# Canted Antiferromagnetism in Polar MnSiN2 with High Néel Temperature

Linus Kautzsch<EMAIL_ADDRESS>Materials Department and Materials Research
Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106
Alexandru B. Georgescu<EMAIL_ADDRESS>Department of Materials Science and
Engineering, Northwestern University, Evanston, Illinois 60208, USA
Department of Chemistry, 800 East Kirkwood Avenue, Indiana University,
Bloomington, Indiana 47405, United States  Danilo Puggioni  Department of
Materials Science and Engineering, Northwestern University, Evanston, Illinois
60208, USA  Greggory Kent  Materials Department and Materials Research
Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106
Keith M. Taddei  Neutron Scattering Division, Oak Ridge National Laboratory,
Oak Ridge, TN, 37831  Aiden Reilly  Materials Department and Materials
Research Laboratory, University of California, Santa Barbara, Santa Barbara,
CA, 93106  Ram Seshadri  Materials Department and Materials Research
Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106
Department of Chemistry and Biochemistry, University of California, Santa
Barbara, Santa Barbara, CA, 93106  James M. Rondinelli
<EMAIL_ADDRESS>Department of Materials Science and Engineering,
Northwestern University, Evanston, Illinois 60208, USA  Stephen D. Wilson
<EMAIL_ADDRESS>Materials Department and Materials Research
Laboratory, University of California, Santa Barbara, Santa Barbara, CA, 93106

###### Abstract

MnSiN2 is a transition metal nitride with Mn and Si ions displaying an ordered
distribution on the cation sites of a distorted wurtzite-derived structure.
The Mn2+ ions reside on a 3D diamond-like covalent network with strong
superexchange pathways. We simulate its electronic structure and find that the
N anions in MnSiN2 act as $\sigma$\- and $\pi$-donors, which serve to enhance
the N-mediated superexchange, leading to the high Néel ordering temperature of
$T_{N}$ = 443 K. Polycrystalline samples of MnSiN2 were prepared to reexamine
the magnetic structure and resolve previously reported discrepancies. An
additional magnetic canting transition is observed at $T_{\mathrm{cant}}$ =
433 K and the precise canted ground state magnetic structure has been resolved
using a combination of DFT calculations and powder neutron diffraction. The
calculations favor a $G$-type antiferromagnetic spin order with lowering to
$Pc^{\prime}$. Irreducible representation analysis of the magnetic Bragg peaks
supports the lowering of the magnetic symmetry. The computed model includes a
10∘ rotation of the magnetic spins away from the crystallographic $c$-axis
consistent with measured powder neutron diffraction data modeling and a small
canting of 0.6 ∘.

##  I Introduction

Ternary nitrides are a burgeoning materials class hosting diverse structures,
compositions, and properties that make them appealing for various applications
[1]. Most of this research focuses on semiconducting and metallic nitrides for
optoelectronic, piezoelectric, lighting, and structural applications,
harnessing their cation-tunable electronic structures and superior mechanical
properties [2, 3, 4, 5, 6]. However, discovery of novel ternary nitrides in
new compositions is challenging owing to difficulty in breaking the strong
molecular dinitrogen bonds and overcoming thermodynamic driving forces that
favor simpler binary nitrides. Nonetheless, recent successful syntheses have
led to the discovery of bulk CaSnN2 and CaTiN2 [7, 8], the thin films MgTiN2,
ZnMoN2, and ZnZrN2 [2, 9, 10] and MgSnN2 and MgZrN22 in both thin-films and
bulk samples [11, 12, 13, 14].

Magnetic nitride-rich transition metal compounds generally remain
underexplored, despite the unique chemical bonding afforded by the N3- ion
with open $d$-shell transition metals. The nitride anion has a higher
oxidation state and lower electronegativity than the oxide O2- anion and
prefers 3-coordinate, triangular bonding preferences[15, 16, 17, 18] rather
than 2-coordinate linear bonding preferences of the oxide anion. The local
bond geometries are further stabilized by stronger $\pi$-bonding interactions,
which because of the lower electronegativity of nitrogen also result in
shorter metal-nitride bond lengths. This increases the covalency of the
magnetic cation $-$ nitrogen ligand bond and can lead to low-spin[15] or non-
magnetic configurations[16]. However, in some cases this increased covalency
favors higher magnetic ordering temperatures.[19, 20, 21] The resulting
magnetic coupling, paired with nitrogen’s preference for 3-coordinate bonding,
may lead to novel magnetic phenomena in low-dimensional crystal structures
forming triangular and hexagonal networks. Such phenomena include noncollinear
spin structures[22] and symmetry-broken ground states or longer range magnetic
interactions which may promote frustration and suppress magnetic order.[23,
24]

