# Dzyaloshinskii-Moriya interaction inducing weak ferromagnetism in
centrosymmetric altermagnets and weak ferrimagnetism in noncentrosymmetric
altermagnets

Carmine Autieri ID<EMAIL_ADDRESS>International Research Centre
Magtop, Institute of Physics, Polish Academy of Sciences, Aleja Lotników
32/46, 02668 Warsaw, Poland  Raghottam M Sattigeri ID
<EMAIL_ADDRESS>International Research Centre Magtop, Institute
of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02668 Warsaw,
Poland  Giuseppe Cuono ID<EMAIL_ADDRESS>International Research
Centre Magtop, Institute of Physics, Polish Academy of Sciences, Aleja
Lotników 32/46, 02668 Warsaw, Poland  Amar Fakhredine  Institute of Physics,
Polish Academy of Sciences, Aleja Lotników 32/46, 02668 Warsaw, Poland

###### Abstract

The Dzyaloshinskii-Moriya interaction (DMI) has explained successfully the
weak ferromagnetism in some centrosymmetric antiferromagnets. However, in the
last years, it was generally claimed that the DMI is not effective in
centrosymmetric systems. We reconciled these views by separating the
conventional antiferromagnets and altermagnets. Altermagnets represent
collinear antiferromagnetic compounds with spin-up and spin-down sublattices
connected only by mirror and roto-translational symmetries. Consequently, the
system shows even-parity wave spin order in the k-space lifting the Kramer’s
degeneracy in the non-relativistic band structure. We emphasize that the DMI
can create weak ferromagnetism in centrosymmetric altermagnets while it is not
effective in centrosymmetric conventional antiferromagnets. Additionally, DMI
can create weak ferromagnetism or weak ferrimagnetism in noncentrosymmetric
altermagnets. Once the spin-orbit coupling is included in an altermagnetic
system without time-reversal symmetry, the components of spin moments of the
two sublattices along the Néel vector are antiparallel but the other two spin
components orthogonal to the Néel vector can be either parallel or
antiparallel for centrosymetric systems. For noncentrosymmetric systems, we
can have different bands showing parallel or antiparallel spin components
resulting in weak ferrimagnetism. We can divide the altermagnetic compounds
into classes based on the weak ferromagnetism or weak ferrimagnetism
properties. In the case of weak ferromagnetism, the Hall vector is orthogonal
to the Néel vector while in the case of weak ferrimagnetism the Hall vector
has a component parallel to the Néel vector too. Therefore, an electric field
breaking the inversion symmetry could generate a magnetization parallel to the
Néel vector. We find a sign change of the magnetization, and possibly of the
anomalous Hall effect, as a function of the band filling and Néel vector. We
describe the dependence of the weak ferromagnetism from the charge doping and
electric field. The weak ferromagnetism and weak ferrimagnetism induced by DMI
is a property exclusively of the altermagnets, not present in either
ferromagnets or conventional antiferromagnets so we propose that altermagnets
should be classified based on this property.

##  I Introduction

It was discovered that non-relativistic spin-splitting in electronic bands can
be a feature in antiferromagnetic materials having two equivalent sublattices
with magnetic moments related by crystal-symmetryMazin  _et al._ (2022);
Autieri (2024). This magnetic phase has been named altermagnetism (AM)Šmejkal
_et al._ (2020); Hayami  _et al._ (2019, 2020); Šmejkal  _et al._ (2022a, b)
or antiferromagnetism with non-interconvertible spin-structure motif pairYuan
and Zunger (2023). The existence of AM necessitates that the electronic charge
of spin-up (down) atoms should be mapped into spin-down (up) atoms not
exclusively via translations, inversions, or their combinations but through
the application of roto-translations or mirrorsŠmejkal  _et al._ (2022a, b).
As a result, AM can be observed in specific space groups, and more precisely,
in the magnetic space groups of type-I and type-IIIGuo  _et al._ (2023).
Numerous transition metal oxidesGuo  _et al._ (2023) and rare-earth compounds
have been documented to present altermagnetismCuono  _et al._ (2023a).
Altermagnets have a wide range of potential applications, they are promising
to enable highly efficient spin-current generationGonzález-Hernández  _et al._
(2021), as well as contributing to the development of giant and tunneling
magnetoresistanceŠmejkal  _et al._ (2022c). It was shown that spin-independent
conductance in altermagnets can be efficiently exploited in spintronics Shao
_et al._ (2021); Bose  _et al._ (2022); Shao  _et al._ (2023). Furthermore,
altermagnets can play a significant role in spincaloritronicsZhou  _et al._
(2023) and can be practical use in Josephson junctionsOuassou  _et al._
(2023).

Figure 1: Origin of weak ferromagnetism in altermagnets. (a) Schematic band
structure of an altermagnet with SOC, Néel vector in the xy plane and Fermi
level in the gap. The red and blue colors represent the negative and positive
Sz component of the total system. (b) Local Sz component for the magnetic atom
as a function of the energy for the energy spectrum (a), in the
centrosymmetric case Sz for the two atoms are equal. (c) Magnetic
configuration without Mz and AHE $\rho_{xy}$. (d) Same as (a) with
metallicity. (e) Sz component as a function of the energy for the energy
spectrum (d). (f) Magnetic configuration with non-zero Mz components and non-
zero AHE $\rho_{xy}$.

