Machine-learning-guided inverse design of lead-free relaxors enabled by multimodal literature mining

Abstract

Lead-free relaxors combining high permittivity with thermal robustness are essential for integrated electronics and power systems operating at elevated temperatures. However, their development remains fundamentally hindered by inherent property trade-offs and the immense chemical search space. Here, we report an inverse-design framework integrating multimodal literature mining with physics-guided descriptor-based deep ensemble learning for systematic relaxor screening. By screening a combinatorial space exceeding 150 million candidates, we identify the (Sr_0.48Na_0.26Bi_0.26)(Ti_1−xSn_x)O₃ (0 ≤ x ≤ 0.02) compositions as a low-complexity composition window that satisfies a sparsely populated tri-target dielectric-property regime. Experimental validation confirms that compositions with x = 0.01 and 0.02 meet the predefined design targets for dielectric stability while maintaining a room temperature permittivity above 3300. This stability is further reflected in the simultaneous satisfaction of the upper-temperature-side X5R/X6R/X7R criteria. Atomic-scale characterization reveals that octahedral framework expansion induces polar heterogeneity, underlying the observed dielectric performance. Our results establish an experimentally validated down-selection framework for navigating complex relaxor design spaces and accelerating the data-driven discovery of temperature-stable lead-free dielectrics.

The authors use literature mining and machine learning to identify lead-free ceramic dielectrics that maintain high permittivity and stable performance across a broad temperature range.

Introduction

Dielectric materials that can maintain stable performance over broad temperature windows are increasingly required for next-generation electronic systems used in electric vehicles, aerospace platforms, and other high-reliability environments^1,2. Relaxor ferroelectrics are particularly attractive in this context because their diffuse phase transitions and nanoscale polar heterogeneity enable broad temperature tolerance, high permittivity, and versatile electroresponsive behavior, which has led to their widespread use in multilayer ceramic capacitors (MLCCs), pulse-power components, and electromechanical devices^3,4. In contrast to normal ferroelectrics that exhibit sharp dielectric anomalies near the Curie temperature, relaxors show gradual polarization dynamics governed by polar nanoregions (PNRs) and local structural disorder^5,6. Despite sustained research efforts, however, the discovery of high-performance lead-free relaxors remains largely empirical, and trial-and-error optimization is poorly suited to navigating the vast compositional landscape while simultaneously satisfying high permittivity and robust thermal stability near room temperature^7,8.

Inverse-design strategies provide a more systematic alternative by first defining a target property window and then identifying chemistries most likely to satisfy it^9,10. In ferroelectrics and dielectric ceramics, data-driven workflows have already accelerated discovery through generative design, high-throughput screening, and machine-learning-guided materials selection^11–13. Recent studies span ML screening for high-T_C ferroelectric perovskites¹¹, literature-scale curation and visualization of dielectric data¹², family-restricted ε–T prediction for BaTiO₃-derived systems¹⁴, NLP-based phase-diagram mining¹⁵, Landau-model-assisted multi-property ferroelectric design⁸, and ML-guided optimization of relaxor dielectrics for capacitive energy storage^16,17. Representative data-driven studies in dielectric materials are summarized in Table 1.

Table 1.

Overview of representative data-driven approaches in dielectric materials research

Study type	Representative study	Target/output	Key distinction
Data infrastructure	Murata et al.12	Literature-scale curation and visualization of dielectric data	Enables data organization; no candidate generation
Single-objective screening	Balachandran et al.11	ML-guided screening for high-TC ferroelectrics	Optimizes a transition metric rather than the full relaxor window
Physics-informed multi-property design	Yuan et al.8	Landau-assisted design of multiple ferroelectric properties	Requires explicit thermodynamic parameterization; not centered on digitized experimental spectra
Application-specific ML discovery	Li et al.16; Zhu et al.17	ML-guided discovery for capacitive energy storage	Focuses on energy-storage figures of merit rather than dielectric temperature-response behavior
Inverse design of ML-selected relaxor compositions(This work)	Literature-mining-enabled uncertainty-aware screening	Uncertainty-aware inverse design using three experimentally digitized relaxor metrics	Open-space screening with experimental validation

Collectively, these efforts have expanded the design toolkit for dielectric materials, but they remain focused on distinct objectives such as data infrastructure, single-transition screening, family-limited prediction, structural phase boundaries, explicit thermodynamic parameterization, or energy-storage figures of merit. An open-space inverse-design framework directed specifically toward the relaxor window itself remains lacking. In particular, to our knowledge, no prior approach has simultaneously targeted sub-ambient freezing behavior, high usable permittivity, and dielectric stability using experimentally grounded dielectric spectra as the basis for discovery. This gap is especially important for lead-free relaxors, where the relevant dielectric evidence is fragmented across the literature and often reported under inconsistent measurement and plotting conventions, making direct comparison difficult even before modeling begins¹².

A practically relevant relaxor candidate for room-temperature operation cannot be judged by a single metric such as T_f or T_C alone. Instead, an effective strategy is to suppress T_f below ambient conditions, retain high permittivity in that regime, and avoid sharp thermal collapse across the operating temperature range^18,19. Accordingly, we formulate relaxor discovery as a tri-target problem defined by T_f, ε(@T_f), and a Lorentzian-derived dielectric stability index. These metrics jointly capture whether a composition can preserve high usable permittivity near room temperature while maintaining a diffuse and thermally tolerant dielectric response. This framing is also physically motivated: relaxor behavior emerges from chemically induced local random fields, lattice distortion, and nanoscale polar heterogeneity, such that predictive models must incorporate descriptors reflecting ionic-size variance, charge distribution, and lattice instability rather than relying solely on statistical correlations^8,20–23.

Here, we report an uncertainty-aware inverse-design framework for lead-free relaxors that integrates multimodal literature mining, physics-guided descriptor engineering, deep-ensemble learning, and experimental validation. Using large language models together with programmatic graph digitization, we curated composition–property relationships from 448 prior studies and converted more than a thousand raw ε–T plots into consistent numerical targets²⁴. A descriptor-based ensemble model was then trained to predict the three relaxor-window metrics along with their associated predictive uncertainties, enabling constrained high-throughput screening over a combinatorial space exceeding 150 million compositions. This workflow identified candidate (Sr_0.48Na_0.26Bi_0.26)(Ti_1-xSn_x)O₃ (x = 0.01, 0.02) compositions satisfying a sparsely populated tri-target dielectric-property regime under low-complexity composition constraints, and experimental synthesis together with PFM, Raman spectroscopy, and Cs-corrected STEM confirmed the targeted balance of permittivity and stability^17,25.

