A Generalizable Machine Learning Framework for Half-Heusler Thermoelectrics

Shoeb Athar; Adrien Mecibah; Philippe Jund

cond-mat.mtrl-sciFeb 1, 2026

A Generalizable Machine Learning Framework for Half-Heusler Thermoelectrics

Authors: Shoeb Athar, Adrien Mecibah, Philippe Jund

Organizations: ICGM, Univ Montpellier, CNRS, ENSCM, 34293 Montpellier, France

Abstract

Machine learning (ML) has emerged as a powerful approach for accelerating the discovery of thermoelectric (TE) materials, yet many models suffer from limited generalizability. Here we present a robust ML framework for predicting the thermoelectric figure of merit (zT) in half-Heusler (hH) compounds, emphasizing prediction of previously unseen compositions. A principal component analysis (PCA)-based train/test splitting strategy is introduced to generate representative partitions of chemical space. Bayesian hyperparameter optimization is combined with correlation-based k-best feature selection for Random Forest, XGBoost, and Neural Network models, together with the interpretable SISSO symbolic-regression method. Relative to the full set of 118 input features, the optimized feature subsets reduce the test RMSE by 16-27%. SHAP and SISSO analyses identify temperature, A-site dopant concentration, and A-site heat of vaporization as the most important descriptors of zT. The framework is validated through prediction of previously unexplored compositions, recovery of literature-reported optimal zT values for established hH systems, and first-principles calculations for the new compounds CeCoBi and CeIrBi, which exhibit favorable electronic transport properties. The framework is further applied to high-throughput screening of the doped hH chemical space, covering about 2.46 million elemental substitution combinations across 631 thermodynamically screened parent matrices under stability constraints. This screening yields a concise set of promising candidates for future study. Overall, the work establishes a generalizable workflow for reliable data-driven thermoelectric materials discovery.

Figures & tables

Figure 1: a) Illustration in 2D of the ordering of the materials by increasing distance from the centroid of the PCA space. b) Distribution of the materials in the training (Tr) or test (Te) sets for all the folds, based on the order defined in a).

Figure 2: Two-dimensional t-SNE projection of each fold’s test set

Figure 3: Ranking of the elemental features by absolute Pearson coefficient

Figure 4: (a) Pearson correlation matrix of five sample elemental features of the A-site elements; (b) Evolution of the number of features $k_{\mathrm{max}}$ retained as a function of the $r_{\mathrm{min}}$ .

Figure 5: a) Optimization of the XGBoost hyper-parameters with the features set by the pair $(r_{\mathrm{min}},k)=(0.4,41)$ ; b) $k$ -best selection: the pair $(r_{\mathrm{min}},k)=(0.4,41)$ leads to a validation RMSE of $0.152$ ; c) Effect of $k_{\mathrm{best}}$ over the use of $k_{\mathrm{max}}$ : decrease of the validation RMSE and of the threshold $r_{\mathrm{min}}$ leading to a decreased number of features.

Figure 6: Evolution of the RMSE of the SISSO model during the iterative variable selection process

Elemental Features (EFs)	Number of occurrences	Fold
T	5	F1, F2, F3, F4, F5
$x_{A^{\prime}}$	3	F2, F3, F5
$HV_{A}$	3	F1, F4, F5
$HF_{C}$	3	F1, F3, F5
$ea_{A}$	2	F1, F3
$HF_{A}$	2	F2, F5

Table 1: Number of occurrences of elemental features in the final SISSO descriptors across all the folds

Metric	Fold	SISSO	NN	XGB	RF
RMSE	1	0.158	0.149	0.132	0.154
	2	0.170	0.198	0.194	0.193
	3	0.156	0.133	0.146	0.145
	4	0.176	0.142	0.168	0.175
	5	0.153	0.183	0.181	0.174
	Overall	0.163	0.164	0.166	0.169

Table 2: Fold-wise and overall predictive performance metrics of the ML techniques evaluated on the recombined test dataset.

Figure 7: Parity plots of predicted $zT$ ( $zT^{\mathrm{pr}}$ ) vs experimental $zT$ ( $zT^{\mathrm{xp}}$ ) for each ML technique.

