# Laser-induced Demagnetization in van der Waals
$XY$\- and Ising-like Antiferromagnets NiPS3 and FePS3

D. V. Kuntu  0000-0003-0372-8331<EMAIL_ADDRESS>Ioffe Institute,
194021 St. Petersburg, Russia  E. A. Arkhipova  0009-0002-7939-9749 Ioffe
Institute, 194021 St. Petersburg, Russia  L. A. Shelukhin  0000-0001-8538-3773
Ioffe Institute, 194021 St. Petersburg, Russia  F. Mertens
0000-0002-7877-0107 Department of Physics, TU Dortmund University, Otto-Hahn-
Straße 4, Dortmund 44227, Germany  M. A. Prosnikov  0000-0002-7107-570X Ioffe
Institute, 194021 St. Petersburg, Russia  I. A. Eliseyev  0000-0001-9980-6191
Ioffe Institute, 194021 St. Petersburg, Russia  A. N. Smirnov
0000-0001-9709-5138 Ioffe Institute, 194021 St. Petersburg, Russia  V. Yu.
Davydov  0000-0002-5255-9530 Ioffe Institute, 194021 St. Petersburg, Russia
S. Mañas-Valero  0000-0001-6319-9238 Instituto de Ciencia Molecular (ICMol),
Universidad de Valencia, Paterna, 46980, Spain  E. Coronado
0000-0002-1848-8791 Instituto de Ciencia Molecular (ICMol), Universidad de
Valencia, Paterna, 46980, Spain  M. Cinchetti  0000-0003-0735-8921 Department
of Physics, TU Dortmund University, Otto-Hahn-Straße 4, Dortmund 44227,
Germany  A. M. Kalashnikova  0000-0001-5635-6186 Ioffe Institute, 194021 St.
Petersburg, Russia

###### Abstract

The critical behaviour of laser-induced changes in magnetic ordering is
studied experimentally in two-dimensional zigzag antiferromagnets XY-like
NiPS3 and Ising-like FePS3. To examine laser-induced dynamics in flakes of
these compounds, we employ time-resolved exchange linear dichroism effect
sensitive to zigzag magnetic ordering and independent of the orientation of
the antiferromagnetic vector. In both compounds laser excitation in the
vicinity of the absorption edge induces partial quenching of the
antiferromagnetic ordering manifested by exchange linear dichroism reduction.
The amplitude of the effect varies with temperature as the derivative of the
antiferromagnetic vector and exhibits a critical behaviour with the exponents
corresponding to XY\- and Ising-models for NiPS3 and FePS3, respectively.
Critical slowing down of the demagnetization in the vicinity of Néel
temperature is found, however, only in FePS3. In contrast, the increase of the
demagnetization time near the ordering temperature in NiPS3 is minor. We show
that the difference in the demagnetization times correlates well with the spin
specific heat in both compounds. Beyond the range of slowing down, the
demagnetization times in NiPS3 and FePS3 are comparable, about 5 – 10 ps, and
are longer than those reported earlier for CoPS3 and considerably shorter than
for MnPS3. This points to the importance of the unquenched angular momentum of
transition-metal ions in laser-induced demagnetization process.

††preprint: APS/123-QED

##  I Introduction

The phenomenon of ultrafast demagnetization, namely quenching of the magnetic
ordering in magnets following excitation with femtosecond laser pulses, has
become a challenging topic for experimental and theoretical studies. It
yielded discoveries of single- and multi-shot all-optical magnetic reversal
[1; 2; 3], generation of ultrashort spin currents [4] and emission of
broadband THz pulses [5]. In metals, demagnetization is governed by electron-
phonon coupling [6], leading to characteristic timescales of the process of
less than 100 femtoseconds [7; 8; 9]. In insulators, this timescale is defined
mainly by spin-lattice coupling, and the demagnetization process develops in a
subnanosecond range [10; 11]. The ultrafast demagnetization process is also
intrinsically sensitive to critical phenomena occurring in the vicinity of
magnetic ordering temperature [12; 13]. Therefore, the type and dimensionality
of the magnetic ordering can play an important role in this process, along
with the electronic band structure and coupling of spins to other energy and
angular momenta reservoirs.

Layered magnetic materials, such as van der Waals materials [14; 15], organic-
inorganic hybrids [16], MAX phases [17], etc., offer an appealing possibility
to investigate ultrafast magnetic dynamics in connection to details of
magnetic ordering. Indeed, while having similar crystallographic structure
within a family, these materials demonstrate a broad variety of spin
arrangements. Another common feature of such compounds is a weak interlayer
exchange coupling, making them quasi two-dimensional (2D) structures. As a
result, these materials are used for examining the Mermin-Wagner theorem and
ways to overcome restrictions on magnetic ordering in the 2D limit [18]. In a
few- and single-layer limit readily achievable for, e.g., van der Waals
materials [19; 20; 21; 22], their magnetic ordering can experience drastic
changes such as a transition between different spin alignments [23] or
quenching of magnetic ordering [24], although the latter remains a matter of
further discussion [25].

Laser-pulse driven spin dynamics in layered magnets has recently become a
subject of active studies. They cover revealing manifestations of a quasi-2D
nature of magnetic ordering [26], as well as examining coupling between
lattice and spins and its effect on laser-driven processes in both subsystems
[27; 28; 29; 30; 31]. Examination of a critical behaviour of the laser-induced
spin dynamics takes a central place in many of these studies [32; 30; 31],
with van der Waals transition metal thiophosphates MPS3 (M=Mn,Fe,Co,Ni)
playing an essential role as the materials offering a variety of
antiferromagnetic (AFM) arrangements [14].

