Machine learning-aided Genetic algorithm in investigating the structure–property relationship of SmFe₁₂-based structures Open Access

Despite the lively discussion in the community,^1–4 density functional theory is still the most popular approach for calculating the electronic structure of molecules and extended materials.^5,6 When the crystal structure is known, many properties can be computed reliably using a first-principle calculation, such as volumes within 1%–3% (LDA, GGA), bandgaps within 30% underestimated (LDA, GGA), and 10% (GW) and somehow unsatisfactory—for van der Waals crystals or systems with localized d- and f-electrons. However, when we do not know detail structure information of the compounds or its thermal decomposition mechanism, finding new stable and outperforming one seemly as infeasible was summarized by the one-word paper by Gavezzotti.⁷ Therefore, it is necessary to investigate the correlation between its physical properties, e.g., stability, as the function of geometrical information, including element site properties and lattice symmetry configuration. We introduce machine learning (ML) based method approaching to this problem and taking the SmFe$12$ compound as the object for investigation.

The SmFe$12$ structure with tetragonal $I4/mmm$ symmetry (space group 139) for the ThMn$12$ prototype structure is expected as the next generation magnet with outperform magnetic properties as high saturation magnetization, magnetocrystalline anisotropy, and Curie temperature.⁸ However, SmFe$12$ bulk crystal have not been synthesized due to the difficulty of stabilizing the material in nature. Modifying element site properties such as substituting a part of the Fe atom with the other atom in the SmFe$12$ structure may derive a bundle of structures in which the difference relies on the occupied position of the new atom. Many first-principle studies have been conducted by replacing Co, Ti, V, Cr, Mo, W, or Ga to the original Fe site of SmFe$12$ or replacing Zr atom to rare-earth sites^9,10 to obtain a stable ThMn$12$-type phase.^11–17 Unrestricted from ternary compounds, recent researchers have emphasized searching for the most potentially stable SmFe$12$-based quaternary compounds using the bi-element substitution method.^9,10,18–24 In Ref. 18, authors report more than 3000 substituted structures generated by substituting up to three Fe sites of the original structure by other atoms. On the other hand, in the attempt to produce magnets, the difficulty comes from controlling correlations between synthesized parameters, e.g., temperature, pressure, diffusion rate, and statistical information of macro property, e.g., the particle grain size.^25–28 The continuous dependence of first-principle calculation properties on the synthesized parameters or the unit-cell geometry could be a promising guideline to accelerate the synthesizing process.

Taking into account perturbation of the lattice parameter as expanding, collapsing the unit cell, and changing its symmetry may increase the number of combinations that may appear in nature. As shown in Ref. 29, the similar ThMn$12$-type structures, YFe$12$ with two monoclinic $C2/m$ (space group 12) phases, rather than the original $I4/mmm$ symmetry, were reported in a metastable state by permutating rare-earth site and $8i$ iron site. These two phases are estimated their total magnetization outperform 4.5% than the original $I4/mmm$ symmetry. There is also an interstitial approach that may also stabilize the desired structure. In Ref. 30, authors reported properties of thin films fabricated SmFe$12$ and SmFe$12$N$x$ with N substituted components. The nitrogenation process was reported to increase the Curie temperature, saturation magnetization, and anisotropy change for all SmFe$12$ and Sm$2$Fe$17$ structures. The number of hypothetical structures will rise significantly to the arbitrary position of the newly added atom.

In this research, we introduce a machine learning-aided genetic algorithm method for efficiently studying SmFe$12$ compounds aiming to optimize three criteria simultaneously: (1) diversifying the symmetry of screening structure, (2) finding optimal stability, and (3) revealing smoothness correlation between the stability and geometrical information. Descriptions about the machine learning method, genetic algorithm and first-principle calculation are shown in Sec. II. In Sec. III, we present our finding in investigating the structure–property relationship of SmFe$12$⁠, Co substituted SmFe$12$⁠, and N interstitial SmFe$11$Co structures using the proposed method.