Results

Multimodal literature mining and dataset construction

Figure 1 presents the workflow developed for the systematic discovery and validation of high-performance lead-free relaxor ceramics. The workflow integrates multimodal data acquisition, uncertainty-aware property prediction via physics-guided descriptor-based machine learning, and experimental verification.

Fig. 1. Workflow of this study.

a Dataset establishment through LLM-powered literature mining and programmatic graph digitization. b Machine-learning-based framework integrating physics-guided descriptor engineering with a deep ensemble model for robust property prediction. c Discovery pipeline including high-throughput virtual screening to identify top candidates, followed by material synthesis and electrical characterization for experimental validation. The crystal-structure schematic in Fig. 1a was visualized using VESTA (Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr. 44, 1272–1276 (2011)).

The foundation of the framework is a curated dataset constructed via multimodal literature mining (Fig. 1a). To consolidate fragmented dielectric data dispersed across various publications, we established an LLM-assisted curation pipeline to process 448 studies and extract 1202 high-fidelity entries. The resulting dataset encompasses stoichiometric compositions alongside key microstructural and geometric parameters, including average grain size and specimen thickness. To complement textual metadata, raw ε-T plots were programmatically digitized. Application of the Vogel–Fulcher–Tammann relation and Lorentzian functions—commonly employed for relaxor analysis^26,27—enabled systematic conversion of digitized dielectric profiles into consistent numerical descriptors. This strategy extends beyond conventional single-value extraction approaches commonly used in literature data compilation^24,28.

Physics-guided descriptors were derived directly from composition to connect statistical correlations with relaxor physics (Fig. 1b; Supplementary Table 1). These descriptors account for ionic-size variance, electronegativity differences, and configurational entropy, which govern the stability of polar nanoregions (PNRs). Indicators such as A-site 6s² lone-pair proxies and B-site d⁰ orbital fractions were included to reflect local lattice instability and polarization tendencies.

Based on this feature space, a deep ensemble model utilizing multiple multilayer perceptrons (MLPs) was trained for multi-objective prediction²⁹. The ensemble output provides both a predictive mean (μ) and a standard deviation (σ). Crucially, this uncertainty-aware capability highlights regions of high predictive variance, thereby mitigating false positives during extrapolation into unseen chemical spaces.

The final stage involves high-throughput virtual screening (HTS) (Fig. 1c). From a combinatorial space exceeding 150 million potential compositions, approximately 940,000 candidates satisfied the charge-neutrality and structural constraints and were subsequently evaluated by the ensemble model. Candidates satisfying the requirements for sub-ambient T_f, high permittivity and strong dielectric stability were prioritized for synthesis via the solid-state reaction method. The workflow culminates in experimental validation, where dielectric measurements and atomic-scale characterization are used to evaluate model predictions.

Property trade-offs and model performance

Stable control of permittivity at operating temperatures is a prerequisite for electronic device reliability. In relaxor systems, cooling below the freezing temperature (T_f) results in a transition into a non-ergodic state^5,30, which is typically accompanied by a sharp reduction in permittivity and a rise in dielectric loss (Fig. 2a). Suppressing T_f below 298 K is therefore a primary design constraint for maintaining high permittivity and low dissipation near room temperature^31,32. In this framework, T_f was used as the primary descriptor for the freezing boundary because T_m shifts with measurement frequency and a sharp T_c is not uniquely defined in diffuse relaxor transitions. Strong frequency dispersion is therefore not treated as intrinsically detrimental; rather, it becomes undesirable when the freezing regime overlaps the operating window, leading to unstable usable permittivity and loss. The design target consequently seeks to keep T_f below room temperature while retaining high permittivity and broad temperature stability.

Fig. 2. Machine-learning-based design and validation workflow.

a Schematic illustration of dielectric responses for compositions with T_f > 298 K and T_f < 298 K. b 3D scatter plot of reported compositions showing T_f, ε(@T_f), and stability. c 2D scatter map of reported compositions with T_f < 298 K, highlighting the design objective of this work. d–f Parity plots comparing predicted and measured values of T_f, ε(@T_f), and stability, respectively, with corresponding R² values shown in each panel.

Property distributions of the 1202 literature-curated entries were analyzed using three performance metrics: T_f, permittivity at the freezing temperature (ε(@T_f)), and a dielectric stability index (Fig. 2b). Values for T_f were determined via Vogel–Fulcher–Tammann fits, while ε(@T_f) and stability metrics were derived from Lorentzian function parameters. Specifically, the stability index was defined as the ratio of the dielectric peak width (Δ) to its baseline-subtracted peak height (A), where a higher value represents a more diffuse and temperature-insensitive dielectric. The overall distributions of T_f, ε(@T_f), and stability are summarized in Supplementary Fig. 1. Mapping the dataset reveals that while high ε(@T_f) and stability are frequently observed at T_f > 298 K, such combinations are notably absent within our compiled dataset for compositions with sub-ambient freezing temperatures. The non-bold rows of Supplementary Table 2 summarize previously reported compositions meeting ε(@T_f) > 3300 and stability > 0.08, underscoring the limited availability of high-performing candidates below 298 K prior to this work. This statistical gap motivated the establishment of design targets—ε(@T_f) > 3300 and stability > 0.08—to identify potential materials in this sparsely populated tri-target property region (Fig. 2c).

The predictive accuracy of the base MLP model was first established via five-fold cross-validation (R² > 0.8, Fig. 2d–f). To assess robustness against data splitting and model initialization, we then trained a deep ensemble of 30 independent MLPs with different random seeds. Ensemble averaging yielded out-of-bag R² values of 0.816, 0.783, and 0.815 for T_f, ε(@T_f), and log(stability), respectively, while the ensemble standard deviation was used as a sample-wise estimate of predictive uncertainty during screening. Six compositions were synthesized and characterized to experimentally assess the framework (red stars in Fig. 2d–f). Among them, three HTS-selected candidates serve as the primary inverse-designed compositions examined in detail below. The remaining three were synthesized as auxiliary benchmarking compositions to test whether the model reproduces dielectric trends across additional regions of the dataset; their dielectric responses are provided in Supplementary Fig. 2. Across all six compositions, the experimental measurements show reasonable agreement with the predicted values. While the broader literature-derived dataset exhibits increased dispersion—which likely reflects heterogeneous reporting practices and intrinsic variability across studies^24,28—the overall predictive trends provide a sufficient statistical basis for rank-ordering candidates during virtual screening^28,33. To mitigate this point-estimate variability and quantify predictive uncertainty, the framework utilized a deep ensemble approach, thereby prioritizing candidate compositions with a high probability of satisfying the design criteria for robust experimental follow-up³⁴.