In this Article, we report on a comparative study of laser-induced
demagnetization in the van der Waals transition metal thiophosphates NiPS3 and
FePS3. Within the MPS3 family, nickel and iron phosphorus trisulfides possess
a number of similarities with each other and are very distinct from the other
two compounds. Namely, FePS3 and NiPS3 both are zigzag antiferromagnets, and
their magnetism is of predominantly spin origin. Yet, FePS3 demonstrates the
2D Ising-like critical properties, while those of NiPS3 are described by the
2D $XY$-model. By studying the time-resolved dynamics of the exchange linear
dichroism (exLD), we reveal that the demagnetization degree in both
thiophosphates increases in the vicinity of the Néel temperature and is
governed by the critical behaviour of the order parameter depending on the
type of 2D magnetic ordering. However, only 2D Ising-like antiferromagnet
FePS3 exhibits a pronounced slowing-down of the demagnetization rate as Néel
temperature is approached, while in 2D $XY$-like NiPS3, such a slowing down in
the vicinity of $T_{\mathrm{N}}$ is minor. Using a two-temperature model for
the lattice and the spin responses to the optical excitation, we identify that
the difference in demagnetization times stems from distinct spin heat capacity
behaviours.

##  II Experimental

###  II.1 Materials and samples characterization

NiPS3 and FePS3 from the family of transition metal thiophosphates MPS3 are
layered van der Waals materials [14]. They belong to a monoclinic crystal
system, space group C$2/m$ (#12) [33]. As shown in Fig. 1(a,b), the transition
metal ions are arranged in a honeycomb structure within the $ab$-plane [34;
35]. The band gap of NiPS3 and FePS3 is 1.7 eV and 1.5 eV, respectively [36],
and is defined by the charge-transfer transitions involving states of P and S
ions. Within a bandgap there are weaker $d\\!-\\!d$ transitions of M^2+ ions
in the near-infrared range [37].

Among MPS3, nickel and iron thiophosphates have the most similarities in terms
of spin arrangement. Furthermore, they take an intermediate place in terms of
quenching the orbital momentum of a magnetic ion between MnPS3 (the zero
orbital momentum of Mn2+) and CoPS3 (the largest orbital momentum) [34]. Thus,
they suit well for the comparative study of laser-induced demagnetization. The
Néel temperatures are $T_{\mathrm{N}}$ = 155 K for NiPS3 and $T_{\mathrm{N}}$
= 118 K for FePS3 [34]. As illustrated in Fig. 1(a), in NiPS3, magnetic
moments form ferromagnetically (FM) coupled zigzag chains. The chains are AFM-
coupled within the $ab$-plane and FM-coupled in the adjacent layers along the
$c$-axis [35].

In samples with a thickness of more than 50 nm, the AFM vector L lies along
$a$-axis with a small deviation from the $ab$-plane [35; 38]. This magnetic
system can be described by the 2D $XY$ or $XXZ$ model [24]. In FePS3, the
zigzag ordering in the $ab$-plane is the same, while the chains are AFM-
coupled in adjacent layers Fig. 1(b). In contrast to NiPS3, the magnetic
properties of FePS3 are described by the 2D Ising model. L is perpendicular to
the layers [39]. The magnetic order is preserved at least down to the bilayer
in NiPS3 [24] and to the monolayer in FePS3 [40; 25].

Figure 1:  (a,b) Magnetic structure of (a) NiPS3 and (b) FePS3, blue spheres
represent metal atoms. $\mathbf{L}$ is the antiferromagnetic vector. The
figure was created using the VESTA [41]. (c,d) Atomic force microscopy images
of (c) NiPS3 and (d) FePS3 flakes on Si/SiO2 substrate. (e) Polarized Raman
spectra of NiPS3 measured at $T=80$ K in parallel [$z(yy)\overline{z}$] and
crossed [$z(yx)\overline{z}$] scattering configurations. Inset shows
temperature dependence of magnetic-order-induced frequency difference
$\Delta\omega$ for phonon lines $P_{1}$ in parallel and cross polarization
configurations. The red line represents fit by the function
$\propto|T-T_{\mathrm{N}}|^{2\beta}$. (f) Non-polarized Raman spectrum of
FePS3 measured at $T=80$ K. Inset shows the integral intensity $I$ of the low-
frequency ZF1 phonon peak (87 cm-1) as a function of $T$. Red line is a guide
for the eye. Blue vertical bars in (e,f) mark $T_{\mathrm{N}}$ extracted from
Raman lines temperature dependences.

Flakes of NiPS3 and FePS3 were obtained from the crystals grown by solid state
reaction, as detailed in [42; 27]. They were mechanically exfoliated using
adhesive tape inside an argon glovebox, and deposited onto Si/285 nm SiO2
substrates. Samples were characterized by atomic force microscopy, as shown in
Fig. 1(c,d). The lateral sizes of NiPS3 and FePS3 flakes are $\approx$ 80 and
$\approx$ 90 $\mu$m, respectively. Average flake thicknesses amounted to 200
nm and 180 nm respectively, i.e. their properties are expected to correspond
to those of the bulk samples.

For a detailed characterization, Raman measurements were performed using a
LabRAM HR Evolution UV-VIS-NIR (Horiba, France) spectrometer equipped with a
confocal microscope. The spectra were measured using continuous-wave
excitation at the 532 nm (2.33 eV) line of a Nd:YAG laser, and the power of
the incident light on the sample was limited to 0.4 mW. The spectra were
recorded using an 1800 lines/mm grating and a nitrogen-cooled charge-coupled
device detector, while a Leica PL FLUOTAR 50× (NA = 0.55) long working-
distance objective lens was used to focus the incident beam into a spot of 2
µm diameter. For both samples, the sets of phonon lines in the obtained
spectra shown in Fig. 1(e,f) are in good agreement with literature data [40;
24; 43]. The temperature dependence of the intensity and frequency of the
selected lines is used to examine the transition to the AFM phase. In order to
control the sample temperature, a Linkam THMS600 (Linkam Scientific
Instruments, UK) temperature control system was used. Polarized Raman spectra
of NiPS3 were measured in parallel $[z(yy)\overline{z}]$ and crossed
$[z(yx)\overline{z}]$ scattering configurations, with $z$ being the normal of
the flake and $x$ being along its horizontal side, as shown in Fig. 1(c). Fig.
1(e) shows the spectra obtained at $T=80$ K. In the paramagnetic phase of
NiPS3, the phonon peak P1 is observed at a frequency of 178.3 cm-1 for both
polarizations. In the AFM state, the spectral positions of P1 in parallel and
crossed polarizations do not coincide due to the splitting of degenerate
$E-$like phonon modes, which dramatically increases upon magnetic ordering
[24]. The difference $\Delta\omega$ of P1 frequencies for two polarizations as
a function of $T$ is shown in the inset of Fig. 1(e). Following [24], the
temperature dependence is well described by
$\Delta\omega\propto|T-T_{\mathrm{N}}|^{2\beta}$, with a critical exponent
$\beta=0.24\pm 0.04$ and $T_{\mathrm{N}}=155\pm 0.5$ K, confirming 2D $XY$
($\beta=0.231$ [44]) ordering and bulk Néel temperature, respectively.

