# High-throughput calculations of antiferromagnets hosting anomalous transport
phenomena

Takuya Nomoto<EMAIL_ADDRESS>Research Center for Advanced Science
and Technology, University of Tokyo, Komaba 153-8904, Japan  Susumu Minami
Department of Mechanical Engineering and Science, Kyoto University, Kyoto
615-8540, Japan  Yuki Yanagi  Liberal Arts and Sciences, Toyama Prefectural
University, Toyama 939-0398, Japan  Michi-To Suzuki  Center for Computational
Materials Science, Institute for Materials Research, Tohoku University, Sendai
950-8577, Japan  Center for Spintronics Research Network, Graduate School of
Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
Takashi Koretsune  Department of Physics, Tohoku University, Sendai 980-8578,
Japan  Ryotaro Arita  Research Center for Advanced Science and Technology,
University of Tokyo, Komaba 153-8904, Japan  Center for Emergent Matter
Science, RIKEN, Wako 351-0198, Japan

###### Abstract

We develop a high-throughput computational scheme based on cluster multipole
theory to identify new functional antiferromagnets. This approach is applied
to 228 magnetic compounds listed in the AtomWork-Adv database, known for their
elevated Néel temperatures. We conduct systematic investigations of both
stable and metastable magnetic configurations of these materials. Our findings
reveal that 34 of these compounds exhibit antiferromagnetic structures with
zero propagation vectors and magnetic symmetries identical to conventional
ferromagnets, rendering them potentially invaluable for spintronics
applications. By cross-referencing our predictions with the existing MAGNDATA
database and published literature, we verify the reliability of our findings
for 26 out of 28 compounds with partially or fully elucidated magnetic
structures in the experiments. These results not only affirm the reliability
of our scheme but also point to its potential for broader applicability in the
ongoing quest for the discovery of new functional magnets.

††preprint: perspective

##  I Introduction

In computer organization, the importance of memory hierarchy and non-von
Neumann architectures has been increasingly recognized, particularly from the
perspective of optimizing computational speed and power consumption. This
entails the development of various devices optimized for diverse purposes, and
innovative material designs are sought after in the field of materials
science. Spintronics is a prominent example, which has emerged as a viable
option by offering new devices that complement or even replace semiconductor-
based devices.

More recently, spintronics utilizing antiferromagnets (AFMs) has gained
significant attention since they offer distinct advantages over conventional
ferromagnets (FMs). These benefits include the absence of stray field, fast
response, and robustness against magnetic field noise Jungwirth  _et al._
(2016); Baltz  _et al._ (2018); Fukami  _et al._ (2020); Xiong  _et al._
(2022). However, the detection and control of magnetic structures that
maintain time-reversal symmetry (TRS) present challenges, and thus,
particularly desirable for device applications are AFMs breaking TRS. The
examples include magnetic toroidal orders Watanabe and Yanase (2018); Godinho
_et al._ (2018); Šmejkal  _et al._ (2022) and altermagnets Naka  _et al._
(2019); Šmejkal  _et al._ (2020, 2022a, 2022b), a third class of magnets with
zero net magnetization and alternating spin splitting in momentum space. These
materials hold promising potential for future applications relying on their
controllability via electric currents and the spin splitter effects,
respectively.

Among these compounds, AFMs with magnetic symmetries identical to those of
conventional FMs are particularly valuable for integration into existing FM-
based spintronics devices. These ’FM-like AFMs,’ as we term in this paper,
feature a small canting moment in the presence of the spin-orbit coupling.
Their antiferromagnetic domains can be detected as the sign of anomalous
transverse transport and manipulated using magnetic fields. A notable example
is Mn${}_{3}X$ ($X=$ Sn and Ge) Nakatsuji  _et al._ (2015); Nayak  _et al._
(2016); Liu and Balents (2017); Higo  _et al._ (2018), where recent studies
have reported magnetization switching via electric current and tunneling
magnetoresistance effects Tsai  _et al._ (2020); Higo  _et al._ (2022); Qin
_et al._ (2023); Chen  _et al._ (2023); Yoon  _et al._ (2023), thereby opening
a new era of AFM-based spintronics. However, for practical implementation,
numerous considerations such as fabrication procedures, operational
temperature ranges, resistance to oxidation, and basic performance metrics
intersect in a complex manner. Consequently, the exploration of new functional
AFMs continues to be a critical focus for advancing AFM-based spintronics.
Indeed, high-throughput calculations for magnetic materials have been
vigorously pursued in recent years Balluff  _et al._ (2017); Horton  _et al._
(2019); Zhang  _et al._ (2023). Many of them have focused on specific groups
of materials, such as Heusler compounds and two-dimensional materials, and
have yet to conduct magnetic symmetry analysis related to the functionality of
the AFMs. Moreover, systematic calculations including non-collinear magnetic
structures are still challenging, which is required to find prototype FM-like
AFM, Mn3Sn.

In this study, we performed high-throughput calculations in exploring
potential AFMs for spintronics applications. Note that our present scheme
takes into account not only collinear but also non-collinear magnetic
structures, which is in clear contrast to the previous studies. Here, we
utilized the cluster multipole (CMP) method and spin density functional theory
(SDFT) calculations to predict stable magnetic structures from input crystal
structures Suzuki  _et al._ (2017, 2019a); Huebsch  _et al._ (2021); Yanagi
_et al._ (2023a). For material pre-screening, we consulted the AtomWork-Adv
database Xu  _et al._ (2011), where AFMs known with specified Néel temperature
are listed. Subsequent application of the CMP+SDFT methodology on selected
candidates allowed us to systematically investigate the most stable and
metastable AFM configurations. From the energy comparison and the magnetic
space group analysis, we identified 8 metallic and 26 insulating FM-like AFMs
as among the most stable magnetic structures. Notably, our findings
corroborate the efficacy of our computational approach, as they include
several known FM-like AFMs such as Mn3Ir, Mn3Sn, and Mn3SnN. While most
identified magnetic structures of FM-like AFMs are already established in the
experiments, we discovered that MnPtAl and Fe[NCN] are promising candidates
for FM-like AFMs, whose magnetic structures are not completely determined by
the experiments. Furthermore, we proposed a range of metastable FM-like AFMs
that could potentially be realized through efforts like chemical doping or
synthesis optimization. Regarding the effects of electronic correlation, we
also performed an analysis using SDFT+$U$ calculations. While we did not
discover any new promising FM-like AFM materials, there was considerable
overlap between the two sets of calculations. This suggests that the
correction from the Hubbard $U$ is not significant, and thus, our scheme
including metastable states might be enough to ensure that no potential FM-
like AFMs are missed. Our results affirm the effectiveness of our methodology
in identifying functional AFMs, even extending to metastable configurations.
Future work encompassing a broader spectrum of crystals and magnetic
structures with finite propagation vectors could reveal an extensive array of
novel functional AFM compounds.

##  II Methods and Workflow

We performed the CMP+SDFT method to identify functional AFMs Huebsch  _et al._
(2021). The specific workflow adopted in this study is outlined as follows:
Initially, we selected input crystal structures for analysis from the
AtomWork-Adv database. Secondly, magnetic structures were generated for each
crystal using the CMP method to serve as initial inputs for SDFT calculations.
These calculations were then executed, and their outputs were subjected to
magnetic symmetry analysis for further scrutiny. Below, we briefly summarize
each part of the workflow.

###  II.1 Input crystal structures

Although the CMP+SDFT method could accommodate any crystal structure,
computational limitation needs a more selective approach to input. For this
study, we confined our analysis to structures cataloged in the AtomWork-Adv
database Xu  _et al._ (2011), provided by the National Institute for Materials
Science, for several reasons. Firstly, the database contains abundant
materials whose Néel temperatures, $T_{N}$, are established, enabling a
targeted focus on high-$T_{N}$ materials, which is crucial for spintronics
applications. Secondly, it includes both AFMs with established magnetic
structures and those where only a Néel transition has been indicated. This
makes the database invaluable both for predicting new AFMs and as a benchmark
for our method.

