# Uncovering the role of the light-induced reduction of the inter-atomic
exchange in the ultrafast demagnetization of Fe, Co, and Ni

Philippe Scheid<EMAIL_ADDRESS>Department of Physics and
Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Université
de Lorraine, LPCT, CNRS, UMR 7019, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex,
France  Université de Lorraine, IJL, CNRS, UMR 7198, BP 70239, 54000 Nancy
Cedex, France  Julius Hohlfeld  Université de Lorraine, IJL, CNRS, UMR 7198,
BP 70239, 54000 Nancy Cedex, France  Gregory Malinowski  Université de
Lorraine, IJL, CNRS, UMR 7198, BP 70239, 54000 Nancy Cedex, France  Michel
Hehn  Université de Lorraine, IJL, CNRS, UMR 7198, BP 70239, 54000 Nancy
Cedex, France  Wei Zhang  Université de Lorraine, IJL, CNRS, UMR 7198, BP
70239, 54000 Nancy Cedex, France  Anders Bergman  Department of Physics and
Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden  Olle
Eriksson  Department of Physics and Astronomy, Uppsala University, Box 516,
SE-75120 Uppsala, Sweden  School of Science and Technology, Örebro University,
SE-701 82, Örebro, Sweden  Sébastien Lebègue  Université de Lorraine, LPCT,
CNRS, UMR 7019, BP 70239, 54506 Vandoeuvre-lès-Nancy Cedex, France  Stéphane
Mangin  Université de Lorraine, IJL, CNRS, UMR 7198, BP 70239, 54000 Nancy
Cedex, France

###### Abstract

Time resolved measurements of the linear magneto-optical Kerr rotation reveal
that, in the 3d ferromagnets Fe, Co and Ni, the amplitude of the
demagnetization increases linearly with the fluence of the light. We
rationalize this phenomenon as being linearly driven by the increase in
temperature, the electron-phonon and electron-magnon scattering, and a
reduction of the inter-atomic exchange. The amplitude of the latter
phenomenon, which until the present study has been widely overlooked, is
obtained through ab initio density functional theory calculations, and is
argued to be the principal source of demagnetization in Ni and Co, while still
contributing largely in Fe.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

More than twenty years ago, Beaurepaire et al.Beaurepaire  _et al._ (1996)
produced the first experimental evidence of the ultrafast quenching of the
magnetic order by femtosecond light pulses. This discovery, and the
perspective it held to allow for the manipulation of magnetization in
unprecedented timescale, and without the use of a magnetic field, triggered an
intense and very fruitful research that is still ongoing. Notably, a few years
later, the all–optical switching, in which femtosecond light pulses allow for
the deterministic manipulation of the magnetic state of metallic thin films
was evidencedStanciu  _et al._ (2007); Radu  _et al._ (2011). Since then, both
the single pulse, all-optical helicity-independent switching, as well as its
helicity-dependent counterpart were found in evermore complex structures and
compoundsMangin  _et al._ (2014); Lambert  _et al._ (2014). However,
regardless of the apparent intricacy of these light–induced studies of
magnetisation dynamics, the common denominator of all these phenomena is the
presence of an initial ultrafast demagnetization.

Naggingly, up to this day, the inner workings of this overarching phenomenon
remain heavily debated, as much from the standpoint of the magnetic
excitations responsible for the quenching, as regarding the types of
mechanisms by which they are triggered. Early photoemission resultsRhie  _et
al._ (2003); Cinchetti  _et al._ (2006); Tengdin  _et al._ (2018); You  _et
al._ (2018), pointed toward a reduction of the exchange splitting, suggesting
induced Stoner excitations, thus inducing a longitudinal decrease of the
atomic magnetic momentsScheid  _et al._ (2022). More recently Turgut et
al.Turgut  _et al._ (2016) and Eich et al.Eich  _et al._ (2017) explicitly
addressed this issue and experimentally showed a predominance of band
mirroring in Co, a consequence of the presence of transversal excitations. As
opposed to a decrease of the magnitude of each individual atomic magnetic
moment, the demagnetization would therefore mainly result from a randomization
of their direction. A similar conclusion has been drawn for bcc FeJana  _et
al._ (2020).