The non-polarized Raman spectrum for FePS3 at $T=80$ K is shown in Fig. 1(f).
In the paramagnetic phase, there is a broad asymmetric phonon peak at 98 cm-1
[45]. Below $T_{\mathrm{N}}$, two distinct peaks at 87 and 93 cm-1 appear due
to the zone-folding (ZF) effect caused by the spin ordering [40]. The
intensity of the ZF1 peak at 87 cm-1 grows with the decrease of $T$ below
$T_{\mathrm{N}}$. As one can see from the inset of Fig. 1(f), the integral
intensity $I$ of this peak drops to zero at $T_{\mathrm{N}}$ = 118 K, which
indicates the transition of FePS3 to the paramagnetic phase. The obtained
$T_{\mathrm{N}}$ value is in good agreement with previous studies of bulk
FePS3 [40; 45].

###  II.2 Pump-probe experiment

Ultrafast magnetization dynamics in NiPS3 and FePS3 were measured with the
femtosecond magneto-optical pump-probe technique. The experimental scheme is
shown in Fig. 2(a). Probe pulses with a duration of 170 fs and a photon energy
$E_{\mathrm{pr}}$=1.2 eV were generated by the Yb:KGW regenerative amplifier
PHAROS (Light Conversion, Lithuania) at a repetition rate of 5 kHz. Optical
parametric amplifier ORPHEUS (Light Conversion, Lithuania) was used as a
source of pump pulses with photon energy in a range $E_{\mathrm{p}}=$1.63–1.97
eV and duration $\sigma\approx 170$ fs. Pump pulses had a fluence of $F=$ 2.5
mJ/cm2 and an angle of incidence of about 5∘. The pump spot diameter at a
sample was about 35 $\mu$m. Probe pulses with a fluence of 7 mJ/cm2 were
focused on a sample into a spot with diameter of 11 $\mu$m at a normal
incidence. The optical response was checked to be linear at the given probe
pulse fluence, as discussed in Appendix A. Both pump and probe spots were
smaller than the lateral sizes of the flakes, and it was possible to focus on
an area without visible defects. Although within the laser spots the flakes
possess some degree of thickness inhomogeneity, the latter is not expected to
affect the magnetic or optical properties because of the overall large
thickness of the flakes and the penetration depth of the pump and probe pulses
exceeding it. They are found to be 300 nm and 4 $\mu$m, respectively, for
NiPS3, and 800 nm and 7 $\mu$m, respectively, for FePS3.

The sample temperature was varied in the range $T$=78–295 K with the
temperature control system Linkam HFS600 (Linkam Scientific Instruments, UK).
Pump-induced probe polarization $\Delta\phi$ and reflectivity $\Delta R$ were
measured using balanced and single photodiode schemes, respectively, as
functions of pump-probe time delay $\Delta t$. In Ref. [46], it was shown that
AFM ordering in FePS3 gives rise to an exLD, i.e., a difference in absorption
and refraction for the light incident along the $c$-axis and polarized along
and perpendicularly to the zigzag chains, i.e. along the $a$\- and $b$-axes,
respectively. This effect stems from the sensitivity of the optical properties
to the spin correlations and can also be seen as a magnetic refraction [47],
which acquires anisotropy because of distinct exchange coupling along and
perpendicular to chains. As a result, exLD gives rise to the rotation of the
polarization of the reflected light in the AFM phase. Since this effect is not
sensitive to the orientation of $\mathbf{L}$, it is expected to be present in
NiPS3 and FePS3 when the probe is incident along the flake normal. This is in
contrast to quadratic magneto-optical Voigt and Kerr effects [48], which, in
the considered geometry, require $\mathbf{L}$ to lie in the sample plane and
thus should vanish in the FePS3 flake.

It was found that both $\Delta\phi(\Delta t)$ and $\Delta R(\Delta t)$ were
temperature-independent above $T_{\mathrm{N}}$ in both flakes (see Appendix A,
Fig. 5 for details). Thus, we used the pump-probe curves $\Delta\phi(\Delta
t)$ measured at $T=173$ K$~{}>T_{\mathrm{N}}$ for both compounds as reference
data, which includes temperature-independent laser-induced optical response of
the flakes, substrate, strain-induced dichroism, etc. Below, all presented
data are the result of subtraction of the reference curve from a signal at a
given temperature, as illustrated in Fig. 2(b) for the pump-induced rotation.

##  III Results

###  III.1 Laser-induced dynamics of exLD

The dynamics of the laser-induced probe polarization rotation in the NiPS3
flake $\Delta\phi$ measured at different sample temperatures and probe
polarizations is shown in Fig. 2(c,d). As seen in Fig. 2(c), below
$T_{\mathrm{N}}$ there is a monotonic change of $\Delta\phi$ at $\Delta t>0$,
followed by a saturation. Relaxation to the initial state takes place at a
longer time scale, which is inaccessible in our experiment. Noticeably, both
the magnitude and characteristic time scale of $\Delta\phi(\Delta t)$ evolve
with temperature, suggesting that the signal originates from the dynamics of
magnetic properties. Furthermore, below $T_{\mathrm{N}}$, $\Delta\phi(\Delta
t)$ demonstrates pronounced dependence on the incident probe polarization
angle $\phi$, with the sign of the laser-induced changes reversed upon
rotation of $\phi$ by 90∘ as shown in Fig. 2(d). The laser-induced changes of
$\Delta\phi$ are absent for any $\phi$ above $T_{\mathrm{N}}$. These
temperature and probe polarization dependencies suggest that the measured
signal is indeed the transient exLD, and, thus, represents the laser-driven
evolution of the AFM ordering in NiPS3.

