UFABC]Federal University of ABC (UFABC), 09210-580, Santo André, São Paulo,
Brazil These authors contributed equally to this work. UFABC]Federal
University of ABC (UFABC), 09210-580, Santo André, São Paulo, Brazil These
authors contributed equally to this work. UFABC]Federal University of ABC
(UFABC), 09210-580, Santo André, São Paulo, Brazil UFABC]Federal University of
ABC (UFABC), 09210-580, Santo André, São Paulo, Brazil ML, 2D, TMDs, DFT, PEC

# Machine Learning Study of the Magnetic Ordering in 2D Materials

Carlos Mera Acosta<EMAIL_ADDRESS>[  Elton Ogoshi
<EMAIL_ADDRESS>[  Jose Antonio Souza  [  Gustavo M. Dalpian
<EMAIL_ADDRESS>[

###### Abstract

Magnetic materials have been applied in a large variety of technologies, from
data storage to quantum devices. The development of 2D materials has opened
new arenas for magnetic compounds, even when classical theories discourage
their examination. Here we propose a machine-learning-based strategy to
predict and understand magnetic ordering in 2D materials. This strategy
couples the prediction of the existence of magnetism in 2D materials using
random forest and the SHAP method with material maps defined by atomic
features predicting the magnetic ordering (ferromagnetic or
antiferromagnetic). While the random forest model predicts magnetism with an
accuracy of 86%, the material maps obtained by the SISSO method have an
accuracy of about 90% in predicting the magnetic ordering. Our model indicates
that 3$d$ transition metals, halides, and structural clusters with regular
transition metals sublattices have a positive contribution in the total weight
deciding the existence of magnetism in 2D compounds. This behavior is
associated with the competition between crystal field and exchange splitting.
The machine learning model also indicates that the atomic SOC is a determinant
feature for the identification of the patterns separating ferro- from
antiferromagnetic order. The proposed strategy is used to identify novel 2D
magnetic compounds which, together with the fundamental trends in the chemical
and structural space, paves novel routes for experimental exploration.

###### keywords:

Two-dimensional materials, Machine Learning, High throughput screening,
magnetism, ferromagnetic, antiferromagnetic

##  1 Introduction

The first synthesis of graphene opened a new era for two-dimensional (2D)
materials 1. These materials, with strong in-plane and weak interlayer
interactions, enclose nearly all electronic and optical phenomena found in
solids. The subsequent experimental realization and theoretical insights of
other 2D materials 2 have influenced and transformed our fundamental
understanding of materials properties. It also envisaged and put forward
several technological applications in many areas of condensed matter physics,
chemistry, and materials science. In this 2D materials revolution, some
breakthroughs were achieved including the discovery of superconductivity in
rotated bilayer graphene 3, the theoretical description of symmetry protected
topological phases in bismutene 4 and the extremely high electron mobilities
in phosphorene5. Although 2D materials physics overlaps many other well-
established areas, historically, it has been anticipated the existence of
mutual exclusion with magnetism, i.e., long-range ordering of spin and orbital
magnetic moments 6. This brings limitations for the integration of 2D
materials in magnetic devices.

The study of magnetism and magnetic materials goes back to the beginning of
the theoretical foundations of physics and chemistry. However, even though
there have been theoretical predictions of magnetic stable monolayers7, a
definitive experimental observation of magnetic order at finite temperature in
2D layered materials was only reported recently8, 9, 10, 11. Specifically,
scanning magneto-optic Kerr microscopy measurements revealed intrinsic long-
range ferromagnetic order in CrGeTe3 Van der Waals (VdW) atomic layers12.
Similarly, Bevin Huang et al., demonstrated for the first time the existence
of ferromagnetism in CrI3 VdW crystals down to the monolayer limit13. The
discovery of 2D magnetism has not only opened a new path for potential
revolutionary spintronic 14 and valleytronic 15 applications but also raised
fundamental questions about the understanding and prediction of ferromagnetic
(FM) and antiferromagnetic (AFM) ordering in 2D compounds.

In three-dimensional bulk materials, the nature of magnetic ordering ground
state is determined by collective mechanisms involving magneto-crystalline
anisotropy arising from the spin-orbit coupling and how magnetic moments (S
and L) are effectively coupled by exchange interactions. Furthermore, the
interplay and coexistence of exchange interactions can give rise to not only
FM and AFM orderings, but also to a very rich variety of different magnetic
states even in compounds with similar crystal structures.16 In the simplest
scenario, two identical transition metal ions with unpaired electrons interact
through the isotropic Heisenberg Hamiltonian $\hat{H}=-JS_{1}S_{2}$, where the
electron spins $S_{1}$ and $S_{2}$ are coupled by the exchange interaction
parameter $J$. In this isotropic magnet, depending on the extent of
delocalization of the magnetic moments, the magnetic order can be described by
four exchange interactions – indirect, itinerant, direct, and super
exchanges7. These exchange interactions can coexist in a given compound,
suggesting that even without magneto-crystalline anisotropy, the magnetic
ordering is determined by competing effects. Indeed, the complexity in
predicting magnetic ordering has been recently recalled: "the prediction of
the magnetic state based solely on chemical and structural information is a
delicate exercise"17. The description of magnetic ordering in 2D materials is
even more complex. Specifically, in 1966, based on the isotropic Heisenberg
model, Mermin and Wagner demonstrated that, unlike bulk compounds, in 2D
materials long-range magnetic order is suppressed by thermal fluctuations.
This suggests that 2D materials can only exhibit magnetic order in the
presence of large magneto-crystalline anisotropy and hence, the magnetism in
2D and bulk materials are supposed to have different physical mechanisms.6
Indeed, in general, there is no rule, as a knowledge base, that a priori
determines the magnetic behavior in 2D materials or a trend of AFM or FM
magnetic orderings in the feature space of atomic properties. For instance, 2D
magnetic semiconductors can violate the Goodenough-Kanamori rules for
superexchange interactions18 that discriminate FM from AFM orders according to
the angles between $d$-orbitals from TM and $p$-orbitals from oxygen anions19,
20. Clear and well-defined patterns would facilitate the search for stable AFM
and FM 2D materials. In the ideal scenario, one wishes to determine a set of
stable AFM and FM configurations for a given combination of atoms with a
specific stoichiometry and crystal symmetry without additional calculations.
Our work thus aims to solve this problem using machine learning algorithms.

In recent years, much effort has been invested in the systematic prediction of
magnetic two-dimensional systems 21, 22, 23, 24, 25, 26, 27, 28, 29, coming
mainly from density functional theory (DFT) calculations. These predictions
are based on a direct approach that involves the calculation of all possible
candidates. This trial-and-error process can be time-consuming, and expensive.
Daniele Torelli et al., used the computational 2D materials database (C2DB)
30, 31 to search for new ferromagnetic 2D materials based on the spinwave gap
as a descriptor that accounts for the role of magnetic anisotropy, finding 17
novel insulating materials that exhibit magnetic order at finite temperatures
32. Similarly, a computational screening for 2D magnetic materials based on
experimental bulk compounds that are exfoliable into 2D derivatives 33 found
85 ferromagnetic and 61 antiferromagnetic materials for which DFT calculated
magnetic exchange and anisotropy parameters are reported 34. Finally, Arnab
Kabiraj et al., developed a fully automated code to perform DFT calculations
followed by Heisenberg model-based Monte Carlo simulations to estimate the
Curie temperature from the crystal structure, predicting 26 materials with a
Curie point beyond 400 K 18.