We introduce a machine learning-aided genetic algorithm in the structure discovery method consisting of five consecutive stages: (i) structure generator based on symmetry transformation operator, (ii) machine learning recommendation system, (iii) genetic algorithm structure generator, and (iv) first-principle calculation to estimate physical properties. Details illustration of the method is summarized in Fig. 1.

Let us assume, we start with an interested symmetry group $G$⁠, e.g., $I4/mmm$ (139), in the original structure of SmFe$12$ with the ThMn$12$ prototype structure. In this structure, there are four Wyckoff positions with 2a-occupied by Sm atom and $8i$⁠, $8j$⁠, and $8f$ occupied by Fe atoms. A new structure with sub-symmetry group $H$ with $H<G$ is created by splitting Wyckoff positions from the original structure with a predefined distortion constant. A simple transformation of the SmFe$12$ structure with the detailed movement of Fe sites in $I4/mmm$ (139) to $Immm$ (71), and $C2/m$ (12) is shown in Sec. III and distortion of Fe network in Sm local is shown in Fig. 1(c). In this work, we use PyXtal library³⁷ for symmetry transformation operators we applied in the structure generator.

In Fig. 1(b), the graph of possible symmetry transformation beginning with the $I4/mmm$ (139) symmetry is shown, limiting up to the $Immm$ (71) symmetry. Other nodes in the middle correspond to the intermediate subgroups between these symmetries. Two nodes in the graph are connected with an edge if there is a group–subgroup relation between the corresponding subgroups. A node is connected with itself if this symmetry group has an isomorphic subgroup. The graph is created by querying information from Bilbao Crystallographic Server.^38–40 More details about sub-symmetry transformation can be found in this Bilbao Crystallographic Server. Figure 1(c) shows example of skeleton of SmFe12 structures with $I4/mmm$ (139), $P4/mmm$ (123), $Immm$ (71), and $C2$ (5) symmetry. Surfaces connected by Fe atoms are shown to visualize distortions among sub-symmetry transformations better. The $C2$ (5) symmetry skeleton as shows a great distortion of Fe nets surrounding the Sm center compared with $P4/mmm$ (123) and $Immm$ (71).

The machine learning recommendation system has two central components: (1) the prediction model and (2) the acquisition function. The prediction method used in our framework is the Gaussian process, in which the formation energy is set as the target variable, and the orbital field matrix-based metric learning vector is used as predicting variable.^18,41,42 Consequently, structures in the sub-symmetry pool with the lowest predicted formation energy and structures with maximum prediction variance are ranked as the most potential structures used in the first-principle calculation step of the next generation. In detail, structures with the minimum prediction formation energy are expected as the most stable structures–exploitation acquisition function, while structures with the highest predicted variance are expected to enrich the exploration ability of the Gaussian process predictor–exploration acquisition function. Details description and discussion are shown in the previous study in Ref. 18. The genetic algorithm used in this study was inherited from USPEX. For each generation, a structure population consists of 164 crystal structures in which 64 structures are generated by the lattice mutation operator and 100 structures are generated using the machine learning recommender system. By repeatedly performing this process, the population evolved with each element in the population diversified gradually from the original symmetry and, consequently, kept the target of optimizing both the stability and its magnetic properties.

The priority is to maximize the prediction ability of the machine learning estimator using the most limited training data because of the expensive computational resource. Querying new labeled data is equivalent to correcting the form of learned function with respect to target property. As an alternative advantage, following the correction process leads to better insight regarding the phenomena of interest. In this work, a method to localize information on the learned function, monitoring its change to improve the querying data process is introduced.

With the target property as a continuous variable, we consider the learned formation energy function $y=f(u)$⁠, which is called interpretable if it is possible to allocate on the representation space $u$⁠, where the function meets a predefined condition $g$⁠. In detail, given a condition $g$⁠, the probability distribution spanning on the embedding space $u$ is defined as follows:

with $p(u|g)$ as the probability density at $u$ under $g$⁠; $ui$ as the location of an observed data point $i$⁠, i.e., $ui=Axi$⁠; and $h$ as a tuning kernel width. The indicator $1[⋅]$ returns 1 if the condition $[⋅]$ is true and 0 otherwise. In the present work, we consider two forms of relevant conditions,