Figure 2:  (a) The time-resolved pump-probe experimental scheme. Grey shaded
area depicts the plane of incidence for the pump pulses. Probe polarization
angle $\phi$ is measured from this plane. (b) Laser-induced probe polarization
rotation $\Delta\phi$ as a function of the delay time $\Delta t$ measured in
NiPS3 flake at $T=173$ K $>$ $T_{\mathrm{N}}$ (red solid symbols) and $T<$
$T_{\mathrm{N}}$ (blue solid symbols) and their difference (open blue
symbols). $\Delta\phi_{d}$ at $\Delta t_{d}=45$ ps is taken as an amplitude of
the laser-induced change of exLD. (c) Laser-induced probe polarization
rotation $\Delta\phi$ as a function of delay time $\Delta t$ between pump
($E_{\mathrm{p}}=$1.88 eV) and probe pulses measured for different sample
temperatures in NiPS3. (d) Transient polarization rotation
$\Delta\phi~{}(\Delta t)$ measured at two probe linear polarizations $\phi=0$
(open symbols) and $\phi=90^{\circ}$ (solid symbols) at $T=$ 133 K (blue
symbols) and 145 K (red symbols). Solid lines are fit according to the Eq. 1.

The same measurements were performed for the FePS3 flake [Fig. 4(c)], and
yielded qualitatively similar trends, with clearly longer times characterizing
the laser-induced change of exLD at some temperatures. In both samples, NiPS3
and FePS3, no dependence of $\Delta\phi(\Delta t)$ on pump polarization was
observed in the whole studied temperature range. This points to the essential
role of the ultrafast heating by a laser pulse in the measured dynamics.

Taking into account all the experimental observations, we conclude that in
both NiPS3 and FePS3 below $T_{\mathrm{N}}$ the optical excitation causes the
transient partial quenching of the AFM ordering, i. e., demagnetization,
induced by the laser heating and manifesting itself via a decrease in exLD. We
note that the laser-induced demagnetization in FePS3 was earlier demonstrated
in [32], while in NiPS3 no data on this process at the picosecond timescale
was reported.

Since the demagnetization is driven by ultrafast laser heating, it can be
sensitive to the pump photon energy $E_{\mathrm{p}}$ due to the spectral
dependence of the light absorption coefficient. To quantify the degree of
demagnetization, we used the value $\Delta\phi_{d}$ at $\Delta t_{d}=45$ ps
[Fig. 2(b)]. At most of the studied temperatures, the transient change of exLD
saturates at this time delay in both NiPS3 and FePS3. The study of the laser-
induced exLD dynamics in NiPS3 as a function of the pump phonon energy
revealed that $\Delta\phi_{d}$ has a maximum at $E_{\mathrm{p}}=$1.88 eV [Fig.
3(b)]. This photon energy is in good agreement with the position of an
absorption peak reported for NiPS3 [49; 50]. Based on this, we chose such a
pump photon energy for measurements of temperature dependences in NiPS3. In
FePS3, no pronounced spectral dependence of the signal was found, and the
measurements were performed at $E_{\mathrm{p}}=$1.97 eV.

To analyze the temperature dependence of the laser-induced change of exLD and
its features, we examined $\Delta\phi_{d}(T)$ and also found characteristic
times by fitting experimental data at each temperature with a function

$\Delta\phi(\Delta t,T)=\sum\limits_{i=1}^{2}B_{i}(T)\left[e^{-\Delta
t/\tau_{i}(T)}-1\right],$ (1)

where $B_{i}$ are the scaling parameters and $\tau_{i}$ are the time
constants. Bi-exponential decay was introduced since in FePS3 one clearly
distinguishes faster ($\tau_{1}$) and slower ($\tau_{2}$) processes as shown
in Fig. 4(c). The physical origin of the bi-exponential decay is discussed
further. We note that the time delay $\Delta t_{d}=45$ ps at which degree of
demagnetization $\propto\Delta\phi_{d}$ is analyzed indeed exceeds $3\tau_{2}$
at most of the studied temperatures in NiPS3 and FePS3.

In NiPS3, one can see that $\Delta\phi_{d}$ increases sharply with temperature
and then abruptly decreases to zero [Fig. 3(a)]. This is, again, a
characteristic behaviour of the ultrafast laser-induced demagnetization, which
is expected to be the largest just below $T_{\mathrm{N}}$ [10; 11]. We ascribe
the temperature at which the abrupt change of $\Delta\phi_{d}$ occurs to an
experimental Néel temperature $T_{\mathrm{N,exp}}$, which is below
$T_{\mathrm{N}}$ determined from the Raman scattering study (see inset in Fig.
1(e)). This is due to unideal thermal contact between the sample and the cold
finger in the cryostat, as well as the excess laser heating not fully
dissipating between the excitation events often present in the pump-probe
experiments. The value of $T_{\mathrm{N,exp}}$ is discussed below. The
approximation of $\Delta\phi(\Delta t)$ using Eq. 1 gave $B_{1}=0$. Thus,
$\tau_{2}$ can be readily ascribed to a characteristic time of the AFM
ordering decrease. $\tau_{2}$ grows only slightly in the vicinity of
$T_{\mathrm{N,exp}}$ and does not exceed 15 ps in the whole studied range, as
shown in Fig. 3(c).

Figure 3:  (a, b) Transient exLD amplitude $\Delta\phi_{d}$ and (c, d)
demagnetization time $\tau_{2}$ in NiPS3 as a function of (a, c) sample
temperature $T$ with a fixed $E_{\mathrm{p}}=$1.88 eV and (b, d) pump photon
energy $E_{\mathrm{p}}$ with a fixed sample temperature $T=$ 123 K. Incident
probe polarization angle $\phi=0$. Solid (a) green and (c) blue lines are fit
according to Eqs. (3; 5) and Eqs. (4; 5), respectively.

