# Magneto-optical anisotropies of 2D antiferromagnetic MPX3 from first
principles

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  Paulo E. Faria
Junior  Institute of Theoretical Physics, University of Regensburg, 93040
Regensburg, Germany  Tomasz Wozniak  Department of Semiconductor Materials
Engineering Faculty of Fundamental Problems of Technology, Wrocław University
of Science and Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland
Pawel Scharoch  Department of Semiconductor Materials Engineering Faculty of
Fundamental Problems of Technology, Wrocław University of Science and
Technology Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland  Jaroslav Fabian
Institute of Theoretical Physics, University of Regensburg, 93040 Regensburg,
Germany  Magdalena Birowska  Institute of Theoretical Physics, Faculty of
Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland
<EMAIL_ADDRESS>Institute of Theoretical Physics, Faculty of
Physics, University of Warsaw, Pasteura 5, 02-093 Warsaw, Poland

(August 28, 2024)

###### Abstract

Here we systematically investigate the impact of the spin direction on the
electronic and optical properties of transition metal phosphorus
trichalcogenides (MPX3, M=Mn, Ni, Fe; X=S, Se) exhibiting various
antiferromagnetic arrangement within the 2D limit. Our analysis based on the
density functional theory and versatile formalism of Bethe-Salpeter equation
reveals larger exciton binding energies for MPS3 (up to 1.1 eV in air) than
MPSe3 (up to 0.8 eV in air), exceeding the values of transition metal
dichalcogenides (TMDs). For the (Mn,Fe)PX3 we determine the optically active
band edge transitions, revealing that they are sensitive to in-plane magnetic
order, irrespective of the type of chalcogen atom. We predict the anistropic
effective masses and the type of linear polarization as an important
fingerprints for sensing the type of magnetic AFM arrangements. Furthermore,
we identify the spin-orientation-dependent features such as the valley
splitting, the effective mass of holes, and the exciton binding energy. In
particular, we demonstrate that for MnPX3 (X=S, Se) a pair of non equivalent
K+ and K- points exists yielding the valley splittings that strongly depend on
the direction of AFM aligned spins. Notably, for the out-of-plane direction of
spins, two distinct peaks are expected to be visible below the absorption
onset, whereas one peak should emerge for the in-plane configuration of spins.
These spin-dependent features provide an insight into spin flop transitions of
2D materials. Finally, we propose a strategy how the spin valley polarization
can be realized in 2D AFM within honeycomb lattice.

first keyword, second keyword, third keyword

##  I Introduction

The subtle interplay between the spin, charge, orbital, and lattice degrees of
freedom driven by the electron correlation is one of the key aspects in
condensed matter physics behind novel electronic phases of matter and
intriguing physical phenomena. In particular, the electronic properties can be
modified whenever the spin direction is altered, as spin-orbit coupling (SOC)
depends on the spin direction. For instance, spin-valley coupling serves as
the fundamental mechanism in optically-controlled valley polarization Zeng
_et al._ (2012), spin-Hall and valley-Hall effects Xiao  _et al._ (2012).
Contrast to most of the findings in non-magnetic 2D materials Tombros  _et
al._ (2007); van ’t Erve  _et al._ (2012); Kamalakar  _et al._ (2015), where
pseudo-spins are involved, 2D magnets exhibit active carrier spins, enabling
studies of magnetism in reduced dimensions Gibertini  _et al._ (2019).

Unlike ferromagnets (FMs), the antiferromagnets (AFMs) are commonly found in
nature and they are permitted in each magnetic symmetry group, however they
are less utilized than FMs Nemec  _et al._ (2018). Currently, AFM materials
are considered as promising candidates for future spintronic applications, due
to unique properties including insensitivity to external magnetic fields, lack
of stray fields, and ultrafast spin dynamics in the terahertz regime Zhang
_et al._ (2019). The spontaneously long-range ordered of microscopic magnetic
moments, resulting in zero net magnetization, which makes the AFMs insensitive
to external magnetic fields. Thus, a control of AFM state requires very high
magnetic fields and demands unconventional means of detection Nemec  _et al._
(2018).

Many properties such as optical, electronic, and vibrational, rely on the
magnetic orderingBirowska  _et al._ (2021); Autieri  _et al._ (2022); Linhart
_et al._ (2023), as well as magnetic moment orientationLiu  _et al._ (2019).
In particular, a giant impact of the spin direction on the band structure,
have been recently demonstrated for 2D ferromagnetic CrI3 material Jiang  _et
al._ (2018). Although, the spin-direction properties are reported for
conventional and layered FMs Sander (2004); Grünberg (2001), the research on
2D AFM materials is very limited and scarce Nemec  _et al._ (2018); Gibertini
_et al._ (2019); Martí  _et al._ (2015); Rahman  _et al._ (2021).

One of essential feature for manipulating AFM state is the magnetocrystalline
anisotropy (MAE) which is evident in spin flip or spin flop transitions. The
latter one requires relatively weak MAE and it is reported in literature for
AFM compounds exhibiting various electronic states such as topological
insulators Sass  _et al._ (2020), layered materials Balamurugan  _et al._
(2014), and conventional semiconductors Machado  _et al._ (2017). In addition,
the experimental techniques including indirect means of magnetic phases such
as anisotropic magnetoresistance (AMR), anomalous Hall effect (AHE) or second
harmonic generation (SGH)Jungwirth  _et al._ (2016); Nemec  _et al._ (2018);
Fina  _et al._ (2014), are employed to detect the AFM orientation in 2D
materials. Interestingly, the magnetic phase transition of controlled
anisotropic phenomenon in layered magnets provides crucial understanding of
fundamental magnetism in reduced dimensions.

The present work is motivated by the lack of systematic studies regarding the
impact of AFM orientation of magnetic moments on the optoelectronic properties
of 2D materials. Hence, in this paper we put particular attention to pinpoint
the magnetic fingerprints in indirect properties that can engineer the AFM
ordering. We conduct a theoretical analysis based on density functional theory
(DFT) and effective Bethe-Salpeter equation (BSE) to identify various spin
direction features, which can be utilised in spin-processing functionalities.
This study focuses on spin angle evolution of the effective masses of
carriers, optically active band edge transitions, exciton binding energies,
which remain almost unexplored in the context of van der Waals (vdW) AFM
crystals.

Here, we present a systematic study of the impact of spin orientation on the
electronic and optical properties of series of the monolayers of MPX3, where
M= Mn, Ni, Fe, and X= S, Se, assuming collinear arrangement of the magnetic
moments. The results are presented as following: first we examine the magnetic
ground state, determining easy and hard axes of magnetization at the level of
PBE+U+SOC approach. Next, we consider the electronic features such as band
extrema, effective masses, valley splitting and excitonic properties including
the excitonic binding energy. Although, the electronic properties have been
widely reported for particular spin arrangement and direction, the impact of
the orientation of the collinearly ordered spins on opto-electronic properties
MPX3 are largely missing. Finally, band edge excitons have been systematically
predicted for this class of materials for the first time.

##  II Computational Details

The calculations were performed in the framework of density functional theory
(DFT) using the generalized gradient approximation within the PBE flavor Paier
_et al._ (2005), as implemented in VASP software Kresse and Furthmüller
(1996). The ion–electron interactions were described by the projector
augmented wave (PAW) method Holzwarth  _et al._ (2001). Plane-wave basis
cutoff and $\Gamma$ centered Monkhorst-Pack Monkhorst and Pack (1976) k-point
grid were set to 500 eV and $10\times 6\times 1$, respectively. A Gaussian
smearing of 0.05 eV was employed for the Brillouin zone (BZ) integration. The
interlayer vdW forces were treated within Grimme scheme using D3 correction
Grimme  _et al._ (2010). A vacuum of the thickness equal to 20 Å was added to
mimic the isolated monolayer. Most of the results were obtained using PBE+U
method based on Dudarev’s approach, with the effective on-site Hubbard U
parameter (Ueff =U-J, where J=1eV) assumed for $3d$ orbitals. Note that Ueff
is hereafter denoted as U. To check its impact on various properties two
values of U= 3, 5 eV were employed. The SOC within the non-collinear treatment
of magnetism was taken into account on the top of the PBE+U scheme. The
position of the atoms and unit cell were fully optimized within the PBE+U
approach. In order to predict the magnetic easy and hard axis, the spins of
the magnetic ions have been rotated from out-of plane to in-plane directions
with polar angle step $\Delta\Theta$=150. For every $\Theta$ the atomic
positions and the lattice parameters have been fully optimized. The effective
masses of carriers at the band edges were examined using finite difference
method Fonari and Sutton (2015). The direct interband momentum matrix elements
were computed from the wave function derivatives using Density Functional
Perturbation Theory (DFPT)Gajdoš  _et al._ (2006) in order to determine the
optically active transitions, as discussed in Woźniak  _et al._ (2020). The
macroscopic optical dielectric constants frequency $\epsilon_{\infty}$ were
obtained using DFTP in the independent particle approach with (DFT-TD) and
without local field effects (IP) Gajdoš  _et al._ (2006)

Exciton calculations were performed within the effective BSE formalismAlbrecht
_et al._ (1998); Rohlfing and Louie (2000); Zollner  _et al._ (2019, 2020,
2023); Birowska  _et al._ (2021) using the effective masses and dielectric
constants obtained from the DFT calculations. We solve the effective BSE
numerically using the parameters given in Ref.Birowska  _et al._ (2021).

