# Ultrafast spin dynamics: role of laser-induced modification of exchange
parameters

Sergiy Mankovsky  Department of Chemistry/Phys. Chemistry, LMU Munich,
Butenandtstrasse 11, D-81377 Munich, Germany  Svitlana Polesya  Department of
Chemistry/Phys. Chemistry, LMU Munich, Butenandtstrasse 11, D-81377 Munich,
Germany  Hubert Ebert  Department of Chemistry/Phys. Chemistry, LMU Munich,
Butenandtstrasse 11, D-81377 Munich, Germany

###### Abstract

Induced by an ultra-short laser pulse, the electronic structure of a material
undergoes strong modifications leading to a fast demagnetization in magnetic
materials. Induced spin-flip transitions are one of the reasons for
demagnetization, that is discussed in the literature as a Stoner-like
mechanism. On the other hand, demagnetization due to transverse spin
fluctuations is usually discussed on the basis of the Heisenberg Hamiltonian
and hardly accounts for the modification of the electronic structure. In this
work we demonstrate a strong impact of the laser-induced electron transitions,
both spin-flip and spin-conserving, on the exchange coupling parameters. For
this, a simple two-step scheme is suggested. As a first step, the electronic
structure time evolution during the ultra-short laser pulse is described
accurately within time-dependent density-functional theory (TD-DFT)
calculations. As a next step, the information on the time-dependent electronic
structure is used for calculations of the parameters of the Heisenberg
Hamiltonian. A strong modification of the exchange coupling parameters is
found in response to the applied ultra-short laser pulse. The most important
reason for this modification is played by the laser induced repopulation of
the electronic states. Although the changes of the exchange parameters are
most prominent during the laser pulse, they may be important also for the
magnetic relaxation. The same concerns the spin-lattice interactions playing a
central role for the relaxation process. A strong impact of the laser-induced
modification of the electronic structure on the spin-lattice coupling
parameters is also shown in this work.

##  I INTRODUCTION

Ultrafast spin manipulation is one of the most exciting research topics in the
latest decades. Since the first report by Beaurepaire et al.Beaurepaire  _et
al._ (1996), demonstrating that magnetic order can be manipulated on a sub-
picosecond timescale using ultrashort laser pulses, many efforts have been
done, both experimental and theoretical, to shed light on the physical
processes behind ultrafast magnetic dynamics on the one hand side and to
adjust these processes to control, e.g. the magnetization dynamics in magnetic
materials Kirilyuk  _et al._ (2010); Bigot and Vomir (2013); Carva  _et al._
(2017). Despite the huge progress made by these activities, the various
microscopic mechanisms responsible for the demagnetization at different time
scales is still under debate.Beaurepaire  _et al._ (1996); Bigot  _et al._
(2004); Fähnle  _et al._ (2015); Koopmans  _et al._ (2010); Hübner and
Bennemann (1996); Fähnle  _et al._ (2015); Dornes  _et al._ (2019) Definitely,
one can see a consensus in the literature concerning the role of the spin-
orbit coupling (SOC) responsible for the laser-induced demagnetization in
metals leading to an angular momentum transfer from the spin subsystem to
other degrees of freedom Zhang and George (2008); Fähnle  _et al._ (2015) via
spin-flip transitions during the laser field induced excitation Zhang and
Hübner (2000); Zhang and George (2008) or Elliott-Yafet-like processes with
spin-flip electron scattering on phonons Koopmans  _et al._ (2010); Carva  _et
al._ (2011); Illg  _et al._ (2013); Carva  _et al._ (2015). The Stoner-type
demagnetization mechanism is described reasonably well within time dependent
density functional theory (TD-DFT) Krieger  _et al._ (2014); Sharma  _et al._
(2022). It should be stressed that such calculations are usually performed for
a non-distorted crystal lattice and a collinear magnetic state. On the other
hand, the impact of the electron-phonon scattering on the time evolution of
the magnetization was reported recently Sharma  _et al._ (2022) by performing
TD-DFT calculations in the presence of preexcited phonon modes in the system.
Apart from TD-DFT, other quantum-mechanical investigations have been done
estimating the rate of angular momentum transfer from the electron to the
lattice and magnetic degrees of freedom via SOC-driven spin-flip electron
scattering Koopmans  _et al._ (2010); Schellekens and Koopmans (2013); Illg
_et al._ (2013); Fähnle  _et al._ (2015, 2017); Dornes  _et al._ (2019).

The experimental observation of a strong laser-induced magnetization decay
observed by Eich et al. Eich  _et al._ (2017) in Co led the authors to the
conclusion, that the ferromagnetic-paramagnetic phase transition is a result
of an extremely efficient ultrafast generation of collective transverse spin
excitations, i.e. magnons, rather than a loss of the exchange splitting.
Tengdin et al. Tengdin  _et al._ (2018), also report about the demagnetization
in Ni which occurs much earlier than the collapse of the exchange splitting.
The authors, however, suggest that the ultrafast demagnetization is driven by
a highly nonequilibrium process, e.g. superdiffusive spin currents.