Here we propose a data-driven strategy to explore the magnetism in 2D
materials: contrary to the commonly employed direct approach, we aim to
explore the material-to-attribute connection to provide the simplest
correlation between magnetic ordering and features, e.g., composition, crystal
symmetry, and atomic properties35, 36, 37. Specifically, we train machine
learning algorithms using a recently created database 18 of 2D magnetic
materials to obtain descriptors that are capable of classifying materials as
non-magnetic, ferromagnetic, or antiferromagnetic. Our strategy is divided
into two main steps, namely: i) we first develop a random forest model to
separate magnetic compounds from non-magnetic ones based on trends in the
crystal structure and atomic composition; and ii) based on the sure
independence screening and sparsifying operator (SISSO) method 38, we find a
mathematical model (i.e., a function of the atomic features) that uses the
composition to provide a materials map with defined regions for AFM and FM 2D
materials. The accuracy in the classification of 2D AFM and FM is about 90% in
the training set. We find that the magnetic ordering is decided by features
involving the lowest- occupied Kohn-Sham eigenvalues for the cations and
anions that constitute the target material, their electron affinity, and
atomic radii of $p$-orbitals. However, different rules involving these
features are observed for materials that belong to different space groups,
clearly indicating the complexity of this problem.

##  2 Machine learning models, results and discussion

Figure 1: Illustration of the proposed machine learning strategy to determine
and predict the existence of magnetism (step I) in a given 2D compound and its
specific magnetic ordering (step II). The starting point for this strategy is
the clustering of atomic species and structures. Atomic species are
intrinsically clustered according to the organization of the periodic table,
while an ML model is performed to group structures according to the possible
composition, crystal point group, and local distortions. This cluster is then
used to identify the tendency of a specific element and structure to form a
magnetic (blue) or non-magnetic (red) 2D structure. If the specific
combination of atoms and structures for the 2D compound tends to be magnetic,
we identify its position in a materials map separating antiferromagnetic from
ferromagnetic ordering. These positions are computed by means of atomic
features of the constituent atoms. Finally, this ML strategy allows us to
predict and select compounds for specific applications in electronics and
spintronics.

In the ideal scenario, one would like to have a model determining the
existence and type of magnetic ordering in an arbitrary combination of Atoms,
in a specific Composition, and Symmetry (ACS). In this section, we present the
proposed strategy to solve the two above-noted problems. First, as presented
in Figure1, the proposed scheme divides the problem into two steps, namely:
(I) classification for magnetic vs non-magnetic materials and (II)
classification for AFM vs FM ordering in 2D compounds. In order to input the
ACS attributes into the ML model, we start by encoding the materials’
compositions and crystal structure information into proper machine-readable
features. In this first step, these features are used to study the
distribution of magnetism on the space of different two-dimensional materials.
Compounds are then classified as magnetic (M) or non-magnetic (NM) according
to the DFT-calculated magnetic ground state. The second step is the machine
learning classification that separates AFM from FM 2D materials to construct a
materials map of compounds with magnetic ordering. The axis of this materials
map (see Figure 1) are given by descriptors that are functions of the atomic
properties $D_{n}=f(\xi_{1},\xi_{2},...,\xi_{n})$, where $n$ is the descriptor
dimension and $\\{\xi_{m}\\}$ are the $m$ atomic properties identifying the
compound. As illustrated in Figure 1, the prediction of novel magnetic 2D
compounds starts by selecting both the atomic elements and structural clusters
with a larger tendency to be magnetic. Then, based on the selected atoms and
crystal point group, the mathematical descriptors $D_{n}$ are calculated for
all possible compositions formed by the selected atoms in the selected
structure; the position in the materials map will indicate if the 2D materials
candidates are either AFM or FM. In ML models for materials, three components
are needed: available data, a numerical representation of materials, and
learning algorithms and their optimization 35.

The starting point for both ML models described in Figure 1 is the C2DB 30
database of bidimensional materials, which contains a total of 3814 entries in
its 2020 version. The C2DB database was built with DFT calculations of
experimentally known 2D structural prototypes combinatorially decorated with
different ions from the periodic table. Each entry in the C2DB database is a
combination of a crystal structure, a composition, and a magnetic order. We
filtered out materials with heat of formation > -25 meV/atom, so the dataset
contains only materials that are thermodynamically stable. Additionally, we
decided to exclude the materials which composition did not include a
transition metal. This series of filters resulted in a dataset with 2205
entries. By grouping the entries by crystal structure and composition, we
ended with a total of 1845 materials, each with (possibly) different magnetic
orderings.

### Random forest classification of magnetic and non-magnetic 2D compounds

The random forest classification scheme of magnetic (M) and non-magnetic(NM)
compounds is discussed below:

Available magnetic and non-magnetic 2D compounds entries: To extract trends
differentiating M and NM, we propose the training of an interpretable ML
algorithm using the 850 M and 2569 NM compounds from the C2DB 30. The DFT
calculated total energy provided by the C2DB for NM ($E_{FM}$) and M
($E_{NM}$) phases allows to identify the ground state for the initial set of
1845 2D compounds. ML models and general trends for the NM and M behavior in
the chemical and structural space requires to eliminate ambiguous entries,
i.e., those with a relatively small energy difference between different
magnetic configurations, $\Delta
E_{\text{M}-\text{NM}}=E_{\text{M}}-E_{\text{NM}}$. We thus exclude compounds
with $|\Delta E_{\text{M}-\text{NM}}|<0.01$ eV/atom, resulting in a dataset
with 1713 compounds, with 474 M and 1239 NM.

Numerical representation of magnetic and non-magnetic compounds: The
descriptors used to characterize the chemical space were simple boolean
features that are set to True (False) when the element is (not) present in the
composition. To get to a structural descriptor, we used a non-supervised
approach to cluster the crystal structures, and then we used these cluster
labels as a categorical feature. The C2DB actually provides a label for
grouping different structures and it has a format that uses the material’s
stoichiometry, crystal space group, and the set of occupied Wyckoff positions.
The H-phase of MoS2 is labeled as AB2-187-ai, for example. Even though this
label is very specific, unfortunately we observed that it was not
qualitatively univocal, i.e. in some cases it grouped different structures
into the same label. In order to bypass that, we used CrystalNN crystal
fingerprint 39 to find an embedding that deeply characterizes the structural
differences of all materials. The CrystalNN is based on the evaluation of
every site of the crystal regarding different local geometrical environments
(e.g octahedral, tetrahedral). The outcome is then a 244-dimensional vector
for each material. Using these vectors in an embedding method such as the
T-SNE 40, and then automatically clustering this embedding using DBSCAN 41, we
were able to get to a set of 51 structural groups. The Supporting Information
provides some further details on this process. For this specific analysis,
this kind of feature has shown to be more efficient than the labeling from
C2DB or simply grouping by space group or point group.

Learning algorithm and its optimization: Random forest classifier. With the
huge variety of models one can use to make classifications42, we chose the
non-linear Random Forest (RF) model due to its simplicity and ease of
interpretation. Another great advantage of RFs is that, by using the out-of-
bag methodology (OOB) 43, it is possible to use 100% of the data for both
training and for validating the model. Details regarding its implementation
and hyperparameters are presented in the Supporting Information. Table 1 shows
the RF accuracy results for the 1684 materials. As can be observed, the out-
of-bag F1 score is 86%, which indicates that by using very simple
compositional and structural features one is able to delineate bidimensional
materials which have magnetic order with good accuracy.

Table 1: Random Forest Out-of-bag (OOB) validation results. The F1 score is the best performance metric as the dataset has two unbalanced classes, with M as the minority class. M/NM Accuracy stands for the RF individual class accuracy, i.e. how many materials of these classes are correctly classified by the model. Number of materials | 1713
---|---
M Proportion | 25.8%
Accuracy | 92.5%
F1 score | 86.0%
M Accuracy | 84.5%
NM Accuracy | 95.9%
Figure 2: Strip plots of SHAP values of compositional features for all TMs.
Each individual plot represents a group, with color encoding the period in the
periodic table. Each dot represents a calculated SHAP value for an
instance/material which contains the TM in its composition. Positive
(Negative) SHAP values, indicated by red (blue) background, mean that the
presence of the specific TM contributes to the model to classify the material
as M (NM). SHAP values concentrated around zero indicates that having the TM
in the composition does not affect the model classification.