Figure 1:  (a) The rotation angle $\Theta$ of collinearly aligned AFM spins
directed towards the (001) plane. (b-d) Polar plots of energy difference
$\Delta E$ between the particular direction of the spins and the lowest energy
configuration for Hubbard U=5 eV. (e) The magnetocrystalline energy (MAE) is
defined here, as a difference between the energies of spins parallel to hard
(highest energy) and easy axes (lowest energy) of magnetization. (f-h)
Structural arrangement of the spins exhibited by the magnetic ground state for
all employed systems. The direction of the spins are parallel to the easy axis
/ plane of magnetization. The $\Delta E$ and the MAE are given in meV per
magnetic ion.

##  III Results

###  III.1 Magnetic ground state $\&$ magnetic easy axis

First, we examine the impact of the rotation of spins on the energy profile to
determine the magnetic ground state and magnetic easy/hard axes for all
employed MPX3 systems. We assume a collinear alignment of the spins and angle
of rotation $\Theta$ (see Fig. 1 (a)) varied from the out-of plane to in-plane
configuration. The rotation of the spins within the basal plane yields up to
two orders of magnitude smaller energy changes compared to out-of plane ones,
hence in-plane rotations are not further considered.

The magnetic ground state exhibits antiferromagnetic Neel (AFM-N) and
antiferromagnetic zigzag (AFM-z) arrangement of the spins for Mn and M=Ni, Fe,
respectively (see Fig. 1), in line with other reports Sivadas  _et al._
(2015); Birowska  _et al._ (2021); Autieri  _et al._ (2022); Budniak  _et al._
(2022). The MPX3 materials are reported to be robust antiferromagnets Autieri
_et al._ (2022) and even high concentration of the substitutional dopants
could not alter the magnetic ordering of the host Basnet  _et al._ (2022);
Autieri  _et al._ (2022). Additionally, changing the angle of spin alignment
$\Theta$ requires at least one order of magnitude lower energy (tens of meV
per magnetic ion) than change of the AFM ordering (at least tens of meV per
magnetic ion). The computed energy difference ($\Delta E$) between the out-of
plane and in-plane directions can be one order of magnitude greater for Se
than S compounds (see Fig. 1 (b-d)), which is expected for heavier atoms,
exhibiting larger SOC coupling Basnet  _et al._ (2022). Surprisingly, FePS3
exhibits a larger MAE than FePSe3 (see Fig 1(e)). The value of U does affect
neither the type of AFM ordering nor the direction of the magnetic easy axis.
However, the $\Delta$E depends on the Hubbard U parameters. Namely, the
smaller values are generally obtained for larger U, except for FePS3 monolayer
(see Fig. S2). For the employed materials, an increase in the effective U
resulted in larger lattice constants by up to 1.5 % and larger magnetic
moments (see Table S2).

Let us now consider the easy axes of magnetization, which are presented in
Fig. 1(f, g) for all employed structures, with the hard axes predicted to be
orthogonal to the corresponding easy axis/plane. In particular, for FePX3 the
magnetic easy axis points along the $c$ crystallographic direction as reported
in previous studies Olsen (2021); Wildes  _et al._ (2015); Kurosawa  _et al._
(1983); Koo  _et al._ (2021); Sivadas  _et al._ (2015, 2015); Wiedenmann  _et
al._ (1981). Hence, this kind of systems are considered to be Ising-type
antiferromagnets, with a strong uniaxial magnetic anisotropy.

For the MnPX3 compounds, we predict that the easy plane coincides with
monolayer plane, unlike the recent experimental report for MnPS3, which has
demonstrated that the spins are slightly tilted from the $c$-axis Ressouche
_et al._ (2010). The direction of the ordered spins for MPX3 compounds result
from the interplay between the single-ion anisotropy and magnetic dipolar
interactions (MDIs), as already discussed in a recent publication Basnet  _et
al._ (2022). In addition, the spin-orbit splitting is negligible for MnPS3,
which manifests itself as the lowest MAE for all employed structures (see Fig.
1(b,e)). Thus, the MDI, generally regarded to be weakJohnston (2016), might be
decisive in determining the direction of the MnPS3 spins. Note, that the MDIs
are not accounted for in our PBE+U+SOC approach, and will be considered
elsewhere. On the other hand, in the case of the MnPSe3, the MDIs could be
neglected due to larger lattice parameters and SOC. Hence, the magnetic easy
axis of MnPSe3 coincides with the monolayer plane, in agreement with other
reports in the literatureSen and Chouhan (2020); Wiedenmann  _et al._ (1981);
Kurosawa  _et al._ (1983), and confirmed by spin flop transition reported upon
the nonmagnetic substitution in MnPS3 thin films Basnet  _et al._ (2022).
Regarding NiPS3, the in-plane position of the magnetic moments are preferable
and the magnetic ordering can be described by the XY Heisenberg
HamiltonianWildes  _et al._ (2015); Olsen (2021); Hwangbo  _et al._ (2021);
Belvin  _et al._ (2021). Our results predict the easy axis of magnetization
within the monolayer frame, without any deviation in the z direction as
reported recently for monolayerKim  _et al._ (2019). However, we do not
exclude the possibility that the inclusion of the MDIs interaction can
facilitate the rotation of the spins towards the out of the plane direction.
For the case of NiPSe3, we determined the easy axis of magnetization to be in
out-of-plane direction in line with recent report Sun  _et al._ (2023).
Similar results regarding the easy axis/plane of magnetization have been
reported for the corresponding bulk systems Basnet  _et al._ (2022).

Figure 2:  Spin-orientation electronic properties. (a)Band extrema for all employed monolayers as a function of spin direction (color scale). (b) Enhancement of the electronic band gaps $\Delta E_{gap}$ for particular spin directions in respect to out-of-plane spins. (c) Valley splitting $\Delta^{VB}$, $\Delta^{CB}$ are defined $\Delta^{VB}$=E${}^{VB}_{K_{-}}-$E${}^{VB}_{K+}$, $\Delta^{CB}$=E${}^{CB}_{K_{-}}-$E${}^{CB}_{K+}$, respectively for Mn and Fe contained monolayers, given in meV. Note, that for the clarity of picture regarding MnPS3, the $-\Delta^{CB}$ is plotted. (d) In-plane components of effective mass tensor for electrons (positive masses) and holes (negative masses) of MnPX3 in respect to spin direction and for two non-equivalent $\pm K$ valleys. Table 1:  Spin-dependent electronic features for direct band edge transition occurring at $\pm K$ valleys (VBM$\rightarrow$CBM) in the presence of SOC (PBE+U(5eV)+SOC approach). The valley splitting of the VBM ($\Delta^{VB}$) and CBM ($\Delta^{CB}$) are defined as $\Delta^{VB}$= E${}^{VB}_{K_{-}}-$E${}^{VB}_{K+}$ and $\Delta^{CB}$ = E${}^{CB}_{K_{-}}-$E${}^{CB}_{K+}$), respectively. The spin degeneracy of the band edges $\delta^{VB}$ and $\delta^{CB}$ are calculated as $\delta^{VB}$=| E${}_{\uparrow}^{VB}-$E${}_{\downarrow}^{VB}|$ and $\delta^{CB}$=| E${}_{\uparrow}^{CB}-$E${}_{\downarrow}^{CB}|$, respectively. We refer intensity (Inten.) to the oscillator strength of of the band edge transitions. The optically active transitions coupled to linear polarization of light (x,y,z) are collected in the last column. The intensity is given by $\left|\frac{\hbar}{m_{e}}\hat{e}\cdot\vec{p}_{cv}\right|^{2}$ in which $\hat{e}=\\{\hat{x},\hat{y},\hat{z}\\}$ is the light polarization (pol.) and $\vec{p}_{cv}$ is the matrix element between CBM and VBM.  MPX3 (magn. state) | spin direction ($\Theta$) | K+/K- ($\Delta^{VB}$, $\Delta^{CB}$) | Spin Deg. ($\delta^{VB}$, $\delta^{CB}$) | Hex. symm. | (pol.) Inten. [$(\textrm{eV\AA})^{2}$]
---|---|---|---|---|---
MnPS3 (AFM-N)) | 00 |  $\checkmark$ (-6.4, -4.4) | X | $\checkmark$ | (z) 0.59
| 900 | X (0, 0) | X | $\checkmark$ | (z) 0.62
MnPSe3 (AFM-N) | 00 |  $\checkmark$ (-17.7, 33.1) | X | $\checkmark$ | (z) 0.13
| 900 | X (0, 0) | X | $\checkmark$ | (z) 0.18; (x,y) 0.05
FePS3 (AFM-z) | 00 | X |  $\checkmark$ (1.8, 9.1) | broken | 0.71 (y)
| 900 | X |  $\checkmark$ (1.5, 7.4) | broken | 0.73 (y)
FePSe3 (AFM-z) | 00 | X |  $\checkmark$ (7.4, 3.1) | broken | (y) 0.27; (x) 0.04
| 900 | X |  $\checkmark$ (9.3, 5) | broken | (y) 0.29; (x) 0.02