To rationalize such dynamics, many theories have been proposed throughout the
years. As reviewed by Scheid et al.Scheid  _et al._ (2022), possible sources
of longitudinal excitations have been proposed; such as Elliot–Yafet electron-
phonon spin-flip scatteringsCarva  _et al._ (2013), a spin-dependent
superdiffusive propagation of the hot-electronsBattiato  _et al._ (2010), a
light–induced increase of the electronic temperatureScheid  _et al._ (2019a),
and a direct light–matter interaction.Krieger  _et al._ (2015); Scheid  _et
al._ (2019b, 2021) However, these theories were not able to explain the
magnitude of the experimentally seen demagnetization, at least within the
confinement of the experimentally absorbed energy. On the other hand, ever
since the pioneering work of Beaurepaire et al.Beaurepaire  _et al._ (1996),
various temperature models have been a mean of understanding the
characteristics of the demagnetization through a rapid increase of the
temperature experienced by the magnetic momentsKoopmans  _et al._ (2010); Roth
_et al._ (2012); Zahn  _et al._ (2021); Pankratova  _et al._ (2022). While
such a framework does not explain how the angular momentum is conserved during
the demagnetization, it provides the most straightforward rationalization of
the latter, as originating from an ultrafast heating. Accordingly, in this
regime, the demagnetization would occur through the generation of transversal
excitations, or magnons, as they are the primary magnetic excitations
responsible for the ferromagnetic-paramagnetic phase transition occuring at
the Curie temperature in the elements studied here (e.g. as pointed out by
GunnarssonGunnarsson (1976)).

In the present, combined experimental and theoretical work, we demonstrate
that the maximal demagnetization amplitude in the elemental transition metal
(TM) ferromagnets Fe, Co and Ni challenges the aforementioned heating
hypothesis. Indeed, it is argued here that only accounting for purely thermal
effects occurring on top of electron-phonon and electron-magnon couplings
hardly allows to simultaneously explain the experimentally seen
demagnetization amplitude across the compounds. However, utilizing ab initio
calculations, we provide significant evidences that considering the light-
induced reduction of the inter-atomic exchange can rationalize the
experimental demagnetization. Hence, rather than driving the magnetic system
toward the Curie temperature, T${}_{\text{C}}$, through heating as explained
within the three temperature model based on rigid spins, T${}_{\text{C}}$ is
suggested here to be transiently decreased.

Glass/Ta(2)/TM(12)/Pt(2) multilayers were grown by RF (TM) and DC (Ta,
Pt)-magnetron sputtering in a deposition tool with base pressure less than
3$\cdot 10^{-8}$ mbar. Float glass was used for the pump/probe geometry, the
Ta(2) buffer layer promotes adhesion of the layers and block any oxygen
diffusion, and Pt(2) capping layer is used to prevent layer oxidation. The
thickness of the films is large enough such that bulk properties are
recovered, while the effect of absorption gradients are mitigated by the rapid
spreading of hot electrons. The dynamics were probed using a magneto-optical
microscope that monitors the longitudinal Kerr effect for spatially uniform
s-polarized probe pulses having a wavelength of $515\,$nm and a duration of
$150\,$fs as function of pump-probe delay. The latter were incident on the
samples at an angle of $45\degree$. The normally incident, linearly polarized
$800\,$nm, $150\,$fs pump pulses were focused on the sample to a $95\,\mu$m
(FWHM) Gaussian. The pump was aligned to the center of the $560\times
360\,\mu$m area of the sample displayed on the camera. The spatial resolution
of the image was $\approx 10\,\mu$m. In-plane magnetic saturation fields were
applied during the measurements to ensure full restoration of the
magnetization between the pump-probe pulse pairs. The pulse repetition rate
was 50 kHz and the exposure time of the camera was of the order of 1s, so that
each image represents the average signal resulting from 50 000 laser pulses.
The fluence dependence of the pump induced magnetization dynamics is derived
from the measured images in three steps: First, potential artifacts (e.g. from
pump induced changes of the non-magnetic sample reflectivity) are suppressed
by subtracting the images measured for opposite magnetization directions at
identical pump-probe delay, and dividing this difference by the one obtained
at negative delay. Second, the spatial variations of the measured
magnetization changes are mapped to the ones of the pump beam. Third, external
fluences are converted to absorbed ones by means of the TMM python package and
the complex indices extracted from Ref.Rumble (2022).

Figure 1: a) Light–induced ultrafast demagnetization dynamics in Fe, Co and Ni
at different absorbed fluences measured using the TR-MOKE. b) Normalized
maximum demagnetization,
m${}_{\text{min}}=\frac{\text{M}_{\text{min}}}{\text{M}_{\text{0}}}$ as a
function of the fluence.

Fig. 1 a) presents the magnetization dynamics for all three samples for
different values of the absorbed fluence. Even though, as expected, all the
dynamics are characterized by a very swift reduction of the magnetization,
these results challenge the existing viewpoints on multiple accounts.
To further investigate these experimental results, in Fig. 1 b), we consider
the maximum relative demagnetization,
m${}_{\text{min}}=\frac{\text{M}_{\text{min}}}{\text{M}_{\text{0}}}$, where
$M_{0}$ and $M_{\text{min}}$ respectively are the magnetization at saturation
and the minimum magnetization reached during the dynamics. Doing so shows that
the amplitude of the demagnetization increases linearly with the absorbed
energy, i.e. can be written as:

$\text{m}_{\text{min}}=1+\alpha_{\text{exp}}\text{E}_{\text{abs}}$ (1)

where $\alpha_{\text{exp}}$ is the slope, which value is displayed in Tab.1
for the different compounds and $\text{E}_{\text{abs}}$ is the absorbed energy
density, computed by dividing the absorbed fluences by the thicknesses of the
absorbing films (16 nm).
Interestingly, this linear decrease of the magnetization with the absorbed
energy contradicts the well established Bloch law stating that, upon thermal
excitation, the variation of the magnetization is proportional to
$T_{m}^{3/2}$, or equivalently $\propto E_{m}^{3/5}$ since $E_{m}\propto
T_{m}^{5/2}$, where $T_{m}$ and $E_{m}$ respectively are the temperature and
the energy of the magnons.
Nonetheless, as suggested by the fact that it holds the lowest
T${}_{\text{C}}$ (627 K) of the three compounds studied here, Ni is by far the
most easily demagnetized element. On the other hand, and quite surprisingly
such a reasoning fails when comparing Co, which has a T${}_{\text{C}}$ of 1388
K, with Fe, that has a significantly lower T${}_{\text{C}}$ of 1043 K, as both
of them present very similar demagnetization amplitudes.
In the framework of temperature models, this unexpected similarity in the
value of $\alpha_{\text{exp}}$ for Co and Fe, could be the consequence of a
larger electron-phonon coupling in Fe than in Co, thus more rapidly evacuating
the absorbed energy from its electronic and spin degrees of freedom, and
mitigating its demagnetization. Along this line of argument, the very
pronounced demagnetization of Ni could originate from a very low electron-
phonon coupling that would allow for the light–absorbed energy to almost
solely affect the spin system and thus enhance the demagnetization. However,
as can be seen in Tab.1 recapitulating ab initio calculated values from
Ritzmann et al.Ritzmann  _et al._ (2020), the electron-phonon coupling in Fe
is more than three times weaker than it is in Co, while in Ni it is located in
between, thus invalidating the aforementioned scenario.
Likewise, in a framework where the demagnetization is mostly due to
transversal excitations, which, as discussed supra, appears to be the case, a
high electron-magnon coupling in Ni and Co as compared to Fe could explain the
higher demagnetization in Ni and Co and the surprising resilience of Fe.
However, as reported in Tab. 1, ab initio calculations from Haag et al.Haag
_et al._ (2014) in Fe and Ni show that the electron-magnon scattering rate in
Ni is one order of magnitude lower than in Fe. While Haag et al.Haag  _et al._
(2014) did not treat the case of Co, the work of Müller et al.Müller  _et al._
(2019) indicates that it is similar to the one in Ni. Indeed, Müller et al.
reported ab initio calculations of the electronic lifetime broadening and band
renormalization in Fe, Co and Ni as a consequence of the electron–magnon
couling and found the strongest signatures of the latter in Fe, followed by Co
and Ni. Consequently the electron-magnon coupling alone cannot explain the
very large demagnetization in Ni and the resilience of Fe either.
One should also note that the demagnetization dynamics in Fe differs from the
one of Co and Ni as, even at low demagnetization amplitudes, it does not
feature the so-called fast recoveryYou  _et al._ (2018), otherwise present in
Ni and particularly in Co. According to You et al.You  _et al._ (2018) the
passage from a slow recovery to a slow recovery regime is due to a transient
crossing of T${}_{\text{C}}$, however the result presented here contradicts
this hypothesis. Indeed, Ni, which has a much lower T${}_{\text{C}}$ than Fe,
recovers more quickly than Fe, while demagnetizing much more than the latter.
Finally, in the framework of the microscopic three temperature modelKoopmans
_et al._ (2010); Roth  _et al._ (2012), the demagnetization rate is governed
by the Elliot-Yaffet spin-flip scattering, and, if the electron-phonon
coupling is low enough such that the electronic temperature stays high long
enough, the initial demagnetization is followed by a remagnetization.
Consequently, in this framework too, Fe, which has the lowest electron-phonon
coupling and seemingly the largest Elliot-Yaffet spin-flip scattering rate of
the three compounds studied hereCarva  _et al._ (2013), should be more prone
to feature a large demagnetization followed by a remagnetization occuring in
less than 1 ps, the duration required for the equilibration of the temperature
of the phonons and the electrons in Fe, as shown by Ritzmann et al.Ritzmann
_et al._ (2020).

| Ni | Co | Fe
---|---|---|---
Curie temperature (K) | 627 | 1388 | 1043
G${}_{\text{ep}}$ (1017W/(m3K)) | 18.9 | 33.4 | 10.5
W${}^{\uparrow\downarrow}_{\text{magnon}}$ (scattering / 100 fs) | 0.15 |  $\approx$ 0.15 | 1.34
$\frac{dj}{dE}$ (% GJ-1 m3) | -48.40 | -15.82 | -7.85
$\alpha_{\text{exp}}$ (% GJ-1 m3) | -58.51 | -15.71 | -19.27
Table 1: Table summarizing the Curie temperature, T${}_{\text{C}}$, and the
electron-phonon scattering rates, G${}_{\text{ep}}$, from Ref. Ritzmann  _et
al._ (2020) and electrons–magnons scattering between two dominant spin states
per 100 fs and per atom, $W^{\uparrow\downarrow}_{\text{magnon}}$ calculated
at an electronic temperature of 900 K in Ref. Haag  _et al._ (2014) for Ni and
Fe. As discussed in the text, based on Ref. Müller  _et al._ (2019) we
assessed the value of $W^{\uparrow\downarrow}_{\text{magnon}}$ in Co to be
close to the one in Ni. In addition, both the slope of the ab initio
calculated inter-atomic exchange reduction, as well as the maximum
demagnetization experimentally measured as a function as the absorbed energy
are recapitulated.

