Finding stable multi-component materials by combining cluster expansion and crystal structure predictions

The study addresses the challenge of predicting low-energy crystal structures for multi-component materials where prior structural prototypes may be unknown and the configurational space is vast. While unary and many binary systems are well charted, only a small fraction of higher-order (ternary, quaternary, quinary) systems have been experimentally explored. Layered materials such as MAX and MAB phases have attracted attention due to their combined metallic/ceramic properties and prospects for chemical exfoliation to 2D derivatives (MXenes and boridenes). Recent discoveries of in-plane chemically ordered i-MAX and i-MAB phases motivate investigation of related boride systems with potential for ordered metal mixing and 2D derivatives. The work focuses on the pseudo-binary quaternary system (MoxSc1−x)2AlB2 (0 ≤ x ≤ 1), motivated by the reported i-MAB Mo4/3Sc2/3AlB2 (R3m) and the need to assess whether cluster expansion (CE) and/or crystal structure prediction (CSP) can efficiently and reliably recover this phase and uncover additional low-energy structures while covering the compositional/structural space more thoroughly and efficiently.

• CSP frameworks (e.g., USPEX) have successfully predicted low-energy/stable phases across binary and ternary borides, but computational cost grows steeply with system complexity (examples include Mo–B, MnB2, W–Cr–B; hexagonal Ti2InB2 discovery).
• CE methods efficiently model configurational energetics on predefined lattices and have been applied to MAX phases for alloy mixing, band gaps, and magnetic interactions. However, CE accuracy depends on appropriate truncation and a suitable parent lattice; previous CE on MAX phases sometimes failed to capture experimentally realized in-plane ordering (e.g., (V2/3Zr1/3)2AlC), likely due to structural differences between input lattices and realized ordered phases and insufficient training.
• For MAB phases, theory-guided mixing on distinct sublattices enabled prediction and subsequent synthesis of out-of-plane ordered o-MABs (e.g., Ti4MoSiB2) and prediction/synthesis of in-plane ordered i-MABs (e.g., Mo4/3Y2/3AlB2, Mo4/3Sc2/3AlB2). Prior work showed structural/lattice rearrangements (e.g., Al Kagomé-like layers) can substantially lower energy, underscoring the need for methods that can capture lattice relaxations beyond a fixed prototype.

Overall approach: Combine CSP and CE with DFT to efficiently search for low-energy structures in (MoxSc1−x)2AlB2.

1. DFT calculations:

• Exchange-correlation: PBE-GGA; PAW method; VASP v5.4.1.
• Plane-wave cutoff: 400 eV.
• Brillouin zone sampling: Monkhorst-Pack scheme (k-point density ~0.04 Å−1).
• Relaxations at 0 K and ambient pressure for all structures.

2. Phase stability evaluation:

• Formation enthalpy ΔHcp at 0 K computed relative to the convex hull of competing phases in the Mo–Sc–Al–B quaternary system.
• Competing phases gathered from OQMD, Materials Project, and Springer Materials. For a given composition, ΔHcp = E(candidate) − E(equilibrium simplex), with stability defined by ΔHcp < 0.

3. Crystal Structure Prediction (USPEX):

• Ternary endpoints (Mo2AlB2, Sc2AlB2): initial 200 random structures; subsequent generations of 40 structures.
• Quaternary variable-composition (MoxSc1−x)2AlB2: initial 800 random structures; subsequent generations of 100 structures; 60% elite selection per generation; variation operators per generation: 40% heredity, 15% lattice mutation, 15% transmutation, 15% soft-mode mutation, 15% random.
• Seeding strategy: incorporate low-energy CE ground-state configurations as seeds into USPEX after a defined number of generations to accelerate convergence toward low-energy basins.

4. Cluster Expansion (CLEASE):

• Parent lattices from CSP-identified low-energy ternaries: Pmmm, P4/mbm, Cmmm, P6m2 M2AlB2 prototypes.
• CE models constructed on Mo/Sc mixing on the M sublattice; Al and B treated as spectator sublattices; lattice constants set to averages of relaxed Mo2AlB2 and Sc2AlB2 for each symmetry.
• Configuration generation methods considered: random, ground-state search (simulated annealing-based), and probe. Ground-state search chosen for efficient discovery of low-energy configurations.
• Training protocol: start with 25 random configurations per model; then iteratively add at least 100 ground-state configurations, generating 25–50 new configurations per generation. Structure sizes up to ~50 atoms (random) and up to ~100 atoms (ground-state-generated).

5. Workflow integration:

• Step 1: CSP on ternary subsystems to identify plausible low-energy parent lattices.
• Step 2: CE on pseudo-binary (MoxSc1−x)2AlB2 using those lattices to explore mixing and identify low-energy configurations.
• Step 3: Use selected low-energy CE configurations as seeds in a variable-composition CSP for the full (MoxSc1−x)2AlB2 space to discover stable structures and reduce computational cost compared to unseeded CSP.
• Stability and structural characterization performed with DFT and ΔHcp analysis.