Model interpretation and high-throughput screening

Shapley additive explanations (SHAP) analysis was performed to interpret the predictive rationale of the model (Fig. 3a–c)³⁵. A beeswarm plot summarizing the distribution of SHAP values is provided in Supplementary Fig. 3. The results reveal that A-site features—specifically the average ionic radius (r_Aavg), electronegativity (EN_Aavg), and the fraction of 6 s² lone-pair electrons (6s_avg)—exert a stronger influence on T_f, ε(@T_f), and stability than B-site variables. Notably, a strong property trade-off is observed, wherein compositional conditions that enhance ε(@T_f) generally degrade dielectric stability. This indicates that compositional tuning must consider multiple coupled variables rather than simple elemental doping.

Fig. 3. Model interpretation and high-throughput screening for composition design.

a–c Feature importance analysis for predicting T_f, ε(@T_f), and stability, respectively. Red and blue bars indicate features that increase and decrease the target values, respectively. d High-throughput screening workflow illustrating sequential filtering steps as shown in the diagram. e Post hoc family distribution of the 37 retained candidate compositions. ML-predicted temperature-dependent permittivity curves for varying NBT/ST ratios (f) and Sn contents (g).

The 6s_avg feature plays the most critical role in determining T_f. An increased concentration of cations with stereochemically active 6s² lone pairs, such as Bi³⁺ and Pb²⁺, drives T_f to higher temperatures (Fig. 3a)^36–39. Physically, the strong hybridization between the 6s states and oxygen 2p orbitals induces significant local structural asymmetry and spontaneous polarization, thereby promoting the transition from a relaxor state to a long-range ferroelectric order. Accordingly, the SHAP trends suggest that achieving T_f < 298 K is associated with sufficient dilution of the A-site with non-lone-pair cations (e.g., Ba²⁺, Sr²⁺, Ca²⁺) to attenuate the coupling between polar nanoregions (PNRs).

Reducing the concentration of 6s cations to suppress T_f inevitably penalizes the permittivity. The SHAP analysis identifies r_Aavg and EN_Aavg as key compensatory features to overcome this dilemma (Fig. 3b). An enlarged r_Aavg expands the perovskite unit cell volume, providing the necessary “rattling space” for B-site cations to off-center, which enhances the dielectric response^40–42. The SHAP analysis also suggests the contribution of A-site electronegativity (EN_Aavg) to permittivity. A higher electronegativity intensifies the covalency of the A–O bonds, leading to local distortion of the electron cloud. This asymmetric charge distribution is likely associated with enhanced local polarizability, ultimately increasing the dielectric response to external fields⁴³. The consistency between the SHAP attributions and established solid-state chemistry trends indicates that the model has captured physically meaningful relationships rather than merely statistical correlations.

Although grain size has been reported to influence permittivity in perovskite dielectrics^44–47, SHAP attributions in the present model assign comparatively lower importance to microstructural variables than to compositional descriptors (Fig. 3b). This relative ranking should be interpreted cautiously, because microstructural metadata are incomplete in the literature-derived dataset and partially imputed. Within these limitations, the model suggests that compositional tuning governs the primary screening trade-offs, whereas grain size acts as a secondary modulatory variable^48–51. Accordingly, grain size was fixed as a representative boundary condition for synthesis and screening rather than treated as a primary design variable. To navigate the remaining multidimensional complexity, the deep ensemble model comprising 30 individual MLPs was deployed for high-throughput virtual screening (HTS) (Fig. 3d). To ensure high predictive accuracy and maintain compositional simplicity, our search initially prioritized systems with the minimal number of A- and B-site elements. However, preliminary screening indicated that compositions containing no more than two A-site species rarely satisfied all target criteria (Supplementary Table 3). This observation motivated the expansion of the search space to three-component A-site systems coupled with up to two B-site cations (3 × 1 and 3 × 2). Lower-order combinations (e.g., 1 × 2, 2 × 2) are inherently included within this compositional framework. While model predictions indicated that Hf⁴⁺ and Zr⁴⁺ are also effective B-site substituents for inducing relaxor behavior (Supplementary Fig. 4), Sn⁴⁺ was strategically selected for the final screening space. This decision was driven by manufacturability constraints: HfO₂ is economically unfavorable for scalable production, and Zr⁴⁺ substitution significantly increases the required sintering temperature^52–54, which exacerbates the volatilization of Bi³⁺ from the A-site^51,55–57.

Within the 3 × 2 space, approximately 150 million hypothetical compositions were generated. Candidates were first screened through a physics-guided prefilter based on strict charge neutrality (|Δq | <10⁻⁴) and structural stability via the Goldschmidt tolerance factor (0.95 <t < 1.05)⁵⁸, narrowing the search space to approximately 940,000 compositions. Subsequent filtering incorporated the ensemble statistics of the predictive distributions relative to the design targets. The screening criteria required 〈T_f〉 ≤ 298 K, 〈ε(@T_f)〉 ≥ 3,300, and 〈stability〉 ≥ 0.08. In addition, only candidates with relative predictive uncertainty below 30% (σ/μ ≤ 0.3) for each target property were retained to ensure robust model confidence. The introduction of a variance threshold enabled prioritization of compositions with both high performance and high predictive confidence. For the virtual screening stage, the grain size and pellet thickness were fixed at representative values of 1 μm and 1 mm, respectively. The chosen thickness corresponds to the most commonly reported pressed-pellet thickness in the curated literature dataset (Supplementary Fig. 5). This procedure isolated 37 retained candidate compositions (Fig. 3e; Supplementary Table 4). These retained candidates were then grouped post hoc according to compositional family to visualize the distribution of surviving chemistries, revealing the (Sr,Na,Bi)(Ti,Sn)O₃ (SNBTS) family as the most frequently represented cluster. Fine-tuning of the elemental ratios within this post hoc dominant family was subsequently performed using the prediction curves of the model to pinpoint the exact stoichiometry for experimental validation (Fig. 3f, g). For the A-site, increasing the Sr²⁺ fraction effectively suppressed T_f but simultaneously penalized permittivity; the model identified a predicted trade-off point at a Sr fraction of approximately 0.48. On the B-site, Sn⁴⁺ substitution enhanced dielectric stability but further degraded permittivity. The prediction curves indicated that exceeding 3 mol% Sn would substantially increase the probability of the permittivity dropping below the 3300 threshold. Therefore, Sn⁴⁺ was not selected to maximize permittivity alone, but to provide a Pareto-type compromise between improved thermal stability and retained high permittivity. The selected 1–2 mol% Sn range represents the low-doping window in which stability is enhanced while the predicted permittivity remains above the target threshold. Guided by these quantitative parameters, the Sr fraction was fixed at 0.48, and three specific compositions with Sn substitution levels of 0, 1, and 2 mol% were selected for experimental synthesis, hereafter denoted as SNBT, SNBTS1, and SNBTS2, respectively.