The Dzyaloshinskii-Moriya interaction (DMI) is a relativistic magnetic
interaction that has successfully explained the weak ferromagnetism described
for the first time in centrosymmetric antiferromagnetsDzyaloshinsky (1958);
Moriya (1960); Camley and Livesey (2023) with space group 167. This weak
ferromagnetism has a relativistic origin and is the only kind of weak
ferromagnetism discussed in this paper. Some compounds of the centrosymmetric
space group 62 exhibit weak ferromagnetism and lead to a significant
ferromagnetic behavior, which was later explained by the mechanism of the
Dzyaloshinskii Moriya exchange based on torque measurementsSalazar-Rodriguez
_et al._ (2022); Treves (1962). The spin canting has been attributed to the
DMI in most cases, although other effects share the same requirements to be
imposed on the crystal symmetryZhou  _et al._ (2020). The weak ferromagnetism
can also be present in complex systems like solids with molecules as
cationsBlake and Hatfield (1979) and biphasic systemsKoval  _et al._ (2017).
In other literature, the weak ferromagnetism from DMI in bulk systems was
associated with noncentrosymmetric systemsWeng  _et al._ (2011a). Furthermore,
in ferroelectric BiFeO3, a subtle DMI which constitutes about 1% of the
antiferromagnetic exchange strength causes a tilting of the two spin
sublattices, consequently giving rise to ”weak ferromagnetism”Camley and
Livesey (2023); Dixit  _et al._ (2015). The DMI as explained by definition,
appears as an effect of antisymmetric interactions, which necessitates a lack
of inversion symmetry between two spins. Nevertheless, a structure can possess
an overall centrosymmetric crystal symmetry while also accommodating a DMI
interactionWeng  _et al._ (2011b). DMI plays an important role in forming
skyrmions with a fixed chirality and vorticity in the case of
noncentrosymmetric magnets since it allows the tilting of adjacent spins under
special rules related to crystal symmetry. In contrast, in centrosymmetric
structures, other mechanisms are believed to be responsible for complex
magnetismTokura and Kanazawa (2020).

The non-relativistic spin-splitting is proportional to the hopping parameters
which tend to increase with the atomic number. Large spin-splittings have been
found in compounds with heavy elements as MnTe, RuO2, CrSb, etc … which are
also the most representative and most studied compounds among the
altermagnetic systems. Beyond the large spin-splitting, these heavy atoms own
a huge spin-orbit coupling (SOC). Therefore, the study of the spin-orbital
effects is extremely relevant, especially for the most representative
altermagnets. Beyond that, the anomalous Hall effect (AHE) strongly depends on
the size of the SOC. In the case of a metallic system, the DMI produces weak
ferromagnetism in AM compounds that display spontaneous AHEŠmejkal  _et al._
(2020); Gonzalez Betancourt  _et al._ (2023) which can be enhanced by Weyl
pointsWu  _et al._ (2023). The orientation of the Hall vector strongly depends
on the Néel vector, which is defined as the difference between the
magnetization vectors of the two distinct magnetic sublatticesTurek (2022);
Shao  _et al._ (2023); Fakhredine  _et al._ (2023). While the direction of the
Hall vector has been explored using symmetry considerations, there is a
notable absence of information in the literature regarding how to establish
this orientation based on fundamental electronic properties through density
functional theory calculations.

In this work, our objective is to have a general understanding of the
appearance of weak ferromagnetism in centrosymmetric systems, while we found
weak ferromagnetism and weak ferrimagnetism in noncentrosymmetric systems. We
discuss why weak ferromagnetism and weak ferrimagnetism are relativistic
effects that can appear only in altermagnets. Once the time-reversal symmetry
is broken and the Kramers degeneracy is lifted, the spin components of the two
sublattices orthogonal to the Néel vector are related by symmetry in
centrosymmetric systems and therefore must be parallel or antiparallel. The
scenario changes in noncentrosymmetric systems where different sets of bands
can generate at the same time parallel or antiparallel spin components in
which we establish that the direction of the Hall vector is orthogonal to the
Néel vector. A schematic representation of our main results is presented in
Fig. 1. We propose that the band structure reported in Fig. 1(a) can be
present in all altermagnets for a suitable Néel vector, as a consequence of
this band structure, the local density of states (DOS) for the component of
the spin orthogonal to the Néel vector have a relativistic spin-splitting as
shown in 1(b). If the Fermi level is in the gap, we still observe a net zero
magnetic configuration Mz and zero anomalous Hall effect in altermagnets owing
to a complete antiparallel spin configuration in two sublattices. However,
when bands cross the Fermi level (as in Fig. 1(d)) which is also evident from
the increase in the contribution from the Sz component (as in Fig. 1(e)), the
SOC creates an effective non-zero magnetic configuration giving rise to
ferromagnetic spins aligned perpendicular to the Néel vector (as in Fig. 1(f))
leading to a non-zero anomalous Hall effect in altermagnetic compounds.
Integrating in energy the spin component orthogonal to the Néel in a realistic
multiband system we will obtain the magnetization as a function of the energy,
this magnetization can be zero at the same energies by compensation of the
spins, however, the compensation of the spins does not mean that we have a
compensation of the Berry curvature. Therefore, a null magnetization is not
associated with a null anomalous Hall effectAsa  _et al._ (2020). The weak
ferromagnetism is difficult to detect especially on the nanoscale level, so
our work provides a guide to theoretically predict the direction of the weak
ferromagnetism to support the experimental investigation. Using density
functional theory calculations, the results of the self-consistent calculation
will report parallel or antiparallel components depending on which of them
minimizes the total energy. We calculate the spin components of the two
sublattices as a function of the Néel vector for the centrosymmetric d-wave
RuO2, the centrosymmetric g-wave CrSb and the noncentrosymmetric g-wave MnSe.
The paper is organized as follows: the second Section describes the
computational framework while the third Section is dedicated to the results
for the three chosen compounds. Finally, in the fourth Section, the authors
draw their conclusions.

