Half-Heuslers (HH) have a valence electron count of 8 or 18, and are generally represented by the formula ABX, where A and B are usually transition metal elements and X is a leading group element. A typical HH compound is MgAgAs, with a face-centered cubic (FCC) crystal structure, as shown in Figure 2A (Larson et al., 1999; Uher et al., 1999; Fang et al., 2018). HH compounds have high mechanical strength, excellent electrical properties, and good thermal stability (Everhart and Newkirk, 2019); they are considered to be the most promising high-temperature TE material candidates studied in recent years (Fu et al., 2015b).

Starting from the available experimental data, Yang et al. (2008) used ab initio calculations and Boltzmann transport theory under the constant relaxation time approximation to screen more than 100 HH compounds taken from the ICSD and studied the electrical transport properties of 36 HH compounds. The p-type and n-type HH compounds with high power factors are shown in Figures 2B,C. The recommended potential HH compounds were Co-, Rh-, and Fe-based HH compounds for p-type prospective materials, and LaPdBi for n-type, with some subsequently confirmed experimentally (Yang et al., 2008). Up to the present day, the experimental ZT values of HH compounds are 1.0–1.5. For example, for the p-type NbFeSb-based HH compound, a maximum ZT of 1.1 at 1100 K was reported (Figure 2D), with the TE properties optimized by a band engineering approach via the tradeoff between the band effective mass and carrier mobility (Fu et al., 2015b). Heavy-band p-type FeNbSb with doped Hf reached a high ZT of 1.5 (Figure 2E) (Fu et al., 2015a). A p-type TaFeSb-based HH compound demonstrated a record high ZT of ∼1.52 at 973 K, which was identified by screening the family of V¹-VIII-V² HH compounds (with V¹ = V, Nb, or Ta; VIII = Fe, Ru, or Os; and V² = As, Sb, or Bi) (Figure 2F) (Zhu et al., 2019). Guo et al. (2019) screened 95 HH compounds using HTP and predicted six p-type and four n-type unreported promising HH compounds, with the finding that the cooperative effects of high band degeneracy, small deformation potential, light band, and large phonon velocity led to a big power factor.

HH compounds have high lattice thermal conductivity; forming solid solutions can effectively reduce the lattice thermal conductivity (He et al., 2014; Tavassoli et al., 2017; Yu et al., 2017; Dylla et al., 2020). As the formation of solid solutions affects the electronic structure, the electronic structure of HH compounds needs to be studied. Dylla et al. (2020) constructed an orbital phase diagram (Figure 3B) according to the atomic orbital composition of HH valence bands; this technique helps to track the changes in electronic structure through the solid solutions. They used HTP calculations and ML to compare the valence band maximum (VBM) of known HH compounds and discover new chemical guidelines for promoting the highly degenerate W-point to the VBM. The k-point of the VBM is different for different electronic structures, and the VBM may occur at the Γ-point (with a band degeneracy Nv of 1), L-point (Nv = 4), or W-point (Nv = 6), where Nv is the total band degeneracy (Yan et al., 2015).

HTP calculations and ML techniques have been increasingly used for the development of HH compounds in recent years. The lattice thermal conductivity of HH compounds (Carrete et al., 2014) was screened from approximately 79,000 HH entries in the AFLOWLIB.org database (Curtarolo et al., 2012; Taylor et al., 2014). All possible compounds in the periodic table of elements, excluding radioactive elements, were considered for HH compounds. By enthalpy calculations, second-order force constant phonon dispersion calculations, and global thermodynamic stability analysis, 75 stable structures of HH compounds were obtained. Three computational methods were proposed to estimate the thermal conductivity of those 75 compounds. Eventually, they concluded that compounds were most likely to have low thermal conductivity if the average atomic radius of the atoms in structural positions A and B was large. Moreover, they proposed that PtLaSb, RhLaTe, and SbNaSr had low lattice thermal conductivity (less than 5 W/m K). After that, Toher et al. (2014) used the automatic Gibbs library (AGL) to analyze these 107 HH compounds (Figure 3A), and demonstrated that the AGL method was much cheaper computationally compared with the full ab initio approach.

