Stable and metastable rare-earth-free permanent magnets from a database of predicted crystal structures

With the recent developments in crystal structure prediction, databases of new (not previously synthesized) materials are being created. One of these databases contains more than a million entries with the distance to the Convex Hull predicted by crystal-graph attention networks. Hence, stable and metastable materials can be extracted and then investigated for any desired properties. A high-throughput and data-mining approach we previously developed to search for rare-earth-free permanent magnets was applied to these compounds. As a result, four promising candidates for novel rare-earth-free permanent magnets were discovered with high magnetization, high uniaxial magnetocrystalline anisotropy, and high Curie temperature - Ta 3 ZnFe 8 , AlFe 2 , Co 3 Ni 2 , and Fe 3 Ge. The materials were investigated in more detail and all were verified to be dynamically stable.


Introduction
One of the pieces in the work against climate change is the transition to renewable energy sources (e.g.wind power) and sustainable transportation (electric vehicles).The resulting increased demand for high-performance permanent magnets (PMs) drives the research into new magnetic materials.All the PMs used in wind-turbine end electric vehicle engines are based on rare-earth (RE) elements [1], many of which are on the list of critical raw materials [2].They are subject to high supply risks, are not recycled efficiently, and are often mined with a high strain on the environment.Hence, there is a growing interest in finding high-performance PMs that are free from or contain smaller amounts of RE elements.Some companies have recently announced the transition to RE-free PMs as one of their goals [3].
We have previously developed [19] and used [20,21] a highthroughput computational approach to the search for RE-free PMs.However, in all the investigations, we were filtering through the materials of the Inorganic Crystal Structure Database (ICSD) [22].In this database, all entries have been previously reported from experiment.Recently, some sieving through the ICSD database were performed by other researchers as well [23].However, looking for new materials (which have never been synthesized before) can provide valuable guidance for future experiments and can be quite useful in the search for novel RE-free PMs.
As computational materials science, particularly approaches based on the ab-initio theory that relies on density functional theory (DFT), can investigate multiple combinations of elements of the Periodic Table in various geometrical arrangements with more freedom than the experiment, there is an increasing interest in stable and metastable crystal structure prediction.This way, theory can guide the experiment by suggesting the plausible new materials that are worthwhile of a synthesis attempt.Different methods have been developed in this quest, such as evolutionary techniques [24][25][26][27], deep-learning [28], machinelearning [29], and others [30][31][32].Moreover, a combination of experimental approaches with computational techniques (e.g.machinelearning) was used in the search for new stable and metastable magnetic materials in [33][34][35].
A database of new (experimentally unknown) crystal structures was recently created using the new graph neural network approach and machine learning [36,37].A large space of almost 1 billion crystalline compounds was explored by scanning the composition space for various crystal structure prototypes.This resulted in a data set of DFT calculations for about 2.6 million crystal structures [38].Most importantly, the data for each material contains the distance to the convex hull of thermodynamic stability predicted by the crystal-graph attention network [37]  A schematic image of a convex hull for Fe-B-Al systems.Purple circles denote stable materials (their formation enthalpies were taken from the Materials Project [41]) and form the convex hull.The surface 50 meV above it limits the metastable materials (which can still be synthesized).Squares symbolize the result of structure generationblue (below hull) and orange (less than 50 meV above it) will pass through our filters, the red will not be considered further.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)can be synthesized experimentally.All the materials are geometrically optimized with DFT.
In this work, we will apply our high-throughput filtering technique reported in Refs.[19][20][21] to the crystal structures predicted in [37].Promising candidates for high-performance permanent magnets found as the result will be further discussed and tested for dynamical stability.

