# Direct Optical Probing of the Magnetic Properties
of the Layered Antiferromagnet CrPS4

Tomasz Fąs  Institute of Experimental Physics, Faculty of Physics, University
of Warsaw, Pasteura St. 5, 02-093 Warsaw, Poland  Mateusz Wlazło  Chemical and
Biological Systems Simulation Lab, Centre of New Technologies, University of
Warsaw, 02-097 Warsaw, Poland  Magdalena Birowska  Institute of Theoretical
Physics, Faculty of Physics, University of Warsaw, Pasteura St. 5, 02-093
Warsaw, Poland  Miłosz Rybak  Department of Semiconductor Materials
Engineering Faculty of Fundamental Problems of Technology, Wrocław University
of Science and Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Małgorzata Zinkiewicz  Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, Pasteura St. 5, 02-093 Warsaw, Poland  Leon Oleschko
Department of Physics, University of Konstanz, Universitätsstraße 10, D-78457
Konstanz, Germany  Mateusz Goryca  Institute of Experimental Physics, Faculty
of Physics, University of Warsaw, Pasteura St. 5, 02-093 Warsaw, Poland
Łukasz Gondek  Faculty of Physics and Applied Computer Science, AGH University
of Kraków, Mickiewicza 30, 30-059 Kraków, Poland  Bruno Camargo  Institute of
Experimental Physics, Faculty of Physics, University of Warsaw, Pasteura St.
5, 02-093 Warsaw, Poland  Jacek Szczytko  Institute of Experimental Physics,
Faculty of Physics, University of Warsaw, Pasteura St. 5, 02-093 Warsaw,
Poland  Adam K. Budniak  Schulich Faculty of Chemistry, Technion – Israel
Institute of Technology, Haifa 3200003, Israel  Yaron Amouyal  Department of
Materials Science and Engineering, Technion – Israel Institute of Technology,
Haifa 3200003, Israel  Efrat Lifshitz  Schulich Faculty of Chemistry, Technion
– Israel Institute of Technology, Haifa 3200003, Israel  Jan Suffczyński
<EMAIL_ADDRESS>Institute of Experimental Physics, Faculty of Physics,
University of Warsaw, Pasteura St. 5, 02-093 Warsaw, Poland

(August 28, 2024)

###### Abstract

Unusual magnetic properties of Van der Waals type antiferromagnetic
semiconductors make them highly attractive for spintronics and
optoelectronics. A link between the magnetic and optical properties of those
materials, required for practical applications, has not been, however,
established so far. Here, we report on a combined experimental and theoretical
study of magnetic, optical, and structural properties of bulk CrPS4 samples.
We find that the magnetic-field-dependent circular polarization degree of the
photoluminescence is a direct measure of the net magnetization of CrPS4.
Complementary, Raman scattering measured as a function of magnetic field and
temperature enables the determination of the magnetic susceptibility curve of
the material. Our experimental results are backed by Our experimental results
are supported by density functional theory calculations that take as input the
lattice parameters determined from temperature-dependent X-ray diffraction
measurements. This allows us to explain the impact of spin ordering on the
spectral position of Raman transitions in CrPS4, as well as anomalous
temperature shifts of selected of them. The presented method for all-optical
determination of the magnetic properties is highly promising for studies of
spin ordering and magnetic phase transitions in single- or a few-layer samples
of magnetic layered materials, for which a poor signal-to-noise ratio
precludes any reliable neutron scattering or magnetometry measurements.

###### Abstract

This file contains supplementary information to the article "Direct Optical
Probing of the Magnetic Properties of the Layered Antiferromagnet CrPS4". It
includes additional information regarding a structural characterisation of the
samples, their magnetic properties, additional results of temperature and
magnetic field dependent Photoluminescence and Raman spectroscopy
measurements, and description and results of theoretical modelling.

first keyword, second keyword, third keyword

2D materials, such as binary transition metal dichalcogenides (TMDs), have
recently been at the forefront of materials research.[1, 2] TMDs offer spin
and valley degrees of freedom, but in general, lack magnetism, which limits
their potential for applications in spintronic or optoelectronic devices. An
alternative is to consider 2D semiconducting materials whose lattice natively
incorporates transition metal ions with a non-vanishing spin moment such as
Mn, Ni, Cr, or Fe.[3, 4, 5, 6, 7] The prominent examples include transition
metal phosphorus chalcogenides such, as NiPS3, MnPS3, FePS3, or CrPS4.[8, 9]
This class of materials exhibits an antiferromagnetic ordering with Néel
temperature (TN) of up to 150 K. A variety of their potential applications in
electronics and optospintronics include spin tunnel field-effect transistors
or electrical switches with carrier mobility controlled by strain.[10, 11, 5,
7] The intrinsic robustness of 2D antiferromagnetic materials against
perturbations induced by an external magnetic field, and the possibility of
controlled directing of the Néel vector along one of the (typically multiple)
magnetic easy axes, are advantageous for their implementation in non-volatile
memory devices.

In the present work, we investigate CrPS4 (chromium phosphorus tetrasulphide
or chromium thiophosphate), which has been the focus of increasing attention.
It is a rare case of a layered A-AFM-type antiferromagnet, where magnetic ions
(Cr3+) within van der Waals planes are organised in a rectangular lattice.
CrPS4 is indicated as a promising material for spintronic, electrocatalytic,
energy-storage, optoelectronic and quantum information applications[12, 13,
14, 15, 16] or even for the generation of a tunable X-ray emission.[17]

Despite the increasing number of studies on CrPS4,[18, 19, 20, 21, 22, 17, 16,
23, 24] a link between its magnetic phase and optical response has not been
established so far. Here, we address this issue by choosing photoluminescence
(PL) and Raman scattering as our optical spectroscopy tools. Both methods have
been proven efficient in studies of spin ordering in conventional bulk[25, 26]
and layered 2D antiferromagnetic materials.[27, 28, 29, 30, 31] The usefulness
of the PL stems from the fact that in many cases the PL signal originates from
internal transitions of the magnetic ions, thus carries the information of the
magnetic state of those ions. As for the Raman scattering, an exchange-type
term contributes to interaction energy between the magnetic cations, which
affects the vibrational characteristics of the lattice. As a result, at least
some of the phonon modes are affected by the type of spin ordering in the
crystal lattice and magnetic phase transitions. Thanks to strong phonon
interactions and large exciton binding energy in 2D materials, even a few-
layer samples provide a sizeable Raman scattering and PL signal. Spectroscopy
of Raman- or PL-type is particularly useful in the case of antiferromagnets
that inherently provide only weak Kerr-rotation or magnetic circular dichroism
signal.[32]

Hitherto, PL and Raman spectroscopy studies of CrPS4 were conducted in the
absence of magnetic field. They concentrated on layer thickness and
polarization dependence of the Raman signal,[19, 21, 20, 33] the issue of the
material stability,[21, 32] and on the origin of the PL signal and its
pressure dependence.[24, 15, 34, 35] As a result, little is known about
magneto-optical properties of this material.

In the present work, to uncover a link between the magnetic and optical
properties of the CrPS4, we combine optical magneto-spectroscopy of bulk and
thin layer samples of CrPS4 with SQUID magnetometry. We support the
interpretation of the experimental data by density functional theory (DFT)
modelling of the CrPS4 phonon structure taking as an input lattice parameters
provided by temperature-dependent X-ray diffraction (XRD).

With the magnetic field and temperature-dependent experiments, we establish
that the circular polarization degree of the PL provides a direct probe of the
magnetization of CrPS4, while Raman scattering enables the determination of
its magnetic susceptibility curve. The calculations taking into account the
spin ordering of Cr3+ cations clarify an energy order of different magnetic
phases of CrPS4, explain their impact on a spectral position of Raman
transitions, and describe the observed anomalous spectral shifts with
temperature. As such, our work provides a comprehensive picture of structural,
magnetic, optical and electronic properties of CrPS4. We envision that the
presented magneto-optical method for determining magnetic susceptibility curve
and magnetic-phase transitions will be highly useful in the case of very thin
samples of magnetic materials down to a single layer level, for which
insufficient signal-to-noise ratio precludes the employment of standard
methods to determine magnetic ordering such as SQUID magnetometry or neutron
scattering.