In practice, we adhere to the following three criteria in selecting crystal
structures for analysis: (1) a Néel temperature exceeding 273 K, (2) a unit
cell containing between 2 and 24 magnetic atoms, and (3) the absence of
disorder or partially occupied sites. The condition (1) and (2) are related to
the limitation of the computational source as mentioned above. Note that, for
the same reason, we consider magnetic structures with zero propagation vector,
and thus, the structures with a single magnetic atom in the unit cell do not
support AFM states. Since the AFMs with finite propagation vectors are
commonly observed in experiments, it is anticipated that in the future, this
issue will be addressed combining with the CMP method and the downfolding
techniques to derive the classical spin models, such as the local force method
Liechtenstein  _et al._ (1984); Gyorffy  _et al._ (1985); Nomoto  _et al._
(2020). The condition (3) is required due to the lack of reliable methods for
handling disorder or partial occupancy, such as the coherent potential
approximation. While lifting this restriction could be valuable, especially
given that chemical substitution is a key strategy for material optimization,
we defer this issue to future research. Another point overlooked in this
scheme is the impact of finite temperature effects because they cannot be
handled in the SDFT calculations. To deal with these effects, we would need to
use methods that require more computing cost, such as the disordered local
moment method Staunton  _et al._ (1984). Alternatively, we can combine our
current scheme with a mapping method to the classical spin models. This
approach allows us to conduct spin-wave analysis, akin to crystal structure
optimization that takes into account the finite temperature effects by
including the phonon contributions Masuki  _et al._ (2022).

###  II.2 Input magnetic structures

In addition to the crystal structures, magnetic structures are required as
input for SDFT calculations. For this purpose, we employ the CMP expansion
method, as outlined in Refs. Suzuki  _et al._ (2019b); Yanagi  _et al._
(2023b). This method produces an orthonormal and complete basis set in a
3$N$-dimensional vector space, where $N$ represents the number of magnetic
atoms considered for the expansion. This basis set can represent any magnetic
structure that arises in a given crystal, as it is classified by multipole
degrees of freedom and the irreducible representations of the point group
symmetry. Notably, as detailed in Ref. Huebsch  _et al._ (2021), this basis
set effectively represents magnetic structures observed in nature. In fact,
more than 90% of magnetic structures cataloged in the MAGNDATA database
Gallego  _et al._ (2016a, b) can be expressed by linear combinations of up to
three CMP basis elements. Thus, in this study, we focus on magnetic structures
that can be represented by linear combinations of up to three CMP basis
components. For computational feasibility, we adhere to two conditions for
these structures: First, as in Ref. Huebsch  _et al._ (2021), we limit
ourselves to linear combinations of CMP bases that belong to the same
multipole rank and irreducible representation. Second, we exclude magnetic
structures where equivalent atoms (in terms of Wyckoff positions) have
different sizes of local moments. Such configurations are often
computationally unstable due to the need for spontaneous symmetry breaking in
the charge density distribution. While such input magnetic structures can give
rise to more plausible AFM structures through the self-consistent iteration,
we choose to simplify our model by not considering this possibility.

###  II.3 SDFT calculations

For a given input of crystal and magnetic structure, we perform SDFT
calculations to obtain the total energies. We use atomate package for the
management of workflow of the high-throughput calculation Mathew  _et al._
(2017), which internally utilizes pymatgen, custodian, and FireWorks libraries
Ong  _et al._ (2013); Jain  _et al._ (2015). For the electronic structure
calculations, we use Vienna ab initio simulation package (VASP) Kresse and
Furthmüller (1996). Here, we employ the exchange-correlation functional
proposed by Perdew, Burke, and Ernzerhof Perdew  _et al._ (1996), and
pseudopotentials with the projector augmented wave (PAW) basis Blöchl (1994);
Kresse and Joubert (1999). The spin-orbit coupling effect is also taken into
account. The other key setting of the VASP inputs are given as follows: ENCUT
= $1.5\times$ (the highest ENMAX in POTCAR file), EDIFF = $10^{-5}$, SIGMA =
0.05, LASPH = True, AMIX = 0.1, AMIX_MAG = 0.4, BMIX = 0.0001, BMIX_MAG =
0.0001, PREC = Accurate, and NELM = 400. We use $\Gamma$ centered
$\bm{k}$-grid generated by pymatgen code with the grid density =
$4000/$(number of atoms in the unit cell). For the unconverged cases, we try
additional calculations with the change guided by the default setting in
custodian code. In the case for SDFT+$U$ calculations, we set $U=3$ eV for all
3$d$ transition metal elements and $0$ eV for the other elements. The other
settings are the same as the SDFT calculations.

In this paper, we deal with all transition elements as magnetic and the other
elements as non-magnetic. Owing to this definition, we treat the rare-earth
elements as non-magnetic although they show magnetic orders in most cases.
This is because we consider only materials with high $T_{N}>273$ K in this
paper, and thus, almost all materials include transition metal elements, as
shown in Sec. III.1. Even in compounds containing both types of elements, it
is a reasonable assumption that only transition metal elements exhibit
substantial magnetic moments at elevated temperatures, as commonly seen in
rare-earth and transition metal oxides. Accordingly, we employ open-core
pseudopotentials for rare-earth elements through the calculations.

###  II.4 Analysis of outputs

Finally, we analyze the total energies and magnetic structures for each
material from the SDFT solutions. Here, we regard a material to be AFM when
the output magnetic structures satisfy,

$\displaystyle\frac{||\sum_{i}\bm{m}_{i}^{\rm
out}||}{\sum_{i}||\bm{m}_{i}^{\rm out}||}<0.05,$ (1)

where $\bm{m}_{i}^{\rm out}$ denotes the magnetic moment of the $i$ site.
Then, we identify potential FM-like AFMs by analyzing their magnetic point
group symmetries using the spinspg package Shinohara  _et al._ (2023), built
upon spglib Togo and Tanaka (2018). Note that a small canting moment is
generally induced in the presence of the spin-orbit coupling if thier
magnegtic point groups are identical to conventional FMs Cheong and Huang
(2024). However, the anomalous Hall transport is allowed regardless of the
existence of net magnetization. In this analysis, we use the input magnetic
structures, rather than the outputs, since the self-consistent calculation
could break the crystal symmetry due to numerical errors. Instead, we only
adopt results where the overlap between the input and output magnetic
structures satisfies,

$\displaystyle\bigg{|}\frac{\sum_{i}\bm{m}_{i}^{\rm in}\cdot\bm{m}_{i}^{\rm
out}}{\sqrt{\sum_{i}||\bm{m}_{i}^{\rm out}||\sum_{i}||\bm{m}_{i}^{\rm
in}||}}\bigg{|}>0.99.$ (2)

Note that not only the most stable but also the metastable magnetic structures
are considered in the analysis since they are potentially accessible through
experimental techniques such as synthesis optimization, chemical doping, and
device-imposed strain. Here, we define the most stable structures as those
with a relative energy per magnetic atom, $\Delta E$, less than 0.1 meV,
reflecting the numerical accuracy of our high-throughput calculations.
Structures with 0.1 meV $<\Delta E<20$ meV are classified as metastable, which
approximates the energy scale of the Néel temperatures under consideration.

##  III Results and discussion

###  III.1 Investigated materials

After applying the selection criteria, 231 material candidates remain for
consideration. As anticipated, a vast majority of these materials include
3$d$-transition metal elements due to their high Néel temperatures. Indeed,
only three exceptions, GeS ($T_{N}=455$K) Kyriakos  _et al._ (1991), GdI2
($T_{N}=313$K) Kasten  _et al._ (1984), and hcp-Gd ($T_{N}=293$K) are listed
in AtomWork-Adv. However, recent literature suggests that GeS may have AFM-
like behavior induced by transition metal impurities Kyriakos  _et al._
(1991), and GdI2 is ferromagnetic with a Curie temperature $T_{c}=276$K Felser
_et al._ (1999); Ahn  _et al._ (2000). Additionally, it is known that hcp-Gd
shows ferromagnetism at the highest transition temperature. Thus, these three
materials are simply excluded from subsequent calculations. A mere two
substances, BaRuO3 ($T_{N}=430$K) and Sr2MgReO6 ($T_{N}=320$K), are found to
contain only 4$d$ and 5$d$ transition metals as magnetic atoms, respectively.
As a result, our following calculations include 226 materials with 3$d$, one
with 4$d$, and one with 5$d$ transition metal magnetic materials, which are
listed in the Appendix.