Overall, the HTP method has yielded some success in both electrical and thermal transport performance, and a set of guidelines has been established for the screening and optimization of HH compounds. Starting from the experimental data, in combination with first-principles calculations and ML, HH compounds have achieved ZT values of 1.0–1.5.

Diamond-like compounds are derived from tetrahedrally bonded diamonds. The anions in these compounds occupy an FCC sublattice, and the cations occupy the tetrahedral center. The cations and anions in the unit cell follow the simple valence-octet rule (Xi et al., 2018). Diamond-like compounds can be divided into two types according to the ratio of the number of anions and cations in the compound: a cation/anion ratio of <1 is referred to as a defect diamond-like compound (e.g., In₂Te₃, CdIn₂Se₄), and a cation/anion ratio = 1 is referred to as a normal diamond-like compound (e.g., ZnS, CuInTe₂) (Li et al., 2020). Owing to the flexibility of the cation sublattice, the diamond-like compounds have many possible compositions, including unitary, binary, ternary, and quaternary components or even higher. Diamond-like compounds play an essential role in the field of photovoltaics and infrared detectors. In 2009, the discovery of quaternary Cu₂MSnQ₄ (M = Zn, Cd; Q = S, Se) as promising TE materials attracted intensive study; the ZT reached 0.65 at 700 K (Liu et al., 2009a). Ternary CuGaTe₂ is a high-efficiency TE material with a high ZT of 1.4 at 950 K (Plirdpring et al., 2012), and the ZT value of CuInTe₂ reached 1.18 at 850 K, and displayed good electrical properties and low thermal conductivity (Liu et al., 2012). The advantages of environmentally friendly constituent elements, relatively large Seebeck coefficient, and low thermal conductivity mean that the diamond-like structures are promising TE materials.

In an initial attempt at HTP, Zhang et al. (2014) used a series of ternary diamond-like compounds (Figure 4A) and proposed a pseudocubic approach to search for and design high-performance non-cubic TE materials. Zhang et al. tuned crystal structures to design pseudocubic or cubic-like structure blocks in non-cubic materials, to lead directly to cubic-like degenerate band-edge electronic states (Figure 4B) (Zhang et al., 2014). The diamond-like structure might show increased randomness of the locally irregular tetrahedra through a rationally designed mixing strategy while maintaining the cubic-like degenerate electronic states at the band edge. The ZT had peak values when the crystal field splitting energy, Δ_CF, was close to zero (Figure 4C). The distortion parameter η (η = c/2a) was also proposed as a means to evaluate and quantify the TE performance. When η was close to 1, Δ_CF was close to 0 (Figure 4D). The unity-η rule provided a systematic strategy for guiding the evaluation and optimization of TE diamond-like structures. The parameter η could be easily obtained from lattice parameters a and c; based on this theory, Liu R. et al. (2016) synthesized a series of Cu_1-x-δAg_xInTe₂ structures, obtaining a high ZT of 1.24 from the Cu_0.75Ag_0.2InTe₂ structure (Figure 4E).

In TE materials, it is difficult to calculate the carrier relaxation time, and the accurate calculation of electron–phonon coupling is time-consuming, which is not suitable for HTP calculation (Noffsinger et al., 2010; Chen et al., 2012; Liu et al., 2017). Deformation potential method (Bardeen and Shockley, 1950) are usually used in HTP calculations. However, their accuracy has always been questioned. In 2018, Xi et al. (2018) screened chalcogenides with a diamond-like structure, space group 225, from the MatHub-3d database (Yao et al., 2021) through HTP calculations (Figure 5A). This work reported a new strategy for computing the electronic relaxation time τ, which was efficient enough for HTP calculation and more accurate than the more advanced electron–phonon coupling method. One type of novel compound with a defect-chalcopyrite structure, such as CdIn₂Te₄ and ZnIn₂Te₄, has been proposed a potentially good TE material, and Cd₂Cu₃In₃Te₈ has been experimentally synthesized and proved to have a high ZT, of above 1 (Figure 5B). Following the work of (Xi et al., 2018), Li et al. (2019) used the HTP method to screen the TE performance of ABX₂ compounds with diamond-like structures (Figure 5C). ZnSnSb₂, AgInS₂, AgGaSe₂, AgInSe₂, and LiInTe₂ were predicted to have high ZT values for both n-type and p-type. The aforementioned work achieves the full flow from material design of HTP screening to experimental validation, which is of great significance for the prediction of the novel high-performance TE materials.

