# Ultrafast behavior of induced and intrinsic magnetic moments in CoFeB/Pt
bilayers probed by element-specific measurements in the extreme ultraviolet
spectral range

Clemens von Korff Schmising<EMAIL_ADDRESS>Somnath Jana  Kelvin Yao
Martin Hennecke  Philippe Scheid  Sangeeta Sharma  Max-Born-Institut für
Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489
Berlin, Germany  Michel Viret  Jean-Yves Chauleau  SPEC, CEA, CNRS, Université
Paris-Saclay, CEA Saclay - 91191 Gif sur Yvette, France  Daniel Schick  Max-
Born-Institut für Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-
Straße 2A, 12489 Berlin, Germany  Stefan Eisebitt  Max-Born-Institut für
Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Straße 2A, 12489
Berlin, Germany  Technische Universität Berlin, Institut für Optik und Atomare
Physik, 10623 Berlin, Germany

###### Abstract

The ultrafast and element-specific response of magnetic systems containing
ferromagnetic 3d transition metals and 4d/5d heavy metals is of interest both
from a fundamental as well as an applied research perspective. However, to
date no consensus about the main microscopic processes describing the
interplay between intrinsic 3d and induced 4d/5d magnetic moments upon
femtosecond laser excitation exist. In this work, we study the ultrafast
response of CoFeB/Pt bilayers by probing element-specific, core-to-valence
band transitions in the extreme ultraviolet spectral range using high harmonic
radiation. We show that the combination of magnetic scattering simulations and
analysis of the energy- and time-dependent magnetic asymmetries allows to
accurately disentangle the element-specific response in spite of overlapping
Co and Fe M2,3 as well as Pt O2,3 and N7 resonances. We find a considerably
smaller demagnetization time constant as well as much larger demagnetization
amplitudes of the induced moment of Pt compared to the intrinsic moment of
CoFeB. Our results are in agreement with enhanced spin-flip probabilities due
to the high spin-orbit coupling localized at the heavy metal Pt, as well as
with the recently formulated hypothesis that a laser generated, incoherent
magnon population within the ferromagnetic film leads to an overproportional
reduction of the induced magnetic moment of Pt.

††preprint: APS/123-QED

##  I Introduction

Combining ferromagnetic 3d transition metals with 4d/5d heavy metals leads to
magnetic systems with new macroscopic functionalities. Co/Pt multilayers or
FePt nanoparticles, for example, exhibit very high magnetocrystalline
anisotropies and are therefore important model systems for data storage
technology. Pt underlayers can be exploited for spin-orbit torque induced
switching [1, 2] with potential for ultrafast applications [3]. Light-driven
processes allowing the manipulation and control of ferromagnetic order are
receiving renewed attention due to the discovery of helicity-dependent, all-
optical switching in thin Co/Pt multilayers and FePt based granular films [4],
with promising new developments based on tailored double-pulse excitation
schemes [5].

The ultrafast response of optically excited ferromagnetic transition/heavy
metal systems is characterized by their very different atomic spin-orbit
coupling strength, their distinct electronic structures and importantly by the
behavior of intrinsic 3d versus induced 4d/5d magnetic moments. The phenomenon
of proximity-induced magnetism is caused by hybridization of the 3d and 4d/5d
bands and leads to parallel spin alignment between intrinsic and induced
moment [6, 7]. The complex interplay of the involved elements after optical
excitation has been studied in a growing number of element-specific
experiments based on resonant X-ray or extreme ultraviolet spectroscopy,
leading, however, to conflicting observations and corresponding competing
theoretical explanations. While experiments with Co/Pt bi- [8] or multi- [9]
layers as well as FePt alloys [10] suggest that the induced Pt moment follows
the dynamics of the ferromagnetic transition metal, later work found clear
evidence for a distinct, element-specific response: both, in an ordered FePt
compound [11] as well as in a Co/Pt multilayer [12]. There, a significantly
slower dynamics of Pt was found and rationalized with a higher mobility of Co
or Fe compared to Pt majority electrons leading to an enhanced demagnetization
rate of the transition metal due to superdiffusive spin currents.
Additionally, ground-state density-of-state calculations have predicted that a
potential generation of incoherent magnons would lead to an overproportional
reduction of the induced compared to the intrinsic magnetic moments: as
canting of the 3d spins changes the average spin alignment between neighboring
atoms, the exchange interactions on the 4d/5d heavy metal is reduced, leading
to a reduction of the induced moment. These calculations showed that if the
response of the transition metal is dominated by Heisenberg-like, transversal
excitation, the induced moment behaves differently and exhibits a strong
reduction of its amplitude [11, 13]. The aforementioned study investigating a
CoPt alloy, qualitatively confirmed this hypothesis, revealing slightly larger
demagnetization amplitudes of Pt compared to Co [13]. An explanation based on
enhanced spin-orbit coupling of the heavy metal Pd and a respective stronger
spin-flip probability was invoked to rationalize an accelerated
demagnetization rate with increasing Pd concentrations in NiPd alloys [14].
Finally, we suggested a scenario where optical intersite spin transfer (OISTR)
[15], i.e., a laser driven transfer of minority carriers from Pt to Co,
competes with spin-orbit driven spin-flips locally enhanced on the Pt atom,
resulting in comparable magnetization dynamics of Co and Pt. Experimentally,
this hypothesis was tested in an experiment investigating a CoPt alloy using a
combination of ultrafast magnetic circular dichroism and helicity-dependent
transient absorption spectroscopy in the XUV spectral range [16]. Also based
on OISTR, ab-initio calculations of the ultrafast magnetization changes of a
FePd3 compound predicted element-specific dynamics with a qualitative
dependence on the laser pulse intensity [17].

However, when evaluating and comparing these results in literature, it is
important to acknowledge the different experimental techniques, different
excitation levels and – most importantly – the different studied sample
geometries, ranging from alloys with different stoichiometries to bi- and
multilayers with different film thicknesses leading to very different local
environments. While in reflection geometry with an enhanced surface
sensitivity, super-diffusive spin currents may lead to a distinct 3d versus
4d/5d response [10, 11, 12, 13], experiments in transmission geometry yield
signals averaged over the entire sample volume and will be less affected by
intralayer spin transport [8, 16, 9]. OISTR is confined to spin transfer among
nearest neighbors [18, 19] and is hence expected to be most pronounced in
alloys with equal concentrations of Co and Pt or ultrathin layered systems.

In this study, we report on distinct, layer-dependent dynamics of a CoFeB(5.3
nm)/Pt(3.5 nm) bilayer observed by element-specific, transverse magneto-
optical Kerr effect (T-MOKE) measurements in the XUV spectral range using
radiation of a high-harmonic generation (HHG) source. We demonstrate that the
combination of magnetic scattering simulations [20, 21] and an analysis of the
ultrafast response at different photon energies allows accurately separating
the element-specific response of Co or Fe and Pt in spite of strongly
overlapping resonances. The experimental results reveal a transient magnetic
state with a significantly faster and more efficient quenching of the induced
Pt moments compared to the response of the transition metal CoFeB alloy. A
systematic comparison with a CoFeB/MgO/Pt system, exhibiting no induced
magnetization of Pt, supports our analysis.