In the selected materials, Mn is the most prevalent among 3$d$ transition
metals, appearing 100 times. It is followed by Fe with 76 occurrences, Co with
22, Cr with 11, and Ni appearing 10 times. This distribution aligns well with
the tendency of Mn and Fe to adopt high-spin configurations in oxide
environments, where ions of 3$d$ transition metals typically have valence
states of 2+ or 3+. Such configurations often lead to strong exchange
interactions and high Néel temperatures. This usually results in strong
exchange interactions and elevated Néel temperatures. It is worth noting that
compounds containing Co, as well as Fe, are renowned for their high Curie
temperatures, especially in the context of permanent magnets Nelson and
Sanvito (2019). However, in our analysis of AFMs, Co appears only about one-
fourth as frequently as Mn. This is likely due to the AFM nature of 3$d$
transition metals located to the left of Mn, compared to the FM nature of
those to the right of Fe Sakuma (1999). Therefore, when searching for AFM
materials with high Néel temperatures, focusing on Mn-based compounds proves
to be the most effective strategy.

Table 1: Summary of FM-like AFMs identified as the most stable magnetic structures found by CMP+SDFT calculations. Substance names follow the notation of Atomwork-Adv database, where polymorphisms are distinguished by the characters following the formula. $G_{\rm SG}$ represents the space group in standard setting and may not be identical to MAGNDATA/Atomwork-Adv in cartain cases. $G_{\rm MSG}$ represents the magnetic space group of the FM-like AFMs. For systems meeting the criterion $\Delta E<0.1$ meV, all corresponding $G_{\rm MSG}$ symbols are listed. The ‘Met./Ins.’ column signifies whether the system is metallic (M) or insulating (I), based on the calculations. A note is added when a system is metallic in calculations but insulating in experimental data (∗M). $T_{N}$ is the experimental Néel temperature, given in Atomwork-Adv. In the ‘MAGNDATA’ column, $\bm{q}=\bm{0}$ and $\bm{q}\neq\bm{0}$ indicate that a magnetic structure with zero and finite propagation vectors is listed in MAGNDATA, respectively. The compounds with ${}^{*}1$-${}^{*}4$ indicate that no identical material is listed, but chemically-substituted materials with the same crystal structure are found. Specifically, ∗1 refers to La0.5Sr0.5FeO2.5F0.5, ∗2 to La0.33Sr0.67FeO3, ∗3 to HoxBi1-xFeO3 ($x=0.15,0.20$), and ∗4 to HoxBi1-xFeO3 ($x=0.05,0.10,0.15,0.20$). The ’conv. AFM’ column specifies whether there is a conventional AFM structure also meets the criterion $\Delta E<0.1$ meV. An asterisk $(^{*})$ indicates that a conventional AFM is indicated in the experiments. The ‘Consistency’ column employs two symbols to denote the following: $\times$ represent that a $\bm{q}=0$ structure is reported in the experiments, but our method fails to reproduce it as the most stable states; an asterisk ($*$) indicates that a $\bm{q}\neq 0$ structure is reported in the experiments, which is beyond the reach of our method. All other entries imply that the magnetic structures have either been partially or fully corroborated by the experiments and are also reproduced by our method. Substance |  $G_{\rm SG}$ |  $G_{\rm MSG}$ |  Met./Ins. |  $T_{\rm N}$ [K] |  MAGNDATA |  conv. AFM |  Consistency
---|---|---|---|---|---|---|---
Mn3Ir rt |  $Pm\bar{3}m$ |  $R\bar{3}m^{\prime}$ |  M |  855 |  $\bm{q}=\bm{0}$ |  No |
LaFeO3 rt |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  746 |  ${}^{*1}\bm{q}=\bm{0}$ |  Yes |
LaFeO3 ht1 |  $R\bar{3}c$ |  $P\bar{1},C2/c$ |  I |  740 |  ${}^{*2}\bm{q}\neq\bm{0}$ |  Yes |  $*$
PrFeO3 orth |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  730 |  - |  Yes |
Ca2Fe2O5 rt |  $Pnma$ |  $Pn^{\prime}m^{\prime}a$ |  I |  723 |  $\bm{q}=\bm{0}$ |  No |
EuFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  689 |  - |  Yes |
SmFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  688 |  $\bm{q}=\bm{0}$ |  Yes |
Sr2Fe2O5 rt |  $Ima2$ |  $Im^{\prime}a2^{\prime}$ |  I |  680 |  - |  No |  $\times$
KFe[MoO4]2 hp3 |  $P\bar{3}c1$ |  $C2^{\prime}/c^{\prime},C2/c$ |  I |  653 |  - |  No |  $*$
FeBiO3 hp V |  $Pnma$ |  $Pn^{\prime}ma^{\prime}$ |  I |  643 |  ${}^{*3}\bm{q}=\bm{0}$ |  No |
YFeO3 rt |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  639 |  - |  Yes |
HoFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  ∗M |  637 |  ${}^{*3}\bm{q}=\bm{0}$ |  Yes |
ErFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  636 |  - |  Yes |
NaFe[MoO4]2 |  $C2/c$ |  $C2/c$ |  I |  635 |  - |  No |  $*$
TmFeO3 orth |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  630 |  - |  Yes |
YbFeO3 orth |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  629 |  - |  Yes |
FeBiO3 rt |  $R3c$ |  $P1,Cc$ |  I |  627 |  ${}^{*4}\bm{q}=\bm{0}$ |  No |
NdFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  609 |  $\bm{q}=\bm{0}$ |  Yes |
FeS ht2 |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  ∗M |  599 |  - |  Yes |
TlFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  ∗M |  560 |  - |  Yes |
DyFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  540 |  $\bm{q}=\bm{0}$ |  Yes |
GdFeO3 |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  530 |  - |  Yes |
Fe3BO6 |  $Pnma$ |  $Pnm^{\prime}a^{\prime}$ |  I |  508 |  - |  No |
La2NiO4 lt |  $P4_{2}/ncm$ |  $P4_{2}/nc^{\prime}m^{\prime}$ |  I |  420 |  $\bm{q}=\bm{0}$ |  Yes |  $\times$
Mn3Sn |  $P6_{3}/mmc$ |  $Cmc^{\prime}m^{\prime},Cm^{\prime}cm^{\prime}$ |  M |  411 |  $\bm{q}=\bm{0}$ |  No |
Mn3SnN |  $Pm\bar{3}m$ |  $R\bar{3}m^{\prime}$ |  M |  395 |  - |  No |
FeF3 rt |  $R\bar{3}c$ |  $P\bar{1},C2/c$ |  I |  364 |  $\bm{q}=\bm{0}$ |  ∗Yes |
Fe[NCN] |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  I |  345 |  - |  Yes |
LaCrO3 ht1 |  $R\bar{3}c$ |  $C2/c$ |  I |  327 |  $\bm{q}=\bm{0}$ |  ∗No |
Nd2NiO4 rt |  $Cmce$ |  $Cm^{\prime}ca^{\prime}$ |  I |  323 |  $\bm{q}\neq\bm{0}$ |  Yes |  $*$
LaCrO3 rt |  $Pnma$ |  $Pn^{\prime}ma^{\prime},Pn^{\prime}m^{\prime}a$ |  I |  291 |  $\bm{q}=\bm{0}$ |  Yes |
MnPtAl |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  M |  290 |  - |  No |
AgMn3N |  $Pm\bar{3}m$ |  $R\bar{3}m^{\prime}$ |  M |  281 |  - |  ∗Yes |
SrMnO3 4H rt |  $C222_{1}$ |  $C22^{\prime}2_{1}^{\prime}$ |  I |  280 |  - |  Yse |

###  III.2 Crystal structure analysis

In our dataset of 228 materials, several share identical crystal structure
prototypes. To understand the structural types more likely to exhibit high
Néel temperatures, we performed a prototype analysis. In this work, two
conditions define an equivalence relation for classifying these prototypes:
(1) the space group number and (2) the set of atom counts in the unit cell,
although exact matching at the Wyckoff positions is not required.