Material	$T$ (K)	$zT^{\mathrm{xp}}$ [Ref.]	SISSO	NN	XGB	RF	Fold
Nb 0.88 Hf 0.12 FeSb 0.88 Sn 0.12	973	0.822 Mohamed et al. (2020)	0.716	0.703	0.976	0.948	1
ZrNiSn	784	0.478 Gong et al. (2019)	0.343	0.349	0.245	0.267	1
TiNiSn	753	0.376 Kim et al. (2007)	0.508	0.485	0.530	0.568	2
Nb 0.88 Hf 0.12 FeSb	1200	1.450 Fu et al. (2015)	0.960	0.726	0.783	0.634	2
Ta 0.84 Ti 0.16 FeSb	970	1.390 Zhu et al. (2019b)	0.934	0.898	0.748	0.747	3
Hf 0.6 Zr 0.4 CoSb 0.8 Sn 0.2	1123	0.856 Hu et al. (2018)	0.817	0.944	0.822	0.832	3

Table 3: Comparison of predicted and literature-reported

zT

at the optimal dopant concentration and temperature for ten previously published half-Heusler systems, one representative composition per cross-validation fold. Predictions are evaluated at the exact (composition,

T

) pair reported as optimal in the cited source; each system was held out of training in the fold indicated.

Fold	Parameters	RMSE (Train/test)	Descriptors
F1	$c_{1}=0.962$ $c_{2}=-0.001$ $A=0.141$	0.158/0.158	$d_{1}=T\times{AR}_{A^{\prime}}\times{HV}_{A}\times\exp\!\left(\dfrac{{HF}_{C}}{{mp}_{A^{\prime}}}\right)$ $d_{2}=\dfrac{T}{{ea}_{A}\times\displaystyle\left(\dfrac{\mathrm{HF}_{C}}{{HV}_{A}}-\operatorname{scd}\!\bigl({ea}_{C^{\prime}}\bigr)\right)}$
F2	$c_{1}=5.04$ $c_{2}=1.13$ $A=0.026$	0.131/0.170	$d_{1}=T\times{AR}_{C}\times{ea}_{C^{\prime}}\,\times\sqrt[3]{\lvert{HF}_{A}-{HF}_{C}\rvert}$ $d_{2}=\dfrac{{mp}_{C}\times\operatorname{scd}\!\left(\dfrac{{x}_{A^{\prime}}}{{ea}_{A^{\prime}}}\right)}{{HF}_{A}-{IE}_{B}}$
F3	$c_{1}=0.104$ $c_{2}=0.314$ $A=0.011$	0.144/0.156	$d_{1}=\left(\dfrac{T}{{n}_{B^{\prime}}}\right)^{3}\times\sqrt[3]{\dfrac{{x}_{A^{\prime}}}{\,{ea}_{A^{\prime}}\,}}$ $d_{2}=\dfrac{1}{\exp\!\left(\dfrac{{bp}_{B}}{T}\right)\,\times\sqrt[3]{\lvert{HF}_{C}-ea}_{A}\rvert}$
F4	$c_{1}=0.398$ $c_{2}=-4.43$ $A=-0.132$	0.143/0.176	$d_{1}=\dfrac{T\times{HV}_{A}\times\bigl({Z}_{C^{\prime}}+\lvert{Z}_{A}-{Z}_{A^{\prime}}\rvert\bigr)}{{bp}_{C}}$ $d_{2}=\dfrac{T\times\lvert{Z}_{C}-{n}_{B}\rvert}{\,{Z}_{C}\big/\log\!\bigl({Z}_{C^{\prime}}\bigr)}$
F5	$c_{1}=1.96$ $c_{2}=-0.942$ $A=0.05$	0.142/0.153	$d_{1}=\dfrac{T\times{HV}_{A}\times\sqrt{\lvert{HF}_{A}-{HV}_{C}\rvert}}{{mp}_{A}}$ $d_{2}=\dfrac{\bigl\lvert{HF}_{C}-\bigl({IE}_{B^{\prime}}-{HV}_{A}\bigr)\bigr\rvert}{\exp\!\left(\dfrac{{x}_{A^{\prime}}}{{k}_{A^{\prime}}}\right)}$

Table 4: Final SISSO descriptors of the form

zT^{\mathrm{pr}}=c_{1}d_{1}+c_{2}d_{2}+A

across all the folds

Figure 8: Comparative SHAP global analysis for the test sets for a) RF, b

Model	Composition	Stability	$zT^{\mathrm{pr}}$ (673 K)
SISSO	CeCoBi	0.611	0.370
	NbFeBi	0.699	0.369
	CeIrBi	0.663	0.331
	LuPdBi	0.831	0.300
NN	TaFeBi	0.658	0.422
	HfCoBi	0.844	0.405

Table 5: Top-ranked half-Heusler compositions identified from the Legrain et al. (2018) Legrain et al. (2018) candidate set using the present ML models. The stability values correspond to the predicted probability of adopting the half-Heusler structure, and

zT^{\mathrm{pr}}

denotes our predicted thermoelectric figure of merit at 673 K.

