Stefanos
Giaremis
a,
Georgios
Katsikas
a,
Georgios
Sempros
a,
Margarit
Gjoka
b,
Charalambos
Sarafidis
a and
Joseph
Kioseoglou
*a
aSchool of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece. E-mail: sifisl@auth.gr
bInstitute of Nanoscience and Nanotechnology, NCSR “Demokritos”, Athens, Greece
First published on 26th March 2022
The use of the mischmetal alloy, comprised of La and Ce in 1:
3 ratio, as a partial substitute for Sm in the CaCu5-type structure is explored, as a means for the search of viable alternatives for permanent magnets that require fewer steps in the rare earth separation processing. The structural and magnetic properties of the introduced stoichiometry, containing 50% less Sm, are compared to the ones of the SmCo5, LaCo5 and CeCo5 binary compounds by means of ab initio simulations. The capability of artificial neural networks to accurately predict the relationship between structure and total magnetization from DFT calculations in the supercell approach that was employed, is also demonstrated. Experimental fabrication and structural and magnetic characterization of the proposed stoichiometry verifies the structural configuration and provides insight for the macroscopic hard magnetic properties of the material. The reduction of magnetic properties was found to be favorable compared to the respective reduction of the raw materials cost, while measurements of the Cure temperature verify that the proposed compound is still suitable for high temperature applications.
Regarding the industrial trends though, there is a growing demand for PMs that are capable to operate at high temperatures for applications such as electric vehicles and turbines. The performance of Nd2Fe14B PMs in such applications is maintained by substitution of Nd by Dy or Tb.6 However, the price of Dy and Tb is several times higher than that of Nd, with Dy and Tb being scarcer than Nd and all the three elements reaching a price peak at the time this paper is written, due to recent geopolitical circumstances.8 SmCo5 magnets, although currently representing a substantially smaller portion of the total market share and having a smaller energy product, are also suitable for high temperature applications due to their thermal stability.6 The remaining one third of the PM market, besides Nd2Fe14B, SmCo5 and the old AlNiCo PMs, is represented by ferrites, which are based on abundant and cheap materials but have a low energy product (<100 kJ m−3). Thus, there is room for development of PMs that are less dependent on commercially critical and expensive elements, with an energy product (BH)max, that lies on the gap between and 100 and 200 kJ m−3, which is the gap between ferrites and Nd2Fe14B based PMs.6,9
Among the family of rare-earth (RE) – transition metal (TM) intermetallic phases, RE-TM5 is of particular interest for the development of new permanent magnets due to their magnetic properties arising from the strongly correlated electrons of the partially filled 3d states in the TM atom network and the anisotropy induced from the 4f electrons of the RE atoms.10 Bonding is dominated by the interaction of the itinerant TM 3d orbitals with the localized 4f ones of the RE atoms.10 They crystallize preferably in the hexagonal, (P6/mmm, No. 191), CaCu5 structure,11 made of alternating hexagonal layers of the TM atoms (2c positions) and ones with the RE atoms in the center of the plane and TM surrounding them (3g positions) arranged in a Kagome network.12 This arrangement of the TM atoms is responsible for the large values of MAE reported for this family of materials, as its associated crystal electric field arising from the itinerant 3d electrons pins the aspherical 4f orbitals at the RE atoms sites.13 The 3d sublattice of the TM atoms is dominant from a structural point of view for the magnetic properties and especially magnetization. This fact implies that, theoretically, the RE atoms are not indispensable for the anisotropy,6 while the use of Co as TM is preferred due to its excellent magnetic properties, arising from its 3d states.14–17
As mentioned above, SmCo5 is currently considered as the most prominent compound of the RE-TM5 family for PMs due to its excellent hard magnetic properties. Regarding the use of other RE atoms in recent studies, CeCo5 is an interesting compound, as it presents high values of uniaxial anisotropy and Ce is the most abundant RE element.18–20 Ce also has one 4f electron which is found to have an itinerant character, leading to a fractional occupation of the 4f states.12 However, these itinerant electrons are found to interact with the 3d electrons of Co, thus reducing their spectral weight at the Fermi level and leading to a drop in the overall magnetization.21 On the other hand, LaCo5 is an interesting compound from a theoretical point of view, as a reference material for the isostructural RE-TM5 compounds, due to its lack of 4f electrons, which, due to their localized nature, are problematic to be treated with traditional one-electron approaches.12 Experimental advances in manufacturing of anisotropic LaCo5 powders,22 along with further substitution of La with Y and Co with Fe and C in La0.5Y0.5Co4.7−yFeyC0.3 ribbons for y = 