##  I Results and discussion

###  I.1 Structural characterization of the CrPS4 samples

CrPS4 single crystals are synthesized from stoichiometric amounts of elements
via physical vapor transport without any transport agent (see Methods).[22]

Figure 1: High-resolution scanning transmission electron microscopy with
atomic resolution of a mechanically exfoliated CrPS4 layer. (a) HAADF-STEM
micrograph. EDS elemental maps presenting the atomic positions of (b) Cr, P,
and S, (c) Cr only, (d) P, and S. The micrographs are overlapped with
simulated structures showing the position of the atom columns of chromium
(blue spheres), phosphorus (red spheres), and sulfur (yellow spheres). Scale
bars are indicated. A schematic view of a CrPS4 layered structure: (e) a
single layer in $a-b$ crystal plane with apparent chain-like atomic
structures, (f) a side view of $a-b$ planes bound by van der Waals forces
along the $c$ axis. Figure 2: (a) Lattice parameters ($a,b$, and $c$), (b)
monoclinic angle and $\beta$ and unit cell volume as a function of temperature
determined by non-ambient X-ray diffraction studies conducted on powdered
samples of CrPS4. An anomalous-type dependence of the $b$ lattice parameter is
observed. A minimum in the dependence of the cell volume clearly marks
position of the Néel temperature at 38 K.

To perform a structural characterization of the CrPS4 samples, we employ a
range of methods, including an aberration-corrected scanning transmission
electron microscopy (STEM) with high-angle annular dark field (HAADF)
investigation and energy-dispersive X-ray spectroscopy (EDS), temperature
dependent X-ray diffraction, as well as scanning electron microscopy (SEM).
Fig. 1a presents a micrograph registered in [001] zone axis of a mechanically
exfoliated CrPS4 crystal using a HAADF detector.

Fig. 1a shows a HAADF micrograph registered in the [001] zone axis of a
mechanically exfoliated CrPS4 crystal. A molecular model[36] is superimposed
on a micrograph, with the chromium atoms (shown by blue spheres) form a
rectangular lattice in the $a-b$ plane. Distances between the Cr atoms along
the $a$ and $b$ axes are determined to be 0.55 $\pm$ 0.02 nm and 0.37 $\pm$
0.02 nm, respectively. Phosphorus atoms ($Z=15$) cannot be distinguished from
sulfur atoms ($Z=16$) by the HAADF detector because of the too small atomic
number difference. In order to resolve signals related to these two atoms, we
switch to atomically-resolved EDS mapping. The result of the mapping is shown
in Fig. 1b, where sulfur atoms are represented by yellow and phosphorus atoms
with red spheres. Fig. 1c shows the result of EDS mapping of the chromium
atoms only. The distances between rectangularly arranged columns, 0.55 $\pm$
0.02 nm and 0.37 $\pm$ 0.02 nm, are in full agreement with the result of the
HAADF measurement. Additionally, Fig. 1d presents the EDS map of phosphorus
atoms. The distances between respective pairs of them are 0.37 $\pm$ 0.02 nm,
0.41 $\pm$ 0.02 nm, and 0.73 $\pm$ 0.02 nm, and agree with previous works.[22]

Fig. 1e shows a schematic view of a single layer of CrPS4 in $a-b$ plane.
Sulfur atoms in CrPS4 are located in puckered layers, while chromium atoms are
coordinated in the middle of distorted octahedrons formed by six sulfur atoms.
Phosphorus atoms are in the middle of tetrahedrons of four sulfur atoms. Since
these phosphorus-sulfur tetrahedrons are placed between columns of chromium-
sulfur octahedrons aligned along the $b$ axis, chromium-chromium distances are
longer along the $a$ axis than the $b$ axis, which is the source of a
structural in-plane anisotropy of the CrPS4 crystal. The van der Waals gap
lies across $a-b$ planes and layers forming CrPS4 are stacked along the $c$
direction, as shown in Fig. 1f.

XRD measurements on powdered samples are conducted as a function of
temperature in the range between 20 K and 300 K. XRD patterns reveal that in
the investigated temperature range all reflections can be indexed by the
monoclinic C2 (No. 5) space group. The measurements reveal complex
dependencies, as depicted in Fig. 2a. Firstly, the lattice parameter $b$ shows
an anomalous decrease with the rise of the temperature. The decrease is sudden
below 38 K and slows down above this temperature, reaching a minimum at 140 K
(see Suppl. Fig. S2 in Supplementary Information). Lattice parameters $a$ and
$c$ rise according to a typical thermal expansion of the lattice, with a
slight increase in the expansion rate around the TN. The monoclinic angle
$\beta$ between $b$ and $c$ crystallographic directions exhibits a monotonic
decrease with temperature in the entire investigated range. In the vicinity of
38 K the change of the $\beta$ first slows down and then undergoes a sudden
drop (see Fig. 2b). The dependence of the unit cell volume on temperature
shows a distinct minimum at 38 K, with a roughly linear temperature dependence
in the vicinity of this anomaly, as apparent from Fig. 2b and Suppl. Fig. S2.

Figure 3: (a) Magnetic phase diagram of CrPS4 constructed from SQUID and
photoluminescence data. Horizontally and vertically filled symbols represent
points obtained from magnetometry measurements with the magnetic field
parallel and perpendicular to the $c$-axis of the sample, respectively (see
text). Crosses represent critical fields of spin-flop and spin-flip
transitions determined for consecutive temperatures from the dependence of the
degree of circular polarization of photoluminescence on the magnetic field
parallel to the $c$-axis. Solid lines show a fit to the points from
magnetometry and indicate a boundary between magnetic phases. (b) A scheme of
Cr3+ ions spin arrangements in the three magnetic phases of the lowest energy:
A-AFM (a magnetic ground state), canted AFM (cAFM) and ferromagnetic (FM)
phase. A calculated difference of energy per Cr3+ ion with respect to A-AFM
phase for cAFM and FM phase is indicated. Cr, P, and S atoms are represented
by blue, orange and yellow spheres, respectively.

Theoretical considerations allow us to explain a strongly anisotropic thermal
expansion of the CrPS4, manifested by, e.g., an anomalous-type variation of
the $b$ lattice parameter with temperature. We ascribe the origin of the
anisotropy to an interplay of magnetostriction, lattice dynamics and/or
crystal electric field effects. The minimum of the cell volume observed at TN
results from a competition in minimizing the overall energy of the crystal
between Cr-related magnetic interactions (that decrease and vanish upon
approaching the TN) and the elastic energy of the lattice (that increases with
the temperature).[37] The crystallographic direction along the $b$ axis is
distinguished by the ferromagnetic interaction as being the strongest between
the nearest neighbouring Cr3+ cations along that direction. In consequence,
chains of Cr3+ ions along $b$-axis exhibit a higher stiffness than the rows of
Cr3+ ions in other crystal directions. Calculations indicate that the
switching from the A-AFM phase to any different AFM type ordering shortens the
lattice parameter $b$ and elongates the lattice parameters $a$ and $c$. Since
the probability of any other magnetic arrangement than the ground state A-AFM
increases with the rise of the temperature, the decrease of the $b$ lattice
parameter is observed. Suppl. Fig. S11 shows the calculated temperature
variation of the $a$ and $b$ crystal parameters, remaining in qualitative
agreement with the experiment.

Results of a further structural characterization of the samples are provided
in Sec. S1 of Supplementary Information.

###  I.2 Magnetometry characterization of the CrPS4 samples

The magnetic susceptibility of CrPS4 determined as a function of temperature
for magnetic field parallel and perpendicular to the sample $c$-axis is
presented respectively in Suppl. Fig. S3a and S3b. Both geometries exhibit a
clear maximum at the Néel temperature of 38 K, in agreement with previous
reports.[36, 23] The position of the maximum keeps the value of 38 K (Néel
temperature) for low magnetic fields ($\leq 1$ T). It is then progressively
suppressed by increasing magnetic fields, reaching 19 K at 7 T. We observe an
additional, smaller peak below the main maximum of the dependence in the
parallel configuration, which we attribute to a phase transition (spin flop)
between the A-AFM and canted AFM phase.[23] For temperatures higher than 60 K,
both parallel and perpendicular configurations exhibit a typical Curie-Weiss
behaviour with the paramagnetic Curie temperature of $\theta=40.0\pm 0.3$ K,
in agreement with previous reports.[23, 16, 32] The positive value of $\theta$
indicates a ferromagnetic coupling between Cr3+ spins in the $a-b$ plane.