Trends for M vs. NM behavior in the space of chemical species: The next step
was to rationalize what the RF models have learned in order to extract trends.
For this task, we used the popular SHAP 44 methodology used to explain tree-
based ML models. This method is based on using game theory’s Shapley values 45
(or SHAP values, in this context) for assigning the impact of features on the
model’s predictions at the instance level. These values are calculated for
each feature value and for each specific instance in the dataset: by using all
possible combinations of the set of features, i.e. the feature power set
containing $2^{n}$ different feature combinations ($n$: number of features),
the method consists in fitting separated one-instance models to the RF model
output, making a total $2^{n}$ fittings for each instance. From this set of
fittings, it is possible to calculate the feature Shapley values, which
evaluate numerically how the feature contributes to the RF output.

In the context of an NM vs M binary classification, SHAP values can be
negative or positive if the feature locally contributes to a classification of
NM or M, respectively. Their absolute values are directly correlated with how
important the feature value was to the model output. Thus one is able to
investigate the general influence of a feature on the model by analyzing the
distribution of all its SHAP values.

We chose the SHAP methodology over traditional model-level importance
features, such as Permutation Feature Importance (PFI), because the latter
does not provide class-specific values. While PFI directly provides a single-
valued weight for each feature (on which the greater the value, the more
important the feature is), it fails on discriminating how a feature helps to
classify the material as NM or M. In general, model-level importance features
lacks the description of how the individual feature values affect the
classification, while SHAP, as an instance-level method, covers this
shortcoming by showing how each feature value contributes locally for the
model output.

Figures 2 and 3 show the distribution of compositional features’ SHAP values.
As the C and N groups are not statically representative, and for the sake of
simplicity, we present only the plots for transition metals (TMs) and for
anion groups (chalcogens and halogens). The Supporting Information provides
the full plots for the cited remaining periodic table groups (Figure S2).

From Figure 2 we can see a clear trend regarding the effect of the period of
the TM. While 4d (orange) and 5d (green) TMs show neutral contributions to the
model, visually denoted by their SHAP values distributions concentrated around
0.0, the 3d (blue) TMs show very positive SHAP values, with the highest values
happening for Mn, Fe, Cr, V, Ni and Co. This means that the presence of these
elements had a huge impact on the model classification of the compounds into
magnetic (M).

The difference between 3d TM and 4d, 5d TMs also happens in tridimensional
materials and can be explained by the crystal/ligand field splitting of
d-levels. 3d TMs usually have a smaller crystal splitting of d-levels, which
are associated with FM or AFM magnetic order, as we discuss in the next
section.

Figure 3: Strip plots of SHAP values of compositional features for anions
(chalcogens and halogens).

The influence of different anions (Figure 3) in classifying a material into M
or NM was also evaluated. We see that compared to the TMs, the anions SHAP
values distributions are concentrated around much lower absolute values,
indicating that they are less determinant in the classification process.
However, we can qualitatively see that the model associates the presence of
chalcogens with NM order, while the presence of halides is associated with M.
This means that the reduced coordination of halides presents a higher
correlation with magnetic ordering in two-dimensional materials.

Figure 4: All structural clusters with SHAP values distributions correlated
with a magnetic classification. They can be divided into four main groups
regarding the transition metal sublattice: Triangular, Square, Honeycomb, and
Distorted Triangular.

Trends for M vs. NM behavior in the space of structural clusters: The
categorical structural feature has a total of 51 labels corresponding to 51
qualitatively different structural groups. From the SHAP distributions plots
for structural clusters, presented in the Supporting Information (Figure S3),
we see that some of them present large absolute SHAP values and thus high
influence on the RF model prediction. Figure 4 shows the general crystal
structure of selected clusters that contributed to an M classification.

The selected structural clusters can be separated into four main groups
regarding their respective TM sublattices: Honeycomb, Square, Triangular, and
Distorted triangular. By verifying the recent literature on magnetism in two-
dimensional materials 46, one can see that most of the compounds where
magnetism has been verified (experimentally or computationally) present a
crystal structure that fits into the general structures of the clusters shown
in Figure 4.

The first experimental observations of magnetic ordering in the monolayer
limit support the structural trend for Honeycomb TM sublattices, for example,
with FePS3 in 2016 47 and CrI3 (cluster 26) in 2017 13 presenting AFM and FM
order, respectively. Other monolayers with the cluster 26 structure and with
composition MX3 (M = Cr, Mn; X = F, Cl, Br, I) also have been studied and
theoretically predicted to present intrinsic ferromagnetism 48, 49.
Calculations of CoBr3 in the cluster 27 crystal structure also showed that it
presents intrinsic FM order 50.

The Triangular TM sublattice result is also supported by experimental and
computational reports, such as the experimental verification of ferromagnetism
at room temperature in VSe2 51 and MnSe2 52, both in the T-phase of TMDCs
(cluster 4); the FM phase prediction by calculations of ScCl, YCl, LaCl in the
cluster 12 structure 24, 53; and the theoretical ground state of AFM for Ta3C2
54 and FM for Cr3C2 55 and Ti3C2 56 in the pristine MXene crystal structure
(cluster 23).

Magnetic two-dimensional materials with Square or Distorted Triangular
sublattices, as far as we know at the moment, remain to be further
investigated. Even though the study of specific compounds with these
structures was not found, the training data from C2DB suggests that these
geometries are viable for magnetic order and for future experimental
verification.

With the exception of the Distorted Triangular TM sublattice, all groups
present a regular transition metal 2D sublattice; they represent the only
three ways one can regularly tile a 2D plane, with honeycomb, triangular or
square geometries.

In contrast to this regularity, the only structural cluster which contributed
effectively to an NM classification was the one with the general crystal
structure of H-TMDC transition metal alloys (cluster 33). These specific
alloys have a stoichiometry of ABX4 or of AB3X8, where A and B are transition
metals from periods 4 and 6, and X is a chalcogen. Something particular of
this structure is that the alloying with A TM breaks the chemical regularity
of the B TM and, in some cases, it even breaks the structural regularity by
breaking the system mirror symmetry, and thus distorting the TM plane.

From these results, the main structural trend suggested is the need for
geometric regularity of the TM sublattice for the emergence of intrinsic
magnetic ordering. This regularity is possibly correlated with long-range
exchange interactions between TM nearest neighbors.

### Machine learning classification of AFM and FM 2D compounds

The tendency in the feature space defined by the atomic properties and crystal
symmetries of the magnetic behavior allows us to understand which atomic
combinations and structures are prone to present a magnetic ordering, however,
it does not indicate what type of order is expected. In this section, we thus
focus on the machine learning distinction of magnetic phases such as FM and
AFM. We first describe the three components that are needed for machine
learning models: available data, a numerical representation of materials, and
learning algorithms.

