Present address: ]Grupo de Investigación en Aplicaciones del Láser y Fotónica,
Departamento de Física Aplicada, University of Salamanca, 37008 Salamanca,
Spain Present address: ]European XFEL GmbH, Holzkoppel 4, 22869 Schenefeld,
Germany

# Ultrafast Opto-magnetic Effects in the Extreme Ultraviolet Spectral Range

Martin Hennecke<EMAIL_ADDRESS>Clemens von Korff Schmising  Kelvin
Yao  Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-
Born-Straße 2A, 12489 Berlin, Germany  Emmanuelle Jal  Boris Vodungbo
Valentin Chardonnet  Sorbonne Université, CNRS, Laboratoire de Chimie Physique
– Matière et Rayonnement, LCPMR, 75005 Paris, France  Katherine Légaré
Institut National de la Recherche Scientifique, INRS-EMT, Varennes, Quebec J3X
1P7, Canada  Flavio Capotondi  Denys Naumenko  Emanuele Pedersoli  Ignacio
Lopez-Quintas  [  FERMI, Elettra-Sincrotrone Trieste, 34149 Basovizza,
Trieste, Italy  Ivaylo P. Nikolov  Lorenzo Raimondi  FERMI, Elettra-
Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy  Giovanni De Ninno
FERMI, Elettra-Sincrotrone Trieste, 34149 Basovizza, Trieste, Italy
Laboratory of Quantum Optics, University of Nova Gorica, 5001 Nova Gorica,
Slovenia  Leandro Salemi  Department of Physics and Astronomy, Uppsala
University, P.O. Box 516, SE-751 20 Uppsala, Sweden  Sergiu Ruta  Department
of Physics, University of York, York YO10 5DD, United Kingdom  College of
Business, Technology and Engineering, Sheffield Hallam University, Howard
Street, Sheffield, S1 1WB, United Kingdom  Roy Chantrell  Department of
Physics, University of York, York YO10 5DD, United Kingdom  Thomas Ostler
College of Business, Technology and Engineering, Sheffield Hallam University,
Howard Street, Sheffield, S1 1WB, United Kingdom  Bastian Pfau  Dieter Engel
Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-
Straße 2A, 12489 Berlin, Germany  Peter M. Oppeneer  Department of Physics and
Astronomy, Uppsala University, P.O. Box 516, SE-751 20 Uppsala, Sweden  Stefan
Eisebitt  Max-Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie,
Max-Born-Straße 2A, 12489 Berlin, Germany  Institut für Optik und Atomare
Physik, Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin,
Germany  Ilie Radu<EMAIL_ADDRESS>[  Max-Born-Institut für Nichtlineare
Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489 Berlin, Germany

###### Abstract

Coherent light-matter interactions mediated by opto-magnetic phenomena like
the inverse Faraday effect (IFE) are expected to provide a non-thermal pathway
for ultrafast manipulation of magnetism on timescales as short as the
excitation pulse itself. As the IFE scales with the spin-orbit coupling
strength of the involved electronic states, photo-exciting the strongly spin-
orbit coupled core-level electrons in magnetic materials appears as an
appealing method to transiently generate large opto-magnetic moments. Here, we
investigate this scenario in a ferrimagnetic GdFeCo alloy by using intense and
circularly polarized pulses of extreme ultraviolet radiation. Our results
reveal ultrafast and strong helicity-dependent magnetic effects which are in
line with the characteristic fingerprints of an IFE, corroborated by ab initio
opto-magnetic IFE theory and atomistic spin dynamics simulations.

