the model, and comparing the fitting results between these two models (i.e.,

81with and without the critical fluence Fc ). In the second model, we model a3 (z) = b03 F (z), in
replacement for Eq. 5.1, so that b1 and b03 are the free fitting parameters in this case. As shown in
Fig. 5.6, both models agree very well with the data in the first 100 fs, highlighting the universal
timescales of the ultrafast demagnetization processes. However, the model without critical fluence(dashed lines in Fig. 5.6) deviates significantly from the re-magnetization data, especially at the
low and high fluences, as shown in Fig. 5.6. Our TMOKE measurement with a full range offluences, spanning from a small reduction to full quenching of the sample magnetization, allows us
to observe this discrepancy that previous measurements with only few fluences may not be able
to detect. This evidence further corroborates that the critical behavior, which drives the magnetic
dynamics through different recovery pathways depending on the laser fluence, is also necessary forthe correct interpretation of the Tr-TMOKE data.

5.1.4

Fluence dependence of the exchange-splitting dynamics

Figure 5.7 displays the exchange splitting dynamics measured by Tr-ARPES for a series ofabsorbed fluence. When the fluence is below Fc (top three curves), the exchange splitting recovers
through a fast channel with a timescale of 537 fs. While for fluences above Fc (bottom three
curves), the exchange splitting recovers through the fast channel and an additional slow channel
with a timescale of 76 ps. Upon fluence increasing, the relative fast recovery amplitude gradually
decreases while the slow one increases, exhibiting a similar competing behavior as described in themain text for TMOKE data. Given the surface sensitivity of the ARPES measurement, this result
suggests that the phase coexistence we proposed is in the transverse direction and different fromthe depth-averaging effect.

5.1.5

Thermal or non-thermal nature of ultrafast demagnetization?

In Fig. 5.4a, we show that the contribution of the fast recovery dynamics (a2) completely
disappears at the laser fluence Fc ’. Interestingly, we find this fluence is very close to the fluencethat drives the lattice temperature above the Curie temperature. This can be estimated using the

82

Figure 5.6: Comparison between models with and without including the critical behavior. The
symbols are the Tr-TMOKE data, the red solid line is model results with Fc and the blue dashedline is model results without Fc .

83

Figure 5.7: The change of exchange splitting measured by Tr-ARPES for a range of absorbed
fluence both below and above Fc .

84
two-temperature model (TTM), by considering the electron-phonon coupling, thermal transport(see chapter 4 and appendix A) and the temperature-dependent heat capacities of the electron-spin
system and the lattice. The required laser fluence obtained from this estimation is 3.7 mJ/cm2 ,which is in good agreement with the experimental value Fc ’ = 3.84 mJ/cm2 . This interestingfinding again remarkably connects the laser-induced magnetic phase transition to its counterpart
under the thermal equilibrium.
This agreement, however, cannot conclusively prove that the variation of sample magnetization in our experiments is only determined by the electron/lattice temperature, and, hence, theultrafast demagnetization is purely a thermal process. A clear evidence is from the linear responseof the demagnetization amplitude at long time (a3 ) to the laser fluence. At long times, if the system(electrons, spins and lattice) has reached the equilibrium, we would expect the sample demagnetization to follow the nonlinear relation: a3 ∝ T 3/2 , similar to the situation under the thermal
equilibrium, and the temperature (T) is proportional to the laser fluence (F), to a good approximation. However, the experimentally extracted a3 as a function of F is obviously deviated fromthis nonlinear relation. In Fig. 5.8, we present this comparison, by showing the experimentally
extracted demagnetization at long time delay (a3 ) and the expected value from thermal equilibrium. The sample magnetization under the thermal equilibrium is extracted from Ref. [6] , and thetemperatures of sample different laser fluences are calculated using the TTM. he linear responseof a3 in our experiment is very important, because it highlights the non-equilibrium nature of thelaser-induced magnetic phase transition in Ni, in which exotic photoinduced states, such as thehighly excited spin system at tens of femtoseconds and the coexisting ferro- and para-magnetic
phases at picoseconds, could be established.

5.1.6

Demagnetization and recovery timescales

In order to further confirm the universal timescales of the dynamics of magnetization, we
also fit the TMOKE data with a commonly used two-exponential function. The resulted demagnetization times is fluence-independent within experimental uncertainties, as shown in Fig. 5.9.

85

Figure 5.8: Comparison between the experimentally extracted demagnetization at long delay time
and the values expected under the thermal equilibrium. The sample magnetization under thethermal equilibrium is extracted from Ref. [6] and the temperatures are calculated using the TTM
for different laser fluences.

Figure 5.9: Demagnetization timescale extracted from a two-exponential function fitting to the
TMOKE data for different pump fluences. The grey line is a guide to eye at the average value toindicate the timescales are fluence-independent within experimental uncertainties.

86
Here we also note that τrecover1 , the fast recovery time constant, is slightly larger than thevalue extracted from the Tr-ARPES results (within the error bar). This difference can be attributed
to the difference in the signal-to-noise ratios (SNR) of these two experiments. In Tr-ARPES, thesignal intensity was intentionally reduced in order to avoid space-charge distortion of the photoelectron signal [5]. In contrast, Tr-TMOKE provides results with a very high SNR, allowing to
extract the time constants and amplitudes of magnetization dynamics more reliably.

Chapter 6

Direct Light-induced Spin Transfer between Elemental Sublattices in a
Spintronic Heusler Material via Femtosecond Laser Excitation

6.1

Introduction
In this chapter, we move to studying a more complex material, half-metal heusler compound

Co2 MnGe, using the same tool as previously: time-resolved transverse magneto optical Kerr effect.
We show that a single ultrafast laser pulse can directly transfer spin polarization from one magnetic
sublattice to another within the time duration of the optical excitation. Using the powerful abilityof the high harmonic comb of light resonant at the 3p edges of cobalt and manganese to measure themagnetization of each element simultaneously and independently, we observe a surprising disparity
between the response of the two magnetic sublattices. The magnetization of cobalt is transiently
enhanced during the laser pump pulse, while that of manganese quenches rapidly. This extremely
fast enhancement of the average magnetization of Co represents the fastest change in magnetization
observed to date and is a unique property that stems from the half-metallic nature of the Heusler
material.
Next, we use density functional theory (DFT) calculations [84, 85], that compute the densityof states for each element in the compound, to show that optical excitations are preferentiallyenhanced in the majority spin channel of Mn and in the minority spin channel of Co. This imbalanceof allowed excitation pathways for each element then induces a direct and instantaneous transferof spin polarization from Mn sites to Co sites that occurs during the entire duration of the laser
pulse. As a control, we observe no enhancement of magnetization or transfer of spin polarization in

88
the disordered non-half-metallic (A2) phase of the same material. This is consistent with electronic
structure theory that does not support any imbalance in the excited spin population of manganesein this phase. The observed transient enhancement of ferromagnetic ordering demonstrates direct
manipulation of spins via light, thus providing a path towards spintronic logic devices such as
switches that can operate on few femtosecond or even faster timescales. It also establishes band
structure engineering as an efficient method of selecting for specific desirable behaviors in a quantum
material.
Figure 6.1A illustrates the ultrafast spin dynamics that occur in the B2 (semi-ordered) halfmetallic phase of Co2 MnGe. Before the pump pulse arrives, the material is ferromagneticallyordered, with a B2 structure (the CsCl structure), where Co occupies the corner sites, and the
body centered position is randomly occupied by Mn or Ge. Note that the Mn atoms carry ≈3x
as much magnetic moment as the Co atoms. As shown in Figs. 6.1 and 6.2, within the timescaleof the laser pump pulse, there is a direct transfer of magnetization from the Mn atoms to the
Co atoms: the magnetization of the Mn atoms decreases, while the magnetic moment of the Co
atoms increases. After the pump pulse, the material continues to demagnetize, dissipating theangular momentum into the lattice within a ps, with a ≈120 fs lag between the response of Mn and
Co. We emphasize that the initial magnetic response of the system is completely dominated bya direct optical excitation process, while rotation of atomic magnetic moments enters at a second
stage. This is corroborated by a theoretical analysis based on the atomistic Landau-Lifshitz-Gilbertequation [86] (presented in the supplementary information), that fails to describe the initial phaseof the magnetic response (< 250 ps) but provides the correct trend for longer time scales (>500 fs).

6.2

Experimental Results
Figure 6.1B shows the experimental setup used to simultaneously measure the response of

Co and Mn after ultrafast excitation. We excite the sample with an infrared laser pulse with aphoton energy of 1.55 eV and 55 fs full width half maximum (FWHM) in duration. To record themagnetic response, part of the laser light is directed into a He-filled hollow waveguide to generate

89

Figure 6.1: (A) Representation of spin dynamics in Co2 MnGe. Before excitation, Mn atoms (orange
arrows) have a 3x larger magnetic moment than Co atoms (blue arrows), which are 2x more abundant in the bcc lattice. The purple arrow represents the net magnetic moment of the compound.
Upon excitation, the Mn moment begins to decrease and the Co magnetic moment immediately
grows by 10%. This occurs within the duration of the pump pulse. Hundreds of femtoseconds later,
the Mn and Co atomic spins become disordered, and the angular momentum begins to transfer tothe lattice. The net magnetic moment begins to decrease. After 1-2 ps, the spins have reachedtheir maximum quenching. (B) Schematic of the experimental apparatus. Ultrafast femtosecond
infrared pulses are used to excite the sample, while the magnetization dynamics are tracked withfemtosecond EUV pulses recorded on a spectrometer. (C) Density of states for each element in thehalf metal. Note that the minority spin channel is gapped, with no available states at the Fermi
level for the minority channel. Critically, this gap is larger for Mn than for Co. After excitation,
the conductions band states are hybridized, as illustrated by the shared red wavefunction.

90
10 fs high harmonic EUV pulses that are simultaneously resonant with the 3p edges of Co (59 eV)
and Mn (47 eV). The EUV transverse magneto optical Kerr effect (TMOKE) signal is constructed+ −I−
by a differential asymmetry [A] measurement with A= II++I− where I+ and I− are the intensitiesof the EUV light that is reflected from the sample for opposite directions of magnetization. The
responses from individual elements are separated via a spectrometer. A more detailed schematicof the experimental layout is given later in the chapter.
Figure 6.2A plots the experimentally measured asymmetry of Co and Mn in the half-metallic
B2 phase as a function of pump probe time delay. During the laser pulse, the Co magnetizationtransiently increases by ≈ 10%, while that of Mn immediately decreases. Although the initial
dynamics are markedly different, the decay rates of the two magnetic sublattices are similar, andoccur on timescales similar to the average values measured by visible MOKE [35]: Mn demagnetizes
with an exponential decay constant of 328 ± 37 fs while Co does so with a decay constant of 323± 86 fs (after a lag due to the initial transient enhancement). As the timescale for this process
is longer than in pure ferromagnets (see appendix B), we can unambiguously decouple the direct
optically generated spin dynamics (that occur over the entire laser pulse duration of 55 fs) fromthe dissipation of angular momentum in the system (that follows in the next ps). As yet anothercontrol to test the time sequence of spin transfer, we increased the pump pulse duration from 55 fs
to 90 fs. As shown in dashed lines in the inset of Fig. 2A, this resulted in an increase of the delayin the demagnetization between the Co and Mn sublattices: from 120 ± 33 fs (55 fs pump pulse)
to 157 ± 27 fs (90 fs pump pulse). This is consistent with a spin transfer process from Mn to Co
that occurs only during the laser pump pulse.
Figure 6.2B plots the element specific dynamics in the metallic A2 phase of the material
driven by a 90 fs excitation pulse. In this phase, the lag in response of the magnetic sublattices(Mn and Co) is much less, only 56 ± 32 fs, and no transient increase in the Co magnetization isobserved. As discussed below and in appendix B, this is because the optical pathways to excitationin the minority band of manganese is not blocked in this phase. Fig. 6.2C shows the elemental
crystal structure of the material in the two phases studied (B2 and A2). The details of sample

