# Predicting the Curie temperature of magnetic materials
with automated calculations across chemistries and structures

Marian Arale Brännvall  Gabriel Persson  Luis Casillas-Trujillo  Rickard
Armiento  Björn Alling  Department of Physics, Chemistry, and Biology (IFM),
Linköping University, 581 83 Linköping, Sweden

(September 3, 2024)

###### Abstract

We develop a technique for predicting the Curie temperature of magnetic
materials using density functional theory calculations suitable to include in
high-throughput frameworks. We apply four different models, including
physically relevant observables and assess numerical constants by studying 32
ferro- and ferrimagnets. With the best-performing model, the Curie temperature
can be predicted with a mean absolute error of approximately 126 K. As
predictive factors, the models consider either the energy differences between
the magnetic ground state and a magnetically disordered paramagnetic state, or
the average constraining fields acting on magnetic moments in a disordered
local moments calculation. Additionally, the energy differences are refined by
incorporating the magnetic entropy of the paramagnetic state and the number of
nearest magnetic neighbors of the magnetic atoms. The most advanced model is
found to extend well into Fe1-xCox alloys, indicating the potential efficacy
of utilizing our model in designing materials with tailored Curie temperatures
by altering alloy compositions. This examination can illuminate the factors
influencing magnetic transition temperatures in magnetic materials and provide
insights into how they can be employed to make quantitative predictions of
Curie temperatures. Our approach is not restricted to specific crystal
structures or chemical compositions. It offers a more cost-effective
alternative, in terms of human time and need for hands-on oversight, to other
density functional theory methods for predicting the Curie temperature. As a
result, it provides a practical strategy for conducting high-throughput
screening for new technologically applicable magnetic materials.
Alternatively, it can complement machine-learning-based screening of magnetic
materials by integrating physical principles into such approaches, thereby
enhancing their prediction accuracy.

††preprint: APS/123-QED

##  I Introduction

For a magnet to be of practical use, it must maintain its magnetic properties
at operational conditions. For ferro- and ferrimagnetic materials, the
temperature at which they transition into a macroscopically non-magnetic
state, due to a magnetic order-disorder transition, is called the Curie
temperature ($T_{\text{C}}$). The applicability of the magnet is therefore
dependent on $T_{\text{C}}$ and the optimal value will depend on the
application. For magnetic sensors in aeroengines or permanent magnets in
electric motors, for example, $T_{\text{C}}$ must significantly exceed room
temperature [1, 2]. Whereas, in the case of magnetic refrigeration, it is
preferable to have a magnetic transition occurring near room temperature [3,
4]. When searching for new magnetic materials for technological use, it is
therefore crucial to be able to predict $T_{\text{C}}$. As Nelson and Sanvito
point out, most magnetic compounds have a $T_{\text{C}}$ below room
temperature. Hence, merely searching for new magnets without information about
$T_{\text{C}}$ runs the risk of mostly finding candidates that are not very
useful [5].

There are both experimental and theoretical approaches to determine
$T_{\text{C}}$, each with their own limitations when considering their
applicability in the exploration of a diverse range of materials. To
experimentally investigate a magnetic material, the alloy must first be
synthesized, and then one can study the magnetic properties using different
measurement methods, e.g., using a vibrating-sample magnetometer, a
superconducting quantum interference device, or a magnetic torsion balance [6,
7]. This procedure is time consuming, costly, and not feasible to employ for
investigating a wide range of materials across diverse chemistries.
Theoretical investigations of $T_{\text{C}}$ typically involve constructing a
Heisenberg Hamiltonian by deriving the exchange interactions between magnetic
moments using the method developed by Liechtenstein et al. [8, 9, 10, 11]. The
Curie temperature can then be obtained through, e.g., Monte Carlo simulations
or a mean-field approximation [8, 12, 13, 14, 15, 16, 17, 18]. The
Liechtenstein method of extracting the exchange interactions is cumbersome to
apply across a diverse set of complex structures in a high-throughput manner.
Furthermore, the need for considerable human hands-on decisions regarding
aspects such as magnetic reference state, number of included coordination
shells, and effects of local lattice relaxations, has limited studies to
treatments of a single or a few systems per study [19, 20, 21, 22]. In
particular, the current methodology for determining $T_{\text{C}}$ faces
challenges in accurately predicting values for compounds characterized by
strongly correlated 4f electrons [23, 24, 25, 26]. Resolving this challenge
often necessitates the adoption of methods such as the integration of density
functional theory (DFT) with dynamical mean-field theory and/or the DFT+U
method [27, 28, 29, 30]. When navigating through wide-ranging materials, this
introduces an additional layer of complexity.

To overcome these difficulties, different machine-learning (ML) approaches
have been developed to predict the critical temperature [31, 32, 33, 5, 34,
35, 36, 37, 38, 39, 40]. The big advantage of these types of methods is that
the predictions are very cheap and more computationally heavy calculations or
experimental work can be done based on these predictions. However, ML models
often operate as black boxes, making it challenging to understand the
underlying mechanisms driving the predictions [41]. Moreover, the
applicability of ML models for predicting critical temperatures remains to be
proven, especially for, e.g., disordered systems and the effects of alloying.
Nelson and Sanvito developed ML models for predicting $T_{\text{C}}$ using a
few different ML algorithms and the experimental data of about 2500 known
ferromagnets [5]. They developed a model which predicts $T_{\text{C}}$ with an
accuracy of approximately 50 K. They also evaluated the ability of the model
to predict $T_{\text{C}}$ of some binary alloys and the ternary Al-Co-Fe
system with not fully convincing results. In a study by Belot et al., they
developed machine-learning models using a random forest and a k-NN model [39].
They found that for binary alloys, the ML model did not manage to interpolate
smoothly between points with known experimental $T_{\text{C}}$, instead, it
had a tendency to drift towards the average $T_{\text{C}}$ of the training
data. To reduce this issue they trained a new model only on materials with
$T_{\text{C}}$ greater than $600$ K, hence raising the average $T_{\text{C}}$.
In both these studies, the models were only based on chemical composition.
Attempts were made to incorporate structural data, giving an increased number
of features at the cost of less available training data. The result was a
model which could not outperform the chemical-composition-only models [5, 39].
The aim of this work is to investigate a different approach of predicting
$T_{\text{C}}$; a method operating at the level of DFT, but without the need
of a detailed modeling of the magnetic interactions. This method can be used
both as an initial screening to identify candidate materials to investigate
with more expensive in-depth calculations, or as a means to incorporate
physics-based insights into a machine-learning model.