Figure 2: Band structure of d-wave RuO2 along the S1-$\Gamma$-S2 low-symmetry
k-path. (a) The positive and negative Sx components are plotted in red and
blue, respectively, while the Néel vector is along the x-axis. (b) The
positive and negative Sy components are plotted in red and blue, respectively,
while the Néel vector is along the x-axis. (c) The positive and negative Sz
components are plotted in red and blue, respectively, while the Néel vector is
along the z-axis. Sx, Sy and Sz represent the spin components for the total
system. The Fermi level is set to zero. The coordinates of the k-points are
S1=(0.5,0.5,0) and S2=(-0.5,0.5,0) in direct coordinates. Figure 3: Band
structure of d-wave RuO2 along the R1-$\Gamma$-R2 low-symmetry k-path with the
Néel vector along (a) the x- and (b) z-axis. The positive and negative Sy and
Sx components are plotted in red and blue, respectively. Sx, Sy and Sz
represent the spin components for the total system. The Fermi level is set to
zero. The coordinates of the k-points are R1=(0.5,0,0.5) and R2=(-0.5,0,0.5)
in direct coordinates.

##  II Computational details

Figure 4: DOS for the spin-components as a function of the energy for the
d-wave RuO2. (a,b) DOS of the Sx and Sy components with the Néel vector along
the x-axis, respectively. (c) DOS of the Sz component with Néel vector along
the z-axis. The other components are antiparallel. The Fermi level is set to
zero. The total DOS is plotted in green and renormalized to fit within the
plot.

The electronic properties were computed within the framework of density
functional theory calculations based on plane wave basis set and projector
augmented wave method using VASPKresse and Hafner (1993); Kresse and
Furthmüller (1996, 1996) package. A plane-wave energy cut-off of 400 eV was
chosen and the generalized gradient approximation of Perdrew, Burke, and
Ernzerhof has been usedPerdew  _et al._ (1996). The Hubbard U effects for the
4d-orbitals of Ru4+, the 3d-orbitals of Cr and 3d-orbitals of Mn2+ have been
adoptedLiechtenstein  _et al._ (1995). We have used the value of U = 2 eV
Ivanov  _et al._ (2016); Basnet  _et al._ (2022) keeping JH = 0.15U. The value
of U strengthens the gap in MnSe and establishes a robust magnetic moment in
RuO2, however, all qualitative results are independent of the value of U. We
have performed the calculations using 20$\times$20$\times$28 and
24$\times$24$\times$18 k-points centered in $\Gamma$ for RuO2 and CrSb,
respectively. A denser k-grid of 28$\times$28$\times$18 was used for wurtzite
MnSe, which is necessary to have an accurate description of the DOS,
especially in the presence of relatively small spin-orbit coupling. The
lattice constants are a=4.4825 Å and c=3.1113 Å for the RuO2, a=4.103 Å and
c=5.463 ÅReimers  _et al._ (2023) for the CrSb and a=4.12 Å with c=6.72 Å for
the MnSeGrzybowski  _et al._ (2023). The experimental Néel vectors are along
the z-axis for RuO2 and CrSbYuan  _et al._ (2020a). All calculations were
performed in the presence of SOC. The information on the individual spinor
components is available for the site’s projected density of states in the VASP
code. The site-projected orbital- and energy-resolved spin density Sx, Sy and
Sz are available. We sum over all orbitals to obtain the total Sx, Sy and Sz
components. The smearing value for the DOS was 50 meV. Regarding the magnetic
space group (MSG), the RuO2 has MSGs 58.398 and 136.499 for the Néel vector
along the x-axis and z-axis, respectively. CrSb has MSGs 63.457, 63.462 and
194.268 for the Néel vector along the x-axis, y-axis, and z-axis,
respectively. Wurtzite MnSe has the magnetic space group 194.268 when the Néel
vector is along the z-axis. The magnetic point group without spin-orbit
couplingTurek (2022); Yuan  _et al._ (2020b) of the wurtzite MnSe is 6′m′m,
different from that of the hexagonal CrSb in the NiAs structure which is
6′/m′m′m. However, both groups belong to the same magnetic Laue class
(6′/m′m′m) proving that the spin splitting of electron bands without SOC is
qualitatively the same in both systems. This could be established using the
formalism of spin groupsŠmejkal  _et al._ (2022a, b).

##  III Results

In a centrosymmetric conventional antiferromagnet, we have Kramers degeneracy
therefore the DOS of the spin-up sublattice and the spin-down sublattice are
always equal for all components. The two sublattices have always antiparallel
spin components and no weak ferromagnetism is possible. In a
noncentrosymmetric conventional antiferromagnet with SOC, there is an
additional Rashba spin-splitting where the spin components belonging to two
sublattices are antiparallel.Yamauchi  _et al._ (2019) Therefore, in
noncentrosymmetric conventional antiferromagnets, there is no weak
ferromagnetism as well.

We move to the study of the electronic properties responsible for weak
ferromagnetism in centrosymmetric and noncentrosymmetric altermagnets. We
study the electronic properties of d-wave centrosymmetric altermagnet RuO2 in
the first subsection. In the second and third subsections, we compare the
g-wave altermagnets CrSb and wurzite MnSe, where the former is centrosymmetric
and the latter is noncentrosymmetric. We will see how SOC acts differently on
centrosymmetric and noncentrosymmetric space groups. All selected compounds
have two atoms per unit cell which we will call M1 and M2 (M= Ru, Cr, Mn) to
represent the properties of the two sublattices with opposite spins.