The results are summarized as follows. The most frequently occurring crystal
prototype is the ThCr2Si2 structure with space group $G_{\rm SG}=$ $I4/mmm$,
appearing 35 times. Most of them are $R$Mn2Si2 and $R$Mn2Ge2 ($R$ is rare-
earth elements), and unfortunately, FM-like AFM structures with zero
propagation vector are hardly realized in this prototype. The second most
common is the HfFe6Ge6 structure with $G_{\rm SG}=$ $P6/mmm$, appearing 27
times. This structure has been the subject of intense recent study due to its
unique properties related to the Kagome net of the Fe layer. The third most
common is the orthorhombic perovskite $AB$O3, appearing 18 times, followed by
the CeCo4B, tetragonal CeFeSi, and HfFe6Sn4Ge2 structures, each appearing 8
times. The orthorhombic TiNiSi appears 7 times, and the NiAs structure 6
times. Among these frequently appearing prototypes, only the HfFe6Ge6 and
HfFe6Sn4Ge2 structures can host non-collinear AFMs based on our CMP+SDFT
analysis. This suggests that high-$T_{N}$, zero-propagation vector, and non-
collinear FM-like AFMs are rarely realized in nature. Therefore, collinear
AFMs breaking TRS, such as altermagnets, would play a significant role in
future spintronics applications. In the following, we will see that the NiAs
structure and its derivatives are such promising crystal structures in the
search for FM-like AFMs.

###  III.3 FM-like AFMs as the most stable structures

Table 1 shows the FM-like AFMs identified as the most stable magnetic
structures in the CMP+SDFT method. The 34 materials are listed, where 8 are
metals and 26 are insulators, although SDFT calculations sometimes fail to
reproduce metallicity due to the approximate treatment of the electronic
correlation effects. For 16 of these 34 AFMs, the magnetic structures are
already established and listed in the MAGNDATA database, as indicated in Table
1. Only two compounds, LaFeO3 ht1 and Nd2NiO3 rt, exhibit AFM structures with
finite propagation vectors, which are beyond the scope of our current study.
However, for 13 out of the remaining 14 compounds, MAGNDATA features at least
one magnetic structure that aligns with one of the most stable AFMs determined
in our calculations. The sole exception is La2NiO4 lt, where a collinear AFM
structure with the magnetic moments aligned parallel to the $a$-axis. In our
CMP+SDFT calculations, the closest initial guess has finite components along
both the $a$-axis and $c$-axis. After self-consistent calculations, this leads
to an AFM structure oriented along the $c$-axis. Since this structure is
different from the one listed in MAGNDATA only in the direction of the
magnetic moments, it could potentially be realized by some efforts including a
chemical doping. It should be noted that although FeF3 rt and LaCrO3 ht1 are
identified as conventional AFMs in MAGNDATA, these are also reproduced in our
calculations as nearly degenerate states.

Even in cases where magnetic structures are not listed in the MAGNDATA
database, there are experimental studies providing direct and indirect
evidence for their magnetic structures in most cases. After examining the
existing literature, it appears that the magnetic structures of Fe[NCN] and
MnPtAl are not fully established in the experiments, in which our results
offer predictive insights into the potential FM-like AFM structures. A brief
discussion of these compounds is presented below.

####  III.3.1 orthorhombic RFeO3

The orthorhombic perovskites $R$FeO3, where $R=$ Pr, Eu, Y, Er, Tm, Yb, and
Gd, are identified as potential FM-like AFMs yet remain unlisted in MAGNDATA.
However, these materials are well-studied, particularly for their
ferroelectric and multiferroic properties. They generally exhibit two types of
magnetic transitions. The first type initially forms a $G_{z}$-type AFM
structure with $G_{\rm MSG}=Pn^{\prime}ma^{\prime}$, followed by a spin-
reorientation transition to a $G_{y}$-type AFM with $G_{\rm
MSG}=Pn^{\prime}m^{\prime}a$. Compounds in this category include $R=$ Pr
Sosnowska and Steichele (1982); Sosnowska and Fischer (1985); Li  _et al._
(2019), Er Gorodetsky  _et al._ (1973); Bazaliy  _et al._ (2004); Deng  _et
al._ (2015), Tm Leake  _et al._ (1968); Kimel  _et al._ (2004), and Yb Bazaliy
_et al._ (2005); Nikitin  _et al._ (2018); Ma  _et al._ (2021). The second
type undergoes a single magnetic transition to the $G_{z}$-type AFM, as in the
cases of $R=$ Y Hahn  _et al._ (2014); Salazar-Rodriguez  _et al._ (2019), Eu
White (1969); Seifu  _et al._ (2006), and Gd White (1969). This fact shows
that the two AFM structures are energetically so close that they can
interchange with a small energy scale of the temperature effects, and our SDFT
calculations correctly capture this feature. Note that although these AFM
states often possess a weak FM component due to the Dzyaloshinskii-Moriya
interaction, which is not accounted for in our initial structures, it appears
during the self-consistent calculation. As these compounds are insulators,
magneto-optical effects are anticipated for their AFM structures, and these
have already been observed experimentally Treves (1965); Kahn  _et al._
(1969). Note that TlFeO3 is also indicated to be an FM-like AFM in our
calculations. While the specifics of its magnetic structure are likely not
well-documented, its crystal structure and systematic variations in $T_{N}$
lead us to expect properties similar to those of $R$FeO3 Kim  _et al._ (2001).

####  III.3.2 Sr2Fe2O5

The compound Sr2Fe2O5 with space group $G_{\rm SG}=Ima2$ is a brownmillerite
oxide. Neutron diffraction experiments suggested that its magnetic structure
is consistent with $G_{\rm MSG}=Im^{\prime}a^{\prime}2$ Takeda  _et al._
(1969); Schmidt and Campbell (2001), however, this result is in contradiction
with our computational prediction of the FM-like AFM with $G_{\rm
MSG}=Im^{\prime}a2^{\prime}$. The discrepancy arises because our initial guess
does not include the experimentally suggested $G$-type AFM structure. The
complexity lies in the fact that such a structure demands a linear combination
of non-equivalent Fe moments, which belong to different irreducible
representations of the point group. To capture such AFM configurations, it
will be necessary to expand the search area by relaxing the present
constraints.

####  III.3.3 AFe[MoO4]2

$A$Fe[MoO4]2, where $A=$ K and Na, are layered molybdates known for their
ferroelastic transition from a high-temperature trigonal phase with $G_{\rm
SG}=P\bar{3}c1$ to a low-temperature monoclinic phase with $G_{\rm SG}=C2/c$
Otko  _et al._ (1978). The frustrated triangular Fe net has prompted interest
in their magnetic properties, and indeed, the multiferroic AFM structures
along with electronic polarization for $R=$ Rb have been intensively discussed
recently Kenzelmann  _et al._ (2007); Mitamura  _et al._ (2014). For the $R=$
K case, coexistence of spin spiral and commensurate AFM structures has been
reported, both with finite propagation vectors Smirnov  _et al._ (2010),
although $T_{N}$ reported in Ref. Smirnov  _et al._ (2010) is $2.5$ K
considerably lower than $653$ K in Ref. Ismailzade  _et al._ (1981) referred
in AtomWork-Adv. In the Na variant, experimental data on magnetic structures
seems to be lacking, but a theoretical investigation suggested a $G$-type AFM
with finite propagation vectors as the most stable state Tamboli  _et al._
(2021), which our current calculations cannot capture.

####  III.3.4 FeS ht2

FeS is characterized as a $p$-type narrow-gap semiconductor and undergoes a
structural transition from a high-temperature hexagonal phase (FeS ht$2$) with
$G_{\rm SG}=P6_{3}/mmc$ to a low-temperature trigonal phase with $G_{\rm
SG}=P\bar{6}2c$ Putnis (1974). Although AtomWork-Adv lists an additional
phase, FeS ht$1$ with $G_{\rm SG}=P6_{3}mc$, we have chosen to exclude it from
consideration in this study since it includes the distortion accompanied by
the Néel transition Li and Franzen (1996). In both high- and low-temperature
phases, FeS manifests collinear AFM order. Within each Fe layer, the local
moments align ferromagnetically, while they stack antiferromagnetically along
the $c$-axis. Additionally, a spin reorientation transition, commonly referred
to as the Morin transition, occurs in the low-temperature phase Coey  _et al._
(1976); Horwood  _et al._ (1976). In this transition, the local moments
reorient from being normal to parallel with respect to the $c$-axis. The easy-
plane AFM structure aligns with the calculated $Cm^{\prime}c^{\prime}m$ phase.
This phase holds the potential for exhibiting anomalous transport phenomena, a
claim recently substantiated through experimental work Takagi  _et al._ .