High-throughput filtering and further calculations
The crystal structure database contains three separate data sets [38].According to the authors of Ref. [38], the first one is quite biased due to the lack of structural and chemical diversity in the available data [36].This issue was later overcome, resulting in creation of two additional data sets [37] with a reduced mean absolute error in the distance to the convex hull.In our search, we will consider the latter two data sets only.
As we are interested in metastable and stable materials, which makes them possible to be synthesized experimentally, the first filtering criterion we employ is the distance to the convex hull.We select those entries where the distance is less than 50 meV, which makes them likely to be synthesized [39,40].Fig. 1 illustrates our selection.
The database created by machine-learning-assisted exploration of the materials space [37,38] already contains the magnetic moments of the crystal structures included in it.However, those magnetic moments were obtained starting with the ferromagnetic (FM) initial orientation of the moments.Hence, we first filter through the database for the materials with sufficiently high saturation magnetization (  ≳ 1 T).Next, however, we need to make sure that the FM state is the ground state of the system.Calculations are performed here for the FM initial configurations of the spins (assumed when the database was produced) and several possible anti-ferromagnetic (AFM) orientations of the magnetic moments to see which state is more stable.We have also excluded all the materials with RE-elements, as well as the elements that are expensive, difficult to work with experimentally, and radio-active (Au, Tc, Ru, Hg, Pt, As, Pd, Ir, Rh, Tl, Os, Pu, Th, Ag).Out of about a million compounds in the database, excluding all the unwanted elements, we were left with 71 stable and metastable materials with   ≳ 1 T. Checking for the magnetic state, we found some to be AFM.Here, we also excluded systems where the FM state was energetically very close to an AFM configuration (  −  < 10 meV).All the materials can be found in the Appendix A, (Table 2).For the systems that were found to be FM, we further calculated the magnetic anisotropy energy (MAE) and Curie temperature (see Methods section for further details).Only those with MAE > 0.8 MJ/m 3 and   > 600 K were investigated further.For the high-throughput step, MAE is calculated as the difference in total energy between the states with the magnetization directed along and perpendicular to the -axis, as the same criteria are applied for a large number of systems.However, for the systems of interest, additional calculations are then performed for spin axes pointing in several directions.The procedure is schematically shown in Fig. 2.

Results
Materials with high enough saturation magnetization, MAE, and Curie temperatures can be further explored with respect to similar phases, stability, and existing compounds.Additional magnetocrystalline anisotropy analysis was also performed for the promising compounds to see the change in the total energy with different orientations of the spin axes.The details can be found in Appendix D.

Ta 3 ZnFe 8
Ta 3 ZnFe 8 found in the database has   = 0.94 T, MAE = 3.02 MJ/m 3 , and   = 630 K with a zero distance to the convex hull and formation energy of −163 meV/at.The unit cell with space group number 160 is shown in Fig. 3. Neither Ta 3 ZnFe 8 nor any other ternaries combining Ta, Zn, and Fe are reported either experimentally or theoretically (as per ICSD [22] and The Materials Project [41]).According to the phase diagram, the solubility of Ta in Fe is 17% at 600 • C, however, it increases with temperature up to 28% for 900 • C [42], which is about the ratio of Ta and Fe in Ta 3 ZnFe 8 (not considering Zn).The necessary solubility of Zn in Fe can be reached around 1100 • C, according to [43].Only two Ta-Zn phases, namely Ta 6 Zn 7 and TaZn 2 , have been reported [44].

Ga 2 Fe 6 B
Ga 2 Fe 6 B was found to be a FM compound with   = 1.49T, MAE = 0.80 MJ/m 3 , and   = 1170 K.The material has hexagonal crystal structure with space group number 189.The distance above the convex hull is 29 meV/at with the formation energy of −173 meV/at against the decomposition into GaFe 3 , Fe 2 B, and Ga 3 Fe.Again, no ternaries with Ga, Fe, and B are in either ICSD or the Materials Project.The As we can see in Appendix D, the more detailed investigation of MAE shows that in Ga 2 Fe 6 B the plane formed by  and -axes is an easy magnetization plane.