A phase diagram of CrPS4 constructed based on the magnetic susceptibility data
collected in both, parallel and perpendicular configurations is presented
respectively in Fig. 3a and 3b. Blue cricles represent the position of the
main peak of susceptibility in Suppl. Fig. S3, while the green ones come from
approximating the high temperature (100 K - 280 K) dependence by Curie-Weiss
law. Orange circles represent the position of the spin flop-related peak in
the susceptibility curve, defining a boundary of the transition between the
AFM and the cAFM phase upon the increase of temperature or magnetic field. The
boundary is described by the phenomenological relation:
$H_{SF}(T)=H_{0}\left[1-\frac{T}{T_{SF}}\right]^{1/2}$. The fitting to the
experimental data returns the values of spin-flop temperature $T_{SF}=34.5\pm
0.2$ K and $H_{0}=1.23\pm 0.05$ T. Upon further increase of the temperature,
cAFM to FM and, ferro- to paramagnetic (PM) phase transitions occur.

In order to support our interpretation of the magnetometry data, we perform
DFT+U calculations (U = 3 eV, see Methods) for various magnetic arrangements
of the Cr3+ spins. Three of the magnetic phases of CrPS4 considered in the
calculations are shown in Fig. 3b, while a full set of them is presented in
Suppl. Fig. S10. Our calculations show that the magnetic ground state of bulk
CrPS4 is A-AFM (with intralayer FM-type and interlayer AFM-type coupling of
Cr3+ spins; see Fig. 3b), which is in line with previous reports.[18, 16, 23]
In view of our calculations, Cr3+ spins are tilted about 10o from the $c$-axis
(see Suppl. Figure S9), with a non-zero component along the $a$ direction, in
agreement with previous neutron diffraction measurements.[38, 23] Spin-flop
transition that rotates the Cr3+ spins from the out-of-plane to the in-plane
direction (cAFM phase) requires an energy of around 0.05 meV per magnetic ion
(see Suppl. Fig. S9). The spin-flip of the Cr3+ spins resulting in FM phase of
the crystal enforces an energy of around 1 meV. These generally small energy
differences between the metastable magnetic phases and the A-AFM ground state
confirm that cAFM and FM phases can emerge through thermally-activated spin-
flop and spin-flip transitions, respectively, upon increasing the temperature
up to the $T_{N}$.

###  I.3 Photoluminescence of CrPS4 layers $vs$ temperature and magnetic field

CrPS4 photoluminescence is measured on thin, mechanically exfoliated flakes
with thicknesses ranging from 12 nm to 105 nm kept at temperatures between 5 K
and 40 K. A magnetic field of up to 10 T is applied either perpendicular or
parallel to the surface of the CrPS4 flake. The PL signal is excited with a
532 nm continuous wave laser beam and collected through a high numerical
aperture lens, in a direction perpendicular to the flake. All studied flakes
exhibit quantitatively comparable magneto-optical properties, except for a
lower overall PL intensity for thinner flakes.

A PL spectrum of a 105 nm thick CrPS4 flake is shown in Fig. 4a. Three
distinct maxima are seen in the 1.350 eV - 1.380 eV range. Three weaker maxima
showing a comparable intensity pattern as the main ones appear in the range
1.310 eV - 1.340 eV. The emission is attributed to internal
${}^{4}A_{2g}\rightarrow\,^{4}T_{2g}$ transitions within the $d$ shell of the
Cr3+ ion.[19] At the same time, electrons from the $d$ shell give rise to the
non-vanishing moment of the Cr3+ ions responsible for the magnetization of the
material. Thus, we anticipate that the observed emission will carry the
information on the magnetic state of the crystal. In particular, we expect
that the spin projection of the emitted photons on the propagation direction
(perpendicular to the CrPS4 flake) will reflect the magnetization component of
the material parallel to that direction. In order to verify this hypothesis,
we investigate the circular polarization of the PL.

Figure 4: (a) Photoluminescence spectrum of a 105 nm thick CrPS4 flake at B =
0 T at the temperature of 10 K. (b) The degree of circular polarization of
photoluminescence for the flake in (a), measured in the spectral range
$1.353\leavevmode\nobreak\ \text{eV}\leq\hbar\omega\leq 1.374$ eV for
temperatures between between 5 K and 40 K. (c) Magnetization curves obtained
from SQUID magnetometry for bulk CrPS4 for temperatures between 2 K and 40 K.
A direct correspondence between the degree of circular polarization and SQUID
magnetization is evidenced.

The degree of circular polarization of the signal is defined as
$DOP=(I_{\sigma+}-I_{\sigma-})/(I_{\sigma+}+I_{\sigma-})$, where $I_{\sigma+}$
and $I_{\sigma-}$ are the intensity of the signal registered in $\sigma+$ or
$\sigma-$ circular polarizations, respectively. In Fig. 4b the $DOP$ is
plotted as a function of the magnetic field applied perpendicular to the CrPS4
flake for consecutive temperatures. Fig. 4c shows corresponding plots of
magnetization of bulk material determined by SQUID measurements. A direct
correspondence between $DOP$ and magnetization obtained in SQUID experiment is
observed. In particular, for temperatures below 10 K, both the DOP and
magnetization are negligibly small for magnetic fields below 0.8 T, as
expected for the antiferromagnetic CrPS4. Spin-flop type transition of Cr3+
ions from A-AFM to cAFM phase is evidenced as an abrupt increase of both the
$DOP$ and the magnetization at around $0.8$ T. The increase in temperature
shifts the spin-flop transition towards lower fields. The inset to Fig. 4b
shows that the $DOP$ saturates at around 8 T consistent with the tendency seen
in the magnetization shown in the inset to Fig. 4c.

Figure 5: Raman spectra of CrPS4 registered in two orthogonal linear
polarizations of the detection at T = 5 K for circularly polarized excitation.
Phonon modes with $A$ or $B$ symmetry representation are labeled with a full
square or an empty dot, respectively. Inset: polar plot of normalized
intensity of Raman transitions against the detected linear polarization angle.
Despite being excited by a circularly polarized beam, the Raman signal is
strongly linearly polarized, with clearly distinguished two perpendicular
directions of the polarization, labeled as $H$ and $V$.

A direct proportionality between the $DOP$ and SQUID magnetization is observed
also for the magnetic field applied in the plane of the CrPS4 flake (see
Suppl. Fig. S4), albeit with a much smaller magnitude of the $DOP$ signal. As
discussed above, in our experimental configuration the $DOP$ is primarily
sensitive to the out-of-plane magnetization component. However, the use of a
high numerical aperture lens to collect the PL signal means that photons
emitted at some angle to the normal are also collected, making the measurement
also sensitive to the in-plane magnetization to some extent. Additionally, any
out-of-plane tilt of the Cr3+ spins gives rise to the non-zero $DOP$ signal,
even if the spins are mostly aligned with the in-plane magnetic field.

The critical fields of spin-flop and spin-flip transitions for consecutive
temperatures determined from the PL data are superimposed to the magnetic
phase diagram shown in Fig. 3a. A perfect agreement between the
magnetospectroscopy and magnetometry data indicates that magneto-
photoluminescence provides a very efficient tool in determining the
magnetization and magnetic phase diagram of thin layers of CrPS4. Such a
direct optical probing of the magnetic properties of this antiferromagnetic
material confirms that $d$-shell carriers defining the spin order of Cr3+ ions
indeed dictate the properties of the optical emission process.

###  I.4 Raman spectroscopy of CrPS4 layers $vs$ temperature in the absence of
magnetic field

The vibrational spectra of CrPS4 yield 36 phonon modes, following symmetry
representation $\Gamma_{acoustic}=A+2B$ and $\Gamma_{optical}=16A+17B$.
According to the selection rules for the non-centrosymmetric space group C2
(No. 5), modes with the irreducible representation $A$ and $B$ are Raman-
active.[39]

Raman scattering experiments are conducted on bulk CrPS4 samples placed in
temperatures between 5 K and 250 K. The signal is excited by a circularly
polarized laser beam propagating along the crystal’s $c$-axis and registered
as a function of a detected linear polarization angle. Fig. 5 shows selected
spectra acquired for two orthogonal linear polarizations denoted as $H$ and
$V$, evidencing well-resolved transitions that represent Raman active modes.
Despite the circular polarization of the excitation, the Raman signal is
strongly linearly polarized. We attribute this observation to the monoclinic
crystal structure of CrPS4.

