Search for Stable and Low-Energy Ce–Co–Cu Ternary Compounds Using Machine Learning

Abstract

Cerium-based intermetallics have garnered significant research attention as potential new permanent magnets. In this study, we explore the compositional and structural landscape of Ce–Co–Cu ternary compounds using a machine learning (ML)-guided framework integrated with first-principles calculations. We employ a crystal graph convolutional neural network (CGCNN), which enables efficient screening for promising candidates, significantly accelerating the material discovery process. With this approach, we predict five stable compounds, Ce₃Co₃Cu, CeCoCu₂, Ce₁₂Co₇Cu, Ce₁₁Co₉Cu, and Ce₁₀Co₁₁Cu₄, with formation energies below the convex hull, along with hundreds of low-energy (possibly metastable) Ce–Co–Cu ternary compounds. First-principles calculations reveal that several structures are both energetically and dynamically stable. Notably, two Co-rich low-energy compounds, Ce₄Co₃₃Cu and Ce₄Co₃₁Cu₃, are predicted to have high magnetizations.

1. Introduction

Ce-based intermetallic compounds are of great interest due to the relative abundance of Ce compared to other rare earths and the potential technological applications of Ce-based materials in clean energy. Currently, the most widely used high-performance magnets for energy generation/conversion and information storage, like generators, motors, mobile machines, and computer hard drives, contain critical rare earth (RE) elements like Nd, Sm, and Dy. , Owing to insecure sources and supply of these critical RE materials, there are considerable interests and efforts in replacing these critical RE elements with a more abundant and cheaper Ce to meet the performance and cost goals for advanced electromagnetic devices. − Ce-based intermetallic compounds are also interesting from the fundamental science point of view, as they can vary in valence from the nonmagnetic Ce⁴⁺ to magnetic Ce³⁺ and form heavy Fermion and mixed-valent materials with exotic quantum criticality, superconductivity, and magnetism. , Because of the technological and fundamental importance of Ce-based intermetallic compounds, comprehensive knowledge about the stable, low-energy, potentially metastable phases of these compounds and alloys is highly desirable.

In this paper, we focus on searching for the stable and low-energy (100 meV above the convex hull and therefore potentially metastable) phases of Ce–Co–Cu ternary compounds as a step toward this goal. This ternary system is chosen on the basis of experimental studies, which suggest that ternary Ce–Co–Cu intermetallic compounds/alloys would be a promising class of materials for the discovery and design of novel strong permanent magnets with light RE elements. For example, SmCo₅ has been known to be a strong permanent magnet since the early 1960s. However, samaritium–cobalt magnets are expensive and subject to supply and price fluctuations. It has been shown that replacing Sm with a much cheaper and abundant Ce and partially substituting Co with the nonmagnetic element Cu can result in Ce–Co–Cu compounds, such as the CeCo₄Cu alloy, which exhibit desirable magnetic properties for permanent-magnet applications. − A recent combination of experimental and first-principles calculation studies on Cu-doped SmCo₅ shows that Cu doping can enhance the stability and magnetic anisotropy of the compound. The study suggests that Cu doping modifies the crystal field splitting (CFS) of the RE atoms and thus the magnetic anisotropy. Cu doping also modifies the exchange coupling between the RE atoms and Co atoms in favor of a stronger magnetic anisotropy. Moreover, Ce₂Co₁₇ has a high magnetization, but the magnetic anisotropy is too low for permanent-magnet application. However, it has also been shown by experiments that replacing some Co with Cu atoms to form Ce₂Co_17‑xCu _x (x = 1) can substantially increase the magnetic anisotropy. Therefore, searching for more stable and low-energy Ce–Co–Cu compounds and alloys is interesting and highly desired.