####  III.3.5 Fe3BO6

Fe3BO6 is an insulator that exhibits a collinear AFM structure with a minor FM
component. Similar to orthoferrites, $R$FeO3, this compound undergoes a spin
reorientation transition between two distinct AFM configurations Wolfe  _et
al._ (1969); Hirano  _et al._ (1974). In the high-temperature phase, the
magnetic moments align antiferromagnetically along the $c$-axis, while two
non-equivalent Fe moments within the same $c$-plane couple ferromagnetically
Nakamura  _et al._ (2017). This magnetic structure matches the $G_{\rm
MSG}=Pnm^{\prime}a^{\prime}$ configuration listed in Table 1. It is
anticipated to exhibit anomalous magneto-optical effects, which is in
alignment with experimental findings Abe  _et al._ (1980).

Table 2: List of FM-like AFMs identified as metastable magnetic structures found by CMP+SDFT calculations. $\Delta E$ is the relative energy to the ground state per one magnetic atom of the FM-like AFM structure. The other notations carry the same meaning as in the Table 1. Substance |  $G_{\rm SG}$ |  $G_{\rm MSG}$ |  $\Delta E$ [meV] |  Met./Ins. |  $T_{\rm N}$ [K] |  MAGNDATA
---|---|---|---|---|---|---
Fe2O3 hem |  $R\bar{3}c$ |  $P\bar{1},C2^{\prime}/c^{\prime},C2/c$ |  0.14, 0.14, 0.15 |  I |  957 |  $\bm{q}=\bm{0}$
CrSb |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  0.11 |  M |  711 |  $\bm{q}=\bm{0}$
FeSe hp1 |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  16.8 |  M |  460 |  -
Ca2MnFeO5 orth1 |  $Pnma$ |  $Pnm^{\prime}a^{\prime},Pn^{\prime}m^{\prime}a$ |  0.4, 0.3 |  I |  407 |  -
Mn3SnC |  $Pm\bar{3}m$ |  $R\bar{3}m^{\prime}$ |  0.3 |  M |  348 |  -
MnTe rt |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  0.2 |  I |  309 |  $\bm{q}=\bm{0}$
CrSb2 |  $Pnnm$ |  $Pnn^{\prime}m^{\prime}$ |  4.4 |  ∗M |  275 |  -

####  III.3.6 Mn3SnN and AgMn3N

Mn3SnN and AgMn3N belong to a family of cubic anti-perovskite nitrides. This
family typically features two types of magnetic structures within the Kagome
(111) plane: two 120-degree triangular spin textures that are rotated by 90
degrees relative to each other Fruchart and F. Bertaut (1978). These
configurations are referred to as the $\Gamma_{4g}$ and $\Gamma_{5g}$
structures, whose magnetic symmetries are $G_{\rm MSG}=R\bar{3}m^{\prime}$ and
$R3m$, respectively. Although the energy levels of these two states are close
due to a result of the six-fold symmetry of magnetic anisotropy and the small
spin-orbit couplings, only the $\Gamma_{4g}$ state exhibits the anomalous Hall
effect attributed to nonzero Berry curvature, which has been experimentally
observed in Mn3SnN thin films Gurung  _et al._ (2019); You  _et al._ (2020).
More precisely, AgMn3N initially exhibits the $\Gamma_{5g}$ structure before
transitioning to a mixed phase of $\Gamma_{4g}+\Gamma_{5g}$ states.
Conversely, Mn3SnN starts with the mixed state and later transitions to a more
complex phase characterized by finite propagation vectors Fruchart and F.
Bertaut (1978). Note that our calculations focus solely on states with zero
propagation vector and exclude mixed states with different irreducible
representations, and thus, an exact match with experimental observations
cannot be achieved. Nevertheless, both the $\Gamma_{4g}$ and $\Gamma_{5g}$
states are identified as one of the ground state magnetic structures in
AgMn3N, and $\Gamma_{4g}$ as the ground state and $\Gamma_{5g}$ as the meta-
stable state in Mn3SnN. Although accurately estimating the stability of these
structures is challenging due to minor differences in energy scales, these
results indicate that our scheme reliably finds potentially useful magnetic
structures. Note that the magnetic structures and anomalous Hall effects of
this Mn${}_{3}X$N group have been the subject of intensive theoretical
investigation in recent years Huyen  _et al._ (2019); Zhou  _et al._ (2019).

Figure 1:  Magnetic configurations of (a) $G_{\rm MSG}=Cm^{\prime}c^{\prime}m$
structure of Fe[NCN], (b) $G_{\rm MSG}=Cm^{\prime}c^{\prime}m$ structure, and
(c) $G_{\rm MSG}=Cmcm$ structure of MnPtAl.

####  III.3.7 Fe[NCN]

Fe[NCN] is recognized as a potent electrode material in Li- and Na-ion
batteries Chen  _et al._ (2020). It crystallizes in a hexagonal lattice with
space group $G_{\rm SG}=P6_{3}/mmc$ at room temperature, featuring Fe2+ planes
interconnected by linear NCN2- chains. While antiferromagnetic coupling of Fe
atoms has been suggested in prior studies Liu  _et al._ (2009); Herlitschke
_et al._ (2014), the exact magnetic structure seems to remain undetermined.
Our results, whose magnetic structure is shown in Fig. 1(a), indicate that
exploring the magneto-optical properties of this material could be quite
interesting.

####  III.3.8 MnPtAl

MnPtAl is an intermetallic compound with a Ni2In-type structure, sharing the
same crystal arrangement as MnPtGa, known to host a Néel-type skyrmion phase
Srivastava  _et al._ (2020). Neutron studies suggested that the Mn moments in
MnPtAl align ferromagnetically within the plane and antiferromagnetically
along the $c$-axis Brück  _et al._ (1996). These magnetic moments are
perpendicular to the $c$-axis, which is in agreement with the $G_{\rm
MSG}=Cm^{\prime}c^{\prime}m$ phase predicted in our calculations, shown in
Fig. 1(b). It is worth noting that, similar to FeS with the NiAs structure,
the in-plane direction of the magnetic moments is critically important; a
30-degree rotation of each spin can transform the FM-like AFM into a
conventional AFM, shown in Fig. 1(c). Thus, the experimental determination of
the local moment directions, as well as the observation of anomalous Hall
effects, is important, although our calculations indicated that the in-plane
magnetic anisotropy is not very weak as no conventional AFM states are found
as the nearly degenerate states Seki  _et al._ .

####  III.3.9 SrMnO3 4H rt

SrMnO$3$ 4H rt is a distorted hexagonal perovskite with $G_{\rm SG}=C222_{1}$,
and is found to have a magnetic structure identical to the $G_{\rm
MSG}=C22^{\prime}2^{\prime}_{1}$ in our calculations. This is consistent with
neutron diffraction studies that describe the AFM ordering along the $c$-axis
both within the Mn2O9 dimers and between the corner-sharing MnO6 octahedra,
with spins parallel to the $a$-axis Søndenå  _et al._ (2006); Daoud-Aladine
_et al._ (2007). Note that a 90-degree rotated AFM structure also falls under
the category of FM-like AFM unlike FeS, Fe[NCN], and MnPtAl structures.

Table 3: Summary of FM-like AFMs identified as the most stable magnetic
structures newly found by CMP+SDFT+$U$ calculations. The notations carry the
same meaning as in the Table 1. Note that CrSb and MnTe rt are listed in TABLE
II.