Since the discovery of p-type NaCo₂O₄ with high TE performance (Terasaki et al., 1997), many oxides, such as the binary oxide ZnO (Ong et al., 2011) and the perovskite-type oxides SrTiO₃ (Muller et al., 2004), CaMnO₃ (Wang et al., 2009), and LaCoO₃ (Vulchev et al., 2012), have been investigated as TE materials. The high-temperature stability, environmentally friendly properties, and low cost make them suitable for various applications, such TE, photoelectric, catalysis, photocatalytic (Sawada and Nakajima, 2018; Nkwachukwu and Arotiba, 2021), and dielectric materials. Unlike the favored materials in the field of TE, such as SnSe (Wang et al., 2015; Zhao et al., 2016), PbTe (Pei et al., 2011; Xiao and Zhao, 2018), and Cu₂Se (Nunna et al., 2017; Olvera et al., 2017), the ZT values of oxides were not very high (Wang H. C. et al., 2011; Wang J. et al., 2017).

With the help of the materials genome, Chen et al. (2016) analyzed the TE properties of over 48,000 compounds from the MP. The dataset in this work was primarily composed of oxygen-containing compounds, and the majority of the compounds performed better as n-type (Figure 6A). The ZT of p-type oxides reached 1; the ZT of n-type oxides was only approximately 0.3. There has been renewed interest in developing new n-type oxide TEs. Chen et al. also compared the power factors of oxides, tellurides, selenides, and sulfides (Figure 6B), which were estimated using a constant relaxation time approximation. The red lines represent the median computed power factor for different anions; the oxides exhibited the lowest power factors. Additionally, the higher band gaps of the oxides complicated doping. Combining high thermal conductivity with excellent TE performance is challenging for oxides (Chen et al., 2016). Additionally, Garrity (2016) employed HTP first-principles calculations to identify promising n-type transition metal oxides, nitrides, and sulfides from ICSD. At 700 K, the Seebeck coefficient and electronic conductivity of 551 oxides, 53 nitrides, and 25 sulfides were estimated using a constant relaxation time approximation. They discovered that a combination of symmetry-enforced degeneracies, low dimensionality, or accidental degeneracies could result in high power factors for these compounds.

The poor TE performance of oxides has become the principal stumbling block to their widespread use. However, recent discoveries in the field have produced oxide TE materials. Peng et al. (2021) employed the ALKEMIE (Wang G. et al., 2021) platform HTP calculation (see Figure 6C) to screen new high-temperature TE oxides. They discovered a new novel oxide (Mg, Ca) Ta₂O₆ with promising TE properties. CaTa₂O₆ exhibited a band structure similar to SrTiO₃, and MgTa₂O₆ had a higher Seebeck coefficient than SrTiO₃ or CaTa₂O₆. The maximum ZT of MgTa₂O₆ was greater than 1 at 1000 K (Figure 6D), which was higher than SrTiO₃. These compounds were promising TE candidates for high-temperature function. Although the current HT method prediction of oxide performance is less than desirable, the accuracy of the method prediction, which is still under investigation, has potential for improvement.

ML usually obtains factors associated with the target properties by evaluating previous experiments or calculated data; these factors are often called features, fingerprints, or descriptors, and can usually be represented by physical models (Zunger, 2018; Recatala-Gomez et al., 2020). ML can be classified into supervised learning and unsupervised learning. Supervised learning learns a model that maps an input (labeled data) to an output based on an existing dataset. The main types of supervised include regression and classification. Unsupervised learning finds previously unknown patterns of given inputs, and no labels are given to the learning algorithm, leaving it on its own to find a model in its input (Wang et al., 2019). Material databases are usually much smaller and sometimes more diverse than in other fields, such as image processing and industrial manufacturing (de Jong et al., 2016; Zhang and Ling, 2018). Therefore, small databases may affect the prediction accuracy more than expected. Active learning can be used to process small databases. Iterative supervised learning is referred to as an active learning model based on Bayesian optimization. Bayesian optimization is shown to significantly reduce the computational cost of discovering the optimal structure compared to finding an optimal structure by building a regression model (supervised learning) to predict the material properties (Bassman et al., 2018).