To the best of our knowledge, there are no known stable Ce–Co–Cu ternaries, even though there are a number of interesting experimentally known compounds with some amount of Co–Cu site disorder such as CeCo_5‑x Cu _x , and Ce₂Co₁₆Cu. − , We want to investigate to see if we can find new stable materials or low-energy materials that might be similarly stabilized by site disorder. We are particularly interested in materials with high Co concentrations, as these are particularly promising for the development of new permanent magnets. Owing to a considerably large number of possible combinations of composition ratio among the three elements and the potential crystal structures they may take, we anticipate that more stable and low-energy Ce–Co–Cu ternary phases remain to be discovered. Nevertheless, the vast combinatorial composition–structure space presents challenges for the discovery of new phases. To overcome this difficulty, recent machine learning (ML) approaches have been employed, − which can greatly accelerate material design and discovery. In this work, we utilize a machine learning (ML) guided framework to efficiently select promising candidates for first-principles calculations. Such an ML-guided ab initio approach greatly accelerates the material discovery process and enables us to predict five new Ce–Co–Cu ternary compounds with formation energies below the currently known convex hull. We also predict 250 low-energy ternary compounds for the Ce–Co–Cu system. We hope these ML-guided computational predictions can motivate experimental efforts in making them. If any of these structures can be experimentally synthesized, they would be the first stable ternary Ce–Co–Cu compounds without a site disorder. In many cases, occupation disorder among the Co and Cu atoms may further stabilize the structure, which would require further investigations.

The rest of the paper is organized as follows. In Section , we describe how ML is used to accelerate the discovery of new compounds. The first-principles calculation results guided by the ML are presented in Section . Finally, a summary is given in Section .

2. Computational Methods

2.1. Efficient Structure Searches Guided by the ML-Guided Framework

A schematic sketch of the ML-guided framework used in this study to search for low-energy, stable Ce–Co–Cu ternary compounds is shown in Figure . A crystal graph convolutional neural network (CGCNN) ML method is used in this framework. In CGCNN, a crystal structure is represented by a crystal graph, which encodes both the atomic information on the atoms and the bonding interactions among them in the crystal. A convolutional neural network is used to process the crystal graphs to optimize the descriptors, followed by another neural network to map out the relationship between the descriptors and the physical properties of the crystals. The training data in CGCNN can be generated by first-principles calculations, which enable a sufficient volume of data for supervision training. In this study, we use a CGCNN ML model to predict the formation energy (E _f ) for Ce–Co–Cu ternary compounds. The CGCNN model for the formation energy predictions of compounds developed by Xie and Grossman was adopted as the first-generation CGCNN (1G-CGCNN) in our framework. This model was trained using the first-principles calculation results of structures and energies of 28,046 binary and ternary compounds from the Materials Project (MP) database. These 28,046 binary and ternary compounds contain a wide range of chemical elements in the periodic table. Therefore, the 1G-CGCNN model is not material-specific and can be used as a good starting point for any compound. After the structure candidates selected by the 1G-CGCNN model had been optimized by first-principles calculations, we obtained 1716 Ce–Co–Cu ternary compounds. Then, we use these 1716 structures to train a second-generation CGCNN model (2G-CGCNN) specifically for predicting the formation energy (E _f) of Ce–Co–Cu compounds, with improved accuracy for this specific system compared to the 1G-CGCNN model. 2G-CGCNN is then applied to search for more promising ternary Ce–Co–Cu compounds. It should be noted that such a CGCNN approach is limited to known structure types in the database. Stable structures whose structural motifs are not present in the existing structural databases will be missed. If a new compound/composition is found experimentally to be stable but missed by the current algorithm (say, a new structure type), then this result changes the hull and predictions.

1.

Schematic flowchart of the ML-guided framework for efficient discovery of stable and low-energy Ce–Co–Cu ternary compounds. In addition to training 1G-CGCNN, a structure pool of hypothetical ternary Ce–Co–Cu compounds is generated by substituting Ce, Co, and Cu on the atomic sites of 28469 ternary compounds extracted from the MP database. For a given ternary template, there are six ways to shuffle the three elements Ce, Co, and Cu at the atomic positions of the structure. We also allow the volume of the unit cell to vary by a scaling factor of 0.96–1.04, in increments of 0.02, to help the CGCNN model differentiate the energetic stability of the same structure with different bond lengths. In this way, a pool of 854,070 hypothetical ternary Ce–Co–Cu compounds is generated.

