Scale-invariant machine-learning model accelerates the discovery of quaternary chalcogenides with ultralow lattice thermal conductivity

The work targets discovery of new quaternary chalcogenides AMM′Q3 with intrinsically low lattice thermal conductivity (κl), important for thermal barrier coatings and thermoelectric applications where reducing κ boosts ZT. Traditional discovery via trial-and-error or high-throughput DFT with prototype decorations often restricts composition space using charge balance and site-occupancy rules, risking omission of unsuspected stable compounds. The research hypothesis is that a structure-based, scale-invariant ML model can accurately predict formation energies and phase stability from unrelaxed structures, enabling exhaustive screening of a vast compositional space to find new stable AMM′Q3 compounds with ultralow κl.

Past advances include HT-DFT searches leveraging databases (OQMD, Materials Project, AFLOW) to assess phase stability and guide synthesis. Prior HT-DFT in AMM′Q3 identified 628 stable and 852 metastable compounds but used restrictive rules (charge balance, site preferences), limiting exploration (Pal et al., 2021). ML has accelerated discovery in multinary spaces (e.g., elpasolites, Heuslers), with successful DFT validations and experimental realizations. Structure-based graph neural networks (CGCNN, iCGCNN) outperform composition-only models. However, iCGCNN performs best with fully relaxed inputs, limiting use when relaxed structures are unavailable. This motivates a model invariant to input structure scale (volume).

- Composition space generation: Consider 66 metallic elements for A, M, M′ sites and chalcogens Q = S, Se, Te, generating 66^3 × 3 = 823,680 compositions. Initial structures are assigned to seven known AMM′Q3 prototypes, starting with KCuZrSe3. - Scale-invariant ML model: Built upon iCGCNN. Crystal graphs from Voronoi-tessellated structures are normalized to set minimum interatomic distance to 1 and associated with a learnable scale factor s updating through convolution steps to predict relaxed volume. Node and edge embeddings encode atomic and local polyhedral features; many-body correlation terms include rescaling by s to ensure formation-energy predictions are invariant to input volume. The model jointly predicts formation energy and relaxed volume. - Training data: ~430,000 unique inorganic entries from OQMD (formation energies < 5 eV/atom) plus 4,659 AMM′Q3 entries from prior HT-DFT (Pal et al., 2021). Graphs generated from relaxed structures; 20% held out for validation. Implemented using PyTorch; Voronoi via pymatgen. - Performance evaluation: Compared against CGCNN and iCGCNN across four input conditions—(1) fully relaxed, (2) unrelaxed, (3) unrelaxed rescaled to relaxed volume, (4) unrelaxed rescaled to Magpie-predicted volume. - Iterative discovery workflow: 1) Apply ML to 823,680 KCuZrSe3-type compositions → 8,370 ML-stable (Ehull = 0) candidates. 2) Remove radioactives and previously known; DFT on 4,199 unique KCuZrSe3-type candidates. 3) Retain ~1,400 low-energy candidates (Ehull ≤ 50 meV/atom). 4) Generate these 1,400 in remaining six prototypes; ML predicts 800 stable across prototypes. 5) DFT on 800 → 99 DFT-stable (Ehull = 0) and 362 metastable (0 < Ehull ≤ 50 meV/atom). - Phase stability analysis: Construct convex hulls using OQMD; classify by hull distance (stable Ehull = 0; metastable Ehull ≤ 50 meV/atom). - Thermal transport calculations: From the 99 stable compounds, select 14 nonmagnetic semiconducting KCuZrSe3-type compounds. Compute phonons (Phonopy) with 2×2×1 supercells; third-order IFCs via CSLD with cutoff to sixth NN; solve PBTE iteratively in ShengBTE with 12×12×12 q-mesh; report in-plane (κ∥) and cross-plane (κ⊥) components. Analyze group velocities, mode Grüneisen parameters, weighted phase space, and scattering rates; compute elastic moduli and average sound velocities from DFT elastic tensors.