91

Figure 6.2: Element resolved ultrafast magnetization dynamics following excitation by femtosecond
laser in the (A) half-metallic B2 phase. Note that the Co magnetization increases as the Mn
decreases, followed by a lag of 100 fs between the demagnetization of the Co and Mn sublattices.
Inset: Dynamics of ultrafast spin transfer. Solid lines are the element specific dynamics after
excitation by a 55 fs (full width half maximum) pump pulse. Dashed lines indicate the dynamics
after excitation by a 90 fs pump pulse. Note that the location of the peak of the enhancement isshifted in time by 25 fs for the enhancement driven by a 90 fs pump (half the difference between
the duration of the two pulses), underlining that this process is a direct optical manipulation. (B)Element resolved ultrafast magnetization dynamics in the non half-metallic A2 phase. There is noenhancement of the Co magnetization. The subsequent lag in demagnetization dynamics between
the Co and Mn sublattices decreases to 57 fs. (C) Atomic structure of compounds studied. In theB2 phase, the Co atoms have ordered so that they occupy sites at the edges of the bcc structure,
while the centers are randomly interspersed between Mn and Ge. In the A2 phase, the material has
formed the ordered bcc structure, but the location of the atoms within the structure are random.

92
growth are given in [87]. In contrast to the B2 phase, the A2 phase is disordered, does not have ahalf-metallic gap for the minority spin state (as shown in Fig. 3B), and serves as a control fromwhich to understand the effect of half-metallicity on the element specific magnetization properties
of Co2 MnGe.
It is clear from a comparison between the dynamics observed in the A2 and B2 phases thatordering and emergence of a half-metallic gap in the minority band play a fundamental role in the
optical magnetic response of Co2 MnGe. The transient enhancement observed is a coherent process
driven directly by the optical excitation pulse, as demonstrated by the shift in the peak of the
Co excitation when driven by a longer laser pulse. Moreover, our measurements also show thatelemental specificity is critical for revealing the underlying magnetic dynamics of this material: as
shown in Fig. 6.2 and appendix B, when the Co and Mn responses are averaged (as is the case forvisible MOKE), no difference in the behavior of the A2 and B2 phases can be detected.

6.3

Computational Results
Figures 6.3A and B plot the element specific spin resolved density of states (DOS) calculatedusing DFT for the two phases of the material. In the B2 phase (Fig. 6.3A) a gap has formed forthe minority carriers, while the majority has full mobility across the Fermi level. Critically, thisgap is larger for the Mn states than the Co states. In contrast, the crystal has a fully metallic
characteristic in the A2 phase (Fig. 6.3B) for both the majority and minority states. When a1.55 eV IR photon is absorbed, the density of states of each phase results in several key differencesin the transition probabilities for the material. As shown in Fig. 6.4 and appendix B, in the B2phase, most of the minority carriers that are excited come from Co states, but very few minority
excitations can take place for Mn states. One must bear in mind that both initial and final states ofthis optical excitation involve wavefunctions that are shared across hybridizing Mn and Co orbitals.
Due to an imbalance in how Co and Mn projected orbitals contribute to the initial and final states,
direct and spin conserving transitions lead to an effective transfer of spin angular momentum from
Mn to Co.

93

Figure 6.3: Density of states for Co2MnGe in the (A) B2 and (B) A2 phases. Note that thehalf-metallic character is only present in the B2 phase.

94
Figures 6.4A and B plot the calculated imbalance of the transition probabilities from eachsublattice that give rise to the spin transfer from Mn to Co. The combination of these effects results
in a direct transfer of spin polarization from Mn sublattices to Co as observed experimentally in
the B2 phase and illustrated in Fig. 1C. In Fig. 6.4C, we illustrate the basic mechanism thatallows for an effective angular momentum transfer from Mn d-states to Co d-states. The initialstate wavefunction that is involved in the optical transition is hybridized and composed of both
Mn and Co d-states, with a larger contribution from the Mn atom (initial state). For the final
state, the situation is reversed, and the Co d-states dominate. Hence, when an electron is optically
excited from the initial to the final state wavefunction, this is associated with a transfer of d-state
population from Mn to Co. A higher probability of transitions for the spin-up electrons comparedto the spin-down electrons in Mn (Fig. 6.4A) then causes an effective spin-transfer from Mn toCo, as observed in the data of Fig. 6.2A. In the A2 phase (Fig. 6.2B), excitation in the minority
valence band of Mn is now optically allowed, and direct optical transfer of spin polarization does
not occur (see appendix B for these calculations).
We emphasize that the transient enhancement of the magnetic signal shown in Fig. 6.2A
for the B2 phase has no contribution from the change in refractive index due to electronic changes
in the reflectivity. We demonstrate this by measuring the change in reflectivity for the B2 andA2 phases at the Co edge, and showing that the magnitude of these changes is equal in the twophases of the material (see appendix B for more information). The mechanism responsible for themagnetization dynamics of Co2 MnGe in the B2 phase involves direct optical transitions, that leads
to an initial transfer of spin moment from Mn to Co. This purely electronic mechanism is necessaryin addition to a description based on the atomistic Landau-Lifshitz-Gilbert (aLLG) equation, thatsometimes is employed to analyze the kind of experiments presented here [88, 89]. As shown inappendix B, the aLLG equation captures some features of the observed magnetization dynamics,
especially at longer timescales (≈250 fs and longer), but it fails to explain the initial phase of thedemagnetization process. We also note that a similar enhancement to that shown in Fig. 6.2A
was predicted via time dependent DFT calculations for the related compound Co2 MnSi, with the95

Figure 6.4: The probability for exciting a spin up (majority) vs. spin down (minority) electron fromthe valence band in the B2 phase for different pump energies in (A) Mn sites. Note that for a 1.55
eV pump, the probability is higher for spin up electrons to be excited from Mn. (B) Probability
for excitations in Co sites. In contrast to the Mn result, the probability is higher for minority
electrons to be excited in Co. (C) Illustration of process that leads to direct optical transfer of spinpolarization from Mn to Co. The initial state wavefunction is hybridized and composed of both
Mn and Co d-states, with a larger contribution from the Mn atom. In the final state, the situationis reversed, and the Co d-states dominate. Hence, when an electron is optically excited from theinitial to the final state wavefunction, this is associated with a transfer of spin polarization from
Mn to Co.

96
mechanism in this case also attributed a transfer of spin polarization between Co and Mn[37].

6.4

Discussion
Given that we now understand how to measure and predict all-optical spin transfer based on

the electronic structure and the density of states of the material, future experiments can explore
how changing the pump photon energy or tailoring the bandgap (i.e. density of states)[33] inother materials can be used to tune the demonstrated spin manipulation. Since the enhancement
is only observed in the half-metallic phase of the material, the presence of the band gap in theminority channel of both elements may help to enable the spin polarization enhancement in Co;
after majority electrons are excited in Mn and shared with Co via hybridized orbitals, these electrons
may not be able to decay via spin-flip scattering into the minority band [38]. We also note herethat the B2 phase of Co2 MnGe, where the large spin-transfer between two atom types is observed,
is characterized by hybridizing states, as well as a low-spin state of one of the atoms (Co) thathas a high spin state available at not too large energies. Indeed, Co occupies this higher spin state
for the A2 phase. We speculate that these material characteristics are important when trying tofind other materials that may have similar characteristics as those of Co2 MnGe and the results
shown in Fig. 2A. The L21 phase shares these characteristics with the B2 phase of Co2 MnGe, andis therefore expected to have similar magnetization dynamics. There are several other materials
that also have these characteristic properties. For instance, the meta-magnetic Laves phase YCo2 ,when doped with Fe or Ni, as well as fcc Fe doped with Co or Ni, are all systems that can becharacterized in this way. Although direct spin transfer is likely the dominant mechanism in thismaterial, we note that other sub 50 fs demagnetization mechanisms have recently been uncovered,and could cause different kinds of spin dynamics that operate on the same timescales as direct spintransfer in these other materials [90, 5]. Finally we note that a similar lag in spin dynamics waspreviously observed in alloys of FeNi [13], therefore our element-resolved measurements of a lag in
a fundamentally different system suggest that these phenomena may be ubiquitous.
Our time- and element- resolved EUV MOKE method will have useful applications for select-

97
ing spintronic materials, to understand which materials are good candidates for spin injection. In
the past several decades, various techniques for spin injection have emerged, but all have degrees ofspin polarization <100%. Pursuing the goal of a pure spin current source, ferromagnetic half-metals
were developed to ideally provide conduction of a single spin state. However, 100% spin polarized
conduction has not yet been demonstrated in these compounds [91]. Although many techniques
are used to characterize the properties of half-metallic materials such as Andreev reflection [92, 93],
spin-polarized tunneling, spin-polarized photoemission [94], and visible MOKE[34], these all rely on
surface measurements. However, device applications require coherence and functionality through afinite nanoscale depth, and bulk and surface properties can be significantly different [95, 96, 97]. A
strong advantage of using EUV MOKE as a probe of these materials is the larger penetration depthof EUV light, more than twice that of visible light, which can in addition to providing elemental
specificity, probe the magnetization state of the material more deeply inside of a device. To utilize
the potential of our findings in a functional spintronics device, the transfer of spin must occur acrossfinite distances. According to DFT calculations, such a transfer process may be possible for a thinfilm stack [43]. Additionally, several new breakthroughs have recently reported miniaturizing spincurrent devices to the nanoscale [98, 99]. Considering these recent developments, we anticipate thatthe femtosecond spin transfer demonstrated in this work could be used for information processing
utilizing spin currents on few fs timescales.
In conclusion, we experimentally demonstrate the first direct optical manipulation of themagnetic moment of an individual element in a compound material. Ultrafast transfer of magnetization takes place over sub-nanoscale spatial dimensions and sub-55 fs timescales. The timescale
for transient enhancement depends on the duration of the excitation pulse, further confirming thatthe light directly transfers the spin. Theory based on atomistic spin-dynamics as well as density
functional theory show that electronic excitations drive the magnetization dynamics in the initial
phase of the demagnetization, and that these excitations directly transfer spin from Mn to Co,
via preferred spin-polarized excitation pathways. Rotation of atomic moments, as calculated byatomistic spin-dynamics and that result in significant non-collinear orientations, become relevant

98
at a later stage of the dynamics. We confirm the predictions of the DFT calculations and importance of the half-metallic (B2) band structure of the material by a further lack of enhancement inthe (A2) phase of the material (that lacks half-metallicity). This work represents the first opticalmanipulation of its kind and demonstrates a route to sub-femtosecond all-optical logical operations
in magnetic recording media.