0.3–0.4, are reported to present improved hard magnetic properties compared to LaCo523 but are still inferior compared to the ones of SmCo5. Regarding other RE elements, Gd as a substituent is found to lead to a high coercivity and remenance, however it is substantially more expensive than Sm.24 NdCo5 presents large magnetization, but also in-plane anisotropy, which is detrimental for the magnetic properties.25 YCo5 also lacks 4f electrons, as LaCo5, but presents interesting properties under pressure due to electron–electron interactions.26 Moreover, advances in mechanochemical synthesis have enabled the fabrication of YCo5 and PrCo5 single crystals with higher coercivity than SmCo5 (although their (BH)max is still smaller than SmCo5).27,28 Guo et al. have recently successfully fabricated the high entropy Sm1/4Nd1/4Y1/4Tb1/4Co5 alloy and investigated the impact of stoichiometry, however they also reported inferior magnetic properties compared to SmCo5.29 The replacement of RE atoms with heavy 4d and 5d TM ones, namely in ZrCo5 and HfCo5, was demonstrated to be thermodynamically viable but without any promising magnetic properties.30
In general, the substitution of RE elements in permanent magnets is the subject of numerous academic, public, and private sector initiatives due economic and political issues.31 However, as the role of RE atoms in atomistic level hard magnetic properties appears to be crucial, the optimization of the performance-cost interplay seems to be a more viable approach. It has also been demonstrated via ab initio high throughput screening methods that there is a trend for better magnetic performance in light RE based magnetic compounds,32 which is fortunate as, besides supply, the processing for the separation of the RE elements and especially of the heavy ones present a heavily monopolistic profile.33 Even for light RE elements though, the process of their separation from raw ores is also complex and related with geopolitical issues and monopolistic practices.8,33 Taking all these into account, the use of less critically endangered RE-based materials could be a viable alternative. In this direction, mischmetal (MM), a light RE alloy initially extracted from the monazite ore, presents an interesting candidate. MM is the least expensive RE based metal as it is mainly produced via fused salt electrolysis of RE chlorides, which is a relatively simple and inexpensive process.34 Typically, Ce-rich MM contains La and Ce in a ratio of 1:
3, which are among the most abundant RE elements, along with smaller compositions of other RE elements such as Pr or Nd and often traces of Mg or Fe, depending on the production processing. However, due to the recent high demand for Pr and Nd, along with advances in commercial processing, a common stoichiometry for modern, commercially available MM is LaCe3.35,36 Therefore, in order to find new compounds of the CaCu5 structure with reduced content of critical and expensive elements that could potentially “plug the gap” regarding magnetic performance,6 one promising route is to explore using cheaper, more abundant and easier to be produced RE substitutes, such as MM with the LaCe3 stoichiometry.
The use of MM as a Nd substituent in the Nd2Fe14B has already been explored very recently by numerous experimental studies and have yielded very promising results, due to advances in sintering techniques and grain refinement.37–40 In the RE-TM5 family, early attempts for the substitution of Sm with MM (50 wt% Ce, 27 wt% La, 16 wt% Nd, 5 wt% Pr, 2 wt% other REs) yielded PMs with high energy product but low coercivity.41 However, in a recent work, Zuo et al. report the fabrication of CeCo5 and La0.35Ce0.65Co5 granular ribbons with improved MR/MS ratio due to strong exchange interactions between the nanograins and coercivity comparable to the Dy-substituted Nd2Fe14B PMs.35 Towards this direction, in this work, we explore the partial substitution of Sm with MM with the LaCe3 stoichiometry as a means of the improvement of performance – cost interplay in the family of RE-Co5 permanent magnets, by a combinatorial approach comprising ab initio calculations, machine learning techniques and experimental synthesis and characterization. The hard magnetic properties and groundstate structural stability of the new candidate compound, MM0.5Sm0.5Co5, with respect to the SmCo5, LaCo5 and CeCo5 stoichiometries were assessed both theoretically and experimentally. The impact of the various ways of distribution of the MM atoms in the Sm sites to the total magnetization was explored with a combination of ab initio and machine learning techniques. We demonstrate that an artificial neural network (ANN) algorithm was found, in principle, to be able to make successful predictions, based on the underlying structure–property relationship that dictates magnetization in the MM0.5Sm0.5Co5 crystal system. Moreover, experimental samples of the proposed stoichiometry were fabricated using the arc-melting technique under Ar atmosphere and treated at high temperatures for homogenization. Structural and magnetic characterization took place to confirm the hexagonal structure and evaluate the main magnetic properties of the system.