Substance |  $G_{\rm SG}$ |  $G_{\rm MSG}$ |  Met./Ins. |  $T_{\rm N}$ [K] |  MAGNDATA |  conv. AFM |  consistency
---|---|---|---|---|---|---|---
CrSb |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  ∗M |  711 |  $\bm{q}=\bm{0}$ |  No |
Mn2As |  $P4/nmm$ |  $P4/nm^{\prime}m^{\prime}$ |  M |  582 |  $\bm{q}\neq\bm{0}$ |  No |  $*$
Sr2Co2O5 |  $Ima2$ |  $Ima^{\prime}2^{\prime}$ |  M |  535 |  $\bm{q}\neq\bm{0}$ |  No |  $*$
Fe2As |  $P4/nmm$ |  $Pmm^{\prime}n^{\prime},Cmm^{\prime}a^{\prime}$ |  M |  353 |  $\bm{q}\neq\bm{0}$ |  No |  $*$
FeGe rt |  $P2_{1}3$ |  $P2_{1}^{\prime}2_{1}^{\prime}2_{1}$ |  M |  310 |  $\bm{q}\neq\bm{0}$ |  Yes |  $*$
MnTe rt |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  I |  309 |  $\bm{q}=\bm{0}$ |  Yes |
NiS ht |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  I |  304 |  - |  ∗Yes |

Table 4: List of FM-like AFMs identified as metastable magnetic structures
newly found by CMP+SDFT+$U$ calculations. The notations carry the same meaning
as in the Table 2. Note that FeS ht2, Fe[NCN], MnPtAl, and AgMn3N are listed
in TABLE I.

Substance |  $G_{\rm SG}$ |  $G_{\rm MSG}$ |  $\Delta E$ [meV] |  Met./Ins. |  $T_{\rm N}$ [K] |  MAGNDATA
---|---|---|---|---|---|---
FeS ht2 |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  0.9 |  I |  599 |  -
Fe[NCN] |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  2.4 |  I |  345 |  -
MnNiGe rt |  $Pnma$ |  $Pnm^{\prime}a^{\prime}$ |  1.3 |  M |  336 |  -
Ba2Y2Co4O11 rt |  $Pmmm$ |  $Pm^{\prime}m^{\prime}m$ |  14.5 |  M |  291 |  -
MnPtAl |  $P6_{3}/mmc$ |  $Cm^{\prime}c^{\prime}m$ |  2.5 |  M |  290 |  -
AgMn3N |  $Pm\bar{3}m$ |  $R\bar{3}m^{\prime}$ |  0.1 |  M |  281 |  -

###  III.4 FM-like AFMs as the metastable structures

An advantage of the CMP method lies in its ability to systematically assess
the magnetic structures with energies of metastable states. Table 2 outlines
FM-like AFM materials that we identified as metastable states; these are
characterized by $\Delta E$ is less than 20 meV. Table 2 shows 7 such
metastable magnetic structures, with experimental data for three of these
materials available in MAGNDATA. Thanks to their small $\Delta E$, we expect
that it is possible to realize them given the experimental tunability, finite
temperature effects, and the limitation of the accuracy of SDFT. For instance,
Fe2O3 hem exhibits a collinear AFM and undergoes a Morin transition, altering
the direction of magnetic moments from in-plane at high temperatures to out-
of-plane at low temperatures Hill  _et al._ (2008). These transitions
respectively align well with the $C2^{\prime}/c^{\prime}$ and $P\bar{1}$
states in Table 2. MnTe rt, which adopts the NiAs structure, serves as another
such example. This compound has been reported to exhibit an antiferromagnetic
(AFM) state with magnetic moments aligned parallel to the $[1\bar{1}00]$
direction, representing a conventional AFM structure Kunitomi  _et al._
(1964). However, recent research suggests that strain induced by epitaxial
lattice mismatch with a substrate alters the in-plane magnetic anisotropy.
Specifically, this leads the magnetic moments to align parallel to the
$[11\bar{2}0]$ direction, corresponding to the $Cm^{\prime}c^{\prime}m$ state
listed in Table 2 Kriegner  _et al._ (2017). In fact, this structure has been
the subject of recent study as a typical example of an altermagnet Gonzalez
Betancourt  _et al._ (2023). These observations demonstrate the effectiveness
of our approach in identifying metastable structures.

For the other candidates, we confirmed that the experimental structures do not
correspond to FM-like AFMs listed in Table 2. Here, we briefly comment on
them. CrSb belongs to the NiAs structure and shows staggered AFM along the
$c$-axis, whose magnetic moments are aligned parallel to the $c$-axis Snow
(1953); Yuan  _et al._ (2020). $Cm^{\prime}c^{\prime}m$ structure in Table 2
is obtained by rotating spins of it to the in-plane. MnSn3C is known to show
complex spin configurations consisting of AFM and AM components with finite
propagation vectors, as suggested in Ref. Lorthioir  _et al._ (1973). Our
results show that $R\bar{3}m^{\prime}$ structure, which is the same as MnSn3N,
also lies as a low energy state. Ca2FeMnO5 orth1 is known to show $G$-type AFM
structure, whose magnetic moment is parallel to the $c$-axis Rykov  _et al._
(2008). Our $Pnm^{\prime}a^{\prime}$ and $Pn^{\prime}m^{\prime}a$ structures
correspond to the same AFM structure with the moments perpendicular to the
$c$-axis like Ca2Fe2O5 Auckett  _et al._ (2015). CrSb2 is known to show
collinear AFM structure with a finite propagation vector Holseth  _et al._
(1970), which cannot be accessible in our current scheme. FeSe hp1 is known to
show both AFM and FM-like behavior, depending on temperature and amount of
self-intercalation of Se Song  _et al._ (2013). Since the origin of the AFM
nature of FeSb has not been clear yet, we may have a chance to realize an FM-
like AFM state in hexagonal FeSe.

###  III.5 FM-like AFMs in SDFT+U calculations

Finally, we comment on the results with the SDFT+$U$ calculations. As widely
recognized, strong electronic correlation effects play a critical role in many
magnetic materials, especially in oxides, and these effects are not
sufficiently captured by standard SDFT. Therefore, we explored the impact of
the correlation effects on magnetic structures by performing the SDFT+$U$
calculations. According to the systematic calculations, we found that while
the inclusion of the Hubbard $U$ does affect the energy levels of various
magnetic states, it does not significantly change their order in most
materials. Indeed, we identified only 7 FM-like AFMs as ground states that are
not found in the SDFT calculations, as shown in TABLE 4. Of these, 2 are also
identified as metastable states in the SDFT calculations. Among the remaining
5 materials, 4 have established AFM structures listed in MANGDATA, all of
which have finite propagation vectors, and thus, fall outside the scope of our
current scheme. The last one, NiS ht with the NiAs-structure, is known in
experiments to be an AFM oriented along the $c$-axis Sparks and Komoto (1963).
This state is indeed found in our calculations as one of the most stable
magnetic structures; however, it is a conventional AFM.

The results for metastable states are shown in Table 4. Our claculations
identify 6 FM-like AFMs not identified as metastable states in the SDFT
calculations, while 4 of them are already listed as the most stable magnetic
structures in the SDFT calculations. The remaining two are MnNiGe rt and
Ba2Y2Co4O11 rt. MnNiGe shares the same crystal structures as NbMnP, in which
the anomalous Hall effect in an AFM state has been recently observed Kotegawa
_et al._ (2023). However, MnNiGe is known to be an AFM with a finite
propagation vector, making it outside what our calculations can handle
Fjellvåg and Andresen (1985). Similarly, Ba2Y2Co4O11, a layered perovskite, is
also known to be an AFM with a finite propagation vector, again exceeding the
bounds of our current approach Aurelio  _et al._ (2007). In summary, while
certain materials have been newly recognized through the SDFT+$U$
calculations, a majority coincide with those identified in the SDFT
calculations, suggesting that the incorporation of $U$ exerts small influence
on the stability of magnetic structures, particularly when metastable states
are also considered in the evaluation.

##  IV Conclusion

In this work, we have introduced a high-throughput computational framework
utilizing CMP theory and SDFT calculations to systematically identify new
functional AFMs, suitable for spintronics applications. We applied this
approach to 228 magnetic compounds cataloged in the AtomWork-Adv database,
renowned for their elevated Néel temperatures. Our systematic investigations
yielded 34 compounds that display AFM structures with zero propagation
vectors, featuring magnetic symmetries identical to conventional FMs. Among
these, 9 are metallic and 25 are insulating, and our results include known FM-
like AFMs such as Mn3Sn and Mn3SnN. Furthermore, we identified MnPtAl and
Fe[NCN] as promising, yet experimentally undetermined, candidates for FM-like
AFM materials. By cross-referencing our theoretical predictions with existing
experimental data from the MAGNDATA database and published literature, we
confirmed the accuracy of our findings for 26 out of the 28 compounds that
have been partially or fully characterized in experiments. This validation not
only attests to the reliability of our computational scheme but also hints at
its broader applicability in the continuous endeavor to discover new
functional magnetic materials. Importantly, our method has also uncovered a
range of metastable FM-like AFM structures that could be experimentally
realized through techniques such as chemical doping or synthesis optimization.