We extract parameters of the transitions in the Raman spectrum by fitting each
line with a Lorentz profile. Experimental transition energies, as well as
those calculated within the DFT+U for A-AFM phase, are collected in Suppl.
Table S3. Their mutually good agreement allows us to assign the calculated
modes to the experimental ones and to determine their symmetry. A polar plot
(see the inset to Fig. 5) shows normalized intensities of the most prominent
Raman lines as a function of the detected linear polarization angle. We find
that there are two groups of orthogonally polarized lines. The lines which
exhibit the same polarization direction have the same representation of their
phonon mode, the symmetric $A$ in the case of $H$-polarized lines and anti-
symmetric $B$ in the case of $V$-polarized lines.

Figure 6: Evolution of (a) X1, (b) X2, (c) X3, (d) X4 transition in Raman
spectrum of CrPS4 with temperature. The intensity in panels (a-d) is
normalized to the intensity of a given transition at 5 K. (e-g) Position of
the lines X1, X2, X3, X4, respectively, as a function of temperature. Full
dots: experiment. Lines: a result of the DFT calculations taking as input
CrPS4 lattice parameters determined in temperature-dependent XRD measurement.
The vertical dashed line marks the Néel temperature ( 38 K).

The temperature evolution of selected transitions in the Raman spectrum is
shown in Fig. 6a-d. Their position, intensity, and linewidth vary with
temperature with a qualitative change in the vicinity of the $T_{N}$, thus
reflecting their susceptibility to the type of magnetic ordering in the
crystal. The most accentuated change is observed for line X1, positioned at
184.5 cm-1 at 5 K (see Fig. 6a). Upon approaching the Néel temperature, the X1
transition weakens and shifts towards lower energies. Above the $T_{N}$, the
shift gets more pronounced, the transition strongly broadens spectrally and
finally vanishes around 150 K. Qualitatively similar, but less pronounced
redshifts are observed for the lines X2 (at 299.5 cm-1, see Fig. 6b) and X4
(at 409.8 cm-1, see Fig. 6d).

On the contrary, the X3 line at 305.7 cm-1 at 5 K shown in Fig. 6c shifts
toward larger energies (blueshifts) upon approaching and crossing TN. This
behavior is qualitatively different with respect to that of lines X1, X2 and
X4, which redshift upon approaching the TN (see Fig. 6e-h). A similar,
anomalous, temperature dependence as for X3 transition has been reported
previously in the case of selected phonon modes in MnPS3.[40] It has been
attributed to a temperature-induced modification of the superexchange paths
between magnetic ions and a resulting change in constants of magnetoelastic
interactions between them.[40] Suitably, our theoretical predictions indicate
that X3 shifts towards higher energies when the magnetic configuration departs
from the ground, A-AFM state (see Fig. 8). An evolution of the selected Raman
transitions in the temperature range of up to 250 K is presented additionally
in Suppl. Fig. S5.

Suppl. Video SV1 shows an atomic vibrational composition of X1, X2, X3, and X4
Raman lines determined by DFT calculations. In the case of lines X1 and X3, a
predominant contribution comes from in-plane oscillations with an opposite
phase of Cr atoms along the $b$ crystal axis. In the case of the X2 line, the
CrPS4 crystal cell undergoes torque-like oscillations with the Cr atoms
shifting in-plane, in a direction parallel to the $a$ axis of the crystal,
yielding a relatively small variation of a pathway length between the Cr
atoms. The contribution of the Cr sub-lattice oscillations is even smaller in
the case of the X4 line. Hence, in the case of X2 and X4 lines, we attribute
the observed spin-phonon coupling to a Cr-S-Cr super-exchange pathway with
pronounced oscillations of the S atom placed in between the nearest Cr atoms.

In order to understand the effect of the structural and magnetic phase changes
on the Raman modes, we perform the DFT+U calculations with lattice parameters
found in the temperature-dependent X-ray diffraction measurement (see Fig. 2).
Previous DFT calculations of the Raman modes in 2D magnetic materials assumed
optimized lattice parameters and neglected their temperature dependence. In
this regard, the structural changes determined by XRD are accounted for by the
respective rescaling of the lengths of the optimized cells (for the details,
see Sec. S5 in Suppl. Information).

As shown in Fig. 6e-g, the calculated temperature evolution for most Raman
modes in the case of the most of the modes generally follows the experimental
trends (see also Suppl. Fig. S12). The calculated and experimental changes are
of the same order of a few cm-1 in the studied temperature range, and the
slopes of the dependencies increase with the temperature. As the calculated
evolution reflects changes in the lattice parameters, the observed
temperature-induced shifts of the presented Raman modes above the TN are
primarily of structural origin. On the other hand, slopes of the calculated
and experimental dependencies qualitatively disagree below TN for the lines X2
and X4. Also, the dependence determined theoretically for the X1 line does not
follow the experimental trend, but shows a surprising insensitivity to
temperature-induced structural changes. In order to give a physical insight
into a plausible mechanism of the observed discrepancies, the impact of
various magnetic arrangements of the Cr3+ spins on the Raman modes is examined
in the next Section.

Figure 7: (a) Spectral position of the X4 transition in Raman spectrum of
CrPS4 as a function magnetic field applied along the $c$ crystal axis for
temperatures 6 K, 35 K and 100 K (points). Lines represent linear fit to the
experimental data. (b) Slopes of the linear dependencies found for consecutive
temperatures in the range from 5 K to 150 K (points) along with the magnetic
susceptibility curve obtained from SQUID magnetometry (orange line) at B =
0.05 T. Shaded regions in both panels encodes uncertainty of the Raman data.
The magnitude of the magnetic-field-induced shift of the X4 line follows the
magnetic susceptibility curve from magnetometry.

###  I.5 Raman spectroscopy of CrPS4 layers $vs$ magnetic field and
temperature

Magneto-Raman measurements are performed in the Faraday configuration, with
the magnetic field applied parallel to the $c$-axis of the crystal. Although
in ordinary semiconductors, magnetic field influences phonon frequencies only
weakly, in the case of the studied material incorporating magnetic Cr3+ ions
in its lattice, the impact of the magnetic field on Raman spectra is sizeable.

Fig. 7a shows the spectral position of the X4 line (points) as a function of
the magnetic field for three selected temperatures: 6 K, 35 K and 100 K. The
X4 line shifts towards higher energies with the increasing field, following an
approximately linear dependence. The linear fits (solid lines in Fig. 7a)
performed for the data registered at temperatures in the range between 6 K and
150 K yield slopes, which are represented by black points in Fig. 7b. The
magnetic susceptibility curve obtained from the magnetometry measurements (see
Suppl. Fig. S3a) at a small field of 0.05 T in the parallel configuration is
also displayed in Fig. 7b with an orange line. As clearly seen, the slopes of
the magnetic-field-induced energy shift of the X4 line follow the magnetic
susceptibility curve of the material. In particular, the maxima of both
dependencies in Fig. 7b overlap, which means that the susceptibility of the
phonon mode X4 energy to the magnetic field is the largest in the vicinity of
the Néel temperature. Moreover, the magnetic-field-induced change in the
degree of circular polarization of the X4 line is strongly reduced above the
Néel temperature (see Suppl. Fig. S6), which marks the magnetic phase
transition taking place at TN.

The results described above demonstrate that, by performing temperature and
magnetic-field-dependent Raman scattering measurement, one can determine the
shape of the magnetic susceptibility curve and state the presence of magnetic
phase transitions of 2D layered antiferromagnet CrPS4. The theoretical
calculations presented below indicate which transitions in the Raman spectrum
of CrPS4 are the best suited for this purpose.

We consider the impact of the in-plane and out-of-plane arrangements of the
Cr3+ ions spins on optically active Raman modes, examining the lowermost
energetically, collinear magnetic phases, _i.e._ A-AFM, FM and Y∗-AFM.
Ferromagnetic order within the $a-b$ crystal plane is maintained in AFM and FM
phases. The Y∗-AFM is a prototype phase, where spins of the Cr3+ first nearest
neighbors (1NN) are antiferromagnetically aligned, leading to a suppression of
the the FM ordering within the $a-b$ plane. In view of our DFT calculations,
the change of the type of out-of-plane magnetic ordering (e.g., A-AFM to FM)
shifts Raman modes only by fractions of cm-1 in relation to unperturbed modes
(see Fig. 8 and Suppl. Table S3). However, we predict pronounced changes in
the spectral position of the Raman modes (of the order of cm-1) when a
magnetic phase transition that alters the in-plane spin ordering occurs (see
Fig. 8 and Suppl. Table S3). In particular, a transition from A-AFM to Y∗-AFM
results in a 9 cm-1 redshift in the case of the X1 line, thus confirming its
strong susceptibility to variation of the magnetic order revealed in the
experiment (see Fig. 6a). Our predictions are in line with the calculated
strength of the exchange coupling (see Suppl. Table S1), which is order of
magnitude stronger (J${}_{1}=2.13$ meV) for in-plane than for out-of-plane
(J${}_{0}=-0.14$ meV) directions. Calculations confirm also the enhanced
proximity between next-neighboring Cr atoms along the $b$-direction in the
crystal. These findings are consistent with the results of temperature-
dependent XRD studies (Fig. 2 and Suppl. Fig. S2), showing that the $b$
lattice parameter of the crystal diminishes with the emperature. This leads to
an increased probability of magnetic states other than ground A-AFM in the
material (see Suppl. Tab. S2).

