The search for materials with extreme thermal transport properties is crucial for various applications, including thermal barrier coatings, waste-heat recovery, and high-performance computing. Semiconducting materials with ultralow lattice thermal conductivity (κ) are particularly important for thermoelectric (TE) applications, where reducing κ enhances TE efficiency (ZT = S²σT/κ). Traditional trial-and-error methods and even high-throughput (HT) density functional theory (DFT) calculations, while beneficial, have limitations in exploring the vast composition space of multicomponent materials. These limitations include adhering to rules based on known compounds, which may overlook potentially valuable “unsuspected” materials. Machine learning (ML) offers a computationally efficient solution by rapidly screening the phase space for potentially stable compounds. Previous ML studies have successfully predicted new stable compounds in various material families, such as bulk metallic glasses and quaternary Heuslers. This work focuses on exploring the phase space of quaternary chalcogenides (AMM′Q3), a family known for exhibiting low κ and diverse structure types. A previous HT-DFT search identified a large number of stable compounds but explored only a subset of possible compositions due to constraints based on known chemical trends. This study aims to overcome these limitations by employing a novel scale-invariant ML model to explore a much larger compositional space without those constraints.

The literature review section extensively cites previous works on low thermal conductivity materials and their applications (references 1-11), highlighting the importance of finding materials with extreme thermal properties, especially for thermoelectrics. The authors discuss the limitations of traditional and high-throughput DFT methods (references 12-17), emphasizing the need for computational approaches capable of efficiently exploring large compositional spaces. Previous successful applications of ML in materials discovery are reviewed (references 18-29), showcasing the power of ML to accelerate the discovery of novel materials with desirable properties, specifically mentioning examples where ML models predicted and led to the experimental synthesis of new stable bulk metallic glasses and quaternary Heuslers. The authors highlight their previous work on quaternary chalcogenides (AMM'Q3) (reference 45), emphasizing the limitations in exploring the full compositional space due to constraints based on known chemical trends. This sets the stage for the current work which aims to overcome those limitations by utilizing a novel machine learning approach.

The researchers developed a scale-invariant ML model based on the improved crystal graph convolutional neural network (iCGCNN) framework. In iCGCNN, crystal structures are represented as graphs, with nodes representing atoms and edges representing bonds. Node and edge embeddings encode elemental and structural information, respectively. The model iteratively updates these embeddings using convolution functions, capturing many-body correlations. The key innovation is the incorporation of a scale factor (s) representing the minimum interatomic distance, which is iteratively updated during the convolution steps. This scale factor is used to normalize the input crystal structures, making the formation energy predictions independent of input volume. The model's performance was compared to CGCNN and iCGCNN using four different conditions: fully relaxed, unrelaxed, unrelaxed but rescaled to relaxed volume, and unrelaxed but rescaled to volume predicted by Magpie. The results (Table 1) show that the scale-invariant ML model significantly outperforms existing ML methods, especially when unrelaxed crystal structures are used as input. The experimentally known AMM′Q3 compounds crystallize in seven structural prototypes (Fig. 2). The authors generated a vast search space (823,680 compositions) by substituting 66 metallic elements at cation sites (A, M, M′) while keeping the Q site fixed to S, Se, or Te (Fig. 2h). The scale-invariant ML model was used iteratively to predict stable compounds, followed by DFT calculations to validate the predictions. A multi-step process (Fig. 3a) was followed involving generating initial compositions in the KCuZrSe3 structure type, then filtering out compounds with radioactive elements and those previously discovered. DFT calculations were then performed on 4199 compounds to validate predictions, resulting in 1400 low-energy compounds. These compounds were then tested in the other six structure types using the ML model again. Finally, DFT calculations identified 99 stable (Ehull = 0) and 362 low-energy metastable compounds. For thermal transport property analysis, 14 semiconducting and non-magnetic compounds (mostly KCuZrSe3 structure type) were randomly chosen. Their lattice thermal conductivities (κ) were calculated using the Peierls–Boltzmann transport equation (PBTE), incorporating three-phonon scattering rates. The average speed of sound was calculated using bulk and shear moduli obtained from DFT calculations.

The scale-invariant ML model demonstrated a 25% improvement over iCGCNN in predicting formation energies using unrelaxed structures. The study discovered 99 new DFT-stable and 362 low-energy metastable quaternary chalcogenides (AMM′Q3). The distribution of these compounds across seven structure types is shown in Figure 3b. Forty of the 99 stable compounds were semiconducting with band gaps ranging from 0.53 eV to 2.63 eV (Fig. 3c). The elemental distributions of these stable compounds show unique chemical trends compared to previously known AMM′Q3 compounds (Fig. 4). In particular, some compounds are not charge-balanced, and certain sites have unique element occupancies. Thermal transport property calculations for 14 randomly selected stable, semiconducting, and non-magnetic compounds (Fig. 5) revealed ultralow κ (≤1.80 Wm⁻¹K⁻¹ and ≤0.50 Wm⁻¹K⁻¹ at T ≥ 300 K), lower than values reported for single-crystalline SnSe. The analysis of KLiZrSe3 (Fig. 6) reveals that the ultralow κ results from soft acoustic phonon branches leading to low sound velocities, strong hybridization between phonon branches at low frequencies, and large phonon anharmonicity evident in high mode Grüneisen parameters. These factors increase the phonon scattering phase space and scattering rates, contributing to ultralow κ. The presence of low-energy, nearly dispersionless optical phonon branches, similar to rattling phonon branches, also reduces phonon lifetimes. Further analysis of the electronic structure of these compounds revealed flat and dispersive bands and multiple band extrema near the valence and conduction-band edges. This makes these compounds promising candidates for thermoelectric applications.

The findings address the research question by successfully demonstrating the effectiveness of the scale-invariant ML model in accelerating the discovery of novel quaternary chalcogenides. The discovery of 99 stable and 362 low-energy metastable compounds significantly expands the known compositional space of this material family. The ultralow lattice thermal conductivity observed in these compounds demonstrates their potential for TE applications and thermal energy management. The high success rate of the ML-guided discovery (~11%) highlights the efficiency and power of this approach compared to traditional and even previous HT-DFT-based methods. The detailed analysis of the phonon properties of KLiZrSe3 provides a mechanistic understanding of the ultralow thermal conductivity in this family of compounds, emphasizing the roles of soft acoustic phonon branches, strong phonon hybridization, and large anharmonicity. The results contribute significantly to the field of materials science by providing a large set of novel compounds with potentially useful properties for TE and thermal energy management. Future research could focus on experimental synthesis and characterization of these materials to validate the theoretical predictions and assess their practical applications.

This study successfully demonstrates the utility of a scale-invariant machine-learning model in accelerating the discovery of new materials. The model's ability to predict formation energies without relying on DFT-relaxed volumes significantly enhances its efficiency for high-throughput screening. The discovery of 99 new stable quaternary chalcogenides with ultralow thermal conductivity expands the landscape of this material family, presenting opportunities for applications in thermoelectrics and thermal energy management. Future work should focus on experimental synthesis and characterization of these compounds and refining the ML model to account for stress and ionic position changes during crystal structure relaxation.

The study's main limitations include the reliance on DFT calculations for validation, which can be computationally expensive, and the fact that the model doesn't fully account for stress and ionic position changes during crystal relaxation. While the model accounts for volume differences, further improvements might be necessary to accurately capture all aspects of structural relaxation. The thermal transport calculations were limited to three-phonon scattering; the inclusion of higher-order interactions might further refine the results. The selection of 14 compounds for thermal transport analysis was random and might not fully represent the properties of the entire set of discovered compounds.