Available 2D compounds with magnetic ordering: The computational 2D materials
database (C2DB) 30 is also the source of the data set used to train the
machine learning model discriminating AFM from FM ordering. In the C2DB, the
three considered magnetic phases (i.e., non-magnetic, ferromagnetic or anti-
ferromagnetic) are not always calculated for a given compound. Indeed, there
are 3052 compounds for which only one of these magnetic phases is reported:
2533 NM, 446 FM, and 73 AFM. Similarly, NM and FM phases are evaluated for 195
compounds, FM and AFM are calculated for 147 2D materials, and only 14
compounds with reported total energy for the NM and AFM phases. However, for
the 875 compounds with non-zero local magnetic moments, the DFT calculations
are based on a one supercell model, i.e., the considered AFM and FM phases are
restricted to the smaller unit cell with at least two transition metals.
Calculating all possible symmetry allowed magnetic orderings, Arnab Kabiraj et
al 18 found that from the 788 DFT calculated FM phases, 368 are actually AFM.
Naturally, it is also possible that some compounds are misclassified as NM,
which can explain the obtained accuracy in the magnetic vs non-magnetic
classification in the previous section. Previous machine learning
classification for magnetic ordering based on the C2DB intrinsically has the
bias of this DFT calculations 57. Here, we restrict the training set to the
database of 525 magnetic order compounds from Ref. 18. Even though at a first
glance this seems to be a relatively small dataset for training, this has
shown to be sufficient for other ML analysis in Materials Science 58. Also,
for the test set, the accuracy of our ML is larger than 90% as discussed
below, indicating a very good model. In order to demonstrate the
transferability of the machine learning descriptor, the test set used to
verify the accuracy of the predictor is constructed from a different database
that has recently obtained 54 2D magnetic ordered materials from bulk
compounds that are potentially exfoliable 33. Finally, in the training process
we consider only compounds with extreme behavior, that is, those that are
clearly defined within the limits of the AFM and FM ordering.

Numerical representation of magnetic 2D compounds: The second component of
machine learning models involves the mapping of the material-to-attribute
connection, i.e., a computer-friendly representation of the materials. These
numerical representations of materials include a generalized representation
that comprises atomic properties along with structural features 59 and also
complex representations compressing electronic properties 60, e.g., electronic
density. In order to find trends in the chemical space that do not depend on
complex features that require a priori knowledge of the electronic properties,
we use features associated with the properties of the constituent atoms of the
compound. In Table 2, the primary atomic features are declared. We
discriminated atomic features related to the cations and anions by the sub-
indexes $c$ and $a$, respectively. The 2D compounds in the original database
are usually formed by a unique transition metal element (i.e., cation) and
different anions (or only one element as an anion). When there is more than
one element as anion we consider statistical functions of the atomic features
59 as the primary features in the ML classification. For a material formed by
$n_{s}$ anions and $N$ atoms in the unit cell, we introduce for each atomic
property $\gamma$ the statistical function
$\bar{\gamma}=\left(\sum^{n_{s}}_{i=1}\gamma_{i}/\right)n_{s}$, which stand
for the average value of the property. This redefinition of the input features
allows unifying the size of the feature space for compounds with a different
number of atoms, which can represent a barrier to simultaneously classify
binary and ternary 2D compounds.

Property |  Description
---|---
$Z$ |  Atomic number
$\chi$ |  Pauling electronegativity
$\mathcal{G}$ |  Periodic group
$v$ |  Valence
ø |  Unfilled valence orbitals
$\mathcal{E}$, $\mathcal{I}$ |  Electron affinity and ionization potential
$\epsilon^{ho}$, $\epsilon^{lu}$ |  Highest-occupied and lowest-unoccupied Kohn-Sham eigenvalue
$\alpha$ |  Polarizability
$r$ |  Atomic non-bonded radius
$r_{v}$ |  Radius of the last occupied valence orbital
$r_{s}$, $r_{p}$ |  Extensions of the $s$ and $p$ orbitals∗
∗ i.e., the radii where the radial probability density of the valence $s$ and
$p$ orbitals are maximal.
Table 2: Atomic and structural properties included for the construction of
primary features by the $\gamma$ statistical operations. Figure 5: Machine
learning classification separating AFM (blue) from FM (red) for compounds with
crystal point groups containing horizontal planes, $D_{nh}$ with $n=$ 2, 3,
and 4. The descriptors $\mathcal{D}^{\sigma_{h}}_{x}$ ($x$ axis) and
$\mathcal{D}^{\sigma_{h}}_{x}$ ($y$ axis) simultaneously discriminate magnetic
ordering in compound with different crystal space groups. Green and purple
regions stand for the classification for compounds formed by TM with low-SOC
and high-SOC, respectively. The $x$ axis stand for average atomic SOC of TM
and anions.

Learning algorithm: For the third component of machine learning, i.e., the
learning algorithms and their optimization, we adopt the sure independence
screening and sparsifying operator (SISSO) approach 38. This method first
automatically constructs a feature space and subsequently selects the feature
that better separates the classes in the training set, e.g., AFM and FM. Based
on the initial set of primary features $\Phi_{0}$, including the statistical
combination of atomic properties (Table 2) for anions, analytical operations
are used to combinatorially associate primary features with the same unit. The
use of the primary features listed in Table 2 (i.e., atomic properties) has
proved to be a powerful way to do machine learning in materials. This kind of
analysis has been used to predict the stability of 2D materials 61,
topological phases 62, and structural configurations 63, 64. Additionally,
this set of features, in general, represented the main trends in the chemical
space for atomic properties 65, 66, 67. The regions for FM and AFM classes are
defined as convex hull. The descriptor, i.e., the combination of atomic
features that minimizes the overlap region between convex hulls, is selected
by systematically computing all overlap regions defined in a two-dimensional
map, where each axis is given by a specific function of atomic properties.

In the training set, there are compounds with the same composition and even
stoichiometry that have different magnetic ordering. These compounds are
undistinguished in the feature space defined only by atomic properties.
However, even including structural features for the representation of the
materials, the random forest algorithm used in the previous section provides
great inaccuracy in classifying the materials with AFM configuration. This
unsuccessful capture of trends in the magnetic ordering is similar to the
reported in previous works. To overcome the intrinsic dependence of the
magnetism with respect to the symmetry, we use a multi-task version of the
SISSO method 68. Specifically, in this novel approach, a machine learning
descriptor classifying the FM and AFM classes is simultaneously optimized for
different groups of compounds (i.e., tasks) that contain both classes. Here,
these groups are all symmetry prototypes in the C2DB. This process arrives at
the best nonlinear features employed in linear models that can differentiate
between classes simultaneously for every symmetry.

Trends for AFM vs FM 2D compounds: For AFM vs FM classifications based on the
multi-task SISSO method, we evaluate different ways to group materials that
have simultaneously the same descriptor. Each class (or group) of 2D materials
is divided into AFM and FM ordering. Since these magnetic configurations can
strongly depend on the symmetry, one can expect that the patterns in the
chemical space for AFM and FM are different for compounds with different point
groups symmetry classes (e.g., polar, non-polar, chiral, non-chiral,
centrosymmetric). Thus, materials are grouped according to the group symmetry
classes and the symmetry operation contained in the crystal point group
symmetry (CPGS). In principle, the multi-task process can potentially capture
the relation between the crystal symmetry and the magnetic ordering, or at
least, identify patterns that are different across the PGS. For instance, we
consider three different strategies to construct classes of 2D compounds
containing AFM and FM ordering, namely, according to: i) the polarity and
chirality of the CPGS, ii) the larger order of rotation symmetries in the
crystal point groups (e.g., compounds with symmetry axis $R_{3}$, $R_{4}$, and
$R_{6}$ form different classes of 2D compounds), and iii) the type of mirror
symmetry planes in the CPGS (e.g., compounds with CPGS containing horizontal
${(\sigma_{h})}$, diagonal ${(\sigma_{d})}$ or vertical ${(\sigma_{v})}$
mirror planes form different groups). We note that the grouping of compounds
based on symmetry operations (groups $i-ii$) leads to a limited SISSO model
accuracy of about 80%, which is larger than the accuracy of 70% of previous
random forest models.

In order to increase the classification efficiency, separations based on
materials properties are also considered. Specifically, 2D compounds are also
grouped according to materials properties such as the band gap, bulk spin-
orbit coupling (i.e., calculated as the average of the atomic spin-orbit
coupling), and the existence of either 3$d$, 4$d$, or 5$d$ transition metals.
These materials’ properties are selected based on the magnetic physical
mechanism in three-dimensional compounds, as discussed in the next section.