###  III.2 Electronic properties

#### electronic band gaps

Our results reveal that the structures containing Mn and Fe exhibit direct
band gaps located at K high symmetry point (see Fig. 2 (a)), in line with
previous reports Birowska  _et al._ (2021); Budniak  _et al._ (2022); Geraffy
_et al._ (2022), while the NiPS3 system is an indirect semiconductor confirmed
by experiments Hwangbo  _et al._ (2021). In particular, for FePS3 and NiPSe3
there is a strong contribution of the $3d$ states to the bands in vicinity of
Fermi level. Hence, for these systems, significant changes of the position of
the band extrema (VBM, CBM) and their curvatures upon the Hubbard U parameter
are visible (see Tables S2 and S3 and discussions therein).

#### U dependence

For all employed compounds, except MnPX3, the U parameter significantly
affects the curvature of the conduction bands as reported previously Budniak
_et al._ (2022), and hence the effective masses of holes. Hence, all features
are examined assuming two Hubbard U parameters U=3, 5 eV). In particular, the
large value of the effective masses of electrons exhibited for Ni contained
systems, indicating a flat character of the conduction bands, which can be
further enhanced by adopting larger values of U (see Table S3). By applying
the Hubbard U to 3d states, one can shift the 3d states away from the Fermi
level. In particular, for MnPX3 structures the conduction bands are mainly
build from p-states, and further shifting 3d states do no affect the band
curvature nor the band gap value, whereas for FePX3 compounds the conduction
bands consist mainly of 3d states, hence further enlargement of U strongly
impacts electronic features around band edges. Additionally, our results
corroborate the previous reports on strong dependence of Hubbard U on
fundamental energy band gap (see Table S2) Budniak  _et al._ (2022).

#### Effective masses

For all of employed monolayers, the heaviest effective mass is obtained for
NiPX3, whereas the lightest one is demonstrated for MnPSe3 (see Table S3 and
Fig. S3). Generally, the lower effective masses are exhibited by the electrons
than holes, and for Se contained structures than corresponding S systems (by
about 30$\%$), except for NiPX3 structures where these trends are opposite.
Additionally, the anisotropic in-plane components of effective masses
(m${}_{1}\neq$ m2) are exhibited by monolayer with AFM-z ground state, whereas
for MnPX3 monolayers with AFM-N ordering, the in-plane components are
isotropic (m1=m2). Note, that the AFM-z magnetic arrangement breaks the
hexagonal symmetry, as it consist of spins ferromagnetically aligned along
zigzag chain (see Fig 1(g)), whereas the AFM-N type of magnetic order is
commensurate with the structural symmetry. Hence, the anisotropic effective
masses, and thus, anisotropic transport properties can be regarded as a
magnetic marker distinguishing different type of antiferromagnetic ordering
within hexagonal lattice. In addition, the inclusion of the SOC does not
affect the curvature of examined systems, except for MnPSe3 where slight
changes are shown for K+ and K- valleys (see Table S4).

#### Spin-orientation-dependent electronic features in MPX3 (M=Mn, Fe; X=S,Se)

Now, let us make closer inspection on the band edges of MLs exhibiting direct
transitions. The electronic features are collected in Table S3. Note, that in
the absence of the SOC, the band extrema are doubly degenerated for all
employed systems. The presence of SOC in the structures containing Mn
preserves the spin degeneracy of the band extrema (VBM, CBM), however causing
the valley splitting at +K and -K, which is expected for the honeycomb lattice
with AFM-N arrangements of the spins with SOC includedLi  _et al._ (2013).
Namely, the SOC preserves the spin degeneracy but leads to a renormalization
of the valleys (+K and -K are not equivalent) for AFM-N magnetic ground state.
The size of the valley splitting ($\Delta$) depend on the chalcogen atoms
(larger values for Se atoms), Hubbard U, and spin directions (see Fig. 2 and
Tables 1, S6). In particular, the largest valley splitting is attained for the
out-of-plane direction of the spins ($\Delta^{CB}=39$ meV for MnPSe3 and
U=3eV) and lower value of U as presented in Fig. 2(c), whereas no polarity of
the valleys are observed for the in-plane direction of the spins. Namely, the
rotation of the spins towards out-of-plane directions results in enhancement
of the valley splitting. Hence, the band gap of MnPSe3 can be changed by up to
35 meV as presented in Fig. 2(b, c), whereas for the rest of the compounds the
minor changes are observed (see Fig. 2(b)). Additionally, the effective masses
of holes for MnPSe3 are sensitive to the direction of the spins as presented
in Fig. 2(d). Interestingly, in the case of MnPS3, the band gap changes its
character from direct to indirect when the spins are rotated from an out-of-
plane to an in-plane directions.

Figure 3:  (a) Large exciton binding energies obtained for all considered
monolayers with magnetic ground state with respect to the dielectric screening
of the surroundings. The values for air/vacuum, SiO2 Berkelbach  _et al._
(2013) substrate, and hBN encapsulation Stier  _et al._ (2018) are denoted by
vertical lines. (b) Evolution of the EB with respect to the spin direction for
two values of Hubbard U (3 and 5 eV). (c) The energy difference between the EB
calculated for $-K$ and $+K$ valleys as a function of the spin direction angle
$\Theta$ (within DFT+U=5eV and DFT-TD approaches), for hBN encapsulated and
freestanding layers.

Regarding the monolayer of FePX3, the presence of SOC and local inversion
symmetry breaking in FeX6 octahedra lift the degeneracy of the band extrema
(few meV), irrespective to the spin direction (see column 4 in Table 1). In
particular, larger spin polarization $\delta$ is obtained for CBM than VBM
($\delta^{CB}$= 9.1 meV, $\delta^{VB}$= 1.8 meV), and for higher values of U
for FePS3. In contrast to the FePS3, the opposite trends are obtained for
FePSe3 (for the details see Table S6). Hence, the spin splitting of the bands
is obtained for hexagonal lattices exhibiting in-plane structural anisotropy,
as reported for MnPS3 (see Fig. 5(d) in Geraffy  _et al._ (2022)). In this
case the spin polarization occurred, alongside with renormalization of the
+K/-K valleys as expected for AFM-N. In addition, one of the consequences of
the local inversion symmetry breaking exploited in Fe monolayers is a lattice
distortion reflected in the ratio of the lattice parameters deviated from
hexagonal symmetry ($b/a\neq\sqrt{3}$) Geraffy  _et al._ (2022); Ghosh  _et
al._ (2023).

###  III.3 Excitonic properties

#### Dielectric properties

We start our discussion by analyzing the dielectric properties of the employed
systems. Generally, the in-plane components of static dielectric constants
($\varepsilon_{xx}$, $\varepsilon_{yy}$) are isotropic for all studied
systems, except for monolayers of (Ni,Fe)PX3, where they differ by up to 4$\%$
(see SI and Tabs. S7, S8). Due to the fact that dielectric tensor is well
defined for the bulk materials, we calculate the 2D polarizability
$\chi_{\parallel}$ (dielectric screening length, for the details see SM). Our
results reveal that the dielectric screening length increases as the atomic
number of the metal also increases
$\chi_{\parallel}^{Mn}>\chi_{\parallel}^{Fe}>\chi_{\parallel}^{Ni}$,
exhibiting the smallest and highest values equal to 3.25 Å (MnPSe3, U=5 eV)
and 6.13 Å (NiPSe3, U=5eV), respectively. In addition, the systems containing
Se exhibit $\sim$40$\%$ larger screening properties than their corresponding S
systems. The screening properties of MPX3 compounds are generally smaller than
for other vdW structures such as MoSe2 (8.23 Å). Note that for the (Ni,Fe)PX3
systems, the dielectric screening lengths strongly depend on the Hubbard U. In
particular, the larger values of $\chi_{\parallel}$ are attained for the
smaller Hubbard U, which correlates with the strong impact of the U on the DOS
(see Fig. S2 (b)).