Figure 5: Band structure of centrosymmetric CrSb with SOC along the
L1-$\Gamma$-L2 low-symmetry k-path. With the Néel vector along the x-axis, we
report the projection of the spin-component (a) Sx, (b) Sy and (c) Sz. With
the Néel vector along the y-axis, we report the projection of the spin-
component (d) Sx, (e) Sy and (f) Sz. With the Néel vector along the z-axis, we
report the projection of the spin-component (g) Sx, (h) Sy and (i) Sz. Sx, Sy
and Sz represent the spin components for the total system. The Fermi level is
set to zero. The coordinates of the k-points are L1=(0.5,0,0.5) and
L2=(-0.5,0,0.5) in direct coordinates.  Figure 6: DOS for the spin-components
Sx, Sy and Sz as a function of the energy for the centrosymmetric CrSb. (a,b)
DOS for the components Sx and Sz with the Néel vector along the x-axis and
along the z-axis, respectively. (c,d) DOS for the spin component Sz with the
Néel vector along the x-axis and along the y-axis, respectively. Both of them
show weak ferromagnetism. (e,f) DOS for the spin component Sx and Sy with the
Néel vector along the z-axis. Both of them show weak ferromagnetism. The Fermi
level is set to zero. The total DOS is renormalized to fit within the plot.

###  III.1 Weak ferromagnetism in centrosymmetric tetragonal altermagnet RuO2

In this subsection, we describe the metallic centrosymmetric d-wave RuO2.
Several authors described the altermagnetic properties of RuO2Šmejkal  _et
al._ (2020); Feng  _et al._ (2022), however, we will focus on the band
structure and the DOS for the spin-components which were never investigated
before. The appearance of the weak ferromagnetism derives from the breaking of
the time-reversal symmetry, therefore, there is the chance for the spins to be
not antiparallel when SOC is included. The self-consistent DFT calculations
provide the energetic ground state for the parallel or antiparallel
configuration of the spin components orthogonal to the Néel vector. Since RuO2
is metallic, we expect that this weak ferromagnetism could be observed in the
DFT calculations. Indeed, the magnetic moment orthogonal to the Néel vector is
0.022 $\mu_{B}$ per Ru atom along the y-axis when the Néel vector is along the
x-axis. In Figs. 2 and 3, we report the band structures of RuO2 projected on
different spin components for the two inequivalent directions of the Néel
vector (x-axis and z-axis) along the S1-$\Gamma$-S2 and R1-$\Gamma$-R2 low-
symmetry k-paths, respectively. When the Néel vector is along the $x$-axis,
the Sx spin component of the total system for every energy has a value along
S1-$\Gamma$ and an equal but opposite value along $\Gamma$-S2 as reported in
Fig. 2(a). As a result, the magnetization component Mx of the entire system
that is the integral over the energy up to the Fermi level of Sx is always
zero. On the contrary, the spin components Sy of the total system show a band
structure where the value of Sy along S1-$\Gamma$ is equal to the value along
$\Gamma$-S2. For every value of the energy, the value of the magnetization My
as integral of Sy will be in general non-zero resulting in a weak
magnetization in the $y$-direction as we can see from Figs. 2(b) and 3(a). The
origin of the canted spin moment derives exclusively from the spin-orbit
coupling but this picture is equivalent to the DMI picture. In the case of
RuO2 with spins along the z-axis, we have a pure antiparallel order without
weak ferromagnetism, as shown in Figs. 2(c) and 3(b). The other results were
not shown because they are either equivalent to the shown results or with null
spin components. The band degeneracy is broken at $\Gamma$ for the Néel vector
along the x-axis, but not for the Néel vector along to z-axis. Therefore, we
have lifting of the degeneracy at $\Gamma$ when the weak ferromagnetism rises.

To further extend our consideration to the entire Brillouin zone, we also show
the density of states for the non-zero and inequivalent spin components in
Fig. 4 for the Néel vector along the x- and z-axis. When the Néel vector is
along the x-axis, the local DOSs of the Sx components for the two sublattices
are equal and opposite as described by the local DOS related to atoms Ru1 and
Ru2 in Fig. 4(a), therefore, the Sx components are equal and opposite for the
two sublattices. From the DOS of the Sy component reported in 4(b), it is
further clear that there is a magnetic moment along the y-axis on the Ru1 and
Ru2 atoms that are equal and in the same direction resulting in a weak
ferromagnetism. The DOSs of the spin component Sy for the atoms Ru1 and Ru2
are perfectly equal, therefore, the size of the local magnetic moments of Ru1
and R2 are equal as expected since the two atoms are connected by roto-
translation symmetries. We can see that we have a sign change in My as a
function of the energy, this could be accompanied by a sign change of the AHE.
Comparing the DOS of the spin-component Sx with the total DOS, we can see a
strong reduction of Sx around -0.4 eV together with the maximum of Sy,
therefore, there is a strong rotation of the Sx component into the Sy
component at -0.4 eV. In the case of the Néel vector along the z-axis, the DOS
of all spin-components relative to Ru1 and Ru2 are equal and opposite,
therefore, the Ru spins are perfectly compensating each other as it is shown
in Fig. 4(c), consequently, there is no weak ferromagnetism. From the DOS we
can also understand that the system presents ferromagnetic or
antiferromagnetic features depending on the orientation of the Néel vector,
and this property in the k-space is the same in the whole Brillouin zone and
for all bands.