AlFe 2
Another structure remaining after the high-throughput filtering is AlFe 2 with   = 1.30T, MAE = 1.61 MJ/m 3 , and   = 660 K (Fig. 3).It should crystallize in space group 187 with the energy of 11 meV/at above the Convex Hull and has a formation energy of −230 meV/at.According to the database, another phase of AlFe 2 will form under the ambient conditions, namely space group 191.There are no compounds with AlFe 2 composition within the previously synthesized materials of ICSD.There are two such systems in the Materials Project, not with the same structure, however.On top of that, there are many synthesized binaries of Al and Fe.It is worth mentioning here, that all the data from the Materials Project as well as all data from AFLOW [46] were used to construct the dataset for machine learning [36].Hence, all the above-mentioned binaries are included in the convex hull.

Co 3 Ni 2
We would also like to highlight Co 3 Ni 2 .It has a slightly lower magnetic anisotropy of 0.89 MJ/m 3 than that required for a highperformance PM.However, the magnetization of with   = 1.31T, and high Curie temperature   = 940 K can make it a promising gap magnet.Its space group number is 166.Energy above the convex hull is 25 meV/at (decomposition into Co 3 Ni and CoNi 3 ) with the formation energy of 2 meV/at.We could find neither synthesized nor theoretically predicted material with the chemical formula Co 3 Ni 2 (Fig. 3), however, there exist several binaries of Co and Ni.One of the complications here might be the fact that Co-Ni exists for the whole temperature range as solid solution [47].

Fe 3 Ge
The last material we report on is Fe 3 Ge.Its magnetic characteristics are   = 1.56 T, MAE = 1.24MJ/m 3 , and   = 840 K, the unit cell is shown in Fig. 3.It has tetragonal crystal structure (139), a distance to the Convex Hull of 24 meV/at and a formation energy of −76 meV/at.Under normal conditions, another crystal structure of Fe 3 Ge will be formed, the stable cubic structure (space group 225).However, several crystal structures of Fe 3 Ge were observed experimentally, though none in the same symmetry group as the one given here.There are many investigations into magnetic properties of various forms of Fe 3 Ge, e.g.[48][49][50].As the distance from the convex hull for the desired material is small, one of the possible solutions for obtaining the correct crystal structure might be growing it on a suitable substrate.

Discussion and conclusions
With skyrocketing interest in machine-learning methods, there is a constant development in crystal structure prediction, resulting in a huge number of new materials.The main interest lies, however, in finding the compounds that can be further produced experimentally.Here we have filtered through about a million entries of the database containing new (not previously synthesized) materials, looking for stable and metastable compounds (as determined by the distance to the Convex Hull) that can be promising candidates for RE-free PMs [38].As mentioned previously, we investigated two of the more reliable (according to the authors [37]) datasets.
As the initial magnetic state in the database was assumed to be FM, we had to check for the stable magnetic ground state configuration, finding several systems to prefer the AFM state.To see the full list of materials we investigated (with   ≳ 1  reported in the database), see Appendix A, (Table 2).The list also contains materials discarded on other bases, such as low or in-plane magnetocrystalline anisotropy and non-collinear magnetic structures at elevated temperatures.
Five materials were found to be interesting candidate phases, their properties are shown in Fig. 4 and listed in Appendix A, ( Table 1).One of them, Ta 3 ZnFe 8 , is stable (with zero distance to the convex hull), while the other four are metastable (distance to the hull below 50 meV).
The ratio between magnetization and MAE is an important characteristic of a permanent magnet, as it determines the hardness  of a magnet and  ≥ 1 is needed for a hard magnet in order to resist the self-demagnetization [51].Two of the systems, Ta 3 ZnFe 8 (magnetic hardness parameter  = 2.07) and AlFe 2 ( = 1.09), have significantly higher coercivity.
As we demonstrated in Appendix D, it is important, especially in the case of crystal structures generated theoretically, to keep in mind that calculating magnetocrystalline anisotropy energy as a difference in the total energy for the spins aligned along the c-axis and perpendicular to it might require an additional investigation.In the case of Ga 2 Fe 6 B, the plane formed by the b-and c-axes was found to be an easy plane.
We also looked into the origin of high magnetocrystalline anisotropy in the five materials discovered, for details see Appendix B. As expected, the high MAE in Ta 3 ZnFe 8 originates in the 5d-element.Iron sources high values of anisotropy in the other three Fe-base materials.The key element in Co 3 Ni 2 is cobalt.
All five systems, with their high Curie temperatures, if synthesized, can be promising permanent magnets.Ta 3 ZnFe 8 , with its magnetization slightly lower than 1  but extremely high MAE, can be used as a gap material.As several crystal structures of Fe 3 Ge have been discovered experimentally, however, it might be difficult to obtain the necessary phase experimentally.
As the materials of the database were predicted theoretically, we calculated the phonon dispersions for the five candidate materials, to test their dynamic stability.The resulting phonon band dispersions and vibrational densities of states are shown in Fig. 5 for Ta 3 ZnFe 8 and in Appendix C Fig. 8 for the other four compounds.We can see the absence of imaginary frequencies in the phonon dispersion for all five materials, hence they are dynamically stable.
In conclusion, in the current investigation, we bridged the modernday structure generation techniques and our high-throughput approach to the search for high-performance permanent magnets.The possibility of synthesizing these materials was one of the search criteria, as we considered both their distance to the Convex Hull and their dynamic stability.We suggest that the four candidate phases, Ta 3 ZnFe 8 , AlFe 2 , Co 3 Ni 2 , and Fe 3 Ge, can be interesting materials to consider for an experimental investigation.