Antiferromagnetic (AFM) semiconductors with ordering temperatures above room
temperature are of increased technological interest owing to advances in AFM
spintronics [25], current-induced switching of AFM [26], noncollinear spin-
splitting [27, 28], and magnetoelectric memories [29, 30]. Promising members
of this class of materials are manganese nitrides, which form a family
exhibiting oxidation states ranging from Mn5+ ($d^{2}$ electronic
configuration) to Mn2+ ($d^{5}$) and can host large magnetic moments due to
their large Hund’s coupling on the Mn $d$ manifold that favors high-spin
states [31]. The manganese nitridosilicate MnSiN2 adopts the
anti-$\beta$-NaFeO2-type structure, distorted from wurtzite, with ordering of
divalent Mn and tetravalent Si cations. The chemical ordering produces a
lattice orthorhombicity [32] within the polar noncentrosymmetric space group
$Pna2_{1}$ (number 33) and three-dimensional (3D) corner connectivity of MnN4
and SiN4 tetrahedra with the crystal chemical formula
${}^{~{}3}_{\infty}[\mathrm{MnN}_{4/4}]^{-}[\mathrm{SiN}_{4/4}]^{+}$ (Fig.
1a).

Semiconducting MnSiN2 exhibits a high magnetic ordering temperature arising
from its Mn2+ ions residing on a 3D diamond-like network. It was first
synthesized in an amonolysis process [33] ; an antiferromagnetic ordering
temperature of $T_{\mathrm{N}}=453$ K was later reported in [20]. Further
characterizing the state, Esmaeilzadeh et al. employed neutron diffraction and
observed a broad magnetic transition [34], which they conjectured arose from
frustration of the collinear spin structure aligned along the $c$ axis and
low-dimensional character in the system. This behavior was suggested to be
alleviated by disordered spin canting of an unknown amount below
$T_{\mathrm{N}}=453$ K [20], and a Curie-Weiss constant much greater than
$T_{\mathrm{N}}$ was reported indicating strong magnetic fluctuations above
$T_{\mathrm{N}}$. Until now, all models of the magnetic structure have only
explored those magnetic configurations compatible with the nuclear symmetry of
MnSiN2 [20].

Here, we reexamine the magnetic structure of MnSiN2 and resolve the magnetic
discrepancies reported previously. We perform density functional theory (DFT)
calculations to guide our magnetic structure refinement of temperature-
dependent neutron diffraction data and also show that the high Néel
temperatures arise from strong Mn-N $\pi$-bonding interactions. We find
$T_{\mathrm{N}}=443$ K in our samples and this initial state corresponds to
AFM collinear spin ordering along the crystallographic $c$ axis with an
orthorhombic magnetic symmetry, $Pn^{\prime}a2_{1}^{\prime}$. Upon cooling, we
then find magnetic canting at $T_{\rm cant}=433$ K which arises from a
magnetic symmetry breaking to the monoclinic $Pc^{\prime}$ symmetry, such that
there are two inequivalent Mn sites with long-range AFM ordering of spins in
the $ac$-plane. Although both polar phases permit weak ferromagnetism (FM)
through a relativistic Dzyaloshinskii-Moriya interaction, only the
noncollinear spin state with canted moments in the $Pc$ structure produce
weak-FM. Owing to unintentional magnetic oxide impurities in our samples, we
are unable to experimentally resolve the weak-FM. Last, we show that the
separation in temperature between $T_{\mathrm{N}}$ and $T_{\rm cant}$ is not a
consequence of a structural symmetry lowering but rather arises from entropic
competition among competing noncollinear states from $d^{5}$ Mn in a distorted
tetrahedral environment.

##  II Methods

###  II.1 Sample preparation

Polycrystalline samples of MnSiN2 were prepared from Mn (Alfa Aesar) 99.95 %
and Si3N4 (Fisher Scientific) 98.5 % in a nitrogen gas flow reaction [34]. The
starting materials were mixed using a Mn:Si ratio of 1.1:1, intimately ground,
pelletized and heated under a nitrogen gas flow (flow rate $\approx$ 10 dm3
min-1) for 24 h at $T=1250$ ∘C. The samples were then re-ground and pelletized
and heated under the same condition for 24 h. Samples with a total mass of 5 g
were prepared. This procedure resulted in well-crystallized samples.

###  II.2 Characterization

The sample quality and purity after each of the two heating steps was verified
by powder X-ray diffraction (XRD) using a Panalytical Empyrean powder
diffractometer operating with Cu-Kα radiation operating in Bragg-Brentano
geometry. High-resolution synchrotron powder X-ray diffraction was collected
at the beamline 11-BM at the Advanced Photon Source (APS), Argonne National
Laboratory, using an average wavelength of $\lambda=0.458118$ Å at $T=295$ K.

Rietveld refinement of the crystal structure was carried out using TOPAS
academic. A Thompson-Cox-Hastings pseudo-Voigt profile function (six
parameters) was used. The Finger model was used to handle peak asymmetry due
to the axial divergence of the beam [35]. Unit cell parameters, atomic
positions, and isotropic displacement parameters ($B_{\rm iso}$) of the Mn and
Si atoms were refined.