We should make clear what we mean by the critical temperature or, more
specifically, the Curie temperature in this work. The critical temperature of
a magnetic material is defined as the temperature above which the long-range
magnetic ordering vanishes [42]. For ferro- and ferrimagnets, this means that
there is no spontaneous bulk magnetization. However, the spontaneous bulk
magnetization also vanishes in a transition from the ferromagnetic to
antiferromagnetic state, which would be an order-order transition. Our
approach gives the magnetic order-disorder transition temperature. Hence, if a
material undergoes a transition from the ferromagnetic to the
antiferromagnetic to the paramagnetic state, our approach will find the
temperature corresponding to the transition to the paramagnetic state, where
in fact the relevant transition for applications may be the lower
ferromagnetic to antiferromagnetic temperature.

Our approach offers a physics-based way of estimating $T_{\text{C}}$ which
illuminates the contributing factors of the transition temperature. The
approach is general, in that it is not constructed around a specific crystal
structure or chemical family of materials. We explore four different models
that, in one way or another, use either the energy difference between the
magnetic ground state and the magnetically disordered paramagnetic state or
the strength of the constraining fields necessary for modeling the
paramagnetic state. To model the paramagnetic phase, we employ the disordered
local moments model [43, 44]. For the ground-state energy, the magnetic ground
state must be determined, which in the general case can be of various types,
such as ferro-, ferri-, and antiferromagnetic, or even more exotic
noncollinear structures. In the work by Horton et al., a workflow for
predicting the collinear magnetic ground state using DFT was developed [45].
By generating multiple plausible magnetic orderings and ranking them based on
symmetry, they investigated several experimentally known magnetic materials
with nonferromagnetic ground-state ordering. The corresponding experimental
magnetic ordering was found in 60 % of the cases and was correctly predicted
as nonferromagnetic in 95 % of the cases. Galasso and Oganov developed a
similar workflow for determining the collinear ground-state structure of
magnetic materials. Additionally, the Hubbard U in DFT+U calculations and the
Curie temperature through Monte Carlo simulations can be determined in their
workflow [46]. Their method was tested on three known collinear
antiferromagnets. The work of Ehn et al. [47] describes a technique for a
ground-state search with noncollinear magnetism. In this approach, the
magnetic moment directions are initialized as randomly distributed but without
any constraints, and the atomic positions as ideal. The relaxations are
subsequently performed, allowing for both atomic positions and the directions
and magnitudes of the magnetic moments to relax. The magnetic moments will
consequently change sizes and rotate toward their preferred configuration. In
this work, we use this latter technique for determining the ground-state
energy, where we focus on materials with experimentally determined ferro- or
ferrimagnetic ground state, however, allowing for DFT to find other ground
states.

Based on mean-field arguments, there is a correlation between the energy
difference and $T_{\text{C}}$ [48, 43, 23, 43, 49, 50]. The relation between
the strength of the constraining fields and $T_{\text{C}}$ is motivated by
their connection to the average magnetic interactions, $J_{0}$, acting on each
magnetic moment. Furthermore, we combine these results with structural
information and information about the magnetic entropy in the paramagnetic
state to draw a physical picture of the factors that play a role in
determining $T_{\text{C}}$. The insights from these investigations are then
used to construct models to predict $T_{\text{C}}$ of ferro- and ferrimagnetic
materials. This approach is a more general way of estimating $T_{\text{C}}$
compared to the theoretical methods described previously and it provides
insights that may get lost in predictions solely made with machine learning
methods.

##  II  Methodological Background

###  II.1 Energy Difference and the Curie Temperature

The Curie temperature can be estimated using both the mean-field approximation
(MFA) and Monte Carlo (MC) simulations. In a mean-field treatment of the
magnetic system we may start from the semiclassical Heisenberg Hamiltonian,

$\mathcal{H}=-\sum_{i\neq j}J_{ij}\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j},$
(1)

where the sum is over every pair of magnetic moments $i$ and $j$;
$\boldsymbol{e}_{i}$ is the direction of the atomic magnetic moment $i$ and
$J_{ij}$ is the pair exchange interaction. In a mean-field treatment, the
exchange interactions of one magnetic moment with the whole crystal is
examined. The effective exchange parameter is then the sum of all pair
exchange interaction parameters between the considered moment and the rest,

$J_{0}=\sum_{j}J_{0j},$ (2)

where the considered magnetic moment is at site 0 [8]. The energy per magnetic
moment can then be expressed in the mean-field framework as

$E^{MFA}=-\frac{2J_{0}}{M_{0}^{2}}\langle\boldsymbol{M}\rangle\cdot\boldsymbol{M},$
(3)

where $\boldsymbol{M}$ is the magnetic moment vector and $M_{0}$ is the
magnitude of the considered magnetic moment [8]. Subsequently, it can be shown
that the Curie temperature, $T_{\text{C}}$, is approximated as

$k_{\text{B}}T_{\text{C}}=\frac{2}{3}J_{0},$ (4)

where $k_{\text{B}}$ is the Boltzmann constant. As a rough estimate, we may
then identify the exchange interaction $J_{0}$ as proportional to the energy
difference between an ideal magnetically disordered paramagnetic state and the
ferromagnetic state through Equation (3). Combining this insight with Equation
(4) gives us an estimate of $T_{\text{C}}$ as [48, 49, 23, 50]

$k_{\text{B}}T_{\text{C}}\propto(E_{\text{PM}}-E_{\text{FM}}).$ (5)

where $E_{\text{PM}}$ and $E_{\text{FM}}$ are the energies of the ideal
paramagnetic state and the ferromagnetic state, respectively.

Another, more accurate way to estimate the Curie temperature is by means of MC
simulations. In this case, the energy can be evaluated based on the
Hamiltonian of Equation (1) and with exchange interactions extracted
beforehand from DFT calculations. The $T_{\text{C}}$ can then be determined,
e.g, from the location of the characteristic peak in the temperature
dependence of the specific heat at the phase transition.