Although the NBT–ST parent family and B-site Sn substitution have been reported previously, to the best of our knowledge, the selected low-Sn Sr–Sn composition window is not represented in our compiled NBT–ST–Sn literature map (Supplementary Fig. 6). Thus, the design objective was not to claim an entirely new parent chemistry, but to identify a simple low-doping composition window that occupies the sparsely populated tri-target dielectric-property region while satisfying the tri-target dielectric criteria.

The value of the workflow is therefore not the chemical complexity of the final formula, but the quantitative down-selection that led to it. The search was first constrained to Pb-free and rare-earth-free perovskite chemistries and then filtered by charge neutrality, tolerance factor, target-property thresholds, and predictive uncertainty. This sequence reduced approximately 150 million hypothetical compositions to 37 retained candidates. Model-guided ratio refinement then identified Sr = 0.48 and Sn = 1-2 mol% as the narrow window that balanced competing trends: Sr lowers T_f but penalizes permittivity, whereas Sn improves stability only at low doping before permittivity falls below the target. Thus, the simple SNBTS compositions are presented as the outcome of constrained, experimentally validated down-selection rather than as an exotic new chemistry.

Experimental validation of structural and dielectric properties

The compositions identified by the inverse-design framework were experimentally validated by synthesizing (Sr_0.48Na_0.26Bi_0.26)(Ti_1-xSn_x)O₃ (0 ≤ x ≤ 0.02) ceramics via a solid-state reaction method. XRD patterns of SNBT, SNBTS1, and SNBTS2 (Fig. 4a) are consistent with a single-phase pseudocubic average perovskite structure, with no detectable secondary phases within the detection limit. To clarify the local diffraction changes induced by Sn substitution, magnified XRD profiles near the pseudocubic (200) reflection are shown in Fig. 4b. All three compositions retain a single pseudocubic-like (200) peak without resolvable peak splitting, while the peak shifts progressively toward lower 2θ with increasing Sn content, indicating lattice expansion caused by substitution of Ti⁴⁺ with the larger Sn⁴⁺ ion. Rietveld refinement of the full-range XRD data further supports the single-phase pseudocubic average structure, with a slight increase in the refined pseudocubic lattice parameter and unit-cell volume upon Sn substitution (Supplementary Fig. 7 and Supplementary Table 5). SEM analysis of SNBTS1 (Fig. 4c) shows a dense microstructure with an average grain size of approximately 1 μm; representative images for other compositions are provided in Supplementary Fig. 8. SEM-EDS mapping shows no apparent elemental segregation at the micrometer scale, supporting a chemically uniform distribution of constituent elements (Supplementary Fig. 9 and Supplementary Table 6).

Fig. 4. Experimental validation of selected compositions.

a XRD patterns of SNBT (0 mol% Sn), SNBTS1 (1 mol% Sn), and SNBTS2 (2 mol% Sn). b Magnified XRD profiles near the pseudocubic (200) reflection, showing a progressive low-angle shift with increasing Sn content. c SEM image of SNBTS1 with the corresponding grain-size distribution histogram. Temperature-dependent permittivity and dielectric loss of (d) SNBT, e SNBTS1, and f SNBTS2 measured at different frequencies; dashed lines indicate ML-predicted permittivities. g Comparison of reported lead-free relaxors in the stability-permittivity space. The dashed curve denotes the p95 upper envelope. Three reported points (labels 1–3) above the envelope correspond to Sr_0.7Bi_0.2TiO₃ (SBT)-containing systems (see Supplementary Note 1). h Temperature-dependent permittivity of present samples compared with synthesized BaTiO₃ (BTO) at 0.1 kHz. i Revised Curie–Weiss law plots for SNBT, SNBTS1, and SNBTS2 at 0.1 kHz, with the diffuseness parameter (γ) indicated.

Measurements of the temperature-dependent permittivity and dielectric loss were performed between 100 Hz and 100 kHz (Fig. 4d–f). All compositions exhibit characteristic relaxor behavior, including frequency-dependent dispersion. At 100 Hz, the dielectric permittivity continues to increase as the temperature decreases toward room temperature, consistent with T_f remaining below room temperature for these compositions. To directly visualize the VFT analysis of the synthesized compositions, we extracted the dielectric maximum temperatures (T_m) from the frequency-dependent permittivity curves and fitted their frequency dependence using the VFT relation (Supplementary Fig. 10 and Supplementary Table 7). The fitted freezing temperatures were −9.8, −11.9, and −13.0 °C for SNBT, SNBTS1, and SNBTS2, respectively, confirming that the freezing process remains below room temperature for all three compositions. The corresponding dielectric metrics are listed in the bold rows of Supplementary Table 2. Reconstruction of the dielectric profiles using the three ML-predicted parameters in a Lorentzian function resulted in profiles that quantitatively agree with the experimental data. This agreement supports the ability of the framework to reproduce the overall temperature-dependent trends of complex relaxor systems. For profile reconstruction, T_f, ε(@T_f), and stability were fixed to the ML-predicted values, while ε_∞ was optimized as a single auxiliary baseline parameter to minimize the residual error between the Lorentzian profile and the measured ε–T curve.

Room-temperature dielectric permittivity and loss spectra were also extracted at 25 °C as a function of frequency from 100 Hz to 100 kHz (Supplementary Fig. 11). At 100 Hz, SNBT, SNBTS1, and SNBTS2 exhibit ε values of 3952, 3422, and 3307 with tan δ values of 0.0129, 0.0178, and 0.0120, respectively. These results confirm that the synthesized compositions retain high room-temperature permittivity with moderate dielectric loss in the low-frequency measurement range.