Light-driven ultrafast magnetic phenomena occurring on ever faster and smaller
time and length scales are at the core of the modern magnetism research [1,
2]. A long sought-after and yet to be realized phenomenon is the coherent and
deterministic control of a macroscopically ordered spin ensemble on the sub-
cycle timescales of the photo-exciting laser field [3, 4]. In this context,
one of the most prominent effects that could lead to large, ultrafast changes
of magnetization is the inverse Faraday effect (IFE). Discovered originally in
paramagnetic solids [5], the IFE was shown to generate an effective magnetic
field or, equivalently, an induced magnetic moment in a medium upon photo-
excitation with circularly polarized light pulses. According to classical IFE
theory [5], the orientation and the magnitude of the light-induced magnetic
moment are respectively determined by the helicity of circularly polarized
light (or its angular momentum) and the combined effect of the magneto-optical
susceptibility of the photo-excited medium and the incident light intensity.
In case of metallic ferro- or antiferromagnetic materials, a fully ab initio
quantum-mechanical treatment of the IFE has revealed that its action on these
materials is significantly different compared to the originally discovered
effect, predicting a complex dependence of the IFE-induced magnetization on
both helicity and wavelength, which is not covered anymore by the conventional
theory [6, 7, 8]. However, the high absorption of the optical excitation in
these materials typically leads to strong heat-induced demagnetization which
hinders the observation of any potential non-thermally induced magnetization
[9]. Thus, the actual influence of an IFE on the sub-picosecond spin dynamics
observed in such systems as, for example, all-optical switching phenomena in
ferrimagnetic rare-earth transition-metal alloys like GdFeCo [10, 11, 12, 13],
remains highly debated and elusive so far [14].

The magnitude of the IFE depends on the light intensity and its polarization,
as well as on the spin-orbit coupling (SOC) of the involved electronic states
via the wavelength- and helicity-dependent opto-magnetic constant
$\mathcal{K}^{\text{IFE}}_{\sigma_{\pm}}(\omega)$ [6, 8]:

$\vec{M}_{\rm ind}(\omega)=i\,\mathcal{K}^{\rm
IFE}_{\sigma_{\pm}}(\omega)\left[\vec{E}(\omega)\times\vec{E}^{*}(\omega)\right],$
(1)

where $\vec{E}(\omega)$ denotes the light electric field, $\sigma_{\pm}$ its
helicity, and $\vec{M}_{\text{ind}}$ the induced magnetization along the
$\vec{k}$ vector of the incident circularly polarized light beam. The IFE,
which is an electronic Raman process, has thus far been investigated
theoretically [6, 8, 15, 16, 17, 18], but a direct comparison with
measurements is lacking. A potential way of generating strong opto-magnetic
interactions in metallic magnets is the excitation of core-level electrons,
involving states with much higher SOC compared to the valence band (e.g.,
$1.1\text{\,}\mathrm{e}\mathrm{V}$ vs.
$65\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$ in case of Fe 3p and 3d electrons,
respectively [19]) that can be accessed by short wavelength radiation in the
extreme ultraviolet (XUV) or soft x-ray range. The availability of free-
electron lasers (FEL) providing femtosecond pulses of very high brilliance
with control over the light polarization has enabled studies of x-ray induced
magnetization dynamics, extending the capabilities from element-specific
probing, employing, e.g., the x-ray magnetic circular dichroism (XMCD),
towards element-specific excitation of highly non-equilibrium states [20, 21,
22, 23, 24]. However, besides first theoretical predictions, there is no
experimental evidence available so far on the existence of opto-magnetic
effects like the IFE at these XUV wavelengths.

In this Letter, we study the ultrafast magnetization dynamics of ferrimagnetic
GdFeCo alloy induced by photo-excitation with intense femtosecond circularly
polarized pulses in the XUV spectral range generated by the free-electron
laser FERMI. Comparing the helicity-dependent dynamics upon on\- and off-
resonant excitation with respect to the Fe $\text{M}_{3,2}$ resonance, we find
strong dynamic helicity- and wavelength-dependent effects that resemble the
expected characteristics of an IFE. Analyzing the magnitude and wavelength-
dependence of the observed effect, we can rule out a thermal origin caused by
dichroic absorption due to the XMCD, this finding being corroborated by
atomistic spin dynamics (ASD) simulations. Instead, we find qualitative
agreement with ab initio calculations of the IFE in the XUV spectral range.
Our results thus identify the existence of an IFE in the so far unexplored XUV
spectral range.