##  II Experiment

Figure 1: Schematic of the T-MOKE geometry with an incident angle
$\theta=45^{\circ}$ of the p-polarized XUV radiation. After reflection, the
XUV pulses are spectrally dispersed by a flat field concave grating and
detected by a CCD camera. An external magnetic field, $B$, is aligned
perpendicularly to the incoming p-polarized XUV light. The sample is excited
by laser pulses at $\lambda=$800\text{\,}\mathrm{n}\mathrm{m}$$ in a nearly
collinear geometry. The inset shows the depth-dependent absorption profiles of
the CoFeB/Pt and CoFeB/MgO/Pt samples.

Si/Co40Fe40B20($5.3\text{\,}\mathrm{n}\mathrm{m}$)/MgO($1\text{\,}\mathrm{n}\mathrm{m}$)/Pt($3.5\text{\,}\mathrm{n}\mathrm{m}$)
as well as
Si/Co40Fe40B20($5.3\text{\,}\mathrm{n}\mathrm{m}$)/Pt($3.5\text{\,}\mathrm{n}\mathrm{m}$)
were deposited on Si substrates by magnetron sputtering. Both samples exhibit
an in-plane magnetization and a square hysteresis loop with a small coercive
field below $2\text{\,}\mathrm{m}\mathrm{T}$. The magnetic films are protected
against oxidation by the Pt layer or MgO/Pt bilayer. The thicknesses as well
as interface roughnesses of below $0.5\text{\,}\mathrm{n}\mathrm{m}$ have been
confirmed by independent X-ray reflectivity measurements.

A schematic of the setup is shown in Fig.1. Intense laser pulses at a
repetition rate of $3\text{\,}\mathrm{k}\mathrm{H}\mathrm{z}$, a central
wavelength of $\lambda=$800\text{\,}\mathrm{n}\mathrm{m}$$ and a pulse length
of $25\text{\,}\mathrm{f}\mathrm{s}$ are focused into a cell filled with neon
gas, leading to higher harmonic radiation. The emitted XUV spectrum consists
of discrete peaks with an approximate spectral width of
$200\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}$. Additionally, we generate
continuous spectra for the static characterization by averaging multiple
acquisitions with varied chirp and peak intensities of the laser pulses as
well as with varying gas pressures. The p-polarized XUV radiation is incident
on the sample at an angle of $\theta=45^{\circ}$, reflected, and then
spectrally dispersed by a flatfield concave grating and detected by a charged
coupled device (CCD). The magnetization of the sample is set by applying an
external magnetic field perpendicular to the incident polarization,
facilitating the T-MOKE geometry [22]. The magnetic asymmetry, $A$, is
calculated as the normalized difference of two spectra, $I$, recorded for
opposite magnetization directions of the CoFeB film, cf. Fig.1. The time-
resolved experiments were performed in a pump-probe geometry using pump pulses
with a central wavelength of $800\text{\,}\mathrm{n}\mathrm{m}$ and a pulse
duration of $<$30\text{\,}\mathrm{f}\mathrm{s}$$ as confirmed by
autocorrelation measurements at the sample position. We estimate the upper
bound for the temporal resolution of the experiment to be
$35\text{\,}\mathrm{f}\mathrm{s}$. The absolute values of incident fluences
between 0 and
$12\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}\mathrm{m}$
are determined by a calibrated power meter and careful measurements of the
beam area (full width at half maximum) using a beam profile camera.
Calculations of the layer-dependent absorption profile based on a multilayer
formalism [20, 21], using literature values of the optical refractive indices
[23, 24], show an almost homogeneous energy deposition witin the CoFeB and Pt
layers (cf. Fig.1). To improve the signal-to-noise ratio, we acquire the
incoming spectrum with a separate spectrometer for normalization and repeat
each time-delay scan up to 50 times. For more details on the experimental
setup please refer to Yao et al. [25].

##  III Results

Figure 2: Measured and calculated static magnetic asymmetry of a) CoFeB/MgO/Pt
and b) CoFeB/Pt as a function of photon energy. The color-coded dots mark the
photon energies of the time-resolved measurements, while the solid lines are
static measurements with a continuous HHG spectrum. In b) we additionally show
the calculated magnetic asymmetry stemming from CoFe (dotted line) and Pt
(dashed line). The largest relative contributions to the magnetic asymmetry of
Pt are not only found around the Pt O2,3 and Pt N7 edges, but also at
$57\text{\,}\mathrm{e}\mathrm{V}$, where contribution of Co and Fe approaches
zero, cf. panel c).

In Fig. 2, we show the magnetic asymmetry in a photon energy range between 45
eV and 72.5 eV for both samples, a) CoFeB/MgO/Pt and b) CoFeB/Pt. We plot the
asymmetry measured with a continuous HHG spectrum and mark the photon energies
of the nine discrete HHG peaks (H31 to H47) used in the time-resolved
experiments with colored dots. The simulations for static asymmetries are
carried out with the udkm1Dsim toolbox [20, 21] numerically calculating the
polarization-dependent wave propagation taking into account refraction and
reflections of the nanolayered sytems. We use published values of the atomic
and magnetic form factors of Fe and Co [26] and extract values for the induced
moment of Pt from measurements of CoPt [8, 27]. Atomic form factors of MgO and
B are taken from Henke et al. [28]. We keep the thicknesses and interface
roughnesses fixed, but allow small variations in the densities as well as in
the absolute values of the elemental magnetic moments. For the CoFeB/Pt
system, we fix the thickness of the magnetized Pt layer at the interface to
$1\text{\,}\mathrm{n}\mathrm{m}$ [7, 29, 30]. The fitting results yield a good
agreement with the experimental measurements, for both samples, we can clearly
note the dominant contributions from the Fe and Co M2,3 edges around
$53.5\text{\,}\mathrm{e}\mathrm{V}$ and $60\text{\,}\mathrm{e}\mathrm{V}$
probed by the HHG emission peaks H35 and H39. In the energy range between
$47\text{\,}\mathrm{e}\mathrm{V}$ and $51\text{\,}\mathrm{e}\mathrm{V}$ both
samples exhibit qualitatively similar deviations between simulation and
experiment. As in the CoFeB/MgO/Pt sample Pt is not magnetic, we can rule out
that they are caused by incorrect magnetic form factors of Pt. For CoFeB/Pt,
the simulation also slightly overestimates the small magnetic asymmetries
around $57\text{\,}\mathrm{e}\mathrm{V}$, probed by HHG peak H37. For the
CoFeB/Pt sample, we extract the contributions of Pt to the magnetic asymmetry
by setting the magnetic moment of Co and Fe to zero. This is shown as a
dashed-dotted line in Fig.2 b). The contribution of the CoFeB layer is
calculated accordingly and shown as a dotted line. While this approach
neglects the influence of the dichroic part of the refractive index of either
Pt and Co or Fe to the reflectivities, we validate this approximation by
finding an excellent agreement between the sum of the individual asymmetries
and the total asymmetry of CoFeB/Pt. In Fig.2 c), we plot the relative
contribution of Pt to the magnetic asymmetry of the CoFeB/Pt sample, which is
determined by the complex spectral overlap of the three magnetic elements Co,
Fe and Pt. The asymmetry of harmonic H35 is dominated by Fe (80%) and H39 is
dominated by Co (85%). The maximum relative contribution of Pt is at the Pt N7
edge at $71.2\text{\,}\mathrm{e}\mathrm{V}$ with close to 90%. The small
asymmetries beyond the Co edge (harmonic peaks H41 to H45) are also dominated
by Pt and yield values $>80\%$. Furthermore, the harmonic peak H33 and H37
exhibit appreciable Pt O2,3 contributions of 30$\%$ and 70$\%$. The system
CoFeB/MgO/Pt is not expected to exhibit a static induced magnetization of Pt,
which – within our signal-to-noise ratio – is also consistent with our
simulation. The subtle differences of the energy-dependent magnetic
asymmetries between both sample systems highlights the importance of dedicated
calculations in order to extract elements-specific information in the XUV
spectral range.