To place the present results in the broader context of reported lead-free relaxors, we constructed an empirical p95 upper envelope to describe the trade-off between permittivity and stability (Fig. 4g). This quantile-based representation reduces sensitivity to extreme observations and provides a more robust estimate of the upper performance boundary⁵⁹. This envelope was obtained by partitioning the log-transformed stability axis into 15 bins, computing the within-bin 95th percentile of ε(@T_f) together with the bin-wise median log-stability, and evaluating the resulting curve on a 400-point x-grid⁶⁰. Within this framework, SNBTS1 and SNBTS2 lie above the p95 upper envelope, while SNBT remains on the envelope, indicating that Sn substitution shifts the system toward the upper tail of the reported distribution.

A small number of reported points (labeled 1–3 in Fig. 4g) are also found above the envelope. These points correspond to Sr_0.7Bi_0.2TiO₃ (SBT)-containing literature systems, whose reported performance appears together with additional structural or processing-related complexity, including A-site deficiency inherited from the SBT-derived component and, in the source studies, impurity formation, local segregation, or process-sensitive microstructural evolution^61,62. By contrast, the present SNBT/SNBTS compositions achieve the targeted dielectric window in a near-stoichiometric and structurally cleaner regime, without detectable secondary phases within the instrumental limit or obvious micrometer-scale elemental segregation, as discussed further in Supplementary Note 1.

SNBTS1 and SNBTS2 satisfy the established design targets of ε(@T_f) > 3300 and stability > 0.08. The observed thermal robustness is further evidenced by the system’s adherence to the simultaneous upper-temperature-side X5R/X6R/X7R criteria, with |Δε/ε₂₅ | <15% at 85, 105, and 125 °C. These results highlight the efficacy of the inverse-design strategy in targeting specific statistical gaps within the established property space.

To benchmark the present compositions against directly related systems, we further compiled 74 reported SNBT/NBT-ST-family entries and ranked them by room-temperature permittivity under identical upper-temperature-side stability criteria (Supplementary Table 8). Values exceeding the corresponding X5R, X6R, or X7R upper-side tolerance were marked in red. This comparison shows that several reported compositions exhibit high room-temperature permittivity but fail one or more of the 85, 105, or 125 °C stability criteria, whereas other temperature-stable compositions show substantially lower room-temperature permittivity. Among the 22 compositions satisfying the simultaneous X5R/X6R/X7R upper-side criteria, SNBTS1 and SNBTS2 provide the highest and second-highest room-temperature permittivity, respectively. This benchmark supports the role of low-level Sn substitution in improving upper-temperature stability while retaining high usable permittivity. The full 74-entry benchmark dataset has been deposited in the Zenodo data repository (10.5281/zenodo.20039899).

Comparison with a BaTiO₃ (BTO) standard illustrates the enhanced temperature stability of the SNBTS system (Fig. 4h). While the synthesized BTO (black solid line) exhibits a sharp dielectric maximum near its Curie temperature (125 °C), the SNBTS samples maintain a stable permittivity near or above 3000 across a significantly broader temperature range. The ML-predicted dielectric profile for BTO, generated using the measured grain size and composition as inputs (red dashed line), shows a high degree of overlap with the experimental BTO response. This indicates that the model captures general dielectric trends even for standard ferroelectric compositions. The larger grain size observed for the BTO reference is consistent with its higher-temperature processing conditions: BTO was calcined at 1100 °C and sintered at 1300 °C for 4 h, whereas the SNBT/SNBTS ceramics were calcined at 900 °C and sintered at 1200 °C for 2 h. Grain size can affect the absolute permittivity of BaTiO₃ through grain-size-dependent domain-wall contributions. However, this difference does not account for the main dielectric contrast observed here: BTO retains a sharp normal-ferroelectric Curie anomaly, whereas SNBT/SNBTS show broad, frequency-dispersive relaxor-like responses arising primarily from compositional disorder, local random fields, and polar heterogeneity. Because BaTiO₃ is widely used as a benchmark dielectric in multilayer ceramic capacitor technologies, it is used here as a representative normal-ferroelectric reference rather than as a grain-size-matched quantitative benchmark.

Finally, analysis using the revised Curie–Weiss law shows an increase in the diffusion exponent (γ) with higher Sn content (Fig. 4i). This trend suggests that Sn⁴⁺ substitution enhances chemical disorder within the lattice, which strengthens the relaxor characteristics of the material^5,63. The diffuseness parameter γ was further compared with representative relaxor ferroelectric systems to assess the degree of diffuse phase-transition behavior (Supplementary Table 9). The extracted γ values of SNBT, SNBTS1, and SNBTS2 are 1.55, 1.65, and 1.72, respectively, lying between the normal-ferroelectric limit of γ ≈ 1 and the ideal diffuse/relaxor limit of γ ≈ 2. This trend confirms that low-level Sn substitution progressively enhances relaxor-like diffuseness. Importantly, γ was not used as a standalone optimization target; the selected compositions retain high usable permittivity while increasing diffuse relaxor character, consistent with the tri-target design objective.

Electric-field-dependent polarization and dynamic domain response

Evaluation of the ferroelectric characteristics was performed by measuring polarization–electric field (P–E) hysteresis loops at various field strengths (20, 40, and 60 kV/cm) and a frequency of 20 Hz. All SNBT and SNBTS compositions exhibited slim hysteresis loops with minimal remanent polarization (P_r) and coercive fields (E_c), consistent with characteristic relaxor ferroelectric behavior (Fig. 5a–c). Comparison with a BaTiO₃ (BTO) standard at 20 kV/cm highlights a clear contrast, as BTO displays the large P_r and E_c typical of a long-range ordered ferroelectric, whereas the SNBTS system maintains a nearly linear response with low hysteresis (Fig. 5d). Increasing the Sn⁴⁺ substitution level resulted in a progressive decrease in the P–E slope, P_r, and E_c (Fig. 5e, f). This trend is consistent with reduced long-range ferroelectric order and enhanced relaxor-like polarization heterogeneity.

Fig. 5. Electric-field-dependent polarization behavior and bias-induced domain response of selected compositions.

Polarization–electric field (P–E) curves of (a) SNBT, b SNBTS1, and c SNBTS2 measured at electric fields of 20, 40, and 60 kV/cm. d P–E curves of SNBT, SNBTS1, SNBTS2, and BaTiO₃ measured at 20 kV/cm for direct comparison. e Remanent polarization (P_r) as a function of electric field for SNBT, SNBTS1, and SNBTS2. f Coercive field (E_c) as a function of electric field for SNBT, SNBTS1, and SNBTS2. Piezoresponse force microscopy (PFM) phase images obtained (left) in the pristine state, (middle) under a DC bias of 10 V, and (right) after bias removal for (g) BTO and h SNBTS2. Phase-angle distribution boxplots for (i) BTO and j SNBTS2 in the pristine state, under 10 V bias, and after bias removal.