#### Band edge excitons

Let us now examine the band edge excitons using the relevant information from
DFT ($m_{eff}$, $\chi_{\parallel}$) combined with the effective BSEAlbrecht
_et al._ (1998); Rohlfing and Louie (2000). We show the exciton binding
energies (EB) in Fig. 3(a) (for the details of calculations see Ref.Birowska
_et al._ (2021)). All employed structures exhibit large EB exceeding 1 eV and
0.8 eV for bare monolayers of MPS3 and MPSe3, respectively, outperforming the
values in TMDCs Birowska  _et al._ (2021) (the exact values are collected in
Tables S9, S10). The excitons have been recently experimentally reported for
few layers of NiPS3 Kang  _et al._ (2020); Hwangbo  _et al._ (2021); Wang  _et
al._ (2021), and FePS3 Geraffy  _et al._ (2022), however, their origin is
still under a hot debate. Generally the EB are larger by about 30$\%$ for S
contained structures than corresponding Se compounds, mainly due to larger
effective masses and smaller dielectric screening for MPS3. The EB decreases
as the effective dielectric constant of the environment increases, preserving
the same trend for all monolayers. This can be explained within the
oversimplified exciton picture (hydrogen model) for which EB is proportional
to effective mass and inversely proportional to the square of dielectric
screening.

#### Optical transitions and selection rules

Besides the binding energy, it is also relevant to determine the selection
rules of the direct band edge transitions, which are summarized in Table S3.
All of the transition are optically allowed (non-zero oscillator strength)
exhibiting linear polarization of light. In particular, for monolayers with
AFM-N phase (MnPX3) the polarization of light is along $z$ direction, whereas
for the AFM-z (FePX3) pointing along $y$ direction. Interestingly, similar
conclusions are observed assuming various AFM metastable phases within the
same magnetic material Birowska  _et al._ (2021). Therefore, the polarization
of light is sensitive to magnetic order irrespective of the type of transition
metal and chalcogen atoms. This is in line with recently reported linear
polarization of the sharp emission, that aligned perpendicular to the spin
orientation Wang  _et al._ (2021). Hence, the polarization of light might be a
tool to distinguish the type of AFM ordering. All of these direct band edge
excitons are optically active transitions and couple to $z$-polarized light.
Comparing with the widely studied TMDCs, these transitions in MPX3 systems
have an intensity two orders of magnitude smaller than the bright (A and B)
transitions but have comparable intensity to the dark (D) transitionsWoźniak
_et al._ (2020); Faria Junior  _et al._ (2022).

Figure 4:  Schematic diagrams (U=3 eV) of direct transitions for (a, b) MnPSe3
and (c, d) MnPS3 exhibiting in-plane and out-of-plane directions of the AFM
aligned spins, as schematically denoted at the bottom of the picture (g, h).
Schematic absorption onsets (c, f) with depicted in coloured exciton peaks A,B
and C. Namely, green (A) red (B) and blue (C), peaks indicate band edge
excitons for in-plane and out-of-plane spin configurations of AFM aligned
spins at +K and -K valleys, respectively. Black arrows close to bands indicate
spin degenerated bands. The position of the peaks are in order but not in
scale. All of the presented direct transition are coupled to "z" polarized
light with the intensities listed in Table S3. (i) The first BZ with labeled
high symmetry k-points.

#### Spin-orientation-dependent excitonic features in MnPX3

Interestingly, the excitonic properties are sensitive to the direction of the
spins. Owing to the spin direction dependent effective masses observed in
MnPX3 presented in Fig. 2 (d), we discuss below only these type of monolayers.
The effect of the spin direction on EB is on the order of 1-2 meV in MnPS3 but
on the order of 10 meV in MnPSe3 monolayer, as presented in Fig. 3 (b),
irrespective of the value used for the Hubbard U parameter. Furthermore,
opposite K-valleys (at $+$K and $-$K) show different effective masses and,
consequently, different values of EB, as shown in Fig. 3(c). Hence, the EB
depends on the orientation of the AFM aligned spins. Particularly, due to the
sizeable valley-dependent curvature of the VBM (see Fig. 2 (d)) observed for
the MnPSe3, the EB of excitons at $\pm K$ valleys differ by up to
E${}_{B}^{-}$ $-$ E${}_{B}^{+}$ = 12 meV (for MnPSe3 in vacuum).

Combining the information of the optical selection rules and exciton binding
energies, we present in Fig.S8 the schematic diagram of optically active band
edge transition considering the in-plane and out-of the plane direction of
spins. Our calculations reveal that the emissions from the opposite $+K$ and
$-K$ valleys are energetically below the onset of absorption spectrum. For the
in-plane directions of the spins, the energy difference between $+K$ and $-K$
is rather small (separated about 1 meV and 0.1 meV for MnPSe3 and MnPS3,
respectively, as shown in Figs. S8(a,d)). On the other hand, for out-of-plane
spins, the exciton emission from $\pm$K valleys are separated by 42 meV and 6
meV for MnPSe3 and MnPS3, respectively. The separation of the exciton peaks at
$\pm K$ valleys (see difference between the position of the B and C peaks in
Fig. S8) is calculated as:

$\displaystyle\Delta_{B}$
$\displaystyle=\left[E_{g}(K-)-E_{B}(K-)\right]-\left[E_{g}(K+)-E_{B}(K+)\right]$
$\displaystyle=\Delta E_{B}+\Delta^{VB}-\Delta^{CB}\,.$ (1)

where $\Delta E_{B}=E_{B}(K-)-E_{B}(K+)$ is the difference between the exciton
binding energies at $\mp K$ valleys and $E_{g}(K\pm)$ the electronic band gap
at a given $K$ valley. We predict that one peak should be observed for in-
plane oriented spins (green peak in Fig. S8) whereas two peaks emerge for the
spins deflected from the monolayer plane (red and blue peaks in Fig. S8).
Therefore, the energy $\Delta_{B}$ is a robust magnetic fingerprint of the AFM
spin directions. Note that the energy separation of $\pm$K valleys depends on
the dielectric environment, as presented in Fig. 3(c), but should be visible
in hBN encapsulated samples, specially in MnPSe3 since the valley splitting is
dominated by the electronic counterparts $\Delta^{CB}$ and $\Delta^{VB}$.

##  IV Conclusions

Here, we systematically examine the MPX3 materials emphasizing the role of
spin reorientation, magnetic arrangement and electron correlation effects in
various properties. In this regard, the chemical trends in respect to the type
of the chalcongen atom as well as transition metal are examined. We have
demonstrated that the electronic features such as band gaps, effective masses,
dielectric screening and exciton binding energies strongly depends on the type
of chalcogen atom. Notably, larger effective masses are attained for S than Se
contained monolayers (by about 30 $\%$), whereas smaller dielectric screening
(up to $\leavevmode\nobreak\ 40\%$) are reached for MPS3. In general, the
larger effective masses and smaller dielectric screening length are obtained
for employed materials than for widely examined TMDs, resulting in larger
exciton binding energies of direct transitions of MPX3 than corresponding
monolayers of TMDcs. Generally, all examined band edge direct transitions of
(Mn,Fe)PX3 monolayers turned out to be optically active coupled to linear
polarized light, with type of polarization sensitive to magnetic arrangement.

Regarding the direct band edge transitions, the MnPX3 are of particular
importance as could be exploited as promising valley electronics materials. In
particular, we have shown that the valley splitting at the $\pm K$ can be
effectively controlled by the direction of the magnetic moments. In
particular, the sizable valley splitting occurs for the out-of plane direction
of the magnetic moments in monolayer of MnPSe3 (smaller effect observed for
MnPS3), resulting in valley dependent gaps. Namely, two distinct peaks are
expected to be visible below the absorption onset for the out-of-plane AFM
aligned spins, whereas one peak for the in-plane case. Hence, the number of
the peaks and the separation of the peaks can be regarded as magnetic
fingerprint of the orientation of AFM aligned spins. The change in the
separation of the peaks points towards the plausible spin rotation.

On the other hand, the $\pm K$ valleys obtained for MnPX3 are spin
degenerated, which might hinder their spintronic applications. Hence, we have
also propose a novel way how the spin valley polarization can be attained. We
have suggested that particular deformation of the hexagonal lattice of MnPX3
could lead to the spin resolved valley splitting. Our results give insight
into the valley splitting realization in 2D antiferromagnets. In addition, the
anistropic effective masses and the type of linear polarization can be
regarded as magnetic markers probing the type of AFM arrangements. Finally,
the spin dependent features have been identified such as valley splitting of
VBM and CBM, the effective mass of holes and exciton binding energies. These
features can be referred as sensitive parameters that provide insight into
spin flop transitions.