The 1G- and 2G-CGCNN energy models are applied, respectively, to the same structure pool of 854,070 ternary Ce–Co–Cu compounds. The formation energy distribution (histogram) from the predictions of the second-generation CGCNN models is shown in Figure . The formation energy E _f per atom is defined relative to the elemental phases of Ce_αCo_βCu_γ with α + β + γ = 1 as

$$E_{\mathrm{f}}=E({\mathrm{Ce}}_{\alpha }{\mathrm{Co}}_{\beta }{\mathrm{Cu}}_{\gamma })-\alpha E(\mathrm{Ce})-\beta E(\mathrm{Co})-\gamma E(\mathrm{Cu})$$

Here, E(Ce _α Co _β Cu _γ ) is the total energy per atom of a Ce _α Co _β Cu _γ structure. Reference energies are the total energies per atom of face-centered cubic Ce, hexagonal close-packed Co, and face-centered cubic Cu. Based on the E _f histograms shown in Figure and after removing the redundancy structures with similarity, we select 3807 and 1912 structures, respectively, from the 1G- and 2G-CGCNN predictions for further evaluation by first-principles calculations.

2.

Distribution of formation energies (E _f) of the hypothetical ternary Ce–Co–Cu ternary compounds predicted from the (a) 1G and (b) 2G-CGCNN energy models. The total number of structures is 854,070. We select 3807 and 1912 structures from 1G- and 2G-CGCNN predictions, respectively, for further optimization by DFT calculations.

The first-principles calculations are performed based on density functional theory (DFT) using the VASP package, , with Perdew–Burke–Ernzerhof (PBE) functionals combined with the projector-augmented wave (PAW) method and a cutoff energy of 520 eV. We use a k-point grid with a mesh size of 2π × 0.025 Å^–1 generated by the Monkhorst–Pack scheme. This mesh size is fine enough to sample the first Brillouin zone for achieving better k-point convergence. The lattice vectors and the atomic positions of candidate structures selected from 1G- and 2G-CGCNN predictions are fully optimized by the DFT calculations until forces on each atom are less than 0.01 eV/atom. In these DFT calculations, spin–orbit interactions are neglected and not expected to have a significant effect on the formation energies.

3. Results and Discussion

The results from the DFT calculations show that 1716 and 662 nonequivalent structures selected from the 1G- and 2G-CGCNN predictions can be fully optimized. Other structures that cannot pass the electronic self-consistent calculations or are duplicate structures are discarded. These discarded structures are most likely far from the realistic structures for Ce–Co–Cu ternary compounds. In total, 2378 structures are identified as new structures for ternary Ce–Co–Cu compounds. The 1716 structures from 1G-CGCNN are used to train the 2G-CGCNN model specifically for Ce–Co–Cu ternaries, as discussed above. To evaluate the thermodynamic stability of these newly predicted ternary compounds, we calculate the formation energies (E _hull) of these structures with respect to the Ce–Co–Cu ternary convex hull at the accuracy level DFT. The E _hull is the decomposition energy of a Ce _α Co _β Cu _γ ternary compound with respect to the nearby three known stable phases, which can be a ternary, binary, or elemental phase. The chemical compositions of these phases are located at the vertexes of the Gibbs triangle that encloses the composition of Ce _α Co _β Cu _γ . The compositions of stable and low-energy (E _hull ≤ 0.1 eV/atom) Ce–Co–Cu ternary phases with respect to the currently known convex hull predicted from our CGCNN+DFT approach are shown in Figure (a,b), respectively. More detailed information about these structures is shown in Table S1 in the Supporting Information.

3.