In summary, our study establishes a solid foundation for future research that
could expand the scope to include a wider variety of crystal structures and
magnetic arrangements with finite propagation vectors, thereby opening new
avenues for the discovery of novel functional AFM materials

##  V Acknowledgments

We are grateful to Dr. Tanno for the technical supports. We also acknowledge
the use of supercomputing system, MASAMUNE-IMR, at CCMS, Tohoku University in
Japan. This work was financially supported by JSPS-KAKENHI (No. JP23H04869,
JP23H01130, JP22K03447, JP22H00290, JP22K14587, JP21H04437, JP21H04990,
JP21H01789, JP21H01031, and JP19H05825), JST-Mirai Program (JPMJMI20A1), JST-
PRESTO (No. JPMJPR20L7), JST-CREST (No. JPMJCR18T3, JPMJCR23O3), and JST-
ASPIRE (No. JPMJAP2317).

## Appendix: List of all materials

This appendix shows the summary of materials we considered in this paper.
These are listed in the AtomWork-Adv database and satisfy (1) a Néel
temperature exceeding 273 K, (2) a unit cell containing between 2 and 24
magnetic atoms, and (3) the absence of disorder or partially occupied sites.
The results are shown in Table LABEL:tab:all_list.

Table 5: List of materials considered in this paper. Remarks for Formula, $G_{\rm SG}$, and $T_{N}$ [K] are the same as in Table 1 Formula | $G_{\rm SG}$ |  $T_{\rm N}$ [K] |  | Formula | $G_{\rm SG}$ |  $T_{\rm N}$ [K] |  | Formula | $G_{\rm SG}$ |  $T_{\rm N}$ [K]
---|---|---|---|---|---|---|---|---|---|---
CsFeO2 ht | $Fd\bar{3}m$ | 1055 |  | TbFe6Ge6 | $Cmcm$ | 489 |  | AuMn2 rt | $I4/mmm$ | 366
RbFeO2 ht | $Fd\bar{3}m$ | 1027 |  | TmMn2Ge2 | $I4/mmm$ | 487 |  | LuMn6Sn4Ge2 | $P6/mmm$ | 366
KFeO2 rt | $Pbca$ | 1001 |  | ScFe6Ge6 | $P6/mmm$ | 485 |  | FeF3 rt | $R\bar{3}c$ | 364
Fe2O3 hem | $R\bar{3}c$ | 957 |  | ThMn2Si2 | $I4/mmm$ | 483 |  | ErMn6Sn4Ge2 | $P6/mmm$ | 362
Mn3N2 ht1 | $I4/mmm$ | 919 |  | TmMn6Ge6 | $P6/mmm$ | 482 |  | CaMnSi | $P4/nmm$ | 360
Dy4Ni6Al23 | $C2/m$ | 900 |  | SrFe12O19 | $P6_{3}/mmc$ | 480 |  | TmMn6Sn4Ge2 | $P6/mmm$ | 358
TiO2 ana | $I4_{1}/amd$ | 880 |  | YbFe6Ge6 | $P6/mmm$ | 480 |  | Sr2FeO3F | $P4/nmm$ | 358
Mn3Ir rt | $Pm\bar{3}m$ | 855 |  | YbMn6Ge6 | $P6/mmm$ | 480 |  | Fe2As | $P4/nmm$ | 353
Mn4N rt | $Pm\bar{3}m$ | 750 |  | YMn6Ge6 | $P6/mmm$ | 476 |  | Au5Mn2 rt | $C2/m$ | 351
LaFeO3 rt | $Pnma$ | 746 |  | ErMn6Ge6 | $P6/mmm$ | 475 |  | ErMn6Sn6 | $P6/mmm$ | 351
LaFeO3 ht1 | $R\bar{3}c$ | 740 |  | DyCo3Ga2 | $P6/mmm$ | 475 |  | ErMn2Si2 | $I4/mmm$ | 350
PrFeO3 orth | $Pnma$ | 730 |  | TmFe6Ge6 | $Immm$ | 474 |  | BaY2CoO5 | $Pnma$ | 350
La2BiO2 | $I4/mmm$ | 723 |  | YMn2Si2 | $I4/mmm$ | 470 |  | GdMn2Ge2 | $I4/mmm$ | 350
Ca2Fe2O5 rt | $Pnma$ | 723 |  | LuFe6Ge6 | $P6/mmm$ | 469 |  | Mn3SnC | $Pm\bar{3}m$ | 348
CrSb | $P6_{3}/mmc$ | 711 |  | LaMn2Si2 | $I4/mmm$ | 468 |  | NdMnAsO | $P4/nmm$ | 347
CaFeO3 rt | $Pnma$ | 710 |  | HoMn6Ge6 | $P6/mmm$ | 466 |  | VO2 rt | $P2_{1}/c$ | 345
Ba2YFe3O8 tet1 | $P4/mmm$ | 690 |  | LuMn2Si2 | $I4/mmm$ | 464 |  | Fe[NCN] | $P6_{3}/mmc$ | 345
EuFeO3 | $Pnma$ | 689 |  | MnIrSi | $Pnma$ | 460 |  | FeGe ht2 | $C2/m$ | 342
SmFeO3 | $Pnma$ | 688 |  | BaMnGe | $P4/nmm$ | 460 |  | GdMn2Si2 | $I4/mmm$ | 340
Sr2Fe2O5 rt | $Ima2$ | 680 |  | FeSe hp1 | $P6_{3}/mmc$ | 460 |  | SmMn2Ge2 | $I4/mmm$ | 340
Fe3O4 rt | $Fd\bar{3}m$ | 679 |  | HfFe6Ge6 | $P6/mmm$ | 457 |  | YMn6Sn6 | $P6/mmm$ | 339
CaMn2Ge2 | $I4/mmm$ | 670 |  | TbCo4B | $P6/mmm$ | 455 |  | Ba2MoFeO6 ht | $Fm\bar{3}m$ | 338
KFe[MoO4]2 hp3 | $P\bar{3}c1$ | 653 |  | Mn3Pd5 rt | $Cmmm$ | 450 |  | MnNiGe rt | $Pnma$ | 336
HfMn6Ge6 | $P6/mmm$ | 650 |  | BaCuYFeO5 | $P4mm$ | 448 |  | YMn6Sn4Ge2 | $P6/mmm$ | 336
FeBiO3 hp V | $Pnma$ | 643 |  | Fe7Se8 ht | $P3_{1}21$ | 447 |  | Ba2MoFeO6 rt | $I4/mmm$ | 334
YFeO3 rt | $Pnma$ | 639 |  | YbMn2Ge2 | $I4/mmm$ | 434 |  | NiS rt | $R3m$ | 330
HoFeO3 | $Pnma$ | 637 |  | TbMn2Ge2 | $I4/mmm$ | 434 |  | LaCrO3 ht1 | $R\bar{3}c$ | 327
ErFeO3 | $Pnma$ | 636 |  | YMn2Ge2 | $I4/mmm$ | 431 |  | MnSn2 | $I4/mcm$ | 324
NaFe[MoO4]2 | $C2/c$ | 635 |  | DyMn2Ge2 | $I4/mmm$ | 431 |  | Nd2NiO4 rt | $Cmce$ | 323
ErFe4B | $P6/mmm$ | 630 |  | BaRuO3 rt | $R\bar{3}m$ | 430 |  | Sr2MgReO6 rt | $I4/mmm$ | 320