- ML model performance (MAE in meV/atom): • CGCNN: (1) 41.3, (2) 72.2, (3) 48.8, (4) 59.6 • iCGCNN: (1) 30.1, (2) 62.3, (3) 40.2, (4) 49.2 • This work: (1) 42.7, (2) 46.5, (3) 46.5, (4) 46.5 The new model reduces error by ~25% vs iCGCNN for unrelaxed inputs (Condition #2) and by ~55% vs CGCNN. - Discovery outcomes: • DFT-validated: 99 stable and 362 low-energy metastable quaternary chalcogenides (overall discovery success ~11% from 4,199 DFTed compositions). • Prototype distribution among 99 stable: 71 KCuZrSe3; 17 TiCuTiTe3; 7 NaCuTiS3; 2 Eu2CuS3; 2 BaAgErS3; none in BaCuLaS3 or Ba2MnS3. • Chemistry: 23 sulfides, 40 selenides, 36 tellurides; several feature atypical site occupancies (e.g., alkali/alkaline-earth/post-transition metals at M site) and lack nominal charge balance. • Band gaps: 40 of 99 are semiconducting (Eg ≈ 0.53–2.63 eV). - Thermal transport (14 selected semiconductors, KCuZrSe3-type): All exhibit ultralow κ at ≥300 K: κ⊥ ≤ ~1.80 W m−1 K−1 and κ∥ ≤ ~0.50 W m−1 K−1 (e.g., KLiHfS3 κ⊥≈1.80, κ∥≈0.50; CsMgPrTe3 κ⊥≈0.67, κ∥≈0.17 at 300 K), below single-crystal SnSe benchmarks. - Mechanistic insights (e.g., KLiZrSe3): Soft acoustic branches, strong acoustic–optical hybridization, large mode Grüneisen parameters (γ up to ~150 for soft modes), enlarged weighted phase space (>0.1 ps/rad−4 up to ~48 cm−1), and high three-phonon scattering rates (>10 ps−1) underpin ultralow κ. Elastic moduli are small (B≈28 GPa, G≈20 GPa), yielding low average sound velocity (~2.48 km s−1).

The scale-invariant, graph-based ML framework enables accurate formation energy predictions from unrelaxed structures, overcoming a key bottleneck where relaxed volumes are unavailable. This facilitates exhaustive screening of an expansive AMM′Q3 composition space, identifying many previously overlooked stable compositions with atypical site chemistries. DFT validation confirms 99 new stable and 362 metastable compounds, many semiconducting, providing a rich set of candidates for thermoelectrics and related applications. Detailed lattice-dynamics analyses connect structural motifs (layered frameworks, low symmetry distortions) to phonon behavior: soft acoustic modes and strong anharmonicity enhance scattering phase space and rates, driving ultralow κ. The anisotropy (κ⊥ > κ∥) reflects weaker interlayer versus intralayer interactions typical of layered structures. Collectively, the results validate the hypothesis that scale-invariant structural ML can efficiently discover low-κ materials and elucidate structure–property relationships governing thermal transport.

This study introduces a scale-invariant iCGCNN-based ML model that jointly predicts formation energy and relaxed volume from normalized crystal graphs, achieving superior accuracy on unrelaxed inputs compared to existing CGCNN variants. Iterative ML–DFT screening over 823,680 AMM′Q3 candidates yields 99 DFT-stable and 362 metastable compounds, substantially expanding the known chemical space with unconventional site chemistries. First-principles PBTE calculations on 14 randomly selected semiconducting candidates reveal universally ultralow lattice thermal conductivity, rationalized by soft elasticity, strong phonon anharmonicity, and enhanced scattering phase space. Future work should extend the model to account for relaxation-induced stress and ionic-position changes and incorporate higher-order phonon interactions and extrinsic scattering to further refine κ predictions. The dataset and predictions open avenues for experimental synthesis and thermoelectric optimization.

- The ML model is invariant to volume but does not capture structural changes due to stress and ionic-position relaxations; performance may degrade when unrelaxed and relaxed structures differ substantially in geometry. - Training used relaxed structures; extrapolation to unrelaxed inputs assumes learned correlations remain valid. - Thermal conductivity calculations considered only three-phonon scattering; neglect of higher-order anharmonicity and extrinsic scattering (e.g., grain boundaries) likely overestimates κ. - κ analysis was limited to 14 nonmagnetic, semiconducting KCuZrSe3-type compounds; magnetic/metallic cases and other prototypes were not assessed for κ due to computational cost. - Some predicted compositions are not nominally charge-balanced, which may affect experimental synthesizability despite DFT stability indications.