Attributing the ultrafast demagnetization to magnetic disorder in a
ferromagnet, a natural question is what is the efficient mechanism responsible
for ultrafast generation of magnons, which will depend on the material. To
find the answer to this question, an efficient way is to combine classical
models used for spin dynamics with first-principles DFT calculations. This may
be for instance the atomistic Landau-Lifshitz-Gilbert (LLG) model based on a
classical spin Hamiltonian Kazantseva  _et al._ (2007) giving access to the
magnetic torque accounting for contributions from isotropic exchange,
Dzyaloshinskii-Moriya interactions (DMI), and magnetic anisotropy calculated
from first principles Deák  _et al._ (2017). The ab-initio parametrized
quantum kinetic approach was reported recently to describe magnon occupation
dynamics due to electron-magnon scattering in ferromagnetic metals excited by
ultrafast laser radiation Weißenhofer and Oppeneer (2023). The results
demonstrate a strong demagnetization of Fe within 200 fs due to an ultrafast
generation of high-energy magnons. The optically induced magnetic torque may
be a consequence of a coupling between the electric field of the laser light
and the electron spins Němec  _et al._ (2012); Subkhangulov  _et al._ (2014);
Mikhaylovskiy  _et al._ (2015); Freimuth  _et al._ (2016); Mentink (2017);
Hamamera  _et al._ (2022), and may originate, e.g., due to the optical spin
transfer torque Núñez  _et al._ (2004); Němec  _et al._ (2012) or due to the
inverse Faraday effect (IFE) Subkhangulov  _et al._ (2014); Mikhaylovskiy  _et
al._ (2015); Freimuth  _et al._ (2016); Mentink (2017); Hamamera  _et al._
(2022). Furthermore, it may be seen as a consequence of the optically-induced
changes of the interatomic exchange interactions Subkhangulov  _et al._
(2014); Mikhaylovskiy  _et al._ (2015). The laser induced demagnetization and
relaxation may also be controlled by spin-lattice interactions, which are
responsible for the angular momentum transfer between the magnetic subsystem
and the crystal lattice leading to ultrafast magnon or phonon generation in
the demagnetization and relaxation processes, respectively Hübner and
Bennemann (1996); Rückriegel  _et al._ (2020); Frietsch  _et al._ (2020).

One should note, however, that the efficiency of the magnon generation should
depend not only on the mechanism of magnon excitation, but also on the
properties of the magnon spectrum determined by the parameters of the spin
Hamiltonian. Ultrafast heating of the electronic subsystem should almost
immediately (i.e. on the time scale of $\sim 10^{-15}$ s) result in a
modification of all parameters entering the equations used to describe spin-
dynamics. Accordingly, all parameters should change with time following the
absorbed energy and corresponding time evolution of the electronic subsystem.
This concerns, in particular, the anisotropy field generated due to the
absorbed laser pulse van Kampen  _et al._ (2002). In general, the theoretical
description of these changes for the non-equilibrium electron density require
to go beyond the standard approach used for the equilibrium, and should rely,
for instance, on the non-equilibrium Green’s-function formalism Secchi  _et
al._ (2013); Freimuth  _et al._ (2016), or exact diagonalization approach
Hübner and Zhang (1998). Alternatively, we suggest in the present work a
simple scheme to demonstrate the impact of the optically excited out-of-
equilibrium electron subsystem on the demagnetization or relaxation processes.
Information about the laser-pulse excited electron system is obtained via TD-
DFT calculations, which are used here to determine the time evolution of the
exchange coupling and spin-lattice coupling parameters. As a result, the spin
dynamic characteristics should follow the changes of all the parameters that
determine the torque on the magnetic moments.

Note that on a longer time scale, when the magnetization is also decreased due
to increasing magnetic disorder, its impact on the exchange coupling
parameters may be significant and should also be taken into account Deák  _et
al._ (2017).

##  II Results and discussion

###  II.1 Magnetic torque due to the optical heating of the electronic
subsystem

We focus first on the impact of the transverse spin fluctuations on the laser-
induced demagnetization in ferromagnets. When the system is in equilibrium,
these fluctuations are reasonably well described using the extended Heisenberg
spin model Hamiltonian ${\cal H}_{\rm Heis}$ with the parameters calculated
from first principles, i.e. the parameters representing symmetric exchange
interactions, Dzyaloshinskii-Moriya interactions, magnetic anisotropy, etc.
The spin evolution out of equilibrium described by the atomistic Landau-
Lifshitz-Gilbert equation Bhattacharjee  _et al._ (2012), is governed by the
torque on the magnetic moments on the sites $i$,
$\vec{T}_{i}=\hat{s}_{i}\times\vec{H}_{{\rm{eff}},i}$. The torque is
determined by the external perturbation field and effective magnetic field,
$\vec{H}_{{\rm{eff}},i}=-\frac{\partial{\cal H}}{\partial\hat{s}_{i}}$, given
in particular, in terms of the spin Hamiltonian ${\cal H}={\cal H}_{\rm
Heis}$.

For a system affected by an ultrafast laser pulse, one can consider the
magnetic torque $\Delta\vec{T}_{\rm{Heis},i}$ created due to changes of the
parameters of the model Hamiltonian, i.e. $\Delta J_{ij}(t)$ and
$\Delta\vec{D}_{ij}(t)$, $\Delta\underline{K}_{i}(t)$ as a result of the
strong laser-induced heating of the electron subsystem. Note that in general,
the torque on the magnetic moment should account for the impact of the spin
evolution during the whole time period starting with the laser pulse (see,
e.g. Ref. [Bhattacharjee  _et al._ , 2012]). However, using a simplified time-
dependence for the magnetic torque, one can derive it from the current state
of the electronic structure evolving in time. Despite the resulting
uncertainty of the numerical results, this allows a qualitative, semi-
quantitative, analysis of the impact of the laser-excited electronic subsystem
on the transverse spin dynamics.