###  III.2 Weak ferromagnetism in centrosymmetric hexagonal altermagnet CrSb

We repeat the investigation for the g-wave centrosymmetric altermagnet CrSb.
In Figs. 5, we report the band structures of CrSb projected on different spin
components (Sx, Sy and Sz) for the three different directions of the Néel
vector along the L1-$\Gamma$-L2 focusing on the region near $\Gamma$ point.
The ground state is altermagnetic phase with the Néel vector along the z-axis,
we have an energy difference of 0.2 meV per formula unit with respect to the
in-plane Néel vector. When the Néel vector is along the x-axis as in Fig.
5(a,b,c), we have weak ferromagnetism only for Sz resulting in a non-zero
value of Mz. We have qualitatively similar results when the Néel vector is
along the y-axis as in Fig. 5(d,e,f). We have weak ferromagnetism only for Sz
resulting in a non-zero value of Mz. When we have the Néel vector along the
z-axis, we have weak ferromagnetism for the Sx and Sy components resulting in
a non-zero Mx and My as shown in 5(g,h,i). Overall, we find a weak
ferromagnetism out-of-plane when the Néel vector is in-plane and a weak
ferromagnetism in-plane when the Néel vector is out-of-plane. Additionally, we
find a strong change in the relativistic band structure as a function of the
Néel vector, especially between in-plane and out-of-plane directions of the
Néel vector. Qualitatively similar results are observed for the high-symmetry
k-path K1-$\Gamma$-K2 (not reported) with the difference that Sz is always
zero when the Néel vector is along the z-axis since the non-relativistic spin
splitting is absent. We check the electronic properties in the DOS as well
where we have information along the entire Brillouin zone and not for a
limited number of k-points. The DOS for Sx and Sz when the Néel vector is
along the x- and z-axis are reported in 6(a) and (b), respectively. We compare
the DOS of the spin components with the total DOS illustrated in green. At
energy 0.2 eV above the Fermi level, the spin-component resolved DOS with spin
along the Néel vectors has less weight with respect to the total DOS. This
could mean that 0.2 eV above the Fermi level there is DOS from non-magnetic
atoms or there is a rotation of the spins with the weight of the DOS moving
towards the spin components orthogonal to the Néel vector. When the Néel
vector is along the x- and y-axis, we have non-zero magnetization along the
z-axis. The values of DOS for Sz are reported in 6(c) and (d) for the Néel
vector along the x- and y-axis. The case with Néel vector along the x-axis
gives the larger weak ferromagnetism for CrSb with a value of -0.002 $\mu_{B}$
per formula unit. We can observe a value of the total DOS above 0.2 eV larger
than the DOS of the spin-component, therefore, the spins in this region are
rotated from the main component by spin-orbit. We can see that the DOS of Sz
is positive or negative depending on whether the Néel is along the y- or
x-axis. This means that by rotating the Néel vector in the xy plane, we will
have a sign change of Mz. This sign change of Mz could also produce a sign
change of the anomalous Hall effect. Recent studies on the sign change of the
AHE in ferromagnets have explained this effect which is relatively rare in
experimental conditionsvan Thiel  _et al._ (2021); Alam  _et al._ (2023). When
the Néel vector is along the z-axis, we observe weak ferromagnetism for both
spin components for Sx and Sy reported in 6(e) and (f).

Figure 7: Band structure of noncentrosymmetric MnSe with SOC along the
L1-$\Gamma$-L2 low-symmetry k-path. With the Néel vector along the x-axis, we
report the projection of the spin-component (a) Sx, (b) Sy and (c) Sz. With
the Néel vector along the y-axis, we report the projection of the spin-
component (d) Sx, (e) Sy and (f) Sz. With the Néel vector along the z-axis, we
report the projection of the spin-component (g) Sx, (h) Sy and (i) Sz. Sx, Sy
and Sz represent the spin components for the total system. Sx, Sy and Sx
represent the spin components for the total system. The Fermi level is set to
zero. The coordinates of the k-points are L1=(0.5,0.0.5) and L2=(-0.5,0,0.5)
in direct coordinates. Figure 8: Schematic band structure of
noncentrosymmetric altermagnets with Néel vector along the z-axis. The red and
blue colors represent the negative and positive Sy components of the total
system in the band structure. DOS of the spin component Sy and magnetic
configuration for three selected cases. a) Weak ferromagnetism in some bands
(B1) of noncentrosymmetric altermagnets. b) Weak antiferromagnetism in some
bands (B2) of noncentrosymmetric altermagnets. c) Weak ferrimagnetism as the
superposition of weak FM and weak AFM from bands B1 and B2.  Figure 9: DOS for
the spin-components Sx, Sy and Sz as a function of the energy for the wurtzite
MnSe. (a,b) DOS for the spin components Sx and Sz with the Néel vector along
the x-axis and along the z-axis, respectively. (c,d) DOS for the spin
component Sz with the Néel vector along the x-axis and along the y-axis,
respectively. (e,f) DOS for the spin component Sx and Sy with the Néel vector
along the z-axis, respectively. The solid and dashed lines represent the local
spin components Sx, Sy and Sz for the two sublattices represented by Mn1 and
Mn2. The Fermi level is set to zero. The total DOS is renormalized to fit
within the plot.

###  III.3 Weak FM and Weak ferrimagnetism in noncentrosymmetric hexagonal
altermagnet MnSe

To understand the effect of the breaking of the inversion symmetry, we analyze
MnSe which is a hexagonal g-wave altermagnet like CrSb. Indeed, MnSe with a
wurtzite crystal structure represents the case of an altermagnet without
inversion symmetry with a small number of atoms in the unit cell. The
structural and electronic properties without SOC for wurtzite MnSe are
reported in the literatureGrzybowski  _et al._ (2023). The non-relativistic
spin-splitting in wurzite MnSe can reach a maximum of 200-300 meV in the Mn-
bands, the SOC of Mn is of the order of 30 meV while the SOC of Se is 140 meV.
Therefore, the SOC would be a perturbation at least in the region where the
spin-splitting is maximum. For MnSe, the top of the valence band at $\Gamma$
and A are mainly px/py with a sizeable contribution from Mn orbitals. When the
Néel vector is moved from in-plane to out-of-plane, the top of the valence
band rises. We have a reduction of the band gap of 80 meV (from 1.72 to 1.64
eV) when we move from AFM with the Néel vector in the plane to AFM with the
Néel vector out-of-plane. The same sensitivity of the band gap from the Néel
vector was found for the centrosymmetric g-wave altermagnet MnTe.Faria Junior
_et al._ (2023)