The enthalpies of formation, ΔH, for the configurations with the MM0.5Sm0.5Co5 stoichiometry were calculated in two ways. The first one corresponds to the enthalpy of formation per formula unit with respect to the metallic phases of Sm, Co, Ce and La:
![]() | (1) |
![]() | (2) |
The implementation of the ANN algorithm on the MM0.5Sm0.5Co5 dataset was performed via the TensorFlow package,50 with further use of tools from the scikit-learn python package.51 The Python Materials Genomics (pymatgen) package52 was also used for the extraction of quantities from the relaxed structures of the dataset. For the ANN regression model, the test set comprised 30% of the total dataset, with the rest being split between training and validation. The interatomic distances between all the different atom types, as calculated with the pymatgen package, and their statistical quantities (mean, mode, standard deviation) where used as input variables (features). The feature vector that was used comprised of the total volume, lattice constants and statistical quantities (mean, mode, standard deviation). The output variable was the total magnetization. Cross-validation was performed via 5 k-folds, stratified based on the magnetization classes that will be described in the “Results and discussion” section. Input and output values were transformed with the standard scaler, ensuring a mean of 0 and a standard deviation of 1. A simple sequential model with 16, 3 and 1 neurons in each layer with “ReLU” activation on all except the last one was used, with the last layer utilizing a linear activation function and a dropout layer of 0.25 rate inserted between the layers with 16 and 3 neurons. The dropout layer was found to prevent overfitting while encouraging every neuron to train and contribute to the final prediction. The network was optimized using the mean squared error (MSE) metric, but the root mean squared error (RMSE) metric is displayed in all graphs as it is more physically meaningful. A grid search was executed in order to perform hyperparameter tuning, where the number of epochs ranged from 25 to 100 with a step of 25, and batch sizes ranged from 2 to 64 incrementing as powers of two. The final model was trained for 50 epochs using a batch size of 4, after achieving best results in the corresponding metrics. For the ANN classification model, the test set comprised 40% of the total dataset and the rest was oversampled using the synthetic minority oversampling technique algorithm (SMOTE) with three neighbors for the balanced representation of the minority classes. For evaluating the performance of the model, 3-fold validation, stratified based on the magnetization classes, was performed. A simple sequential model was also used here, with 16, 8 and 3 neurons in each layer and “ReLU” activation in all except the last one, in which the “softmax” activation function was used. The model was also trained for 50 epochs using a batch size of 4.
The simplification of the SRM is sensible for SmCo5, considering that there is evidence that bonding and energetic properties in RE-TM systems are governed by the interaction between the 3d electrons of the TM and the 5d electrons of the RE.61 Moreover, X-ray photoelectron spectroscopy studies have shown that the 4f orbitals of Sm and the 3d orbitals of Co do not overlap and their respective peaks fall almost 6 eV apart from each other.62 Both photoelectron spectroscopy measurements62 and DMFT – H1A calculations59 demonstrated that 4f orbitals lie almost 6 eV below the Fermi level. The DMFT-H1A approach was found to be in better agreement to the experimental photoemission spectrum at deeper energy levels, while the SRM curve falls closer to the experimental photoemission spectrum near the Fermi level.59
In Table S1,† the results for the total average magnetization values per atom type and the total magnetization per formula unit are presented for the two tested pseudopotential configurations. The respective orbital contributions are presented in Table S2.† The values of Ueff = U − J for Sm and Co were set to 4.7 and 2.22 eV, respectively, as explained in the “Computational method” section. The total magnetization of 8.56 μB per f.u. as predicted by the 11-valence e− pseudopotential configuration (SRM) is in good agreement with the experimental value 8.97 μB per f.u.63 and slightly higher than the experimental value of 7.8 μB per f.u.64 In contrast, the 16-valence e− model predicts a total magnetization value of 3.38 μB per f.u., which is less than the half compared to the experimental values. This decrease of the total magnetization is caused by the elevated magnitude of the antiparallel magnetic moment of Sm. The SRM also predicts an antiferromagnetic interaction between Sm and Co atoms, but with a smaller magnitude of the magnetic moments of Sm, which yields a total magnetization closer to the experimental values. Both models yield similar results with previous computational studies using a similar approach.59,65