As Fig. 8 reveals, a relatively strong redshift assisting the AFM to Y∗-AFM
transition is predicted in the case of the X4 line. Since this line, in
contrast to the line X1, is still observable above the $T_{N}$, we have
selected it for the magneto-optical Raman study presented in Fig. 7 and Suppl.
Fig. S6. A sizeable spectral shift predicted for X3 (see Fig. 8) suggests that
the magnetic susceptibility curve can be retrieved also from this line’s
magnetic field and temperature spectral dependencies. Suppl. Fig. S7 shows
that this is indeed the case. We note, however, that magnetic field-induced
spectral shifts of X3 provide an inverted temperature dependence with respect
to that observed for the X4 line (compare Suppl. Fig. S7 for the X3 line with
Fig. 7 for the X4 line). This observation is consistent with the theoretical
considerations of Fig. 8, for which opposite spectral shifts are expected for
X3 (blueshift) and X4 (redshift) at the AFM to Y∗-AFM transition. In turn, the
X2 line shows no susceptibility to a magnetic order change, as can be seen in
Fig. 8. Accordingly, a magneto-optical Raman study of the X2 line does not
bring any meaningful dependencies (not shown).

Figure 8: Calculated relative spectral shift of the X1, X2, X3 and X4 Raman
modes upon a transition from the ground-state A-AFM to out-of-plane-spin-
flipped FM and in-plane-spin-flipped Y*-AFM magnetic phase of CrPS4. A
schematic representation of the spin order of Cr3+ ions in each magnetic phase
is shown below the horizontal axis of the plot.

The consistency between the presented set of magneto-optical and DFT results
shows that the calculations enable the identification of phonon modes in the
Raman spectrum, which are best suited for determination of a susceptibility
curve of the material.

##  II Conclusions

We have performed a multidimensional study on the layered antiferromagnet
CrPS4 involving magnetometry, structural and magneto-optical measurements and
complementary DFT theoretical calculations.

We find that the degree of circular polarization of photoluminescence of CrPS4
mirrors magnetization determined in SQUID measurements. This direct
correspondence can be traced back to the $d$-shell electrons responsible for
both the emission and magnetization of the material. Moreover, Raman
scattering measured as a function of magnetic field and temperature enables
the determination of the magnetic susceptibility curve of the material. The
temperature evolution of the optically active Raman modes above the Néel
temperature agrees well with theoretical predictions taking into account
structural changes of the crystal determined in temperature-dependent XRD
measurements. The discrepancies observed at temperatures at or below $T_{N}$
are a fingerprint of the susceptibility of the Raman modes to the magnetic
arrangement of the Cr3+ ions. The calculations confirm a strong magnetoelastic
coupling in CrPS4 and allow us to explain anomalous dependencies in the
thermal expansion of the crystal lattice and temperature shifts of selected
Raman lines. The theoretical considerations allow us also to predict the
influence of the type of spin arrangement of Cr3+ ions on the Raman modes in
CrPS4, and to identify Raman modes of the material that are the most
susceptible to the applied magnetic field.

With our work, we show that optical magnetospectroscopy, _i.e._ , a purely
optical measurement, is capable of yielding magnetometry-type information on
the properties of an antiferromagnetic semiconductor, including its
magnetization or the magnetic phase diagram. We envision that the presented
approach will be useful in the case of thin layers of other 2D or ordinary-
type magnetic materials, for which standard methods for the determination of
magnetic susceptibility curves do not provide a sufficient signal-to-noise
ratio.

##  III Methods

###  III.1 Synthesis of CrPS4 samples

Bulk CrPS4 samples are synthesized by the physical vapor transport - without
any transporting agent such as iodine (see Ref.[22] for details). Around 1
gram of stoichiometric mixture of elements, with the ratio Cr : P : S = 1 : 1
: 4 is sealed in a quartz ampule previously evacuated by a turbomolecular
pump. The ampule is kept for 4 days in a furnace with a temperature gradient
of 750°C/710°C (substrate zone/deposition zone) and after cooling down a few
mm by a few mm CrPS4 crystals with a thickness of 10-500 $\mu$ are collected.

###  III.2 XRD Investigation

X-ray diffraction measurements are conducted on CrPS4 samples of two types:
single crystals and powdered crystals. Diffractograms of single crystals are
acquired on samples placed in the preferential orientation (001) in a parallel
beam mode by Rigaku SmartLab 9 kW, Cu K$\alpha$ radiation 1.5406 Å, equipped
with a Ge 220 monochromator.

Temperature-dependent non-ambient X-ray diffraction measurements on powdered
samples are performed using MalvernPanalytical Empyrean diffractometer
equipped with Oxford Instruments PheniX helium close-cycle refrigerator.
Measurements are made in 20 - 300 K temperature range with the step of 2 K (in
the range $20-60$ K) or 20 K ($60-300$ K). Thermal displacement of the sample
stage is automatically corrected by reference measurements of a high purity
tungsten (6N) powder. During the measurements high-vacuum conditions (10-7
mbar) are maintained. Collected patterns are refined using FullProf Suite
package.

###  III.3 Electron Microscopy

HR-SEM Zeiss Ultra-Plus FEG-SEM working on 4 keV is used to acquire a picture
with a secondary electron detector. EDS spectrum is recorded by FEI E-SEM
Quanta 200 with 20 kV acceleration voltage. In both cases, the sample is not
coated.

CrPS4 is mechanically exfoliated directly onto a TEM grid by the protocol
developed in Ref.[22] In the present work, the use of the tape: “Ultron
systems, INC. Silicon-Free Blue Adhesive Film P/N 1009R-6.0“, increases the
entire protocol efficiency compared to the previous works.[41]

In the first step, the mechanically exfoliated CrPS4 is measured by cTEM: FEI
Tecnai G2 T20 S-Twin TEM working on 200 keV. The average background
subtraction filter (ABSF) is applied for the micrograph presented in the
Suppl. Fig. S1d. Then, the previously prepared TEM grid with CrPS4 is mapped
by high angle annular dark-field (HAADF) and EDS detectors in STEM mode with
60 kV acceleration voltage on FEI Titan Cubed Themis G2 60–300 equipped with a
Dual-X detector (Bruker) with an effective solid angle of 1.76 sr for fast and
precise local atomic chemical analysis.

The micrograph presented in Fig. 1 is based on 20 averaged and aligned frames
acquired by HAADF detector. EDS maps in Fig. 1 are based on 200 frames. All
EDS maps are pre-filtrated by pixel averaging (7 px) and post-filtrated by
radial Wiener filtering (highest frequency 15.0 and edge smoothing 5.0) with
Velox Software (Thermo Fisher Scientific). Low voltage (60 kV) for electron
acceleration is applied to minimize damage of the material.

###  III.4 Magnetometry measurements

Magnetometry measurements are performed using a SQUID magnetometer on a CrPS4
bulk crystal, with approximate dimensions 3 mm $\times$ 2 mm $\times$ 0.5 mm
and density of 2.64 g/cm3. The sample dipolar magnetic moment is measured
along the applied magnetic field $B$. Measurements are performed in two
configurations: in a parallel one, with $B\parallel$ to the $c$-axis of the
crystal and in a perpendicular one, with $B\perp c$.

###  III.5 Optical magnetospectroscopy

Photoluminescence from CrPS4 layers of a thickness between 12 nm and 105 nm is
excited with a beam of 561 nm laser. The Raman spectroscopy measurements on
bulk CrPS4 samples are performed using a linearly polarized 514.5 nm line of
the Ar-ion laser or an unpolarized 442 nm line of the He-Cd laser, at
temperature varied from 5 K to 250 K.

