Interpretable machine-learning strategy for soft-magnetic property and thermal stability in Fe-based metallic glasses

Soft-magnetic Fe-based metallic glasses are important for electrical power applications because they offer low core loss and high saturation flux density. However, a longstanding challenge is the trade-off between soft-magnetic properties (e.g., high Bs) and thermal stability (high Tx), largely because glass-forming elements tend to be anti-magnetic and the mechanisms coupling glass formation with magnetic interactions are complex. Traditional alloy discovery via trial-and-error is inefficient and lacks clear design rules. This study aims to develop an interpretable machine-learning framework that uses intrinsic elemental descriptors to accurately predict Bs and Tx across Fe-based MGs, identify the most influential features, and uncover physically grounded composition–structure–property relationships that can guide the rational design of Fe-based MGs with both high Bs and high Tx.

Prior work established commercial Fe-based soft-magnetic materials and composites such as FINEMET, NANOPERM, and HITPERM developed largely through trial-and-error. Machine learning has accelerated materials discovery in ceramics, high-entropy alloys, and oxides. For MGs, ML has been used mainly to predict glass-forming ability and composition spaces (e.g., SVM models and other classifiers/regressors). A key gap is interpretability: many ML models function as black boxes and do not directly yield mechanistic insights or simple design rules. Recent efforts introduced physically meaningful features and symbolic regression to enhance interpretability in other materials systems. Building on these advances, the present work targets interpretable ML for Fe-based MGs to pinpoint decisive parameters linking intrinsic elemental properties to magnetic performance and thermal stability.

Dataset: 252 Fe-based metallic glass compositions with reported saturation flux density (Bs) and crystallization onset temperature (Tx) were compiled from literature without filtering by processing route. Features: 30 descriptors were constructed from intrinsic elemental properties and composition, including valence electron concentration (VEC), valence electron concentration excluding Fe (VEC1), electronegativity (χ), averaged atomic radius difference (δ), melting point (Tm), and element-type indicators for 25 elements (e.g., B, C, Al, Si, P, Cr, Mn, Fe, Co, Ni, Cu, Zr, etc.). Feature definitions (per alloy, with Ci, r i, Tmi, χi, Ni the concentration, atomic radius, melting point, electronegativity, and valence electron count of element i; CFe, NFe, rFe those of Fe): δ = Σi (ri − rFe); Tm = Σi Ci Tmi; χ = Σi Ci χi; VEC = Σi Gi Ni; VEC1 = (Σi Gi Ni) − CFe NFe. Data scaling was applied to bring features to similar ranges (e.g., Tm' = (Tm − 800)/1000; δ' = δ × 100). Model: An XGBoost regression model (scikit-learn/XGBoost) was trained to predict Bs and Tx. Five-fold cross-validation was used. Hyperparameters were optimized over Test Size (0.1–0.5) and Max Depth (2–5). A weighted performance metric R^2 = 0.6 R^2_Bs + 0.4 R^2_Tx emphasized Bs. Optimal settings were Test Size = 0.2 and Max Depth = 3. To reduce dimensionality and enhance interpretability, recursive feature elimination was performed by iteratively removing the least important feature and retraining; the best performance was achieved using 14 features. Overfitting was mitigated via XGBoost’s regularization, shrinkage, and feature subsampling. Feature importance from the trained model was used to identify decisive descriptors for each target. Model equations and training objective followed standard XGBoost formulations with regularized objective, second-order Taylor approximation for loss, and gain-based split scoring. Experimental validation: Guided by ML-derived rules, Fe-based amorphous ribbons (e.g., Fe–B–Si–Zr and Fe–B–Si–Ta–Zr systems) were produced by arc melting followed by melt spinning (Cu wheel ~40 m/s). Structures were confirmed amorphous by XRD (Cu Kα). Tx was measured by DSC at 0.33 K s−1. Magnetic hysteresis (−800 to 800 kA m−1) and Bs were measured by VSM; densities were obtained by Archimedes’ method; samples were stress-relief annealed 100 K below Tx prior to VSM.