The low remanent polarization does not indicate a purely non-polar dielectric state. Rather, it reflects relaxor-like polarization dynamics in which field-induced polar order is largely reversible after removal of the electric field. To further examine the electromechanical response, strain–electric field (S–E) curves were measured for SNBT, SNBTS1, and SNBTS2 at 60 kV/cm (Supplementary Fig. 12). The compositions exhibit narrow, nearly parabolic electrostrictive-like responses without pronounced butterfly-shaped hysteresis, further indicating the absence of stable long-range ferroelectric domain switching. The maximum strains are approximately 0.095%, 0.068%, and 0.070% for SNBT, SNBTS1, and SNBTS2, respectively, corresponding to effective large-signal d₃₃^* values of approximately 158, 114, and 116 pm V⁻¹.

Microscopic visualization of these domain dynamics was achieved through piezoresponse force microscopy (PFM) phase imaging (Fig. 5g, h). The BTO reference sample exhibited a pronounced memory effect: application of a 10 V DC bias shifted the phase contrast toward a positive phase-angle state, and this switched state was largely retained after bias removal (Fig. 5g). This retained phase shift is consistent with stable ferroelectric domain switching in a material with a comparatively large coercive field. In contrast, the SNBTS2 sample exhibited a highly reversible phase response (Fig. 5h). Under the 10 V bias, the phase contrast similarly shifted toward a positive phase-angle state, but it rapidly recovered its initial contrast after removal of the external field, consistent with reduced coercive-field stabilization and the absence of persistent long-range ferroelectric domain switching. Quantitative phase-angle boxplots further clarify this contrast (Fig. 5i, j). For BTO, the median phase angle shifts from −7.5° in the pristine state to 50.5° under 10 V bias and remains at 43.0° after bias removal, indicating substantial retention of the switched polar state. For SNBTS2, the median phase angle shifts from −110.5° to 51.3° under bias but returns to −109.3° after bias removal, confirming that the bias-induced polar response is largely reversible and weakly retained after removal of the external field.

The initial phase offset of approximately −90° observed in the pristine SNBTS2 sample is attributed to a potential internal bias field. This internal field may arise from the work function mismatch between the Ag bottom electrode and the cantilever tip. While BTO exhibits a neutral phase (0°) in its pristine state, presumably due to its significantly higher coercive field (E_c), the SNBTS2 composition displays greater susceptibility to this subtle internal bias. The combination of a relatively low E_c and enhanced polarization sensitivity in SNBTS2 facilitates the alignment of dipoles with the bias field, resulting in the observed phase shift even in the absence of an external poling process. These PFM results provide visual confirmation of the reversible polarization switching observed in the macroscopic P–E loops and demonstrate a polar response that readily recovers after removal of the external field (Fig. 5h). Phase angle histograms corresponding to the PFM measurements are provided in Supplementary Fig. 13.

Atomic-scale origin of polar heterogeneity

Aberration-corrected HAADF-STEM imaging along the [001] zone axis was performed to probe atomic-scale structural features associated with the macroscopic relaxor characteristics (Fig. 6a, e). Lattice fringes corresponding to the (100) and (110) planes were clearly resolved in the HAADF-STEM images, and the corresponding reflections were also identified in the fast Fourier transform (FFT) patterns. Complementary selected-area electron diffraction (SAED) patterns are provided in Supplementary Fig. 14. Atomic column positions were determined using two-dimensional (2D) Gaussian fitting, a protocol previously demonstrated to achieve picometer-level precision in local polarization analysis^64–66. While HAADF-STEM captures a 2D projection, prior multislice simulations have shown that cation displacement vectors observed along the [001] projection reflect the projected polarization orientation within the three-dimensional perovskite lattice¹⁷.

Fig. 6. Atomic-scale displacement mapping and Raman analysis of selected compositions.

SNBT: a HAADF-STEM image with the corresponding FFT pattern (inset). b Map of the A-site cation displacement magnitude obtained from atomic-column position analysis. c In-plane displacement orientation (polar-angle) map, color-coded according to the angle legend. d Histogram of displacement orientation angles measured from the [100] direction. e–h Corresponding HAADF-STEM image/FFT, A-site displacement magnitude map, displacement orientation map, and angle histogram for SNBTS2. i Raman spectra of SNBT, SNBTS1, and SNBTS2; the 400–700 cm^–1 region used for peak fitting is highlighted. j–l Multi-peak deconvolution of the 400–700 cm^–1 Raman band for (j) SNBT, k SNBTS1, and l SNBTS2, showing the measured spectra, overall fits, and individual peak components.

In the SNBT control sample, A-site cations exhibited an average displacement magnitude of approximately 17.1 pm, corresponding to an off-centering of 4.4% relative to the lattice constant (3.9 Å) (Fig. 6b). Polar maps and orientation histograms reveal that the displacements in SNBT possess a preferred alignment near the [$\overline{1}10$] direction, suggesting partial short-range correlation among polar regions (Fig. 6c, d). In contrast, the Sn-doped SNBTS2 sample showed a reduced average displacement magnitude of approximately 14.8 pm (3.8%) and increased directional disorder (Fig. 6f, g). The orientation histogram for SNBTS2 shows a broad distribution of angles with no dominant orientation, indicating that Sn substitution is associated with reduced directional correlation of local polarization (Fig. 6h). Although the analysis is based on a two-dimensional projection along the [001] zone axis, the systematic reduction in displacement magnitude and the broadening of the orientation distribution indicate a statistically meaningful difference between SNBT and SNBTS2. This atomic-scale randomization is consistent with the nearly zero remanent polarization and minimal coercive fields observed in the macroscopic P–E measurements.

To compare the local HAADF-STEM displacement maps with the crystallographic average structure, we evaluated a nominal point-charge polarization using the Rietveld-constrained pseudocubic average model, as described in Supplementary Note 2. The full-pattern refinement supports a pseudocubic average perovskite structure within the resolution of laboratory XRD. Trial refinements allowing polar atomic off-centering did not yield a physically robust average-structure solution; therefore, the atomic coordinates were retained at ideal centrosymmetric pseudocubic positions for the point-charge analysis. Under this conservative average-structure model, the calculated point-charge polarization is negligible by symmetry. This result contrasts with the finite local displacements resolved by HAADF-STEM, indicating that nanoscale polar distortions exist locally but are spatially disordered and largely cancel in the crystallographic average.