Compositions of (a) stable and (b) low-energy (E _hull ≤ 0.1 eV/atom) Ce–Co–Cu ternary phases with respect to the currently known convex hull predicted from our CGCNN+DFT approach. The compositions in the convex form are colored by the lowest E _hull for this composition. The E _hull values shown in the color bars are in units of eV/atom. The predicted stable structures are enclosed by a red circle, as shown in the bottom-left region in (a).

3.1. Predicted Stable and Metastable Structures

From the results shown in Figure and Table S6, we can see that many structures are predicted to be stable or have very low energy (with formation energies under the convex hull or within 100 meV/atom above the convex hull) based on the currently known stable binary and elementary phases. We note that the stability presented here is based on the DFT energies at T = 0 K, and no partial site occupancies are considered. Allowing partial site occupancies may further lower the free energies at finite temperatures due to the entropy contribution. Additionally, many stable compounds from our predictions cluster in the Ce-rich region of the ternary convex hull, as indicated by the red circle in Figure (a). This suggests that these predicted stable phases may compete during phase selection and stability in synthesis. When we include the energies of all newly predicted structures to construct a new convex hull for the Ce–Co–Cu ternary system, only five phases remain stable: Ce₃Co₃Cu, CeCoCu₂, Ce₁₂Co₇Cu, Ce₁₁Co₉Cu, and Ce₁₀Co₁₁Cu₄, as shown in the new convex hull plot in Figure (a). Notably, these five stable phases have formation energies of 42, 35, 75, 62, and 35 meV/atom below the previously known convex hull, respectively. However, the low Co content in these compounds makes them less suitable for applications in magnetic materials. However, these Ce-rich compounds still might be interesting if Ce is mixed valent.

4.

(a) New convex hull for Ce–Co–Cu ternaries after the new structures predicted from the present study are included. (b) Low-energy (E _hull ≤ 0.1 eV/atom) Ce–Co–Cu ternary phases with respect to the new convex hull predicted from our CGCNN + DFT calculations. For a given composition, there may be several stable or low-energy structures. The colors in (b) are shown according to the lowest E _hull for the given composition. The E _hull shown in the color bars is in the unit of eV/atom. The two low-energy Co-rich compounds (shown later in Figure ) are enclosed by a red circle in the bottom-right corner of (b).

Accordingly, the low-energy (E _hull ≤ 0.1 eV/atom) phases are also redefined as shown in Figure (b). There are two Co-rich phases with E _hull within 0.02 eV/atom above the new convex hull as indicated by the red circle in the bottom-left corner of Figure (b). These Co-rich low-energy structures might be interesting for magnetic materials.

The structures of the five stable structures and the two Co-rich low-energy structures obtained from our predictions are plotted in Figures and , respectively. Detailed information on these structures is given in Tables S1–S6.

5.

Crystal structures of the five stable Ce–Co–Cu ternary compounds from our CGCNN+DFT calculations: (a) Ce₃Co₃Cu, (b) CeCoCu₂, (c) Ce₁₂Co₇Cu, (d) Ce₁₁Co₉Cu, and (e) Ce₁₀Co₁₁Cu₄. The Ce, Co, and Cu atoms are presented as green, dark blue, and magenta balls, respectively.

6.

Crystal structures of the two Co-rich low-energy Ce–Co–Cu ternary compounds from our CGCNN+DFT calculations. Both can be viewed as slight distortions of Ce₂Co₁₇ with (a) 1 Cu and (b) 3 Cu atoms substituted for Co. The Ce, Co, and Cu atoms are presented as green, dark blue, and magenta balls, respectively. J_s is the magnetization of the compound (T).

The crystal structure of the Ce₃Co₃Cu compound shown in Figure (a) has an orthorhombic lattice with a Cmcm space group symmetry. There are two Wyckoff positions for the Ce and Co atoms. The first Ce atom is bonded to one Co with a bond length of 2.23 Å and three Cu atoms at distances of 2.86 and 3.18 Å. The second Ce atom is bonded to one Co with a similar distance of 2.24 Å and two Cu atoms with a distance of 3.43 Å. Cu is bonded to eight Ce atoms and four equivalent Co atoms to form a mixture of distorted edge-, face-, and corner-sharing CuCe₈Co₄ cuboctahedra.