We find that the SOC is a determinant factor for high accuracy in the
classifier separating AFM from FM in a materials map. In Figure 5, the
classification model for groups formed by the interpolation of crystalline SOC
and different types of mirror symmetry planes is presented. For each group of
materials, AFM and FM compounds are illustrated in red and blue, respectively.
For instance, in Figure 5 compounds with point groups symmetry containing at
least one horizontal mirror plane (i.e., $D_{2h}$, $D_{3h}$, $D_{4h}$, and
$D_{6h}$) are used to train the multi-task ML classifier. The SOC effect is
introduced by dividing compounds into those containing TM with high SOC (i.e.,
4$d$ and 5$d$ atoms) and those with low-SOC (i.e., 3$d$ atoms), which are
illustrated in purple and green regions, respectively. Compounds formed by
low-SOC TM are also divided in terms of the atomic SOC of anions as 2D
compounds with low-SOC and high-SOC anions (Figure 5). The axes determining
the predicted material map are given by the two-dimensional descriptor
$\vec{\mathcal{D}}^{(\sigma_{h})}$ with the components
$\mathcal{D}^{\sigma_{h}}_{x}$ and $\mathcal{D}^{\sigma_{h}}_{y}$ given by the
functional mathematical form:

$\mathcal{D}^{\sigma_{h}}_{x}=\left({\epsilon^{lu}_{c}+\epsilon^{lu}_{a}}\right)\mathcal{G}_{c}/r_{p,c}$
(1)
$\mathcal{D}^{\sigma_{h}}_{y}=r_{v,a}/r_{c}\left(\mathcal{E}_{c}+\mathcal{E}_{a}\right),$

where the superindex ${\sigma_{h}}$ stand for horizontal mirror planes,
$\epsilon^{lu}_{c}$ and $\epsilon^{lu}_{a}$ are the lowest-occupied Kohn-Sham
eigenvalues for the cations (i.e., TMs) and anions, $\mathcal{G}_{c}$ is the
periodic group of the cation, $r_{p,c}$ is the radii for the $p$-orbitals of
the cations, $r_{v,a}$ is the radius of the last occupied valence orbital for
anions, $r_{c}$ is the atomic non-bonded radius for cations, and
$\mathcal{E}_{c,a}$ are the electron affinity for anions and cations. In the
training set, the accuracy of the SISSO descriptor is larger than 98%, while
the accuracy in the test set is larger than 90%. Importantly, even with the
same descriptor $\vec{\mathcal{D}}$, the tendency for every symmetry is almost
unique, i.e., the boundary line defining AFM and FM regions in the materials
maps (dashed lines in Figure 5) is different for each map. For instance, for
the crystal point group $D_{2h}$, the slope of the boundary line is positive
regardless of the atomic SOC of the constituent atoms (Figure 5), but its
position in the materials map changes as the atomic SOC changes. In contrast,
for the CPGS $D_{3h}$, the slope of the boundary lines drastically increases
when the average atomic SOC increases. Also, for compounds with CPGS $D_{4h}$,
a low average atomic SOC directly indicates an AFM phase, with a clear
demarcated tendency to forbid the FM order.

Besides the ML classifier for compounds with CPGS containing horizontal mirror
planes, we also find a functional rule $\vec{\mathcal{D}}^{\sigma_{v}}$
leading to a separation between AFM and FM in compounds with CPGS containing
mirror vertical planes. The component of the two-dimensional descriptor
$\vec{\mathcal{D}}^{\sigma_{v}}$ are given by:

$\mathcal{D}^{\sigma_{v}}_{x}=\chi^{2}_{c}\mathcal{G}_{c}/\mathcal{E}_{a}$ (2)
$\mathcal{D}^{\sigma_{v}}_{y}=\mathcal{G}_{c}v_{c}\left(\mathcal{G}_{a}+v_{c}\right),$

where $\chi_{c}$ is the Pauli electronegativity of TMs, $\mathcal{G}_{c,a}$
are the periodic group for cations and anions, $\mathcal{E}_{a}$ is the
electron affinity of anions, and $v_{c}$ is the valence of the cation. The
descriptor $\vec{\mathcal{D}}^{\sigma_{v}}$ provides a reasonable good
classification with an accuracy of 92% in the training set and 87% in the test
set. This accuracy is significantly higher than that of 70% in previous
machine learning models 18 for magnetic ordering. Although we are mainly
motivated by obtaining a rule that allows us to classify 2D materials into AFM
and FM, rather than interpreting a physical mechanism, one can find some
relationships between atomic properties that could reveal in the future the
physical mechanism determining the magnetic ground state.

### Theoretical trends and captured mechanism

In bulk magnetic materials, the effective spin magnetic moment is usually
different from zero due to the presence of ions with partially filled
electronic $d$\- (transition metals) or $f$-shells (rare earths). In this
case, Hund’s rules lead to a finite spin (S) and orbital (L) magnetic moments
- the total magnetic moment is J = S + L. Usually, the orbital magnetic
moments (L) are quenched for transition metal elements and only spin magnetic
moments (S) are considered. When the magnetic ions have completely filled (or
empty) electronic $d$ or $f$-shell, the effective spin magnetic moment is zero
leading to diamagnetic property — the so-called non-magnetic materials.
Molecular field theory can explain the observation of a net magnetization due
to the competition between the exchange interactions among magnetic ions
leading to a macroscopic ordering of the system and the crystal field
splitting of $d$-orbitals of TMs. The analogy with three-dimensional compounds
paves the route for the interpretation of the obtained ML models for the
existence of magnetism and classification of the magnetic ordering. To do so,
the materials maps for magnetic bulk materials is briefly discussed (see
Figure 6).

Notwithstanding its inherent complexity, there are clear trends for the
magnetic order in bulk materials, as illustrated in Figure 6. These trends are
represented as a material map in the feature space defined by the metallic
behavior of the compounds and the localization of the atomic magnetic moments
7. Four quadrants can thus be identified in Figure 6: (I.) metallic compounds
formed by 5$d$ TM, where the itinerant exchange tends to be predominant; (II.)
In metallic compounds formed by 3$d$ TM, the magnetism is mainly mediated by
the indirect exchange; (III.) For non-metallic materials formed by 3$d$ TM,
the direct exchange is typically dominant; and (IV.) For non-metallic with
5$d$ TM, the super exchange usually explains the magnetic ground state. This
materials map also reveals a trend in magnetic ordering that separates FM
materials in quadrants I and II, while locating AFM materials in quadrants III
and IV. Naturally, this is not a general rule, since in practice the symmetry
of the crystal plays a role in the description of the magnetic state and there
is no atomic composition that a priori dictates a specific exchange
interaction. The description of magnetic order in 2D materials is even more
complex.

Figure 6: Materials map for exchange interaction in bulk materials7. The four
quadrants I-IV correspond to the indirect exchange region (green), itinerant
exchange region (purple), super exchange region (blue), and direct exchange
region (red), respectively.

One of the bottlenecks for understanding the trends of magnetism in 2D
materials, and then proposing functional applications with them, is due to
only a few experimental realizations and their contradiction with the
materials map for Bulk compounds (Figure 6). This includes the more recent
discovery of room-temperature ferromagnetism in monolayer VSe2 51 and MnSe2 52
stabilized by the strong magnetocrystalline anisotropy, which suppresses the
Mermin–Wanger restriction. Both compounds, as well as the reported strong out-
of-plane magnetic anisotropy in Fe3GeTe 69, have itinerant ferromagnetism.
These compounds belong to the purple region of the materials map in Figure 6,
but it contradicts the illustrated trend since they are formed by 3d TM. In
contrast, the strong direct ferromagnetic exchange (red region in Figure 6) is
modeled as an Ising-type magnetism, which describes the AFM with an ordering
temperature of 118 K in FePS3 47. Additionally, the interlayer magnetic
coupling in bilayers of 2D materials can give rise to novel physical
properties 70, 71, 72, 73, 74, 75, 76, 77, which can not be directly
accommodated in the materials map of Figure 6. However, the ML analysis of
theoretical predicted and hypothetical compounds in previous sections provide
clear trends that are actually related to a theoretical explanation for
magnetism in 2D compounds, as discussed below.