###  II.2 Disordered Local Moment

The disordered local moment (DLM) method is an approach for simulating the
paramagnetic state in DFT calculations. The method has been implemented both
within the coherent potential approximation by Gyorffy et al. [43] and within
a supercell approach by Alling et al. [44]. Within the DLM method, the local
magnetic moments are fully disordered, describing an ideal case with no short-
range order (SRO). It is assumed that describing the magnetization density as
locally collinear in the region around an atom is a valid approximation [51].
This assumption leads to the atomic moment approximation, where a magnetic
moment vector is assigned to each atom, defined as

$\mathbf{m}_{i}=\int_{\Omega_{i}}\mathbf{m}(\mathbf{r})d\mathbf{r},$ (6)

where $\Omega_{i}$ describes a region around atom $i$ and
$\mathbf{m}(\mathbf{r})$ is the magnetization density. As mentioned, in the
DLM approach, these atomic magnetic moments are distributed to obtain maximum
disorder with minimal SRO and ideally,

$\frac{1}{N}\sum_{i,j\in\alpha}\mathbf{e}_{i}\cdot\mathbf{e}_{j}=0,\;\forall\alpha,$
(7)

where $N$ is the number of magnetic pairs in the summation; $i$ and $j$ denote
the atoms in question belonging to a specific coordination shell $\alpha$, and
$\mathbf{e}_{i}$ and $\mathbf{e}_{j}$ are the unit vectors in the direction of
the magnetic moments.

Two ways of implementing DLM within a supercell approach have been developed
to ensure results corresponding to a true ideal paramagnetic state. One may
either use the special quasirandom structure (SQS) method [52] applied to
magnetic systems or the magnetic sampling method (MSM) [44]. In MSM, the
magnetic moment directions are randomly distributed (either collinearly or
noncollinearly) but not necessarily fulfilling Equation (7). Averages of
physical properties can then be extracted from multiple of these magnetic
samples. These averages are considered to be representative for the disordered
state.

###  II.3 Constraining Fields

An arbitrary noncollinear magnetic configuration is not a ground state,
therefore, when running noncollinear DFT calculations, the magnetic moments
tend to rotate towards the magnetic ground state during the self-consistency
cycle. By applying constraining fields to each of the magnetic moments to
force them in a certain direction, we can however, simulate the disordered
high temperature paramagnetic state. Ma and Dudarev presented a technique for
constraining the directions of the magnetic moments, allowing them to change
in size but not sign [53]. In this approach, the total energy functional is,

$E=E_{0}+E_{p}=E_{0}+\sum_{i}\lambda_{i}(\absolutevalue{\mathbf{M}_{i}^{F}}-\mathbf{e}_{i}\cdot\mathbf{M}_{i}^{F}),$
(8)

where
$\mathbf{M}_{i}^{F}=\int_{\Omega_{i}}\mathbf{m}(\mathbf{r})F_{i}(\absolutevalue{\mathbf{r}-\mathbf{r}_{i}})d^{3}r$
and $F_{i}(\absolutevalue{\mathbf{r}-\mathbf{r}_{i}})=\sin{(x)}/x$ with
$x=\pi\absolutevalue{\mathbf{r}-\mathbf{r}_{i}}/R_{i}$ to ensure that the
magnetization density is smoothly decreased to zero at the atomic sphere
boundary. The term $E_{0}$ is the DFT energy, and $E_{p}$ is the penalty
energy which tends towards zero as $\lambda_{i}\rightarrow\infty$. The
parameter $\lambda$ must be set high enough and often incremented in steps for
the energy to be converged. The constraining fields are given by the variation
of $E_{p}$ with respect to the magnetic moment direction [53],

$\boldsymbol{b}_{p}(\boldsymbol{r})=-\frac{\delta
E_{p}}{\delta\boldsymbol{m}(\boldsymbol{r})}.$ (9)

The strength of these constraining fields is connected to the effective
exchange parameter, $J_{0}$, of Equation (2). This can be understood by
considering that the effective magnetic field, in a mean-field treatment, is
proportional to $J_{0}$ and the constraining field strengths are connected to
the effective magnetic field [54]. The magnitude of $J_{0}$ essentially
describes the energy cost of rotating a magnetic moment from its equilibrium
direction, thereby influencing how strongly the constraining fields need to
act to keep the magnetic moment in this nonequilibrium direction.

##  III Method

###  III.1 Effect of number of nearest magnetic neighbors

The presence of additional (or fewer) magnetic neighbors should have an impact
on $T_{\text{C}}$ through differing short-range-order effects and,
consequently, influence the relation between the energy difference, $\Delta
E=E_{\text{DLM}}-E_{\text{FM}}$, and $T_{\text{C}}$. This relation can be
investigated with MC simulations. By keeping $\Delta E$ fixed and only varying
the number of magnetic interactions included in the sum of Equation (2) we
investigate the effect this has on the position of the specific heat peak.
Each pair interaction in Equation (2) is equally strong, independent of the
distance between atoms. The number of magnetic interactions is varied from 4
up to 320. The high end of the scale does not represent a realistic number of
equally strong magnetic interactions in a real system, but is included to
highlight the contrast between a few strong and many weak interactions. Figure
1 shows the ratio between the $T_{\text{C}}$ from the MC simulations based on
the peak of the specific heat curve ($T_{\text{C}}^{\text{MC}}$) and
$T_{\text{C}}^{\text{model}}$ as a function of the number of magnetic
interactions. The temperature $T_{\text{C}}^{\text{model}}$ is the Curie
temperature of the structure with the highest number of magnetic interactions,
since in the limit of infinitely many nearest magnetic neighbors, the result
should go towards the mean-field solution. We clearly see that with fewer
nearest neighbors, the discrepancy will be larger between the MC and modeled
$T_{\text{C}}$.