Figure 1:  (a) Crystal structure of the polar orthorhombic nitride MnSiN2
(space group $Pna2_{1}$), showing a network of corner-sharing MnN4 and SiN4
tetrahedra. (b) Synchrotron x-ray diffraction at $T=295$ K and Rietveld fit to
the data using a model consisting of MnSiN2 and the 3 % impurity phase Si2N2O.
(c) Powder neutron diffraction at $T=1.5$ K and Rietveld fit including the
crystallographic phases MnSiN2 and Si2N2O and the DFT-calculated magnetic
$Pc^{\prime}$ structure.

Variable temperature powder XRD measurements were collected on a Bruker KAPPA
APEX II diffractometer equipped with an APEX II CCD detector using a TRIUMPH
monochromater with a Mo Kα X-ray source ($\lambda=0.71073$ Å). Finely ground
MnSiN2 was sealed in a Kapton tube (0.75 mm diameter) with epoxy and the
sample temperature was controlled using an Oxford nitrogen gas cryostream
system. Each powder XRD pattern was collected using two phi scans set at
$-15^{\circ}$ and $-30^{\circ}$ $2\theta$ (Mo Kα), with a detector distance of
100 mm, and utilizing a $t=60$ s scan rate. Scans were collected between
$T=420$ K and $T=462$ K in 3 K steps, and subsequently merged and integrated
using the APEXII software suite.

Powder neutron diffraction data were collected on the HB-2A beamline at the
High Flux Isotope Reactor (HFIR), Oak Ridge National Laboratory. A Ge(113)
monochromator and a wavelength of $\lambda=2.4067$ Å was used. Diffraction
data were collected at temperatures of 1.5 K, 5 K, 20 K, 30 K, and 60 K.
Rietveld refinements of the neutron diffraction data were carried out using
using the Fullprof program [36]. The software package SARAh was used to assess
the propagation vector and the irreducible group representations from the
observed magnetic scattering [37].

DC magnetic measurements between 2 K and 400 K were performed using a Quantum
Design MPMS3 SQUID vibrating sample magnetometer. Samples with masses of
$\approx$ 5 mg were measured in polypropylene capsules. Data at temperatures
above 400 K were collected with the oven-stick option and samples of $\approx
10$ mg were cemented to the oven-stick. DC magnetization measurements were
collected under a constant magnetic field while sweeping temperature at a rate
of 7 K min-1.

###  II.3 Density functional theory

We performed DFT calculations using the Vienna Ab Initio Simulation Package
(VASP) [38, 39, 40, 41] using the recommended projector augmented wave (PAW)
pseudopotentials, [42, 43] and the GGA-PBE exchange correlation functional
[44] with the following valence configurations: Mn ($3p^{6}4s^{2}3d^{5}$), Si
($3s^{2}3p^{2}$), and N ($2s^{2}p^{3}$). We used a 520 eV energy cutoff for
the plane-wave expansion and an energy convergence of $10^{-8}$ eV/16 atom
formula unit (f.u.) energy difference between consecutive self-consistent
steps. For $k$-point integrations, we used Gaussian smearing ($\sigma=0.05$
eV), and sampled the Brillouin zone with a $9\times 8\times 7$ $k$-point mesh.
The atomic positions were relaxed until forces were less than 0.05 meV/Å.

Multiple collinear and noncollinear spin configurations were explored by
including spin-orbit interactions. We performed a sensitivity analysis of the
potential magnetic ground states and band gap with alternative exchange-
correlation functionals, including the strongly constrained and appropriately
normed (SCAN) meta-GGA [45] and PBE plus a Hubbard $U$ correction
($\textrm{PBE}+U$ using both the Dudarev [46] and Anismov [47]. Minor
quantitative changes were found; in all cases, MnSiN2 remains an AFM
semiconductor, indicating the insulating gap arises from the strong Hund’s
coupling for the half-filled Mn2+ $d^{5}$ manifold. For clarity, we focus on
results obtained using $U(\textrm{Mn})=4$ eV and $J(\textrm{Mn})=1.5$ eV
within the Anisimov $\textrm{PBE}+U+J$ approximation, which provides
quantitative agreement with the experimental optical band gap.

Table 1: Results of Rietveld refinement of scattering data. Lattice parameters $a$, $b$, and $c$; magnetic moments $\mu_{\rm Mn}$ on the Mn site; $R$-factors $R_{\rm wp}$ and $R_{\rm exp}$; carried out based on high temperature $Pna2_{1}$ and DFT-calculated $P_{c}^{\prime}$ structure. The magnetic structure for the $Pna2_{1}$ crystallographic group comprises a combination of basis vectors from two Shubnikov groups: $(Pna^{\prime}2_{1}^{\prime},33.147)$ and $(Pn^{\prime}a2_{1}^{\prime},33.146)$. | X-ray | neutron
---|---|---
T (K) | 295 | 60 | 20 | 1.5
symmetry | $Pna2_{1}$ | $Pna2_{1}$ | $Pc^{\prime}$ | $Pna2_{1}$ | $Pc^{\prime}$ | $Pna2_{1}$ | $Pc^{\prime}$
a (Å) | 5.266 | 5.268 | 5.075 | 5.268 | 5.075 | 5.268 | 5.075
b (Å) | 6.519 | 6.518 | 5.268 | 6.518 | 5.268 | 6.518 | 5.268
c (Å) | 5.073 | 5.075 | 8.257 | 5.075 | 8.253 | 5.075 | 8.253
$\beta$ (∘) | 90 | 90 | 127.86 | 90 | 127.82 | 90 | 127.82
$\mu_{\mathbf{Mn}}$ ($\mu_{\mathbf{B}}$) | $-$ | $-$ | 4.37 | $-$ | 4.37 | $-$ | 4.37
Rwp | 6.28 | 13.6 | 13.4 | 14.2 | 13.7 | 14.6 | 14.5
Rexp | 4.41 | 6.94 | 7.02 | 6.93 | 5.82 | 6.94 | 6.96