Considering bcc Fe with a non-distorted crystal lattice ($T_{lat}=0$ K) as an
example, we will focus on the laser induced changes of the isotropic exchange
interactions represented by the parameter $\Delta J_{ij}(t)$ and the
corresponding magnetic torque
$\Delta\vec{T}_{i}=\sum_{j}(\hat{s}_{i}\times\hat{s}_{j})\Delta J_{ij}$, while
the DMI parameters should vanish because of the inversion symmetry.
Furthermore, the parameters of the magnetocrystalline anisotropy (MCA) are
very weak in the bulk.

The torque determined this way, however, does not give access to the angular
momentum transfer to the spin subsystem resulting in the generation of new
magnons responsible for the transverse demagnetization. As discussed above,
magnon generation can be driven by the optically induced magnetic torques that
stem from the direct laser light interaction with the magnetic moments
Mikhaylovskiy  _et al._ (2015), or from the phonon-magnon scattering. The
latter mechanism is determined by spin-lattice coupling (SLC) DMI-like,
$\vec{\cal D}^{\mu}_{ij,k}$, and MCA-like, $\underline{K}_{i,k}^{\mu}$
parameters Mankovsky  _et al._ (2022, 2023) entering the spin-lattice
Hamiltonian ${\cal H}_{\rm{sl}}$ (see Appendix B), and require a finite
temperature of the lattice subsystem leading to a broken local symmetry at the
atomic sites due to a finite amplitude of the lattice fluctuations. As a
consequence, magnetic torques created by the effective fields
${{H}^{\alpha}_{\rm{sl},i}}=-\frac{\partial{{\cal H}}_{\rm{sl}}}{\partial
s_{i}^{\alpha}}$ Mankovsky  _et al._ (2022, 2023); Rückriegel  _et al._ (2020)
may change in the case of a strong laser-induced excitation of the electronic
subsystem, leading to a corresponding induced torque
$\Delta\vec{T}_{\rm{sl},i}(t)$ determined by the change of the SLC parameters
following the time evolution of the excited electronic system.

In order to demonstrate the impact of such a modification of the exchange
coupling parameters, we consider the system at a certain time step instead of
performing an integration of the atomistic equations of motion. The electronic
structure for any time step found within a TD-DFT calculations (here using the
Elk code ELK ) can be used later on to calculate the exchange coupling
parameters for this time step. We focus in this work on two types of changes
in the electronic subsystem, the spin-dependent part of the electronic
potential (in particular, the exchange field $B_{\rm
xc}=\frac{1}{2}(V^{\uparrow}-V^{\downarrow})$) and the re-population of the
electronic states, which in our opinion has the main impact on the exchange
parameters. The goal is to demonstrate a significant role of the laser induced
modifications of these parameters for spin dynamics, in particular for laser-
induced demagnetization dynamics.

To account for the re-population when calculating the exchange parameters, one
has to adopt the expression for the exchange coupling parameters worked out
previously Liechtenstein  _et al._ (1987); Udvardi  _et al._ (2003); Ebert and
Mankovsky (2009) for the case of a reasonably low temperatures ($\lesssim
T_{C}$), when the Fermi-Dirac occupation function $f(E)$ can be well
approximated by the step function. Taking into account the temperature or
laser-pulse induced modifications of $\underline{f}(E)$ which become
significant in the high temperature (or strong pulse fluence) regime, one
obtains an expression for the elements of the $\underline{J}_{ij}$ tensor as
follows (see Appendix A)

$\displaystyle J_{ij}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bigg{[}\frac{1}{\pi}\,\mbox{Im}\mbox{Tr}\int^{\infty}dE\,\frac{d\,\underline{f}(E)}{dE}\,(E-\mu(T))\,\langle\Delta
V\rangle_{i}^{\alpha}\,\underline{\tau}_{ij}(E)\,\langle\Delta
V\rangle_{j}^{\beta}\,\underline{\tau}_{ji}(E)$ (1)
$\displaystyle+\frac{1}{\pi}\,\mbox{Im}\mbox{Tr}\int^{\infty}dE\,\underline{f}(E)\,\,\langle\Delta
V\rangle_{i}^{\alpha}\,\underline{\tau}_{ij}(E)\,\langle\Delta
V\rangle_{j}^{\beta}\,\underline{\tau}_{ji}(E)\bigg{]}\,,$

with $\underline{f}(E)$ the spin-dependent occupation function. A similar
expression can be derived also for the spin-lattice parameters. However, below
we focus on the properties of the isotropic exchange coupling parameters,
$J_{ij}=\frac{1}{2}(J_{ij}^{xx}+J_{ij}^{yy})$, and the DMI-like SLC
parameters.

###  II.2 Exchange coupling parameters for out of equilibrium situation

As a first step we consider the system being at a finite temperature $T$. This
allows us to simulate the effect of the optically induced re-population of the
electronic states, which can be seen as a heating of the electronic structure
to a temperature in the order of $10^{4}$ K (see below). The resulting impact
for the heated electronic subsystem with temperature $T_{\rm el}$ is described
by the corresponding Fermi-Dirac distribution function. In contrast to this,
we do not account for any changes of the electron potential, as well as
thermal atomic displacements and a deviation of the spin moments from perfect
FM ordering, i.e., we neglect at the moment their impact on the exchange
parameters. This simplification allows to demonstrate the impact of the heated
electronic subsystem on the magnetic properties of the system in the thermal
equilibrium represented by the Heisenberg model.