Regarding the antiferromagnetic interaction between RE and TM atoms, according to the model proposed by Campbell, the 4f electrons of the RE lead to a positive 5d moment, thus the 4f–3d coupling with the TM is indirect and occurs via the direct interaction between the 5d orbitals of the RE and the 3d orbitals of the TM.61 This effective treatment of RE as elements at the beginning of the TM series predicts that interactions between d and f orbitals at the RE sites will be ferromagnetic, while interactions between RE and TM atoms at the opposite end of the TM series (such as Co) will be antiferromagnetic.61 Besides the d–d interaction model of Campbell,61 the antiferromagnetic interaction between RE and TM was also proposed by Wallace via the polarization of the s conduction orbitals.66 According to this study, the s–3d interaction is considered ferromagnetic and the s–4f interaction at the RE-TM distance is considered antiferromagnetic, so the 4f–3d coupling is antiferromagnetic.66 The resulting antiferromagnetic behavior, shown in Tables S1 and S2,† is in accordance with the two aforementioned theoretical descriptions and also with experimental results.67 The parallel alignment of spins between Sm and Co was found to be a metastable state which lies 4 to 7 mRy per atom above the groundstate, by using full-potential linearized augmented plane wave (FPLAPW) and FPLAPW LDA + U simulations.65
Regarding the orbital magnetic moment of the f electrons, for the light RE, it is expected to align parallel to the magnetic moments of the TM elements and almost cancel out with the spin moment (as shown in Fig. 4, Brooks et al.68). The results of the 16-valence e− model agree with this description. In the SRM, the almost negligible orbital moment predicted for Sm is due to the 5d valence electrons. Moreover, the orbital moments of Co atoms in the 2c sites are found to be greater than in the 3g sites for the SRM calculations, reproducing the experimental findings of Streever, according to which, the magnetic anisotropy of SmCo5 is mostly due to the orbital moment of Co atoms in 2c positions.69
To further investigate how the introduction of the Hubbard parameter, Ueff, affects the description provided by the 16-valence e− model, in Tables S3 and S4,† we present the results of simulations without the Hubbard model (USm = UCo = 0 eV) and with a slightly enhanced value of Ueff but only in the 4f orbitals of Sm (USm = 6 eV, UCo = 0 eV). It is evident that without using the Hubbard model, the resulting value of total magnetization (7.14 μB per f.u.) is quantitively closer to the experimental values of 7.864 and 8.97 μB per f.u.63 but the orbital magnetic moment of Sm lies antiparallel to the ones of Co, in contrast to.68 By using a value of U only for the 4f orbitals of Sm, the total magnetic moment of Sm lies parallel to the ones of Co, leading to a value of total magnetization per formula unit that exceeds the experimental values. Moreover, the orbital magnetic moment of Sm also lies antiparallel to the ones of Co, as in the previous case.
The resulting electronic Density of States (DOS) corresponding to the 16-valence e− configuration with the three aforementioned set of U parameters, and the 11-valence e− configuration (SRM) with USm = 4.7 eV and UCo = 2.22 eV, are presented in Fig. S1.† The spurious pinning of states corresponding to f orbitals of Sm to the Fermi level is evident for all the cases using the 16-valence e− configuration. These orbitals interact with the respective 3d states of Co at the Fermi level. The location of the rest of the peaks associated with the 4f orbitals of Sm are in good agreement with,70 in which the LDA + U method was employed with U = 6.8 eV and J = 0.8 eV. The value of U parameter for Sm seems to affect the spread of the f orbitals around the Fermi level, but the pinned f orbitals are observed in all cases. The introduction of U parameter is also necessary for Co, as in the case where UCo = 0 eV, the 3d orbitals of Co are also observed to be pinned to the Fermi level and strongly interacting with the also pinned 4f orbitals of Sm. The SRM with USm = 4.7 eV and UCo = 2.22 eV successfully predicts an almost similar total DOS with the 16-valence e-model, with the only exception being the sharp peaks corresponding to the strongly localized 4f states of Sm close to −4 eV.