##  III Results and Discussion

###  III.1 Crystal structure and chemical bonding

Fig. 1b shows our Rietveld refinement of the synchrotron X-ray diffraction
data collected at $T=295$ K and confirms MnSiN2 crystallizes in the previously
reported orthorhombic $Pna2_{1}$ (Fig. 1a) space group [34]. We find sharp
Bragg peaks indicating excellent crystallinity of the sample. A small (3 %
mass) impurity of Si2N2O was included in the fit and the fitting parameters
are presented in Table 1. The lattice parameters obtained from Rietveld
refinement of the X-ray data and the neutron data are in excellent agreement
across the two techniques for space group $Pna2_{1}$ at all temperatures,
indicating negligible thermal contraction from $T=295$ K to $T=1.5$ K.

The experimentally determined 295 K structure is in close agreement with the
DFT-relaxed structure, which has a 1.65 % larger unit cell volume. A deviation
of the lattice parameters of this magnitude is typical for calculations using
the GGA-PBE functional.[48] The local bonding structure, comprising SiN4 and
MnN4 tetrahedra, is also in good quantitative agreement; the average
experimental (DFT) Si-N and Mn-N bond distances are 1.758 Å (1.758 Å) 2.144 Å
(2.143 Å), respectively and both exhibit distorted tetrahedra with bond
distortion indices of the order 10-3. This lowers the ideal tetrahedral
$T_{d}$ symmetry to $C_{1}$. The corner-connectivity of the distorted
tetrahedra produces buckled metal-N-metal bonds angles away from that expected
for ideal tetrahedra in the wurtzite structure ($\sim 109.5\,^{\circ}$). This
distortion along with the Si and Mn chemical order creates three distinct Si-
N-Mn angles nearly along the principal axes ($a$, $b$, $c$): 104.6 ∘ (106.8
∘), 112.8 ∘ (114.9 ∘), and 106.3 ∘ (106.2 ∘). The Mn-N-Mn bond angles that
mediate the superexchange are 93.59 ∘ (93.72 ∘), and the average Mn-Mn
distance is 3.110 Å (3.128 Å).

Figure 2:  (a) Noncollinear density of states (DOS) of MnSiN2 with the
magnetic $Pc^{\prime}$ symmetry (_vide infra_) with a band gap of
$E_{\mathrm{gap}}=2.33$ eV showing the overlap of Mn-$d$ and N-$p$ states.
(inset) 3D ELF (isocontour of 0.84) of the NMn2Si2 tetrahedron showing the
bonding localization basins comprise covalent $\sigma$ and $\pi$ Si-N bonding
and primarily Mn-N $\pi$ bonding. Band overlap of (b) Si-$p$ and Si-$s$
states, showing Si $sp^{3}$ hybridization, and (c) N-$p$ and N$-s$ states
throughout the valence band. Two-dimensional ELF plot in the (100) plane
showing variations in covalent bonding across the SiN4 and MnN4 tetrahedral
network.

Fig. 2a shows the relativistic electronic density of states obtained for
MnSiN2 in its low-temperature magnetically ordered state, exhibiting
$Pc^{\prime}$ symmetry, and discussed in detail in the next sections. It is
essentially a collinear G-type antiferromagnet (AFM-G), i.e., all adjacent
Mn2+ spins are anti-aligned to satisfy superexchange principles for the
$d^{5}$ ions, with small spin canting. The charge state is insulating with a
calculated band gap $E_{\rm gap}=2.33$ eV, consistent with the red color of
the synthesized material. The calculated gap is reduced to approximately 0.5
eV when no Hubbard correction is used.