Chapter 7

Concluding Remarks and Future Directions

In the first two chapters of this thesis, I briefly described the motivation for this work, andthen moved to the physics of ferromagnetic materials from an atomic, molecular, and crystalline
lattice picture. I showed that the solutions to the equations for a many-body system such as the
one relevant for magnetic effects quickly becomes too complicated to solve analytically. Instead,
we rely on approximations that can give us a qualitative understanding and also a quantitative
one, such as is the case for density functional theory. Next, I outlined the origin for resonant
magneto-optical effects by deriving the transition matrix element for resonant interaction between
X-ray radiation and matter in an atomic picture. I showed that the true transition matrix elements
for a solid can be constructed from combinations of the atomic ones. In this way, I show that using
magneto-optics, we can directly probe both the charge and spin dynamics using light resonant at
the 2p or 3p edges, which in our case is generated from high harmonic generation.
Next I outline several experiments that represent a significant step towards the ultimate goalof direct manipulation of ferromagnetic spins with single pulses of light. In chapters 4-5, we showthat the ultrafast demagnetization process discovered over 20 years ago is actually driven by anonequilibrium spin distribution that occurs within the first 20 fs after excitation by a femtosecond
laser pulse, an order of magnitude faster than previously suspected. After this surprising discovery,
we studied spin dynamics in another system, one that proved to have a favorable band structure
for the direct manipulation of spins via light. We showed that within the duration of the driving
laser pulse, the magnetic moment of one element is transiently enhanced, while that of other is100
decreased. The observation was made possible by the element specificity of the tools used to
study the dynamics of the material. Using extreme ultraviolet light resonant with the shallow corelevels of Co and Mn allows us to simultaneously and independently track the dynamics of the twoelements as the material responds to ultrafast laser excitation. In a separate simulation based on
the classical Landau-Lifshitz Gilbert (LLG) equations, we show that these surprising short time
dynamics cannot be captured in a LLG simulaton, and thus must be due to quantum mechanicaleffects. These results are presented in Appendix B. In fact, the LLG equation predicts that the Co
decay should precede that of the Mn, which is the opposite of what we observed experimentally.
In the future, the magnetics and ARPES teams in the KM group will attempt to undertakea number of important further experiments based on the results described here. First, a seriesof experiments are planned with single crystal iron and cobalt, to experimentally test whether
the spin excitation observed in nickel can be generalized to other simple ferromagnets. The Curie
temperature in cobalt is the highest of the elemental ferromagnetic materials, and so measurements
will first investigate Fe and then Co, in hopes that the electrons can be driven above the Curie
temperature, and the true ferromagnetic-paramagnetic phase transition induced. We also plan tocarry out experiments using a longer wavelength pump, such as 1.3 µm, in order to drive the phasetransition without burning the sample.
Separately, future experiments on the Co2 MnGe will be carried out by the magnetics team toinvestigate several outstanding questions arising from the new results outlined in this thesis. First,
we will attempt to shorten the driving laser pump pulse using a plate pulse compression techniquethat has been previously used successfully for low power femtosecond pulses [100]. We will shorten
the pump to 10 fs, to test whether the transient enhancement of Co is correspondingly shortened.
Additionally, the transition probabilities based on the DFT calculated band structure and driving
laser energy allow us to make some predictions about the behavior of the material when excited bylaser pump pulses with different photon energies. We will investigate these predictions by pumping
the sample with 3 eV photons generated via frequency doubling in a BBO crystal, and then moveto pumping with other wavelengths.

101
There are also other interesting half-metallic heusler systems that can be studied using theexperimental apparatus outlined in this thesis. Particularly, Ni2 MnSb is a promising candidate, forwhich predictions have been made that the enhancement could be even stronger than in Co2 MnGe[37]. We also have plans to acquire single crystal Co2 MnGa so that it can be studied using both
Tr-TMOKE and ARPES.
Finally, we conclude by remarking that the results presented in this thesis would never havebeen possible without an enormous number of contributions from previous and current membersof the KM Group in Boulder. The technical contributions made in this dissertation were primarily
to improve the data acquisition and stability of the experimental apparatus so that acquiring theresults presented here became possible. It is important to note that the experimental magnetic
asymmetry signal present in the heusler half-metal system investigated in chapter 6 is only 10
percent that of the signal available in bulk conventional ferromagnetic materials, and even only
5 percent of the signal possible from optimally chosen ferromagnetic thin films. Technical improvements made to the setup such as active beam stabilization, chirped mirrors, and better dataaquisition methods, were critical to improving the long term stability and brightness of the highharmonic source. With this capability, statistics about the dynamics could be acquired over thecourse of multiple days, with fully repeatable and sufficiently low-noise results.
These improvements to the experimental apparatus make additional experiments possible,
and improve the usability of the setup. Following this first demonstration of critical behavior
in simple ferromagnets and the EUV Tr-TMOKE method on the complex compound half-metalmaterial Co2 MnGe, there are many more avenues for further research using this same experimental
apparatus. There are truly no limits in sight.

Bibliography[1] J. Stohr and H. C. Siegmann.
Springer, 2006.

Magnetism: From Fundamentals to Nanoscale Dynamics.

[2] Peter J Meschter, James W Wright, Charlie R Brooks, and Thomas G. Kollie. PhysicalContributions to the Heat Capacity of Nickel. J. Phys. Chem. Solids, 42(9):861–871, 1981.
[3] V. Korenman and R.E. Prange. Temperature Dependence of the Angle-Resolved Photemission in Itinerant-Electron Ferromagnets. Physical Review Letters, 44(19):1291–1294, 1980.
[4] J. K. Chen, D. Y. Tzou, and J. E. Beraun. A semiclassical two-temperature model for ultrafast
laser heating. International Journal of Heat and Mass Transfer, 49(1-2):307–316, 2006.
[5] Phoebe Tengdin, Wenjing You, Cong Chen, Xun Shi, Dmitriy Zusin, Yingchao Zhang, Christian Gentry, Adam Blonsky, Mark Keller, Peter M Oppeneer, Henry C Kapteyn, Zhensheng
Tao, and Margaret M Murnane. Critical behavior within 20 fs drives the out-of-equilibrium
laser-induced magnetic phase transition in nickel. Science Advances, 4(eaap9744), 2018.
[6] A. Fognini, G. Salvatella, R. Gort, T. Michlmayr, A. Vaterlaus, and Y. Acremann. The influence of the excitation pulse length on ultrafast magnetization dynamics in nickel. Structural
Dynamics, 2(2), 2015.
[7] E. Beaupaire, J.-C. Merle, A. Daunois, and J-Y Bigot. Ultrafast Spin Dynamics in Ferromagnetic Nickel. Physical Review Letters, 76(22):1–4, 1996.
[8] B. Koopmans, J. J.M. Ruigrok, F. Dalla Longa, and W. J.M. De Jonge. Unifying ultrafast
magnetization dynamics. Physical Review Letters, 95(26):1–4, 2005.
[9] H.-S. Rhie, H. A. Dürr, and W. Eberhardt. Femtosecond Electron and Spin Dynamics in
Ni=W/(110) Films. Physical Review Letters, 90(24):247201, 2003.
[10] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu,
E. F. Aziz, M. Wietstruk, H. A. Dürr, and W. Eberhardt. Femtosecond modification of electron localization and transfer of angular momentum in nickel. Nature Materials, 6(10):740–
743, 2007.
[11] E. Carpene, H. Hedayat, F. Boschini, and C. Dallera. Ultrafast demagnetization of metals:
Collapsed exchange versus collective excitations. Physical Review B - Condensed Matter andMaterials Physics, 91(17):1–8, 2015.

103[12] Chan La-O-Vorakiat, Mark Siemens, Margaret M. Murnane, Henry C. Kapteyn, Stefan Mathias, Martin Aeschlimann, Patrik Grychtol, Roman Adam, Claus M. Schneider, Justin M.
Shaw, Hans Nembach, and T. J. Silva. Ultrafast demagnetization dynamics at the M edgesof magnetic elements observed using a tabletop high-harmonic soft X-ray source. PhysicalReview Letters, 103(25):1–4, 2009.
[13] S. Mathias, C. La-O-Vorakiat, P. Grychtol, P. Granitzka, E. Turgut, J. M. Shaw, R. Adam,
H. T. Nembach, M. E. Siemens, S. Eich, C. M. Schneider, T. J. Silva, M. Aeschlimann,
M. M. Murnane, and H. C. Kapteyn. Probing the timescale of the exchange interaction ina ferromagnetic alloy. Proceedings of the National Academy of Sciences, 109(13):4792–4797,
2012.
[14] T Roth, A J Schellekens, S Alebrand, O Schmitt, D Steil, B Koopmans, M Cinchetti, and
M Aeschlimann. Temperature Dependence of Laser-Induced Demagnetization in Ni : A Key
for Identifying the Underlying Mechanism. Physical Review X, 2(021006), 2012.
[15] Robert Carley, Kristian Döbrich, Björn Frietsch, Cornelius Gahl, Martin Teichmann, Olaf
Schwarzkopf, Philippe Wernet, and Martin Weinelt. Femtosecond Laser Excitation Drives
Ferromagnetic Gadolinium out of Magnetic Equilibrium. Physical Review Letters, 109(5):1–4,
2012.
[16] Steffen Eich, Moritz Plötzing, Markus Rollinger, Sebastian Emmerich, Roman Adam, Cong
Chen, Henry Cornelius Kapteyn, Margaret M Murnane, Lukasz Plucinski, Daniel Steil, Benjamin Stadtmüller, Mirko Cinchetti, Martin Aeschlimann, Claus M Schneider, and Stefan
Mathias. Band structure evolution during the ultrafast ferromagnetic-paramagnetic phasetransition in cobalt. Science Advances, e1602094 2(3), 2017.
[17] C. Boeglin, E. Beaurepaire, V. Halté, V. López-Flores, C. Stamm, N. Pontius, H. A. Dürr,
and J.-Y. Bigot. Distinguishing the ultrafast dynamics of spin and orbital moments in solids.
Nature, 465(7297):458–461, 2010.
[18] Emrah Turgut, Dmitriy Zusin, Dominik Legut, Karel Carva, Ronny Knut, Justin M. Shaw,
Cong Chen, Zhensheng Tao, Hans T. Nembach, Thomas J. Silva, Stefan Mathias, Martin
Aeschlimann, Peter M. Oppeneer, Henry C. Kapteyn, Margaret M. Murnane, and Patrik
Grychtol. Stoner versus Heisenberg: Ultrafast exchange reduction and magnon generation
during laser-induced demagnetization. Physical Review B - Condensed Matter and Materials
Physics, 94(22):1–6, 2016.
[19] Dmitriy Zusin, Phoebe M Tengdin, Maithreyi Gopalakrishnan, Christian Gentry, Adam Blonsky, Michael Gerrity, Dominik Legut, Justin M Shaw, Hans T Nembach, T J Silva, Peter M
Oppeneer, Henry C Kapteyn, and Margaret M Murnane. Direct measurement of the staticand transient magneto-optical permittivity of cobalt across the entire M -edge in reflection
geometry by use of polarization scanning. Physical Review B, 024433(97), 2018.
[20] Cong Chen, Zhensheng Tao, Adra Carr, Piotr Matyba, Tibor Szilvási, Sebastian Emmerich,Martin Piecuch, Mark Keller, Dmitriy Zusin, Steffen Eich, Markus Rollinger, Wenjing You,
Stefan Mathias, Uwe Thumm, Manos Mavrikakis, Martin Aeschlimann, Peter M. Oppeneer,
Henry Kapteyn, and Margaret Murnane. Distinguishing attosecond electronelectron scattering and screening in transition metals. Proceedings of the National Academy of Sciences,
114(27):E5300–E5307, 2017.