Figure 1: The ratio between the Curie temperature based on specific heat peak
from Monte Carlo simulations ($T_{\text{C}}^{\text{MC}}$) and the Curie
temperature of the simulation with the highest number of magnetic interactions
(the mean-field limit, $T_{\text{C}}^{\text{model}}$) as a function of the
number of interactions. The inset shows the region where diamond, simple-cubic
(SC), body-centered-cubic (BCC), and face-centered-cubic (FCC) structures are
found. Note that all MC simulations are done for systems having the same
$\Delta E$ between the DLM and ground state energies.

This observation suggests that one can adjust $T_{\text{C}}^{\text{model}}$
based on the number of magnetic interactions to better approximate
$T_{\text{C}}^{\text{MC}}$. In this work, interpreting the shape of the curve
of Figure 1, we adjust $T_{\text{C}}^{\text{model}}$ as

$T_{\text{C}}^{\text{adj.}}=T_{\text{C}}^{\text{model}}\Big{(}1-\frac{A}{NN^{B}}\Big{)},$
(10)

where $NN$ is the number of nearest magnetic neighbors of the magnetic atoms,
and the parameters $A$ and $B$ are determined by minimizing the difference
between adjusted and experimental $T_{\text{C}}$. This adjustment ensures that
the different impact of short-range order effects for different crystal
structures and alloy compositions are reflected in the predicted
$T_{\text{C}}$.

###  III.2 Disordered Local Moment Calculations

For the DLM calculations we employ the supercell approach described in section
II.2, however, we use neither the traditional SQS- nor the MSM implementation.
The number of time-consuming calculations must be kept to a minimum. Since we
perform noncollinear magnetic DFT calculations, it is not feasible to conduct
calculations for a large number of magnetic samples. For this reason, we
generate multiple configurations of randomly directed magnetic moments and
choose one where the SRO of the first coordination shell is close to zero (of
the order of 0.001 or smaller) for the DFT calculations. The SRO is defined
as,

$SRO=\Bigg{\langle}\frac{1}{N_{\text{at}}}\sum_{i=1}^{N_{\text{at}}}\frac{1}{N_{\text{NN}}}\sum_{j=1}^{N_{NN}}\boldsymbol{e}_{i}\cdot\boldsymbol{e}_{j}\Bigg{\rangle},$
(11)

where, $N_{\text{at}}$ is the total number of atoms of the cell and
$N_{\text{NN}}$ is the number of magnetic neighbors in the first coordination
shell.

From the DLM calculations, both the constraining fields and the magnitudes of
the magnetic moment are collected. The magnetic entropy of the magnetically
disordered state is approximated as,

$S^{\text{mag}}=k_{\text{B}}\Bigg{[}\sum_{i=1}^{N_{\text{con}}}\ln{(m_{i}+1)}\Bigg{]}/N_{\text{con}},$
(12)

where $k_{\text{B}}$ is the Boltzmann constant, $N_{\text{con}}$ is the number
of atoms with constrained magnetic moments, and $m_{i}$ is the magnetic moment
magnitude of the $i$th constrained magnetic moment.

###  III.3 Experimental Curie Temperatures

The experimental Curie temperatures used in this work are presented in Table
1, with references to the corresponding studies. One should remember that for
many systems there are multiple reported Curie temperatures, which signals the
sensitivity of the exact value of $T_{\text{C}}$ on the exact composition or
presence of defects. Hence, choosing which value to use when developing
predictive models is not obvious. Both Ref. [5] and [39] used the median Curie
temperature when selecting between multiple values in their machine learning
model constructions. This would ensure that the value used is an actual
measured value, rather than the mean value. In our work, we have chosen one
temperature, and for most cases, the differences found are not more than a few
kelvin. In all cases the different temperatures are well within the mean
absolute errors of our developed models.

###  III.4  Workflow

Figure 2 illustrates the workflow implemented to collect the energy difference
between the ground state and the paramagnetic state described within the DLM
approach. We also obtain the constraining fields from the DLM simulations.

Figure 2: The workflow of collecting the energy difference and constraining
fields of ferri- and ferromagnetic systems. “Bernoulli test” refers to the
check made of the distribution of ground-state (GS) energies as described in
the text.

A system to be investigated, from e.g. the Materials Project [55], is fed into
the workflow. If the original structure contains less than 12 magnetic atoms,
a supercell is constructed by multiplying the cell in $x$, $y$, and $z$
direction until the supercell contains enough magnetic atoms. The supercell is
necessary when simulating the magnetically disordered state using the DLM
method. The first ground-state search is subsequently conducted on the
original unit cell fed into the workflow. The ground-state search is done
using the technique described by Ehn et al. where the directions of the
magnetic moments are initially set randomly and are allowed to rotate to their
preferred directions during the structural relaxation [47]. The choice of unit
cell in the ground-state search plays an important role in determining the
outcome of the search. A unit cell with only one magnetic atom will, of
course, always lead to a ferromagnetic ordering. However, a unit cell with
multiple magnetic atoms allows for the exploration of diverse magnetic
configurations, such as antiferromagnetic, ferrimagnetic, or complex
noncollinear ones.

The first step is to perform five ground-state searches. Based on these, the
lowest energy is identified, and the other energies are expressed relative to
this minimum. The distribution of the results within 0.025 eV/magnetic atom
from the minimum energy determines whether additional ground-state searches
should be conducted. If the probability that at least one of the ground-state
searches (excluding the minimum) is within 0.025 eV/atom from the minimum
value is 90 % or more, the ground-state search is considered to be finished
and the magnetic structure with the lowest energy taken as the ground-state
structure. The probability is determined based on the Bernoulli trial
framework and the binomial probability function, where a success is an energy
within 0.025 eV/magnetic atom from the minimum [56]. The number of ground-
state searches is limited to a maximum of 14. The outcome of this procedure is
the energy of the ground-state structure.