The valence band (VB) spans approximately from -10 eV to 0 eV with a pseudogap
near -5.5 eV that separates the Mn-dominated states at the band edge from
those of Si further below (Fig. 2a). The N $2p$ states are found throughout
and the deep N $2s$ states appear below -10 eV (not shown). The VB region
comprises hybridized Si $3p$ and $3s$ states (Fig. 2b) forming $sp^{3}$
orbitals overlapping with N $2p$ states (Fig. 2c) to form single $\sigma$
bonds. The distribution of states is similar to the electronic structure
reported for the thermodynamically stable $\beta$-Si3N4 nitride with
tetrahedral bonding (average Si-N bond distance of 1.738 Å) [49], indicating
strong covalent bonding in the SiN4 tetrahedra. Both $\sigma$\- and
$\pi$-bonding interactions along the Si-N bonds are visible in the 3D electron
localization function (ELF) presented as an inset to Fig. 2a, where ELF values
greater than 0.7 are indicative of bonding regions. The covalent two-electron
bond is highly localized between the atoms. Rather than exhibiting a spherical
shape Fig. 2a (inset), the bond density forms a puckered dome from the $\pi$
back-bonding interaction through an orthogonal $p$ orbital that reduces the
large effective nuclear charge of the nominal N3- anion. These perturbations
to the spherical $\sigma$ bond along the Si-N bond direction are further
discernible in the two-dimensional ELF in the (100) plane (Fig. 2d).

The low-energy portion of the valence band comprises primarily Mn $3d^{5}$
electrons (nominally $e^{2}t_{2}^{3}$ majority spins) hybridized with N $2p$
states from -6 eV to 0 eV, while the minority spin states are unoccupied as
described in a nonrelativistic description. The high-spin insulating state is
robust to the details of our DFT simulations and the strong Mn-N interactions
arise from the ligand character of the N3- anion. While forming $dp\sigma$
antibonding states with the Mn $3d$ orbitals, we also find substantial $dp\pi$
interactions that suppress the number of states at the valence band edge. We
find additional covalent $\pi$ interactions in the 3D ELF along the Mn-N bond
(Fig. 2a, inset), where a torus of bonding electrons are found transverse to
the bond direction. This appears as increased localization in the 2D (100)
projection as well in the form of a peanut-shape perpendicular to the Mn-N
bond axis (Fig. 2d). The covalency reduces the magnetic moments for Mn, which
we calculate as 4.37 $\mu_{\mathrm{B}}$ per Mn atom in the collinear AFM-G
configuration. As a consequence, the nitrogen-mediated superexchange should be
enhanced in MnSiN2 owing to nitrogen acting as both a $\sigma$-donor and
$\pi$-donor. Although the O2- anion acts as a similar ligand, the N3- anion is
a stronger $\pi$-donor owing to the larger size and lower electronegativity of
nitrogen compared to oxygen. These features allow better overlap with the
metal orbitals, partial donation of lone pair of electrons into empty
$\pi^{*}$ orbitals of the metal, and shorter metal-nitrogen bond lengths,
which enhance the $\pi$ contributions to the superexchange.

###  III.2 Magnetic properties

Temperature-dependent magnetic susceptibility measurements were performed and
reveal a high Neél ordering temperature of $T_{\rm N}=443$ K (Fig. 3a). An
additional magnetic transition also appears at $T_{\rm cant}=433$ K (Fig. 3b),
which we attribute to the canting of the magnetic moments forming the AFM-G
ordered state (_vide infra_). Both critical temperatures are in good agreement
with previously reported transition temperatures [34]. A Curie-Weiss (CW) fit
to the high-temperature magnetic susceptibility in the range of $T=500$ K to
$T=800$ K result in $\theta_{\rm CW}=-832$ K and $\mu_{\rm s}=6.79$ $\mu_{\rm
B}$ (Fig. 3b). The negative value of $\theta_{\rm CW}$ indicates dominant
antiferromagnetic interactions and weak magnetic frustration with
$|\theta_{\rm CW}|\approx 2T_{\rm N}$. The spin-only moment is larger than
expected for a high-spin free Mn2+ ion, $\mu_{\rm s}=5.92\,\mu_{\rm
B}=\sqrt{4S(S+1)}$ ($S=5/2$), but this is unlikely due to an orbital
contribution as our DFT calculations show the orbital moment is nearly fully
quenched as expected for a half-filled $d$ manifold in a (distorted)
tetrahedral environment. The discrepancy is anomalous, and the local moment
from our CW analysis is more consistent with a $J=3$ moment, which implies
either unaccounted orbital effects or an invalid thermal regime of CW analysis
owing to the large exchange field. Esmaeilzadeh et al.[34] find a similarly
large moment of $\mu_{\rm s}=6.87\,\mu_{\rm B}$ in their CW fit and attribute
the discrepancy to an invalid thermal regime for the CW analysis.

Fig. 3c shows the magnetic moment $\mu$ as a function of the applied magnetic
field $H$. The linear curves at $T=300$ K and $T=2$ K indicate the presence of
canted antiferromagnetic order that is stable up to magnetic fields of
$\mu_{0}H=7$ T. It is possible that a small manganese oxide impurity, such as
Mn3O4 or other dilute fraction of paramagnetic impurities lead to the slight
s-shape of the curve at $T=2$ K. Additionally, a small coercive field of
$H_{\rm c}=12$ mT at $T=2$ K suggests the presence of a magnetic impurity.
This impurity is too small to be detected in synchrotron x-ray or neutron
diffraction (Fig. 1).