Captured mechanism for the existence of magnetism:

Figure 7: SHAP decision plots of two distinct materials showing how specific
compositional/structural features values (0: False; 1: True) contribute to the
model classification (NM: left; M: right). Here it is shown only the material-
specific top-20 features with respect to the absolute SHAP values; a) FeCl2 in
the cluster 25 crystal structure (TMDC-H): The presence of Fe alone shows to
be sufficient for the M classification in this case; b) IrClS in the cluster
34 crystal structure (TMHC): The absence of many 3d TM in the composition,
denoted by 0, contributes to an NM classification.

To better understand the physical mechanism captured by our ML model, we first
illustrate how the prediction is constructed based on the SHAP scheme. For two
different 2D materials, figure 7 shows the resulting total weight defining a
non-magnetic and magnetic tendency, which are represented by negative values
(left side in blue) and positive values (right side in red), respectively.
Note that both the existence (represented by 1) and the absence (represented
by 0) of a given structure or chemical species contribute to defining the
existence of magnetism. For instance, in the first two lines of Figure 7a, one
can see that the existence of Fe atoms in the composition of the compound
positively contributes to the classification of the compound as magnetic, but
the absence of Mn atoms negatively contributes to the classification as
magnetic (i.e., the contribution to the total weight is negative, and the
absence of Mn tends to make the compounds non-magnetic). Another important
consideration in these trends is that, if the existence of a specific atomic
specie contributes with a positive weight $\xi_{X}$ to the magnetism, it does
not imply that the contribution of the absence of X is $-\xi_{X}$. For
instance, in Figure 7a, the existence of Fe atoms leads to high positive
weight, however, in Figure 7b (fourth line), the absence of the same atom has
a small negative weight. This means that the total weight determining if a
given compound is magnetic or non-magnetic is not defined by a few atoms in
the composition or a specific structure, but by the collective weights of the
absence and existence of atoms and structural prototypes (i.e., symmetries and
Wyckoff positions).

The random forest and SHAP models indicated that the existence of 3$d$ TMs
strongly contributes with a positive weight to the existence of magnetism
(Figure 2), where the V, Cr, Mn, and Fe have the largest positive
contributions. In contrast, besides the existence of Y atoms that positively
contribute magnetism, 4$d$ and 5$d$ TMs do not have a predominant effect in
the existence or absence of magnetism, having then a minor, but non-zero
contribution. For instance, while Rh and Hg atoms have a small positive
contribution to the existence of magnetism, Ir atoms have a relatively small
contribution to the absence of magnetism.

Figure 8: Schematic representation of the d-orbitals splitting including the
exchange splitting $\Delta_{x}$ (yellow region) and the crystal field
splitting $\Delta_{c}$ (blue region). a) the case where $\Delta_{x}$ <
$\Delta_{c}$. b) the case where $\Delta_{x}$ > $\Delta_{c}$. Situations a) and
b) illustrate respectively a low-spin configuration and a high spin
configuration (yellow dots indicate electrons)

These trends can be interpreted in terms of the competition between the
exchange splitting (ES) $\Delta_{x}$ and the crystal field splitting (CFS)
$\Delta_{c}$. In figure 8, the successive effect of $\Delta_{x}$ and
$\Delta_{c}$ are represented for the case of $d$-orbitals. The introduction of
the exchange splitting separates spin up from spin-down; subsequently, the
symmetry-induced crystal field splitting further separates these levels. As an
example, the Td crystal field effect is shown in Figure 8, which splits the
five-degenerated $d$ levels into $t_{2}$ and $e$ levels. We can draw two
different scenarios, namely: $\Delta_{x}$ < $\Delta_{c}$ (Figure 8a) and
$\Delta_{x}$ > $\Delta_{c}$ (Figure 8b). These two situations can be directly
associated with the type of $d$-orbitals. Specifically, 3$d$-orbitals are
usually related to relatively small crystal fields, while 4$d$-orbitals and
5$d$-orbitals typically lead to large crystal field splitting. Thus, the
connection between the $d$-orbital type and the existence of magnetization has
an intrinsic correlation to the competition among CFS and ES. Specifically,
the local magnetic moment depends on the number of electrons; however, a
simple empirical model based on $\Delta_{x}$ and $\Delta_{c}$ usually gives a
reasonable intuition of the magnetization. For instance, for $\Delta_{x}$ <
$\Delta_{c}$ (Figure 8a), both spin up (red) and down (blue) can be
simultaneously populated, favoring a spin configuration with zero or
relatively small local magnetic moment (i.e., compound tending to have zero
magnetization) or having a relatively small magnetic moment. In contrast, when
$\Delta_{x}$ > $\Delta_{c}$ (Figure 8b), only states of one spin type are
populated. Consequently, the spin configuration tends to have non-zero (and
relatively large) local magnetic moment (i.e., FM or AFM compounds). In this
last scenario, the final magnetic configuration depends on other factors
(e.g., the SOC), as we analyze in the next section. Although the schematic
model illustrated in Figure 8 focused on the $T_{d}$ symmetry, the conclusions
are valid for any other general crystal field splitting since the physical
mechanism depends on the order of the levels and not on the specific final
symmetry. These empirical observations are in agreement with the trends
captured by our model: i) 4$d$\- and 5$d$ orbitals tend to be delocalized,
which results in a relatively large CFS ($\Delta_{x}$ < $\Delta_{c}$) and
hence, a local magnetic configuration with small or zero local magnetic
moments, and ii) 3$d$-orbital have a strong localization (compared to 4$d$\-
and 5$d$-orbitals) and consequently, relatively smaller CFS ($\Delta_{x}$ >
$\Delta_{c}$), leading then to a magnetic configuration with large local
magnetic moments. Summarizing, our ML model captures the relation between
3$d$-orbitals (large values in the SHAP plot of Figure 2) and the fact that
4$d$\- and 5$d$-orbitals) cannot directly be related to the absence or
existence of magnetism (near zero values in Figure 2).

The distinction of magnetic and non-magnetic 2D compounds is based on the
existence and absence of specific atoms, as well as specific symmetries,
which, naturally, is intrinsically related to the atomic properties (e.g.,
unpaired electrons, exchange strength, bond length, coordination number) and
symmetry point groups. We have two types of features: structural and atomic.
Structural features can directly correlate to symmetry, coordination number
and exchange strength. On the other hand, atomic features correlate to the
number of unpaired electrons, exchange strength and bond length. At the end,
we are able to probe all important features for magnetism. Specifically, we
find that 2D compounds with non-zero local magnetic moments are typically
formed by 3d transition metals, which are mainly differentiated with respect
to 4d and 5d by the high orbital-localization degree and relatively small
atomic radii. These atomic features result in a high exchange splitting 78,
which as described in Fig. 8, leads to a high spin configuration or in other
words, a tendency to be magnetic.

The previously noted relation between atomic orbital localization and local
magnetic moments in 2D materials can also be used to extract information from
the SHAP plots for anions (Figure 3). Our model shows that the existence of
halides in a compound implies a positive weight for the existence of
magnetization. Indeed, we verify that the $p$-orbitals in halides tend to be
more localized than chalcogen $p$-orbitals. It is then expected that Halides
also lead to relatively small CFS ($\Delta_{x}$ > $\Delta_{c}$), and hence, a
non-zero local magnetic moment.