The time-resolved studies were performed at the DiProI end-station of the
FERMI FEL [25], using a pump-probe geometry as schematically shown in Fig. 1a.
The sample was excited at normal incidence using $\approx
90\text{\,}\mathrm{f}\mathrm{s}$ (FWHM) XUV pulses of variable polarization
[circular right/left ($\sigma_{\pm}$) and linear horizontal (lin. hor.)],
tuned to photon energies around the Fe $\text{M}_{3,2}$ resonance. The XUV-
induced dynamics were probed under $45\text{\,}\mathrm{\SIUnitSymbolDegree}$
by linearly polarized $\approx 90\text{\,}\mathrm{f}\mathrm{s}$ (FWHM) optical
pulses at $400\text{\,}\mathrm{n}\mathrm{m}$ wavelength, generated by a
frequency-doubled Ti:sapphire laser system. A polarization-sensitive balanced
photodetection scheme was employed to measure both the magneto-optical Faraday
rotation in transmission and the Kerr rotation in reflection. The magnetic
contrast was obtained upon flipping an out-of-plane magnetic field of $\pm
8\text{\,}\mathrm{m}\mathrm{T}$ that was magnetically saturating the sample.
The experimental time resolution was evaluated to be $\approx
280\text{\,}\mathrm{f}\mathrm{s}$, using the method described in Ref. 26. Both
pumped and unpumped states of the sample were recorded in an interleaved
measurement. Sorting the data after FEL pulse energy allows retrieving a
fluence-dependence (for more details about the experimental methods, see
Supplemental Material (SM)
[[SeeSupplementalMaterial][formoredetailsabouttheexperimentalmethods,
Faradayvs.Kerrmeasurements, abinitioIFEtheory, andASDsimulations.]supp]).

Figure 1: a) Schematic of the time-resolved pump-probe experiment implemented
at FERMI. b) Static absorption (red, black for opposite magnetic fields) and
XMCD asymmetry (blue) spectra of the GdFeCo sample measured across the Fe
$\text{M}_{3,2}$ resonance. The yellow dashed lines indicate the photon
energies used for XUV pumping (51.0, 54.1, 56.1 and
$64.0\text{\,}\mathrm{e}\mathrm{V}$). The static spectroscopy was carried out
with an energy resolution of $\leq 0.01\text{\,}\mathrm{e}\mathrm{V}$ using
the ALICE reflectometer [28] at the PM3 beamline of BESSY II [29].

The studied sample was a $20\text{\,}\mathrm{n}\mathrm{m}$ thin amorphous film
of a ferrimagnetic $\textrm{Gd}_{24}\textrm{Fe}_{67}\textrm{Co}_{9}$ alloy
with an out-of-plane magnetic anisotropy. For static absorption spectroscopy,
it was deposited on a $30\text{\,}\mathrm{n}\mathrm{m}$
$\textrm{Si}_{3}\textrm{N}_{4}$ membrane, allowing transmission of the XUV
radiation. For the time-resolved measurements, a
$1\text{\,}\mathrm{m}\mathrm{m}$ thick glass substrate was used in order to
allow simultaneous detection of the Faraday and Kerr signals. In both cases,
the magnetic sample was sandwiched between Ta seed and capping layers to
protect it from oxidation. The exact sample compositions are $\rm
Ta($2\text{\,}\mathrm{n}\mathrm{m}$)/GdFeCo($20\text{\,}\mathrm{n}\mathrm{m}$)/Ta($5\text{\,}\mathrm{n}\mathrm{m}$)/Si_{3}N_{4}($30\text{\,}\mathrm{n}\mathrm{m}$)$
and $\rm
Ta($2\text{\,}\mathrm{n}\mathrm{m}$)/GdFeCo($20\text{\,}\mathrm{n}\mathrm{m}$)/Ta($5\text{\,}\mathrm{n}\mathrm{m}$)/Glass($1\text{\,}\mathrm{m}\mathrm{m}$)$,
respectively.

Fig. 1b shows the XUV absorption spectra (XAS) measured on the GdFeCo sample
across the Fe $\text{M}_{3,2}$ resonance for opposite magnetic fields and the
resulting XMCD spectrum. The absorption peak at around
$56.1\text{\,}\mathrm{e}\mathrm{V}$ corresponds to the resonant excitation of
3p electrons of Fe into 3d states; the second and less pronounced peak around
$62.0\text{\,}\mathrm{e}\mathrm{V}$ arises due to resonant excitation of the
small fraction of Co atoms. Note that the vanishing XMCD at this photon energy
results from the superposition of bipolar XMCD features of the Fe and Co
$\text{M}_{3,2}$ resonances weighted by their different concentrations in the
alloy (see SM [27]). To study the wavelength-dependence of the XUV excitation
across the Fe $\text{M}_{3,2}$ resonance, the photon energy of the FEL pulses
was tuned to fixed positions in the spectrum at
$51.0\text{\,}\mathrm{e}\mathrm{V}$ (below resonance),
$54.1\text{\,}\mathrm{e}\mathrm{V}$ (largest XMCD),
$56.1\text{\,}\mathrm{e}\mathrm{V}$ (highest absorption) and
$64.0\text{\,}\mathrm{e}\mathrm{V}$ (above resonance).