The antiferromagnetic behavior of the RE-Co5 compounds, resulting from the corresponding atomic magnetic moments and total magnetization per formula unit for each compound, are demonstrated in Table 2. Their respective orbital moments per atom and per formula unit are shown in Table 3. CeCo5 appears to have a lower value of total magnetization by almost 1 μB per f.u. compared to SmCo5, due to the very pronounced antiferromagnetic contribution of Ce in its structure. This effect is attributed to the itinerant character of the one 4f electron of cerium, which leads to a fractional occupation of the 4f states. These itinerant 4f electrons of Ce create additional bonds with the 3d orbitals of Co, leading to a mixed valency of Ce (between Ce3+ and Ce4+) and reducing magnetization.74–76 By examining the results of the magnetization charge density distributions in Fig. 2, it is evident that the antiferromagnetic contribution of Ce is more spatially pronounced than the respective one for La and Sm. The inclusion of SOC in these cases did not show any different results, both quantitatively and qualitatively for SmCo5 and LaCo5. However, DFT + U with SOC predicted more pronounced values for the magnetic moments of Ce and small differences in the shape of its magnetization density distribution in CeCo5. The total magnetic moments of Co atoms at 2c sites for the binaries which include 4f orbitals (SmCo5, LaCo5) are found to be smaller than the ones in 3g sites.
m/atom (μB per atom) | DFT + U | DFT + U (SOC) | Ref. |
---|---|---|---|
SmCo 5 (SRM) | |||
1a (1 × RE) | −0.316 | −0.301 | −0.2165 |
2c average (2 × Co) | 1.748 | 1.736 | 1.86,77 1.7665 |
3g average (3 × Co) | 1.792 | 1.783 | 1.75,77 1.765 |
Total (μB per f.u.) | 8.557 | 8.520 | 8.97,63 8.0265 |
LaCo 5 | |||
1a (1 × RE) | −0.279 | −0.295 | −0.3078 |
2c average (2 × Co) | 1.736 | 1.712 | 1.673 |
3g average (3 × Co) | 1.764 | 1.761 | 1.7673 |
Total (μB per f.u.) | 8.391 | 8.412 | 8.4679 |
CeCo 5 | |||
1a (1 × RE) | −1.527 | −0.952 | −0.45721 |
2c average (2 × Co) | 1.691 | 1.666 | 1.260,80 1.42921 |
3g average (3 × Co) | 1.769 | 1.728 | 1.260,80 1.37021 |
Total (μB per f.u.) | 7.163 | 7.563 | 7.179 |
SmCo5 | LaCo5 | CeCo5 | |||||
---|---|---|---|---|---|---|---|
m L/atom (μB per atom) | DFT + U (SOC) (16-valence e−) | DFT + U (SOC) (SRM) | Ref. | DFT + U (SOC) | Ref. | DFT + U (SOC) | Ref. |
1a (1 × RE) | 2.115 | 0.087 | 3.2665 | 0.084 | 0.02421 | 0.693 | 0.23221 |
2c average (2 × Co) | 0.315 | 0.437 | 0.2265 | 0.410 | 0.28681 | 0.440 | 0.15021 |
3g average (3 × Co) | 0.424 | 0.352 | 0.1865 | 0.400 | 0.25281 | 0.335 | 0.10921 |
From the projected density of states (DOS) plot in Fig. 3, we observe that in the case of CeCo5, the electronic states attributed to Ce are either pinned (no SOC) or very close (SOC) to the Fermi level. These results, along with Table 2 demonstrate that the impact of Ce to the RE-Co5 compound is detrimental for the total magnetization, with this effect being depicted as less pronounced with the inclusion of SOC to the DFT + U calculations. The 4f orbitals of La are empty.
The most pronounced impact of the size effect though is observed in the values of the total magnetization per formula unit. In the 2 × 2 × 2 dataset, there is a peak close to 7.9 μB per f.u., which corresponds to an almost 7% reduction of the total magnetization. There are also two outlier cases at approximately 8.30 and 8.45 μB per f.u., which are close to the values of total magnetization per formula unit for LaCo5. On the other hand, the magnetization per formula unit of the vast majority of the 2 × 2 × 4 supercell configurations was found close to 7.46 μB per f.u., which corresponds to a 12.8% reduction compared to the respective value for SmCo5. There are also two other groups of cases with respect to their value of total magnetization with distinct values of total magnetization but fewer members. The first one is centered close to 7.37 μB per f.u. and the second one close to 7.6 μB per f.u., which correspond to a 13.8% and 11.1% reduction, respectively, compared to the calculated total magnetization for SmCo5.