where $gy(ui)$ and $gxj(ui)$ intuitively represent a region of interest with potentially stable materials and regions spanned by structures incorporating the maximum value of OFM element $xj$⁠. Since OFM elements represent information about coordination numbers, the condition $gxj(ui)$ reflect the region of structures owing the corresponding high coordination number. The terminology $argmax$ and $argmin$ are abbreviation of indicating the top 10% highest and lowest value, respectively. Then, we measure the Bhattacharyya coefficient⁴³ between a pair of $(gy,gxj)$ as

with the integral taken over the space spanned by $u$⁠. The Bhattacharyya coefficient $BC(gy,gxj)$ measures the probability of joint occurrence between two conditions $gy$ and $gxj$⁠. Higher $BC$ values indicate a higher possibility to obtain correlation between conditions $gy$ and $gxj$ and vice versa; this makes it easier to understand the meaning of the BC coefficient in identifying overlapping distributions. In the discussion of the results provided in 4.2, we characterize any distribution $p(u|g)$ using a single-level contour representation.

All of the DFT calculations in this study were performed by using the projector augmented wave (PAW) method,⁴⁴ as implemented in the Vienna ab initio simulation package (VASP),⁴⁵ within the Perdew–Burke–Ernzerhof (PBE) generalized-gradient approximation (GGA).⁴⁶ The 4f electrons for all the RE elements are frozen in the core assuming trivalent configuration (Sm$_$3 pseudopotential. In the first three optimization steps, the lattice constants and atomic positions of all of the study were optimized with a kinetic-energy cutoff for the plane wave expansion 300 eV; the break condition of the electron self-consistent step is set at 1e–4 eV and the smallest allowed spacing between k points is set at 0.5 A$−1$⁠. In the final optimization step, these three values are set at 400 eV, 1e–5 eV and 0.2 A$−1$⁠, respectively. In all steps, the partial occupancies is estimated using Gaussian smearing method with 0.01 eV smearing width.

The formation energy of a given structure $s$ is defined as follows:

where $ΔE[s]$⁠, $E[s]$⁠, and $E[si]$ are the formation energy, total energy of structure $s$ per formula unit, and simple substance $si$ per atom, respectively. $N$ is the number of atoms in an unit cell. The simple substances were chosen as Im-3m with Fe, R-3m with Sm, Fm-3m with Co with the most converged calculation data from the Open Quantum Materials Database.^47,48 A structure whose formation energy lies below or lower than zero, that is, $ΔE≤0$⁠, is a potentially formable material in nature, whereas a structure associated with $ΔE>0$ could be considered unstable. For the competing phases, the stability of the structure should be discussed using the hull distance. In this study, we use the formation energy defined in Eq. (5) as an index for simplicity. The relationship between the experimental material and the hull distance at $T=0K$ has been summarized in Refs. 49 and 50. The stability of the magnets at finite temperature can be found in Ref. 51.

In addition, the total magnetic moment of these materials $μ[s]$ was recalculated because we used an open-core approximation to treat the 4f electrons of Sm, as follows:

where $m[si]$ is the magnetic moment of atom $i$⁠, $J4fgJ4f[sk]$ is the correction term with $gJ4f$ as the Lande factor, and $J4f$ is the angular momentum of lanthanide $sk$⁠. Index $i$ represents all atoms, and index $k$ represents all lanthanide atoms in the structure. The contribution of the 4f electrons of Sm to the magnetization is $JgJ=0.714$⁠. In this paper, this value is finally converted to magnetization per formula unit, $M$(T).

In this session, we discuss the structure–stability relationships of SmFe$12$-based structures extracted using the machine learning-aided genetic algorithm. Table I summarized structures with the minimum formation energy in each corresponding symmetry of SmFe$12$⁠, SmFe$11$Co, SmFe$11$CoB, and SmFe$11$CoN founded by the system. Overall, our calculated properties show reasonable deviations compared to existing references, e.g., 5% deviation of $M$⁠, while maintaining the consistent correlations between the substitution–interstitial effects to corresponding physical properties among structure families.