• CSP on ternary endpoints identified low-energy structures: • Mo2AlB2: P4/mbm (ΔHcp = 5 meV/atom, close to stable) as the lowest-energy; Cmmm at 16 meV/atom; P6m2 at 89 meV/atom; Pmmm at 148 meV/atom. • Sc2AlB2: Pmmm (ΔHcp = 22 meV/atom) and Cmmm (42 meV/atom) close to stable; P6m2 at 98 meV/atom; P4/mbm at 139 meV/atom. • Identified two low-energy ternary MAB structures not previously established for these compositions: Sc2AlB2 (Pmmm) and Mo2AlB2 (P4/mbm).

• CE exploration using four parent symmetries (P4/mbm, Pmmm, Cmmm, P6m2): • P6m2 and Pmmm show mixing tendencies (ΔHiso < 0) for Mo/Sc on M sites; P4/mbm and Cmmm mostly do not. • When evaluated against competing phases (ΔHcp), P4/mbm and Cmmm yield even fewer near-stable configurations; P6m2 yields multiple low-energy cases, with 6 structures stable in 0.5 ≤ x ≤ 0.67. • Applying a ΔHcp < 10 meV/atom cutoff selected 15 structures for further analysis: 1 P4/mbm (x ≈ 1) and 14 derived from P6m2. • Three lowest-ΔHcp P6m2-derived structures: x = 0.50 (ΔHcp = −12 meV/atom), x = 0.56 (−11 meV/atom), x = 0.67 (−30 meV/atom). The x = 0.67 structure shows in-plane Mo/Sc ordering and Al rearranged into a Kagomé lattice, identified as the known i-MAB motif driver for energy lowering.

• Variable-composition CSP on (MoxSc1−x)2AlB2 and impact of CE seeding: • Unseeded CSP at x = 0.67 after 135 generations found orthorhombic Amm2 with ΔHcp = +29 meV/atom (metastable). • Seeding with CE ground-state configurations (introduced after 5, 25, or 50 generations) led to discovery of a stable R3m structure at x = 0.67 with ΔHcp = −30 meV/atom (Mo4Sc2Al3B6), matching the experimentally reported i-MAB (Mo4/3Sc2/3AlB2) ordering. • A second stable phase was predicted at x = 0.33: Sc4Mo2Al3B6 with space group R3 and ΔHcp = −18 meV/atom. • Across the composition range, several additional low-energy structures close to stability were identified, demonstrating the efficacy of the combined CE+CSP approach.

• Overall, the combined workflow successfully verified the known i-MAB phase and predicted additional stable/near-stable structures while significantly reducing CSP search cost through informed seeding.

The findings show that CE and CSP have complementary strengths: CE efficiently explores configurational energetics on a specified lattice to generate low-energy candidates, while CSP can discover structurally diverse minima without prior structural assumptions but at higher computational cost. CE alone can miss low-energy basins if the true ground states involve lattice rearrangements (e.g., Al rearranging into a Kagomé lattice coupled to in-plane metal ordering), and CSP alone may require many generations/relaxations to reach the correct minima. By first using CSP on ternary endpoints to establish plausible parent lattices, then CE to explore Mo/Sc mixing and identify low-energy configurations, and finally using those as seeds in variable-composition CSP, the authors efficiently navigated the energy landscape of the quaternary (MoxSc1−x)2AlB2 system. This strategy recovered the experimentally known i-MAB at x = 0.67 (R3m) and predicted a new stable phase at x = 0.33 (R3), with several additional compositions close to stability. The results demonstrate that lattice relaxations in spectator sublattices (e.g., Al layer transforming from hexagonal to Kagomé-like due to Sc displacement) are critical contributions to stability and can be captured when CSP is guided by CE-generated seeds. The approach is broadly relevant for multi-component materials discovery where configurational, structural, and sublattice-relaxation effects interplay.

This work establishes an efficient, generalizable framework that combines cluster expansion and crystal structure prediction to identify low-energy crystal structures in complex, multi-component systems. Applied to (MoxSc1−x)2AlB2, the approach: (1) identified low-energy ternary endpoints (Mo2AlB2 P4/mbm; Sc2AlB2 Pmmm), (2) mapped mixing tendencies via CE and predicted P6m2-derived low-energy configurations including an in-plane ordered x = 0.67 structure with Kagomé Al, and (3) verified, via seeded variable-composition CSP, the experimentally known i-MAB Mo4/3Sc2/3AlB2 (R3m) and predicted a new stable Sc4Mo2Al3B6 (R3), along with additional near-stable structures. The CE-seeded CSP significantly reduced computational effort versus unseeded CSP while improving discovery of correct ground states. Future work can apply this iterative CE+CSP strategy to other n-component systems by first exploring n−1 subsystems, constructing CE models on low-energy prototypes, and then running seeded variable-composition CSP on the full n-component space to uncover stable and synthesizable materials.

• CE requires an a priori parent lattice and may miss low-energy basins involving substantial lattice rearrangements or symmetry changes in spectator sublattices (e.g., Al Kagomé formation), unless appropriately captured in the training set.
• CSP without seeding is computationally expensive for high-dimensional composition/structure spaces and may converge slowly to true ground states.
• Stability assessments are performed at 0 K and ambient pressure based on DFT-PBE energetics; finite-temperature effects (vibrational/entropic), zero-point energy, and kinetic factors are not included.
• Competing phases are drawn from databases; any incompleteness or DFT inaccuracies in those references may affect convex-hull construction and ΔHcp values.
• The CE models averaged lattice constants from endpoints and considered Al and B as spectator sublattices; this approximation may not capture all coupling between sublattices across compositions.