CeCoCu₂ crystallizes in an orthorhombic Pnma space group symmetry, as shown in Figure (b). The Ce atom is bonded in a distorted geometry to one Co and 10 Cu atoms with a Ce–Co bond length of 2.16 Å and the Ce–Cu bond length ranging from 2.92 to 3.12 Å.

Figure (c) shows the crystal structure of Ce₁₂Co₇Cu. It has a tetragonal lattice with an I4̅ /mcm space group, with a large unit cell of 80 atoms. There are three inequivalent Ce and Co sites, and a single Cu site. Cu atoms are bonded in a body-centered cubic geometry to eight equivalent Ce atoms, with a bond length of 3.10 Å. The bond length between Ce and Co atoms ranges from 2.36 to 2.95 Å.

Ce₁₁Co₉Cu forms in an orthorhombic Iba2̅ space group with 84 atoms in the unit cell, as shown in Figure (d). There are six Wyckoff sites for Ce, five for Co, and one for Cu. Cu is bonded in a 5-coordinate geometry to four Ce atoms and two equivalent Co atoms. The bond lengths are 2.94 and 3.43 Å for Ce–Cu, respectively, and 2.47 Å for Co–Cu.

Finally, Ce₁₀Co₁₁Cu₄ has a monoclinic lattice with a C2̅ /m space group, as shown in Figure (e). There are five inequivalent Ce sites, six Co sites, and two Cu sites. In both Cu sites, Cu is bonded to eight Ce atoms and four Co atoms to form a mixture of distorted-face and edge-sharing CuCe₈Co₄ cuboctahedra. The bond lengths range from 2.89 to 3.32 Å for Ce–Cu, 2.50 to 2.51 Å for Co–Cu, and 2.24 to 2.29 Å for Ce–Co.

The two Co-rich low-energy compounds Ce₄Co₃₃Cu and Ce₄Co₃₁Cu₃ both have monoclinic lattices with C2̅ /m and P-1̅ space groups, respectively. Both compounds can be viewed as distorted Cu-doped variants of binary Ce₂Co₁₇. Despite Ce₂Co₁₇’s very high Curie temperature and large saturation moment, it remains a rather inferior magnet due to its small uniaxial magnetic anisotropy energy (MAE). Previous studies have shown that the poor MAE stems from the negative contribution of the Co atoms occupying the “dumbbell” 4f Wyckoff site in the rhombohedral structure. Various attempts have been made to introduce dopants such as Fe, Mn, Al, Zr, etc., into this system to enhance the MAE with the aim of developing potential permanent magnets. − Similarly, a doped Cu atom in the “dumbbell” site may also help enhance the MAE. Notably, the experimentally synthesized Ce₂Co₁₆Cu has the same hexagonal lattice with the P6̅ ₃ /mmc space group as the binary Ce₂Co₁₇, with the Co 4f Wyckoff site half-occupied by Cu. Previous computational work has also evaluated the MAE of Ce₂Co₁₅Cu₂, indicating an enhanced magnetocrystalline anisotropy constant K ₁ of 2 MJ/m³. It will be interesting to investigate the MAE for these two newly predicted structures, where we expect that Ce₄Co₃₁Cu₃ will have a higher MAE, as it has one Cu on a dumbbell site. Moreover, the higher configurational entropy in the disordered structure with partial dumbbell site occupancy, as in Ce₂Co₁₆Cu, can help to stabilize the structures at finite temperatures.