Figure 3: Magnetic asymmetry as a function of time after optical excitation.
Harmonics H35 and H39 in panel a) and b) are dominated by the dynamics of Fe
and Co, respectively. For the CoFeB/Pt system, HHG peaks H33 and H37 (panel c)
are sensitive to both layers, CoFeB and Pt and are characterized by a sign
change of the time-resolved asymmetry. The asymmetries of the HHG peaks H41
(panel c), H43, H45 and H47 (panel a) are dominated by Pt contributions and –
differently to H35 and H39 – exhibit a 100% reduction of $A$. For the
CoFeB/MgO/Pt sample system all shown harmonics H33 to H47 show the same
relative demagnetization amplitude and represent the response of Fe or Co.
Harmonic H47, close to the Pt N7 edge, shows no evidence of an increase, i.e.
no evidence for tunneling of spin-polarized carriers from the CoFeB to Pt via
the MgO layer, cf. panel f).

A summary of time-resolved data is shown in Fig. 3 for very early times after
optical excitation. We plot data for two exemplary fluences:
$10\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}\mathrm{m}$
and
$4\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}\mathrm{m}$
for the samples CoFeB/Pt and CoFeB/MgO/Pt, respectively. The solid lines are
fits using a single-exponential decay convolved with a Gauss function, $G(t)$,
reflecting the temporal resolution of the experiment:

$A(t)=G(t)\ast\left[A_{0}+\Theta(t)\left(B\left(e^{-t/\tau}-1\right)\right)\right].$
(1)

$\Theta(t)$ is the Heavyside function, $A_{0}$ is the static asymmetry and $B$
the amplitude of the demagnetization, $\tau$ denotes the demagnetization time
constant. We differentiate the ultrafast response of the CoFeB/Pt sample
measured at the different harmonics according to their element specificity:
H35 and H39 are dominated by Fe and Co, respectively and are shown in panel
a), H33, H37 and H41 display a significant admixture with Pt an are summarized
in panel c) and finally H43, H45 and H47 predominately reflect the ultrafast
dynamics of Pt and are shown in panel e). The response of the CoFeB/MgO/Pt
sample at corresponding photon energies is shown in panels b), d) and f). For
the CoFeB/Pt sample, we find demagnetization time constants for H35
$\tau_{\mathrm{H35}}=(115\pm 7)$$\mathrm{f}\mathrm{s}$ and for H39
$\tau_{\mathrm{H39}}=(105\pm 6)$$\mathrm{f}\mathrm{s}$, for the CoFeB/MgO/Pt
sample the values amount to $\tau_{\mathrm{H35}}=(140\pm
20)$$\mathrm{f}\mathrm{s}$ and $\tau_{\mathrm{H39}}=(100\pm
19)$$\mathrm{f}\mathrm{s}$. Within our signal-to-noise ratio, we do not find
any systematic variations of the demagnetization constants neither for the two
different samples nor as a function of fluence. However, the slightly faster
demagnetization constant of H39 (Co) compared to H35 (Fe) is consistently
reproduced and is also in agreement with previous observations that $\tau$
scales with the magnetic moment of the element [31, 32]. The average of the
extracted demagnetization time constants of harmonics H41 to H47 amounts to
$\tau_{H41-H47}=(70\pm 20)$ fs, significantly smaller than what we find for
H35 (Fe) and H39 (Co). Furthermore, the magnetic asymmetries measured for
CoFeB/Pt at photon energies dominated by Pt, i.e. H41, H43, H45 and H47 shown
in Fig. 3 c) and e) are completely quenched or even show small negative values
at $t=$400\text{\,}\mathrm{f}\mathrm{s}$$. This is in contrast to asymmetry
values of harmonics H35 and H39, which are only reduced by 35% at this time
delay.

The demagnetization amplitudes for different photon energies are more easily
compared by inspection of Fig. 4, where we display the normalized magnetic
asymmetries, $\widehat{A}=\Delta A/A_{0}$, as a function of the incident
fluence at three selected time delays of $t=1,5$ and 500
$\mathrm{p}\mathrm{s}$. Data for harmonics H35 and H39 show very similar
values over the entire fluence range and for the three different times. This
also applies for both sample systems: the red lines mark $\Delta
A(t=1\mathrm{ps})/A_{0}=0.5$, which is reached for an incident fluence of
$6.5\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}\mathrm{m}$
for both samples. In Fig. 4 b), we additionally show the magnetic asymmetry of
harmonic H47 measured for sample CoFeB/Pt. Clearly, here $\Delta A/A_{0}$ is
reduced more efficiently and already reaches zero for a fluence of
$\approx$$8\text{\,}\mathrm{m}\mathrm{J}\mathrm{/}\mathrm{{}^{2}}\mathrm{c}\mathrm{m}$.
Also note that even after a time delay of $500\text{\,}\mathrm{p}\mathrm{s}$,
when the CoFeB layer has recovered a significant fraction of its magnetization
by energy dissipation, the magnetization of the Pt layer remains out of
equilibrium with CoFeB. With the assignment of the different harmonic peaks to
the different magnetic elements within the sample, this presents strong
evidence that the interfacial Pt layer loses its induced magnetization
significantly more efficiently than Co and Fe.

The element-specific dynamics allows us to use the time-resolved data to
further verify and refine the relative elemental contributions to the magnetic
asymmetry discussed in Fig. 2 c). Accordingly, we describe the normalized
nonlinear least square fits of the data (solid lines in Fig. 3) by a sum of
equation 1 representing separately the response of Co/Fe and Pt.