##  V Acknowledgment

M.B. acknowledges financial support from the University of Warsaw under the
"Excellence Initiative - Research University" project. P.E.F.J. and J.F.
acknowledge the financial support of the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) SFB 1277 (Project-ID 314695032, projects B07 and
B11), SPP 2244 (Project No. 443416183), and of the European Union Horizon 2020
Research and Innovation Program under Contract No. 881603 (Graphene Flagship).
T.W. acknowledges financial support of National Science Centre, Poland under
grant no. 2021/41/N/ST3/04516. Access to computing facilities of the
Interdisciplinary Center of Modeling (ICM), University of Warsaw are
gratefully acknowledged. We gratefully acknowledge the Polish high-performance
computing infrastructure PLGrid (HPC Centers: ACK Cyfronet AGH) for providing
computer facilities and support within computational grants no.
PLG/2022/015685 and PLG/2023/016571. We acknowledge ACK Cyfronet AGH (Poland)
for awarding this project access to the LUMI supercomputer, owned by the
EuroHPC Joint Undertaking, hosted by CSC (Finland) and the LUMI consortium
through Pl-Grid organization (Poland), under the grant entitled: "Electronic,
optical and thermoelectric properties of selected layered materials and
selected heterostructures".

Supplementary Material:
"Magneto-optical anisotropies of 2D antiferromagnetic MPX3 from first
principles"
Miłosz Rybak,3 Paulo E. Faria Junior,2 Tomasz Woźniak,3 Paweł Scharoch,3
Jaroslav Fabian,2 and Magdalena Birowska1,∗

1University of Warsaw, Faculty of Physics, 00-092 Warsaw, Pasteura 5, Poland
2Institute for Theoretical Physics, University of Regensburg, 93040
Regensburg, Germany
3Department of Semiconductor Materials Engineering Faculty of Fundamental
Problems of Technology Wrocław University of Science and Technology Wybrzeże
Wyspiańskiego 27, 50-370 Wrocław, Poland
∗Electronic address<EMAIL_ADDRESS>
Magdalena Birowska

## Determination of Hubbard U from Ab initio

Figure S5: Response functions found from a linear fit of the number of
d-electrons as a function of the additional potential V: (a) for
$MPX_{3}(M=Mn,Ni$, (b) for $M^{\prime}PX_{3}(M^{\prime}=Co,Fe)$

To estimate the Hubbard $U_{eff}$ parameter, we adopt the linear response
method proposed by method of Anisimov and co-workers in the framework of a
plane-wave pseudopotential approach implemented in VASP by Cococcioni et. al
Cococcioni and de Gironcoli (2005). The U parameter is found from the linear
fits for both non-selfconsistent (NSCF) as and selfconsistent (SCF) response
functions as defined $U\approx\left(\frac{\partial N_{I}^{SCF})}{\partial
V_{I}}\right)^{-1}-\left(\frac{\partial N_{I}^{NSCF})}{\partial
V_{I}}\right)^{-1}$. The results of U parameters for d electrons of transition
metal ions are collected in Table S2.

Table S2: Calculated Hubbard $U_{eff}$ of d electrons of transition metals for various MPX3 materials. The values are given in eV. $MPX_{3}:$ | $MnPS_{3}$ | $MnPSe_{3}$ | $FePS_{3}$ | $NiPS_{3}$ | $NiPSe_{3}$ | $NiPS_{3}-bulk$ | $CoPS_{3}$
---|---|---|---|---|---|---|---
$U_{eff}[eV]:$ | 4.9 | 5.3 | 5.1 | 5.6 | 5.6 | 5.6 | 5.7