TmFeO3 orth | $Pnma$ | 630 |  | TbMn6Sn4Ge2 | $P6/mmm$ | 428 |  | MnAs rt | $P6_{3}/mmc$ | 318
YbFeO3 orth | $Pnma$ | 629 |  | MnNiO3 | $R\bar{3}$ | 427 |  | LaMnAsO | $P4/nmm$ | 317
FeBiO3 rt | $R3c$ | 627 |  | SmMn2Si2 | $I4/mmm$ | 425 |  | Cr | $Im\bar{3}m$ | 312
GdCo3 | $R\bar{3}m$ | 615 |  | TbMn6Sn6 | $P6/mmm$ | 422 |  | FeGe rt | $P2_{1}3$ | 310
TmFe4B | $P6/mmm$ | 610 |  | DyMn6Ge6 | $P6/mmm$ | 421 |  | KFeSe2 | $C2/c$ | 310
NdFeO3 | $Pnma$ | 609 |  | CuEu2O4 rt | $I4/mmm$ | 421 |  | BaCoS2 rt mon | $Pmmn$ | 310
FeS ht2 | $P6_{3}/mmc$ | 599 |  | NdMn2Ge2 | $I4/mmm$ | 420 |  | TbMn5Ge3 | $Pnma$ | 310
FeS tro | $P\bar{6}2c$ | 593 |  | La2NiO4 lt | $P4_{2}/ncm$ | 420 |  | LaMnSi | $P4/nmm$ | 310
HoCr2Si2 | $I4/mmm$ | 590 |  | DyCo4B | $P6/mmm$ | 420 |  | MnTe rt | $P6_{3}/mmc$ | 309
FeS ht1 | $P6_{3}mc$ | 588 |  | CaMnGe | $P4/nmm$ | 420 |  | NdCo2P2 | $I4/mmm$ | 309
Mn2As | $P4/nmm$ | 582 |  | TbMn6Ge6 | $P6/mmm$ | 419 |  | ScMn6Ge6 | $P6/mmm$ | 308
ZrMn6Sn6 | $P6/mmm$ | 580 |  | CoRh2S4 | $Fd\bar{3}m$ | 418 |  | CrAs rt | $Pnma$ | 306
CuFeS2 hp | $I\bar{4}$ | 575 |  | PrMn2Ge2 | $I4/mmm$ | 418 |  | MnRu2Ge | $Fm\bar{3}m$ | 305
HfMn6Sn6 | $P6/mmm$ | 575 |  | Sr2MoFeO6 rt | $I4/m$ | 416 |  | Cr2O3 | $R\bar{3}c$ | 305
NbFe6Ge6 | $P6/mmm$ | 561 |  | GdTiGe ht | $P4/nmm$ | 412 |  | PrCo2P2 | $I4/mmm$ | 304
TlFeO3 | $Pnma$ | 560 |  | Mn3Sn | $P6_{3}/mmc$ | 411 |  | NiS ht | $P6_{3}/mmc$ | 304
NdMnSiH | $P4/nmm$ | 560 |  | DyMn6Sn4Ge2 | $P6/mmm$ | 411 |  | BaCuLuFeO5 | $P4mm$ | 303
ErFe6Sn6 | $Cmcm$ | 560 |  | TbMnSi rt | $Pnma$ | 410 |  | SmCo2P2 | $I4/mmm$ | 302
HoFe6Sn6 orth | $Immm$ | 559 |  | Ca2MnFeO5 orth1 | $Pnma$ | 407 |  | BaY2NiO5 | $Immm$ | 300
DyFe6Sn6 rt-orth1 | $Cmcm$ | 559 |  | HoCo4B | $P6/mmm$ | 404 |  | La3Mn4Sn4 | $Immm$ | 300
GdFe6Sn6 | $Cmcm$ | 554 |  | FeGe ht1 | $P6/mmm$ | 404 |  | BaNdMn2O6 ht | $P4/mmm$ | 300
CuMn3As2 | $Pnma$ | 550 |  | GdTiSi orth | $P4/nmm$ | 400 |  | HoMn2Ge2 | $I4/mmm$ | 300
Mn ht2 stab | $I4/mmm$ | 540 |  | LaMn2Ge2 | $I4/mmm$ | 398 |  | BaMn3O6 | $C2/m$ | 295
DyFeO3 | $Pnma$ | 540 |  | ScMn6Sn4Ge2 | $P6/mmm$ | 397 |  | CuTb2O4 | $I4/mmm$ | 295
Ca2ReFeO6 lt | $P2_{1}/c$ | 540 |  | DyMn6Sn6 | $P6/mmm$ | 395 |  | LaCrO3 rt | $Pnma$ | 291
Sr2Co2O5 | $Ima2$ | 535 |  | Mn3SnN | $Pm\bar{3}m$ | 395 |  | Ba2Y2Co4O11 rt | $Pmmm$ | 291
GdFeO3 | $Pnma$ | 530 |  | Mn3B4 | $Immm$ | 394 |  | MnPtAl | $P6_{3}/mmc$ | 290
NiO ht | $Fm\bar{3}m$ | 526 |  | CeMn2Ge2 | $I4/mmm$ | 393 |  | ErMn2Ge2 | $I4/mmm$ | 290
Ba4Fe9O14[OH]6 | $C2/m$ | 520 |  | Cr2As rt | $P4/nmm$ | 393 |  | Ba2Dy2Co4O11 ht | $Pmmm$ | 290
YbMn2Si2 | $I4/mmm$ | 520 |  | HoMn6Sn4Ge2 | $P6/mmm$ | 391 |  | SrMnO3 4H | $P6_{3}/mmc$ | 290
LuMn6Ge6 | $P6/mmm$ | 518 |  | MnRhSi | $Pnma$ | 389 |  | CoO ht | $Fm\bar{3}m$ | 289
Ho6Fe23 | $Fm\bar{3}m$ | 515 |  | ScMn6Sn6 | $P6/mmm$ | 388 |  | CoO rt | $I4/mmm$ | 289
HoCo3Ga2 | $P6/mmm$ | 514 |  | ErCo4B | $P6/mmm$ | 387 |  | TbTiSi | $P4/nmm$ | 287
TiFe6Ge6 | $P6/mmm$ | 513 |  | GdMnGe | $Pnma$ | 382 |  | CeCo4B | $P6/mmm$ | 287
TbMnGe | $Pnma$ | 510 |  | FeSn2 rt | $I4/mcm$ | 381 |  | TbTiGe rt | $P4/nmm$ | 287
TbCo3 ht | $R\bar{3}m$ | 509 |  | Sr2Mn2O5 orth1 | $Pbam$ | 380 |  | CrN lt | $Pmmn$ | 285
TbMn2Si2 | $I4/mmm$ | 509 |  | MnCoSi rt | $Pnma$ | 380 |  | CrN rt | $Fm\bar{3}m$ | 283
ZrFe6Ge6 | $P6/mmm$ | 509 |  | Sr3Fe2Cl2O4 | $I4/mmm$ | 378 |  | AgMn3N | $Pm\bar{3}m$ | 281
Fe3BO6 | $Pnma$ | 508 |  | CeMn2Si2 | $I4/mmm$ | 378 |  | SrMnO3 4H rt | $C222_{1}$ | 280
LuMn2Ge2 | $I4/mmm$ | 508 |  | NdMn2Si2 | $I4/mmm$ | 377 |  | TmMn6Sn6 | $P6/mmm$ | 280
DyMn2Si2 | $I4/mmm$ | 506 |  | HoMn6Sn6 | $P6/mmm$ | 376 |  | Fe2RuSi | $Fm\bar{3}m$ | 280
GdCo4B | $P6/mmm$ | 505 |  | LuMn6Sn6 | $P6/mmm$ | 373 |  | TiFe2 rt | $P6_{3}/mmc$ | 278
TmMn2Si2 | $I4/mmm$ | 498 |  | V2WO6 | $P4_{2}/mnm$ | 370 |  | NiBiO3 | $P\bar{1}$ | 278
DyFe6Ge6 | $Cmcm$ | 497 |  | PrMn2Si2 | $I4/mmm$ | 369 |  | CuHo2O4 hp | $I4/mmm$ | 278
Fe2O3 orth | $Pna2_{1}$ | 495 |  | BaTbMn2O6 rt | $P2_{1}/m$ | 367 |  | CrSb2 | $Pnnm$ | 275
Sr3Cu2Fe2Se2O5 | $I4/mmm$ | 490 |  | FeSn | $P6/mmm$ | 367 |  | YFe2Si2 | $I4/mmm$ | 275

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