We also find that the structure plays an important role in the existence of
magnetism in two-dimensional materials. By analyzing the structural cluster
which contributes to an M classification, it is possible to delineate that the
regularity of transition metals in the crystal structure directly reflects on
the exchange interactions and, thus, favors the emergence of magnetic order.
While crystal structures with regular TM sublattices highly favor magnetism,
structures with chemical or structural irregularities seem to disturb the
long-range interactions between ions. However, the ground state magnetic
configuration depends on other features, as captured by the ML model and
explained in the next section.

Captured mechanism for magnetic ordering: As previously discussed, we find
that in 2D-materials, the SOC is a determinant factor for the classification
accuracy of AFM and FM compounds. This is a strong result relating the
patterns of the magnetic ordering classification with the SOC strength, i.e.,
when the atomic average of the SOC for the constituent elements increases in
the 2D compounds, the distribution of AFM and FM compounds change in material
maps. Previous calculations indicate that as the atomic combination forming
the compounds has larger SOC, the magnetic anisotropy increases, which is
fundamental to allow 2D magnetism 79, 80. The ML SISSO classification (see
Figure 5 for the specific case of CPGS $D_{n}h$ with $n=2,3,4$ and 6)
indicates that material maps and boundaries defining AFM and FM regions can
strongly depend on the magnetic anisotropy. On the other hand, our ML model
also indicates that the magnetic order configuration depends on the symmetry,
i.e., higher symmetries favor the AFM order. For instance, compounds with more
symmetric PGS tend to be AFM (see PGS $D_{4h}$ in Figure 5). As the PGS have
lower symmetry, the proportion of FM compounds with respect to AFM increases.

Even though SISSO is a very powerful ML model, it is certainly not the best-
suited model to interpret the results. The descriptors that are built can be
complex and do not have a direct interpretation. However, some hints can
appear in the features defining the descriptors. Indeed, equations 1 and 2
give us some hints of what is important to classify a compound as FM or AFM.
Compounds formed by atoms with large (small) valence tend to have AFM (FM)
ordering, and compounds with a large (small) $r_{pM}$ tend to be FM (AFM). In
the opposite way, for the space group P-3m1, compounds formed by atoms with
large (small) lowest-occupied Kohn-Sham level tend to have AFM (FM) ordering.
Although the interpretation of this descriptor is not obvious, one can infer
that the value of the radii of the $p$-orbitals for cations is related to the
size of the atoms that separate the transition metals. When the size of TM
increases (with the symmetry as a constant), the compound tends to be FM.
Similarly, the descriptor also reveals that the classification of AFM and FM
ordering depends on the electron affinity, which is related to how the
electrons are transferred in atoms to form atomic bonds in the 2D compound.

Additionally, for $\vec{\mathcal{D}}^{\sigma_{v}}$, the valence orbitals $v$,
the Pauling electronegativity, and the periodic groups appear to play the main
role in defining the magnetic ordering. Indeed, the simplest interpretation of
the exchange interaction is based on the unfilled valence orbitals and Pauli
electronegativity. There is no direct relation between these atomic features
and the physical mechanism defining the magnetic ordering. However, with the
predicted material maps, one can infer the magnetic ordering by means of the
atomic composition in a defined symmetry. When the considered symmetry
changes, the classification can also be different, which illustrates the
robustness of the method and determined descriptors.

### Machine learning model validation and potential applications

In this section, we use the proposed scheme to predict the existence of
magnetism and magnetic structure of novel 2D materials candidates, i.e.,
compounds that are not included in the initial database used to train and test
the ML model. The ML prediction is followed by a theoretical total energy
calculation of the magnetic ground states based on ab inition theory. DFT+SOC
calculations are performed using the Vienna Ab initio Simulation Package
(VASP) with the projector-augmented wave (PAW) method 81, 82 and GGA-PBE 83
parametrization for the exchange-correlation functional. As an illustrative
example, we describe here the ML prediction and DFT verification for compounds
with $D_{3h}$ PGS. The complete list of novel predicted compounds for all
symmetries can be found in the Supporting Information. The ML prediction
process starts with the generation of novel compounds based on trends for 2D
magnetism defined in the features space formed by chemical composition and
structural clusters. Afterward, we make the proposed materials go through
steps of the prediction scheme I (classification for magnetic vs non-magnetic
materials) and II (classification for AFM vs FM ordering in 2D compounds).

Generation of novel magnetic candidates: In previous sections, we learned that
compounds containing 3$d$-orbitals TM and halides tend to host 2D magnetism.
the construction of atomic combinations is based on the selection of 3$d$ TMs
V, Mn, Cr, Fe, Ni, Co, and Ti (the highest positive weight) and as anions, the
halides Cl, Br, and I. These atomic species lead to 21 atomic combinations,
which can be accommodated in the 51 structural prototypes (i.e., 1071
compounds). Since the weight of the atomic composition is usually larger in
determining the existence of magnetism, all structural prototypes are
considered for generating 2D magnetic candidates. These compounds
intrinsically account for different stoichiometries and compositions (e.g.,
AB, AB2, and A2B3). The hypothetical generated materials are always in the
green region of the materials maps classifying AFM and FM materials since they
are only formed by low SOC TMs.

Figure 9: Prediction of novel candidates with $D_{3h}$ crystal point group
symmetry with low-SOC (top) and medium-SOC (bottom). The red and blue squares
stand for the ferromagnetic and antiferromagnetic compounds, respectively.
Novel 2D magnetic candidates in the $d_{3h}$ point group are represented in
cyan.

ML model applied to novel candidates: The list of generated compounds is
compared with the initial list of compounds contained in the C2DB database and
only a novel combination of ACS attributes is considered. Using the random
forest model, compounds are classified as magnetic and non-magnetic. After
considering compounds with a 100% of probability to be magnetic (i.e.
classified as M by 100% of the RF’s decision trees), the filtering process
leads 169 novel 2D magnetic candidates. For these 169 prototypes, we use the
SISSO models to identify the magnetic ordering, which results in 137 AFM and
32 FM 2D compounds. This prediction is performed by identifying the position
of the compounds in the materials maps by means of the calculation of the
functional form of the descriptor in terms of the atomic properties. For
instance, Figure 9 show the AFM (blue), FM (red), and novel 2D magnets (cyan)
the CPGS $D_{3h}$. The boundaries (dashed lines in Figure 9) are defined by
the SISSO model in Figure 5. Here, The position of the novel candidates
indicates that they are classified as AFM materials (Figure 9).

DFT verification: Our DFT calculation for the selected compounds in the
$D_{3h}$ PGS reveals that all compounds in composition A2B3 (crystal structure
of MXenes, cluster 23) are not mechanically stable in this structure, i.e.,
the compound tends to dissociate. On the other hand, materials with the
composition AB (crystal structure of ABBA InSe, cluster 17) are energetically
stable with respect to the free atoms and solids formed by the constituent
atoms. The magnetic ground states for the six remaining atomic combinations
are studied by means of DFT calculations. Remarkably, we find that the
compounds AB predicted as AFM by the ML scheme in Figure 1 are confirmed to be
AFM by PBE DFT+SOC total energy calculations, as shown in Table 3. These AB
compounds can be understood as a unit cell containing two quasi layers, where
each quasi layers has one TM. The DFT calculation indicates that the most
stable AFM magnetic configuration for CrCl, CrBr, CrI, and MnCl contains only
two TM atoms in the unit cell, i.e., TMs at different planes have opposite
local magnetic moments along the $z$ axis. On the other hand, the AFM unit
cell for MnBr and MnI contain four TMs, where subsequent TMs in the same plane
have opposite local magnetic moment orientation, which is inverted from one
plane to another.