## Tables of collected electronic features.

Table S3:  Valence and conduction band edges (VBM, CBM) and electronic band gaps for various monolayer systems obtained for magnetic groundstate denoted as magn. state. The $\mu_{m}$ denotes the spin magnetic moment on metal atom (M= Mn, Ni, Fe). ML of MPX3 | Ueff [eV] | magn. state | lattice constant [Å] |  $\mu_{m}$ [$\mu_{B}$] | Band gap [eV] | VBM CBM
---|---|---|---|---|---|---
MnPS3 | 5 | AFM-N | 6.108 | 4.610 | 2.502 (D) | K
| 3 | AFM-N | 6.098 | 4.491 | 2.242 (D) | K
MnPSe3 | 5 | AFM-N | 6.424 | 4.606 | 1.846 (D) | K
| 3 | AFM-N | 6.385 | 4.475 | 1.662 | K
NiPS3 | 6.45 | AFM-zig | 5.856 | 1.621 | 2.288 | KY
| 5 | AFM-zig | 5.839 | 1.522 | 1.954 | KY
| 3 | AFM-zig | 5.819 | 1.378 | 1.572 | KY
NiPSe3 | 6.45 | AFM-zig | 6.184 | 1.566 | 1.883(D) | K
| 5 | AFM-zig | 6.168 | 1.457 | 1.6564 | KY
| 3 | AFM-zig | 6.146 | 1.291 | 1.258 | (0.282;0.5;0)Y
FePS3 | 5.3 | AFM-zig | 5.988 | 3.704 | 2.423(D) | K
| 2.6 | AFM-zig | 5.963 | 3.422 | 1.736 |  $\Gamma$(0.273;0;0)
FePSe3 | 5.3 | AFM-zig | 6.304 | 3.687 | 1.825(D) | K
| 3 | AFM-zig | 6.274 | 3.527 | 1.482(D) | K
Figure S6: Projected density of states (PDOS) assuming various exchange-correlation functional for MnPS3 (right side) and FePS3(left side).  Figure S7: In-plane components of the effective mass tensor for (left picture) all employed monolayers, (right picture) various Hubbard U. Table S4:  In-plane components of the effective mass tensor ($m_{1}$, $m_{2}$). The $m_{1}$ and $m_{2}$ are given in $m_{e}$ unit and they are determined along principal axes (square brackets). The direction of the principal axes are given in Cartesian coordinates in reciprocal space. ML of MPX3 | U${eff}$ [eV] | eff. mass of holes m1 [princ. axis], m2 [princ. axis] | eff. mass of electrons m1 [princ. axis], m2 [princ. axis]
---|---|---|---
MnPS3 | 5 | -0.957 [0.97 0.23 0] -0.955 [-0.23 0.97 0] | 0.577 [-0.39 0.93 -0.1] 0.588 [0.93 0.39 0.1]
| 3 | -1.244 [0.97 -0.23 0] -1.246 [0.23 0.97 0] | 0.691 [0.99 -0.13 0] 0.689 [0.13 0.99 0]
MnPSe3 | 5 | -0.767 [-0.39 0.93 0] -0.768 [0.93 0.39 0] | 0.546 [0.93 -0.39 0] 0.546 [0.93 0.39 0]
| 3 | -1.023 [0.97 -0.23 0] -1.026 [0.23 0.97 0] | 0.625 [ 0.95 -0.32 0] 0.623 [0.32 0.95 0]
NiPS3 | 5 | -1.183 [0 1 0] -1.722 [1 0 0] | 4.724 [0 1 0] 4.262 [1 0 -0.01]
| 3 | -1.550 [0 1 0] -16.592 [1 0 -0.1] | 2.577 [1 0 -0.1] 2.447 [0 1 0]
NiPSe3 | 5 | -1.293 [0 1 0] -2.209 [1 0 0] | 3.865 [0.1 1 0] 1.718 [1.0 -0.1 -0.1]
| 3 | -1.380 [1 0 0] -4.628 [0 1 -0.1] | 1.869 [0.1 1 0] 1.031 [1 -0.1 -0.1]
FePS3 | 5.3 | -0.927 [0 1 0.1] -0.969 [1 0 0] | 1.053 [0.1 1 0] 0.938 [1 -0.1 0]
FePSe3 | 5.3 | -0.772 [0 1 -0.1] -0.788 [1 0 0] | 0.578 [-0.39 0.93 -0.1] 0.577 [0.93 0.39 0.1]
| 3 | -1.056 [-0.01 1 0] -1.334 [1 0.01 -0.01] | 1.087 [0.04 1 0] 1.045 [1 -0.04 0]
Table S5: MnPSe3 with and without SOI included. Note that the collinear spins are assumed. The SO offset is the energy difference between the results obtained with SOC and without it. method | band edge | k-point | holes: m1 [principle axis] |  m2 [principle axis] | SO offset [meV]
---|---|---|---|---|---
without SO | VB | K/K’ | -0.767 | [-0.38 | 0.92 | 0.00] | -0.768 | [0.92 | 0.38 | 0.00] | -
with SOC: in-plane spins | VB | K | -0.752 | [0.52 | 0.86 | 0.00] | -0.825 | [0.86 | -0.52 | 0.00] | 0
| VB | K’ | -0.767 | [0.74 | -0.67 | 0.00 ] | -0.807 | [0.67 | 0.74 | 0.00] | 0
with SOC: out-of-plane | VB | K | -0.752 | [0.52 | 0.86 | 0.00] | -0.825 | [0.86 | -0.52 | 0.00] | -17.9
| VB | K’ | -0.767 | [0.74 | -0.67 | 0.00] | -0.807 | [0.67 | 0.74 | 0.00] | 0.21
method | band edge | k-point | electrons: m1 [principle axis] |  m2 [principle axis] | SO offset [meV]
without SO | CB | K/K’ | 0.547 | [0.92 | -0.38 | 0.00] | 0.546 | [0.38 | 0.92 | 0.00] | -
with SOC: in-plane spins | CB | K | 0.551 | [0.83 | -0.56 | 0.00] | 0.542 | [0.56 | 0.83 | 0.00] | -6.39
| CB | K’ | 0.551 | [0.59 | 0.81 | 0.00] | 0.541 | [0.81 | -0.59 | 0.00] | -6.39
with SOC: out-of-plane spins | CB | K | 0.551 | [0.83 | -0.56 | 0.00] | 0.542 | [0.56 | 0.83 | 0.00] | -3.50
| CB | K’ | 0.551 | [0.59 | 0.81 | 0.00] | 0.541 | [0.81 | -0.59 | 0.00] | -36.60
Table S6:  MAE is the energy difference between the AFM aligned spins along out-of plane (along easy axis) and in-plane directions (along hard axis), given in meV per magnetic ion. Note, that for in-plane directions the electronic structure is identical as without inclusion of the SOC. The $\Delta_{in-out}^{VB,CB(K\pm)}$ denotes the difference between the VBM (CBM) at given K point. The $E_{g}^{in}$ ($E_{g}^{out}$) electronic band gap for the in-plane (out-of-plane) direction of the AFM aligned spins.  | U [eV] | MAE |  $\Delta_{in-out}^{VB(K+)}$ [meV] |  $\Delta_{in-out}^{CB(K+)}$ [meV] |  $\Delta_{in-out}^{VB(K-)}$[meV] |  $\Delta_{in-out}^{CB(K-)}$ [meV] |  $E_{g}^{in}$ [meV] |  $E_{g}^{out}$ [meV]
---|---|---|---|---|---|---|---|---
MnPS3 | 3 | 0.041 | 0 | 0 | 5.5 | 9.3 | 2.203 | 2.194
| 5 | 0.028 | 0 | 0 | 6.5 | 3.9 | 2.483 | 2.479
| 7 | - | 0 | 0 | 6.9 | 1.9 | 2.667 | 2.665
MnPSe3 | 3 | 0.315 | 0 | 34.4 | 17.8 | 0 | 1.689 | 1.654
| 5 | 0.223 | 0 | 27.9 | 15.4 | 0 | 1.853 | 1.825
| 7 | - | 0 | 16.4 | 12.6 | 0 | 1.960 | 1.944
Table S7:  Electronic band edge features such as valley splitting ($\Delta^{VB}$=E${}^{VB}_{K_{-}}-$E${}^{VB}_{K+}$, $\Delta^{CB}$=E${}^{CB}_{K_{-}}-$E${}^{CB}_{K+}$), and spin degeneracy ($\delta^{VB}$=E${}^{VB}_{K_{-}}-$E${}^{VB}_{K+}$, $\delta^{CB}$=E${}_{B}^{K_{-}}-$E${}_{B}^{K+}$, for monolayers of MnPX3 and FePX3, respectively for in-plane and out-of-plane directions of AFM aligned spins.  valley splitting | MnPSe3 | for U=3eV | MnPSe3 | for U=5eV | MnPS3 | for U=3eV | MnPS3 | for U=5eV
---|---|---|---|---|---|---|---|---
| $\Delta^{VB}$ | $\Delta^{CB}$ | $\Delta^{VB}$ | $\Delta^{CB}$ | $\Delta^{VB}$ | $\Delta^{CB}$ | $\Delta^{VB}$ | $\Delta^{CB}$
in-plane spins | -5.7 | 0 | 0 | 0 | 0 | 0 | 0 | 0
out-of-plane spins | -21.5 | 38.6 | -17.7 | 33.1 | -6.6 | -10.3 | -6.4 | -4.4
Spin degeneracy of the band edges | FePS3 | for U=2.6eV | FePS3 | for U=5.3eV | FePSe3 | for U=3eV | FePSe3 | for U=5.3eV
$\delta$ | $\delta^{VB}$ | $\delta^{CB}$ | $\delta^{VB}$ | $\delta^{CB}$ | $\delta^{VB}$ | $\delta^{CB}$ | $\delta^{VB}$ | $\delta^{CB}$
in-plane spins | 0 | 7.5 | 1.8 | 9.1 | 7.4 | 5.5 | 7.4 | 3.1