The lens of 3.5 mm focal length mounted above the sample on piezo-translators
ensures the spatial resolution of 1.5 $\mu$m. Magnetic field of up to 10 T
produced by a superconducting coil is applied in Faraday or Voigt
configuration. The excitation and detection is polarization resolved thanks to
the use of a relevant combination of a linear polarizer, a half-wave plate
with a controlled angle and a quarter-wave plate. The signal is detected using
a 0.75 m grating spectrometer (2400 grooves/mm) with a Peltier cooled CCD
camera on its output (0.1 nm of spectral resolution).

###  III.6 Computational details

Planewave DFT calculations have been performed using VASP package[42] with PAW
pseudopotentials[43] and the PBE functional[44, 45]. On-site Coulomb repulsion
correction is introduced via the DFT+U approach introduced by Dudarev et
al.[46] U = 3 eV and J = 1.4 eV (U${}_{eff}=1.6$ eV) parameters have been
chosen to account for the repulsion between 3d electrons of Cr atoms.[47] We
note that a comparable value of U (U = 1.8 eV) selected previously in the case
of exfoliated MnPS3 layers has been corroborated by the recent ARPES
measurements.[48] Each magnetic phase of CrPS4 is modeled with a
1$\times$1$\times$2 supercell of the conventional cell (8 formula units). The
unit cell shape, volume and atomic positions are relaxed until forces acting
on each atom are smaller than 0.001 eV/Å. The electron density is optimized
until 10-7 eV self-consistency criterion is met. A $\Gamma$-centered
8$\times$8$\times$4 k-point grid is used for all calculations.

The effect of temperature is modeled using experimental lattice parameters
obtained via refinement of XRD data obtained in the temperature range of 20 –
300 K. The various magnetic arrangements have been adopted as presented in
Suppl. Fig. S10. For each of the employed magnetic orderings the position of
the atoms have been fully relaxed. For each magnetic ordering independently,
relaxed lattice parameters $a,b,c$ and $\beta$ are scaled at each temperature
by such factor as found in XRD experiment. $\Gamma$-point phonon frequencies
are calculated using the finite displacement method with a 2$\times$1$\times$2
supercell of the primitive unit cell (8 formula units). The self-consistent
convergence criterion is increased to 10-11 eV to accurately evaluate ionic
forces. The phonopy[49] package is used to generate displacements, build the
dynamical matrix from VASP-calculated forces, and calculate the phonon
frequencies.

##  IV Acknowledgments

M. B. acknowledges financial support from the University of Warsaw under the
"Excellence Initiative - Research University" project. J. S. and M. G.
acknowledge support within the "New Ideas 2B in POB II” IDUB project financed
by the University of Warsaw. We acknowledge Polish high-performance computing
infrastructure PLGrid for awarding this project access to the LUMI
supercomputer, owned by the EuroHPC Joint Undertaking, hosted by CSC (Finland)
and the LUMI consortium through PLL/2022/03/016435. A. K. B. and E. L. were
supported by the European Commission via the Marie Skłodowska-Curie action
Phonsi (H2020-MSCA-ITN-642656). Research leading to these results has also
received funding from the Norwegian Financial Mechanism 2014 to 2021 under
Grant No. 2020/37/K/ST3/03656 and from the Polish National Agency for Academic
Exchange within Polish Returns program under Grant No. PPN/PPO/2020/1/00030.

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Supplementary Information to "Direct Optical Probing of the Magnetic
Properties of the Layered Antiferromagnet CrPS4"

##  S1  Structural characterization

The results of a general structural characterization of bulk CrPS4 are
presented in Suppl. Fig. S1. A photograph of "as-synthesized" chromium
thiophosphate crystals with a size of the order of half of a centimeter and a
thickness of the order of 10 $\mu$m - 100 $\mu$m is shown in Suppl. Fig. S1a.
One of the synthesized crystals is placed on a flat surface to register a
diffractogram using a powder X-ray diffractometer (PXRD), which is shown with
a red curve in Suppl. Fig. S1b. Only a family of 00I planes is registered,
which is expected in the case of the analysis conducted on a single crystal of
a layered material, in which van der Waals planes are perpendicular to the
scattering vector. In contrast, the powder diffractogram presented via the
green line reveals all planes present in CrPS4. The position of Bragg peaks in
both diffractograms remains in agreement with the literature.[36]

Energy-dispersive X-ray spectrum (EDS) shown in Suppl. Fig. S1c confirms the
presence of all three elements, that are chromium, phosphorus, and sulfur.
Results of a quantitative analysis are provided in the table in the same
panel. They corroborate that the atomic composition of the studied crystal is
close to the intended Cr:P:S = 1:1:4. The inset of Suppl. Fig. S1c presents a
scanning electron microscope (SEM) picture of CrPS4, with its clearly
evidenced layered structure. In addition, it can be noticed that the crystal
cleaves along crystallographic directions.

Suppl. Fig. S1d presents a transmission electron microscope (TEM) micrograph
of the crystal mechanically exfoliated onto a grid. The image registered in
the [001] zone axis confirms a good structural quality of the studied sample.

Figure S1: (a) An image of "as-synthesized" bulk CrPS4 crystals. (b) Results
of X-ray studies of CrPS4: a diffractogram of a single CrPS4 crystal aligned
in a preferential orientation that is with the $c$-axis parallel to the X-ray
beam propagation (red curve) and of powdered CrPS4 crystal (green curve).
Vertical black dotted lines describe CrPS4 peak positions according to PDF
04-009-0826.[36] (c) EDS spectrum of a bulk CrPS4 crystal. The obtained atomic
composition shown in the table confirms Cr:P:S atomic ratio close to 1:1:4.
The inset: HR-SEM micrograph of a bulk CrPS4 sample recorded with a secondary
electron detector shows a layered structure of the crystal. (d) A high-
resolution, filtered micrograph of a mechanically exfoliated CrPS4 layer
registered with a conventional TEM.

Suppl. Figure S2 shows the results of XRD measurements as a function of
temperature in the range from 20 K to 300 K on a powdered CrPS4. Suppl.
Figures S2a and b show the variation of the lattice parameters. The lattice
parameters at 300 K are $a$ = 10.8596(1) Å, $b$ = 7.2501(2) Å, $c$ = 6.1382(2)
Å, $\beta$ = 91.886(1)∘, and cell volume is 483.02(1) Å3, in agreement with
the results of our STEM measurements and previous reports.[50] Suppl. Fig. S2c
presents additionally a map composed of consecutive XRD spectra containing
selected two maxima (reflections 002 and 220) as a function of temperature in
the range between 20 K and 300 K.

Figure S2: Results of non-ambient XRD studies on powdered samples of CrPS4 in
the entire investigated temperature range (20 K - 300 K): (a) $a$, $b$ and $c$
lattice parameters (b) $\beta$ angle and unit cell volume as a function of
temperature (c) Variation of the 002 and 220 reflections with the temperature.

##  S2 Magnetometry characterisation

Measurements of magnetic characterization of the CrPS4 bulk sample correspond
to the longitudinal magnetic moment along the applied field. The results
acquired with the magnetic field applied in the direction parallel and
perpendicular to the $c$-axis of the crystal are presented respectively in
Suppl. Fig. S3a and Suppl. Fig. S3b. Phase diagram constructed from the data
collected in both parallel and perpendicular configurations is shown in Fig. 3
in the main text.

Figure S3: Magnetic susceptibility $vs$ temperature of a bulk CrPS4 sample for
the magnetic field applied in a direction (a) parallel and (b) perpendicular
to the crystal $c$-axis. The Néel temperature marked as the maximum of the
curve is 38 K in the absence of magnetic field and shifts noticeably towards
lower temperatures for magnetic fields exceeding 1.1 T. An additional maximum
below the main peak observed at low field values is attributed to a transition
from A-AFM to canted AFM magnetic phase of the crystal.

##  S3 Mangeto-Photoluminescence

Figure S4: The degree of circular polarization ($DOP$, blue curves) of
photoluminescence of the 105 nm thick CrPS4 flake, determined for the spectral
range $1.353\leavevmode\nobreak\ \text{eV}\leq\hbar\omega\leq 1.374$ eV eV at
the temperature of 10 K in two configurations of magnetic field: (a) normal to
the crystal surface, and (b) in-plane of the crystal. Light emitted from the
sample is detected in the direction perpendicular to the surface of the flake.
Orange dashed curves represent the magnetization curves obtained from SQUID
magnetometry for bulk CrPS4 at the temperature of 10 K for the same
configurations of the magnetic field. Note that the $DOP$ signal in panel (b)
has a significantly smaller amplitude than in the case of panel (a), as in our
experimental configuration it is primarily sensitive to the out-of-plane
magnetization component. However, the use of a high numerical aperture lens to
collect the PL signal means that photons emitted at some angle to the normal
are also collected, making the measurement also sensitive to the in-plane
magnetization to some extent. Additionally, any out-of-plane tilt of the Cr3+
spins gives rise to the non-zero $DOP$ signal, even if the spins are mostly
aligned with the in-plane magnetic field.