The next step is a DLM run performed without structural relaxations and with
the directions of the magnetic moments constrained. The constrain is enforced
by a starting $\lambda$-value set to $1$ (see Equation (8)). The directions of
the magnetic moments are set randomly as described in section III.2. The
constraint is only enforced on atom types where the average magnetic moment is
larger than 0.75 Bohr magneton in the ground-state run, smaller magnetic
moments will therefore be free to rotate to whichever direction they prefer.
Hence, very itinerant systems can not be described in the current workflow;
this would require a constraint on the magnitudes of the magnetic moments in
the DLM step. The DLM runs continue with the parameter $\lambda$ set to 5, and
then 10. Subsequent $\lambda$-steps use the wave functions and charge density
of the previous step. Beyond $\lambda=10$, a check is made of the energy
difference between the different $\lambda$-steps. The convergence with respect
to $\lambda$-value is examined by calculating the slope

$\frac{\absolutevalue{E_{new}-E_{old}}}{\absolutevalue{\lambda_{new}-\lambda_{old}}}.$
(13)

The energy with respect to the $\lambda$-parameter is considered converged if
the slope is less than 0.1 meV. If it is not converged, the
$\lambda$-parameter is raised in steps of 5 until convergence is reached.

###  III.5  Model development

Four different models for estimating $T_{\text{C}}$ are investigated, denoted
as models A, B, C, and D. In model A, the energy difference between the
paramagnetic state and the magnetic ground state is used on its own; model B
uses the average strength of the constraining fields necessary for modeling
the paramagnetic state; model C uses the energy difference divided by the
magnetic entropy of the paramagnetic state as given by Equation (12); and
model D adjusts the energy difference divided by the magnetic entropy
according to Equation (10) to incorporate the effect of the number of strong
magnetic interactions, and thus the relative effect of short-range order. In
each case, a linear model is fitted between the DFT output data and the
experimental $T_{\text{C}}$. The linear regression is implemented using
Scikit-Learn [57]. For model D, an additional step is needed to incorporate
the effect of the number of strong magnetic interactions. The energy
difference divided by the magnetic entropy (corresponding to
$T_{\text{C}}^{\text{model}}$) is adjusted according to Equation (10). The
parameters A and B are optimized using the Nelder-Mead method [58] by
minimizing the mean absolute error of the adjusted model compared to the
experimental Curie temperature,
$\langle\absolutevalue{T_{C}^{\text{adj.}}-T_{C}^{\text{exp.}}}\rangle$.

##  IV Computational details

To obtain the energies, magnetic moments, and magnetic constraining fields, we
use DFT calculations as implemented in the Vienna Ab initio Simulation Package
[59, 60, 61, 62] using projector augmented-wave potentials (PAW) [63, 64] and
the Perdew-Burke-Ernzerhof (PBE) generalized gradient approximation (GGA) for
approximating the exchange-correlation functional [65]. The energy cutoff is
set to 450 eV and a Monkhorst-Pack k-point grid with a density of 20
k-points/Å is used [66]. The ground-state search as suggested in Ref. [47] is
conducted on the unit cells obtained from the Materials Project [55].

In the ground-state search, the magnetic moments are initially set to 3 Bohr
magneton ($\mu_{B}$) with random, noncollinear, directions. The magnetic
moments are then allowed to relax in both magnitude and direction, along with
the structural relaxation.

To simulate the paramagnetic state, the DLM approach is used. A supercell of
at least 12 magnetic atoms is constructed. The calculations are spin-
polarized, utilizing noncollinear magnetism. The magnitudes are allowed to
relax in the DLM simulations, but are initially set to 110 % of the relaxed
average ground-state size of the magnetic moments of the specific atomic
species. The directions of the magnetic moments are constrained using the
method presented by Ref. [53].

As a test case, we examine the binary Fe1-xCox alloys with different Co
contents to investigate how well our models estimate their Curie temperatures.
For these calculations, the chemical disorder in the bcc and fcc Fe1-xCox
alloys is implemented using the SQS method [52].

##  V Results

###  V.1 Magnetic Ground State Structures

Table 1 presents both the experimentally observed magnetic ground state and
the theoretically obtained ground state from the ground-state search for
comparison. Most of the ground-state searches reproduce the experimentally
found ferro- and ferrimagnetic ground states of the materials considered here.
However, some differences can be observed. For hcp Gd, an antiferromagnetic
ordering is obtained. It has been shown that treating the 4f electrons of bulk
hcp Gd using the GGA exchange correlation functional gives an incorrect
antiferromagnetic ground state instead of the correct ferromagnetic one. One
solution for this issue is to use GGA+U to treat the 4f electrons [67].
However, in this work we have not explored GGA+U to improve this aspect of the
DFT results.

Both CrBr3 and CrI3 should be ferromagnetically ordered [68, 69, 70]. These
systems are layered and our ground-state searches result in ferromagnetic
alignment of the Cr atoms within the layers, however, not between the layers.
This is most likely due to weak magnetic interactions between magnetic moments
of different layers.

In line with experiments [71, 72], our calculations results in a ferrimagnetic
ground state for Fe3O4 where 1/3 of the Fe magnetic moments are
antiferromagnetically aligned with the remaining 2/3 (the unit cell contains
24 Fe atoms). The ferrimagnetic structure is not obtained by a simple
collinear calculation, which instead results in a ferromagnetic structure with
a significantly higher energy.

Mn3Ge in an L12-type structure, as investigated here, is reported to have a
ferromagnetic ground state [73], however, our ground-state search do not find
this. Instead, we find a magnetic state that can be described as a ferrimagnet
or close to an antiferromagnet with total magnetic moment of only 0.90
$\mu_{B}$ for a unit cell with 3 Mn atoms. A collinear ferromagnetic
calculation gives an energy higher than for the structure obtained with the
ground-state search. The DFT study of Mn3Ge by Arras et al. [74] found the
ground state to be ferrimagnetic for the L12 structure, which suggests that
some effect is not accounted for in the DFT simulations, but this is not
further investigated in this work.

Cr2CuO4 is reported to be ferrimagnetic [75], and our ground-state search
gives this magnetic ordering with a total magnetic moment of 10 $\mu_{B}$ for
a unit cell with 8 Cr atoms. Out of the 8 Cr moments, 6 are ferromagnetically
aligned and the remaining 2 are slightly larger and antiferromagnetically
aligned with the rest.

Mn4N is ferrimagnetic [76], and our ground-state search finds a structure
similar to the ferrimagnetic structure found in the DFT study by Adhikari et
al. [77], with a total magnetic moment of 1.16 $\mu_{B}$ for the unit cell
containing four Mn atoms. However, in our found structure, the face-centered
Mn moments are not completely ferromagnetically aligned with each other.