Similarly, in FePS3, a sharp increase of $\Delta\phi_{d}$ is observed as
$T_{\mathrm{N,exp}}$ is approached from below, followed by a sharp drop of the
signal above it [Fig. 4(a)]. In contrast to NiPS3, the approximation of
$\Delta\phi(\Delta t)$ using Eq. 1 revealed the bi-exponential decay. The
temperature dependencies of $\tau_{1}$ and $\tau_{2}$ are shown in Fig. 4(b).
These time constants demonstrate distinct temperature dependences. $\tau_{2}$
shows a pronounced increase up to 100 ps in the vicinity of
$T_{\mathrm{N,exp}}$, and we ascribe the temperature-dependent $\tau_{2}$ to
the demagnetization time [32], whereas $\tau_{1}\approx 1.5$ ps is independent
of $T$. Furthermore, the corresponding magnitude of the exLD change,
$\Delta\phi(\Delta t=2$ ps), gradually decreases as the temperature is swept
across $T_{\mathrm{N,exp}}$ towards the paramagnetic phase [Fig. 4(a)]. We
note that the overall laser-induced polarization rotation $\Delta\phi(\Delta
t)$ is order of magnitude larger in FePS3 as compared to NiPS3. Thus, the
faster contribution to $\Delta\phi(\Delta t)$ in NiPS3, if present, could be
hindered by the sensitivity of the experimental technique.

Recently, fast demagnetization with a characteristic time of 1–5 ps was
reported for another crystal from the MPS3 family CoPS3 [30]. It was suggested
that the unquenched orbital momentum of Co^2+ ions and strong spin-orbit
coupling facilitate a faster demagnetization process in this compound.
However, the degree and characteristic time of demagnetization in CoPS3
demonstrated a pronounced increase near $T_{\mathrm{N}}$. Thus, we can rule
out such an origin of the fast contribution to $\Delta\phi(\Delta t)$ with
$\tau_{1}$ in FePS3, although Fe2+ has unquenched orbital momentum as well.
The demagnetization process with two faster and slower stages was also
reported for a van der Waals ferromagnet Fe3GeTe2 [31]. In contrast to
metallic Fe3GeTe2, spin-flip electron-phonon scattering responsible for the
fast demagnetization [6] is negligible in non-metallic FePS3, and the two-
stage demagnetization in FePS3 can be ruled out.

To summarize the experimental findings, ultrafast demagnetization triggered by
laser-induced heating is observed by means of the time-resolved exLD in both
NiPS3 and FePS3. The demagnetization degree in both compounds possesses a
sharp increase as the Néel temperature is approached. Demagnetization times,
however, demonstrate distinct behaviour, slowing down in the vicinity of the
transition temperature found only in FePS3. The underlying physical picture of
the differences in the observed critical behaviour of demagnetization dynamics
in NiPS3 and FePS3 is discussed in the following section. In FePS3, there is
an additional faster process leading to changes in exLD with an origin
distinct from that responsible for the demagnetization.

Figure 4:  (a) Transient exLD amplitude $\Delta\phi_{d}$ (open symbols) and
(b) demagnetization time $\tau_{2}$ (open symbols) in FePS3 as a function of
sample temperature with a fixed $E_{\mathrm{p}}=$1.97 eV. Also shown in (a,b)
are the amplitude $\Delta\phi(\Delta t=2$ ps) and characteristic time
$\tau_{1}$ (solid symbols) of the optical contribution to $\Delta\phi$. (c)
Laser-induced probe polarization rotation $\Delta\phi$($\Delta t$) measured in
FePS3 flake at $T=173$ K $>$ $T_{\mathrm{N}}$ (red solid symbols) and $T=94$ K
$<$ $T_{\mathrm{N}}$ (blue solid symbols) and their difference (open blue
symbols). (d) Transient polarization rotation $\Delta\phi$ measured in FePS3
at different sample temperatures. Incident probe polarization angle $\phi=0$.
Solid (a) green and (b) blue lines are fit according to Eqs. (3; 5) and Eqs.
(4; 5), respectively. Solid lines in (d) are fit according to Eq. 1.

###  III.2 Numerical simulations of laser-induced dynamics of order parameter

ExLD scales as a squared order parameter $\mathbf{L}$ with temperature [46;
51]. According to the Landau phase transition theory [52], the order parameter
near $T_{\mathrm{N}}$ varies with temperature following the power law:

$\displaystyle\left\\{\begin{split}L\propto\left(T_{\mathrm{N}}-T\right)^{~{}\beta},T<T_{\mathrm{N}}\\\
L=0,T\geqslant T_{\mathrm{N}}\end{split}\right.$ (2)

where $\beta$ is the critical exponent. It characterizes the magnetic phase
transition and is specific to a model describing the system. Consequently, an
exact expression for the laser-induced change of exLD $\Delta\phi_{d}$ can be
written as

$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\Delta\phi_{d}(T)}$
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}=D(L^{2}-L_{0}^{2})}$
(3)
$\displaystyle{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}=D\left[\left(T_{\mathrm{N}}-(T+\Delta
T_{s})\right)^{2\beta}-\left(T_{\mathrm{N}}-T\right)^{2\beta}\right],}$

where $\Delta T_{s}$ is the laser-induced change of the effective spin
temperature and $D$ is the scaling coefficient. In order to show a relation
between the dynamics of laser-induced heating of the spin subsystem and
$\Delta\phi_{d}(T,\Delta t)$ explicitly and to separate an effect of the
critical behaviour described by $\beta$ from the effect of the spin
temperature increase, we introduce the linear approximation of Eq. (3). The
latter is valid under the assumption of relatively small laser-induced changes
of $L$, i.e., at temperatures $T$ and $T+\Delta T_{s}$ being far enough from
$T_{\mathrm{N,exp}}$. Then one gets $\Delta\phi_{d}\propto L\Delta L$, and the
expression Eq. (3) is reduced to a form

$\displaystyle\Delta\phi_{d}(T)$ $\displaystyle=DL\frac{dL}{dT}\Delta
T_{s}(T)$ (4) $\displaystyle=-2D\beta(T_{\mathrm{N}}-T)^{2\beta-1}\Delta
T_{s}(T).$

As readily seen from Eq. 4, demagnetization time $\tau_{2}$ correlates with
the time required for the heating of the spin subsystem $\tau_{s}$. The
temperature evolution of $\Delta\phi_{d}$, in turn, depends on both $\beta$
and the heating of the spin system. To analyze the response of the spins to
optical excitation we employ a two-temperature model [53], considering that in
laser-driven dielectrics energy and angular momentum transfer to spins is
realized from the phonon subsystem [for a review, see Ref. [54]] and, thus, it
is sufficient to consider only these two reservoirs with effective
temperatures $T_{s}$ and $T_{p}$, given the electron-phonon relaxation is fast
and is of order 2 ps (see App. A for details). Here we assume that the energy
delivered by the pump pulse is absorbed by electrons through $d\\!-\\!d$ and
charge-transfer transitions and rapidly deposited to the lattice.