Figure 3: (a) High temperature susceptibility measured at an applied field of
$H=0.1$ T showing the Néel ordering and canting temperatures. ZFCW indicates a
zero field cooled measurement carried out upon warming. FCW indicates a field
cooled measurement upon warming. (b) Enlarged inverse susceptibility
highlighting $T_{\rm N}$ and $T_{\rm cant}$. (c) Curie-Weiss fit to high
temperature susceptibility from FCW measurement. The $R^{2}=0.9998$ value
indicates an excellent fit and a spin-only moment of $\mu_{\rm s}=6.79$
$\mu_{\rm B}$ and a Curie temperature of $\theta_{\rm CW}=-832$ K are found.
(d) Magnetic moment $\mu$ as a function of the applied magnetic field $H$ at
$T=300$ K and $T=2$ K.

###  III.3 Magnetic structure

Fig. 1c shows the powder neutron diffraction data collected at $T=1.5$ K,
which we acquired to fully determine the magnetic spin structure of MnSiN2.
The software package SARAh was used to begin the initial refinement as it can
determine the irreducible representation (irreps) and basis vectors for a
commensurate magnetic structure consistent with measured magnetic reflections.
The space group $P1$ was initially used for the magnetic phase and basis
vectors from individual irreps $\Gamma_{1}$ to $\Gamma_{4}$ were tested. The
intensities of the magnetic reflections were poorly captured ($R_{\rm
wp}>27$). To improve the magnetic structure refinement, a combination of basis
vectors from multiple irreps was necessary. This constraint indicates that the
symmetry of the magnetic phase is likely lower than that of the nuclear
symmetry and led us to perform a series of noncollinear DFT calculations,
allowing for multiple spin configurations, to guide the magnetic refinements.

Figure 4: Spin orientations for Mn spins $S_{1}$ and $S_{2}$ in the (a)
orthorhombic magnetic group with $\theta=6.2\,^{\circ}$, and (b) monoclinic
magnetic group with $\theta_{1}=5.6\,^{\circ}$ and
$\theta_{2}=12.2\,^{\circ}$, where $\theta$ is defined as the angle between
the Mn-N bond director and large AFM-G spin component from the noncollinear
DFT calculations. The local variation in $\theta$ from the orthorhombic to
mononclinic structure reflects the rotation of the AFM component of the spins
into the $bc$ plane; the canting angle present in the $Pc^{\prime}$ phase is
given by $\theta-\theta_{1}=0.6\,^{\circ}$.

We first computationally explored various magnetic configurations and found
that the AFM-G order is the most stable and strongly favored over other AFM
configurations. The AFM-C order is 67 meV per Mn higher in energy and the
AFM-A order is 99 meV per Mn higher in energy as shown in Ref. 50. The
ferromagnetic configuration is strongly disfavored (200 meV per Mn higher in
energy). We then surveyed the orthorhombic magnetic symmetries compatible with
the crystallographic symmetry to determine the easy direction of the spins.
Three possible magnetic groups compatible with $Pna2_{1}$ are:
$Pn^{\prime}a^{\prime}2_{1}$ (number 33.148), $Pna^{\prime}2_{1}^{\prime}$
(33.147), and $Pn^{\prime}a2_{1}^{\prime}$ (33.146), and can be found in Ref.
50. Each of these magnetic groups permits a noncollinear antiferromagnetic
alignment in the $ab$, $bc$, and $ac$ planes, respectively, with a
ferromagnetic component orthogonal to the AFM direction. After relaxing
various magnetic moment amplitudes compatible with each magnetic symmetry, we
found a 17 meV/Mn lower energy orthorhombic magnetic structure, much smaller
than the energy differences between collinear spin arrangements, for the
collinear AFM-G spin structure with the spins along the $c$ axis (Fig. 4a),
i.e., oriented along the vertex of the MnN4 tetrahedra, and described by
$Pn^{\prime}a2_{1}^{\prime}$. The two Mn spins, $S_{1}$ and $S_{2}$, are
collinear and there is no spin canting; the angle $\theta$ formed between the
spin direction and the Mn-N bond are identical for the two spins. In addition,
the AFM spin component in the $a$ direction is essentially zero following
relaxation; therefore, although the magnetic symmetry permits a weak FM
component, the Dzyaloshinskii-Moriya interaction (DMI) proportional to
$S_{1}\times S_{2}=0$; thus, there is no net FM along the $b$ axis. The other
orthorhombic magnetic groups were slightly higher in energy (of the order
70-85 $\mu$eV/Mn).