Fe3BO6 is a weak ferromagnet (or canted antiferromagnet) [78, 79], and our
ground-state search results in a magnetic structure that can be described as
ferrimagnetic, resulting in a total magnetic moment of 4.4 $\mu_{B}$ for a
unit cell with 12 Fe atoms, with a lower energy than for the ferromagnetically
ordered structure.

###  V.2 Assessment of Model Parameters

Table 1:  Data for all investigated systems. The reference to where the experimentally data can be found is given next to the system formula. The magnetic ground states are given both from experiment and from our DFT ground-state search where FM = ferromagnetic, FiM = ferrimagnetic, and AFM = antiferromagnetic. The experimental $T_{\text{C}}$ and the $T_{\text{C}}$ estimated by model A, B, C, and D. The energy difference between the ground state and disordered paramagnetic state ($\Delta E$) the magnetic entropy ($S^{\text{mag}}$), and the average constraining field ($B_{\text{con}}$) are given per constrained atomic magnetic moment. | Magnetic Ordering |  | Model $T_{\text{C}}$ (K) | DFT data
---|---|---|---|---
Material | Exp. | DFT | T${}_{C}^{\text{exp.}}$ (K) | $T_{\text{C}}^{\text{A}}$ | $T_{\text{C}}^{\text{B}}$ | $T_{\text{C}}^{\text{C}}$ | $T_{\text{C}}^{\text{D}}$ |  $\Delta E$ (meV) |  $S^{\text{mag}}$ (meV/K) | B${}_{\text{con}}$ (meV)
Co3V2O8 [80] | FM111FM to AFM at 6 K, AFM to paramagnetic at 10 K | FM | 10 | 425 | 440 | 398 | 366 | 73.8 | 0.108 | 28.7
CrBr3 [68] | FM | FM 222 FM within layers but not between layers. | 33 | 139 | 242 | 133 | 132 | 4.6 | 0.118 | 4.5
CrI3 [69, 70] | FM | FM 2 | 58 | 103 | 189 | 131 | 136 | 8.9 | 0.177 | 0.5
GdN [81] | FM | FM | 68 | 211 | 244 | 220 | 197 | 21.4 | 0.120 | 4.9
Cr2CuO4 [75] | FiM | FiM | 122 | 351 | 291 | 313 | 311 | 58.1 | 0.108 | 10.3
GdCd [82] | FM | FM | 262 | 172 | 221 | 164 | 169 | 24.3 | 0.177 | 4.6
GdZn [83] | FM | AFM | 269 | 157 | 237 | 173 | 153 | 8.1 | 0.177 | 4.1
MnP [84, 85] | FM333FM from $T>$ 50 K, helical order below 50 K | FM | 291 | 537 | 803 | 680 | 604 | 101.3 | 0.068 | 72.7
Gd (hcp) [42] | FM | AFM | 293 | 176 | 222 | 148 | 149 | 14.0 | 0.178 | 2.1
CrO2 [86] | FM | FM | 398 | 508 | 578 | 505 | 449 | 98.1 | 0.092 | 44.9
Mn3Ge (L12) [73] | FM | AFM/FiM | 400 | 461 | 469 | 435 | 459 | 82.6 | 0.104 | 32.2
Fe3Pt [87] | FM | FM | 430 | 505 | 639 | 441 | 476 | 97.4 | 0.110 | 52.2
Fe3C (cementite) [88] | FM | FM | 488 | 529 | 539 | 642 | 589 | 104.1 | 0.076 | 45.5
Cu2MnIn [89] | FM | FM | 500 | 617 | 347 | 473 | 531 | 123.7 | 0.132 | 20.7
Fe3BO6 [78] | weak FM | FiM | 508 | 917 | 678 | 669 | 624 | 194.5 | 0.134 | 57.6
MnB [90] | FM | FM | 579 | 683 | 666 | 771 | 678 | 142.6 | 0.079 | 55.4
FeB [91] | FM | FM | 580 | 579 | 691 | 734 | 719 | 115.1 | 0.071 | 65.0
MnSb [92] | FM | FM | 600 | 465 | 386 | 377 | 340 | 87.1 | 0.122 | 21.7
Fe3P [93] | FM | FM | 716 | 706 | 627 | 810 | 600 | 143.7 | 0.079 | 56.7
Fe3Sn [94] | FM | FM | 725 | 873 | 763 | 812 | 722 | 183.8 | 0.099 | 67.8
FePd [95] | FM | FM | 729 | 669 | 774 | 552 | 544 | 133.7 | 0.119 | 69.2
Mn4N [76] | FiM | FM | 740 | 450 | 458 | 479 | 538 | 86.3 | 0.091 | 35.0
Co3B [96] | FM | FM | 750 | 517 | 774 | 949 | 899 | 101.3 | 0.047 | 75.6
Fe3Ge [97] | FM | FM | 755 | 727 | 732 | 698 | 762 | 153.6 | 0.096 | 63.4
Fe4N [98] | FM | FM | 761 | 566 | 734 | 564 | 642 | 112.8 | 0.092 | 63.6
FeNi (L$1_{0}$) [99] | FM | FM | 805 | 977 | 581 | 830 | 812 | 204.0 | 0.109 | 50.8
CoPt [87] | FM | FM | 813 | 651 | 642 | 664 | 659 | 134.3 | 0.089 | 52.5
Fe3O4 [100] | FiM | FiM | 838 | 581 | 575 | 460 | 508 | 112.2 | 0.130 | 45.1
FeNi3 [87] | FM | FM | 863 | 1008 | 776 | 799 | 830 | 210.9 | 0.118 | 75.9
Co2MnSi [101] | FM | FM | 985 | 838 | 1141 | 668 | 765 | 181.9 | 0.120 | 112.4
Fe (bcc) [42] | FM | FM | 1043 | 996 | 718 | 959 | 1037 | 208.3 | 0.095 | 68.4
Co (fcc) [42] | FM | FM | 1388 | 792 | 594 | 1209 | 1429 | 163.9 | 0.055 | 47.4
Table 2: The three initial models and their DFT input variables, final expressions, mean absolute errors, and $R^{2}$ values for the data set of materials discussed in Sections V-VI. A perfect agreement would have $R^{2}=1$. The DFT input of models A and B are given per constrained atomic magnetic moment. Model | DFT input | Model (K) | Mean Absolute Error (K) | $R^{2}$
---|---|---|---|---
A |  $x=\Delta E$ [eV] | $T_{\text{C}}^{\text{A}}=4147x+104$ | 160 | 0.64
B |  $x=B_{\text{con}}$ [eV] | $T_{\text{C}}^{\text{B}}=8126x+198$ | 176 | 0.51
C |  $x=\Delta E/S^{\text{mag}}$ [K] | $T_{\text{C}}^{\text{C}}=0.365x+130$ | 157 | 0.67
D |  $x=\Delta E/S^{\text{mag}}$ [K] | $T_{\text{C}}^{D}=0.85x\Big{(}1-\frac{0.69}{NN^{0.14}}\Big{)}+124$ | 126 | 0.77