104[21] B Koopmans, G Malinowski, F Dalla Longa, D Steiauf, M Fähnle, T Roth, M Cinchetti, and
M Aeschlimann. Explaining the paradoxical diversity of ultrafast laser-induced demagnetization. Nature materials, 9(3):259–265, 2009.
[22] B. Y. Mueller, T. Roth, M. Cinchetti, M. Aeschlimann, and B. Rethfeld. Driving force ofultrafast magnetization dynamics. New Journal of Physics, 13(0001), 2011.
[23] B. Y. Mueller, A. Baral, S. Vollmar, M. Cinchetti, M. Aeschlimann, H. C. Schneider, and
B. Rethfeld. Feedback effect during ultrafast demagnetization dynamics in ferromagnets.
Physical Review Letters, 111(16):1–5, 2013.
[24] Michael Krauß, Tobias Roth, Sabine Alebrand, Daniel Steil, Mirko Cinchetti, Martin Aeschlimann, and Hans Christian Schneider. Ultrafast demagnetization of ferromagnetic transition
metals: The role of the Coulomb interaction. Physical Review B - Condensed Matter andMaterials Physics, 80(18):1–4, 2009.
[25] M. Battiato, K. Carva, and P. M. Oppeneer. Superdiffusive spin transport as a mechanismof ultrafast demagnetization. Physical Review Letters, 105(2):1–4, 2010.
[26] A Eschenlohr, M Battiato, P Maldonado, N Pontius, T Kachel, K Holldack, R Mitzner,
A Föhlisch, P M Oppeneer, and C Stamm. Ultrafast spin transport as key to femtosecond
demagnetization. Nature materials, 12(4):332–336, 2013.
[27] Jean-Yves Bigot, Mircea Vomir, and Eric Beaurepaire. Coherent ultrafast magnetism inducedby femtosecond laser pulses. Nature Physics, 5(7):515–520, 2009.
[28] W Töws and G M Pastor. Many-Body Theory of Ultrafast Demagnetization and Angular Momentum Transfer in Ferromagnetic Transition Metals. Physical Review Letters,
217204(115):1–6, 2015.
[29] G P Zhang and W Hübner. Laser-Induced Ultrafast Demagnetization in Ferromagnetic Metals. Physical Review Letters, 85(14):3025–3028, 2000.
[30] Y Zhang, T Chuang, Kh Zakeri, and J Kirschner. Relaxation Time of Terahertz Magnons
Excited at Ferromagnetic Surfaces. Physical Review Letters, 109(087203), 2012.
[31] A B Schmidt, M Pickel, M Donath, P Buczek, A Ernst, V P Zhukov, P M Echenique, L M
Sandratskii, E V Chulkov, and M Weinelt. Ultrafast Magnon Generation in an Fe Film on
Cu ( 100 ). Physical Review Letters, 105(197401), 2010.
[32] Shikun He, Yifan Liu, Yuhong Zheng, Qing Qin, Zhenchao Wen, Qingyun Wu, Yi Yang, Yupu
Wang, YuanPing Feng, Kie Leong Teo, and Christos Panagopoulos. Tunable magnetization
relaxation of Fe2Cr(1-x)CoxSi half-metallic Heusler alloys by band structure engineering.
Physical Review Materials, 1(064401):064401, 2017.
[33] P. Klaer, M. Kallmayer, C. G.F. Blum, T. Graf, J. Barth, B. Balke, G. H. Fecher, C. Felser,and H. J. Elmers. Tailoring the electronic structure of half-metallic Heusler alloys. PhysicalReview B - Condensed Matter and Materials Physics, 80(14):1–13, 2009.
[34] Georg M. Müller, Jakob Walowski, Marija Djordjevic, Gou Xing Miao, Arunava Gupta,
Ana V. Ramos, Kai Gehrke, Vasily Moshnyaga, Konrad Samwer, Jan Schmalhorst, Andy

105
Thomas, Andreas Hütten, Günter Reiss, Jagadeesh S. Moodera, and Markus Münzenberg.
Spin polarization in half-metals probed by femtosecond spin excitation. Nature Materials,
8(1):56–61, 2009.
[35] Andreas Mann, Jakob Walowski, Markus Münzenberg, Stefan Maat, Matthew J. Carey, Jeffrey R. Childress, Claudia Mewes, Daniel Ebke, Volker Drewello, Günter Reiss, and Andy
Thomas. Insights into ultrafast demagnetization in pseudogap half-metals. Physical Review
X, 2(4):1–9, 2012.
[36] Gary A. Prinz. Spin-Polarized Transport. Physics Today, 48(4):58–63, 1995.
[37] P. Elliott, T. Müller, J. K. Dewhurst, S. Sharma, and E. K. U. Gross. Ultrafast laser induced
local magnetization dynamics in Heusler compounds. Scientific Reports, 6(1):38911, 2016.
[38] Daniel Steil, Sabine Alebrand, Tobias Roth, Michael Krau, Takahide Kubota, Mikihiko
Oogane, Yasuo Ando, Hans Christian Schneider, Martin Aeschlimann, and Mirko Cinchetti.
Band-structure-dependent demagnetization in the Heusler alloy Co 2Mn1-xFexSi. PhysicalReview Letters, 105(21):17–20, 2010.
[39] K C Kuiper, R R J C De Wit, B Koopmans, and A J Schellekens. Ultrafast spin-transfer
torque driven by femtosecond pulsed-laser excitation. Nature Communications, 5(4333), 2014.
[40] C. H. Lambert, S. Mangin, B. S.D.Ch S. Varaprasad, Y. K. Takahashi, M. Hehn, M. Cinchetti,
G. Malinowski, K. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fullerton. All-opticalcontrol of ferromagnetic thin films and nanostructures. Science, 345(6202):1337–1340, 2014.
[41] S. Mangin, M. Gottwald, C. H. Lambert, D. Steil, V. Uhlı́, L. Pang, M. Hehn, S. Alebrand,
M. Cinchetti, G. Malinowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton. Engineeredmaterials for all-optical helicity-dependent magnetic switching. Nature Materials, 13(3):286–
292, 2014.
[42] Jan Peter Wüstenberg, Daniel Steil, Sabine Alebrand, Tobias Roth, Martin Aeschlimann,and Mirko Cinchetti. Ultrafast magnetization dynamics in the half-metallic Heusler alloy
Co2Cr0.6Fe0.4Al. Physica Status Solidi (B) Basic Research, 248(10):2330–2337, 2011.
[43] John Dewhurst, Peter Elliott, Sam Shallcross, Eberhard K. U. Gross, and Sangeeta Sharma.
Laser induced inter-site spin transfer. Nano Letters, 18:18421848, 2018.
[44] J.J. Sakuri. Modern Quantum Mechanics, Revised Edition. Pearson, 1994.
[45] J. C. Slater. The ferromagnetism of nickel. Physical Review, 49(7):537–545, 1936.
[46] J. C. Slater. The ferromagnetism of nickel. II. Temperature effects.
49(12):931–937, 1936.

Physical Review,[47] E C Stoner. Collective Electron Specific Heat and Spin Paramagnetism in metals. Proc. Roy.
Soc.,A,, 152(1935):656–678, 1936.
[48] Edmund C. Stoner. Collective electron ferromagnetism. Proc. Roy. Soc.,, 1937.
[49] N. F. Mott. A discussion of the transition metals on the basis of quantum mechanics.
Proceedings of the Physical Society, 47(4):571–588, 1935.

106[50] R. Wu and A.J. Freeman. Limitation of the Magnetic-Circular-Dichroism Spin Sum Rulefor Transition Metals and importance if the mangetic dipole term. Physical Review Letters,
73(14):1994–1997, 1994.
[51] G. P. Zhang, W. Hübner, Georgios Lefkidis, Yihua Bai, and Thomas F. George. Paradigm of
the time-resolved magneto-optical Kerr effect for femtosecond magnetism. Nature Physics,
5(7):499–502, 2009.
[52] Chan La-O-Vorakiat, Emrah Turgut, Carson A. Teale, Henry C. Kapteyn, Margaret M.
Murnane, Stefan Mathias, Martin Aeschlimann, Claus M. Schneider, Justin M. Shaw, Hans T.
Nembach, and T. J. Silva. Ultrafast demagnetization measurements using extreme ultraviolet
light: Comparison of electronic and magnetic contributions. Physical Review X, 2(1):1–7,
2012.
[53] Dennis Rudolf, Chan La-O-Vorakiat, Marco Battiato, Roman Adam, Justin M. Shaw, Emrah
Turgut, Pablo Maldonado, Stefan Mathias, Patrik Grychtol, Hans T. Nembach, Thomas J.
Silva, Martin Aeschlimann, Henry C. Kapteyn, Margaret M. Murnane, Claus M. Schneider,and Peter M. Oppeneer. Ultrafast magnetization enhancement in metallic multilayers driven
by superdiffusive spin current. Nature Communications, 3:1037, 2012.
[54] Boris Vodungbo, Bharati Tudu, Jonathan Perron, Renaud Delaunay, Leonard Mueller, Magnus H. Berntsen, Guillaume Lambert, Philippe Zeitoun, Christian Gutt, Emmanuelle Jal,
Alexander H. Reid, Patrick W. Granitzka, Nicolas Jaouen, Georgi L. Dakovski, Stefan
Moeller, Michael P. Minitti, Ankush Mitra, Sebastian Carron, Bastian Pfau, Clemens vonKorff Schmising, Michael Schneider, Stefan Eisebitt, and Jan Lüning. Indirect excitation ofultrafast demagnetization. Scientific Reports, 6(1):1–9, 2016.
[55] Christian Illg, Michael Haag, and Manfred Fähnle. Ultrafast demagnetization after laser
irradiation in transition metals: Ab initio calculations of the spin-flip electron-phonon scattering with reduced exchange splitting. Physical Review B - Condensed Matter and Materials
Physics, 88(21):1–11, 2013.
[56] J. L. Erskine and E. A. Stern. Calculation of the M23 magneto-optical absorption spectrumof ferromagnetic nickel. Physical Review B, 12(11):5016–5024, 1975.
[57] G. van der Laan, B. T. Thole, G. Sawatzky, J.B. Goedkoop, J.C. Fuggle, J.-M. Esteva,
R. Karnatak, J.P. Remeika, and H.A. Dabkowska. Experimental proof of magnetic x-ray
dichroism. Physical Review B, 34(9):6529–6531, 1986.
[58] G. Schuetz, W. Wagner, W. Wilhelm, P. Kienle, R. Zeller, R. Zeller, R. Frahm, and G. Materlik. Absorption of Circularly Polarized X Rays in Iron. Physical Review Letters, 58(7):737–740, 1987.
[59] A. McPherson, G. Gibson, H. Jara, U. Johann, T. S. Luk, I. McIntyre, K. Boyer, and C. K.
Rhodes. Studies of multiphoton production of vacuum-ultraviolet radiation in the rare gases.
Journal of the Optical Society of America B, 4(4):595 – 601, 1987.
[60] M Ferray, A L’Huillier, X F Li, L A Lompre, G Mainfray, and C Manus. Multiple-harmonic
conversion of 1064 nm radiation in rare gases. Journal of Physics B: Atomic, Molecular andOptical Physics, 21(3):L31–L35, 2002.