We solve the following system of equations with the initial condition
$T_{p}=T_{s}=T$:

$\displaystyle\left\\{\begin{split}\frac{\rho}{\mu}C_{p}\left(T_{p}\right)\frac{dT_{p}}{dt}&=g_{sp}\left(T_{s}-T_{p}\right)+P(t),\\\
\frac{\rho}{\mu}C_{s}\left(T_{s}\right)\frac{dT_{s}}{dt}&=g_{sp}\left(T_{p}-T_{s}\right),\end{split}\right.$
(5)

where $C_{p}$ and $C_{s}$ are lattice and magnetic molar specific heat,
respectively, and $g_{sp}$ is the spin-lattice coupling constant. $C_{p}(T)$
was determined by an approximation of experimental data on the total specific
heat $C(T)$ reported in Ref. [42] with the Debye function [55]. Magnetic heat
capacity was then obtained as $C_{s}(T)=C(T)-C_{p}(T)$. $\rho$ and $\mu$ are
the density and the molar mass, respectively. $P(t)$ is the absorbed laser
power density found for the flake with a thickness $d\approx 200$ nm as [56]

$\displaystyle P(t)=AF\frac{\left(1-e^{-\alpha d}\right)}{d}\frac{2\sqrt{\ln
2}~{}e^{-4\ln 2~{}t^{2}/\sigma^{2}}}{\sigma\sqrt{\pi}},$ (6)

where $A$ and $\alpha$ are absorptance and absorption coefficient,
respectively. Absorptance $A=1-R$ was calculated using the Fresnel formula for
reflection $R$ at the air-MPS3 interface with a complex refractive index $n$.
Contributions from the reflections at other interfaces do not exceed 10 % and
are not taken into account. Other parameters used in calculations are
presented in Table 1.

Table 1:  Material parameters used in calculations, spin-phonon coupling
constant $g_{sp}$ and critical exponent $\beta$ obtained from the fit for
NiPS3 and FePS3.

Parameter |  NiPS3 |  FePS3
---|---|---
$\rho$ (kg m-3) |  3200 [57] |  2900 [57]
$\mu$ (kg mol-1) |  0.186 |  0.183
$n$ |  $3.7+0.5i$ [58] |  $2.9+0.3i$ [59]
$\alpha$ (m-1) |  $3.2\cdot 10^{6}$ [60] |  $1.2\cdot 10^{6}$ [61; 36]
$g_{sp}$ (Wm-3K-1) |  $(3.2\pm 0.8)\cdot 10^{16~{}\dagger}$ |  $(1.2\pm 0.4)\cdot 10^{16~{}\dagger}$
$\beta$ |  $0.21\pm 0.06^{\dagger}$ |  $0.10\pm 0.03^{\dagger}$

†This work

Solving Eq. 5, we find both the temporal evolution of the spin temperature
$\Delta T_{s}(\Delta t)$ and the heating of the spin system $\Delta T_{s}$ at
time delay $\Delta t_{d}$. In NiPS3 and FePS3 the increase in the spin
temperature $\Delta T_{s}$ is below 20 and 10 K, respectively, and has a
minimum in the vicinity of $T_{\mathrm{N,exp}}$. This justifies a direct
comparison of the characteristic time of the spin temperature increase
$\tau_{s}$ and the demagnetization time $\tau_{2}$ according to the linearized
expression [Eq. (4)]. By fitting $\Delta T_{s}(\Delta t)$ with a function
$1-e^{-\Delta t/\tau_{s}(T)}$, we obtain $\tau_{s}$ and compare it with the
demagnetization time $\tau_{2}$ at different initial temperatures. Note that
the fitting procedure also involved correction for $T_{\mathrm{N,exp}}$ $<$
$T_{\mathrm{N}}$ due to unideal thermal contact in the cryostat and excess
laser heating. It was found to be $\approx 13$ K in both samples, which
yielded $T_{\mathrm{N,exp}}$ $=142$ K for NiPS3 and $T_{\mathrm{N,exp}}$
$=105$ K for FePS3. Since $\tau_{s}$ is essentially the spin-lattice
thermalization time and is determined by the spin-lattice coupling constant
$g_{sp}$, this allows us to find values of $g_{sp}$. Figs. 3(c) and 4(b) show
the calculated thermalization time $\tau_{s}(T)$ plotted along with $\tau_{2}$
for NiPS3 and FePS3, respectively. Reasonable agreement with the
experimentally obtained temperature dependence of the demagnetization time is
obtained for both compounds at the values of $g_{sp}$ listed in Table 1.

As a next step, we fitted the temperature dependence of $\Delta\phi_{d}(T)$
below $T_{N,exp}$ with the function [Eq. 3] using scaling coefficient $D$ and
critical exponent $\beta$ as parameters [see solid green lines in Figs. 3(a)
and 4(a)]. We obtained $\beta=0.21\pm 0.06$ for NiPS3 close to the values
0.231 corresponding to the 2D $XY$ model [44; 62] and $0.24\pm 0.04$ obtained
from the temperature-dependent Raman scattering lines [inset in Fig. 1(e)].
For FePS3, the critical exponent amounts to $\beta=0.10\pm 0.03$ in agreement
with the 2D Ising model ($\beta=0.125$ [44]) and data obtained from Mössbauer
measurements ($\beta=0.16\pm 0.01$ [63]).

We note that in a range of $\approx 10$ K above $T_{\mathrm{N,exp}}$, there is
still a weak nonzero transient exLD signal whose time constant $\tau_{2}$
follows the spin temperature rise time $\tau_{s}$ [see Figs. 3(c) and 4(b)].
This signal can originate from the presence of short-range magnetic ordering,
which is not accounted for in the model, and, thus, the amplitude of the
transient exLD in this range is not described by Eq. 3 [see Figs. 3(a) and
4(a)].