The parameters $J_{ij}$ calculated for bcc Fe for five different temperatures
are represented in Fig. 1. We focus here only on these parameters, first of
all because they are responsible for the FM order and characterize the Curie
temperature in the system, while the DMI parameters are equal to zero for
symmetry reason.

The top panel in Fig. 1 shows the temperature dependent Fermi function
$f(E,T_{\rm el})$ for the temperatures (a) 0 K, (b) 3000 K, (c) 6000 K, (d)
9000 K, and (e) 12000 K. The corresponding exchange coupling parameters
$J_{ij}$ for bcc Fe are shown in the bottom panel. One can clearly see their
significant decrease with increasing electronic temperature. This in turn
should lead to a decrease of the Curie temperature, as can be demonstrated by
Monte Carlo simulations for a finite temperature $T_{\rm Heis}$ of the spin
subsystem. It is worth noting that this implies thermalization of the spin
subsystem, that occurs at the timescale of $\sim 10^{-12}$ s, i.e. much longer
than the duration of a typical laser pulse considered here Kirilyuk  _et al._
(2010); Carva  _et al._ (2017). This point has to be discussed in addition,
when dealing with a realistic situation.

(a)  (b)  (c)  (d)  (e)

Figure 1:  The Fermi-Dirac distribution corresponding to temperatures (a) 0 K,
(b) 3000 K, (c) 6000 K, (d) 9000 K, (e) 12000 K, together with corresponding
results for the temperature dependent exchange coupling parameters $J_{ij}(T)$
for bcc Fe.

To get more insight into the behavior of the $J_{ij}$ parameters, Fig. 2(a)
shows the energy dependence of the integrand in the second term of Eq. (1)
which determines the interatomic nearest-neighbor (nn) and next-to-nearest-
neighbor (nnn) exchange coupling parameters $J_{01}$ and $J_{02}$,
respectively, for bcc Fe in the ground state, i.e. $T=0$ K. Note that the
first term in Eq. (1) vanishes in this case because of the distribution
function $f(E,T_{\rm el}=0)=\theta(E-E_{F})$. For both curves in Fig. 2(a),
one can see a pronounced variation in the energy region close to the Fermi
energy. The corresponding integrals in Eq. (1), using the energy $E_{\rm occ}$
as an upper limit of integration, represent the parameters $J_{01}(E_{\rm
occ})$ and $J_{02}(E_{\rm occ})$ as a function of occupation of the electronic
states. As can be seen in Fig. 2(b), these parameters exhibit a strong
dependence on the occupation of the electronic states (see, e.g., discussions
in Ref. [Mankovsky  _et al._ , 2020]). A crucial consequence of the strong
energy dependencies shown in Fig. 2(a) occurs for the out-of-equilibrium
electron subsystem with the laser-induced deoccupied states below $E_{F}$, and
occupied states above $E_{F}$. In this case the main changes of the exchange
parameters occur as a result of a convolution of the integrand plotted in Fig.
2(a) with the occupation function $f(E)$ shown in Fig. 4. This can be seen in
Fig. 2(b), dashed line, representing the $J_{01}(E_{\rm occ},T)$ and
$J_{02}(E_{\rm occ},T)$ parameters obtained using the occupation function
$f(E,T_{\rm el}=12000K)$ (see Fig. 1).

(a)  (b)

Figure 2: (a) The integrand in the second term of Eq. (1) for the interatomic
nn and nnn exchange coupling parameters of bcc Fe in the ground state ($T_{\rm
el}=0$ K) as a function of energy; (b) interatomic nn and nnn exchange
coupling parameters, $J_{01}$ and $J_{02}$, respectively, represented as a
function of occupation of the electronic states characterized by the threshold
energy $E_{\rm occ}$ below or above the Fermi energy $E_{\rm F}$. Solid lines
represent results obtained for the electronic temperature $T_{\rm el}=0$ K,
while dashed lines correspond to $T_{\rm el}=12000$ K.

###  II.3 Exchange coupling parameters under laser pulse and ultrafast
demagnetization

As a next step, we focus on the magnetic properties of a material determined
by laser induced changes of the exchange interactions, considering also bcc Fe
as an example and making use of Eq. (1). In this case we use as an input the
exchange field $B_{\rm xc}=\frac{1}{2}(V^{\uparrow}-V^{\downarrow})$ and spin-
resolved occupation function $f_{\sigma}(E)$, delivered by TD-DFT calculations
performed with the Elk electronic structure code ELK , for a laser pulse
polarized along the $x$ axis, a fluence of 24 mJ/cm2, a frequency of 413 THz
and full width at half maximum (FWHM) of 4.8 fs.

(a)  (b)

Figure 3: (a) Laser-pulse induced magnetization for bcc Fe. TD-DFT
calculations using a laser pulse polarized along the $x$ axis, fluence 24
mJ/cm2, FWHM of 4.8 fs. (b) Radial dependence of the effective exchange field
$B_{\rm xc}(r)=\frac{1}{2}(V^{\uparrow}-V^{\downarrow})$ for Fe within the
muffin-tin sphere, plotted for different time steps during the laser pulse.

The TD-DFT calculations have been performed for the time interval up to 22 fs.
As one can see in Fig. 3 (a), the magnetization decreases during the laser
pulse by $\sim 4\%$ due to the SOC-induced spin-flip transitions. It continues
to decrease also after the laser pulse down to $\sim 8\%$, due to the
relaxation within the electronic subsystem accompanied by SOC-induced spin-
flip transitions between the ’up’ and ’down’ spin channels characterized by
different out-of-equilibrium occupation functions $f_{\sigma}(E,t)$.