Raman spectroscopy was employed to identify potential structural drivers of this displacement randomization by analyzing the vibration modes of the BO₆ octahedra (Fig. 6i). The 400–700 cm^–1 band is associated with the O–B–O stretching and breathing modes of the octahedral framework^67,68. Multi-peak deconvolution reveals a systematic redshift in the centroids of Peaks B and C with increasing Sn content (Fig. 6j–l). This mode of softening is consistent with expansion of the BO₆ octahedra and the lengthening of B–O bonds^19,67, resulting from the substitution of Ti⁴⁺ (0.605 Å) with the larger Sn⁴⁺ cation (0.690 Å)⁶⁹. The resulting expansion of the octahedral framework likely destabilizes the energetic preference for A-site alignment along specific crystallographic axes, thereby facilitating polar heterogeneity. These observations are statistically consistent with Sn-associated octahedral modification and increased polarization disorder, providing a plausible structural basis for the enhanced dielectric stability observed in the SNBTS system.

Overall, these results indicate that the relaxor nature of SNBT, SNBTS1, and SNBTS2 arises from coupled A-site chemical disorder and B-site octahedral perturbation. The mixed Sr²⁺/Na⁺/Bi³⁺ A-site sublattice introduces charge disorder and ionic-size mismatch, producing local random electric and strain fields that disrupt coherent long-range ferroelectric ordering^6,70. Bi³⁺-centered local off-centering can still generate nanoscale polar distortions, but these distortions remain directionally disordered and therefore cancel in the pseudocubic average structure. Low-level Sn⁴⁺ substitution further expands and perturbs the Ti–O octahedral network because Sn⁴⁺ is larger than Ti⁴⁺ and has different B–O bonding character, which broadens local polar environments without stabilizing remanent ferroelectric domains^68–70. The resulting state is therefore a chemically disordered relaxor with high local polarizability but weak macroscopic remanence, consistent with the frequency-dependent dielectric dispersion, sub-ambient VFT freezing, slim P–E loops, electrostrictive-like S–E response, reversible PFM behavior, HAADF-STEM displacement disorder, and Raman mode softening.

Discussion

This work established a machine-learning-driven inverse-design framework that screened a combinatorial space exceeding 150 million candidates, narrowed the search to 37 retained compositions, and identified (Sr_0.48Na_0.26Bi_0.26)(Ti_1−xSn_x)O₃ (0 ≤ x ≤ 0.02) through post hoc family analysis and model-guided refinement. Application of this data-centric approach navigated the multidimensional property trade-offs inherent in perovskite dielectrics by specifically targeting statistical gaps in the literature—particularly the achievement of high permittivity and stability at sub-ambient freezing temperatures. Within this family, the x = 0.01 and 0.02 compositions (SNBTS) satisfied the predefined design objectives, illustrating how ensemble machine learning can direct discovery toward a sparsely populated tri-target composition–property region in the curated literature-derived space.

In addition, the mechanism underlying the performance of the designed system was investigated through a multi-scale analysis. SHAP attributions indicate that the model captures physical trends consistent with established solid-state chemistry, identifying A-site ionic radius, electronegativity, and 6s² lone-pair concentration as dominant features influencing dielectric response. Specifically, the captured relationship between A-site electronegativity and permittivity supports the physical plausibility of the framework, as it aligns with the role of covalency in enhancing local polarizability. The macroscopic relaxor behavior observed in the SNBTS system is further linked to atomic-scale heterogeneity in local cation displacements. Reversible domain switching and low hysteresis observed in P–E measurements are consistent with the polar heterogeneity identified through STEM displacement mapping. Raman spectroscopy suggests that these effects originate from the expansion of the BO₆ octahedral framework upon Sn⁴⁺ substitution, which is associated with improved dielectric stability through suppression of long-range A-site alignment.

The present framework is designed to down-select temperature-stable dielectric candidates from literature-mined dielectric-temperature spectra. Accordingly, the primary model targets were restricted to quantities that are broadly available and consistently extractable across the curated literature corpus: freezing behavior, usable permittivity, and temperature stability. Device-level capacitor metrics such as breakdown strength, DC-bias dependence, energy-storage density, and long-term reliability were not used as primary screening targets because they are reported less consistently and depend strongly on electrode geometry, sample thickness, processing history, and test protocol. Therefore, the present work should be viewed as an experimentally validated materials down-selection step for temperature-stable dielectric response, rather than as a full capacitor qualification. Incorporating standardized device-level metrics into future datasets will enable extension of the framework toward application-oriented capacitor screening.

Methods

Materials database establishment and preprocessing

A comprehensive database of lead-free relaxor dielectrics was established through a multimodal literature mining framework, beginning with a systematic search on Google Scholar using keywords such as “relaxor,” “SEM,” “VFT,” and “ferroelectrics” to select 448 domain-relevant publications. This process yielded a curated dataset of 1,202 entries, where data extraction was facilitated by NotebookLM (Gemini 2.0) using two specialized prompt protocols to ensure high-fidelity representation of compositional and microstructural metadata.

Consistent parsing of complex A₁A₂…B₁B₂…O₃ perovskite stoichiometries was ensured by a fixed protocol that converted reported formulas into a standardized triplet representation—element, partial coefficient, and global mixing coefficient—as detailed in the parsing rules provided in Supplementary Note 3; the resulting frequency distributions of the top 10 A-site and B-site elements in the curated dataset are shown in Supplementary Fig. 15. Simultaneously, microstructural parameters including grain size (GS) and pellet thickness were extracted based on their association with specific figures within each paper using a separate structured prompt protocol; this approach strictly distinguished pressed from sintered thickness and prioritized explicitly reported numerical values over estimates, with the detailed extraction rules described in Supplementary Note 4. This curation workflow was executed between March and August 2025 under consistent prompting conditions, and no discrepancies were identified in an audit of 50 randomly selected papers.

A strict fold-wise preprocessing protocol was implemented to prepare the dataset for machine learning, ensuring that critical missing microstructural entries were explicitly flagged and treated while preventing information leakage. Missing GS values were imputed using the median computed from the training subset of each cross-validation fold, while missing thickness values were replaced with the mean of the corresponding training subset. Furthermore, both the standardization of the processing-feature block and target scaling were fitted exclusively on the training portion of each fold before being applied to the validation data, ensuring rigorous evaluation of the model’s predictive performance.