107[61] C. L. Gordon J. J. Macklin, J. D. Kmetec. High-Order Harmonic Generation Using Intense
Femtosecond Pulses. Science (New York, N.Y.), 70(6):766–769, 1993.
[62] Anne L’Huillier and Ph Balcou. High-order harmonic generation in rare gases with a 1-ps
1053-nm laser. Physical Review Letters, 70(6):774–777, 1993.
[63] Paul B. Corkum. Plasma perspective on strong field multiphoton ionization. Physical Review
Letters, 71(13):1994–1997, 1993.
[64] W. Li, A. A. Jaron-Becker, C. W. Hogle, V. Sharma, X. Zhou, A. Becker, H. C. Kapteyn, andM. M. Murnane. Visualizing electron rearrangement in space and time during the transition
from a molecule to atoms. Proceedings of the National Academy of Sciences, 107(47):20219–
20222, 2010.
[65] K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and E. K.U. Gross. Laser-Induced Demagnetization at Ultrashort Time Scales: Predictions of TDDFT. Journal of Chemical Theory
and Computation, 11(10):4870–4874, 2015.
[66] David L. Miller, Mark W. Keller, Justin M. Shaw, Ann N. Chiaramonti, and Robert R. Keller.
Epitaxial (111) films of Cu, Ni, and Cu x Ni y on αAl 2 O 3 (0001) for graphene growth bychemical vapor deposition. Journal of Applied Physics, 112(064317), 2012.
[67] T. Greber, T. Kreutz, and J. Osterwalder. Photoemission above the Fermi Level: The Topof the Minority d Band in Nickel. Physical Review Letters, 79(22):4465–4468, 1997.
[68] R Knorren, K H Bennemann, R Burgermeister, and M Aeschlimann. Dynamics of excitedelectrons in copper and ferromagnetic transition metals: Theory and experiment. PhysicalReview B, 61(14):9427–9440, 2000.
[69] Wenjing You, Phoebe Tengdin, Cong Chen, Xun Shi, Dmitriy Zusin, Yingchao Zhang, Christian Gentry, Adam Blonsky, Mark Keller, PeterM. Oppeneer, Henry Kapteyn, Zhensheng Tao,and Margaret Murnane. Revealing the Nature of the Ultrafast Magnetic Phase Transitionin Ni by Correlating Extreme Ultraviolet Magneto-Optic and Photoemission Spectroscopies.
Physical Review Letters, 121(7):077204, 2018.
[70] C. W. Nicholson, C. Monney, R. Carley, B. Frietsch, J. Bowlan, M. Weinelt, and M. Wolf.
Ultrafast Spin Density Wave Transition in Chromium Governed by Thermalized ElectronGas. Physical Review Letters, 117(13):1–5, 2016.
[71] C Stamm, N Pontius, T Kachel, M Wietstruk, and H A Dürr. Femtosecond x-ray absorption
spectroscopy of spin and orbital angular momentum in photoexcited Ni films during ultrafast
demagnetization. Physical Review B, 81(104425):1–4, 2010.
[72] Thomas Gilbert. Classics in Magnetics: A Phenomenological Theory of Damping in Ferromagnetic Materials. IEEE TRANSACTIONS ON MAGNETICS, 40(6):3443–3449, 2004.
[73] Mikihiko Oogane, Takeshi Wakitani, Satoshi Yakata, Resul Yilgin, Yasuo Ando, Akimasa
Sakuma, and Terunobu Miyazaki. Magnetic damping in ferromagnetic thin films. Japanese
Journal of Applied Physics, Part 1: Regular Papers and Short Notes and Review Papers,
45(5A):3889–3891, 2006.

108[74] J. Walowski, M. Djordjevic Kaufmann, B. Lenk, C. Hamann, J. McCord, and M. Münzenberg.
Intrinsic and non-local Gilbert damping in polycrystalline nickel studied by Ti : Sapphire laser
fs spectroscopy. Journal of Physics D: Applied Physics, 41(164016), 2008.
[75] R. M. White. Quantum Theory of Magnetism. Springer-Verlag, 2007.
[76] S. Eich, A. Stange, A. V. Carr, J. Urbancic, T. Popmintchev, M. Wiesenmayer, K. Jansen,
A. Ruffing, S. Jakobs, T. Rohwer, S. Hellmann, C. Chen, P. Matyba, L. Kipp, K. Rossnagel,
M. Bauer, M. M. Murnane, H. C. Kapteyn, S. Mathias, and M. Aeschlimann. Time- and angleresolved photoemission spectroscopy with optimized high-harmonic pulses using frequencydoubled Ti:Sapphire lasers. Journal of Electron Spectroscopy and Related Phenomena,
195(2014):231–236, 2014.
[77] Avinoam Kornblit and Guenter Ahlers. Heat capacity of EuO near the Curie temperature.
Physical Review B, 11(7):2678–2688, 1975.
[78] P.B. Johnson and R.W. Christy. Optical constants of transition metals: Ti, V, Cr, Mn, Fe,
Co, Ni, and Pd. Physical Review B, 9(12):5056–5070, 1974.
[79] Zhibin Lin, Leonid V. Zhigilei, and Vittorio Celli. Electron-phonon coupling and electron
heat capacity of metals under conditions of strong electron-phonon nonequilibrium. PhysicalReview B - Condensed Matter and Materials Physics, 77(7):1–17, 2008.
[80] Emrah Turgut, Chan La-O-Vorakiat, Justin M. Shaw, Patrik Grychtol, Hans T. Nembach,Dennis Rudolf, Roman Adam, Martin Aeschlimann, Claus M. Schneider, Thomas J. Silva,
Margaret M. Murnane, Henry C. Kapteyn, and Stefan Mathias. Controlling the competition between optically induced ultrafast spin-flip scattering and spin transport in magnetic
multilayers. Physical Review Letters, 110(19):1–6, 2013.
[81] Zhensheng Tao, Cong Chen, Tibor Szilvási, Mark Keller, Manos Mavrikakis, Henry Kapteyn,
and Margaret Murnane. Direct time-domain observation of attosecond final-state lifetimes inphotoemission from solids. Science, 6793(June):aaf6793, 2016.
[82] N. Kazantseva, U. Nowak, R. W. Chantrell, J. Hohlfeld, and A. Rebei. Slow recovery of themagnetisation after a sub-picosecond heat pulse. Epl, 81(2), 2008.
[83] A. Traeger, G., Wenzel, L., Hubert. Computer experiments on the information depth andthe figure of merit in magnetooptics. Phys. Stat. Sol. (a), 131(1):201–227, 1992.
[84] P Hohenberg and W. Kohn. The Inhomogeneous Electron Gas. Phys. Rev., 136(3B):B864–
B871, 1964.
[85] W. Kohn and L. J. Sham. Self-Consistent Equations Including Exchange and Correlation
Effects. Physical Review, 140(4A):A1133–A1138, 1965.
[86] O.
Ericksson,
A.
Berman,L.
Bergqvist,
andJ.
Hellsvik.
Atomistic spin-dynamics; foundations and applications. Oxford Scholarship, 2017.
[87] Justin M Shaw, Erna K Delczeg-czirjak, Eric R J Edwards, Yaroslav Kvashnin, Danny
Thonig, Martin A W Schoen, Matt Pufall, Michael L Schneider, Thomas J Silva, Olof Karis,
Katherine P Rice, Olle Eriksson, and Hans T Nembach. Magnetic damping in sputterdeposited Co 2 MnGe Heusler compounds with A2 , B2 , and L21 orders : Experiment andtheory. Phys Rev B, 97:094420, 2018.

109[88] R. F.L. Evans, U. Atxitia, and R. W. Chantrell. Quantitative simulation of temperaturedependent magnetization dynamics and equilibrium properties of elemental ferromagnets.
Physical Review B - Condensed Matter and Materials Physics, 91(14):1–7, 2015.
[89] M. Hofherr, S. Moretti, J. Shim, S. Häuser, N. Y. Safonova, M. Stiehl, A. Ali, S. Sakshath,J. W. Kim, D. H. Kim, H. J. Kim, J. I. Hong, H. C. Kapteyn, M. M. Murnane, M. Cinchetti,
D. Steil, S. Mathias, B. Stadtmüller, M. Albrecht, D. E. Kim, U. Nowak, and M. Aeschlimann.
Induced versus intrinsic magnetic moments in ultrafast magnetization dynamics. PhysicalReview B, 98(17):174419, 2018.
[90] R. Gort, K. Bühlmann, S. Däster, G. Salvatella, N. Hartmann, Y. Zemp, S. Holenstein,
C. Stieger, A. Fognini, T. U. Michlmayr, T. Bähler, A. Vaterlaus, and Y. Acremann.
Early Stages of Ultrafast Spin Dynamics in a 3d Ferromagnet. Physical Review Letters,
121(8):087206, 2018.
[91] X. Y. Dong, C. Adelmann, J. Q. Xie, C. J. Palmstrøm, X. Lou, J. Strand, P. A. Crowell,
J. P. Barnes, and A. K. Petford-Long. Spin injection from the Heusler alloy Co2 MnGe into
Al0.1 Ga0.9 AsGaAs heterostructures. Applied Physics Letters, 86(10):1–3, 2005.
[92] A.F. Andreev. The Thermal Conductivity of the Intermediate State in Superconductors.
Soviet Physics JETP, 19(5):1228–1231, 1964.
[93] Joern N. Kupferschmidt and Piet W. Brouwer.
Andreev reflection at halfmetal/superconductor interfaces with nonuniform magnetization. Physical Review B,
83(1):014512, 2011.
[94] Z. Sun, Q. Wang, J. F. Douglas, H. Lin, S. Sahrakorpi, B. Barbiellini, R. S. Markiewicz,
A. Bansil, A. V. Fedorov, E. Rotenberg, H. Zheng, J. F. Mitchell, and D. S. Dessau.
Minority-spin t2g states and the degree of spin polarization in ferromagnetic metallic La22xSr1+2xMn2O7(x = 0.38). Scientific Reports, 3(3167), 2013.
[95] J. J. Attema, L. Chioncel, C. M. Fang, G. A. de Wijs, and R. A. de Groot.
Half-Metals: Challenges in Spintronics and Routes Toward Solutions, volume 678. Springer,
2005.
[96] M E Fisher. The theory of equilibrium critical phenomena. Rep. Prog. Phys., 30:615–730,
1967.
[97] T Pincelli, V Lollobrigida, F Borgatti, A Regoutz, B Gobaut, C Schlueter, T Lee, D J Payne,
M Oura, K Tamasaku, A Y Petrov, P Graziosi, F Miletto Granozio, M Cavallini, G Vinai,
R Ciprian, C H Back, G Rossi, M Taguchi, H Daimon, G Van Der Laan, and G Panaccione.
Quantifying the critical thickness of electron hybridization in spintronics materials. Nature
Communications, 8(16051):1–8, 2017.
[98] V. V. Kruglyak, S. O. Demokritov, and D. Grundler. Magnonics. Journal of Physics D:
Applied Physics, 43(264001), 2010.
[99] K. Wagner, A. Kákay, K. Schultheiss, A. Henschke, T. Sebastian, and H. Schultheiss.
Magnetic domain walls as reconfigurable spin-wave nanochannels. Nature Nanotechnology,
11(5):432–436, 2016.