(a) (b) (c) (d) (e)

Figure 4: (Top) spin-dependent electronic occupation function
$f_{\sigma}(E,t)$ for five time steps during the laser pulse, calculated using
the Elk TD-DFT code, with the laser pules parameters: FWHM = 4.8 fs, pulse
fluence of 24 mJ/cm2, light frequency 413 THz. (Middle) The exchange coupling
parameters $J_{ij}$ (open symbols) calculated for several time steps with the
SPR-KKR band structure code using corresponding occupation functions
$f_{\sigma}(E,t)$ (top panel) and effective exchange field $B_{\rm xc}$
obtained within TD-DFT calculations. Full symbols represent the results for
the ground state parameters. (Bottom) Spin-wave dispersion relations
calculated for different time steps (red symbols) using the exchange
parameters shown in the middle panel. Black symbols represent the results for
the results for the ground state.

The decrease of the Fe magnetic moment is a consequence of the modified spin
density due to the re-population of the electronic states, induced by the
laser pulse. This in particular results in a change of the effective exchange
field $B_{\rm xc}(r)$ shown in Fig. 3 (b), plotted within the muffin-tin
sphere for different time steps. One can clearly see a decrease of the
magnitude of the B-field during the laser pulse, while the changes are almost
invisible when the pulse is finished.

The laser-induced spin-dependent occupation function $f_{\sigma}(E,t)$ plotted
in Fig. 4, top panel, is represented for five time steps during the laser
pulse. One can see a considerable depopulation of the electronic states below
$E_{F}$ due to the optical excitation to high-energy states. Moreover, in
contrast to finite-temperature considerations discussed above, the occupation
functions are different for the majority- and minority-spin states, and the
difference increases during the laser pulse.

The quantities $B_{\rm xc}(r)$ and $f_{\sigma}(E,t)$ have been used in a
quasi-stroboscopic way for calculations of the Green function varying in time
under the influence of the laser pulse, to be used later on for calculation of
time-dependent exchange coupling parameters. The calculations have been
performed by means of the SPR-KKR band structure method using $B_{\rm xc}(r)$
at every time step to calculate the corresponding scattering path operator and
the perturbation potential $\langle\Delta V\rangle_{i}^{\alpha}$. The exchange
coupling parameters, $J_{ij}$, modified under the impact of the laser pulse
are given in the middle panel of Fig. 4. As one would expect, the $J_{ij}$
parameters decrease with time during the laser pulse. The most pronounced
changes concern the nearest-neighbor and next-nearest-neighbor parameters.
After the pulse, the exchange parameters remain practically unchanged (see
Fig. 4(e), middle panel).

To illustrate, which impact on the magnetic properties can be expected due to
such a modification of the exchange interactions, one can consider the system
at every time step as one being electronically in a frozen state. As can be
seen in Fig. 4 (bottom), the modification of the exchange interactions shown
in Fig. 4 (middle) leads to a softening of the magnon spectrum, reflecting the
fact that the magnetization gets less robust. In line with this, we find a
distinct decrease of the effective Curie temperature during the absorption of
the laser pulse (see Fig. 5(a)), which was obtained from Monte Carlo
simulations using the fixed exchange coupling parameters. Furthermore,
considering the initial temperature for the spin subsystem, $T_{\rm Heis}$, to
be finite and fixed during the laser pulse, which seems to be a reasonable
approximation for the ultrashort time scale, one can check the time dependent
behavior of the net magnetization using different exchange parameters
calculated for different time steps. Corresponding results for the
magnetization evolution obtained making use of the parameters shown in Fig. 4
are plotted in Fig. 5(b) for three different temperatures $T_{\rm Heis}$,
demonstrating a strong magnetization decrease during the laser pulse which can
lead even to the break of the magnetic order if $T_{\rm Heis}$ is high enough.
Obviously, these conclusions on the magnetic properties have essentially semi-
qualitative character because of the approximations discussed above.
Nevertheless, we would like to stress that they are based on the electronic
structure with its time evolution coherently treated within the TD-DFT
formalism.

(a)  (b)

Figure 5: (a) Curie temperature for bcc Fe as a function of time calculated
using the exchange coupling parameters changing in time due to the laser pulse
induced evolution of the electronic structure. (b) Time-dependent evolution of
the magnetization for different temperature of the spin subsystem taken
unchanged at the time interval shown in the plot. Even neglecting the energy
transfer from the electronic subsystem in the ultrafast regime, the
temperature can be seen as an initial temperature of the sample being in
thermodynamic equilibrium.

In a more realistic scenario, $T_{\rm Heis}$ is non-zero and can change also
during and after the laser pulse, that happens however on a longer time scale.
This implies a magnetic disorder in the system varying in time until the full
relaxation of the excited system. As the exchange parameters depend also on
the magnetic configuration, this situation can also be taken into account
considering the laser induced demagnetization and relaxation processes. Such
investigations have been performed recently combining first principles
calculations and spin dynamics simulationsDeák  _et al._ (2017), where the
finite temperature effect after the laser pulse has been taken into account
for the electron and spin subsystems. However, in that work, both effects were
small because of a weak laser power used in the calculations (for comparison,
the maximal effective temperature of the electron gas considered in this work
was about 1000 K, far below the effective temperature $\sim 10^{4}$ K
estimated for the laser pulse used in the present case.) Furthermore, the
thermalization of the spin and lattice subsystems at later times results in an
increase of the magnetic disorder as well as increase of the amplitude of
atomic displacements. One can compare the impact of these factors on the
exchange coupling with the impact of the laser-induced strong excitation of
the electron subsystem. For this we consider the limiting case of $T=T_{C}$
with thermally induced magnetic disorder treated here making use of the
relativistic disordered local moment (RDLM) theory Staunton  _et al._ (2006);
Ebert  _et al._ (2015). Corresponding exchange coupling parameters $J_{ij}$
are plotted by squares in Fig. 6. As one can see, the nnn parameters decrease
significantly when compared to the FM system ($T=0$K), however at the same
time one can see an increase of the nn parameters. Obviously, this effect is
very different when compared to the effect caused by the laser-excited out-of-
equilibrium electron subsystem.