Figure 2: XUV-induced magnetization dynamics driven by fs XUV pulses with
variable polarization (circular polarization with opposite helicities
$\sigma_{\pm}$ and linear horizontal polarization) for two different fluences
1.3 and $6.7\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$
at $54.1\text{\,}\mathrm{e}\mathrm{V}$ photon energy, i.e., at the maximum
XMCD effect – see Fig. 1b. The magnetization is normalized to the equilibrium
magnetization in the unexcited state ($M/M_{0}$). The solid lines correspond
to double-exponential fits – see main text and SM [27]. The data sets measured
for the full range of XUV fluences are shown in the SM [27]. The inset depicts
how the helicity-dependent demagnetization amplitudes and their difference
$\Delta M$ shown in Fig. 3 are obtained.

Fig. 2 shows the temporal evolution of magnetization after resonant excitation
by linearly and circularly polarized XUV pulses at a photon energy of
$54.1\text{\,}\mathrm{e}\mathrm{V}$, as measured by the Faraday signal; for
the Kerr measurements leading to similar results, see SM [27]. The data was
fitted using a double-exponential fit function convoluted with a Gaussian
(solid lines in Fig. 2), taking into account the ultrafast demagnetization,
the subsequent relaxation process and the experimental time resolution. For
clarity, the pump-probe scans are displayed only for two incident XUV
fluences, showing the dynamics upon low and high fluence excitation. The data
reveals a clear influence of the XUV polarization on the demagnetization
amplitudes, emerging upon excitation with increasing incident fluence (compare
1.3 and $6.7\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$).
Above $1.3\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$,
the XUV pump helicity drives either an enhancement or a decrease of the
demagnetization effect that takes place within the first hundreds of
femtoseconds; this helicity-dependent difference in demagnetization magnitudes
lasts for tens of picoseconds, i.e., also during the magnetization relaxation
process, emphasizing the strength and robustness of the transient XUV-induced
magnetization. Such a helicity-dependent behavior is at variance with the case
of linearly polarized XUV excitation that generates demagnetization based on
purely thermal effects, i.e., demagnetization driven by ultrafast electronic
heating.

Figure 3: Demagnetization amplitudes ($D$) measured upon pumping with
circularly polarized XUV pulses with opposite helicities ($\sigma_{\pm}$,
upper panels) and their difference $\Delta M$ (lower panels) as a function of
incident fluence and XUV photon energy. All values are normalized to the
magnetization in the unexcited state and $\Delta M$ was obtained as shown in
the inset of Fig. 2. The solid lines in the upper panels correspond to sigmoid
functions fitted as guide to the eye; the shaded areas denote the experimental
uncertainty by a $90\text{\,}\mathrm{\char 37\relax}$ confidence interval. The
dashed lines in the lower panels indicate linear trends fitted to the
different fluence regimes (white and grey shaded background). For clarity,
only the fluence-dependence of the $\sigma_{\pm}$-polarized excitation is
shown (for linear polarization, see SM [27]).