3.2. Electronic and Magnetic Properties

We next explored the electronic and magnetic properties of these seven ternary Ce–Co–Cu compounds at the DFT-PBE level. Prior research has shown that Ce is nonmagnetic in tetravalent CeCo₅, , while antiferromagnetic behavior is observed around 4 K in trivalent CeCu₅. Doping Cu into CeCo₅ enhances stability, magnetic anisotropy, and coercivity, which can be attributed to the transition from Ce⁴⁺ to Ce³⁺. ,, Additionally, X-ray absorption spectroscopy suggests a Ce valence between 3.0 and 3.3 in Ce₂Co₁₇, indicating weak mixed valency. Our DFT-PBE calculations show nonmagnetic Ce and Co behavior in the five stable Ce–Co–Cu compounds, where both exhibit negligible magnetic moments, ranging from 0 to 0.03 μ _B in both the spin-polarized calculations and multiple antiferromagnetic configurations. Note that the Ce magnetization may not be correctly evaluated through DFT due to its limitation in describing Ce 4f electrons, as discussed below. In contrast, both Co-rich compounds Ce₄Co₃₃Cu and Ce₄Co₃₁Cu₃ display significant magnetization, with the ferromagnetic configuration being favored. Ce possesses a magnetic moment of approximately 0.9 μ _B, oriented opposite to the Co atoms with moments ranging from 1.4 to 1.6 μ _B.

The spin-polarized electronic density of states (DOSs) for the ferromagnetic state obtained from our DFT-PBE calculations are depicted in Figure . The results show that the electronic bands near the Fermi level are predominantly contributed by the d-electrons of Co, with the f-orbitals of Ce located slightly above the Fermi level in all seven compounds. Experimental evidence suggests that Ce⁴⁺, Ce³⁺, and mixed valence are all possible in this ternary system. Current DFT-PBE calculations do not treat the Ce interactions appropriately, and so we cannot draw any conclusions about the valence of the Ce for any of the compounds. The 0th order guess is that the Ce-rich compounds are all nonmagnetic (Ce⁴⁺ or mixed valent), as none of the magnetic structures indicated Ce moments, in contrast to the Co-rich compounds. Further investigation using advanced methods, such as dynamic mean-field theory (DMFT), is highly recommended.

7.

Electronic density of states (DOS) of the five stable (a–e) and two low-energy (f, g) Ce–Co–Cu ternary compounds. The Fermi level is shifted to zero. The red line represents the projected DOS of Ce f-orbitals; the blue line represents the Co d-orbitals, and the green line represents the Cu d-orbitals.

3.3. Dynamic Stability

To assess the dynamic stabilities of the five predicted stable phases and the two Co-rich low-energy phases, we performed phonon calculations for these seven compounds. The Phonopy package , was used with the finite displacement method. Supercells are used so that the lattice parameters in each direction are around 14 Å, and the forces were calculated with displacement along different directions. The DFT calculations adopted the same setups used in the structure optimization, as well as the formation energy calculations discussed above. Force constants were then calculated from the set of forces. Dynamic matrices were built from the force constants; thus, phonon frequencies and eigenvectors were obtained at the specified q points. No imaginary vibrational frequencies were found for three out of five predicted stable structures and the two low-energy structures, indicating that these five structures are dynamically stable at T = 0 K. The phonon dispersions of these five dynamically stable structures are shown in Figure .

8.

Phonon dispersions of the five stable and low-energy Ce–Co–Cu ternary compounds that are dynamically stable: (a) Ce₃Co₃Cu, (b) CeCoCu₂, (c) Ce₁₁Co₉Cu, (d) Ce₄Co₃₃Cu, and (e) Ce₄Co₃₁Cu₃.

The Ce₁₀Co₁₁Cu₄ and Ce₁₂Co₇Cu structures did have imaginary frequencies in the harmonic phonon calculation at T = 0 K. To see if these two structures can be stabilized with anharmonic interactions included at finite temperatures, we performed ab initio molecular dynamics (AIMD) at a constant temperature of 500 K and constant zero pressure using the NPT (constant number of particles, pressure, and temperature) ensemble to study their stabilities. A 1 x 4 x 1 supercell (200 atoms) for Ce₁₀Co₁₁Cu₄ and a 1 x 1 x 1 supercell (80 atoms) for Ce₁₂Co₇Cu were used. Only the Γ point was used for sampling the Brillouin zone, with a smaller plane-wave cutoff energy of 300 eV. Both structures were calculated over at least 60 ps. The total energy, pressure, and volume as a function of time at 500 K are shown in Figure . The total energy, pressure, and volume of the two crystals fluctuated around their average values for more than 60 ps. These AIMD simulation results indicate that the structures do not collapse or transform into other structures. We also take the atomic position average of the last 10,000 ionic steps (30 ps) and find that the average structures are the same as the original structures. These results suggest that these two structures can maintain stability at 500 K.