Figure 6: Exchange coupling parameters $J_{ij}$ for bcc Fe calculated for the
FM reference state (full circles), for the magnetically disordered states
treated within the RDLM approch (opened squres) and accounting both magnetic
disorder and atomic diaplacements at $T=T_{C}$. The amplitude of the lattice
vibrations at given temperature is estimated using the Debye model Ebert  _et
al._ (2015); Mankovsky  _et al._ (2020).

The lattice vibrations at this temperature Mankovsky  _et al._ (2020) also
lead to some changes of $J_{ij}$, which however mainly concern the nn
parameters.

###  II.4 Spin-lattice coupling parameters

Finally we will shortly discuss another type of interaction, namely the spin-
lattice interactions responsible for the energy and angular momentum transfer
between the magnon and phonon subsystems. These interactions are discussed, in
particular, for a long time scale when considering the mechanisms responsible
for the magnetic relaxation Mankovsky  _et al._ (2022).

The impact of the lattice vibrations on the exchange coupling parameters is
demonstrated in Fig. 6. Discussing this in terms of the spin-lattice
interaction, it was shown Mankovsky  _et al._ (2023) that it can be associated
with the SLC parameters of second order with respect to displacements,
$J^{\alpha\alpha,\mu\nu}_{ij,kl}$, leading to corrections of the exchange
coupling $\Delta J_{ij}\sim\langle u^{2}\rangle_{T}$. Note that the amplitude
of the atomic displacement is directly connected to the lattice temperature
$T_{\rm lat}$, and can be reasonably well estimated for a given temperature
$T_{\rm lat}$ within the Debye model approach. A more pronounced effect is
expected due to the SLC parameters linear w.r.t. the atomic displacements.
These parameters characterize the rate of spin angular momentum transfer due
to the torque on the magnetic moments, $\vec{\cal
T}_{i}=\hat{e}_{i}\times\vec{H}_{\rm eff}$, caused by the effective field
induced by the displacements $u_{k}^{\mu}$ of the atoms on sites $k$, or
alternatively, due to the mechanical torque on the atom $k$,
$\vec{\mathfrak{T}}^{ph}_{k}=\vec{u}_{k}\times\vec{\cal F}_{k}$, created by
the forces induced by spin tiltings $\hat{e}^{\alpha}_{i}$ on sites $i$
Mankovsky  _et al._ (2022, 2023); Rückriegel  _et al._ (2020).

Similar to the exchange coupling parameters, the SLC parameters are determined
by the electronic structure and one can expect their changes caused by the
laser induced strong modification of the electron subsystem. We will focus
here only on the so-called DMI-like SLC parameters $\vec{\cal D}^{\mu}_{ij,k}$
Mankovsky  _et al._ (2022, 2023), characterizing the change of the DMI
parameters $\Delta\vec{D}_{ij}=\sum_{k,\mu}\vec{\cal
D}^{\mu}_{ij,k}(u_{k}^{\mu}-u_{i}^{\mu})$ caused by atomic displacements
$\vec{u}_{k}$ (note that similar arguments hold also when considering the
properties of the MCA-like SLC parameters, ${\cal K}^{\alpha z,\mu}_{i,k}$
characterizing local changes of the MCA parameters $\Delta{K}^{\alpha
z}_{i}=\sum_{k,\mu}{\cal K}^{\alpha z,\mu}_{i,k}(u_{k}^{\mu}-u_{i}^{\mu})$).

As an example, Fig. 7 shows the parameters ${\cal D}^{z,\mu}_{ij,j}\,u_{D}(T)$
represented as a function of time for three different temperatures of the
crystal lattice $T_{\rm lat}$: 300 K, 600 K and 1040 K. The lattice
temperature is assumed to be constant during the full time range. One can see,
that the SLC parameters (and corresponding contributions to $D^{z}_{ij}$) are
of the order of $0.1$ meV in the equilibrium, and change in amplitude and sign
during the laser pulse reaching a value up to $\sim 0.4$ meV (depending on the
lattice temperature). Similar arguments as in the case of the exchange
coupling parameters may be used also to explain the time dependent change of
the ${\cal D}^{z,\mu}_{ij,j}$ parameters. This is a consequence the strong
energy dependent changes of the integrand (see Fig. 7(b)) in the expression
for the SLC parameter (see Refs. [Mankovsky  _et al._ , 2022, 2023]), leading
in particular to a strong dependence of the SLC on the occupation of the
electron states shown in Fig. 7(c).