All properties needed for the four models described in section III.5 are
calculated for 32 different systems, and results from the DFT calculations
along with the predicted $T_{\text{C}}$ of the four models are presented in
Table 1. Figure 3 shows the experimental $T_{\text{C}}$ against the predicted
$T_{\text{C}}$ for each of the four models. In Table 2, the expressions of the
respective models are found along with their mean absolute error (MAE) and
$R^{2}$-value. The $R^{2}$-value is given by

$R^{2}=1-\frac{\sum_{i}[y^{i}-f(\boldsymbol{x}^{i})]^{2}}{\sum_{i}[y^{i}-\mu]^{2}}$
(14)

where $y^{i}$ are target values, with mean $\mu$, and $f(\boldsymbol{x}^{i})$
are predicted values. The $R^{2}$ value and MAE are determined using a 3-fold
cross validation scheme for the mean value of the respective errors. For model
D, the resulting adjustment after optimization gives A $=0.69$ and B $=0.14$
in Equation (10). The final model D is given by,

$T_{\text{C}}^{\text{D}}=0.85x\Bigg{(}1-\frac{0.69}{NN^{0.14}}\Bigg{)}+124\>\text{K},$
(15)

where $x=\Delta E/S^{\text{mag}}$. The distribution of the absolute errors
$\absolutevalue{T_{\text{C}}^{\text{exp.}}-T_{\text{C}}^{\text{D}}}$ and the
relative errors
$(T_{\text{C}}^{\text{exp.}}-T_{\text{C}}^{\text{D}})/T_{\text{C}}^{\text{exp.}}$
are shown in Figure 4 for model D.

Figure 3:  Experimental versus predicted Curie temperatures based on the four
models from Table 2.

(a)  (b)

Figure 4:  The absolute errors
$\absolutevalue{T_{\text{C}}^{\text{exp.}}-T_{\text{C}}^{\text{D}}}$ and the
relative errors
$(T_{\text{C}}^{\text{exp.}}-T_{\text{C}}^{\text{D}})/T_{\text{C}}^{\text{exp.}}$
of model D. The inset in each subplot shows the systems with errors outside of
the intervals on the $x$-axes.

###  V.3  Fe1-xCox Alloy

Estimating the Curie temperature of various compositions in binary alloys is
of great interest as it enables the selection and optimization of materials
tailored to specific applications. Here, we apply model D of Equation (15) to
the disordered Fe1-xCox alloys to investigate its effectiveness in making
predictions for disordered binary alloys over compositions of different ratios
of Fe and Co, for both the bcc and fcc structures. Figure 5 shows predicted
$T_{\text{C}}$ as a function of number of electrons per atom, going from pure
bcc Fe to pure fcc Co. The experimental $T_{\text{C}}$ from Nishizawa and
Ishida, and Rajeevan et al. are also included [102, 103]. The experimental
results from Refs. [104, 105] were compiled into a phase diagram of the
magnetic transition in Fe1-xCox by Nishizawa and Ishida. The experimental
point from Rajeevan et al. is for a bcc structure [103]. For the Fe1-xCox
alloy, the magnetic transition coincides with the bcc-fcc transition at
approximately 31 at.% Fe and 78 at. % Fe in the Co- and Fe-rich regions,
respectively [102].

##  VI Discussion

###  VI.1 Model Performance

A general correlation of co-variation is found between the different input
factors of the four different models and the experimental $T_{\text{C}}$. As
expected, higher values correspond to higher experimental $T_{\text{C}}$,
however, there is some significant noise. Mostly, model D performs better than
the remaining three models, as can be seen in Figure 3 and Table 2. For some
of the materials, the different models give vastly different predictions. Pure
face-centered-cubic (fcc) Co is such a case. The fcc Co has relatively small
magnetic moments (approximately 1 $\mu_{B}$) in the DLM simulations, and
including the effect of magnetic entropy, as in models C and D, makes the
estimations significantly better. A similar tendency is seen for FeNi and
Cu2MnIn where, in both cases, model A gives an estimate of $T_{\text{C}}$ that
is 100 K too large, and model B more than 100 K too small. Models C and D
estimate $T_{\text{C}}$ to be less than 50 K from the experimental value.
There are some cases where model A gives a significantly better estimate, for
example for FeB and MnP. However, the estimate for MnP is still quite poor
when using either of the models. Model B gives a better estimate for hcp Gd
and GdCd which could be due to the need for GGA+U to treat Gd correctly. When
looking at the average strength of the constraining fields, the impact of not
going beyond GGA in the calculations may not be as prominent since it is not
dependent on the ground-state energy. We can also see that in general,
materials with very low $T_{\text{C}}$ are difficult for all of the models.
Estimations for low $T_{\text{C}}$ materials are also difficult for ML models
[5, 39]. The reason for these difficulties could be attributed to these
systems having more complicated magnetic transitions to a larger extent than
robust ferromagnets.