To sort out the apparent inconsistencies discussed above, we investigate the
involvement of a previously overlooked phenomenon, which we find to be
strongly correlated to the magnitude of the demagnetization: the ultrafast
light-induced quenching of the inter-atomic exchange. We use ab initio density
functional theory calculations to compute the dependence of the inter-atomic
exchange on the light–absorbed energy. As in our previous workScheid  _et al._
(2019a), as well as in the framework of the temperature models, the increased
electronic energy is accounted for as a rise of the electronic temperature,
T${}_{\text{e}}$, such that the occupation of the different Kohn–Sham states
obey the Fermi-Dirac distribution. In this framework, the T${}_{\text{e}}$
-dependent magnons are obtained within a formalism consisting in a mapping of
the energy obtained from density functional theory spin-spiral calculations on
the Heisenberg HamiltonianAntropov  _et al._ (1995, 1996); Halilov  _et al._
(1998). In the latter, the energy of interaction of an atom indexed by $i$
with all of the other atoms indexed by $j$ writes as:

$\hat{H}_{i}(t)=-\frac{1}{2}\sum_{i\neq
j}J_{ij}\hat{\bm{M}}_{j}(t)\cdot\hat{\bm{M}}_{i}(t).$ (2)

Neglecting the spin correlations and in the adiabatic approximation, one
retrieves the dampingless Laudau–Lifshitz equation for the motion of the
atomic magnetic momentsAntropov  _et al._ (1996)
$\frac{d\left\langle\hat{\bm{M_{i}}}\right\rangle(t)}{dt}\approx-\frac{g\mu_{B}}{\hbar}\sum_{i\neq
j}J_{ij}\left\langle\hat{\bm{M}}_{i}\right\rangle(t)\times\left\langle\hat{\bm{M}}_{j}\right\rangle(t)$,
which eigenstates in crystal lattices, are collective excitations called
magnons. When the latter only contains one atom per Bravais lattice point, as
it is the case of this work considering bcc-Fe, fcc-Co and fcc-Ni, the magnon
states are fully indexed by their wavevector, $\bm{q}$, and their
eigenenergies simply write as:

$\omega_{\bm{q}}=\frac{Mg\mu_{B}}{\hbar}\tilde{J}^{\bm{q}},$ (3)

where $\tilde{J}^{\textbf{q}}=J^{\textbf{q}}-J^{\textbf{q}=\bm{0}}$, $M$ is
the atomic magnetic moment and $J^{\textbf{q}}$ is the Fourier transform of
the real-space Heisenberg exchange $J_{ij}$:

$J^{\textbf{q}}=-\sum_{j}(1-\delta_{ij})J_{ij}e^{i\textbf{q}(\textbf{R}_{j}-\textbf{R}_{i})}.$
(4)

$\tilde{J}^{\textbf{q}}$ can readily be assessed in density functional theory
calculations using the spin-spiral ansatzKurz  _et al._ (2004), as it relates
to the energy, Eq of such a state through:

$E_{\bm{q}}=\frac{1}{2}M^{2}\left(J^{0}+\tilde{J}^{\textbf{q}}\sin^{2}\theta\right),$
(5)

where E0 is the electronic energy of the ferromagnetic state, $E_{\textbf{q}}$
is the electronic energy when the atomic magnetic moments are forming a spin-
spiral of wavevector q and are are tilted away from the parallel,
ferromagnetic axis by $\theta$.
Finally, in order to obtain the exchange interaction between pairs of atoms
indexed by $i$ and $j$, we performed the inverse Fourier transform of
$\tilde{J}^{\textbf{q}}$, writing as:

$J_{ij}=-\frac{1}{\Omega_{BZ}}\int_{BZ}d\textbf{q}e^{-i\textbf{q}\cdot(\textbf{R}_{i}-\textbf{R}_{j})}\tilde{J}^{\textbf{q}}$
(6)

Ground state energies were obtained from density functional theory
calculationsHohenberg and Kohn (1964); Kohn and Sham (1965) using the non-
colinear full potential augmented planewave method (APW) and the LSDAPerdew
and Wang (1992) for the exchange correlation potential, as implemented in the
ELK code. The APW basis is expanded in 10 spherical harmonics,
$\left|k+G\right|_{max}R=8$, where $R$ is the radius of the muffin–tin, and
the Brillouin zone is sampled with a 28x28x28 mesh for all the compounds. A
large grid is particularly important in order to insure the convergence at low
T${}_{\text{e}}$. The self–consistence is achieved when the total energy
changes by less than 10-8 Ha in two successive electronic loops. The integrals
over the first Brillouin zone required in the computation of the density of
states (DOS) of the magnons, as well as the real-space inter-atomic exchanges
are performed on a 14$\times$14$\times$14 q-point sampling grid. The DOS are
obtained by using a Gaussian smearing of 0.02 eV to replace the Dirac deltas.
All of these tasks are performed in a separate, home-made, python code.