out-of-plane spins | 0 | 6.4 | 1.5 | 7.4 | 11.5 | 8.2 | 9.3 | 5
Table S8:  Static dielectric constants $\varepsilon_{ij}$ (electronic contribution) and 2D polarizability $\chi^{\parallel}$ calculated for the monolayer systems and U=3 eV. The static dielectric properties are calculated by means of density functional perturbation theory in the independent particle approach (IP) neglecting local field effects and including local fields effect (TD-DFT)  Magn. | U [eV] | method |  $\varepsilon_{xx}$ $\varepsilon_{yy}$, |  $\chi^{\parallel}_{xx}$ $\chi^{\parallel}_{yy}$ [Å]
---|---|---|---|---
MnPS3 | 5 | IP | 2.35 2.35 | 3.25 3.25
| 5 | TD-DFT | 2.16 2.16 | 2.80 2.80
MnPSe3 | 5 | IP | 2.60 2.60 | 4.50 4.50
| 5 | TD-DFT | 2.42 2.42 | 3.99 3.99
NiPS3 | 5 | IP | 2.84 2.82 | 4.28 4.23
| 5 | TD-DFT | 2.70 2.68 | 3.94 3.90
NiPSe3 | 5 | IP | 3.31 3.26 | 6.13 6.00
| 5 | TD-DFT | 3.18 3.13 | 5.76 5.65
FePS3 | 5.3 | IP | 3.20 3.21 | 3.51 3.52
| 5.3 | TD-DFT | 2.96 2.95 | 3.13 3.10
FePSe3 | 5.3 | IP | 2.81 2.78 | 4.90 4.84
| 5.3 | TD-DFT | 2.63 2.61 | 4.43 4.38
Table S9:  Static dielectric constants $\varepsilon_{ij}$ (electronic contribution) and 2D polarizability $\chi^{\parallel}$ calculated for the monolayer systems and Hubbard U=3eV. The static dielectric properties are calculated by means of density functional perturbation theory in the independent particle approach (IP) neglecting local field effects and including local fields effect (TD-DFT)  Magn. | U [eV] | method |  $\varepsilon_{xx}$ $\varepsilon_{yy}$, |  $\chi^{\parallel}_{xx}$ $\chi^{\parallel}_{yy}$ [Å]
---|---|---|---|---
MnPS3 | 3 | IP | 2.44 2.44 | 3.45 3.45
| 3 | TD-DFT | 2.26 2.26 | 3.02 3.02
MnPSe3 | 3 | IP | 3.02 3.02 | 4.82 4.82
| 3 | TD-DFT | 2.82 2.82 | 4.33 4.33
NiPS3 | 3 | IP | 3.15 3.10 | 5.12 5.01
| 3 | TD-DFT | 3.02 2.98 | 4.82 4.72
NiPSe3 | 3 | IP | 4.14 4.02 | 7.51 7.21
| 3 | TD-DFT | 3.02 2.98 | 4.82 4.72
FePSe3 | 3 | IP | 3.06 3.03 | 5.56 5.49
| 3 | TD-DFT | 2.91 2.89 | 5.15 5.09
Table S10:  Exciton binding energies at different K valleys and environments (vacuum, hBN) as a function of spin direction $\Theta$ for MnPX3 (X=S, Se).  system | U [eV] |  spin direction $\Theta$ [0] |  E${}_{b}^{K+}$ in vacuum |  E${}_{b}^{K+}$ in hBN |  E${}_{b}^{K-}$ in vacuum |  E${}_{b}^{K-}$ in hBN
---|---|---|---|---|---|---
MnPS3 | U=3 | 0 | 1032.19 | 344.46 | 1034.46 | 346.02
|  | 50 | 1032.47 | 344.67 | 1034.39 | 345.83
|  | 90 | 1033.37 | 345.34 | 1033.48 | 345.45
| U=5 | 0 | 1023.98 | 322.49 | 1025.60 | 323.21
|  | 50 | 1023.98 | 322.49 | 1025.60 | 323.21
|  | 90 | 1024.69 | 322.79 | 1024.54 | 322.85
MnPSe3 | U=3 | 0 | 770.25 | 266.92 | 787.82 | 278.42
|  | 50 | 773.76 | 269.31 | 785.75 | 276.89
|  | 90 | 780.62 | 273.32 | 779.96 | 273.13
| U=5 | 0 | 773.71 | 254.23 | 785.49 | 261.16
|  | 50 | 775.00 | 254.61 | 781.97 | 259.01
|  | 90 | 778.86 | 256.97 | 779.77 | 257.45
Table S11:  Exciton binding energies for MLs considering different values of the effective dielectric constant of the surroundings $\varepsilon$, calculated using two approaches (IP and TD-DFT) for dielectric screening length of ML.  ML | $\varepsilon$ | U=3 eV, IP | U=3 eV, TD-DFT | U=5 eV, IP | U=5 eV, TD-DFT
---|---|---|---|---|---
| 1.0 | 1028.58 | 1132.13 | 1019.06 | 1132.42
| 1.5 | 814.46 | 890.75 | 797.34 | 879.03
| 2.0 | 673.53 | 732.35 | 652.41 | 714.90
| 2.5 | 571.60 | 618.59 | 549.11 | 598.30
MnPS3 | 3.0 | 494.27 | 532.73 | 470.56 | 510.31
| 3.5 | 433.05 | 464.87 | 409.35 | 442.17
| 4.0 | 383.39 | 410.52 | 360.20 | 387.07
| 4.5 | 342.54 | 365.24 | 319.61 | 342.01
| 5.0 | 308.07 | 327.64 | 285.84 | 304.97
| 1.0 | 778.82 | 843.61 | 778.93 | 850.94
| 1.5 | 623.37 | 671.57 | 616.20 | 668.73
| 2.0 | 519.65 | 557.50 | 508.78 | 549.51
| 2.5 | 444.61 | 475.08 | 431.41 | 463.95
MnPSe3 | 3.0 | 386.72 | 412.18 | 372.29 | 398.80
| 3.5 | 340.69 | 362.07 | 325.94 | 347.92
| 4.0 | 303.63 | 321.54 | 288.12 | 306.50
| 4.5 | 272.23 | 287.72 | 256.96 | 272.58
| 5.0 | 246.07 | 259.44 | 230.95 | 244.25
| 1.0 | 985.67 | 1031.65 | 1073.15 | 1140.95
| 1.5 | 825.85 | 862.46 | 886.32 | 939.24
| 2.0 | 716.31 | 746.68 | 759.50 | 802.73
| 2.5 | 634.46 | 660.29 | 665.62 | 701.86
NiPS3 | 3.0 | 569.94 | 592.46 | 592.25 | 623.30
| 3.5 | 517.45 | 537.29 | 532.90 | 559.77
| 4.0 | 473.66 | 491.15 | 483.65 | 507.35
| 4.5 | 436.31 | 451.94 | 442.10 | 462.95
| 5.0 | 404.01 | 418.21 | 406.42 | 424.93
| 1.0 | 670.36 | 940.83 | 805.38 | 844.93
| 1.5 | 562.14 | 775.12 | 673.12 | 704.36
| 2.0 | 487.23 | 662.72 | 582.05 | 608.00
| 2.5 | 430.83 | 579.42 | 513.95 | 536.08
NiPSe3 | 3.0 | 386.65 | 514.41 | 460.50 | 479.60
| 3.5 | 350.42 | 462.08 | 416.84 | 433.64
| 4.0 | 320.06 | 418.51 | 380.60 | 395.41
| 4.5 | 294.47 | 381.96 | 349.65 | 362.93
| 5.0 | 272.19 | 350.44 | 323.07 | 334.86
| 1.0 | 1038.18 | 1132.84 | - | -
| 1.5 | 825.91 | 896.20 | - | -
| 2.0 | 685.68 | 740.31 | - | -
| 2.5 | 584.24 | 628.15 | - | -
FePS3 | 3.0 | 506.71 | 543.07 | - | -
| 3.5 | 445.41 | 475.55 | - | -
| 4.0 | 395.42 | 420.84 | - | -
| 4.5 | 354.01 | 375.72 | - | -
| 5.0 | 319.04 | 338.00 | - | -
| 1.0 | 766.45 | 812.40 | 741.87 | 798.61
| 1.5 | 626.08 | 661.47 | 590.19 | 632.78
| 2.0 | 530.94 | 559.68 | 489.66 | 522.38
| 2.5 | 461.05 | 484.65 | 416.52 | 443.24
FePSe3 | 3.0 | 406.60 | 426.59 | 360.88 | 382.75
| 3.5 | 362.81 | 379.87 | 316.85 | 334.78
| 4.0 | 326.70 | 341.66 | 281.01 | 296.30
| 4.5 | 296.51 | 309.55 | 251.24 | 264.49
| 5.0 | 270.58 | 282.16 | 226.68 | 237.58
Figure S8:  Schematic diagrams (U=5 eV) of direct transitions for (a, b)
MnPSe3 and (c, d) MnPS3 exhibiting in-plane and out-of-plane directions of the
AFM aligned spins, as schematically denoted at the bottom of the picture (e,
f). On the right side of the pictures schematic absorption onsets with
depicted in coloured exciton peaks. Namely, blue, red and green peaks indicate
band edge excitons for out-of-plane spin configurations at +K and -K valleys,
and in-plane configuration of AFM aligned spins, respectively. Black arrows
close to bands indicate spin degenerated bands. The position of the peaks are
in order but not in scale. All of the presented direct transition are coupled
to "z" polarized light with the intensities listed in Table S3. (g) The first
BZ with labeled high symmetry k-points.