We calculate the band structure for the three different spin components along
the low symmetry k-path L1-$\Gamma$-L2 where we have the non-relativistic
spin-splitting. The top of the valence band at the $\Gamma$ point without SOC
is four-fold degenerate. Starting from the low-symmetry k-path L1-$\Gamma$-L2
reported in Fig. 7, we find a relativistic spin-splitting in all cases. When
the Néel vector is in the plane, the eigenvalues at the $\Gamma$ point split
into two eigenvalues twofold degenerate with zero magnetization in the plane
(see Figs. 7(a,b,d,e)) and non-zero magnetization along the z-axis (see Figs.
7(c,f)). When the Néel vector is along the z-axis, there is a strong splitting
with the two highest eigenvalues single-degenerate and the one eigenvalue
double-degenerate at the $\Gamma$ point. The single-degenerate highest
eigenvalues produce non-zero Mx and My as shown in Fig. 7(g) and 7(h),
respectively. However, the double degenerate bands show antiparallel spin
components. Without SOC, there is a fourfold degeneracy in the top of the
valence bandGrzybowski  _et al._ (2023). When the system is noncentrosymmetric
as in the wurzite structure, we can have a combination of parallel and
antiparallel spin orientation despite both sets of bands coming from the same
fourfold degenerate point without SOC. As a consequence in the DOS, we will
see that we do not have a completely parallel or antiparallel DOS weight.
Finally, 7(i) shows the sign and weight of Sz on the two sublattices that are
equal and opposite when the Néel vector is along the z-axis. Qualitatively
similar results are observed for the K1-$\Gamma$-K2 path (not reported) with
the difference that Sz is always zero when the Néel vector is along the z-axis
since the non-relativistic spin splitting is absent. We find weak
ferromagnetism for g-wave compounds along the same direction with or without
inversion symmetry. The main difference is the coexistence of both parallel
and antiparallel magnetic configurations for the different bands of the in-
plane spin-components when the Néel is along the z-axis. Basically, we have
the superposition of weak ferromagnetism [see Fig. 8](a)] and weak
antiferromagnetism [see Fig. 8](b)]. The superposition of both kinds of
magnetism as reported schematically in Fig. 8(c) ends up forming weak
ferrimagnetism (weak FiM). In this case, there is no exact compensation of the
magnetization along the Néel vector and the Hall vector has a component along
the Néel vector too.

To further check the results from the band structure, we report in Fig. 9 the
resolved DOS for the spin-sublattice and spin components Sx, Sy and Sz for
different directions of the Néel vector, namely along the x-, y- and z-axis.
The DOSs for the spin components along the Néel vector are reported in Fig.
9(a,b). The two sublattices, represented by the Mn1 and Mn2 local DOS, have
equal and opposite DOS of the spin components since the spins are
antiparallel. Summing the DOSs of the components along the Néel vector for the
two sublattices, we would obtain zero. Now, we focus on the components
orthogonal to the Néel vector. The size of these DOSs is 3-4 orders of
magnitude smaller than the size of the DOSs for the spin-components along the
Néel vector. When the Néel vector is along the x-axis, we observe from Fig.
9(a,c) that the in-plane Sx components of the two sublattices are
antiparallel, while the spin components Sz are parallel. Since the filling was
completed and the MnSe is an insulator, the weak ferromagnetism was not
obtained in the computational details but in the case of p-doping, the weak
ferromagnetism would be observable. Assuming a p-doping, the magnetization Mz
obtained integrating the spin density up to the Fermi level will be non-zero
and the system will host a weak ferromagnetism along the z-direction. The same
happens for the Néel vector along the y-axis, the in-plane components are
antiparallel while we have weak ferromagnetism along Sz as shown in Fig. 9(d).
When the Néel vector is along the z-axis, we have both Mx and My components
different from zero, as shown in Fig. 9(e,f). When the Néel vector is in the
xy plane, the weak ferromagnetism is along the z-axis. When the Néel vector is
along the z-axis, the weak ferromagnetism is in the xy plane. This weak
ferromagnetism generates AHE with the Hall vector along the direction of the
weak ferromagnetism. We observe that the Hall vector is always orthogonal to
the Néel vector. Even if some band structures resemble the Rahsba-like
dispersions, the spin textures of the altermagnets are different and more
complicated than the Rashba effectGuo  _et al._ (2023). Of course, a Rashba
spin texture rises on top of the altermagnetic spin texture when the system
lacks inversion symmetry like in the wurtzite structure. For the wurtzite with
Néel vector along the z-axis, we have a coexistence of bands with parallel and
antiparellal components regarding Sx and Sy as shown in Fig. 7(g) and (h),
respectively.

##  IV Discussion and Conclusions

In this paper, we propose that all compounds hosting weak ferromagnetism
induced by the relativistic Dzyaloshinskii-Moriya interactionDzyaloshinsky
(1958); Moriya (1960) belong to the family of the altermagnets. While
Dzyaloshinskii and Moriya focused on the interplay between magnetic
interactions and symmetries, the recent studies on altermagnetism provided a
deeper knowledge of the crystal symmetries, extended the investigation on the
magnetic properties and discovered new effects on the electronic
propertiesMazin  _et al._ (2022); Hayami  _et al._ (2019, 2020); Šmejkal  _et
al._ (2022a, b); Fakhredine  _et al._ (2023). The connection between these two
fields emphasized in this paper paves the way to simplify the physical
scenario and merge the two fields giving the chance to connect the literature
developed from two different perspectives which until now had a minor overlap.