Methods
When the stable and meta-stable materials with high magnetic moment were filtered from the database, the energies of the states with FM and several AFM initial magnetic configurations were calculated using Vienna Ab Initio Simulation Package (VASP) [52,53] within the Projector Augmented Wave (PAW) method [54].Generalized Gradient Approximation (GGA) in Perdew, Burke, and Ernzerhof (PBE) form [55] was employed.Only systems with FM ground state were kept.
Full-potential linear muffin-tin orbital method (FP-LMTO), including spin-orbit interaction as implemented in the RSPt code [56,57], with the PBE functional [55] for exchange and correlation was used further.Tetrahedron method with Blöchl correction for the Brillouin zone integration [58,59] was employed.The initial magnetic state for each compound was set to be ferromagnetic (FM).In the highthroughput step, magnetic anisotropy energy (MAE) was calculated as the energy difference  =   −   between the states with the magnetization directed along (  ) and perpendicular (  ) to the -axis.A positive sign corresponds to the uniaxial magnetic anisotropy.The converged k-point Monkhorst-Pack meshes [60] 20 × 20 × 20 were used for the calculations.
The Curie temperature and magnetic state at higher temperatures were determined using Metropolis Monte Carlo (MC) and Atomistic Spin Dynamics (ASD) simulations, as implemented within the Uppsala Atomistic Spin Dynamics (UppASD) software [61].The simulations were performed with a 25 × 25 × 25 supercell with periodic boundary conditions.Exchange parameters were calculated with the RSPt code within the first seven coordination shells [62][63][64].
The magnetic hardness parameter was calculated as  = √ ∕ 0  2  [65], where  is MAE,   is saturation magnetization and  0 is the vacuum permeability.
PHONOPY code [66] and a finite displacement method were used to produce phonon dispersion curves.3 × 3 × 3 supercell and the displacement of 0.01 Å was employed to calculate forces with VASP.Crystal structures were relaxed with respect to the ions positions but not the cell shape or volume.
In addition, the Sumo package [67] and VESTA code [68] were utilized for visualization.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A. Stable and metastable materials with the high magnetic moment found in the database
Five promising candidates were found as the result of the highthroughput search, their detailed information is given in Table 1.Distances to the convex hull and the formation energies are extracted from the database, magnetic moment, MAE, and   were calculated.Some of the materials with magnetic moment close to or higher than 1.0  (as found in the database) were then discarded due to either the non-FM ground state, low or planar MAE, or low Curie temperature.These materials are listed in Table 2 along with the calculated values of   , MAE,   , and the energy difference between the FM and AFM states.Note, that when a value was found to be below the required, other characteristics were not calculated.