## References

  * Zeng  _et al._ (2012) H. Zeng, J. Dai, W. Yao, D. Xiao, and X. Cui, Nature Nanotechnology 7, 490 (2012).
  * Xiao  _et al._ (2012) D. Xiao, G.-B. Liu, W. Feng, X. Xu, and W. Yao, Phys. Rev. Lett. 108, 196802 (2012).
  * Tombros  _et al._ (2007) N. Tombros, C. Jozsa, M. Popinciuc, H. T. Jonkman, and B. J. van Wees, Nature 448, 571 (2007).
  * van ’t Erve  _et al._ (2012) O. M. J. van ’t Erve, A. L. Friedman, E. Cobas, C. H. Li, J. T. Robinson, and B. T. Jonker, Nature Nanotechnology 7, 737 (2012).
  * Kamalakar  _et al._ (2015) M. V. Kamalakar, C. Groenveld, A. Dankert, and S. P. Dash, Nature Communications 6, 6766 (2015).
  * Gibertini  _et al._ (2019) M. Gibertini, M. Koperski, A. F. Morpurgo, and K. S. Novoselov, Nature Nanotechnology 14, 408 (2019).
  * Nemec  _et al._ (2018) P. Nemec, M. Fiebig, T. Kampfrath, and A. V. Kimel, Nature Physics 14, 229 (2018).
  * Zhang  _et al._ (2019) W. Zhang, P. K. J. Wong, R. Zhu, and A. T. S. Wee, InfoMat 1, 479 (2019).
  * Birowska  _et al._ (2021) M. Birowska, P. E. Faria Junior, J. Fabian, and J. Kunstmann, Phys. Rev. B 103, L121108 (2021).
  * Autieri  _et al._ (2022) C. Autieri, G. Cuono, C. Noce, M. Rybak, K. M. Kotur, C. E. Agrapidis, K. Wohlfeld, and M. Birowska, The Journal of Physical Chemistry C 126, 6791 (2022).
  * Linhart  _et al._ (2023) W. M. Linhart, M. Rybak, M. Birowska, P. Scharoch, K. Mosina, V. Mazanek, D. Kaczorowski, Z. Sofer, and R. Kudrawiec, J. Mater. Chem. C 11, 8423 (2023).
  * Liu  _et al._ (2019) J. Liu, S. Meng, and J.-T. Sun, Nano Letters 19, 3321 (2019).
  * Jiang  _et al._ (2018) P. Jiang, L. Li, Z. Liao, Y. X. Zhao, and Z. Zhong, Nano Letters 18, 3844 (2018), pMID: 29783842.
  * Sander (2004) D. Sander, Journal of Physics: Condensed Matter 16, R603 (2004).
  * Grünberg (2001) P. Grünberg, Physics Today 54, 31 (2001), https://pubs.aip.org/physicstoday/article-pdf/54/5/31/11151061/31_1_online.pdf .
  * Martí  _et al._ (2015) X. Martí, I. Fina, and T. Jungwirth, IEEE Transactions on Magnetics 51, 1 (2015).
  * Rahman  _et al._ (2021) S. Rahman, J. F. Torres, A. R. Khan, and Y. Lu, ACS Nano 15, 17175 (2021).
  * Sass  _et al._ (2020) P. M. Sass, J. Kim, D. Vanderbilt, J. Yan, and W. Wu, Phys. Rev. Lett. 125, 037201 (2020).
  * Balamurugan  _et al._ (2014) K. Balamurugan, S.-H. Lee, J.-S. Kim, J.-M. Ok, Y.-J. Jo, Y.-M. Song, S.-A. Kim, E. S. Choi, M. D. Le, and J.-G. Park, Phys. Rev. B 90, 104412 (2014).
  * Machado  _et al._ (2017) F. L. A. Machado, P. R. T. Ribeiro, J. Holanda, R. L. Rodríguez-Suárez, A. Azevedo, and S. M. Rezende, Phys. Rev. B 95, 104418 (2017).
  * Jungwirth  _et al._ (2016) T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich, Nature Nanotechnology 11, 231 (2016).
  * Fina  _et al._ (2014) I. Fina, X. Marti, D. Yi, J. Liu, J. H. Chu, C. Rayan-Serrao, S. Suresha, A. B. Shick, J. Železný, T. Jungwirth, J. Fontcuberta, and R. Ramesh, Nature Communications 5, 4671 (2014).
  * Paier  _et al._ (2005) J. Paier, R. Hirschl, M. Marsman, and G. Kresse, The Journal of Chemical Physics 122, 234102 (2005).
  * Kresse and Furthmüller (1996) G. Kresse and J. Furthmüller, Computational Materials Science 6, 15 (1996).
  * Holzwarth  _et al._ (2001) N. A. W. Holzwarth, A. R. Tackett, and G. E. Matthews, Comput. Phys. Comm. 135 (2001).
  * Monkhorst and Pack (1976) H. J. Monkhorst and J. D. Pack, Phys. Rev. B 13 (1976).
  * Grimme  _et al._ (2010) S. Grimme, J. Antony, S. Ehrlich, and H. Krieg, The Journal of Chemical Physics 132, 154104 (2010).
  * Fonari and Sutton (2015) A. Fonari and C. Sutton, Effective Mass Calculator (EMC) (2015).
  * Gajdoš  _et al._ (2006) M. Gajdoš, K. Hummer, G. Kresse, J. Furthmüller, and F. Bechstedt, Phys. Rev. B 73, 045112 (2006).
  * Woźniak  _et al._ (2020) T. Woźniak, P. E. Faria Junior, G. Seifert, A. Chaves, and J. Kunstmann, Phys. Rev. B 101, 235408 (2020).
  * Albrecht  _et al._ (1998) S. Albrecht, L. Reining, R. Del Sole, and G. Onida, Phys. Rev. Lett. 80, 4510 (1998).
  * Rohlfing and Louie (2000) M. Rohlfing and S. G. Louie, Phys. Rev. B 62, 4927 (2000).
  * Zollner  _et al._ (2019) K. Zollner, P. E. Faria Junior, and J. Fabian, Phys. Rev. B 100, 085128 (2019).
  * Zollner  _et al._ (2020) K. Zollner, P. E. Faria Junior, and J. Fabian, Phys. Rev. B 101, 085112 (2020).
  * Zollner  _et al._ (2023) K. Zollner, P. E. Faria Junior, and J. Fabian, Phys. Rev. B 107, 035112 (2023).
  * Sivadas  _et al._ (2015) N. Sivadas, M. W. Daniels, R. H. Swendsen, S. Okamoto, and D. Xiao, Phys. Rev. B 91, 235425 (2015).
  * Budniak  _et al._ (2022) A. K. Budniak, S. J. Zelewski, M. Birowska, T. Woźniak, T. Bendikov, Y. Kauffmann, Y. Amouyal, R. Kudrawiec, and E. Lifshitz, Advanced Optical Materials 10, 2102489 (2022).
  * Basnet  _et al._ (2022) R. Basnet, K. M. Kotur, M. Rybak, C. Stephenson, S. Bishop, C. Autieri, M. Birowska, and J. Hu, Phys. Rev. Res. 4, 023256 (2022).
  * Olsen (2021) T. Olsen, Journal of Physics D: Applied Physics 54, 314001 (2021).
  * Wildes  _et al._ (2015) A. R. Wildes, V. Simonet, E. Ressouche, G. J. McIntyre, M. Avdeev, E. Suard, S. A. J. Kimber, D. Lançon, G. Pepe, B. Moubaraki, and T. J. Hicks, Phys. Rev. B 92, 224408 (2015).
  * Kurosawa  _et al._ (1983) K. Kurosawa, S. Saito, and Y. Yamaguchi, Journal of the Physical Society of Japan 52, 3919 (1983).
  * Koo  _et al._ (2021) H.-J. Koo, R. Kremer, and M.-H. Whangbo, Molecules 26 (2021), 10.3390/molecules26051410.
  * Wiedenmann  _et al._ (1981) A. Wiedenmann, J. Rossat-Mignod, A. Louisy, R. Brec, and J. Rouxel, Solid State Communications 40, 1067 (1981).
  * Ressouche  _et al._ (2010) E. Ressouche, M. Loire, V. Simonet, R. Ballou, A. Stunault, and A. Wildes, Phys. Rev. B 82, 100408 (2010).
  * Johnston (2016) D. C. Johnston, Phys. Rev. B 93, 014421 (2016).
  * Sen and Chouhan (2020) P. Sen and R. K. Chouhan, Electronic Structure 2, 025003 (2020).
  * Hwangbo  _et al._ (2021) K. Hwangbo, Q. Zhang, Q. Jiang, Y. Wang, J. Fonseca, C. Wang, G. M. Diederich, D. R. Gamelin, D. Xiao, J.-H. Chu, W. Yao, and X. Xu, Nature Nanotechnology 16, 655 (2021).
  * Belvin  _et al._ (2021) C. A. Belvin, E. Baldini, I. O. Ozel, D. Mao, H. C. Po, C. J. Allington, S. Son, B. H. Kim, J. Kim, I. Hwang, J. H. Kim, J.-G. Park, T. Senthil, and N. Gedik, Nature Communications 12, 4837 (2021).
  * Kim  _et al._ (2019) K. Kim, S. Y. Lim, J.-U. Lee, S. Lee, T. Y. Kim, K. Park, G. S. Jeon, C.-H. Park, J.-G. Park, and H. Cheong, Nature Communications 10, 345 (2019).
  * Sun  _et al._ (2023) H. Sun, L. Qiu, Y. Han, E. Yi, J. Li, M. Huo, C. Huang, H. Liu, M. Li, W. Wang, D.-X. Yao, B. A. Frandsen, B. Shen, Y. Hou, and M. Wang, Materials Today Physics 36, 101188 (2023).
  * Geraffy  _et al._ (2022) E. Geraffy, S. Zuri, M. M. Rybak, F. Horani, A. K. Budniak, Y. Amouyal, M. Birowska, and E. Lifshitz,  (2022), arXiv:2208.10890.
  * Li  _et al._ (2013) X. Li, T. Cao, Q. Niu, J. Shi, and J. Feng, Proceedings of the National Academy of Sciences 110, 3738 (2013).
  * Berkelbach  _et al._ (2013) T. C. Berkelbach, M. S. Hybertsen, and D. R. Reichman, Phys. Rev. B 88, 045318 (2013).
  * Stier  _et al._ (2018) A. V. Stier, N. P. Wilson, K. A. Velizhanin, J. Kono, X. Xu, and S. A. Crooker, Phys. Rev. Lett. 120, 057405 (2018).
  * Ghosh  _et al._ (2023) A. Ghosh, M. Birowska, P. K. Ghose, M. Rybak, S. Maity, S. Ghosh, B. Das, K. Dey, S. Bera, S. Bhardwaj, S. Nandi, and S. Datta, Phys. Rev. B 108, L060403 (2023).
  * Kang  _et al._ (2020) S. Kang, K. Kim, B. H. Kim, J. Kim, K. I. Sim, J.-U. Lee, S. Lee, K. Park, S. Yun, T. Kim, A. Nag, A. Walters, M. Garcia-Fernandez, J. Li, L. Chapon, K.-J. Zhou, Y.-W. Son, J. H. Kim, H. Cheong, and J.-G. Park, Nature 583, 785 (2020).
  * Wang  _et al._ (2021) X. Wang, J. Cao, Z. Lu, A. Cohen, H. Kitadai, T. Li, Q. Tan, M. Wilson, C. H. Lui, D. Smirnov, S. Sharifzadeh, and X. Ling, Nature Materials 20, 964 (2021).
  * Faria Junior  _et al._ (2022) P. E. Faria Junior, K. Zollner, T. Woźniak, M. Kurpas, M. Gmitra, and J. Fabian, New Journal of Physics 24, 083004 (2022).
  * Cococcioni and de Gironcoli (2005) M. Cococcioni and S. de Gironcoli, Phys. Rev. B 71, 035105 (2005).