In Fig. 3, the demagnetization amplitudes obtained from the double-exponential
fits are shown as a function of XUV photon energy, polarization and incident
excitation fluence, evaluated for the full range of used XUV fluences. The
data reveals a strong helicity-dependent effect that is present over the whole
range of photon energies used for excitation, be it on- (54.1 and
$56.1\text{\,}\mathrm{e}\mathrm{V}$) or off-resonant (51.0 and
$64.0\text{\,}\mathrm{e}\mathrm{V}$) with respect to the Fe $\text{M}_{3,2}$
resonance. The splitting between either enhanced or attenuated demagnetization
amplitudes upon $\sigma_{\pm}$-polarized excitation
($D=1-\text{min}[M/M_{0}]$, upper panels) and accordingly the magnitude and
sign of the difference ($\Delta M=D[\sigma_{-}]-D[\sigma_{+}]$, lower panels)
show that the effect is wavelength-dependent, clearly changing its polarity
along the resonance. The data further reveals that the effect is strongly
fluence-dependent; within the experimental uncertainty, two fluence regimes
become apparent (illustrated by the white/grey shaded background in the lower
panels of Fig. 3) where the magnitude of $\Delta M$ initially increases with
XUV fluence and then either reaches a saturated state at a constant $\Delta M$
value or decreases again as the fluence is further elevated. In case of below-
and above-resonant excitation (51.0 and $64.0\text{\,}\mathrm{e}\mathrm{V}$),
$\Delta M$ scales linearly with the fluence until the saturated state is
achieved. It thereby reaches up to $32.2\pm 0.7\text{\,}\mathrm{\char
37\relax}$ difference in pump-induced change with respect to the equilibrium
magnetization (largest $\Delta M$ observed for
$5.3\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ at
$64.0\text{\,}\mathrm{e}\mathrm{V}$). In case of resonant excitation (54.1 and
$56.1\text{\,}\mathrm{e}\mathrm{V}$), the data reveals a more complex fluence-
dependence, indicating either a threshold-like behavior
($54.1\text{\,}\mathrm{e}\mathrm{V}$) or even a fluence-dependent sign change
of $\Delta M$ ($56.1\text{\,}\mathrm{e}\mathrm{V}$). For excitation between
54.1 and $64.0\text{\,}\mathrm{e}\mathrm{V}$, the demagnetization reaches
amplitudes of up to $\approx 100\text{\,}\mathrm{\char 37\relax}$ at high
fluences (depending on the helicity), corresponding to full demagnetization of
the magnetic layer. This leads to the observed saturation of $\Delta M$ in the
high fluence regime, as the increasing thermal load and demagnetization starts
to mitigate the helicity-dependence.

The most striking observation is the presence of strong helicity-dependent
differences even for the off-resonant excitation at photon energies where the
XUV spectroscopy shows very small or even zero XMCD (compare Fig. 1b for 51.0
and $64.0\text{\,}\mathrm{e}\mathrm{V}$). Based on this finding, a purely
thermal mechanism due to dichroic absorption can already be ruled out as the
driving force behind the observed helicity-dependence at these two photon
energies, which strongly indicates the existence of an opto-magnetic effect
such as the IFE. This interpretation is further corroborated by the linear
scaling with the fluence (XUV intensity) observed at 51.0 and
$64.0\text{\,}\mathrm{e}\mathrm{V}$, resembling a typical IFE feature, since
the magnetization induced by the IFE depends linearly on the incident light
intensity via the opto-magnetic, wavelength-dependent constant (see Eq. (1)
and Ref. 8). A possible explanation for the more complex fluence-dependence
and the deviation from a purely linear scaling observed at 54.1 and
$56.1\text{\,}\mathrm{e}\mathrm{V}$ can be given by the steep slope of the XAS
and XMCD spectra (see Fig. 1b) at these photon energies, significantly
changing the response of the system when the spectrum is slightly reshaped
while the system is driven out of equilibrium, depending on the strength of
the excitation [30, 31].

The generation of helicity-dependent effects that are longer-lived compared to
the duration of the driving pump pulse (see Fig. 2) have been previously
reported in various experimental studies of the IFE on different materials
[32, 33, 9, 15, 13, 34]. Although the IFE is active only during the pump pulse
duration, the subsequent magnetization evolution is affected on much longer
time scales, especially, when the impulsively IFE-induced magnetization is
large (here we reach a maximum of $\approx$30$\%$ of the ground-state $M_{0}$
values) and, consequently, the system needs longer times to accomodate such
large magnetic changes.