##  IV Discussion

The performed analysis accounting for the laser-induced heating and spin-
phonon coupling captures the main experimental finding. Transient exLD indeed
measures the demagnetization due to laser-induced heating and is governed by
the increase in spin temperature. Because of the thermal nature of the laser-
induced changes, transient exLD and demagnetization $\Delta L$ itself possess
critical behaviour at $T_{\mathrm{N}}$ defined by the critical exponent
$\beta$. Criticality agrees well with the one predicted by 2D $XY$ or 2D Ising
models.

The demagnetization characteristic time is, in turn, defined by the spin-
lattice coupling constant $g_{sp}$ and the temperature dependence of the spin
specific heat, $\tau_{2}=\tau_{s}=C_{s}/{g_{sp}}$. Pronounced slowing down of
the demagnetization near the Néel temperature observed in FePS3 is in
agreement with previous works. In contrast, it is negligible in NiPS3. This
correlates well with the features of the specific heat in these materials
reported earlier in [25; 64; 42]. The heat capacity of NiPS3 does not have a
well-pronounced feature at $T_{\mathrm{N}}$, while that of FePS3 demonstrates
a pronounced lambda-peak in $C_{s}(T)$.

We note that the difference in the critical behaviour of demagnetization times
in NiPS3 and FePS3 resembles the recently reported critical behaviour of
characteristic times of restoration of the equilibrium magnetic ordering [29].
These times are much longer than the demagnetization times and can be
associated with the low interplanar thermal conductivity [65] governing the
cooling down process. In the vicinity of $T_{\mathrm{N}}$ in FePS3, this time
shows a pronounced slowing down, reaching 1000 ps, and appears to be two
orders of magnitude longer than in NiPS3.

Spin-phonon coupling constants $g_{sp}$ are found to be of the same order of
magnitude for NiPS3 and FePS3, see Table 1. This, in particular, stems from
comparable demagnetization times of around 10 ps in these compounds, apart
from the temperature range in the vicinity of $T_{\mathrm{N}}$. Note that in
[32], similar demagnetization time was reported for FePS3 in a bulk limit. As
for the other two MPS3 compounds, MnPS3 and CoPS3, the demagnetization times
well below the corresponding $T_{\mathrm{N}}$ were reported to be 30 ps [66]
and 2 ps [30], respectively. Adding our data for NiPS3 and FePS3 to this set
reveals that the demagnetization time below $T_{\mathrm{N}}$ in the MPS3
family decreases in the row Mn–Fe, Ni–Co. One notes that this trend does not
have a direct correlation with the trend in the Néel temperatures, since NiPS3
has the highest $T_{\mathrm{N}}$, FePS3 and CoPS3 have very close
$T_{\mathrm{N}}$ [67], while the $T_{\mathrm{N}}$ for MnPS3 is as low as 78 K
[50]. Also, there is no direct correlation with the magnitudes of magnetic
moments in these compounds either [50]. This is in contrast to demagnetization
in metals [6]. We could speculate that in MPS3, unquenching of orbital
momentum, the most present in CoPS3 and absent in MnPS3, affects spin-phonon
coupling and, thus, demagnetization times.

To address dependences of the demagnetization degree and time on pump photon
energy, we note that they demonstrate no features, apart from the local
maximum in $\Delta\phi_{d}$ at $E_{\mathrm{p}}=1.88$ eV in NiPS3 [Fig. 3(b)].
Demagnetization time does not show any pronounced trend within the precision
of our experiments [Fig. 3(d)]. On the other hand, in the studied spectral
range, optical absorption steadily grows with photon energy due to increased
contribution from charge-transfer transitions [36], resulting in an increase
of the absorbed pump power. If the initial temperature is far enough below
$T_{\mathrm{N}}$, an increase in absorbed laser power $P$ should lead to a
slowdown of demagnetization by $\approx 20$ % with a concomitant increase in
its amplitude, as can be shown based on Eqs. 3 and 5. Such a slowing down
cannot be resolved in our experiments [Fig. 3 (c,d)]. We note, however, that
the demagnetization degree in NiPS3 appears to be sensitive to the localized
$d\\!-\\!d$ transition at $E_{\mathrm{p}}=1.88$ eV in NiPS3 [49; 50]. This may
indicate that there can be an efficient channel of energy dissipation from
electrons excited to $3d$ sublevels of $\mathrm{Ni}^{2+}$ to phonons involved
in spin-phonon coupling $g_{sp}$. These can be, e.g., shear phonons, whose
particular role in demagnetization was revealed recently [29]. Noticeably, no
resonant enhancement is found in FePS3, and there are no reports in the
literature on any pronounced features in the absorption spectrum of this
material in the studied spectral range. This can be due to a narrower bandgap
in FePS3 as compared to NiPS3 (1.5 vs. 1.7 eV [36]), and, thus, a less
pronounced contribution to absorption from $d-d$ transitions. These
observations, however, require further studies.

Finally, we discuss possible reasons for the transient decrease of exLD with
the characteristic time $\tau_{1}$ observed in FePS3. Temperature dependences
$\Delta\phi(\Delta t=2~{}\mathrm{ps})$ and $\tau_{1}$ [Fig. 4(a,b)] suggest
that the increase in spin temperature and the corresponding decrease of
$\mathbf{L}$ cannot be responsible for this signal, as discussed in Sec.
III.1. On the other hand, in optical pump-probe measurements, changes in the
measured signal may also be caused by the laser-induced changes of the
proportionality coefficient $D$ between the polarization rotation and the
magnetic characteristic [68], that is
$\Delta\phi\propto\Delta{\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}D}L^{2}$.
Such a contribution to $\Delta\phi$ would follow the temperature dependence of
static exLD $\phi$ and decrease in the vicinity of $T_{\mathrm{N,exp}}$, as we
observed in the experiment [Fig. 4(a)]. In the case of exLD, $D$ includes
optical parameters as well as the exchange coupling $J$. Measurements of the
laser-induced dynamics of reflectivity $\Delta R(\Delta t)$ in FePS3 revealed
that it is characterized by a fast drop with a time constant of 1.5 ps [see
Appendix A and Fig. 5] which is close to $\tau_{1}$.