Material	$a$ (Å)	$d_{\mathrm{A-B}}$ (Å)	$d_{\mathrm{A-Bi}}$ (Å)	$V$ (Å 3 )	$E_{g}$ (eV)
CeCoBi [This work]	6.527	2.826	3.264	278.06	0.2
ZrCoBi (2020)	6.190	2.700	3.120	236.96	0.98
CeIrBi [This work]	6.763	2.929	3.381	309.38	0
HfIrBi Hu et al. (2023)	6.460	2.798	3.230	269.58	0

Table 6: Calculated structural parameters of

AB

Bi (

A

: Ce, Hf, Zr;

B

: Co, Ir) half-Heusler alloys.

Figure 9: Calculated total and projected density of states of (a) CeCoBi and (b) CeIrBi; calculated electronic band structures of (c) CeCoBi and (d) CeIrBi.

Figure 10: Calculated Seebeck coefficient, electrical conductivity ( $\tau=10^{-14}$ s), and power factor as a function of temperature for CeCoBi and CeIrBi.

Technique	Composition	$zT_{\mathrm{pr}}$
SISSO a	$\mathrm{Hf}_{0.55}\mathrm{Li}_{0.45}\mathrm{Ni}_{1-x}\mathrm{Zn}_{x}\mathrm{Sb}_{1-y}\mathrm{Bi}_{y}$ ( $0.01\leq x,y\leq 0.5$ )	1.24
	$\mathrm{Ta}_{0.55}\mathrm{Li}_{0.45}\mathrm{Co}_{1-x}\mathrm{Fe}_{x}\mathrm{Sb}_{1-y}\mathrm{Sn}_{y}$ ( $x,y\geq 0.01$ , $x+y<0.2$ )	1.19
	$\mathrm{Nb}_{0.5}\mathrm{Li}_{0.5}\mathrm{Co}_{1-x}\mathrm{Pt}_{x}\mathrm{Sb}_{1-y}\mathrm{Sn}_{y}$ ( $0.01\leq y<x\leq 0.5$ )	1.18
	$\mathrm{Ti}_{0.5}\mathrm{Mg}_{0.5}\mathrm{Ni}_{1-x}\mathrm{Zn}_{x}\mathrm{Ge}_{1-y}\mathrm{Si}_{y}$ ( $0.01\leq x,y\leq 0.5$ )	1.08
NN	$\mathrm{Ta}_{0.75}\mathrm{Li}_{0.25}\mathrm{Fe}_{0.95}\mathrm{Ag}_{0.05}\mathrm{Sb}$	0.77
	$\mathrm{Nb}_{0.8}\mathrm{Li}_{0.2}\mathrm{Fe}_{0.95}\mathrm{Cu}_{0.05}\mathrm{Sb}$	0.75

Table 7: Highest-ranked HTS candidates at 673 K obtained from within-technique and across-technique inverse-error-weighted ensemble predictions.

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Beyond the Black Box: An Interpretable Machine Learning Framework for Predicting Electronic Structure Microdescriptors and Structure-Performance Relationships in Fe-based Catalytic Systems

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The current catalyst discovery and development pipeline for energy-intensive applications like methane conversion remains bottlenecked by expensive trial-and-error experimentation, irreproducible chemical intuition, and a lack of frameworks linking complex catalytic design spaces to performance. This work presents an interpretable machine learning framework that integrates SHAP-based feature importance analysis (Explainable AI) with tree-based ensembles (Random Forest and Bayesian-optimized CatBoost) to characterize Fe-zeolite and oxide-supported catalysts for the partial oxidation of methane (POM). Despite limited data, the framework decodes complex structure-performance relationships by identifying and ranking thermodynamic, structural, and geometric microdescriptors that influence the electronic band gap and govern macroscale performance metrics such as selectivity, activity, and stability. This work explicitly demonstrates that thermodynamic lattice stability and geometric factors are the primary drivers of electronic band gap (a critical proxy for redox reactivity) rather than bulk stoichiometry. Non-linear models achieve an R2 of 0.61 - 0.77, significantly outperforming traditional linear baselines (R2 = 0.32). This workflow provides both a light-weight generalizable methodology and a prioritized list of physical features for accelerated catalyst screening - and these features can subsequently be integrated into microkinetic and reaction engineering models to create digital twins of complex reactor systems and to enable predictive optimization in autonomous R&D laboratories.

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