In order to compare our experimental results to theoretical predictions of an
opto-magnetic effect in the XUV spectral range, we carried out first ab initio
calculations of the IFE [8, 6] in vicinity of the Fe $\text{M}_{3,2}$
resonance. To reduce the computational complexity, the system was modeled as a
stoichiometric $\text{GdFe}_{2}\text{Co}$ unit cell, focusing on the static
opto-magnetic response for left ($\sigma_{-}$) and right ($\sigma_{+}$)
circularly polarized excitation (for more details, see SM [27]). The
difference
$\Delta_{\text{IFE}}=\mathcal{K}^{\text{IFE}}_{\sigma_{-}}(\omega)-\mathcal{K}^{\text{IFE}}_{\sigma_{+}}(\omega)$
between the calculated opto-magnetic constants is shown in Fig. 4 in
comparison to the largest helicity-dependent effects that were experimentally
observed for the studied excitation photon energies (see lower panels of Fig.
3). Note that this comparison is purely qualitative, as the constants
$\mathcal{K}^{\text{IFE}}_{\sigma_{\pm}}(\omega)$ are obtained from stationary
IFE calculations, so far not taking into account the fluence-dependent, non-
equilibrium state present upon femtosecond excitation. In order to obtain a
qualitative agreement with the experimental data, the calculated spectrum of
the IFE response had to be shifted by $+2.5\text{\,}\mathrm{e}\mathrm{V}$,
which can be attributed to the effect of the core holes that is not included
in the calculations (see SM [27] and Ref. 35). The calculations support our
experimental findings regarding the existence of an opto-magnetic effect at
these wavelengths by predicting a highly wavelength-dependent IFE in vicinity
of the $\text{M}_{3,2}$ resonance. The calculated $\Delta_{\text{IFE}}$
reverses its sign going through the resonance, which is again in qualitative
agreement to the different polarity of $\Delta M$ observed in the experiment
for below- and above-resonant excitation. This sign reversal occurs because
the IFE spectra $\mathcal{K}^{\text{IFE}}_{\sigma_{-}}(\omega)$ and
$\mathcal{K}^{\text{IFE}}_{\sigma_{+}}(\omega)$ display an energy shift with
respect to each other at the $\text{3p}\to\text{3d}$ resonance, generated by
the exchange splitting of the 3d states (see SM [27]). Furthermore, it has to
be emphasized that the calculations show a finite response at the photon
energies 51.0 and $64.0\text{\,}\mathrm{e}\mathrm{V}$, corroborating the
existence of a helicity-dependent interaction at photon energies where a
thermal mechanism due to the XMCD effect could already be ruled out. However,
the calculations are so far not in quantitative agreement with the experiment
regarding the magnitude of the observed helicity-dependence. Although the XAS
show only a minor contribution due to the small amount of Co in the alloy
(compare Fig. 1b at $\approx 62.0\text{\,}\mathrm{e}\mathrm{V}$), the IFE
calculations show a finite extension of the opto-magnetic response up to the
Co $\text{M}_{3,2}$ resonance, which can qualitatively explain the observation
of helicity-dependent effects at $64.0\text{\,}\mathrm{e}\mathrm{V}$, but not
yet quantitatively the magnitude of the effect compared to the Fe response.

Figure 4: Comparison of the experimentally observed $\Delta M_{\text{exp}}$
(yellow diamonds, left scale) to the difference $\Delta_{\text{IFE}}$
(turquoise line, right scale) between the calculated opto-magnetic constants
$\mathcal{K}^{\text{IFE}}_{\sigma_{\pm}}(\omega)$ obtained from ab initio IFE
theory. The red circles (left scale) correspond to the output of ASD
simulations, simulating the expected demagnetization difference $\Delta
M_{\text{sim}}$ induced by dichroic absorption due to the XMCD effect. At
$56.1\text{\,}\mathrm{e}\mathrm{V}$, two values are shown for $\Delta
M_{\text{exp}}$, reflecting its opposite polarity in the low vs. high fluence
regime.