$\widehat{A}(t)=c_{\mathrm{Pt}}\widehat{A}_{\mathrm{Pt}}(t)+c_{\mathrm{Co/Fe}}\widehat{A}_{\mathrm{Co/Fe}}(t)\\\
$ (2)

We use the extracted demagnetization time constants measured for H35, H39 as
well as the average of H41 to H47 and assign them to Co, Fe and Pt,
respectively. The normalized demagnetization amplitude, $\widehat{B}$, of Pt
is assumed to be 100%. We then describe our normalized, time dependent data,
$\widehat{A}(t)$, by Eqn.2 and only treat the demagnetization amplitude,
$\widehat{B}$, of Co or Fe as well as the relative weights $c_{\mathrm{Pt}}$
and $c_{\mathrm{Co/Fe}}$ as free parameters. Additionally the following
condition has to hold: $c_{\mathrm{Pt}}+c_{\mathrm{Co/Fe}}=1$. The results are
shown as dotted lines in Fig. 3 a),c) and e). Analysis of the response of H35
and H39 yield demagnetization amplitudes for Fe and Co of $(40\pm 5)\%$. The
extracted relative Pt contributions are shown next to the time-resolved
transients. We find a good agreement with the values determined from the
static simulation, except for H33 and H37, where we see a substantial
deviation. We attribute this to the very low values of the absolute asymmetry
values in this energy range. A remarkable feature of the time traces of H33
and H37 is the sign change of the asymmetry. This is caused by the opposite
sign of the respective asymmetries as well as the very different ultrafast
response of the FeCoB and Pt layers. As Pt completely demagnetizes and CoFeB
does not, the asymmetries reverse sign, which is quantitatively reproduced by
Eq.2. The same argument holds for photon energies H43 to H47, where we also
observe a zero crossing of $A$. Importantly, the observed sign change of the
asymmetry is not related to a magnetization reversal. Another potential
explanation for the sign change of $A$ of H33 and H37 could be related to a
spectral reshaping of $A$, as these photon energies probe the system very
close to a zero crossing of the static asymmetry spectra. Such spectral
reshaping, effectively shifting the resonant absorption and circular dichroism
to smaller energies have been recently postulated for the XUV spectral range
and are understood to be caused by laser driven changes of electron
occupations [33, 34, 35]. However, as the slope of $A$ at energies H33 and H37
are very different, a laser induced spectral shift would result in
significantly different dynamic changes, which is contrary to our observation.
Combining the static and time-resolved analysis, we estimate the uncertainty
in assigning the relative elemental contribution for HHG peaks H33 and H37 of
up to 15%, for all other peaks to $<5\%$.

Figure 4: Normalized asymmetry $\Delta A/A_{0}$ as a function of excitation
fluence for three selected time delays, $t=1,5,500$ $\mathrm{p}\mathrm{s}$. In
panel a), we show H35 (Fe) and H39 (Co) for the CoFeB/MgO/Pt and in b)
additionally H47 (Pt) for the CoFeB/Pt system. Pt demagnetizes significantly
more efficiently and remains out of equilibrium with the CoFeB layer for up to
$500\text{\,}\mathrm{p}\mathrm{s}$.

A closer inspection of the ultrafast response of CoFeB/MgO/Pt in Fig. 3
b),d),f) reveals comparable ultrafast dynamics with demagnetization amplitudes
of $\Delta A/A_{0}=(30\pm 5)\%$ at $t=$0.4\text{\,}\mathrm{p}\mathrm{s}$$ for
all HHG peaks. The magnetic asymmetry measured at the HHG peak H47, probing
the Pt N7 edge, does not show any evidence of a magnetization increase, but
stays constant within our signal-to-noise ratio. This implies that we can
exclude significant ultrafast spin tunnelling through the MgO barrier into Pt,
in agreement with a very recent study that found a exponential decay length of
only $2\text{\,}\mathrm{\SIUnitSymbolAngstrom}$ for the tunneling spin current
through MgO [36]. Just as for the CoFeB/Pt system, we observe similar
demagnetization amplitudes of Fe and Co for longer timescales (cf. Fig. 4 a),
pointing towards an identical temperature dependence of the equilibrium
magnetization of Co and Fe. This differs from previous observations on
Co60Fe20B20/SiO2 systems, where a difference in the asymptotic limit of the
remagnetization process was observed [10].

##  IV Discussion and Conclusions

Our observations of distinct ultrafast dynamics of intrinsic vs. induced
moments can be reconciled with our earlier work on Co50Pt50 alloys, where we
identified two competing processes: spin-orbit induced spin flips and OISTR
[26]: we argue again that the more efficient demagnetization of Pt may be a
consequence of the significantly stronger spin-orbit coupling localized at the
Pt atoms compared to Co or Fe [37, 38]. At the same time, with fewer Co or Fe-
Pt nearest neighbors in the studied bilayer, competing OISTR processes are
less likely to occur than in an alloy, leading to distinct dynamics between Co
or Fe and Pt in this work.

Qualitatively, our results of a larger demagnetization amplitude of the Pt
layer may also support the hypothesis that the induced moment of Pt is very
sensitive to the orientation of the intrinsic CoFe moments and could hence be
interpreted as an indicator for the generation of incoherent magnons in the
CoFeB layer. In Yamamoto et al. [11], however, it was postulated that magnons
are predominantly generated after relaxation of the non-equilibrium electron
distribution and will therefore only start to dominate after
$\approx$1\text{\,}\mathrm{p}\mathrm{s}$$, i.e. on a slower timescale compared
to our observations. On the other hand, a number of recent studies have found
evidence for an ultrafast generation of incoherent magnons upon laser
excitation [39, 40, 41, 42, 43, 44]. Also, our observation that the
magnetization of the Pt layers remains out of equilibrium with the adjacent
CoFeB layer for delays larger than $500\text{\,}\mathrm{p}\mathrm{s}$, i.e.
much longer than heat equilibration times, may count as a further indication
that induced moments of Pt are very sensitive to long-lived magnon populations
of the polarizing Co or Fe moments. We anticipate that the combination of
time-dependent density of state calculations with experiments reaching a
significantly improved temporal resolution may be able to disentangle the
different processes in the time domain and identify the time scale on which
such a potential magnon population emerges.
Our results do not show any clear signatures of superdiffusive spin currents,
in spite of also using a reflection geometry with depth-dependent sensitivity
[45]. A potential interlayer spin transport would be characterized by
injection of majority carriers from CoFeB to the interface region of the Pt
layer, leading to an increase of the demagnetiation rate of the CoFeB layer
while simultaneously decreasing the demagnetization rate in the Pt layer,
which is different to our observation. Also, the almost identical
demagnetization rate of the CoFeB/Pt and CoFeB/MgO/Pt system (cf. red lines in
Fig. 4) rules out significant interlayer superdiffusive spin currents, as
transport would be inhibited in the latter sample by the insulating MgO layer.
In conclusion, we have presented a systematic comparison of two sample systems
with and without an induced magnetization of Pt, CoFeB/Pt and CoFeB/MgO/Pt,
and demonstrated that a combination of calculated magnetic asymmetries taking
into account the geometry of the sample structure and an analysis of time-
resolved magnetization traces allows to accurately disentangle the element-
specific magnetic response in spite of strongly overlapping Co or Fe M2,3 and
Pt O2,3 as well as N7 resonances. We find a significantly more efficient
demagnetization rate of the induced moment of Pt compared to the intrinsic
moment of the transition metals in a CoFeB/Pt bilayer. Our results are in
agreement with the presence of enhanced spin-orbit driven spin flips localized
on Pt as well as with the recently postulated hypothesis of a strong
sensitivity of an induced 4d/5d magnetic moment to the potential ultrafast
emergence of an incoherent magnon population.

###### Acknowledgements.

C. v. K. S., S. S. and S. E. acknowledge financial support by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID
328545488 – TRR 227, project A02 and A04. J.-Y. C. acknowledges funding from
the French National Research Agency ANR JCJC through the SPINUP project ref.
ANR-21-CE24-0026-01.