Compound | NO SOC | Weak FM
---|---|---
(No. space group) | splitting | from DMI
(Sr,Ca)3Ru2O7(36) | - | Exp.Ikeda  _et al._ (1998)
Ca3Mn2O7 (36) | Th.Guo  _et al._ (2023) | Exp.Sahlot and Awasthi (2020)
Mn(N(CN)2)2 (58) | Th.Guo  _et al._ (2023) | Exp.Manson  _et al._ (1999)
Ca2RuO4 (61) | Th.Cuono  _et al._ (2023b) | -
YVO3 (62) | Th.Cuono  _et al._ (2023b) | Exp.Ren  _et al._ (2000)
LaMnO3 (62) | Th.Sattigeri  _et al._ (2023) | Exp.Skumryev  _et al._ (1999)
CaMnO3 (62) | Th.Guo  _et al._ (2023) | Exp.MacChesney  _et al._ (1967)
YCrO3 (62) | Th.Guo  _et al._ (2023) | Exp.Jüdin and Sherman (1966)
CaCrO3 (62) | Th.Nguyen and Yamauchi (2023) | Th.Nguyen and Yamauchi (2023)
Ba2CoGe2O7 (113) | Th. Guo  _et al._ (2023) | Exp.Sato  _et al._ (2003)
RuO2 (136) | Th.Šmejkal  _et al._ (2022a) | Th.*, Exp.Feng  _et al._ (2022)
La2NiO4 (138) | Th.Guo  _et al._ (2023) | Exp.Batlle  _et al._ (1992)
LiFeP2O7 (140) | Th.Guo  _et al._ (2023) | Exp.Liang  _et al._ (2012)
KMnF3 (140) | Th.Guo  _et al._ (2023) | Exp.Bogdanova  _et al._ (1997)
MnTiO3 (161) | Th.Guo  _et al._ (2023) | Exp.Arévalo-López and Attfield (2013)
$\alpha$-Fe2O3 (167) | Th.Guo  _et al._ (2023) | Th.Moriya (1960), Exp.Morin (1950)
MnCO3 (167) | Th.Guo  _et al._ (2023) | Th.Moriya (1960), Exp.Maslen  _et al._ (1995)
CoCO3 (167) | Th.Guo  _et al._ (2023) | Th.Moriya (1960), Exp.BOROVIK-ROMANOV and OZHOGIN (1961)
CrF3 (167) | Th.Guo  _et al._ (2023) | Th.Moriya (1960), Exp.Tamine (2000)
CoF3 (167) | Th.Guo  _et al._ (2023) | Exp.Lee  _et al._ (2018)
FeF3 (167) | Th.Guo  _et al._ (2023) | Exp.Lee  _et al._ (2018)
FeBO3 (167) | Th.Guo  _et al._ (2023) | Exp.Dmitrienko  _et al._ (2014)
Ni1/3NbS2 (182) | Th.Sattigeri  _et al._ (2024) | -
Ni1/3TaS2 (182) | Th.Sattigeri  _et al._ (2024) | -
MnSe (186) | Th.Grzybowski  _et al._ (2023) | Th.*
MnTe (194) | Th.Šmejkal  _et al._ (2022a), Exp.Krempaský  _et al._ (2023) | Th.Kluczyk  _et al._ (2023), Exp.Kluczyk  _et al._ (2023)
CrSb (194) | Th.Šmejkal  _et al._ (2022a), Exp.Reimers  _et al._ (2023) | Th.*, Exp.Dohnomae  _et al._ (1992)

* Present work.

Table 1: List of selected altermagnetic compounds where the non-relativistic
spin-splitting appears with weak ferromagnetism induced by DMI. The words
”Th.” and ”Exp.” are reported for the theoretical and experimental
verification of the non-relativistic spin splitting or weak ferromagnetism.
The space groups 36, 182 and 186 lack inversion symmetry.

In Table 1, we report several altermagnetic compounds with different space
groups where the non-relativistic spin-splitting and the weak ferromagnetism
were theoretically predicted or experimentally detected. All the compounds
described in the seminal papers of Dzyaloshinskii and Moryia are altermagnets
present in this tableMoriya (1960). Due to the lifting of the Kramers
degeneracy, all altermagnetic systems could potentially show weak
ferromagnetism, however, for particular Néel vector orientations the DMI is
zero as in the case of the RuO2 with Néel vector along the z-axis. Once
separating the centrosymmetric and noncentrosymmetric compounds, we propose to
divide the centrosymmetric altermagnets into two classes as shown in Fig. 10.
The g-wave CrSb falls into the first category which exhibits weak
ferromagnetism for every Néel vector. In the second category, are some
altermagnets that do not display weak ferromagnetism for a given Néel vector,
such as the d-wave CaCrO3Nguyen and Yamauchi (2023) and RuO2 where we have
shown that the Néel vector is oriented along the z-axis. In the DMI picture,
the symmetries do not allow the presence of the DMI vector for particular Néel
vector orientation. The compounds belonging to this second class are probably
a limited number and seem to have higher symmetry with respect to the first
class. As a consequence of this proposed classification, the family of
altermagnets is wider than the family of weak ferromagnets since there are
altermagnets that do not exhibit weak ferromagnetism. For the
noncentrosymmetric systems, we have found that weak FM or weak FiM are present
depending on the Néel vector orientation. To extend this classification,
further studies are needed especially for the noncentrosymmetric systems. In
this prospect, interesting compounds are the metallic d-wave Ca3Ru2O7 and the
metallic g-wave Ni1/3MS2 (M=Nb, Ta). These results will be presented
elsewhere.