##  S4 Raman spectroscopy

Figure S5: Evolution of selected transitions in the Raman scattering spectrum
of bulk CrPS4 with the increase of temperature from 5 K to 250 K: (a) line at
214.9 cm-1, (b) lines at 299.5 cm-1 (X2 in the main text) and 305.7 cm-1 (X3
in the main text), (c) line at 409.8 cm-1 (X4 in the main text). (d) The
amplitude, (e) spectral shift, and (f) linewidth of the transitions shown in
panels a-c as a function of temperature. The normalized intensities and
spectral shifts are relative to the values respective for each line at 5 K.
Néel temperature is marked with the line.

We register a series of Raman spectra as a function of temperature from 5 K up
to 250 K with the detection of two mutually perpendicular linear
polarizations, $H$ and $V$. An evolution of four selected spectra registered
in the $H$ polarization is shown in Suppl. Fig. S5a-c. To extract an
intensity, energy position, and linewidth a Lorentzian curve is fitted to each
of the lines. Suppl. Fig. S5d shows intensities of the lines normalized to the
intensity value at 5 K. The observed intensity dependencies are comparable for
the signal registered in both linear polarizations (not shown). A general
trend for all the lines is a decrease in intensity with the increasing
temperature. After an initial quick drop, the decrease in intensity lowers its
rate above the T${}_{N}=38$ K, marking the position of the magnetic phase
transition to a paramagnetic phase. The oscillations of intensity superimposed
on a general trend are discussed in the further part of the text.

A relative shift shown in Suppl. Fig. S5e represents a spectral position of a
given line as a function of temperature after subtraction of its position at 5
K. While all lines shift towards lower energies with the increase of
temperature in the full considered temperature range, anomalous behaviour is
observed for the X3 line below the TN. This is consistent with the theoretical
considerations predicting spectral a blueshift of the X3 when a probability of
other arrangements than the A-AFM increases. The X4 line (at around 409.8
cm-1) shows a sudden change in energy upon approaching the Néel temperature.
Suppl. Fig. S5f shows the linewidth of the respective lines. Each of the lines
broadens with the increase of the temperature.

The results shown in Suppl. Fig. S6 are obtained from the same set of data,
which served for the preparation of Fig. 7 discussed in the main text. Suppl.
Fig. S6a shows a degree of circular polarization of the X4 line as a function
of the magnetic field for three selected temperatures. As we can see, the
degree of polarization varies with the increase of the magnetic field in an
approximately linear way. The slopes of the linear dependencies fitted to the
data points (solid lines in Suppl. Fig. S6a) are plotted as a function of
temperature in Suppl. Fig. S6b. The position of the Néel temperature is
clearly marked on the dependence by a change of the value of the slopes, which
decrease in the range approximately from 5 K to 38 K and increase slightly
above the TN. Presented results further confirm the usefulness of the Raman
magnetospectroscopy in the studies of magnetic phase transitions in
antiferromagnetic semiconductors.

Figure S6: (a) Degree of circular polarization of the X4 line in the Raman
spectrum as a function of temperature and magnetic field for temperatures of 5
K, 36 K, and 90 K. (b) Slopes of the magnetic field induced change of the
degree of circular polarization of the X4 line as a function of temperature in
the range between 5 K and 150 K.

The results shown in Suppl. Fig. S7 are obtained from the same measurement as
those in Fig. 7 in the main text. The Suppl. Fig. S7a presents a plot of the
Raman shift of line X3 (305.7 cm-1 at 5 K) as a function of applied magnetic
field for selected temperatures. Also in the case of this line, the strongest
spectral shift induced by magnetic field is observed in the vicinity of Néel
temperature. We note, however, this shift occurs in an opposite direction with
respect to that observed for the X4 line discussed in Fig. 7 in the main text.
Consistently, the slopes plotted in Suppl. Fig. S7b follow the negative of the
magnetic susceptibility $\chi$.

Figure S7: (a) The spectral position of the X3 line in the Raman spectrum as a
function of magnetic field for temperatures of 6 K, 35 K, and 150 K. A shift
with the field is the most pronounced for the temperatures in the Néel
temperature vicinity. (b) Slopes of the magnetic field induced shifts of X3 as
a function of temperature in a range between 5 K and 150 K. We note that the
shift occurs in an opposite direction to the line X4 discussed in the main
text. The dependence exhibits the same shape as the magnetic susceptibility
curve determined for the parallel configuration in SQUID, but is flipped
vertically.

In Suppl. Fig. S8a we plot the intensity of three selected lines of the Raman
spectrum of CrPS4 as a function of temperature. The line between the
experimental points is approximated with a spline. As we can see, oscillations
of the intensity superimposed in the dependence are evidenced in the case of
two lines. Temperature positions of maxima of the dependence (indicated by red
dots) are plotted as a function of the order number of the maximum in Suppl.
Fig. S8b. We can see that the temperature position of the maximum is related
to the maximum number by a square root function, $T\sim\sqrt{n}$ (red points).
To further solidify this claim, we plot also a square of the position in
temperature as a function of the maximum number (blue points in Suppl. Fig.
S8b). We see that those follow a straight line. The observed result suggests
the presence of a kind of temperature-dependent, resonant effect in excitation
of the selected transitions in the Raman spectrum, but its interpretation
requires further analysis.

Figure S8: (a) The intensity of the three selected lines in the Raman spectrum
of CrPS4 as a function of temperature (positions of the lines at 5 K provided
in the legend). Oscillations of the intensity are evidenced in the case of two
lines. The maxima of intensity occur at the same temperature for these lines.
(b) The temperature corresponding to each intensity maximum from panel (a)
(red points) and the square of these values (blue points) plotted as a
function of the order number of the maximum. The plot proves that values of
the temperatures for which the maxima occur follow the square root of their
order number: $T\sim\sqrt{n}$.

##  S5 Theoretical modelling

Magnetocrystalline energy. We start our calculations by examining the
magnetocrystalline energy (MAE), including the spin-orbit coupling (SOC)
within the DFT+U approach. The MAE is defined here, as a difference between
the energies of spins parallel to hard (highest energy) and easy axes (lowest
energy) of magnetization. We rotate the spins from out-of-plane to in-plane
configurations with an angle step equal to $10^{o}$. The energy results are
presented in the polar plot shown in Suppl. Fig. S9a. The MAE is equal to 0.05
meV per Cr3+ ion (0.4 meV per a bulk supercell containing 8 Cr atoms), which
is a similar value like obtained for other 2D magnetic crystals such as MnPSe3
and NiPSe3 as reported recently.[51] The magnetic easy axis is tilted $10^{o}$
from the crystallographic c-axis, and the magnetic ground state is A-AFM
presented in Suppl. Fig. S9b. Due to the tiny difference between the energies
of the spins lying along the hard and easy axes, we assume only spin-polarized
case of the Cr3+ ions in further considerations.

Figure S9: (a) Polar plot of energy difference $\Delta$E between the
particular direction of the Cr spins and the lowest energy, that is A-AFM
configuration. (b) The magnetic ground state configuration is A-AFM phase with
the magnetic easy axis tilted 10o from the $c$-axis. (c) Schematic picture of
the exchange coupling strengths between 1NN (J1), 2NN (J2) and 3NN (J3), as
well as out-of-plane exchange coupling (J0).