We next lowered the magnetic symmetry further while maintaining the atomic
coordinates of the orthorhombic crystal structure by starting with large AFM-G
spins aligned along $bc$ and a FM component along $a$. We then relaxed the
spin structure and found two nearly degenerate monoclinic magnetic structures:
$Pc$ (7.24) with $9.5\,^{\circ}$ rotation of the spins relative to the $z$
direction along the $y$ axis, which is 18 $\mu$eV/Mn and a minor FM spin
component along the $x$ axis, and $P{c^{\prime}}$ (7.26) with a
$10.0\,^{\circ}$ rotation relative to the $z$ direction; however, with the
spins along $y$ axis (20 $\mu$eV/Mn higher in energy). (Here we use Cartesian
directions to facilitate comparison of the monoclinic symmetries with the
orthorhombic magnetic groups, i.e., $z$ in the monoclinic structure is the
same as the $c$ direction of the orthorhombic structure (Fig. 4.) The DMI
permits a weak-FM component along the $x$ direction of the order $5\times
10^{-3}\,\mu_{\mathrm{B}}$ per Mn in the $Pc$ phase owing to low spin-orbit
coupling strength in the system. In contrast, $P{c^{\prime}}$ is strictly zero
owing to a permutation in the minor component of the AFM order of $S_{1}$ and
$S_{2}$ in the Mn-Mn pairs (Fig. 4b), i.e., one pair contributes a w-FM
component along $+x$ while the other gives a $-x$ component. Both monoclinic
spin structures remove symmetry operations formerly present in the
orthorhombic groups. Although the Mn ions with the same sign of the $z$ spin
projection are related by symmetry, adjacent Mn spins $S_{1}$ and $S_{2}$ can
no longer be mapped onto one another. This inequivalence arises from canting
of the two spins, such that the angles between the spin direction and the Mn-N
bonds are no longer the same, i.e., $\theta_{1}\neq\theta_{2}$ (Fig. 4b). In
addition, if $\theta_{1}=\theta_{2}\neq 0$ then there remains no
crystallographic symmetry linking the two spins. The magnetic symmetry is then
monoclinic; however, if $\theta_{1}=\theta_{2}=0$ as in Fig. 4a and measured
with respect to the orthorhombic $c$ axis ($b$ monoclinic), then the
orthorhombic symmetries are compatible with $Pn^{\prime}a2_{1}^{\prime}$.
These magnetic structures are available from Ref. 50.

Figure 5: (a) Powder x-ray ($T=295$ K) and neutron diffraction ($T=1.5$ K)
pattern at low $Q$ showing the additional magnetic Bragg peaks (marked by *)
emerging from the magnetic phase. (b) Powder neutron diffraction patterns at
low $Q$ and difference between data at $T=60$ K and data at $T=1.5$ K.

We then performed a Rietveld fit of the powder neutron diffraction data at 1.5
K using the DFT-calculated crystallographic structure of MnSiN2 (space group
$Pna2_{1}$) and the $P_{c^{\prime}}$ magnetic symmetry with significant spin
projections along the $z$ and $y$ axes Fig. 1c. We found excellent agreement
between our model and the measured 1.5 K data without refining the magnetic
vectors but allowing the lattice parameters and peak shapes to refine. Fig. 5a
compares the low-$Q$ region of the neutron diffraction data to the x-ray
diffraction data and clearly identifies three peaks (marked by *) as magnetic
reflections. This procedure is necessary as the neutron diffraction experiment
did not access the paramagnetic state due to the high ordering temperature of
MnSiN2. We experimentally confirm that the ground state exhibits nominally
$S=5/2$ spins moments aligned along the orthorhombic $c$ axis with a rotation
angle of 10${}^{\circ}\pm$1∘ away from the $c$-axis towards $b$. The magnetic
moment obtained from the neutron refinements in space group $Pc^{\prime}$ is
$\mu_{\mathrm{Mn}}=4.37$ $\mu_{\mathrm{B}}$ (Table 1). Although consistent
with our DFT calculations, this value is smaller than expected for high-spin
Mn2+ ions and deduced from our CW fit. The measured moment is also larger than
the value reported in Ref. 34 with $\mu_{\mathrm{Mn}}=3.55$
$\mu_{\mathrm{B}}$. One possible reason for this smaller ordered moment value
is the enhanced Mn-N bond covalency and a modified form factor for moment
distribution. A second possibility is the presence of enhanced fluctuation
effects due to unaccounted frustration arising from longer-range interactions
across the Mn sublattice. Future inelastic neutron scattering measurements on
single crystals capable of characterizing these fluctuations and performing a
total moment sum rule analysis are an appealing next step.

We further focus on the low-$Q$ region of the neutron powder diffraction data
at multiple temperatures below the critical temperatures (Fig. 5b). The
difference between the data at $T=60$ K and $T=1.5$ K indicates that no
additional continuous canting of the antiferromagnetic spin structure is
observed when lowering the temperature to $T=1.5$ K. This, in combination with
the absence of magnetic transitions below the canting temperature of $T_{\rm
cant}=433$ K in susceptibility measurements, indicates that the low-
temperature spin structure of MnSiN2 is formed at $T_{\rm cant}=433$ K and
remains unchanged to $T=1.5~{}K$. This is a remarkable result highlighting the
strong magnetic superexchange resulting from the nitrogen-mediated
superexchange coupling.