Table 1
Promising stable and metastable systems with their point symmetry group, saturation magnetization, MAE, Curie temperature, distance to the convex hull (a positive value means the material is above Hull), formation energy (negative sign points to table structures), and magnetic hardness coefficient.

Appendix B. The origin of magnetic anisotropy in the promising compounds
Similar to our of previous works [20,21] and the analyses presented in Refs.[69][70][71][72][73][74], to determine the origin of MAE we can calculate the spin-orbit coupling energy (SOC) for each of the atoms with their spins oriented along the z and x directions.The differences in the energies,  so =   so −  so , are given in Table 3 (the negative sign marks contribution to uniaxial magnetic anisotropy).For each of the atoms in the compound, we show the largest contribution only, leaving an empty space if that contribution is negligible.
For each of the materials, we can analyze the dominant contributions to MAE.In the case of Ta 3 ZnFe 8 (see Table 3),   of tantalum significantly exceeds that of Fe, as expected from a 5d-element.If analyzing the d-orbital-resolved difference in SOC energy, one can see (for the details see [20,21])   2 − 2 ← ← →   (same spin channel) to be the main source of easy-axis magnetocrystalline anisotropy.The densities of states (DOS) of all five systems are shown in Figs. 6  and 7.In all the materials but Ta 3 ZnFe 8 , the majority spin channels of

Table 2
Systems from the database with a distance to the convex hull < 50 meV and magnetic moment higher than 1.0 T, with their ground magnetic state, the energy difference between FM and the lowest in energy AFM state (positive sign means the FM state is more stable), saturation magnetization, magnetocrystalline anisotropy, and Curie temperature.NC stands for non-collinear.

Material
Ground

Fig. 1 .
Fig. 1.A schematic image of a convex hull for Fe-B-Al systems.Purple circles denote stable materials (their formation enthalpies were taken from the Materials Project[41]) and form the convex hull.The surface 50 meV above it limits the metastable materials (which can still be synthesized).Squares symbolize the result of structure generationblue (below hull) and orange (less than 50 meV above it) will pass through our filters, the red will not be considered further.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 2 .
Fig. 2. The steps of the high-throughput search with the number of systems remaining at each step.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Alena Vishina :
Initiated the research, Performed the highthroughput search & initial data analysis, Drafted the manuscript, Review & edited, Contributed to discussions & analysed the data.Olle Eriksson: Initiated the research, Review & edited, Contributed to discussions & analysed the data.Heike C. Herper: Initiated the research, Review & edited, Contributed to discussions & analysed the data.

Fig. 6 .
Fig. 6.Spin-polarized DOS (without SOC interaction) of Ta 3 ZnFe 8 (top left panel), Ga 2 Fe 6 B (top right), AlFe 2 (bottom left), and Co 3 Ni 2 (bottom right).For some of the materials, it is separated into the contributions from the   2 − 2 and   2 (  set);   ,   , and   ( 2 set) orbitals.Fermi energy is set at zero.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 7 .
Fig. 7. Spin-polarized DOS (without SOC interaction) of Fe 3 Ge.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 .Fig. 9 .
Fig. 8. Phonon dispersions and vibrational densities of states for Ga 2 Fe 6 B (top left panel), AlFe 2 (top right), Co 3 Ni 2 (bottom left), and Fe 3 Ge (bottom right).(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) . It can be used to evaluate if the material https://doi.org/10.1016/j.actamat.2023.119348Received 10 July 2023; Received in revised form 30 August 2023; Accepted 13 September 2023

Table 3
The difference in SOC energy (VASP) with spin orientation along z and x axis (negative sign corresponds to uniaxial magnetic anisotropy) for the elements of five promising candidates.3d-states are essentially fully occupied.In Ta 3 ZnFe 8 , Ta and Fe are considerably hybridized, there are empty states in both spin channels. the