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$\displaystyle={\mathrm{T}}^{\ast}\\{g\\}({\boldsymbol{x}})=\int_{\mathbb{R}}\int_{\mathbb{S}^{d-1}}h_{{\boldsymbol{x}}}(t,{\boldsymbol{\xi}})\;g(t,{\boldsymbol{\xi}}){\rm
d}{\boldsymbol{\xi}}{\rm d}t$

which, with the help of Hölder’s inequality, yields the bound

$\displaystyle\big{|}(1+\|{\boldsymbol{x}}\|)^{-\alpha}f({\boldsymbol{x}})\big{|}$
$\displaystyle\leq(1+\|{\boldsymbol{x}}\|)^{-\alpha}\|h_{{\boldsymbol{x}}}\|_{L_{p}(\mathbb{R}\times\mathbb{S}^{d-1})}\;\|g\|_{L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})}$
$\displaystyle\leq\sup_{{}_{{\boldsymbol{x}}\in\mathbb{R}^{d},\;{\boldsymbol{\xi}}\in\mathbb{S}^{d-1}}}\left((1+\|{\boldsymbol{x}}\|)^{-\alpha}S_{d}\;\|h_{{\boldsymbol{x}}}(\cdot,{\boldsymbol{\xi}})\|_{L_{p}(\mathbb{R})}\right)\|g\|_{L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})}$
$\displaystyle\leq
S_{d}\;C_{p,\alpha}\;\|g\|_{L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})},$

where $S_{d}=\frac{2\pi^{d/2}}{\Gamma(d/2)}$ is the surface of the unit
hypersphere $\mathbb{S}^{d-1}$. This proves the desired result with
$\|{\mathrm{T}}\|=\|{\mathrm{T}}^{\ast}\|\leq S_{d}C_{p,\alpha}$.

∎

###  5.2 Characterization of the Predual Space
${\mathcal{X}}_{{\mathrm{L}}_{\rm R}}$

The application of the general representer in [53] requires that the native
space ${\mathcal{X}}^{\prime}_{{\rm L}_{\rm R}}=({\mathcal{X}}_{{\rm L}_{\rm
R}})^{\prime}$ be identifiable as the topological dual of some primary space
${\mathcal{X}}_{{\rm L}_{\rm R}}$ endowed with the Banach property. We achieve
this constructively through a completion process that ensures that
${\mathcal{X}}_{{\rm L}_{\rm R}}$ is a complete normed space.

######  Theorem 9 (Construction of the predual space ${\mathcal{X}}_{{\rm
L}_{\rm R}}$).

Let ${\rm L}$ be an isotropic LSI operator that is spline-admissible with a
polynomial null space ${\mathcal{P}}$ of degree $n_{0}$. Then, ${\rm L}_{\rm
R}^{\ast\dagger}={\mathrm{R}}{\mathrm{L}}^{-1\ast}(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}}):L_{1,n_{0}}(\mathbb{R}^{d})\xrightarrow{\mbox{\tiny\
\rm c. }}{\mathcal{X}}_{\rm Rad}$ is bounded, and admits the unique extension
${\mathcal{{\rm L}}}_{\rm R}^{\ast\dagger}:{\mathcal{X}}_{{\rm L}_{\rm
R}}(\mathbb{R}^{d})\xrightarrow{\mbox{\tiny\ \rm c. }}{\mathcal{X}}_{\rm Rad}$
with the properties

$\displaystyle\forall v\in{\mathcal{X}}_{\rm Rad}:$ $\displaystyle\quad{\rm
L}_{\rm R}^{\ast\dagger}{\rm L}_{\rm R}^{\ast}\\{v\\}=v$ (83)
$\displaystyle\forall p_{0}^{\ast}\in{\mathcal{P}}^{\prime}:$
$\displaystyle\quad{\rm L}_{\rm R}^{\ast\dagger}\\{p_{0}^{\ast}\\}=0$ (84)
$\displaystyle\forall f\in{\mathcal{X}}_{{\rm L}_{\rm R}}(\mathbb{R}^{d}):$
$\displaystyle\quad{\rm L}_{\rm R}^{\ast}{\rm L}_{\rm
R}^{\ast\dagger}\\{f\\}=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})\\{f\\}=\mathrm{Proj}_{\mathcal{U}}\\{f\\},$
(85)

where ${\rm L}_{\rm R}^{\ast}={\rm
L}^{\ast}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}$ and ${\mathcal{X}}_{{\rm
L}_{\rm R}}(\mathbb{R}^{d})={\rm L}^{\ast}_{\rm R}({\mathcal{X}}_{\rm
Rad})\oplus{\mathcal{P}}^{\prime}$ is equipped with the norm
$\|f\|_{{\mathcal{X}}_{{\rm L}_{\rm
R}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\max(\|{\rm L}_{\rm
R}^{\ast\dagger}\\{f\\}\|_{{\mathcal{X}}},\|\mathrm{Proj}_{{\mathcal{P}}^{\prime}}\\{f\\}\|_{{\mathcal{P}}^{\prime}})$.
The space ${\mathcal{X}}_{{\rm L}_{\rm R}}$ is complete and isomorphic to
${\mathcal{X}}_{\rm Rad}\times{\mathcal{P}}^{\prime}$ with $f={\rm
L}^{\ast}_{\rm R}\\{v\\}+p_{0}^{\ast}\mapsto(v,p_{0}^{\ast})=({\rm L}_{\rm
R}^{\ast\dagger}\\{f\\},\mathrm{Proj}_{{\mathcal{P}}^{\prime}}\\{f\\})$.
Moreover, ${\mathcal{S}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}L_{1,n_{0}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{X}}_{{\mathrm{L}}_{\rm
R}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ with the embedding being
continuous and dense.

Theorem 9 obviously also applies to scenarios where the null space is trivial
by setting ${\mathcal{P}}^{\prime}=\\{0\\}$ and only retaining (83), in which
case ${\mathcal{X}}_{{\rm L}_{\rm R}}(\mathbb{R}^{d})={\rm L}^{\ast}_{\rm
R}({\mathcal{X}}_{\rm Rad})$ and
${\mathcal{S}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}L_{1}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{X}}_{{\mathrm{L}}_{\rm
R}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$.

###### Proof.

Since ${\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}\big{(}{\mathcal{S}}_{\rm
Rad}\big{)}={\mathcal{S}}(\mathbb{R}^{d})$, the image of ${\mathcal{S}}_{\rm
Rad}$ under ${\rm L}_{\rm R}^{\ast}$ is a vector space denoted by
${\mathcal{S}}_{{\rm L}_{\rm R}^{\ast}}(\mathbb{R}^{d})={\rm L}_{\rm
R}^{\ast}\big{(}{\mathcal{S}}_{\rm Rad})={\rm
L}^{\ast}\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}$. The spline-
admissibility of ${\rm L}$ implies that ${\rm L}^{\ast}$ is injective on
${\mathcal{S}}(\mathbb{R}^{d})$ which, in turn, translates into the
injectivity of ${\rm L}_{\rm R}^{\ast}$ on ${\mathcal{S}}_{\rm Rad}$. The
latter statement is equivalent to the existence of a linear operator
$\left.{\rm L}_{\rm R}^{\ast-1}\right|_{{\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}}={\rm L}_{\rm R}^{\ast-1}$ (for short) such that, for any $u={\rm
L}_{\rm R}^{\ast}\\{\phi\\}\in{\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}(\mathbb{R}^{d})$ with $\phi\in{\mathcal{S}}_{\rm Rad}$, it holds
that