Figure 2 shows the dependence of formation energy on unit-cell volume and magnetization of compounds SmFe$12$ generated using the proposed machine learning-aided genetic algorithm. This figure shows SmFe$12$ structures own the formation energy lower than 0.1 (eV/atom) and only space group index larger than 10. There are four notable space group symmetries $I/4mmm$ (139), $Immm$ (71), $Fmmm$ (69), and $C2/m$ (12). Overall, there is a parabolic dependency between the formation energy and the unit-cell volume, as well as the magnetization. The optimal structure owing space group symmetry $I/4mmm$ (139) will get the formation energy 0.07 (eV/atom) at unit-cell volume 168.5 (⁠$A3$⁠) and magnetization 1.98 (T). Similar trends and the optimal structure appeared with three other symmetry space groups $Immm$ (71), $Fmmm$ (69), and $C2/m$ (12). In this figure, we present the sample skeleton of $I/4mmm$ (139), $Immm$ (71), and $C2/m$ (12). The original $I/4mmm$ (139) appeared with three Fe symmetry sites denoted by $8i$⁠, $8j$⁠, and $8f$ Wyckoff index, the SmFe$12$ structure with $Immm$ (71) symmetry is created by breaking symmetry two Fe $8i$ sites more than 0.1 (Å)—denoted by white-orange atoms. In the SmFe$12$ structure with $C2/m$ (12) symmetry, two Fe $8i$ sites and four Fe $8j$ sites are broken their symmetry comparing to the original structure.

In Fig. 2, it should be noted that there are some atom arrangements in the unit cell of $I/4mmm$ (139) structure, e.g., structure A sharing a similar formation energy achieve 0.07 (eV/atom) but expanding the unit-cell volume to 174.3 (⁠$Å3$⁠)– 3.4$%$ larger than the optimal formation energy structure with the unit-cell volume at 168.5 (⁠$Å3$⁠), structure B. With 2.13 (T), the magnetization of the structure A shows 7.5% higher than the B’s magnetization.

We pickup four SmFe$12$ structures with the formation energy at 0.07 (eV/atom) denoted by A, B, C, and D for studying the difference of x-ray diffraction patterns with Cu-K$α$⁠, $λ=1.5406$ (Å) radiation. Two structures, A and B, share a similar symmetry $I4/mmm$ (139), but the magnetization as $M=2.132$ (T) and 1.989 (T), respectively. Two other structures, C and D, are associated with the $Immm$ (71) and $C2/m$ (12) symmetries, respectively. One might observe that x-ray diffraction of structure A appears with an approximately single peak in range 2$θ$ from $42°$ to $44°$⁠, while three other structures appear with two separated peaks. Therefore, a slight distortion of the original SmFe$12$ may also cause the minor peak appearing in $44.5°$⁠, which is similar to the peak of $α−$Fe in XRD spectra of synthesized structures as previous reports.⁵² Also, the relative intensity of two peaks at 37 and 38 of structure A shows more consistent with report in Refs. 53 and 20 than the other.

Figure 3 shows the dependence of formation energy on unit-cell volume and magnetization of SmFe$11$Co structures. By limiting the space group symmetry of founded structures, there are three separate correlations of the formation energy on the unit-cell volume for three substituted Co atom positions. Overall, all substituted structures are more formable than the original SmFe$12$ structure, even the original space group symmetry $I/4mmm$ (139) broken down to $C2/m$ (12) and $Imm2$ (44). For a given value of unit-cell volume, the SmFe$11$Co structure with the Co atom occupied Fe $8f$ site always shows the lowest formation energy, while the structure with Co atom substituted to Fe $8i$ always has the highest formation energy. However, while there is no difference between magnetization values for structures with Co atom substituted to Fe $8i$ and $8j$ sites, the Co atom substitute to $8f$ site can achieve significantly larger magnetization.¹⁸ Regarding the results of magnetization dependence on substituted elements, almost all substituted structures acquired lower magnetization values than the original SmFe$12$ structure, except for the substitution of Co at the $8f$ position of the Fe site. This result is consistent with other first-principle calculation code Quantum Materials Simulator (QMAS);^31,35 OpenMX¹⁹ or synthesis reports of Sm(Fe$0.9$Co$0.1$⁠)$12$ and Sm(Fe$0.8$Co$0.2$⁠)$12$ in Refs. 32 and 36. From experimental results, authors in Ref. 32 reported that Sm(Fe$0.8$Co$0.2$⁠)$12$ had not yet been stabilized in the bulk form, which is consistent with the positive formation energy appearing with Co substitution from our calculation, although they had lower energy than the host structures.