The relationship between the relative energy per atom, defined as the energy difference per atom of each configuration with respect to the energetically preferable configuration of each dataset, with the total magnetization per formula unit for the two datasets, is shown in Fig. 5. The Pearson's correlation coefficient, ρ, is a measurement of correlation between two variables and ranges between −1 and 1. Values close to 1 (−1) indicate a positive (negative) correlation between the two variables, while values close to 0 indicate a weak dependence of one variable to the other. In both datasets, these two variables are found to be very weakly correlated, meaning that there is no obvious trend between the total energy of each configuration with its total magnetization. In the 2 × 2 × 2 dataset, most values appear in the lowest left corner of the plot, with the most preferable cases close to 7.95 μB per f.u. However, in the 2 × 2 × 4 dataset, we observe a slightly more complex situation, with three clusters of values close to 7.37, 7.46 and 7.6 μB per f.u., respectively, as also presented in the corresponding histogram for the values of total magnetization in Fig. 4.
The full scatter matrix plots for a, c/a ratio, total magnetization per formula unit and relative energy per atom for the two datasets, along with their respective correlation matrices can be found in Fig. S2, S3 and Tables S5, S6.† The lattice constant, a, and c/a ratio are also found to be very weakly correlated with the total magnetization per formula unit in both datasets, with their relationship between them and the relative energy per atom and the total magnetization for both datasets being shown in Fig. S3.†
Regarding the enthalpies of formation, as defined by eqn (1) and (2), all configurations in both datasets were found stable compared to the metallic phases of Sm, Co, La and Ce. However, two configurations of the 2 × 2 × 2 dataset and twelve configurations of the 2 × 2 × 4 dataset were found unstable with respect to the SmCo5, LaCo5 and CeCo5 binary phases. The total magnetization per formula unit, lattice constant, a and c/a ratio for these configurations can be found in Tables S7 and S8.† It is also worth mentioning that all of the twelve configurations of the 2 × 2 × 4 dataset also belong to the middle magnetization cluster that was mentioned above.
To provide a more critical insight about the 2 × 2 × 4 dataset, we examine the following four particular cases: the energetically preferable configuration, the configuration with a total magnetization value closest to the middle of the average magnetization cluster and the configurations with the minimum and maximum magnetization of the entire dataset. These configurations will be referred to as “energetically preferable”, “average magnetization”, “minimum magnetization” and “maximum magnetization”, respectively. The lattice constant, a, c/a ratio, enthalpy of formation with respect to the Sm, Co, La and Ce metallic phases and to the SmCo5, LaCo5 and CeCo5 binary phases and total magnetization per formula unit for the cases mentioned above are presented in Table 4. The lattice constant, a, for the energetically preferable case is found to be larger than the value for the other three cases, with its total magnetization belonging in the middle magnetization cluster. The c/a ratio of the energetically preferable case, though, is closer to the respective value of the case with the smallest magnetization. All of the four cases are found to be stable both against the metallic phase of Sm, Co, La and Ce and the SmCo5, LaCo5 and CeCo5 binary phases, as indicated by the negative values for the respective enthalpies of formation in Table 4.
Energetically preferable | Average magnetization | Minimum magnetization | Maximum magnetization | |
---|---|---|---|---|
a (Å) | 5.096 | 4.896 | 4.787 | 5.012 |
c/a | 0.802 | 0.841 | 0.850 | 0.788 |
ΔHmetallic/f.u. (eV) | −0.698 | −0.657 | −0.682 | −0.553 |
ΔHbinary/f.u. (eV) | −0.196 | −0.155 | −0.179 | −0.050 |
m (μB per f.u.) | 7.47 | 7.46 | 7.35 | 7.69 |
Regarding the magnetization of each atom in these four cases in Table 5, the strong impact of Ce is also observed in a similar manner as in our previous results for the binary RE-Co5 compounds. The large value of the total magnetization in the last case can be attributed to the low value of atomic magnetization for the Ce atoms. The magnetic moment of La atoms is also similar to the one of Sm, again as in the cases of the binary compounds. The different topological distributions of the magnetization densities for these four cases are presented in Fig. 6.