As mentioned, model D performs significantly better than the other models in
most cases. The adjustment based on the number of strong magnetic
interactions, which incorporates structural information, clearly has a notable
impact on the performance of the model. For model D, the $R^{2}$ value is
improved to 0.77 and the MAE to 126 K. Figure 4 shows the absolute and
relative errors of model D. More than 60 % of the systems have an absolute
error equal to or less than 126 K. As the MAE indicates, the concentration of
the absolute errors is not towards zero, but rather around 126 K. The systems
with the less accurate estimates have an absolute error of about 300 K. The
relative errors are the largest for the systems with low $T_{\text{C}}$, where
Co3V2O8 in particular stands out. This system has the highest absolute error.
However, in general, the relative errors are less than 50 % for a large
majority of the systems, and approximately 60 % of the systems have a relative
error of less than or equal to 25 %. For comparison, the best model of Nelson
and Sanvito, which considers only chemical composition, has an MAE of
approximately 50 K and $R^{2}$ value of 0.81[5].

###  VI.2  Fe1-xCox Alloy

From Figure 5, we see that our model D captures the overarching trend well in
both disordered bcc and fcc Fe1-xCox alloys. A clear correlation is shown
where a higher Co content in the bcc structure gives a higher $T_{\text{C}}$
when moving towards the bcc-fcc phase boundary. Similarly, for the fcc
structure, more Fe results in a lowering of $T_{\text{C}}$. The results
suggest the feasibility of obtaining a good estimate of the disordered binary
alloy $T_{\text{C}}$ through our model, which could be used as a basis for
making more complex calculations or experiments. For comparison with other
methods for $T_{\text{C}}$ estimations, we have included the predictions of
the same system by Nelson and Sanvito, and Belot et al. [39, 5] in Figure 5.
In both of these cases, unphysical drifts of the predicted Curie temperatures
are observed. As pointed out by Belot et al., these drifts occur between
points of known experimental data included in the training data. This suggests
a lack of transferability in the ML methods where the model does not capture
the connections between different compositions of a binary alloy. In both
studies, they attempted to include structural information into the
descriptors, however this requires a larger feature space and leads to a
smaller dataset due to the limited amount of structural data. As a result, the
inclusion of structural information did not improve the model. The failure of
ML methods in this case puts a critical perspective on the good agreement with
experiments reported when considering ordered structures. For disordered
structures such as these, a method based on DFT calculations and
straightforward physical assumptions, as used in the models of the present
work, clearly has an advantage.

Figure 5: Experimental and predicted Curie temperatures for the binary
Fe1-xCox alloy as a function of electrons per atom. Estimates using our model
D (red triangles, dashed line) for the disordered bcc structure for high Fe
content (left) and the fcc structure for low Fe content (right). The
experimental Curie temperatures compiled by Nishizawa and Ishida [102] (dark
blue circles) with solid lines for their interpolation between points. The
experimental point from Rajeevan et al. [103] (turquoise circle) is for
Fe40Co60. ML predictions (dashed blue and magneta lines) reproduced from Belot
et al. [39] and Nelson and Sanvito [5]. The crystal structues are indicated in
the plot (bcc or fcc).

###  VI.3 Computational Cost

Both the ground-state search and the DLM calculations are noncollinear
magnetic calculations which are more computationally demanding than collinear
magnetic calculations. Computations were performed using compute nodes with
Intel Xeon Gold 6130 CPUs. The main part of the computational cost is the DLM
step for almost all materials, which requires, on average, approximately 200
core hours; for the most costly calculations, up to 900 core hours are needed.
One ground-state search takes on average 48 core hours which we may compare
with the ferromagnetic collinear calculation which takes, on average, 4 core
hours for the same structure. The noncollinear magnetic ground-state search is
therefore, on average, 12 times more time-consuming than a straightforward
collinear ferromagnetic calculation. For a few materials, it is even several
hundred times more time-consuming. This is a significantly larger
computational cost, but as we have seen, noncollinear spins are necessary to
ensure finding the DFT ground state in some cases.

##  VII Conclusions

In this work we have investigated how the Curie temperature can be estimated
based on the energy difference between the paramagnetic state and the ground
state obtained using first-principles DFT calculations. We find that by
introducing the effect of the magnetic entropy in the paramagnetic state,
effectively also accounting for the magnetic moment sizes of different
systems, we manage to predict $T_{\text{C}}$ with an $R^{2}$ value of 0.67 and
a mean absolute error of 157 K. By also including the number of nearest
magnetic neighbors, influencing the effect of magnetic SRO on $T_{\text{C}}$,
we can further improve the model to get an $R^{2}$ value of 0.77 and a mean
absolute error of 126 K.

In this study, we have also explored the ability of our approach to describe
the change in Curie temperature in a disordered binary alloy as a function of
the composition. We find that our model D estimates the general trend of
$T_{\text{C}}$ well across the range of compositions for the Fe1-xCox
disordered alloy. The ability to predict such trends can be used, e.g., to
predict the tuning of the Curie temperature of an alloy for specific
applications. It may be possible for an ML model trained for $T_{\text{C}}$
predictions for the materials in this work to give a lower MAE and higher
$R^{2}$-value, however, the substantially improved estimations for various
compositions of the Fe1-xCox alloy underscore the necessity of incorporating
physics-based insights when predicting the Curie temperature of a magnetic
material.

Our approach can be used as an early screening step in high-throughput
workflows. The model can efficiently identify candidates for further
exploration which are then investigated using computationally heavy
calculations or experiments. Furthermore, the method can be combined with ML
models to improve their performance in e.g., a delta learning approach where
one trains a model to predict the differences between one of our models and
the true value. This approach may help the physical insights of the DFT-based
model to be integrated in the ML models and act as a substitute for
experimental training data where it is lacking. This integration can
potentially lead to enhanced predictive capabilities in an ML-based screening
for new magnetic materials.

###### Acknowledgements.

The computations were enabled by resources provided by the National Academic
Infrastructure for Supercomputing in Sweden (NAISS) partially funded by the
Swedish Research Council through grant agreement no. 2022-06725. B.A.
acknowledges financial support from the Swedish Research Council (VR) through
Grant No. 2019-05403, and 2023-05194 and from the Swedish Government Strategic
Research Area in Materials Science on Functional Materials at Linköping
University (Faculty Grant SFOMatLiU No. 2009-00971). R.A. acknowledges
financial support from the Swedish Research Council (VR) Grant No. 2020-05402
and the Swedish e-Science Research Centre (SeRC).

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