- Predictive performance: Test-set R^2 reached 0.934 for Bs and 0.947 for Tx with optimized XGBoost and 14 key features. - Key features: For Bs, VEC1 (valence electron concentration excluding Fe) was the most important feature, followed closely by VEC. For Tx, δ (averaged atomic radius difference) and VEC were dominant. - Design rules derived from data: • For Fe-based MGs without Co and Ni, Bs shows a strong linear relationship with VEC1: Bs = 2.32 − 0.998 × VEC1 (R^2 ≈ 0.70 for that subset). The intercept (~2.32) aligns with the magnetic moment of pure Fe (~2.2 μB), consistent with magnetic valence and charge-transfer models. • Tx exhibits a planar linear dependence on δ and VEC across the dataset: Tx = 1518.5 + 27.1 × δ − 123.7 × VEC. Thus, larger atomic size mismatch (higher δ) increases Tx, while higher VEC decreases Tx. - Physical interpretation: • Bs dependence arises from charge transfer from metalloid s/p electrons to Fe minority-spin bands, reducing Fe magnetic moments (consistent with Slater–Pauling and magnetic valence theory). • Tx increases with δ due to stabilization of dense random packing and hindered diffusion; Tx decreases with VEC due to stronger chemical short-range order promoting heterogeneous nucleation. - Experimental validation: Newly designed alloys achieved high performance, e.g., Fe73.8B15.79Si6.9Ta0.75Zr2.76 with Bs = 1.34 T and Tx = 865 K; Fe82.55B13.79Si0.9Zr2.76 with Bs > 1.6 T and Tx = 738 K. Predicted values using the derived equations showed errors <10% for Bs and <5% for Tx. - Model optimization insights: Weighting metric R^2 = 0.6 R^2_Bs + 0.4 R^2_Tx prioritized magnetic performance; optimal hyperparameters were Test Size = 0.2, Max Depth = 3; recursive elimination indicated 14 features maximize predictive accuracy.

The study addresses the central challenge of simultaneously achieving high Bs and high Tx in Fe-based MGs by providing interpretable, physics-informed design criteria derived from ML. The dominance of VEC1 and VEC for Bs connects composition to magnetic moment via charge-transfer and Slater–Pauling principles: metalloids donate s/p electrons to Fe minority-spin bands, decreasing Fe magnetic moments and thus Bs. The simple linear rule Bs = 2.32 − 0.998×VEC1 captures this effect for Fe-based MGs without Co/Ni, enabling rapid screening for high Bs by minimizing VEC contributions from non-Fe elements. For thermal stability, δ enhances topological packing and suppresses diffusion, raising Tx, while higher VEC strengthens chemical short-range order that acts as heterogeneous nucleation sites, reducing Tx. The plane Tx = 1518.5 + 27.1×δ − 123.7×VEC quantifies the competing effects and provides a practical selection guideline: choose alloying elements that increase atomic size mismatch (e.g., small B, C, Si, P or large Hf, Ta, Y) and reduce VEC (e.g., low-valence elements) to elevate Tx. The experimentally validated compositions confirm that these interpretable rules can guide alloy design beyond black-box prediction, balancing magnetic and thermal performance efficiently.

This work introduces an interpretable XGBoost-based framework that accurately predicts Bs and Tx in Fe-based metallic glasses and, crucially, distills simple, physics-consistent design rules: Bs correlates linearly with VEC1 (for Co/Ni-free Fe-based MGs), and Tx depends linearly on δ (positive) and VEC (negative). Using these rules, new Fe-based MGs with high Tx (>800 K) and competitive Bs (>1.4 T) were designed and experimentally validated, with prediction errors below 10% (Bs) and 5% (Tx). The approach demonstrates how ML can bridge composition–property relationships and fundamental mechanisms, accelerating the design of high-performance magnetic amorphous materials. Future work could expand datasets, explore broader composition spaces (including controlled Co/Ni additions), and integrate high-throughput experiments to further refine and generalize the descriptors and models.

- The linear Bs relation (Bs = 2.32 − 0.998×VEC1) is validated for Fe-based MGs without Co and Ni; applicability to Co/Ni-containing systems is not established. - The dataset (252 alloys) is compiled from literature without harmonizing processing histories, which may introduce variability affecting generalizability. - Validation experiments covered a limited set of alloy systems (primarily Fe–B–Si–Zr/Ta–Zr ribbons); broader experimental verification would strengthen conclusions.