(a)  (b)  (c)

Figure 7: (a) DMI-like spin-lattice coupling parameters as a function of time,
represented by a contribution to the DMI parameters due to atomic
displacement, ${\cal D}^{z,\mu}_{01,1}(u_{j}^{\mu}-u_{i}^{\mu})$ for
$\mu=x,y,z$ characterizing the directions of displacements of the atom in the
position $\vec{R}_{01}=(-0.5,-0.5,-0.5)a_{\rm lat}$. The results are presented
for three different temperatures of the crystal lattice, 300 K, 600 K and 1040
K ($T_{C}$), described by the displacement amplitude given by rms displacement
$u_{D}(T)$ estimated within the Debye model, i.e. $0.18$, $0.28$ and $0.39$
a.u., respectively. (b) The integrands for the components of the DMI-like
spin-lattice coupling parameters, ${\cal D}^{z,x}_{01,1}$ and ${\cal
D}^{z,y}_{01,1}$, as a function of energy. (c) The components of the DMI-like
spin-lattice coupling parameters, ${\cal D}^{z,x}_{01,1}$ and ${\cal
D}^{z,y}_{01,1}$, represented as a function of occupation of the electronic
states characterized by the threshold energy $E_{\rm occ}$ below or above the
Fermi energy $E_{\rm F}$.

##  III Summary

In summary, we find a distinct impact of the optically overheated electron
subsystem on the inter-atomic exchange coupling parameters. While a simplified
scheme is used to demonstrate this effect, the calculations of the exchange
parameters rely on the electronic structure calculated using the TD-DFT method
which ensure an accurate description of the electronic structure modifications
under the influence of an ultrafast laser pulse. We have demonstrated a
crucial role of the laser-induced deoccupation of the electronic states
varying in time and dependent on the power of the laser pulse, leading to a
strong weakening of the exchange parameters. Similarly, the laser-induced
changes of the spin-lattice parameters can influence the efficiency of magnon
generation and in turn the rate of demagnetization. At a longer time scale the
laser-induced changes of the exchange parameters as well as spin-lattice
parameters can influence the magnetization relaxation process and should
correct the results obtained so far using equilibrium parameters.

##  IV Appendix

##  Appendix A Exchange coupling parameters out of equilibrium: impact of non-
equilibrium occupation

Considering the FM state as a reference state, the deviation of some magnetic
moments from the collinear FM state is described by the perturbation potential

$\displaystyle\Delta V(\vec{r})$ $\displaystyle=$ $\displaystyle\sum_{i}\delta
v(\vec{r}-\vec{R}_{i})$ (2) $\displaystyle=$
$\displaystyle-\sum_{i}\beta(\vec{\sigma}\cdot\delta\hat{m}_{i})B_{i}^{xc}(r)\;.$

In the presence of a perturbation, the Green function is modified as follows

$\displaystyle G$ $\displaystyle=$ $\displaystyle G_{0}+G_{0}\Delta
VG_{0}+G_{0}\Delta VG_{0}\delta VG_{0}+...=G_{0}+\Delta G\;.$

Accounting only for single particle contribution to the free energy, omitting
the $\vec{r}$ arguments, the perturbation results in a free energy change

$\displaystyle\Delta F$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{\pi}\right)\mbox{Im}\mbox{Trace}\int^{\infty}_{E_{\rm
bottom}}dEf(E)(E-E_{F})\Delta G(E)\;.$ (3)

Keeping the second-order free energy term, one can use the sum rule for the
Green functions $\frac{dG(E)}{dE}=-G(E)G(E)$ to transform this expression and
taking into account the property of invariance of the trace under cyclic
permutations of matrices, the expression above can be transformed as follows

$\displaystyle{\cal F}^{(2)}$ $\displaystyle=$
$\displaystyle\left(-\frac{1}{\pi}\right)\mbox{Im}\mbox{Trace}\int^{\infty}_{E_{\rm
bottom}}dE\,f(E)\,(E-E_{F})\,\,G_{0}(E)\Delta V\,G_{0}(E)\,\Delta V\,G_{0}(E)$
(4) $\displaystyle=$
$\displaystyle\left(\frac{1}{\pi}\right)\mbox{Im}\mbox{Trace}\int^{\infty}_{E_{\rm
bottom}}dE\,f(E)\,(E-E_{F})\,\Delta V\,G_{0}(E)\,\Delta
V\,\frac{d}{dE}G_{0}(E)\,.$

By performing then the integration by parts, we arrive at

$\displaystyle{\cal F}^{(2)}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\bigg{[}\left(\frac{1}{\pi}\right)\mbox{Im}\mbox{Tr}\int^{\infty}_{E_{\rm
bottom}}dE\,\frac{d\,f(E)}{dE}\,(E-E_{F})\,\Delta V\,G_{0}(E)\,\Delta
VG_{0}(E)$ (5)
$\displaystyle+\left(\frac{1}{\pi}\right)\mbox{Im}\mbox{Tr}\int^{\infty}_{E_{\rm
bottom}}dE\,f(E)\,\,\Delta V\,G_{0}(E)\,\Delta V\,G_{0}(E)\bigg{]}\,.$

The first term should vanish for the ground state as
$\frac{d\,f(E)}{dE}=\delta(E-E_{F})$.