Beyond the crystal symmetries and the Néel vector orientation, the only other
requirement for weak FM or weak FiM is to have a non-compensated magnetization
as happens in metallic systems or doped semiconductors. As shown in Fig. 1,
when the system is insulating no weak ferromagnetism is observable since there
is a spin compensation, therefore, a small doping is necessary as in the case
of MnTeKluczyk  _et al._ (2023) and other semiconductive altermagnets. From
the DOS of the spin-components, we can understand that doping can change the
Fermi level, strongly modify the weak ferromagnetism, and can even reverse the
sign of the DMI. This explains the dependence of the weak ferromagnetism from
electron- or hole-doping concentration in several casesKhomchenko  _et al._
(2008); Beutier  _et al._ (2017). What was defined as ”weak altermagnetism” is
the band structure effect that gives rise to the DMI in centrosymmetric
altermagnets.Krempaský  _et al._ (2023) The electric field increases or
activates the non-relativistic spin-splitting depending on the field
orientationSattigeri  _et al._ (2023); Mazin  _et al._ (2023), therefore, the
electric field can increase and manipulate the DMI in altermagnetsFennie
(2008). We can make another observation regarding centrosymmetric
altermagnetic systems that can host an active DMI. In the case of magnetic
systems with frustrated magnetic interactions, the presence of an active DMI
in altermagnetic crystal structure can further push the system towards helical
magnetic structures, spin spiral and complex magnetism as observed for
instance in transition metal monopicnitidesWang  _et al._ (2016); Campbell
_et al._ (2021). The component of the weak ferromagnetism is proportional to
the spin-orbit, therefore, to look for a large canting angle we need to search
among systems with a strong spin-orbit and a small magnetic moment. In the
weak ferromagnets, an extremely relevant role is given by the
magnetocrystalline anisotropy (MCA) that decides a key ingredient as the
direction of the Néel vector. Additionally, the size of the MCA determines how
easy it is for the spin to rotate, therefore a low MCA is fundamental to favor
the weak ferromagnetism. Even if there are altermagnets where the weak
ferromagnetism is zero in the bulk as the compounds belonging to the second
class, the weak ferromagnetism can be obtained by breaking crystal symmetries
on the surfaces. Further studies are necessary for the evolution of the DMI
and weak ferromagnetism on the surfaces of the altermagnets.Pylypovskyi  _et
al._ (2023); Sattigeri  _et al._ (2023)

Figure 10: Altermagnets classified into centrosymmetric and
noncentrosymmetric. We further classify the centrosymmetric systems into two
classes; the first contains compounds in which the weak FM is always present,
and the second contains compounds in which the weak FM is present for suitable
directions of the Néel vector. For the noncentrosymmetric MnSe, we have weak
ferromagnetism with the Néel vector in the xy plane and weak ferrimagnetism
with the Néel vector out-of-plane. Likely, this classification would be
improved and extended in coming years.

In conclusion, we propose that weak ferromagnetism induced by DMI can appear
in centrosymmetric and noncentrosymmetric altermagnetic compounds but not in
centrosymmetric and noncentrosymmetric conventional antiferromagnets. Weak FiM
can appear in noncentrosymmetric altermagnets. This weak ferromagnetism
originates exclusively from the spin-orbit coupling in altermagnets and this
picture is equivalent to the DMI picture. The weak ferromagnetism and the AHE
are always orthogonal to the Néel vector for centrosymmetric systems. The weak
ferromagnetism depends on the filling, can be tuned by an electric field and
can change its sign by rotating the Néel vector in hexagonal systems. From our
results, the weak ferromagnetism is absent for insulators and in lightly doped
semiconductive systems would be proportional to the carriers. We provide a
recipe to establish the presence of weak ferromagnetism and the components of
the Hall vector from a simple relativistic density of state and band structure
calculations without calculating the DMI vector. There is a correlation
between the rise of weak FM and weak FiM with the lifting of the electronic
degeneracy at the $\Gamma$ point. We have the centrosymmetric altermagnetic
and the noncentrosymmetric altermagnets. We further divided the
centrosymmetric altermagnets into two classes, the first class always hosts
weak ferromagnetism for every direction of the Hall vector while the other
class hosts weak ferromagnetism for suitable directions of the Néel vector.
For noncentrosymmetric systems, we have weak ferromagnetism in the x-y plane
while parallel and antiparallel spin components for different bands can
coexist at the same energy resulting in weak ferrimagnetism and a component of
the Hall vector along the Néel vector. Therefore, an external electric field
breaking the inversion symmetry could generate a weak ferromagnetic component
and an AHE along the Néel vector. We have predicted weak ferromagnetism in
CrSb and weak ferrimagnetism in wurztite MnSe. The size of weak
ferromagnetism, weak ferrimagnetism and AHE strongly depends on the spin-orbit
coupling, therefore, further research directions should focus on heavy
elements. In summary, the family of the altermagnets is mainly composed of
weak ferromagnets with a minor number of compounds that are weak ferrimagnets
(MnSe with Néel vector along the z-axis) and zero magnetization compounds
(RuO2 with Néel vector along the z-axis). It was shown that altermagnetism
displays properties that either belong to ferromagnets or conventional
antiferromagnetsMazin  _et al._ (2022), the weak FM and weak FiM induced by
DMI are properties belonging exclusively to altermagnets not present neither
in ferromagnets or in conventional antiferromagnets. If we want to classify
altermagnets based on their macroscopic physical properties, we can define
altermagnets as magnetic systems with spin-up and spin-down sublattices
connected by symmetry where the DMI allows the rise of weak ferromagnetism for
a suitable Néel vector orientation.

###### Acknowledgements.

C. A., R. M. S. and G. C. contributed equally to this work. We acknowledge M.
Cuoco, K. Výborný, M. Gryglas-Borysiewicz and M. Grzybowski for useful
discussions. The work is supported by the Foundation for Polish Science
through the International Research Agendas program co-financed by the European
Union within the Smart Growth Operational Programme (Grant No. MAB/2017/1).
A.F. was supported by the Polish National Science Centre under project no.
2020/37/B/ST5/02299. We acknowledge the access to the computing facilities of
the Interdisciplinary Center of Modeling at the University of Warsaw, Grant
g91-1418, g91-1419 and g91-1426 for the availability of high-performance
computing resources and support. We acknowledge the CINECA award under the
ISCRA initiative IsC99 ”SILENTS”, IsC105 ”SILENTSG” and IsB26 ”SHINY” grants
for the availability of high-performance computing resources and support. We
acknowledge the access to the computing facilities of the Poznan
Supercomputing and Networking Center Grant No. 609.

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