In recent years, ML methods, known for their data analysis capabilities, have been successfully applied in TE materials research. Iwasaki et al. (2019) applied an interpretable ML for spin-driven TE materials with anomalous Nernst effects, and discovered a novel spin-driven TE material with large thermopower. Oliynyk et al. (2016) used a random forest algorithm (Tin Kam, 1998) to predict Heusler compounds vs. non-Heusler compounds. To support research, ML algorithmic software to determine the best descriptors has also been proposed. The proposed software includes OMP (orthogonal matching pursuit) (Tropp and Gilbert, 2007) and commercial software EUREQA (Schmidt and Lipson, 2009), LASSO (least absolute shrinkage and selection operator) (Tibshirani, 1996; Ghiringhelli et al., 2015; Ghiringhelli et al., 2017), and SISSO (Sure Independence Screening and Sparsifying Operator) (Ouyang et al., 2018).

Sheng et al. (2020) employed active learning to predict the p-type power factors of diamond-like materials based on the original 158 entries cited previous work (Xi et al., 2018; Li et al., 2019). Then they created a search space of diamond-like materials with 482 entries by exhausting all possible combinations of the cations and anions mentioned above, as shown in Figure 7A. 324 uncalculated materials were used to build a model by ML algorithms. According to the model, two Sigle-model strategies (Top and Random) and one several-model strategy (Query by Committee, including Support Vector Regression (Smola and Schölkopf, 2004), Gradient Boosting Regression (Friedman, 2001), Random Forest Regression (Breiman, 2001), Adaptive Boosting Regression (Freund and Schapire, 1997), and Kernel Ridge Regression (Robert, 2014) ML model) strategies were carried out, as shown in Figure 7A. Top strategy selected the 15 candidates with the highest predicted PFs. Random strategy just recommended 15 candidates at random. Query by Committee strategy selected 15 candidates with large ambiguity, then measured the ambiguity by the variance of the 5 ML models. The recommended 15 candidates in each strategy were verified by HTP autoflow based on TransOpt. The unsatisfactory extrapolation should be added to the DFT database, the convergence conditions are Pearson R is > 0.9 or the number of iterations reaches the set maximum 10. Finally, they found that Query by Committee has the highest extrapolation accuracy; Binary pnictides, vacancy, and small atom-containing chalcogenides were predicted to have large PFs, as shown in Figure 7B. The active learning architecture updates the ML model through external verification, and uses as few verification samples as possible to improve the extrapolation ability of the ML model to the greatest extent. The application of active learning architecture is not only limited to TE materials, but also can be used in other functional materials, which is of great significance to accelerate the discovery of high-performance materials.

The above description is based on the nature of the materials, which can be defined as digital data. However, ML image segmentation is more difficult than digital data. In the field of medical images, ML image segmentation has proven to be effective (Dolezal et al., 2021), but it has not been studied in the TE field. Interestingly, Sheng et al. (2021) first applied an active learning loop with a fully connected neural network aimed at fast and automated image segmentation. The researchers took promising ternary TE materials, using Cu-Sn-S as an example, and combined the HTP synthesis, HTP characterization, and HTP analysis, and post-processed 99 backscattered electron images of these compounds. Two ML image segmentation strategies were proposed. One strategy was to quickly and efficiently post-process the batch of Bayesian images based on supervised learning (Figure 8A), as active learning greatly reduced the workload of manual annotation, and Cu₇Sn₃S₁₀ was identified to have good TE potential. The other strategy was unsupervised learning (Figure 8B), by which the unreported compound Cu_1.6S was found to be a promising TE material.