In Fig. 4, we show the results of SmFe$11$CoB and SmFe$11$CoN in comparison with the original formation energy and magnetization of the SmFe$12$ structure with $I/4mmm$ (139) symmetry. While the B interstitial method— (gray shade) may create the structure with lower formation energy—0.025 (eV/atom), this method also decreases the optimal magnetization value from 2.0 (T) of the SmFe$11$Co to 1.8 (T). On the other hand, a part of N interstitial structures with $Cm$ (8) symmetry shows remaining formation energy at 0.015 (eV/atom) for a wide range of magnetization from 1.5 to 1.81 (T). However, the other part of N interstitial structures with $Imm2$ (44) and $C2/m$ (12) symmetries can achieve negative formation energy up to $−0.06$ (eV/atom) but keep the optimal magnetization at a high value from 1.85 to 1.98 (T). In the lower panel of Fig. 4, we show formation energy and magnetization of SmFe$11$CoX structures with X as Al, Ti, Cu, Co, Zn, Mo, and Ga collected from Ref. 18 as reference. All SmFe$10$CoX structures appear on the left region of the graph—the region with low magnetization compared with the original SmFe$12$ structure. To summarize, SmFe$11$CoN structures with $Imm2$ (44) and $C2/m$ (12) symmetries surpass all other structures in the ability to maintain low formation energy and high magnetization.

In this part, we show results in interpreting the structure–stability relationship of the most prominent SmFe$11$CoN structure family using the embedding space by metric learning for the kernel regression (MLKR) method. Figure 5 shows the space of SmFe$11$CoN structures learned by embedding information on the formation energy. Appropriated kernel density estimation parameters are applied for smoothing interpolation in the embedding space. As the nature of the metric learning method, the formation energy shown in the background in Fig. 5(b) smoothly transverse through calculated structures in which neighbor structures share both similar geometrical information and formation energy. In Fig. 5(c), we show a similar space, but the background color is filled by the calculated magnetization $M$⁠.

In the embedding space, we represent the condition of Bhattacharyya measure with the red contour indicating the region of minimized formation energy—Fig. 5(b), and maximized the magnetization—Fig. 5(c). Other contours indicating regions of structures with the maximum of the corresponding coordination number. For example, $(f6,p3)$ contour indicates regions of structures owning with the maximum value of coordination number of N atom—$p3$⁠, surrounding Sm atom—$f6$⁠. In Fig. 5(b), there are three separated regions with minimum formation energy that represent for three different geometrical arrangements of substituted Co site and N interstitial site that cause similar stability level.

In Fig. 5(a), we show the Bhattacharyya coefficient score measuring overlapping between regions of SmFe$11$CoN structures with the maximal OFM condition and the region of minimum formation energy. As the most appearance in this metric, highly stable SmFe$11$CoN structures are associated with a high coordination number of Fe/Co surrounding the Fe site. Also, there is a higher possibility of stability for structures with an N atom surrounding the Sm center rather than the Co or Fe atom, i.e., the Bhattacharyya coefficient for $(f6,p3)$ significantly large than $(d7,p3)$ and $(d6,p3)$⁠, respectively. Apparently shown in the embedding map in Fig. 5(b), the $(f6,p3)$ contour show higher overlapping to stable structure region compared to other contours. The region allocated by $(f6,p3)$ also show the maximal calculated magnetization.