9.

Total energy, pressure, and volume as a function of time at 500 K for (a) Ce₁₀Co₁₁Cu₄ and (b) Ce₁₂Co₇Cu obtained from ab initio molecular dynamics (AIMD) simulations using an NPT ensemble.

4. Summary

In summary, by combining the CGCNN ML approach with first-principles DFT calculations, we predict five new Ce–Co–Cu ternary compounds (Ce₃Co₃Cu, CeCoCu₂, Ce₁₂Co₇Cu, Ce₁₁Co₉Cu, and Ce₁₀Co₁₁Cu₄) that are both thermodynamically and dynamically stable. These compounds are rich in Ce and thus likely not suitable for permanent-magnet applications. However, these compounds might be interesting if the Ce is mixed valent, a problem that cannot be examined at the level of DFT calculations and instead requires further investigation with more advanced electronic structure methods like DMFT. Moreover, our ML-guided approach identifies two Co-rich low-energy compounds (Ce₄Co₃₃Cu and Ce₄Co₃₁Cu₃) that exhibit high magnetizations, and where Ce₄Co₃₁Cu₃ has Cu occupying a Co-dumbbell that likely leads to enhanced magnetic anisotropy. The composition region circled by the red line that encloses these two compounds in Figure might be a promising place to look experimentally for compounds suitable for permanent-magnet applications.

The search for stable and low-energy ternary compounds presents a vast and complex space due to the numerous possible combinations of compositions and crystal structures. Compared to the traditional crystal structure search methods like CALYPSO or genetic algorithms, the primary advantage of our ML approach is computational efficiency. The CGCNN ML approach in our ML-guided framework efficiently navigates this vast space. In just four hours on a standard GPU, CGCNN evaluated the energies and magnetic moments of 854,070 hypothetical Ce–Co–Cu compounds, leading to the identification of 5,719 (∼0.7% of the total) viable structures spanning over 2,522 compositions for further DFT analysis. The efficiency of the ML approach enables us to explore a much wider compositional and structural space than feasible with CALYPSO or the genetic algorithm. The ML approach significantly accelerates the discovery process. However, it is important to note that the CGCNN+DFT approach does not guarantee exhaustive coverage, as some stable and low-energy phases may be overlooked if their structural templates are absent from the CGCNN database. Traditional methods might have advantages in exploring novel structural motifs not represented in the training data. In addition, future studies should also consider disordered structures with partial occupancy. Nonetheless, this ML-guided framework can be applied to any combination of three or four chemical elements and represents a powerful tool for rapidly identifying stable compounds across vast chemical and structural spaces, marking a new paradigm in material design and discovery in the digital era, poised for broad adoption and impact.

While the CGCNN ML prediction and DFT energy calculations in our present study can determine the stability of the predicted phases at T = 0 K, they do not account for temperature-dependent effects, such as melting or phase transitions. The issues of temperature effects on the phase stability and the kinetics of phase selection and growth are important for the experimental synthesis of the predicted phase. Although accurate computational predictions of these properties remain challenging, addressing these issues should be an interesting and important research topic for future computational studies.

A preliminary version of this work was previously made available as a preprint.

The data leading to the findings in this paper and the ML models are available from the authors upon reasonable request.

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.inorgchem.5c00899.

• Crystallographic data of the predicted stable Ce–Co–Cu phases and all structure databases of the Ce–Co–Cu system (PDF)

The authors declare no competing financial interest.