Compound | GS | $\Delta_{\text{AFM-NM}}$ | $\Delta_{\text{AFM-FM}}$
---|---|---|---
CrCl | AFM | -75 | -2
CrBr | AFM | -75 | -15
CrI | AFM | -73 | -62
MnCl | AFM | -9 | -25
MnBr | AFM | -8 | -36
MnI | AFM | -7 | -49
Table 3: Magnetic ground state (GS) for compounds with the composition AB in
Figure 9. For each combination, the energy difference for the non-magnetic
($\Delta_{\text{AFM-NM}}$) and ferromagnetic ($\Delta_{\text{AFM-FM}}$) phases
with respect to the AFM ground state are provided in meV/atom.

##  3 Conclusions

The design and the discovery of novel functional materials have been
historically sustained by an Edisonian trial-and-error approach. This
expensive and time-consuming approach has been recently substituted by an
intelligent data-based approach. The basic idea is to rely on computational
databases of compounds, where the physical properties of thousands of
materials have been calculated using quantum mechanical methods. In the
current manuscript, we have used this approach to search for magnetic 2D
compounds. This is a recent field of research with huge potential for
functional applications.

Our study was divided into two steps. We have used the C2DB database as the
source of information regarding structural, electronic, and magnetic
properties of a large variety of synthesized and computer-designed compounds.
These compounds were classified as non-magnetic (NM - diamagnetic),
ferromagnetic (FM), and antiferromagnetic (AFM). It is well-known and
established, however, that there are several other magnetic configurations
that were not included in this list, including ferrimagnetic, helimagnet, and
compounds with frustrated interactions. More complex than that, the AFM
configurations can have a large variety of different accommodations for the
spins, usually labeled as type-A, C, D, G, etc. The C2DB database only
accounted for the more simple configuration, where neighboring spins are
aligned in an anti-parallel configuration. Even though there is much
discussion about the outcomes of this approach 31, the available data should
be sufficient for our machine learning purposes.

The random forest algorithm used in this work was able to successfully
differentiate magnetic from non-magnetic compounds with great accuracy. Non-
magnetic compounds were predicted with an accuracy of $\approx 96$%, while for
magnetic compounds this accuracy was around 85%. The SHAP analysis allowed us
to understand why a material was classified as M or NM, indicating that 3d TMs
greatly impact the classification into M in comparison to 4d and 5d TMs.
Anions showed lower impact in comparison, but a qualitative analysis showed
that halides contribute to magnetic order, in contrast to chalcogens.

An analysis of the crystal structure of the magnetic 2D compounds revealed
that the transition metal sublattice regularity is a common characteristic in
all structural clusters favoring magnetic ordering. This general trend is
supported by the many experimental and theoretical studies of magnetism in
monolayers in recent years.

The classification of FM and AFM compounds is more subtle than the previous
one and demands a different approach. We have used the SISSO method which was
able to successfully find a descriptor that is able to classify FM and AFM
compounds. We have used a large set of atomic features to classify these
compounds, and the final descriptor included information regarding the
electron affinity, the number of valence electrons, and the electronegativity
of the atoms that constitute the compound. More importantly, our results
indicate that it is necessary to separate the compounds according to their
space groups and spin-orbit coupling in order to identify patterns that
classify AFM and FM 2D compounds. Consequently, the magnetic order of material
involves a complex competition among structural and electronic properties.

The information provided here can be useful in the search for novel 2D
magnetic materials, expanding the possibilities of this expanding and exciting
area to compounds that can be more stable and easy to synthesize than the
candidates observed so far.

The authors acknowledge the National Laboratory for Scientific Computing
(LNCC/MCTI, Brazil) for providing HPC resources of the SDumont supercomputer.
The authors also thank the Brazilian funding agencies FAPESP (Grants
17/02317-2, 18/11856-7 and 18/11641-0) and CNPq (309384/2020-6) for financial
support. In the Supporting Information we provide extra information regarding
details of the random forest classification scheme, the details about the
structural labels and detailed SHAP graphs for different features. The
complete list of novel 2D magnets candidates classified as AFM and FM by the
ML model is also provided.

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See pages - of SI2D.pdf


Hydrodynamics is one of the oldest research areas in physics, with applications across all macroscopic scales in the Universe.
Despite the long history of successes, however, fluid modelling still presents severe conceptual and computational challenges.
Not surprisingly, the hurdles become even more formidable for relativistic flows, and new issues come to the fore too.
This work is concerned with advancing multi-fluid models in General Relativity, and in particular focuses on the modelling of dissipative fluids and turbulent flows.
Such models are required for an accurate description of neutron star phenomenology, and binary neutron star mergers in particular.
In fact, the advent of multi-messenger astronomy—started with the first detection of a binary neutron star coalescence in 2017—offers exciting prospects for exploring the extreme physics at play during such cosmic fireworks.

In this work we first focus on modelling dissipative fluids in relativity, and explore the arguably unique model that is ideally suited for describing dissipative multi-fluids in General Relativity.
Modelling single fluids in relativity is already a hard task, but for neutron stars it is easy to argue that we need to understand even more complicated settings: the presence of superfluid/superconducting mixtures, for example, means that we need to go beyond single-fluid descriptions.
We then consider turbulent flows and focus on how to perform “filtering” in a curved spacetime setting.
We do so as most recent turbulent models in a Newtonian setting are based on the notion of spatial filtering.
As the same strategy is beginning to be applied in numerical relativity, we focus on the foundational underpinnings and propose a novel scheme for carrying out filtering, ensuring consistency with the tenets of General Relativity.
Finally, we discuss two applications of relevance for binary neutron star mergers. We focus on the modelling of ($\beta$-)reactions in neutron star simulations, and provide a discussion of the magneto-rotational instability that is suited to highly dynamical environments like mergers. We focus on these two problems as reactions are expected to source the dominant dissipative contribution to the overall dynamics, while the magneto-rotational instability is considered crucial for sustaining the development of turbulence in mergers.

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Even though at times it feels like yesterday, almost four years have passed since I first set foot in Southampton.
All the experiences I have had during this time have massively shaped the person I have become, and if I can proudly stand where I am today I owe it to the great people I shared parts of this journey with.
Since the list is too long to acknowledge everyone personally, I will only thank here those that deserve special mentions and without whom this would not have been possible.

First of all, I am hugely indebted to my supervisors Nils Andersson and Ian Hakwe, and to Greg Comer—a.k.a. the fantastic trio.
I have indeed been extremely lucky to have had not two, but three superb mentors.
You have been an endless source of inspiration and guidance, and I did my best to learn as much as I could from you. Working with giants like you has been an immense pleasure and honour, and I hope our collaboration will continue for a long time.

I must also express my sincere thanks to the members of the Gravity Group, particularly those working on neutron star physics. I am honoured to have been part of such a group of brilliant scientists, and grateful for all I have learnt from you.

Next, I want to thank my mother Anna. You were the first one to show me how fascinating physics is, and I strongly believe you were the real trigger for everything that came later. I can never repay you for all the sacrifices you have made, which have allowed me to be here today. I must also thank my brother and sisters Agostino, Eleonora and Giuditta—rigorously in alphabetic order. I truly came to appreciate during my time here how much you all mean to me. Knowing that I can always count on you is priceless.

My friends also played a key role in this endeavour. Both the new ones I met in Southampton, and those who have known me for longer. Without you, I do not think I would have made it this far.

Last, but by no means least, I wish to thank my partner in crime Elisa.
You are the one that has been closest to me throughout the emotional roller coaster of these years.
I know I have been difficult way too often, and I cannot find the words to express how grateful I am for your love, support and faith in me.
You should know that this achievement is also yours, and I look forward with indescribable excitement to the next steps of our journey.

Dissipative (multi-)fluids in General Relativity

A covariant approach to large-eddy filtering

Binary neutron-star merger applications