$\displaystyle{\rm L}_{\rm R}^{\ast-1}\\{u\\}={\rm L}_{\rm R}^{\ast-1}{\rm
L}_{\rm R}^{\ast}\\{\phi\\}=\phi.$

Hence, if $\phi\mapsto\|\phi\|_{{\mathcal{X}}}$ is a valid norm on
${\mathcal{S}}_{\rm Rad}$, then the same holds true for
$u\mapsto\|u\|_{{\mathcal{U}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|{\rm
L}_{\rm R}^{\ast-1}\\{u\\}\|_{{\mathcal{X}}}$ on ${\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}(\mathbb{R}^{d})$. This means that the normed spaces
$({\mathcal{S}}_{\rm Rad},\|\cdot\|_{{\mathcal{X}}})$ and
$({\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}(\mathbb{R}^{d}),\|\cdot\|_{{\mathcal{U}}})$ are (isometrically)
isomorphic. Moreover, since ${\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}(\mathbb{R}^{d})\subset L_{1,n_{0}}(\mathbb{R}^{d})$ (Condition 4 in
Definition 2) and ${\mathcal{S}}_{\rm Rad}\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{X}}_{\rm Rad}$, the inverse operator $\left.{\rm L}_{\rm
R}^{\ast-1}\right|_{{\mathcal{S}}_{{\rm L}_{\rm R}^{\ast}}}$ coincides on
${\mathcal{S}}_{{\rm L}_{\rm R}^{\ast}}$ with ${\rm L}_{\rm
R}^{\ast\dagger}:L_{1,n_{0}}(\mathbb{R}^{d})\xrightarrow{\mbox{\tiny\ \rm c.
}}{\mathcal{X}}_{\rm Rad}$ whose impulse response is characterized in Theorem
7. The well-posedness and boundedness of ${\rm L}_{\rm R}^{\ast\dagger}$ on
$L_{1,n_{0}}(\mathbb{R}^{d})$ for $n_{0}=\lceil\gamma_{0}-1\rceil$ follows
from Theorem 8 and (68) in Theorem 7, which provides the required stability
condition. The other fundamental ingredient is $\langle
m_{\boldsymbol{k}},u\rangle=\langle m_{\boldsymbol{k}},{\rm L}_{\rm
R}^{\ast}\\{\phi\\}\rangle=\langle{\rm L}_{\rm
R}\\{m_{\boldsymbol{k}}\\},\phi\rangle_{\rm Rad}=0$ for all
$u\in{\mathcal{S}}_{{\rm L}^{\ast}}(\mathbb{R}^{d})$ and
$|{\boldsymbol{k}}|\leq n_{0}$, which implies that
$\mathrm{Proj}_{{\mathcal{P}}^{\prime}}\\{u\\}=0$. Consequently, we have that

${\rm L}_{\rm R}^{\ast\dagger}\\{u\\}={\rm L}_{\rm
R}^{\ast-1}(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})\\{u\\}={\rm
L}_{\rm R}^{\ast-1}\\{u\\}={\rm L}_{\rm R}^{\ast-1}{\rm L}_{\rm
R}^{\ast}\\{\phi\\}=\phi,$

which confirms the equivalence of ${\rm L}_{\rm R}^{\ast\dagger}$ and
$\left.{\rm L}_{\rm R}^{\ast-1}\right|_{{\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}}}$.

So far, we have shown that the operator ${\rm L}_{\rm
R}^{\ast\dagger}:\big{(}{\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}},\|\cdot\|_{{\mathcal{U}}}\big{)}\to{\mathcal{X}}_{\rm Rad}$ is an
isometry. As next step, we invoke the bounded linear transformation (BLT)
theorem [42, Theorem I.7, p 9] to uniquely extend the operator to the
completed space ${\mathcal{U}}=\overline{\big{(}{\mathcal{S}}_{{\rm L}_{\rm
R}^{\ast}},\|\cdot\|_{{\mathcal{U}}}\big{)}}$ which, by construction, is the
Banach space equipped with the norm $\|\cdot\|_{{\mathcal{U}}}$. This
extension argument also applies the other way around: Since
${({\mathcal{S}}_{{\rm
L}^{\ast}}(\mathbb{R}^{d}),\|\cdot\|_{{\mathcal{U}}})}\xhookrightarrow{\mbox{\tiny\rm
iso.}}{\mathcal{U}}$, the operator ${\rm L}_{\rm R}^{\ast}:({\mathcal{S}}_{\rm
Rad},\|\cdot\|_{{\mathcal{X}}})\to{\mathcal{U}}$ has a unique (isometric)
extension ${\rm L}_{\rm R}^{\ast}:{\mathcal{X}}_{\rm Rad}\to{\mathcal{U}}$
with ${\mathcal{X}}_{\rm Rad}$ being the closure ${\mathcal{X}}_{\rm
Rad}=\overline{({\mathcal{S}}_{\rm Rad},\|\cdot\|_{{\mathcal{X}}})}$. This
proves that the spaces ${\mathcal{X}}_{\rm Rad}$ and ${\mathcal{U}}$ are
isometrically isomorphic with ${\mathcal{U}}={\rm L}_{\rm
R}^{\ast}\big{(}{\mathcal{X}}_{\rm Rad}\big{)}$ and ${\mathcal{X}}_{\rm
Rad}={\rm L}_{\rm R}^{\ast\dagger}\big{(}{\mathcal{U}}\big{)}$. In addition,
we have that ${\mathcal{U}}\perp{\mathcal{P}}$, which means that
$\mathrm{Proj}_{{\mathcal{P}}^{\prime}}\\{u\\}=0$ for all $u\in{\mathcal{U}}$.
Since ${\mathcal{U}}$ and ${\mathcal{P}}^{\prime}$ are both Banach spaces, the
direct-sum space ${\mathcal{X}}_{{\rm L}_{\rm
R}}={\mathcal{U}}\oplus{\mathcal{P}}^{\prime}$, equipped with the composite
norm $\|f\|_{{\mathcal{X}}_{{\rm L}_{\rm
R}}}=\max(\|\mathrm{Proj}_{{\mathcal{U}}}\\{f\\}\|_{{\mathcal{U}}},\|\mathrm{Proj}_{{\mathcal{P}}^{\prime}}\\{f\\}\|_{{\mathcal{P}}^{\prime}})$,
is complete (Banach property) and isomorphic to ${\mathcal{X}}_{\rm
Rad}\times{\mathcal{P}}^{\prime}$. The final element is to recognize that
$\|\mathrm{Proj}_{{\mathcal{U}}}\\{f\\}\|_{{\mathcal{U}}}=\|{\mathrm{L}}_{\rm
R}^{\ast\dagger}\mathrm{Proj}_{{\mathcal{U}}}\\{f\\}\|_{{\mathcal{X}}}=\|{\mathrm{L}}_{\rm
R}^{\ast\dagger}\\{f\\}\|_{{\mathcal{X}}}$, where
$\mathrm{Proj}_{{\mathcal{U}}}=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})$.
This direct-sum decomposition has an equivalent description in terms of
operators, which is the form given in the statement of Theorem 9.
Specifically, the isomorphism between ${\mathcal{U}}$ and ${\mathcal{X}}$ is
expressed by (83) and (85) for $f\in{\mathcal{U}}$. This is complemented by
the null-space property (84), which ensures that the components of $f$ that
are in ${\mathcal{P}}^{\prime}$ are annihilated by ${\rm L}_{\rm R}^{\ast}$.