To quantify the effect of the finite XMCD present in case of fully resonant
excitation at 54.1 and $56.1\text{\,}\mathrm{e}\mathrm{V}$ on the helicity-
dependent dynamics, atomistic spin dynamics (ASD) simulations were carried out
[11, 36, 37, 38], simulating the influence of the dichroic absorption on the
helicity-dependent demagnetization amplitudes (see Fig. 4 and SM [27]). For
best comparability, the excitation fluence was calibrated between experiment
and simulation by matching the demagnetization amplitudes obtained upon
linearly polarized excitation. While the absence of any helicity-dependence in
the XMCD-driven ASD simulations is obvious for photon energies without XMCD
(51.0 and $64.0\text{\,}\mathrm{e}\mathrm{V}$), the simulation shows that even
in the fully resonant case (54.1 and $56.1\text{\,}\mathrm{e}\mathrm{V}$), the
expected dichroic demagnetization effects would be smaller by at least a
factor of six. Thus, a significant contribution of a purely thermal mechanism
(via dichroic absorption) on the observed helicity-dependent dynamics can be
ruled out, strengthening our conclusions regarding the existence of an opto-
magnetic effect like the IFE for both on- and off-resonant excitation in
vicinity of the Fe $\text{M}_{3,2}$ resonance.

In conclusion, we studied the impact of femtosecond XUV photo-excitation of a
ferrimagnetic GdFeCo layer using circularly polarized FEL pulses at the Fe
$\text{M}_{3,2}$ resonance. The systematic investigation of time-resolved
magneto-optical Faraday rotation data obtained as a function of XUV excitation
fluence, photon energy and polarization reveals a strong and long-lived
helicity-dependent effect on the ultrafast demagnetization dynamics. The
analysis of its wavelength- and fluence-dependence in comparison to static
XMCD spectroscopy and time-resolved ASD simulations shows that the observed
effect cannot be thermally induced by dichroic absorption. Instead, it has the
characteristic fingerprints of an opto-magnetic effect, which is further
corroborated by first ab initio theory on the IFE at the Fe $\text{M}_{3,2}$
resonance, showing qualitative agreement between the calculated IFE spectrum
and the experimental data. Further experimental studies and theoretical
efforts are needed for obtaining a quantitative understanding of the
experimental results, especially with respect to the non-trivial fluence-
dependence observed for excitation on the XUV resonance. This requires to
consider the IFE also in the highly non-equilibrium state upon femtosecond
pulsed excitation, which is currently not included in the ab initio opto-
magnetic theory. Complementary measurements upon resonant excitation of the Fe
L edges, exciting 2p core-level electrons with even larger SOC compared to the
3p states, would provide further insights into the actual scaling of the IFE
with the SOC and photon energy. Modern x-ray free-electron lasers are able to
provide the required highly intense, circularly polarized femtosecond pulses
of soft x-rays [20, 39, 40]. The observation of large opto-magnetic effects in
the XUV spectral range also implies a significant, yet to be explored impact
on ultrafast magnetization switching phenomena in addition to the MCD-induced
helicity-dependence known from the visible-light regime [41]. Moreover, the
different nature of the Gd 4f and Fe 3d magnetic moments with respect to their
SOC and predominantly localized vs. itinerant character suggests that
comparing element-specific excitation of the rare-earth and transition metal
sublattices will contribute to fully revealing the microscopic mechanisms
behind the XUV and soft x-ray IFE.

###### Acknowledgements.

I.R. acknowledges funding from the Federal Ministry of Education and Research
(BMBF) through project 05K16BCA (Femto-THz-X) and the European Research
Council through project TERAMAG (Grant No. 681917). C.v.K.S. and S.E. would
like to thank the German Research Foundation (DFG) for funding through CRC/TRR
227 project A02 (project ID 328545488). S.R. gratefully acknowledges the
support of ARCHER UK National Supercomputing Service via the project e733.
S.R. and T.O. gratefully acknowledge the financial support from the EPSRC
TERASWITCH project (project ID EP/T027916/1). This work has been supported by
the Swedish Research Council (VR) and through the European Union’s Horizon
2020 Research and Innovation Programme under grant agreement No. 737709
(Femtoterabyte). Part of the calculations were provided by the Swedish
National Infrastructure for Computing (SNIC), funded by VR through grant No.
2018-05973. The authors acknowledge Elettra Sincrotrone Trieste for providing
access to its free-electron laser facilities and thank all members of the
different departments at FERMI for their outstanding assistance during the
preparation and realization of the experiment. The authors also thank the
Helmholtz-Zentrum Berlin für Materialien und Energie for the allocation of
synchrotron-radiation beamtime.

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