Thus, it is reasonable to assume that the faster dynamics of exLD is a trivial
result of changes in optical properties leading to $\Delta D$. Alternatively,
the faster contribution to $\Delta\phi$ may stem from laser-induced
perturbation of $J$ due to rapid energy transfer to phonons, which, in
general, can modulate Fe2+–S1- bonds responsible for exchange coupling. A fast
change of $\Delta R$ can serve a measure of this energy transfer rate. The
feasibility of such phonon-modulated exchange and related transient magneto-
optical response was investigated in Ref. [11] for the case of a dielectric
ferrimagnet Y3Fe5O12 subjected to THz pump pulses. We also note that the
magnitude of $\Delta R$ was found to gradually increase as the temperature was
changed across $T_{\mathrm{N,exp}}$ towards lower temperatures, which may
point to the sensitivity of the electron-phonon coupling to the magnetic
ordering.

##  V Conclusions

A comparative experimental study of the laser-induced exLD in 2D zigzag
antiferromagnets NiPS3 and FePS3 with XY\- and Ising-like magnetic ordering
revealed that ultrafast demagnetization in these compounds can be well
understood in terms of laser-induced heating of the spin system mediated by
spin-phonon coupling. The demagnetization degree shows critical behaviour as
Néel temperature is approached, with exponents being in good agreement with 2D
XY\- and 2D Ising-like ordering in NiPS3 and FePS3, respectively. The spin-
phonon coupling parameter is found to be close for two compounds. Similarly,
the demagnetization times well below $T_{\mathrm{N}}$ appear to be close and
are of the order of 10 ps. Previously, demagnetization times were reported for
all antiferromagnets from the MPS3 family except for NiPS3. Our data complete
this row and show that the time decreases with a change of transition metal Mn
– Fe, Ni – Co. This highlights that unquenched orbital momentum, which is the
strongest in Co and is absent in Mn, plays an important role in ultrafast
demagnetization in MPS3. Apart from similarities, a difference in
demagnetization times in NiPS3 and FePS3 was found in the vicinity of their
respective $T_{\mathrm{N}}$. The critical slowdown is observed in FePS3 only
and is nearly absent in NiPS3. In agreement with the thermal nature of the
laser-induced demagnetization, this difference is found to be related to the
different behaviour of spin specific heat in the vicinity of $T_{\mathrm{N}}$.

##  VI Acknowledgements

We thank R. M. Dubrovin for the help with experiments. We thank D. Bossini for
insightful discussions. The work of D.V.K., V.Yu.D. and A.M.K. was supported
by Russian Foundation for Basic Research Grant No. 19-52-12065. M.A.P.
acknowledges support of Russian Science Foundation under Grant No.
22-72-00039. L.A.Sh. acknowledges Presidential Scholarship Grant No.
SP-4623.2022.5. M.C. and F.M. acknowledge support by the Deutsche
Forschungsgemeinschaft through the International Collaborative Research Centre
160 (Projects B9 and Z4). E.C. and M.C. acknowledge funding from the European
Union’s Horizon 2020 Research and Innovation Programme under Project SINFONIA,
Grant No. 964396. E.C. acknowledges support from the European Union
(ERC-2017-ADG-788222), from the Spanish Government (2D-HETEROS
PID2020-117152RB-100, co-financed by FEDER, and Excellence Unit “María de
Maeztu” CEX2019-000919-M) and the Generalitat Valenciana (PROMETEO Program and
PO FEDER Program, IDIFEDER/2018/061). E.C. acknowledges the Advanced Materials
program, supported by MCIN with funding from European Union NextGenerationEU
(PRTR-C17.I1) and by Generalitat Valenciana.

##  Appendix A Laser-induced reflectivity dynamics

In Fig. 5, we present the raw signals $\Delta\phi(\Delta t)$ and $\Delta
R(\Delta t)$ in NiPS3 and FePS3 at temperatures above and below
$T_{\mathrm{N}}$. The dynamics of the reflectivity change is characterized by
the time constant amounting to 2 ps in NiPS3 and 1.5 ps in FePS3, which is
related to the electron-phonon relaxation time and depends on temperature only
weakly.

In both compounds at $T>$ $T_{\mathrm{N}}$, $\Delta\phi(\Delta t)$ and $\Delta
R(\Delta t)$ show similar dynamics [Fig. 5(a,c)]. Thus, we associate transient
probe polarization rotation at these temperatures with the changes in the
Fresnel coefficients of the flakes, as well as laser-induced changes in the
substrate optical constants, strain-induced effects, etc.

Figure 5:  Comparison of raw experimental traces for laser-induced probe
polarization rotation $\Delta\phi(\Delta t)$ (red) and laser-induced change of
reflectivity $\Delta R(\Delta t)$ (black) measured in NiPS3 at (a) $T=173$ K
$>$ $T_{\mathrm{N}}$ and (b) $T=123$ K $<$ $T_{\mathrm{N}}$, and in FePS3 at
(c) $T=173$ K $>$ $T_{\mathrm{N}}$ and (d) $T=93$ K $<$ $T_{\mathrm{N}}$.
Solid lines are guides for the eye.

At $T<$ $T_{\mathrm{N}}$, transients $\Delta\phi(\Delta t)$ and $\Delta
R(\Delta t)$ show distinct behaviour [Fig. 5(b,d)]. In particular, probe
polarization rotation possesses an additional slower contribution as compared
to the dynamics of reflectivity. This is the contribution from transient exLD,
which can be obtained by subtraction of $\Delta\phi(\Delta t)$ measured at
$T=$ 173 K from the data measured below $T_{\mathrm{N}}$, as discussed in the
main text.

In order to verify that the measurements of the laser-induced dynamics are
performed in the linear regime, we have measured pump-induced changes in
reflectivity $\Delta R$ at different probe fluences. As shown in Fig. 6, there
is a linear dependence of $\Delta R$ magnitude measured at $\Delta t=2$ ps on
the probe fluence in both FePS3 and NiPS3 samples.

Figure 6:  Magnitude of laser-induced change of reflectivity $\Delta R$ as a
function of probe pulse fluence (symbols) for (a) NiPS3 at $T=123$ K $<$
$T_{\mathrm{N}}$, and (b) FePS3 at $T=93$ K $<$ $T_{\mathrm{N}}$. The red
solid line shows a linear fit with a confidence interval.

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