Embeddings: The denseness of ${\mathcal{S}}$ in
${\mathcal{X}}_{{\mathrm{L}}_{\rm R}}$ follows from the observation that
${\mathcal{S}}(\mathbb{R}^{d})=(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})\big{(}{\mathcal{S}}(\mathbb{R}^{d})\big{)}\oplus{\mathcal{P}}^{\prime}$.
Since, by construction, one has that
$(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})({\mathcal{S}})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{U}}$ and
$(\mathrm{Id}-\mathrm{Proj}_{{\mathcal{P}}^{\prime}})(L_{1,n_{0}})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{U}}$, one also has that
${\mathcal{S}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}L_{1,n_{0}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{X}}_{{\mathrm{L}}_{\rm R}}(\mathbb{R}^{d})$.

As for the relation ${\mathcal{X}}_{{\mathrm{L}}_{\rm
R}}={\mathcal{U}}\oplus{\mathcal{P}}^{\prime}\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, we already have that
${\mathcal{P}}^{\prime}\xhookrightarrow{}{\mathcal{S}}(\mathbb{R}^{d})\xhookrightarrow{}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$,
by construction. To show that
${\mathcal{U}}\xhookrightarrow{}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, we
invoke the intertwining relation and express ${\mathcal{U}}={\rm
L}^{\ast}{\mathrm{R}}^{\ast}{\mathrm{K}}_{\rm rad}({\mathcal{X}}_{\rm Rad})$
as ${\mathcal{U}}={\mathrm{R}}^{\ast}{\mathrm{Q}}_{\rm rad}({\mathcal{X}}_{\rm
Rad})$, where ${\mathrm{Q}}_{\rm rad}={\mathrm{L}}_{\rm rad}{\mathrm{K}}_{\rm
rad}:{\mathcal{X}}_{\rm
Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$. Our
hypotheses on $\widehat{L}_{\rm rad}$ ensure that ${\mathrm{Q}}_{\rm
rad}\\{\phi\\}$ is well-defined for every $\phi\in{\mathcal{X}}$, with the
operator being continuous in the weak topology of
${\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})$. (As the latter is
a complete nuclear space, the weak (sequential) convergence also ensures
continuity in the strong topology [50].) Since
${\mathrm{R}}^{\ast}:{\mathcal{S}}^{\prime}(\mathbb{R}\times\mathbb{S}^{d-1})\xrightarrow{\mbox{\tiny\
\rm c. }}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$, as implied by (25), the
composed operator ${\mathrm{R}}^{\ast}{\mathrm{Q}}_{\rm
rad}:{\mathcal{X}}_{\rm Rad}\to{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ is
continuous as well, which proves that ${\mathcal{U}}$ and
${\mathcal{X}}_{{\mathrm{L}}_{\rm R}}$ are both continuously embedded in
${\mathcal{S}}^{\prime}(\mathbb{R}^{d})$. The latter embedding is also dense
by transitivity since
${\mathcal{S}}(\mathbb{R}^{d})\subset{\mathcal{X}}_{{\mathrm{L}}_{\rm R}}$ and
${\mathcal{S}}(\mathbb{R}^{d})\xhookrightarrow{\mbox{\tiny\rm
d.}}{\mathcal{S}}^{\prime}(\mathbb{R}^{d})$ [48, 19].

∎

##  Appendix A Appendix: Duality Maps

The generalization of the Cauchy-Schwarz inequality for any dual pair
$({\mathcal{X}},{\mathcal{X}}^{\prime})$ of Banach spaces is the so-called
duality inequality

$\displaystyle\forall(f,x)\in{\mathcal{X}}^{\prime}\times{\mathcal{X}}:\langle
f,x\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}\leq\|f\|_{{\mathcal{X}}^{\prime}}\|x\|_{{\mathcal{X}}},$
(86)

which is tightly linked to the definition of the dual norm given by

$\displaystyle\|f\|_{{\mathcal{X}}^{\prime}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\sup_{x\in{\mathcal{X}}:\|x\|_{{\mathcal{X}}}\leq
1}\langle f,x\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}.$ (87)

######  Definition 4 (Strict convexity).

A Banach space ${\mathcal{X}}$ is said to be strictly convex if, for all
$x_{1},x_{2}\in{\mathcal{X}}$ such that
$\|x_{1}\|_{{\mathcal{X}}}=\|x_{2}\|_{{\mathcal{X}}}=1$ and $x_{1}\neq x_{2}$,
one has that $\|\theta x_{1}+(1-\theta)x_{2}\|_{{\mathcal{X}}}<1$ for all
$\theta\in(0,1)$.

It is obvious from (87) that (86) is sharp. Moreover, when ${\mathcal{X}}$ is
reflexive and strictly convex, there is a single element
$x^{\ast}\in{\mathcal{X}}^{\prime}$ (the Banach conjugate of $x$) such that
$\|x^{\ast}\|_{{\mathcal{X}}^{\prime}}=\|x\|_{{\mathcal{X}}}$ (isometry) and
$\langle
x^{\ast},x\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}=\|x^{\ast}\|_{{\mathcal{X}}^{\prime}}\|x\|_{{\mathcal{X}}}$
(sharp duality bound) [13]. This leads to the definition of the corresponding
duality map
${\mathrm{J}}_{\mathcal{X}}:{\mathcal{X}}\to{\mathcal{X}}^{\prime}$ as

$\displaystyle{\mathrm{J}}_{{\mathcal{X}}}\\{x\\}=x^{\ast}.$ (88)

Since the dual of ${\mathcal{X}}^{\prime}$ is strictly convex as well, we have
that
${\mathrm{J}}_{\mathcal{X}}^{-1}={\mathrm{J}}_{{\mathcal{X}}^{\prime}}:{\mathcal{X}}^{\prime}\to{\mathcal{X}}$
with ${\mathrm{J}}_{{\mathcal{X}}^{\prime}}\\{x^{\ast}\\}=x$, where
$(x^{\ast})^{\ast}=x\in{\mathcal{X}}^{\prime\prime}={\mathcal{X}}$ is the
unique Banach conjugate of $x^{\ast}\in{\mathcal{X}}^{\prime}$.

A relevant example of reflexive and strictly convex Banach space is
${\mathcal{X}}=L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})$ for $q\in(1,\infty)$.
Its topological dual is
${\mathcal{X}}^{\prime}=L_{p}(\mathbb{R}\times\mathbb{S}^{d-1})$ with
$p=q/(q-1)$ being the conjugate exponent of $q$. For that particular pair,
(86) reduces to the Hölder inequality for hyper-spherical functions. From [13,
Chapter 4], the corresponding duality map
${\mathrm{J}}_{q}:L_{q}(\mathbb{R}\times\mathbb{S}^{d-1})\to
L_{p}(\mathbb{R}\times\mathbb{S}^{d-1})$ is

$\displaystyle{\mathrm{J}}_{q}\\{\nu\\}({\boldsymbol{z}})=\nu^{\ast}({\boldsymbol{z}})=\frac{\left|\nu({\boldsymbol{z}})\right|^{q-1}}{\|\nu\|^{q-2}_{L_{q}}}{\rm
sign}\big{(}\nu({\boldsymbol{z}})\big{)},$ (89)

which establishes a one-to-one isometric mapping between the spaces $L_{q}$
and $L_{p}=(L_{q})^{\prime}$ with the property that
${\mathrm{J}}^{-1}_{q}={\mathrm{J}}_{p}$.