m/atom (μB per atom) | Energetically preferable | Average magnetization | Minimum magnetization | Maximum magnetization |
---|---|---|---|---|
1a (8 × Sm average) | −0.325 | −0.322 | −0.330 | −0.305 |
1a (2 × La average) | −0.317 | −0.310 | −0.323 | −0.301 |
1a (6 × Ce average) | −1.285 | −1.278 | −1.291 | −0.617 |
2c average (2 × Co) | 1.637 | 1.600 | 1.612 | 1.652 |
3g average (3 × Co) | 1.626 | 1.645 | 1.604 | 1.603 |
The projected DOS plot for the above four cases of the 2 × 2 × 4 dataset is shown in Fig. 7. We observe that in the case with the minimum magnetization, electronic states that are associated with Ce are more delocalized and are extended over the vicinity close to the Fermi level. The localized states of La close to 3 eV correspond to the empty 4f orbitals.
From the correlations heatmap of the interatomic distances with the total magnetization in Fig. S4,† it is evident that the Co–Co distance is correlated with total magnetization, with this value of correlation being more pronounced compared to the interatomic distances between the other atom types. This result indicates the importance of the Co sublattice in the Kagome network arrangement for the total magnetization of the RE-Co5 compound.
The result of the implementation of an ANN for the prediction of total magnetization of each compound as a regression task are presented in Fig. 8, with the values of the coefficient of determination, R2, for this model shown in Table 6. The choice of parameters of our final model are presented in the “Computational method” section. Despite the satisfactory performance of our model in the training set, R2 was found to drop to 0.501 in the test set.
Dataset | R 2 |
---|---|
Training | 0.822 |
Validation | 0.665 |
Testing | 0.501 |
The results of the ANN model were improved by transforming the problem to a classification task. The classification ANN uses a slightly larger hidden layer when compared to the regression ANN. This choice was made so as to alleviate the information bottleneck when transitioning from the 16 neurons to the 3 output neurons. This increase in the number of trainable parameters is further justified because the dataset has been artificially augmented for the purpose of balancing the underrepresented classes. A similar increase in the regression model led to increased overfitting of the model, which rendered it less accurate.
For the ANN classification model, we tried to predict the magnetization cluster of each data point, based on the three discrete magnetization clusters of Fig. 5, instead of directly predicting the value of total magnetization. The details of the parameters of our final model for the classification task are also presented in the “Computational details” section. We evaluated our classifier based on its accuracy, precision and recall, as defined by:
![]() | (3) |
![]() | (4) |
![]() | (5) |
The results of accuracy, precision and recall of the model are presented in Table 7. The confusion matrices for each of fold of the cross-validation can be found in Fig. S6.† The results presented in the confusion matrix make it abundantly clear that the network can correctly predict whether a given arrangement of molecules would result in a typical magnetization. Despite the fact that, because of the uneven distribution of classes, the prediction for these classes suffers, the network may still be used as a supplementary tool to gauge how likely the given configuration is to result in less typical value for the magnetization. With respect to that test, the number of false positive results is much smaller than that of the false negative results, resulting in a partial anomaly detector that may save researchers valuable time.
accuracy | precision | recall | |
---|---|---|---|
Training | 1.000 | 1.000 | 1.000 |
Validation | 0.982 | 0.982 | 0.982 |
Test | 0.735 | 0.927 | 0.927 |
Overall, we observe a significantly better performance compared to our regression model, with an accuracy value of 0.735 and a precision and recall value of 0.972. Therefore, we demonstrate that an ANN algorithm can become capable of predicting the trends in the values of magnetization based on the values of the interatomic distances.
In Fig. 10a, a typical magnetic hysteresis loop of Sm0.5MM0.5Co5 sample at room temperature is presented. Saturation magnetization was determined by extrapolating the data of the high field region and its value was 61.3 Am2 kg−1; which corresponds to 4.9 μB per f.u.. This value is lower than the theoretical calculations presented before and pure SmCo5 phase being closer to Ce and La isostructurals.64,82,83 Curie temperature (TC) of the sample under study, as seen in Fig. 10b, was determined at 828 K, lower than pure SmCo5 compound64 but larger than CeCo5.84Table 8 summarizes the aforementioned magnetic properties, MS and TC.