110[100] Yu-chen Cheng, Chih-hsuan Lu, and Yuan-yao Lin. Supercontinuum generation in a multiplate medium. Optics Express, 24(7):462–465, 2016.
[101] D A Shirley. High-Resolution X-Ray Photoemission Spectrum of the Valence Bands of Gold.
Physical Review B, 5(12):4709–4714, 1972.
[102] G. Salvatella, R. Gort, K. Bühlmann, S. Däster, A. Vaterlaus, and Y. Acremann. Ultrafast
demagnetization by hot electrons: Diffusion or super-diffusion? Structural Dynamics, 3(5),2016.
[103] H. Ebert, D. Ködderitzsch, and J. Minár. Calculating condensed matter properties using the
KKR-Green’s function method - Recent developments and applications. Reports on Progress
in Physics, 74(9):096501, 2011.
[104] Paul Soven. Coherent-potential model of substitutional disordered alloys. Physical Review,156(3):809–813, 1967.
[105] G. M. Stocks, W. M. Temmerman, and B. L. Gyorffy. Complete solution of the KorringaKohn-Rostoker coherent-potential- approximation equations: Cu-Ni Alloys. Physical Review
Letters, 41(5):339–343, 1978.
[106] V. Antropov, M. Katsnelson, B. Harmon, and M. van Schilfgaarde. Spin dynamics in magnets:
Equation of motion and finite temperature effects. Physical Review B - Condensed Matterand Materials Physics, 54(2):1019–1035, 1996.
[107] B. Skubic, J. Hellsvik, L. Nordström, and O. Eriksson. A method for atomistic spin dynamics simulations: Implementation and examples. Journal of Physics Condensed Matter,
20(31):315203, 2008.
[108] J. H. Mentink, J. Hellsvik, V. Afanasiev, B. A. Ivanov, A. Kirilyuk, A.V. Kimel, O. Eriksson,
M. I. Katsnelson, and Th. Rasing. Ultrafast Spin Dynamics in Multisublattice Magnets.
Physical Review Letters, 108(057202), 2012.
[109] G. Malinowski, F. Dalla Longa, J. H.H. Rietjens, P. V. Paluskar, R. Huijink, H. J.M. Swagten,
and B. Koopmans. Control of speed and efficiency of ultrafast demagnetization by directtransfer of spin angular momentum. Nature Physics, 4(11):855–858, 2008.

Appendix A

Details of the Angle Resolved Photoemission Experiments

A.1

Experimental Details of the ARPES Experiments
We use a multi-pass Ti:Sapphire laser system to generate 28 fs pulses at a wavelength of

780 nm (1.6 eV) with pulse energy of 2.0 mJ and a repetition rate of 4 kHz. Most of the laser
energy (95%) is used for high-harmonic generation (HHG), while a small portion (5%) is used toexcite the sample as the pump. To generate HHG, we send 95% of the laser power into a 200 µm
thick β-phase barium borate (BBO) crystal, which produces pulses at a wavelength of ≈390 nm
with pulse energy ≈ 0.3 mJ. The residual 780 nm light is removed from the beam path using two
dichroic mirrors after the BBO. The 390 nm light is then focused using a lens with 50 cm focal
distance into a 1-cm-long capillary waveguide. The waveguide has an inner diameter of 150 µm
and is filled with Xe at a pressure of ≈20 Torr or Kr at a pressure of ≈60 Torr. The HHG beam is
focused onto the nickel sample in the UHV chamber using a toroidal BK7 mirror to a spot size of≈100 m in full width of half maximum (FWHM). Any residual driving laser light is blocked by a
200 nm thick polycrystalline Al filter. The excited photoelectrons are collected by a hemisphericalelectron analyzer. The energy resolution is calibrated using the photoemission spectrum from a
polycrystalline Cu at the room temperature and yields ≈160 meV. For ARPES, the atomicallyclean Ni(111) surface is obtained in-situ using repeated cycles of Argon ion sputtering (0.5 keV) atroom temperature, followed by annealing to 900 K for 15 minutes.
The linearly polarized 780 nm pump beam is recombined collinearly with the HHG beamusing an annular silver mirror and focused onto the sample surface. The pump fluence is varied by

112
the combination of a half waveplate and a polarizer, as shown in Fig. 4.5A. The pulse durationsof the pump-probe experiment is directly measured in-situ using the laser-assisted photoemission(LAPE) method. In the LAPE measurement, p-polarized pump light is focused onto the sample
surface. When the pump pulse temporally overlaps with the HHG-probe pulse, a side band can beclearly observed at an energy 1.6 eV away from the main photoemission peak, which allows us to
extract the cross-correlation of the pump and probe pulses. The side band intensity as a functionof td obtained from LAPE measurement is shown in fig. S1B. We note that in order to avoid
the influence of LAPE signals on our measurements of ultrafast charge and exchanging splitting
dynamics, we switch to s-polarized IR pump in the other two measurements.

A.2

Dynamics of electron temperature
The photoelectron spectra of Ni(111) along direction are measured using ≈16 eV photons

from HHG. After Shirley background subtraction [101], the photoemission intensity across EF canbe described by the Fermi-Dirac (FD) distribution multiplied by the density of states of the material.
In order to extract the time-dependent electron temperature (Te ), we fit the photoemission intensity(I) integrated from k≈ 0.85 Å−1 to k≈ 1.3 Å−1 as shown in Fig. A.1A to the function,
I(td ) = {A(td ) × F D[EF (td ), Te (td )] × DOS[EF (td ) − EB , σL ]} × Gauss(σG )(A.1)

where A is the amplitude parameter, FD the Fermi-Dirac function. DOS in Eq. A.1 is the densityof-states function, modeled as a Lorentzian peak with a peak center at EF -EB and a peak width ofσL . Here, EB is the binding energy of the state. The function is finally convolved with a Gaussianfunction (Gauss) with a width σG , which represents the energy resolution of ≈160 meV in our
experiments. The fitting error of the photoemission intensity to Eq. A.1 is defined as σf it .
We note that with pump fluences F>2 mJ/cm2 , the space charge effects caused by surface
charging and multi-photon photoemission induced by the IR pump laser are inevitable, and inducetime-dependent energy shifts of the photoemission spectra in the Tr-ARPES experiments [101].
At the same time, because we are only interested in the photoemission intensity within ≈1 eV

113

Figure A.1: Electron temperature fitting. (A) Photoemission spectrum of Ni(111) excited by 16 eV
HHG photons at room temperature. The white dash-dotted lines represent the momentum range
for the photoemission intensity in the analysis for the transient electron temperature. (B) and (C)Photoemission intensity and fitting results at td = -500 fs and td = 24 fs for F ≈ 6 mJ/cm2 . The redsolid lines are the fitting results using Eq. A.1 and the blue dashed lines represent the DOS functionused in the fitting. The black dashed-dot lines are the Fermi energy obtained from the fitting. (D)Photoemission intensity as a function of pump-probe time delay from the raw experimental data.
(E) Same as (D), but the EF shifts due to pump-induced space charge effects are corrected.

114
across EF , the space-charge induced energy shifts in this small energy window can be assumed
to be an energy-independent constant. As a result, in Eq. A.1, we model the Fermi energy EF
as a time-dependent parameter, while the binding energy of the state (EB ) is a time independent
parameter. Because the electron temperature Te is determined by the slope of the FD function
across EF , this fact allows us to mostly decouple Te from the space-charge induced energy shifts inthe fitting procedure.
When fitting the photoemission intensities at different td to eq. S1, we first determine thetime independent parameters (EB , σL ) with the spectrum obtained at td before the pump-probe
time zero (e.g. td = -500 fs). Then we can fix these parameters in the following fitting proceduresfor the other td . Typically, EB ≈ 0.15 eV and σL ≈ 0.4 eV are used for all the data sets. The timedependent parameters EF and Te can be reliably extracted using the fitting procedure describedabove. The fitting results for F ≈ 6 mJ/cm2 at td = -500 fs and 24 fs are shown as an example inFig. A.1B and C. The extracted values of EF also allow us to compensate the pump-induced spacecharge effect by assuming EF is constant after pump excitation. Indeed, the chemical potential ofthe majority and minority electrons can be different and shift as the electron temperature varies,
but the amount of energy shift is only several meV even with the electron temperature elevated by≈ 2000 K [22]. The photoemission intensity at different td are plotted without the space-charge
correction in Fig. A.1D and with the correction in Fig. A.1E.
Moreover, the energy resolution of ≈160 meV in our experiments places a lower limit on theelectron temperature we can extracted from the experiments and this limit has to be considered aspart of the errors in our data analysis. In order to determine how much the errors from the energy
resolution contribute, we model a FD distribution with a given temperature Treal and convolve itwith the energy resolution ≈160 meV, which is represented by a Gaussian function. A random
noise of 2% of the maximum intensity is added to model the realistic data. Then the generated
distribution is fit to the same fitting function as Eq. A.1, but with DOS = 1.0. The electron
temperature extracted from this fitting procedure is defined as Tf it . We found when the Treal <200K, the fitting procedure we used failed to extract the correct electron temperature, defining the115lower limit of the electron temperature we can extract from the experiment with ≈160 meV energy
resolution. But this uncertainty is decreased to only <10 K when the electron temperature is above
the room temperature (300 K). The energy-resolution-induced uncertainty of electron temperatureis also taken into consideration as σres = |Treal − Tf it | in the final data plotting. The total error ofq
2 + σ 2 . We note that in theelectron temperature (Fig. 4.3B and 4.4A) is defined as σtot = σresf ittemperature range in our experiments (>300 K), σtot has a primary contribution from σf it .

A.3

Dynamics of the electron population at 1.6 eV
The electron population change below and above EF can be directly measured using Tr-

ARPES. The change of photoemission intensity taking the ground-state intensity (td = -500 fs)
as the reference is plotted in Fig. A.2A. We extract the electron population dynamics in two
different regions with ∆E = 0.2 eV as the size of energy window at: (a) E = 0.1 eV and (b) E =1.6 eV, as shown in Fig. A.2A. The change of electron population is measured by the integrated
photoemission intensity in each energy window, taking the values before pump-probe time zero asRR
the reference:∆n(td ) ∝ ∆E I(E, td )dE − ∆E I(E, td < 0)dE. We also normalize n to the electronR
population of nickel in the conduction band before the time zero: n0 ∝ ∆E I(E = −0.1, td , 0)dE.
∆n/n0 for regions (a) and (b) with pump fluence F≈6 mJ/cm2 is plotted in fig. A.3B. We findwhile the population change at E=0.1 eV reaches 8% of the ground state band electron population
density, the increment at E=1.6 eV is only ≈0.8% in maximum.
The electron density directly excited by the IR pump laser can be calculated by ∆n = F (1−R)δ~ω
assuming single-photon excitation, where F is the pump fluence, R the reflectivity, δ the skin depthand ~ω =1.6 eV the photon energy. The values of R and δ are given in table S1. We find the electron
density directly excited by pump photons is ≈4.5 nm−3 for the pump fluence of ≈6 mJ/cm2 . Thisvalue corresponds to ≈0.55% of the d band electron density in nickel, if we consider there are 9
d electrons per nickel atom, which is in good agreement with the value we obtained at E = 1.6
eV above EF . From Fig. A.2B, we find that the electron population at E = 1.6 eV rises at thesame time as those at E=0.1 eV and its population is only ≈0.8% of the band electrons, excluding

116
strong influence directly from the photo-excited non-equilibrium electron population to our electron
temperature measurement. Our results suggest the electron bath in nickel thermalizes in a very
short time after pump excitation, consistent with strong electron-electron interactions reported inprevious works [68, 20].

Figure A.2: Electron population dynamics. (A) The relative change of photoemission intensity as
a function pump-probe time delay, taking the intensity before pump excitation (td = -500 fs) as
the reference. The yellow dashed boxes represent the two regions of energy where the changes ofelectron population are extracted. (B) Normalized electron population as a function of pump-probe
time delay extracted from (a) and (b) regions in (A).