######  Proposition 4 (Banach isometries).

Let $({\mathcal{X}},{\mathcal{X}}^{\prime})$ be a dual pair of reflexive and
strictly convex Banach spaces with corresponding duality map
${\mathrm{J}}_{\mathcal{X}}:{\mathcal{X}}\to{\mathcal{X}}^{\prime}$. We
consider two generic types of linear isometries.

1) One-to-one map: Let
${\mathrm{T}}:{\mathcal{X}}\to{\mathcal{Y}}={\mathrm{T}}({\mathcal{X}})$ be an
injective operator whose inverse is denoted by ${\mathrm{T}}^{-1}$ with
${\mathrm{T}}^{-1}{\mathrm{T}}=\mathrm{Id}$ on ${\mathcal{X}}$. Then,
${\mathcal{Y}}={\mathrm{T}}({\mathcal{X}})=\\{y={\mathrm{T}}\\{x\\}:x\in{\mathcal{X}}\\}$
equipped with the norm
$\|y\|_{{\mathcal{Y}}}\stackrel{{\scriptstyle\vartriangle}}{{=}}\|{\mathrm{T}}^{-1}\\{y\\}\|_{{\mathcal{X}}}$
is a Banach space. Its continuous dual is the Banach space
${\mathcal{Y}}^{\prime}={\mathrm{T}}^{-1\ast}({\mathcal{X}}^{\prime})$ with
$\|y^{\prime}\|_{{\mathcal{Y}}^{\prime}}=\|{\mathrm{T}}^{\ast}\\{y^{\prime}\\}\|_{{\mathcal{X}}^{\prime}}$,
while the corresponding duality map is
${\mathrm{J}}_{{\mathcal{Y}}}=({\mathrm{T}}^{\ast})^{-1}\circ{\mathrm{J}}_{{\mathcal{X}}}\circ{\mathrm{T}}^{-1}:{\mathcal{Y}}\to{\mathcal{Y}}^{\prime}$.

2) Projection: Let
${\mathrm{P}}:{\mathcal{X}}\to{\mathcal{U}}\xhookrightarrow{\mbox{\tiny\rm
iso.}}{\mathcal{X}}$ be a continuous projection operator on ${\mathcal{X}}$
with $\|{\mathrm{P}}\|=1$. Then,
$\big{(}{\mathrm{P}}({\mathcal{X}}),{\mathrm{P}}^{\ast}({\mathcal{X}}^{\prime})\big{)}=\big{(}{\mathcal{U}},{\mathcal{U}}^{\prime}\big{)}$
is a dual pair of Banach subspaces with corresponding duality map
${\mathrm{J}}_{{\mathcal{U}}}={\mathrm{P}}^{\ast}\circ{\mathrm{J}}_{{\mathcal{X}}}\circ{\mathrm{P}}:{\mathcal{U}}\to{\mathcal{U}}^{\prime}$.

###### Proof.

We first recall that the dual of a reflexive and strictly convex Banach space
is reflexive (by definition) and strictly convex as well.

1) Injective operator: For the first property, we refer to [53, Proposition
1]. The key observation is that the operators
${\mathrm{T}}:{\mathcal{X}}\to{\mathcal{Y}}$ and
${\mathrm{T}}^{-1}:{\mathcal{Y}}\to{\mathcal{X}}$, as well as their adjoints,
are isometries with $({\mathrm{T}}^{\ast})^{-1}={\mathrm{T}}^{-1\ast}$. The
argument then primarily rests upon the duality inequality

$\displaystyle\langle
y^{\prime},y\rangle_{{\mathcal{Y}}^{\prime}\times{\mathcal{Y}}}=\langle
y^{\prime},{\mathrm{T}}{\mathrm{T}}^{-1}\\{y\\}\rangle_{{\mathcal{Y}}^{\prime}\times{\mathcal{Y}}}$
$\displaystyle=\langle{\mathrm{T}}^{\ast}\\{y^{\prime}\\},{\mathrm{T}}^{-1}\\{y\\}\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}$
$\displaystyle\leq\|{\mathrm{T}}^{\ast}\\{y^{\prime}\\}\|_{{\mathcal{X}}^{\prime}}\;\|{\mathrm{T}}^{-1}\\{y\\}\|_{{\mathcal{X}}},$
(90)

which is sharp if and only if $x={\mathrm{T}}^{-1}\\{y\\}$ and
$x^{\prime}={\mathrm{T}}^{\ast}\\{y^{\prime}\\}$ (resp., $y^{\prime}$ and $y$)
are Banach conjugates, so that
$x^{\prime}={\mathrm{J}}_{{\mathcal{X}}}\\{x\\}$.

2) Projection operator: The first part is obtained by using a standard
argument with (projected) Cauchy sequences. For the second part, let
$u\in{\mathcal{U}}\xhookrightarrow{\mbox{\tiny\rm iso.}}{\mathcal{X}}$ with
Banach conjugate $u^{\ast}\in{\mathcal{X}}^{\prime}$. Then,

$\displaystyle\|u^{\ast}\|_{{\mathcal{X}}^{\prime}}\|u\|_{{\mathcal{X}}}=\langle
u^{\ast},u\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}$
$\displaystyle=\langle
u^{\ast},{\mathrm{P}}^{2}u\rangle_{{\mathcal{X}}^{\prime}\times{\mathcal{X}}}=\langle{\mathrm{P}}^{\ast}u^{\ast},{\mathrm{P}}{\mathrm{u}}\rangle_{{\mathcal{U}}^{\prime}\times{\mathcal{U}}}$
$\displaystyle\
\leq\|{\mathrm{P}}^{\ast}\|\|u^{\ast}\|_{{\mathcal{U}}^{\prime}}\|{\mathrm{P}}\|\|u\|_{{\mathcal{U}}},$
(91)

from which we deduce that
$\langle{\mathrm{P}}^{\ast}u^{\ast},{\mathrm{P}}{\mathrm{u}}\rangle_{{\mathcal{U}}^{\prime}\times{\mathcal{U}}}=\|u^{\ast}\|_{{\mathcal{U}}^{\prime}}\|u\|_{{\mathcal{U}}^{\prime}}$.
We conclude that $u={\mathrm{P}}u$ and
$u^{\ast}={\mathrm{P}}^{\ast}u^{\ast}={\mathrm{P}}^{\ast}\circ{\mathrm{J}}_{{\mathcal{X}}}\\{{\mathrm{P}}u\\}$
are (unique) Banach conjugates of each other. ∎

### Acknowlegdments

The research was supported by the European Commission under Grant ERC-2020-AdG
FunLearn-101020573 . The author is thankful to Sebastian Neumayer for very
helpful discussions.

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