Magnetic interactions. The exchange coupling constants have been derived from
the classical spin Heisenberg model in a similar manner, as described in
Ref.[6]. To achieve this, we fix the lattice parameters and position of the
atoms to the magnetic ground state (A-AFM), and we calculate the total energy
for five different magnetic arrangements within the spin-polarized case of
DFT+U assuming the same lattice parameters and position of the atoms. Next,
these energies are mapped onto the effective spin Heisenberg model, which
enables us to obtain the nearest neighbor (NN) exchange coupling constants
collected in Suppl. Table S1. The obtained values are in good agreement with
the literature.[38]

Jij (dij) |
---|---
$J_{1}$ ($d_{1}$=3.59Å) | 2.131 meV
$J_{2}$ ($d_{2}$=5.44Å) | 0.496 meV
$J_{3}$ ($d_{3}$=6.52Å) | 0.248 meV
$J_{o}$ ($d_{0}$=6.20Å) | -0.141 meV
Table S1: Exchange coupling strengths between 1NN (J1), 2NN (J2) and 3NN (J3)
nearest neighbors, as well as out-of-plane nearest neighbors (J0). Negative
and positive $J_{i}$ values denote FM and AFM couplings, respectively.  Figure
S10: Various spin arrangements of Cr3+ sites employed in this study. The
dotted lines indicate a supercell used (Cr8P8S32).

Magnetic phases. We consider all non-equivalent spin arrangements for the
employed supercell within DFT+U (see Suppl. Fig. S10). For each of the spin
configurations, the lattice parameters and position of atoms have been fully
optimized and collected in Suppl. Table S2. In the view of the calculations,
the A-AFM phase is the lowest-energy state followed closely by the FM phase.
The energy difference between these two phases is around 1 meV per each
magnetic ion. The next lowest phase, X-AFM lies 12 meV above the energy
minimum set by the A-AFM phase, whereas the highest-energy phase is G-AFM,
with the energy around 26 meV per magnetic ion.

Notably, going from the A-AFM to the Y-AFM phase (17 meV above A-AFM) involves
a transition between out-of-plane to in-plane antiferromagnetic arrangements.
Compared to a fully ferromagnetic system, the A-AFM phase represents the
activation of out-of-plane antiferromagnetism with in-plane ferromagnetism
preserved, while the opposite is true for Y-AFM – it remains ferromagnetic
with respect to the nearest out-of-plane neighbors, while the nearest-neighbor
interaction is antiferromagnetic in the in-plane direction (see Suppl. Fig.
S10). This observation makes it interesting to compare the properties of these
three systems: the ferromagnetic phase as a reference, and the two AFM phase
as prototype out-of-plane and in-plane antiferromagnets. Tracking the
temperature evolution of the simulated Raman modes allows us to partially
decouple the influence of structural and magnetic degrees of freedom in CrPS4.
This can be achieved by analyzing which vibration modes are sensitive to the
change in magnetic order, and which are influenced mostly by structural
changes in the form of simulated temperature increase.

Impact of structural changes determined in PXRD measurements on Raman modes.
Finally, we examine how the changes in the lattice parameters obtained by PXRD
measurements affect the optically active Raman modes. In this regard, we used
DFT+U approach and we took into account directly the changes of the lattice
parameters for a given temperature in accordance to the strategy described
below. First, geometry relaxation was performed for the 8-formula-unit
supercell with different magnetic arrangements presented in Suppl. Fig. S10
and the results of the optimized lattice parameters for each employed spin
configuration are collected in the Suppl. Table S2 and Suppl. Fig. S11.

The data collected in Suppl. Table S2 for $a$ and $b$ lattice parameters of
CrPS4 are plotted in Suppl. Fig. S11. The plot shows values of the $a$ and $b$
lattice parameters calculated as a function of a relative energy difference
$\Delta E$ per Cr atom with respect to the magnetic A-AFM groundstate (bottom
horizontal axis). Corresponding magnetic arrangements for respective $\Delta
E$ values are indicated in the top horizontal axis of the plot. Taking into
account that the probability of deviation from A-AFM state and thus of
transition to a higher energy magnetic arrangement increases with the
increasing temperature, the calculation predicts in an accurate way the
evolution of the $a$ and $b$ lattice parameters with the temperature observed
in the XRD experiment reported in the main text.

Magnetic phase | $\Delta E$ | a (Å) | b (Å) | c (Å) |  $\alpha$ (deg) |  $\beta$ (deg) |  $\gamma$ (deg)
---|---|---|---|---|---|---|---
(I) | A-AFM ($d_{1}$=3.589 Å) | 0 | 10.885 | 7.321 | 6.202 | 90.00 | 92.38 | 90.00
FM ($d_{1}$=3.590 Å) | 0.98 | 10.884 | 7.322 | 6.202 | 90.00 | 92.39 | 90.00
(II) | X-AFM ($d_{1}$=3.588 Å) | 12.08 | 10.902 | 7.287 | 6.217 | 90.00 | 92.19 | 90.00
X*-AFM ($d_{1}$=3.587 Å) | 12.09 | 10.901 | 7.287 | 6.217 | 90.00 | 92.19 | 90.00
E-AFM ($d_{1}$=3.572 Å) | 13.86 | 10.897 | 7.298 | 6.208 | 90.00 | 92.26 | 90.00
Y*-AFM ($d_{1}$=3.573 Å) | 17.09 | 10.894 | 7.299 | 6.201 | 90.00 | 92.30 | 90.00
Y-AFM ($d_{1}$=3.573 Å) | 17.15 | 10.895 | 7.299 | 6.202 | 90.00 | 92.30 | 90.00
(III) | C-AFM ($d_{1}$=3.573 Å) | 25.09 | 10.908 | 7.281 | 6.207 | 90.00 | 92.16 | 90.00
G-AFM ($d_{1}$=3.574 Å) | 26.14 | 10.908 | 7.281 | 6.21 | 90.00 | 92.17 | 90.00

Table S2: Optimal lattice parameters for various magnetic arrangements (see
Suppl. Fig. S10) determined by DFT+U calculations. Energy difference $\Delta
E$ calculated with respect to the magnetic ground state is given in meV per
Cr3+ ion. The (I), (II) and (III) indicate the type of the in-plane coupling
between the Cr3+ spins: FM, mixed AFM and FM, and AFM couplings (only the
first and second nearest neighbour considered), respectively. Figure S11:
Calculated optimal lattice parameters $a$ and $b$ of CrPS4 as a function of
relative energy difference of various magnetic arrangements of Cr3+ spins
(indicated in the top horizontal axis) with respect to the magnetic A-AFM
groundstate (bottom horizontal axis). The plot shows the data collected in
Suppl. Table S2. The presented results indicate that the lattice parameter $a$
and $b$ respectively enlarges and diminishes with the increasing energy per Cr
atom (hence with the increasing temperature).

Then, temperature effects are taken into account by scaling the $a,b,c$, and
$\beta$ parameters according to the data obtained from temperature-dependent
PXRD measurements. Each arrangement is initially allowed to relax to its
ground state, before applying a dimensionless temperature factor in the form
of a${}^{i}_{T}$/a${}^{i}_{20K}$, where a${}^{i}_{T}$ is the value lattice
parameter ai (where i=1,2,3, a1=a, a2=b, a3=c) at temperature T, and
a${}^{i}_{20K}$ is the value of parameter ai at 20 K, the lowest temperature
considered in the PXRD measurements. In other words, the temperature is taken
indirectly into consideration, by the factor defined by the lattice
parameters. Note that temperature evolution of the optically active Raman
modes for the three prototype magnetic phases presented in Suppl. Fig. S12
exhibits a qualitatively similar trend for the full temperature range. Thus,
the evolution of Raman shifts (Raman shift denotes here a difference between
the frequencies at the given temperature and ground state (lowest temperature
for the same magnetic arrangement) is rather independent of magnetic ordering,
and reflects structural changes in lattice parameters. A direct comparison
between the corresponding frequencies of the Raman modes in CrPS4 for various
magnetic phases is collected in Suppl. Table S3.

Figure S12: The calculated impact of the temperature-induced lattice parameters variation on the frequency of Raman modes (with a spectral position in cm-1 at 5 K provided in the legend) of CrPS4. Raman shift denotes the difference between the phonon frequencies calculated for a particular temperature and phonon frequencies obtained for the optimized system (e.g., for FM phase: (X${}^{FM}_{T}$ \- X${}^{FM}_{20K}$)). Note, that for each magnetic phase, a different optimized system might be obtained. Table S3: Experimental and calculated Raman mode frequencies of bulk CrPS4. Mode frequencies are calculated for A-AFM ground state, as well as for FM, Y*-AFM and G-AFM magnetic arrangements. A shift with respect to the A-AFM ground state is provided for FM, Y*-AFM and G-AFM arrangements. Blue- and red-shifted modes are indicated by the font color. Bold formatting of the text highlights the modes with a significant (above 0.1 cm-1) shift from A-AFM. The first column provides information of the irreducible representation of the mode. The X1, X2, X3 and X4 transitions considered in the main text are indicated. |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
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