Using the real space multiple scattering representation of the Green function
$G_{0}(\vec{r},\vec{r}^{\prime},E)$

$\displaystyle G(\vec{r},\vec{r}\,^{\prime},E)$ $\displaystyle=$
$\displaystyle\sum_{\Lambda_{1}\Lambda_{2}}Z^{n}_{\Lambda_{1}}(\vec{r},E){\tau}^{nn^{\prime}}_{\Lambda_{1}\Lambda_{2}}(E)Z^{n^{\prime}\times}_{\Lambda_{2}}(\vec{r}\,^{\prime},E)-\sum_{\Lambda_{1}}\Big{[}Z^{n}_{\Lambda_{1}}(\vec{r},E)J^{n\times}_{\Lambda_{1}}(\vec{r}\,^{\prime},E)\Theta(r^{\prime}-r)$
(6)
$\displaystyle\;\;\;\;\;+\;J^{n}_{\Lambda_{1}}(\vec{r},E)Z^{n\times}_{\Lambda_{1}}(\vec{r}\,^{\prime},E)\Theta(r-r^{\prime})\Big{]}\delta_{nn^{\prime}}\;$

and the perturbation $\Delta V$ associated with a change of the spin-dependent
potential due to a rotation of the spin magnetic moment, given by the
expression

$\displaystyle\Delta V(r)$ $\displaystyle=$ $\displaystyle
V_{\hat{s}}(r)-V_{\hat{s}_{0}}(r)=\beta(\vec{\sigma}\cdot\delta\hat{s})B(r)\;,$
(7)

(where
$\delta\hat{s}=(\delta\hat{s}^{x},\delta\hat{s}^{y},\delta\hat{s}^{z})$), one
obtains the expression for the exchange coupling parameters as follows

$\displaystyle J_{ij}^{\alpha\beta}$ $\displaystyle=$
$\displaystyle\frac{\partial^{2}{\cal
F}^{(2)}}{\partial\hat{s}^{\alpha}_{i}\hat{s}^{\beta}_{j}}$ (8)
$\displaystyle=$
$\displaystyle-\frac{1}{2\pi}\bigg{[}\mbox{Im}\mbox{Tr}\int^{\infty}_{E_{bottom}}dE\,\frac{d\,f(E)}{dE}\,(E-E_{F})\,\langle\Delta
V\rangle_{i}^{\alpha}\,\tau_{ij}(E)\,\langle\Delta
V\rangle_{j}^{\beta}\,\tau_{ji}(E)$
$\displaystyle+\mbox{Im}\mbox{Tr}\int^{\infty}_{E_{bottom}}dE\,f(E)\,\,\langle\Delta
V\rangle_{i}^{\alpha}\,\tau_{ij}(E)\,\langle\Delta
V\rangle_{j}^{\beta}\,\tau_{ji}(E)\bigg{]}\,,$

where Green function in Eq. (6) in terms of the wave functions
$Z^{n}_{\Lambda}(\vec{r},E)$ and $J^{n}_{\Lambda}(\vec{r},E)$ and the
scattering path operator ${\tau}^{nn^{\prime}}_{\Lambda\Lambda^{\prime}}(E)$
Ebert  _et al._ (2016) was used.

Using naive arguments, the time dependent behavior of ${\cal E}^{(2)}$ and as
a result of the exchange coupling parameters may be calculated following the
scheme used for ground state, but with the Green function $G(E,t)$, calculated
for each time step using potentials obtained within the TD-DFT, and using
occupation function, $f(E,t)$, calculated for each time step within TD-DFT
calculations.

##  Appendix B Spin-lattice coupling parameters

Concerning the responsibility of the spin-lattice interactions for the
transverse spin excitations, we focus next on the impact of the laser pulse on
the behavior of the SLC. The torque on the magnetic moments due to the SLC,
$\vec{\cal T}_{i}^{SLC}=\hat{s}_{i}\times\vec{H}_{SLC}$, responsible for the
creation of transverse spin fluctuations, may be associated with different
types of contributions, e.g. by the MCA-like and DMI-like SLC Mankovsky  _et
al._ (2023). Corresponding effective fields originating from the MCA-like term
in the spin-lattice Hamiltonian are given in terms of the parameters ${\cal
K}^{\alpha z,\mu}_{i,k}$ as follows

$\displaystyle{{H}^{\alpha}_{\rm{sl-MA},i}}$ $\displaystyle=$
$\displaystyle\sum_{k}{{K}_{i,k}^{z\alpha,\mu}}(u^{\mu}_{k}-u^{\mu}_{i})\,,$
(9)

while the DMI-like interactions give rise to the effective field

$\displaystyle{{H}^{\alpha}_{\rm{sl-DMI},i}}$ $\displaystyle=$
$\displaystyle\sum_{j,k,\mu}[\hat{s}_{j}\times{\cal
D}^{\mu}_{ij,k}]_{\alpha}(u_{k}^{\mu}-u_{i}^{\mu})\,.$ (10)

On the other hand, the SLC-driven force on atom $k$ due to a spin tilting on
site $i$ in the case of MCA-like SLC is given by the expression

$\displaystyle{{\cal F}^{\mu}_{\rm{sl-MA},k}}$ $\displaystyle=$
$\displaystyle\sum_{\alpha}{{K}_{i,k}^{z\alpha,\mu}}\delta\hat{s}^{\alpha}_{i}$
(11)

and in the case of DMI-like SLC the force looks as follows

$\displaystyle{{\cal F}^{\mu}_{\rm{sl-DMI},k}}$ $\displaystyle=$
$\displaystyle\sum_{j,\alpha}[\hat{s}_{j}\times\vec{\cal
D}^{\mu}_{ij,k}]_{\alpha}\delta\hat{s}^{\alpha}_{i}.$ (12)

Note that the expressions for the SLC parameters ${\cal K}^{\alpha
z,\mu}_{i,k}$ and $\vec{\cal D}^{\mu}_{ij,k}$ derived for the ground state
Mankovsky  _et al._ (2022, 2023) should be modified taking into account re-
population of the laser-perturbed electronic states, that can be done in line
with the corresponding modifications given in the Appendix for isotropic
exchange coupling parameters.

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