A.4

Dynamics of Exchange Splitting
The exchange splitting can be clearly resolved at k≈ 1.05Å−1 in the photoemission spectra

excited by ≈16 eV HHG photons as shown in Fig. A.1C of the main text. In order to extract theexchange splitting and improve the signal to noise ratio in the data analysis, the photoemission
intensity is extracted along the direction normal to the band dispersion (≈ 10 deg tilted from theconstant momentum direction) as shown in Fig. A.3A and B. The photoemission intensity (I) after
Shirley-background subtraction [101] is hence fitted to the function
0GL
I(E) = Σm=1,2,3 Am × V (E; Em
, σm
, σm
)

(A.2)

117
0 , σG , σL )where E is the photoelectron energy, m represents the index of the Voigt functions, V (E; Emm m
0 the center energy,σ G the Gaussian width and σ L theis the normalized Voigt function with Emmm

Lorentzian width of the mth Voigt function, and Am is the amplitude of the mth Voigt function.
The typical fitting results of photoemission intensity are shown in Fig. A.3C - F for fluence below
and above Fc and at different times. As shown in Fig. A.3C, the first Voigt peak (m=1) represents a
low energy d band with a binding energy of ≈0.6 eV, which does not change after pump excitation.
The second and third Voigt peaks (m=2, 3), on the other hand, correspond to the majority andminority d bands that cross the Fermi energy at k≈ 1.05Å−1 [102]. When fitting the results to extract the exchange splitting change as a function of pump-probe time delay, we 1) fix the Gaussian
width of all Voigt peaks considering the energy resolution of ≈160 meV; 2) fix the parameters of
the third Voigt function (m=3) and the width of the second Voigt function (m=2), using the values
obtained from the ground-state fitting (td = -500 fs). As a result, we have 5 fitting parameters(A1 , E10 , σ1L , A2 , E20 ) and the exchange splitting is defined and extracted as Eex = E10 − E20 . The
extracted Eex is 240 meV in the ground state, which is consistent with previous works at a similar
Brillouin-zone (BZ) point [102].
We note that our analysis shows that Eex does not vanish after pump excitation. (Sameas the sample is heated up across the Curie temperature, see below.) It has been shown thatthe absolute value of exchange splitting measured in the spin-integrated photoemission depends
on the model and the fitting strategy [3]. In our work, the 160 meV energy resolution, as wellas the fitting model in which two peaks (majority and minority bands) are always assumed toexist, post the limit of our ability to resolve the collapse of the exchange splitting beyond ∆Eex
= 60 meV. In order to extract reliable dynamics of the exchange splitting, we consistently used
the same fitting strategy as described above for different time delays as well as different fluences.
The fact that the extracted dynamics of exchange splitting can agree with the demagnetization
process measured using Tr-TMOKE method (Fig. 4.2A and D) also validates our fitting strategy
and strongly suggests that the dynamics of Eex can represent the dynamics of magnetization after
pump excitation. At the same time, the difference of exchange splitting for fluence below and above

118

Figure A.3: Analysis of exchange splitting. (A) and (B) Photoemission spectra of Ni(111) before(td = -500 fs) and after (td = 500 fs) pump excitation. The solid black boxes represent the regions
the photoemission intensities are extracted for the analysis on the exchange splitting. (C)-(D) The
photoemission intensities and the Voigt function fitting (Eq. S2) results for different pump fluencesand different pump-probe time delays. The red lines are the overall fitting results. The Cyan, blueand green curves represent the extracted Voigt peaks for m=1, 2 (majority band) and 3(minorityband).

119
Fc at td = 2 ps can be clearly resolved from the raw experimental data as shown in Fig. A.3D andF.
The change of the exchange splitting (∆Eex ) as a function of td under different pump fluence
are fit to Eq. 3.2. We find the results can be fit with a set of universal time constants: τdemag =176±27 fs, τrecover1 =537±173 fs and τrecover2 =26±11 ps, as plotted in Fig. A.4.

A.5

Temperature dependence of exchange splitting in static ARPESThe static photoelectron spectrum along direction measured using He Iα photons (hν =21.218 eV) from helium discharge lamp at room temperature is shown in Fig. A.5A. The exchange
splitting (Eex ) between the majority and minority bands of Ni can be clearly observed at the
momentum ≈ 1.05 Å−1 where its d band crosses the Fermi energy (EF ). The values of Eex can bequantitatively extracted using the same fitting procedures described in the previous section. We
find Eex reduces as the sample temperature approaches the Curie temperature ( 631 K) as shownin Fig. A.5B, indicating the quenching of the local magnetic moments [1, 102].

120

Figure A.4: Global fitting of the exchange splitting dynamics. The global fitting results of theexchange splitting dynamics to Eq. 4.2 for a range of pump fluences (F = 1.17 mJ/cm2 to F =14.12 mJ/cm2 ).

Figure A.5: Collapse of exchange splitting at the Curie temperature. (A) Static photoemission
spectrum excited by He Iα photons (hν = 21.218 eV) at the room temperature. Right panel:
Energy distribution curve extracted at the momentum where the red dashed line is located. The
exchange splitting (Eex ) can be clearly extracted using fitting procedure described in the previous
section. (B) Eex change reduces as the sample temperature increases.

Appendix B

Details of the Heusler dynamics and corresponding DFT calculations

B.1

Further details of the experimental setup
Figure B.1 plots the experimentally measured static asymmetry for the A2 and B2 phases.

Note that in the B2 phase, the magnetic signal from Co is smaller than the asymmetry signal in
the A2 phase, while the Mn signal is larger in the B2 phase than the A2 phase. In Fig. S2B, weshow that the predicted magnetic moment per atom agrees with experimental measurements forboth the Mn and Co magnetic asymmetry signal, which validates the use of our DFT calculations
to explain the dynamical measurements (see also section 6 of this appendix).

Figure B.1: (A) Static asymmetry measurements for the two phases of the material. The magnitudeof the magnetic signal in Co is larger in the A2 phase, while that of Mn is larger in the B2 phase.
(B) DFT calculation of the magnetic moment for each element in compound. The increase inmagnetic moment for Co and decrease in the magnetic moment for Mn in the A2 phase (comparedto the B2 phase) is consistent with these values.

122

B.2

Sample preparation
Samples were dc magnetron sputter deposited at room temperature with the following thinfilm structure: SiO2 / 5 nm Ta / 10 nm Co2 MnGe (CMG) / 2.8nm Ta. The CMG layer was formedby co-sputtering from a Co2 Mn target and the pure Ge target. The CMG is quasi-amorphous in
the as-deposited state as confirmed by x-ray diffraction. The samples are ex situ annealed in highvacuum at or above 240 C to form a crystalline structure. Below 240 C, the structure remains
quasi-amorphous with high resistivity and low magnetic moment. We previously showed in Ref.
[87] that the A2 structure, which is metallic, dominates at annealing temperatures at or below
280 C. However, above 300 C, the structure is B2 with a half metallic bandstructure. Both the
crystalline structure and half-metallic electronic structures were verified through x-ray diffraction,
magnetometry and ferromagnetic resonance spectroscopy [87].

B.3

Change in reflectivity due to optical excitation in both phases
To further verify that the transient enhancement observed in Fig. 6.2A does not arise fromany optical artifacts such as a change in the refractive index due to electronic contributions, wemeasured the change in the reflectivity of the sample with s-polarized light at the Co M-edge andcompared this signal in the B2 phase (where the transient enhancement occurs), to the A2 phase(which has identical chemical composition but no transient enhancement). When measuring in ourgeometry (as shown in Fig. 3.2) with s-polarization, there are no magnetic contributions to thechange in reflectivity. This transient reflectivity is plotted in Fig. B.2. Note that the magnitudeof the signals (to within error) are identical. This confirms that the findings in the main text arise
from changes in the magnetic moment of Co in the B2 phase.

B.4

Element averaged response of the A2 and B2 phases
Given that the element resolved MOKE signal has all information on the total magnetizationof the sample (to within 3 orders of magnitude), we can use it to construct the element averaged

123

Figure B.2: Change in reflectivity measured for both the A2 and B2 phases. Note that to within
experimental error, the signal is identical for the two samples. This confirms that the non-magnetic
contribution to the signal is the same, while the magnetic signal is the cause of the transientenhancement.

124
signal (that would be measured by an element averaging probe such as visible MOKE). This signal
is plotted in Fig. B.3 along with the element specific response. Note that for the element averaged
signal, no transient enhancement would be detected.

B.5

Dynamics of Co2 MnGe on α-Al2 O3
To confirm that the transient enhancement of Co in Co2 MnGe in the B2 phase is ubiquitousfor the material, and not influenced by the choice of substrate, we grew samples of CMG on sapphirerather than SiO2 . The composition was: 2.8 nm Ta / 10 nm CMG / 5 nm Ta / α-Al2 O3 . The results
are shown in Fig. B.4. Due to a more transparent stack with less total fluence absorbed by the
CMG, both the quenching and magnitude of the transient enhancement observed was significantly
smaller than observed on the sample deposited on SiO2 . Note that the enhancement is still veryclearly noticeable.

B.6

Method for calculating density of states and magnetic moments fromdensity functional theory
The density of states and magnetic moments of B2 and A2 phases of Co2 MnGe has been

calculated by means of density functional theory [84, 85] as implemented in the spin-polarized
relativistic Korringa-Kohn-Rostoker (SPR-KKR) code [98]. The chemical disorder were treatedwithin the coherent potential approximation (CPA) [103, 104] We used the Vosko-Wilk-Nusair [105]version of the local spin density approximation for the exchange-correlation functional. The shapeof the potential was considered by using both the atomic sphere approximation (ASA) solving
the full relativistic Dirac equation (to obtain orbital and spin magnetic moments) and the fullpotential (FP) scheme combined with scalar relativistic approximation (to calculate DOS). We
sample the irreducible wedge of the Brillouin zone with 1500 k-points for the magnetic moment
calculations and 16000 for the density of states (DOS) calculations. For all simulations the s, p,
d, f orbitals have been included in the basis set (lmax = 4). All parameters were calculated fora cubic Heusler structure at the experimental lattice parameter of Co2 MnGe = 5.814 angstroms.

125

Figure B.3: Total magnetization of sample plotted with element resolved signal. Total signal
constructed from the weighted contributions of the Mn and Co magnetizations measured with
TMOKE.

Figure B.4: Element resolved dynamics of Co2MnGe (B2 phase) on sapphire.

126
Neither the tetragonal distortion and change in lattice parameter nor the 2 percent Ge excess has
significant effect on the estimated parameters.