We then used this framework to assess the impact of the light on the inter-
atomic exchanges within bcc Fe, fcc Co and fcc Ni by computing the latter at
different values of the electronic temperature, T${}_{\text{e}}$, and thus of
the electronic energy, $E(T_{e})$. As in these compounds, long range exchange
interactions significantly contribute to the magnetic orderingHalilov  _et
al._ (1998); Kudrnovský  _et al._ (2001), we computed the effective value of
the inter-atomic exchange, $j$, felt by an atomic site labeled by $\alpha$, by
including all the interactions $J_{\alpha i}(E)$, comprised from the first, up
to its 20th neighbors as a function of $E(T_{e})$:

$j(E)=\frac{\sum_{i}J_{\alpha i}(E)}{\sum_{i}J_{\alpha i}(E(T_{e}=300K))}$ (7)

where the electronic energy, Ee, is readily extracted from the
T${}_{\text{e}}$ dependent ab initio calculations. Note that, as we are only
interested in the rate of reduction of the inter-atomic exchange and not in
its absolute value, in Eq. 7, we normalized by the same quantity computed at
T${}_{\text{e}}$ $=300$K.

Figure 2: Normalized inter-atomic exchange in Ni, Co and Fe, as a function of
the electronic energy density obtained through ab initio spin-spiral
calculations.

Fig. 2, featuring $j(E)$, shows that the inter-atomic exchange decreases
linearly with the increased electronic energy. Quite remarkably, this
reduction is far from being negligible in the range of the experimental
fluences. This fact is better seen in Tab. 1, gathering the slopes of both the
maximum demagnetization as a function of the absorbed energy density, shown in
Fig. 1 b), and the slope of the decrease of $j$ as a function of the absorbed
energy shown in Fig. 2. Strikingly, in Ni and Co both rates of decrease are
matching quite closely, while in Fe the rate of the decrease of the exchange
is half the one of the maximum demagnetization. This apparent correlation
between the maximum demagnetization and the ab initio calculated reduction of
the exchange points toward a novel mechanism where, instead of, or on top of
being the consequence of an increase of the temperature, the demagnetization
could be due to a quenching of interaction responsible for the magnetic
ordering itself. Indeed, upon reduction of the inter-atomic exchange, the
ferromagnetic ordering is expected to decrease through the creation of
magnons. While the rate at which magnons are generated may be tied to the
damping, the reduction of the inter-atomic exchange is due to its parametric
dependence on the electronic state and thus occurs instantaneously with the
perturbation brought by the light-pulse. Such a phenomenology is in direct
contrast with a transfer of heat from the electrons to the magnons as assumed
by temperature models.

Finally, we propose to investigate correlations existing between temperature
effects, the reduction of the inter-atomic exchange, the electron-phonon
coupling ($G_{ep}$) and the electron-magnon coupling ($G_{em}$) in the
experimental amplitude of demagnetization (slope of the curves in Fig. 1 b)).
To this end, we assume that, as the maximum demagnetization scales linearly
with the absorbed energy, it may scale linearly with the amplitude of the four
aforementioned phenomena. Assuming that the amplitude of thermal effects
scales as the inverse of T${}_{\text{C}}$, we get:

$\begin{split}\alpha^{\text{C}}_{\text{fit}}(x_{T},x_{j},x_{G_{ep}},x_{G_{em}})=\\\
x_{T}\frac{1}{T_{C}^{\text{C}}}+x_{j}\frac{dj^{\text{C}}}{dE}+x_{G_{ep}}G_{ep}^{\text{C}}+x_{G_{em}}G_{em}^{\text{C}},\end{split}$
(8)

where $x_{T}$, $x_{j}$, $x_{G_{ep}}$ and $x_{G_{em}}$ are the fitting
coefficients and $\alpha^{\text{C}}_{\text{fit}}$ is the fitted slope for the
different compounds indexed by C.