Figure 5(d) shows four reference structures, EA1526, EA1593, EA628, and EA1402, as the representative structures for geometrical information in the three most stable regions in Fig. 5(b). The region of maximal $(f6,p3)$ represented by EA628 and EA1402 structures own relatively higher magnetization—larger than 1.8 (T) compared to two other stable regions. It is the region with structures showing N atoms placed in the middle of two Sm atoms (⁠$2p$⁠), compared with the other regions where the N atom interstate among other Fe and Co atoms. It is exclusively the region with symmetry group found by $Imm2$ (44) with Co site substituted in $8i$ position—EA628 structure, and $C2/m$ (12) with Co site substituted in $8f$ position—EA1402 structure. The preferred position of the N interstitial site in the center of the Fe and Sm octahedron inducing strong magnetocrystalline anisotropy on Sm is consistent with previous reports.^34,54 Further investigation of the geometrical shape of other stable structures suggests that interstitial positions of N rather than the middle of two Sm atoms cease more considerable distortion to the original SmFe$12$ skeleton with the $I4/mmm$ (139) space group symmetry.

In Fig. 6, we summarize the correlation between formation energy, magnetization, and the density of three structure families investigated SmFe$12$⁠, SmFe$11$Co, and SmFe$11$CoN. Regardless of the elemental constituent of each family, there is a general trend that by reducing the atomic density of the structure, the corresponding magnetization linearly increases while the formation energy decrease until a stable position.

In this work, we introduce a machine learning-aided genetic algorithm structure generation method to investigate the correlation between geometrical information, stability, and magnetization of SmFe$12$-based structures. For every population of genetic algorithm generation, a pool of structures is created using the sub-symmetry perturbation method. The gaussian process as predictor taking the embedded orbital field matrix representation as structure fingerprint has been applied to ranking the most potential stability structures for first-principle calculation. As a result, the original structure SmFe$12$⁠, well-known with tetragonal $I4/mmm$ skeleton structures, is investigated with distortions of the unit-cell lattice parameter and individual sites. An anomaly atom arrangement of the $I4/mmm$ SmFe$12$ structure is found out of the parabolic dependence between the formation energy and magnetization. Structures with nitrogen interstitial position in the middle of two Samarium sites show outperform all other structures in both ability of stabilization and remaining high magnetization order of the original structure. Lastly, metric learning embedding space brings fruitful insight into the correlation between this structure family’s geometrical arrangement, stability, and magnetization.

This work was supported by the Ministry of Education, Culture, Sports, Science, and Technology of Japan (MEXT) with the Program for Promoting Research on the Supercomputer Fugaku (DPMSD), JSPS KAKENHI Grants 20K05301 and JP19H05815 (Grants-in-Aid for Scientific Research on Innovative Areas Interface Ionics), 21K14396 (Grant-in-Aid for Early- Career Scientists), and 20K05068, Japan.

The authors have no conflicts to disclose. Hereby, I, D.-N.N consciously assure that the manuscript “Machine learning-aided genetic algorithm in investigating the structure–property relationship of SmFe$12$-based structures” is the authors’ own original work, which has not been previously published elsewhere. The paper reflects the author’s own research and analysis in a truthful and complete manner. Besides, the paper properly credits the meaningful contributions of co-authors and co-researchers and is appropriately placed in the context of prior and existing research.

Informed consent was obtained from all individual participants included in the study. The participant has consented to the submission of the case report to the journal.

Duong-Nguyen Nguyen: Conceptualization (lead); Data curation (lead); Formal analysis (lead); Funding acquisition (lead); Investigation (lead); Methodology (equal); Validation (lead); Visualization (lead); Writing – original draft (lead); Writing – review & editing (lead). Hieu-Chi Dam: Conceptualization (equal); Methodology (equal); Project administration (equal); Resources (equal); Supervision (equal).

Dataset of SmFe$12$⁠, SmFe$11$X with X = ${$Co, CoB, CoN$}$ including vasp calculations, OFM descriptor is openly available in Zenodo at https://doi.org/10.5281/zenodo.7318435, Ref. 55. Dataset of SmFe$12−α−βXαYβ$ structures with $X,Y$ as Mo, Zn, Co, Cu, Ti, Al, Ga, and $α+β<4$ including vasp calculations, OFM descriptor is openly available in Zenodo at https://doi.org/10.5281/zenodo.5763325, Ref. 56.