In Fig. 10a, reference values for remanence and coercivity are also presented. These values correspond to the measurement of the powdered alloy, which consists of grains in the range of 10–50 μm. A permanent magnet based on our material could be prepared by the traditional processing paths. Assuming a square loop, from the formula BHmax = (μ0Ms/2)2, the theoretical maximum energy product would be 85.14 kJ m−3, which is close to other proposed materials for permanent magnet applications and sufficient for “plugging the gap”, as discussed in.6
To examine the various phenomena associated with the supercell approach, we constructed and analyzed two datasets with DFT + U calculations in 2 × 2 × 2 and 2 × 2 × 4 supercell structures. The breakdown of symmetry by the random substitution was found to introduce a variance to the calculated quantities, as presented by the respective histograms. The distribution of the structural parameters in the two datasets was found to be similar. However, the total magnetization in the configurations of the 2 × 2 × 2 dataset is centered around 7.9 μB per f.u., which corresponds to a 7% reduction compared to SmCo5, but in the case of the 2 × 2 × 4 dataset, there are three distinct clusters observed in the values for the total magnetization, centered at 7.37, 7.46 and 7.6 μB per f.u., which is a 11.1, 12.8 and 13.8% reduction, respectively, compared to SmCo5. Most configurations were found to belong in the second cluster. Also, the energetically preferable case for each dataset is not the one with either the smallest or the largest magnetization of its corresponding dataset. The same was found to hold also for a and c/a. The energetically preferable case of the 2 × 2 × 4 dataset was found to lie in the majority magnetization cluster. Moreover, for the case with the largest value of magnetization, the antiferromagnetic impact of Ce to the total magnetization was less pronounced than in the other cases, while in the case with the smallest magnetization, the electronic states of Ce were found closer to the Fermi level in the projected DOS plot.
Finally, we investigated the possibility of training an ANN machine learning algorithm in the cases of the 2 × 2 × 4 dataset in order to predict the total magnetization of a given compound, by using the interatomic distance between each atom type as features. By looking the correlation of these features with the total magnetization, we observed a degree of correlation between the Co–Co interatomic distance and the total magnetization, result that indicates the pronounced impact of the Co network to the total magnetization of the structure. Regarding the ANN model, by using an ANN algorithm for regression, in order to predict the values of magnetization in a continue manner, our best model achieved an R2 score of 0.822 in the training set, which was decreased to 0.501, however, for the test set. The problem was subsequently treated as a classification task, in which we trained an ANN algorithm to predict the magnetization cluster of each case, based on the three magnetization clusters that were mentioned above. Our best model achieved a value of 0.735 for accuracy and 0.927 for recall and precision, improving by far the predicting capability of the model. We therefore conclude that these results can act as a proof of concept that an ANN algorithm can in fact be capable of making predictions about the trends of total magnetization, trained on features based on the structural properties of the compounds, as also previously demonstrated.85 This underlying structure-to-property relationship is proven to be complex, thus the implementation of predictive statistics algorithms such as machine learning and ANNs in particular, can possibly circumvent theoretical obstacles and yield practical results.
Experimentally prepared Sm0.5MM0.5Co5 samples confirm the theoretical approach as far as it concerns the stability and the structural parameters. The compound retains the typical hexagonal CaCu5-type structure of the SmCo5 system. Magnetic measurements showed a relatively high Curie temperature of 828K and saturation magnetization of about 61 Am2 kg−1 (4.9 μB per f.u.), the latter being lower than theoretical prediction. Possible microstructural effects or undetected minor phases with very low concentration in grain boundaries may account for the difference.
Overall, by exploring the possibility of using MM, an ore that has not yet been fully separated to its constituents, we aim to the production of permanent magnets that may require simpler processing by avoiding the separation of RE elements. Our ab initio simulations show that the replacement of half of Sm with MM leads to a 7 to 13.8% drop in total magnetization. Experimental characterization of single-phase samples with this stoichiometry also revealed a drop in overall magnetic strength of the material. However, as MM is almost 100 times less expensive than Sm at the time that this work is written, we achieve an almost 50% reduction in expenses, at least regarding raw materials. The extensive knowledge of the system may also draw a path to other possible modifications. These results will hopefully act as a baseline for a process which in step will possibly aid to the design of more affordable and commercially accessible devices.
Footnote |
† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d2nr00364c |
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