Figure 3: Contribution to the demagnetization of: (1) thermal fluctuation,
$x_{T}\frac{1}{T_{C}^{\text{C}}}$, in blue (2) reduction of $j$ as a function
of the absorbed energy, $x_{j}\frac{dj^{\text{C}}}{dE}$, in orange (3)
electron-phonon, $x_{G_{ep}}G_{ep}^{\text{C}}$, in green and (4) electron-
magnon coupling, $x_{G_{em}}G_{em}^{\text{C}}$, in red to
$\alpha_{\text{fit}}^{\text{C}}$, the fitted slope of the maximum
demagnetization in purple for Ni, Co and Fe. $\alpha_{\text{fit}}^{\text{C}}$
is computed in three different cases: (Top) minimizing the norm of
$\bm{x}=\left(x_{T},x_{j},x_{G_{ep}},x_{G_{em}}\right)^{\text{T}}$, (Middle)
setting $x_{j}=0$, (Bottom) setting $x_{G_{em}}=0$. For comparison the value
of the experimental slope, $x^{\text{C}}_{\text{exp}}$ is displayed in brown.

Fig. 3 shows $\alpha_{\text{fit}}^{\text{C}}$ (purple bar) as well as the
different contributions to it (blue, orange, red and green), for Fe, Co and
Ni. The figure also shows $\alpha^{\text{C}}_{\text{exp}}$ (brown) for all
elements investigated here. As the system of equations is under-determined,
i.e. there are four variables,
$\bm{x}=\left(x_{T},x_{j},x_{G_{ep}},x_{G_{em}}\right)^{\text{T}}$, and only
three constraints given by the experimental slopes of three elements, we look
for the solution minimizing $\|\bm{x}\|$. Doing so provides the simplest
explanation of the experimental slopes. Interestingly, as shown in Fig. 3,
such an analysis completely excludes a contribution originating from a rise of
the temperature as the explanation for the amplitude of the demagnetization
and, instead, attributes it mainly to the quenching of the inter-atomic
exchange in Ni and Co. On the other hand, in Fe, and in agreement with the
work of Carpene et al.Carpene  _et al._ (2008), the electron–magnon scattering
explains more than half of the demagnetization, while the reduction of the
inter-atomic exchange explains the remaining contribution. Supposing that the
“quick recovery” regime is, in fact, due to a recovery of the inter-atomic
exchange, this may explain why Fe does not feature this phenomenon as strongly
as Ni and Co do. Here we also notice that the model correctly attributes a
positive sign to $x_{G_{ep}}$, which is in accordance with the aforementioned
physical insight that the electron-phonon coupling reduces the amplitude of
the maximum demagnetization.

The case when $x_{j}$ is imposed to be zero is also considered here, i.e.
placing ourselves in the usual framework in which the reduction of exchange is
neglected. In its absence, the model provides inconsistent results: the
contribution of the electron-magnon scattering is positive, i.e. tends to
limit the amplitude of the demagnetization. This emphasizes the necessity of
accounting for the involvement of the reduction of the inter-atomic exchange
in the demagnetization. Lastly, to provide a range in which this model
correctly predicts that $x_{G_{em}}>0$, we now impose $x_{G_{em}}=0$. In such
a case the temperature has to play a significant role in the explanation of
the experimental slopes, and now amounts for more than a third of the
amplitude in Ni and Co and almost entirely takes over in Fe. We expect the
physical reality to be located in between the graph on the top, minimizing
$\|\bm{x}\|$, and this case, i.e. $x_{G_{em}}=0$. That would mean that both
the electron-magnon scattering and a rise of the temperature play a role in
regulating the amplitude of the demagnetization. However, this model really
emphasize that in any case, the reduction of the inter-atomic exchange plays a
major role in the explanation of the experimental demagnetization amplitude
and cannot be overlooked.

To summarize, the reduction of the inter atomic exchange provides a novel
mechanism through which transversal excitations, or magnons can be generated
on the ultrafast timescales. Indeed, assuming that some notion of temperature
could be defined at such short timescales, rather than simply getting closer
to T${}_{\text{C}}$, the initial part of the demagnetization can, in a
significant part, be due to a transient reduction of T${}_{\text{C}}$ itself,
as shown by the linear relation between T${}_{\text{C}}$ and the first-
neighbors inter-atomic exchange in classical atomistic spin modelsGaranin
(1996); Antropov  _et al._ (1996). As this phenomenon is particularly strong
in Ni, this could notably explain the critical behavior occurring within 20 fs
of the ultrafast demagnetization evidenced by Tengdin et al.Tengdin  _et al._
(2018). This work is therefore in line with recent experimental work
attributing the main part of the demagnetization as originating from
transversal excitationsTurgut  _et al._ (2016); Eich  _et al._ (2017); Scheid
_et al._ (2022).
Furthermore, as evidenced by the fact that the reduction of the inter-atomic
exchange occurs by raising the electronic temperature in ab initio
calculations, this mechanism does not rely on a direct light–matter
interaction. This is in agreement with the experimental work of Bergeard et
al.Bergeard  _et al._ (2016), showing that the presence of hot electrons is
sufficient to induce an ultrafast demagnetization.
In conclusion, this work opens up new perspectives by considering the
involvement of the light-induced inter-atomic exchange reduction in the
demagnetization dynamics. While such a phenomenon has been widely overlooked,
we find it to be directly correlated to the amplitude of the initial
demagnetization. This provides an explanation of the paradoxically high
demagnetization amplitude found in Ni, as compared to Co and Fe, which could
not be rationalized otherwise.