To understand why MnSiN2 orders as a collinear AFM then transforms into a
canted AFM, we performed a computational experiment whereby we cooperatively
modulated the nitride ligands reducing the amplitude of their distortion that
buckles the Mn-N-Si network. Upon reducing the equilibrium bond angles
obtained for the stable $Pc^{\prime}$ structure, i.e., reducing the buckling
by 90 % to make the tetrahedra more regular, we found that the
$Pn^{\prime}a2_{1}^{\prime}$ and $Pc^{\prime}$ become energetically
degenerate. Thus, we hypothesize that the separation in $T_{\rm N}=443$ and
$T_{\rm cant}$ could be due to temperature-dependent variations in the
nitrogen Wyckoff site. To that end, Fig. 6 shows our high-temperature PXRD
across the magnetic transitions. Although the Mo X-ray source and geometry
broaden the peaks, we find there is no crystallographic orthorhombic to
monoclinic symmetry lowering. There are slight changes to the intensities, and
we fit the data to find very minimal changes in the atomic positions between
420 K and 462 K. There were more significant but still small displacements
between 1.5 K and 420 K, and the Mn-N-Mn bond angle decreased from 97.5 ∘ to
96.7 ∘. These changes are much smaller than predicted to be necessary by our
simulation, suggesting weak magnetostructural coupling and unlikely that new
exchange paths are activated.

Spin canting in an antiferromagnet below its Néel temperature can also be
caused by the presence of an external magnetic field, structural defects
within the crystal lattice, or higher-order exchange interactions. We have
minimized contributions from these effects in our experiments and they are
controlled or nonexistent in our simulations. Therefore, we contend that the
transition from the collinear to canted AFM state is due to thermal
fluctuations. The energy scale between competing ordered states is comparable
to the energy gain obtained from the spin canting. Thermal energy thus acts to
reduce the macroscopically observable spin canting and it becomes more
pronounced as dynamical spin fluctuations decay.

Figure 6: Temperature dependent powder x-ray diffraction spanning the magnetic
ordering transition regime from $T=420$ K to $T=462$ K.

##  IV Conclusion

We have synthesized the polycrystalline material MnSiN2 to reexamine its
magnetic structure using powder neutron diffraction. Our magnetic measurements
reveal a Néel ordering temperature of $T_{\rm N}=443$ K and an additional
magnetic canting transition at $T_{\rm cant}=433$ K. Irreducible
representation analysis showed that the magnetic reflections could not be well
captured using basis vestors form one of the individual irreducible
representations $\Gamma_{1}$ to $\Gamma_{4}$. The magnetic refinement required
a combination of basis vectors from multiple representations, indicating
symmetry of the magnetic phase is lower than the nuclear symmetry. A series of
noncollinear DFT calculations show a symmetry reduction to a canted
antiferromagnetic G-type spin configuration with the spins rotated in the $bc$
plane is favored ($Pc^{\prime}$ symmetry). This model of the magnetic
structure fits the magnetic Bragg peaks observed at $T=1.5$ K well. We
conclude that the ground state magnetic structure of MnSiN2 is a G-type
antiferromagnet with a $10\,^{\circ}$ rotation of the spins away from the
crystallographic $c$ direction (orthorhombic setting) and 0.6 ∘ canting.

Our density functional theory calculations support an experimental bandgap of
$E_{\rm gap}=2.33$ eV, consistent with the red color of the material. The
nitrogen $2p$ states in MnSiN2 span a large portion of the valence band and
overlap with silicon $sp^{3}$ hybridized orbitals forming strong $\sigma$
bonds. This creates SiN4 tetrahedra with strong convalent bonding. The
manganese $3d^{5}$ electrons hybridize with the nitrogen $2p$ states and
strong Mn-N interactions arise from the $N^{3-}$ ligand acting as a strong
$\pi$-donor. The increased covalent $\pi$ interactions along the Mn-N bond
result in a slightly reduced magnetic moment of 4.37 $\mu_{\rm B}$ per Mn
atom, which is consistent with the neutron diffraction data and previous
reports in literature. The nitrogen anion acting as $\sigma$\- and
$\pi$-donors enhances the strength of the nitrogen-mediated superexchange
pathways and leads to the high Néel ordering temperature in MnSiN2.

###### Acknowledgements.

This work was supported by the Air Force Office of Scientific Research under
award number FA9550-23-1-0042. The research reported here made use of the
shared facilities of the Materials Research Science and Engineering Center
(MRSEC) at UC Santa Barbara: NSF DMR–2308708. The UC Santa Barbara MRSEC is a
member of the Materials Research Facilities Network (www.mrfn.org). Use of the
Advanced Photon Source at Argonne National Laboratory was supported by the US
Department of Energy, Office of Science, Office of Basic Energy Sciences,
under Contract No. DE-AC02-06CH11357. This research used resources at the High
Flux Isotope Reactor, a DOE Office of Science User Facility operated by the
Oak Ridge National Laboratory. The computational work was supported by the
National Science Foundation (NSF) Grant no. DMR-2011208 and used resources of
the National Energy Research Scientific Computing Center (NERSC), a U.S. DOE
Office of Science User Facility located at Lawrence Berkeley National
Laboratory, operated under Contract no. DE-AC02-05CH11231.

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