B.7

Probabilities for laser-driven transitions in the B2 and A2 phases
To gain insight into the laser-induced electronic excitations, we calculated the transition

probabilities, shown in Fig. 6.4. These results were obtained from the DOS curves obtained for
the B2 and A2 phases, shown in Fig. 6.3. This information was deduced by employing the Fermi’s
Golden rule within the dipole approximation:I(ω) =

2π X
hΨf,k |del|Ψi,k i δ(Ef,k − Ef,k − ω)
~

(B.1)

k,i,f

where I(ω) stands for the absorption intensity of the incident light with energy ω and |Ψi,k i (|Ψf,k i)
are the occupied (unoccupied) Bloch states, characterized by wavevector k and the corresponding
eigenenergy Ei,k (Ef,k ).
By projecting the Bloch states onto the atomic site a and angular momentum l and spin σ(|a, l, i), one can decompose the intensity given by Eq. B.1 into the atom-resolved contributions:I(ω) ≈

2π X X X
hΨf,k |a, l, σi ha, l, σ|del|a0 , l0 , σ 0 i ha0 , l0 , σ 0 |Ψi,k i δ(Ef,k − Ei,k − ω)h0 0 0α,l,σ α ,l ,σ

(B.2)

k

Furthermore, we can neglect the spin-flip transitions due to the smallness of spin-orbit coupling
and take advantage of the fact that the matrix elements of the gradient operator for Co and Mn
states are similar to each other (Mn 3d-4p transition matrix elements are 6 percent larger than thatfor Co). Then we end up with the following expression:Z
2π X X EF
occ
unocc
0
I(ω) ≈dω 0 Nα,l0 ,ω,k Ṅα,l0 ,σ,k (ω + ω)δl−l0 −1
h
−
inf
0

(B.3)

α,l,l ,σ k

The DOS of occupied states is multiplied with a Fermi-Dirac distribution fF −D (ω) at room temperature and that of unoccupied with (1-fF −D (ω)).
As one can see, the DOSs for the same k enter the summation, meaning that only k-conserving
transitions are taking place, as usual for optical excitations. We have also verified that averaging

127
the k-resolved DOS before summing them up marginally changes the results. This will be shown in
Section B8 below. Thus, we employed a further simplified formula for the analysis of experimental
data:

Z EF
2π X
occ
unocc 0I(ω) ≈dω 0 Na,l,σ(ω 0 )Ṅa,l0 ,σ (ω + ω)δl−l0 −1
h
−
inf
0

(B.4)

α,l,l ,σAn analysis based on such a simple expression allows for a straightforward interpretation of theexperimental data at least on a qualitative level. Most importantly, it correctly captures thedifferences in the behavior of Co and Mn magnetic moments after the laser irradiation.
We have found that the main contribution comes from the transitions between d and p states(s-p and d-f transitions are less intense). Thus, to calculate the transition probabilities, we used
the projected DOS for the B2 and A2 phases, shown in Fig. B.5.

Figure B.5: Element and orbital specific DOS for (a) B2 and (b) A2 phases. Note that the porbitals are an order of magnitude smaller than the d orbitals, however transitions take place fromp to d or d to p orbitals.

The obtained atom-, orbital-, and spin-resolved probabilities for the B2 phase are shown in
Fig. B.6. Here one can see that around the experimental pump energy of 1.5 eV the Mn d-ptransitions are dominantly of spin-up character. The same behavior is seen for Co d-p transitions,but it is less pronounced and further compensated by the p-d transitions, which favor spin-down

128
excitations. To set the net effect, we combined these probabilities and showed the total averagespin-resolved intensities for Co- and Mn-derived excitations. These results are given in Fig. 6.4.
The intensity of the transitions from the initial to the final state are provided by the electric dipole
transitions and are allowed if a sufficient amount of d → p and p → d transitions are allowed.
Hence, Figs. 6.4C and B.5 suggests that p-states of Mn and Co that hybridize with d-states arevery important in order to enable sufficiently many transitions (illustrated by the red wiggly line in
Fig. 6.4C). These considerations enable tailoring the optimal condition for spin-moment transferfrom one sublattice to another, and could help to find new compounds with even more extreme
effects compared to what is reported here in this work.
Next we compute the transition probabilities in the A2 phase. Here both Co and Mn can occupy multiple sites, and probabilities for all site are computed. Note that here the spin polarization
at the Fermi level is lost, and as a result transitions in the Mn minority channel become equallyprobable as those in the majority. The atom-, orbital-, and spin-resolved probabilities are plottedin Fig. B.7 The total minority and majority band transitions for both Co and Mn are plotted inFig. B.8. The probabilities have been normalized by the magnetic moments computed with DFT
for both phase (so that there can be direct comparison between A2 and B2 phase probabilities). In
the case of Cobalt, transitions from the minority band remain preferred as in the B2 phase. Becauseof this, we note that the dominant mechanism for spin transfer must come from the blocked spinchannel in Mn.

B.8

Results of the calculations of k-conserving transition probabilities for
the L21 phase
Here we compare the simulated transition probabilities, calculated using Eq.B.3 and Eq. B.4.

The difference between them is that in the former case only k-conserving transitions are calculated,
whereas in the latter case also non k-conserving contributions enter, due to the two independent
k-summations of DOSs. In Fig. B.9 we compare the results for the L21 phase obtained with these
two approaches. As one can see, the results are quite similar and are practically identical around the129

Figure B.6: Transition probabilities for the dipole allowed transitions as a function of photon energy
in the B2 phase of Co2 MnGe.

130

Figure B.7: Transition probabilities for the dipole allowed transitions in the A2 phase.

Figure B.8: Transition probabilities for the dipole allowed transitions in the A2 phase.

131

Figure B.9: Transition probabilities for the dipole allowed transitions in the A2 phase.

132
energy of interest, which is 1.5 eV. Since Eq. B.4 is easier to implement, we used it for computing
the intensities for the A2 and B2 phases.

B.9

Results of atomistic Landau-Lifshitz-Gilbert simulations

B.9.1

Numerical method

We consider classical atomic magnetic moments mi =mi (ei ) at site i. The dynamics of mi isgoverned by the atomistic Landau-Lifshitz-Gilbert (aLLG) equation [106]
∂mi
α δmi
= m × (−γ(Bi + bi
))
∂t
m ∂t

(B.5)

∂H
is the effective precession field related to the
Where γ is the gyromagnetic ratio and Bi = ∂mispin-Hamiltonian, employing a Heisenberg model where
H=−

X

Ji,j mi ṁj − µB B

i,j

X

mi

(B.6)

i

Here, the magnetic moments at site i and j are coupled by the exchange parameter Ji,j . The valuesof the Heisenberg exchange parameters were obtained from first principles electronic structure
theory. In the expression above, B is the external magnetic field. From the fluctuation-dissipation
theorem, thermal fluctuations enter by a stochastic field, bi , that fulfills white noise properties, suchas hbi i=0 and hbµi (t)bνk (t0 )i = Dδi,j δµν δ(t − t0 ), with the fluctuation amplitude D=2αkb T/γ m .
The dissipation part enters the equation of motion via a viscous damping part scaled by the Gilbert
damping constant α. We address in these simulations the demagnetization as a pure thermal effectin which the temperature of the magnetic sub-system is determined by
T = T0 + (Tp − T0 )(1 − e

− τt

1

) + (Tf − T0 )(1 − e

− τt

2

)

(B.7)

In this expression, T0 is the initial temperature, where Tp is the peak temperature and Tf is the
final temperature. Furthermore, τ1 and τ2 are the relaxation times. Note that T0 , Tp , Tf , τ1 and τ2
are parameters that determine the profile of the temperature of the spin system. For simplicity, we
set T0 = Tf = 300 K, since the simulations were done in order to capture experiments performedat room temperature.

133
We calculate the dynamics of classical magnetic moments in a simulation box of 10 x 10 x
10 (1000) atoms and with 10 replica, using the UppASD software [107]. To guaranty numericalstability, we use a step width of dt = 0.1 fs. Element specific properties are obtained by performing
chemical specific averages. In order to reduce even further the numerical noise, an averaging intime is performed for a short period of 100 steps. This period is short enough to not violate
non-ergodicity of the process.

Figure B.10: Element resolved demagnetization in Co2MnGe for the L21, A2 and B2 phase. The
peak temperature of the simulations is set to 1000 K, whereas the relaxation times of the temperature profile are τ1 = 0.01 ps and τ2 = 1.5 ps.

The results of the simulations are shown in the tables following this section. In contrast to
the experimental findings, Co demagnetizes faster than Mn. This is because the magnetic momentof the Co atom is smaller than the Mn moment, in all three crystal phases (L21: µCo = 0.94 µB
and µMn = 3.13 µB, A2: µCo = 1.53 µB and µM n = 2.54 µB and B2: µCo = 0.93 µB and µM n =3.16 µB), and it can be argued that larger moments have slower dynamics according to the aLLG
equation [108]. We have furthermore made a fit of the simulated data with a double exponentialfunction (also used in [109]);M (t)= A0 − A1 e−t/τi − A2 e−t/τf
M (0)

(B.8)

The resulting fitting parameters are listed in table B.1 The relaxation time, τi , differs strongly
between the different phases and is lowest for the L21 phase. In the A2 and B2 phase, the relaxation

Phase
L21
A2
B2

Co
Mn
Co
Mn
Co
Mn

A0
1.0
1.0
0.99
0.99
1.0
1.0

A1
2.13
6.11
0.43
0.40
1.08
1.11

τi (ps)
2.13
1.28
4.92
6.38
5.23
5.17

A2
-2.03
-6.11
-0.46
-.44
-0.87
-1.10

τf (ps)
0.77
1.06
0.48
0.68
0.84
1.03

134

Table B.1: Fitting parameters of the double exponential function to fit M(t)/M(0).

time is comparable. The relaxation time for remagnetization, τf , is also similar for all three phases.
However, both relaxation times depend very strongly on the dissipation term in the equation of
motion. In order to apply a complete model, we used type-resolved Gilbert damping parameters,
which are listed in tables B.1 - B.4 for the particular different phases. The overall results of Fig.
B.10 show that the experimental features discussed in the main part of this report, in particular
the initial increase of magnetic moment of the Co atoms when the B2 phase demagnetizes, are not
reproduced by the aLLG equation. Although some features of the magnetization dynamics seem tobe captured by theory at longer time-scales (250 fs and longer), e.g. reaching a minimum at 1.5-2 pswith M/M(0) being close to 0.5-0.6, there is a marked difference between theory and experiments forthe initial phase of the demagnetization process. This finding is in contrast to earlier calculations forfcc Ni [88] and CoFeB as well as FePt [89], where theory of this level of approximation reproduces
experiments with rather good accuracy. The deviation between experiment and aLLG theory
highlights the effects of electronic excitations, that are discussed in chapter 6 of this thesis.

Site
1
2
3
4

Element
Co
Co
Mn
Ge

Conc
1.0
1.0
1.0
1.0

α
0.008019
0.008019
0.011883
0.001777

Table B.2: Site resolved Gilbert damping parameter in the L21 phase.

135
Site
1
1
1
2
2
2
3
3
3
4
4
4

Element
Co
Mn
Ge
Co
Mn
Ge
Co
Mn
Ge
Co
Mn
Ge

Conc
0.5
0.25
0.25
0.5
0.25
0.25
0.5
0.25
0.25
0.4
0.25
0.25

α
0.0080190.003537
0.003033
0.000539
0.003537
0.003033
0.000539
0.003537
0.003033
0.000539
0.003537
0.003033
0.000539

Table B.3: Site resolved Gilbert damping parameter in the A2 phase.

Site
1
2
3
4
4
4

Element
Co
Co
Mn
Ge
Mn
Ge

Conc
1.0
1.0
0.5
0.5
0.5
0.5

α
0.012380
0.012380
0.006437
0.009406
0.006437
0.009406

Table B.4: Site resolved Gilbert damping parameter in the B2 phase.

L21
A2
B2

Element
Co
Mn
Co
Mn
Co
Mn

a
0.45
0.57
0.68
0.70
0.37
0.40

b
0.55
0.44
0.33
0.31
0.58
0.63

C
0.19
0.37
0.39
0.59
0.24
0.55

Table B.5: Fitting parameters for fits performed to the functions used in chapter 6. Note that inthis fitting, the numerical values for the A2 and B2 phases are very close (within error bars) of theexperimentally measured values. This corroborates with our analysis that the LLG equations cancapture the later >200 fs time dynamics of the system.