This work is supported by the ANR-20-CE09-0013 UFO. Philippe Scheid is
grateful for the many fruitful discussions with Q. Remy, A. Szilva, O. Grånäs,
I. P. Miranda, and M. Pankratova.

## References

  * Beaurepaire  _et al._ (1996) E. Beaurepaire, J. C. Merle, A. Daunois, and J. Y. Bigot, Physical Review Letters 76, 4250 (1996).
  * Stanciu  _et al._ (2007) C. D. Stanciu, F. Hansteen, A. V. Kimel, A. Kirilyuk, A. Tsukamoto, A. Itoh, and T. Rasing, Physical Review Letters 99, 47601 (2007).
  * Radu  _et al._ (2011) I. Radu, K. Vahaplar, C. Stamm, T. Kachel, N. Pontius, H. A. Dürr, T. A. Ostler, J. Barker, R. F. Evans, R. W. Chantrell, A. Tsukamoto, A. Itoh, A. Kirilyuk, T. Rasing, and A. V. Kimel, Nature 472, 205 (2011).
  * Mangin  _et al._ (2014) S. Mangin, M. Gottwald, C. H. Lambert, D. Steil, V. Uhlíř, L. Pang, M. Hehn, S. Alebrand, M. Cinchetti, G. Malinowski, Y. Fainman, M. Aeschlimann, and E. E. Fullerton, Nature Materials 13, 286 (2014).
  * Lambert  _et al._ (2014) C. H. Lambert, S. Mangin, B. S. S. Varaprasad, Y. K. Takahashi, M. Hehn, M. Cinchetti, G. Malinowski, K. Hono, Y. Fainman, M. Aeschlimann, and E. E. Fullerton, Science 345, 1337 (2014).
  * Rhie  _et al._ (2003) H. S. Rhie, H. A. Dürr, and W. Eberhardt, Physical Review Letters 90, 4 (2003).
  * Cinchetti  _et al._ (2006) M. Cinchetti, M. Sánchez Albaneda, D. Hoffmann, T. Roth, J. P. Wüstenberg, M. Krauß, O. Andreyev, H. C. Schneider, M. Bauer, and M. Aeschlimann, Physical Review Letters 97, 177201 (2006), arXiv:0605272 [cond-mat] .
  * Tengdin  _et al._ (2018) P. Tengdin, W. You, C. Chen, X. Shi, D. Zusin, Y. Zhang, C. Gentry, A. Blonsky, M. Keller, P. M. Oppeneer, H. C. Kapteyn, Z. Tao, and M. M. Murnane, Science Advances 4, 1 (2018).
  * You  _et al._ (2018) W. You, P. Tengdin, C. Chen, X. Shi, D. Zusin, Y. Zhang, C. Gentry, A. Blonsky, M. Keller, P. M. Oppeneer, H. Kapteyn, Z. Tao, and M. Murnane, Physical Review Letters 121 (2018), 10.1103/PhysRevLett.121.077204.
  * Scheid  _et al._ (2022) P. Scheid, Q. Remy, S. Lebègue, G. Malinowski, and S. Mangin, Journal of Magnetism and Magnetic Materials 560, 169596 (2022).
  * Turgut  _et al._ (2016) E. Turgut, D. Zusin, D. Legut, K. Carva, R. Knut, J. M. Shaw, C. Chen, Z. Tao, H. T. Nembach, T. J. Silva, S. Mathias, M. Aeschlimann, P. M. Oppeneer, H. C. Kapteyn, M. M. Murnane, and P. Grychtol, Physical Review B 94, 220408(R) (2016).
  * Eich  _et al._ (2017) S. Eich, M. Plötzing, M. Rollinger, S. Emmerich, R. Adam, C. Chen, H. C. Kapteyn, M. M. Murnane, L. Plucinski, D. Steil, B. Stadtmüller, M. Cinchetti, M. Aeschlimann, C. M. Schneider, and S. Mathias, Science Advances 3, 1 (2017).
  * Jana  _et al._ (2020) S. Jana, R. S. Malik, Y. O. Kvashnin, I. L. Locht, R. Knut, R. Stefanuik, I. Di Marco, A. N. Yaresko, M. Ahlberg, J. Åkerman, R. Chimata, M. Battiato, J. Söderström, O. Eriksson, and O. Karis, Physical Review Research 2, 1 (2020), arXiv:1908.02872 .
  * Carva  _et al._ (2013) K. Carva, M. Battiato, D. Legut, and P. M. Oppeneer, Physical Review B - Condensed Matter and Materials Physics 87, 184425 (2013), arXiv:1305.3511 .
  * Battiato  _et al._ (2010) M. Battiato, K. Carva, and P. M. Oppeneer, Physical Review Letters 105, 1 (2010), arXiv:1106.2117 .
  * Scheid  _et al._ (2019a) P. Scheid, G. Malinowski, S. Mangin, and S. Lebègue, Physical Review B 99, 174415 (2019a).
  * Krieger  _et al._ (2015) K. Krieger, J. K. Dewhurst, P. Elliott, S. Sharma, and E. K. Gross, Journal of Chemical Theory and Computation 11, 4870 (2015).
  * Scheid  _et al._ (2019b) P. Scheid, G. Malinowski, S. Mangin, and S. Lebègue, Physical Review B 100, 1 (2019b).
  * Scheid  _et al._ (2021) P. Scheid, S. Sharma, G. Malinowski, S. Mangin, and S. Lebègue, Nano Letters 21, 1943 (2021).
  * Koopmans  _et al._ (2010) B. Koopmans, G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fähnle, T. Roth, M. Cinchetti, and M. Aeschlimann, Nature Materials 9, 259 (2010).
  * Roth  _et al._ (2012) T. Roth, A. J. Schellekens, S. Alebrand, O. Schmitt, D. Steil, B. Koopmans, M. Cinchetti, and M. Aeschlimann, Physical Review X 2 (2012), 10.1103/PhysRevX.2.021006.
  * Zahn  _et al._ (2021) D. Zahn, F. Jakobs, Y. W. Windsor, H. Seiler, T. Vasileiadis, T. A. Butcher, Y. Qi, D. Engel, U. Atxitia, J. Vorberger, and R. Ernstorfer, Physical Review Research 3, 1 (2021), arXiv:2008.04611 .
  * Pankratova  _et al._ (2022) M. Pankratova, I. P. Miranda, D. Thonig, M. Pereiro, E. Sjöqvist, A. Delin, O. Eriksson, and A. Bergman, Physical Review B 106, 1 (2022).
  * Gunnarsson (1976) O. Gunnarsson, Journal of Physics F: Metal Physics 6, 587 (1976).
  * Rumble (2022) J. R. Rumble,  _CRC Handbook of Chemistry and Physics_, CRC Handbook of Chemistry and Physics (CRC Press, 2022).
  * Ritzmann  _et al._ (2020) U. Ritzmann, P. M. Oppeneer, and P. Maldonado, Physical Review B 102, 214305 (2020), arXiv:1911.12414 .
  * Haag  _et al._ (2014) M. Haag, C. Illg, and M. Fähnle, Physical Review B - Condensed Matter and Materials Physics 90, 14417 (2014).
  * Müller  _et al._ (2019) M. C. Müller, S. Blügel, and C. Friedrich, Physical Review B 100, 1 (2019), arXiv:1809.02395 .
  * Antropov  _et al._ (1995) V. P. Antropov, M. I. Katsnelson, M. Van Schilfgaarde, and B. N. Harmon, Physical Review Letters 75, 729 (1995).
  * Antropov  _et al._ (1996) V. Antropov, M. Katsnelson, B. Harmon, and M. van Schilfgaarde, Physical Review B - Condensed Matter and Materials Physics 54, 1019 (1996).
  * Halilov  _et al._ (1998) S. Halilov, H. Eschrig, and A. Perlov,  _Physical Review B - Condensed Matter and Materials Physics_, Tech. Rep. 1 (1998).
  * Kurz  _et al._ (2004) P. Kurz, F. Förster, L. Nordström, G. Bihlmayer, and S. Blügel, Physical Review B - Condensed Matter and Materials Physics 69, 24415 (2004).
  * Hohenberg and Kohn (1964) P. Hohenberg and W. Kohn, Physical Review 136, B864 (1964).
  * Kohn and Sham (1965) W. Kohn and L. J. Sham, Physical Review 140, A1133 (1965).
  * Perdew and Wang (1992) J. P. Perdew and Y. Wang, Physical Review B 45, 13244 (1992).
  * Kudrnovský  _et al._ (2001) J. Kudrnovský, I. Turek, M. Pajda, P. Bruno, and V. Drchal, Physical Review B - Condensed Matter and Materials Physics 64, 174402 (2001), arXiv:0007441 [cond-mat] .
  * Carpene  _et al._ (2008) E. Carpene, E. Mancini, C. Dallera, M. Brenna, E. Puppin, and S. De Silvestri, Physical Review B - Condensed Matter and Materials Physics 78 (2008), 10.1103/PhysRevB.78.174422.
  * Garanin (1996) D. A. Garanin, Phys. Rev. B 53, 11593 (1996).
  * Bergeard  _et al._ (2016) N. Bergeard, M. Hehn, S. Mangin, G. Lengaigne, F. Montaigne, M. L. Lalieu, B. Koopmans, and G. Malinowski, Physical Review Letters 117 (2016), 10.1103/PhysRevLett.117.147203.