Standardization of property reporting across heterogeneous studies was achieved by programmatically digitizing raw dielectric-temperature (ε-T) plots. These digitized profiles were transformed into consistent numerical descriptors using two fundamental physical models. The characteristic frequency dispersion and the determination of the freezing temperature (T_f) were governed by the Vogel–Fulcher–Tammann (VFT) relation:

$$f=f_0\exp \left(\frac{-E_a}{k_B\left(T_m-T_f\right)}\right)$$ 1

where f represents the measurement frequency, T_m is the temperature of the maximum permittivity, E_a is the activation energy, and k_B is the Boltzmann constant. The parameters T_f, E_a, and the pre-exponential factor f₀ were determined through non-linear fitting to the experimental frequency-dependent data.

The diffuse phase transition behavior above T_f was further modeled using a Lorentzian function to capture the dielectric profile:

$$\epsilon \left(T\right)={\epsilon }_{\infty }+\frac{A}{1+{\left(\frac{T-T_f}{\Delta }\right)}^2}$$ 2

In this model, A denotes the fitted amplitude relative to the asymptotic baseline permittivity ε_∞, while ε(@T_f) is obtained from the fitted Lorentzian profile. Because the asymptotic baseline permittivity (ε_∞) is not directly fixed from all digitized dielectric profiles, it was treated as an auxiliary fitting parameter during Lorentzian analysis. ε_∞ was iteratively optimized by non-linear least-squares fitting, and the solution yielding the highest goodness-of-fit (equivalently, the lowest residual error) was adopted. Δ represents the characteristic width, or diffuseness, of the dielectric response. Evaluation of thermal stability was performed using a stability index defined as the ratio of these fitted parameters, Δ/A. This curation process finalized each material entry into three primary target variables— T_f, ε(@T_f), and stability—forming the statistical basis for subsequent machine learning analysis.

Physics-guided descriptor engineering

Compositional features were mapped onto the ABO₃ perovskite framework by normalizing site occupancies to satisfy stoichiometric constraints, where A = 1, B = 1, and O = 3. Elemental species were classified into A-site or B-site categories based on a predefined master database containing fundamental atomic properties—including ionic radii, electronegativity (EN), and valence charges—as provided in the Zenodo repository described in the Data Availability section. This structured classification enabled the calculation of site-specific descriptors that govern the stability and dynamics of polar nanoregions (PNRs). This representation addresses the limitations of simple elemental concentration vectors by incorporating the local chemical environment and lattice site symmetry.

The geometric and thermodynamic stability of the perovskite lattice was captured through descriptors derived from ionic radii (r) and configurational entropy. The weighted average ionic radius and its standard deviation for both A and B sites were calculated to reflect lattice distortion and local random fields. Structural stability was further quantified using the Goldschmidt tolerance factor (t):

$$t=\frac{r_{A_{avg}}+r_o}{\sqrt{2}\left(r_{B_{avg}}+r_O\right)}$$ 3

where r_O represents the ionic radius of oxygen (1.40 Å). The complexity of chemical disorder, which is a prerequisite for relaxor behavior, was represented by the configurational entropy (S_config) for each sublattice:

$$S_{\mathrm{config}}=-R\sum p_i\log p_i$$ 4

where $p_i$ is the molar fraction of the i-th element on a specific site.

Electronic and chemical descriptors were engineered to account for polarization tendencies and local bond covalency. Weighted averages and standard deviations for electronegativity (EN) and valence charge were calculated for each site to represent the influence of local chemical environments on polar nanoregion stability. Specifically, A-site 6s² lone-pair proxies and B-site d⁰ orbital fractions were included as critical features, as these electronic configurations are known to drive local lattice instability and spontaneous polarization through hybridization with oxygen 2p states.

The final input feature matrix (X) was constructed by concatenating the normalized composition vectors, 22 high-order physical descriptors, and microstructural parameters (grain size and thickness). To prevent biased learning from varying units, a processing-specific scaling protocol was applied where only the continuous physical descriptors and microstructural variables were standardized, while categorical flags and stoichiometric ratios remained in their original space. This integration of domain-specific chemical knowledge with statistical features ensures that the machine learning model captures physically plausible trends within the complex relaxor landscape.

Machine learning framework

Predictive modeling was performed using a deep ensemble framework designed for multi-objective regression of the three target variables: T_f, ε(@T_f), and the stability index. To ensure numerical stability and account for the wide distribution of experimental values, the stability index was modeled in the logarithmic domain, log(stability), during training, while a safe-exponential transformation was applied for final property recovery.

The core architecture comprised an ensemble of 30 independent multilayer perceptrons (MLPs), each featuring a hidden layer configuration of (128, 64, 32) with Rectified Linear Unit (ReLU) activation functions. Robustness was integrated into the training pipeline through a bagging (bootstrap aggregating) strategy, where each ensemble member was trained on a unique resampled subset of the 1202 entries. During training, the AdamW optimizer was utilized with a weight decay of 10^-4 to mitigate overfitting, further supported by dropout layers and an early stopping criterion based on validation loss.

Reliability of the model was rigorously evaluated through five-fold cross-validation, where the performance was quantified using the coefficient of determination (R²). Five-fold cross-validation was performed at the entry level, with individual database entries randomly assigned to folds irrespective of source paper or compositional family. The ensemble approach enabled the derivation of both the predictive mean (μ) and standard deviation (σ) for each target property, providing a quantitative measure of model uncertainty. This uncertainty-aware prediction was critical for identifying compositional regions where the model maintained high confidence, thereby ensuring the success of subsequent experimental validation for the prioritized candidates.

Virtual high-throughput screening (HTS)

High-throughput virtual screening (HTS) was executed within a combinatorial search space exceeding 150 million hypothetical compositions to identify optimal lead-free relaxor candidates. The screening process targeted 3 × 2 systems, involving combinations of three A-site species (Ba, Sr, Ca, Bi, Na, K) and two B-site species (Ti, Mn, Sn, Fe, Zn, Nb), with stoichiometry generated via a simplex grid algorithm at a resolution of 0.01. Grain size and pressed-pellet thickness were fixed at 1 μm and 1 mm, respectively, to represent the most common conditions in the curated dataset. Initial candidate reduction was performed through a multi-stage physics-guided prefilter, which enforced strict charge neutrality (|Δq | <10^-4) and structural stability via the Goldschmidt tolerance factor (0.95 <t < 1.05). This preliminary screening isolated approximately 940,000 candidates for subsequent property evaluation.

Property inference for the filtered candidates was conducted using the trained ensemble of 30 multilayer perceptrons, providing both predictive means (μ) and standard deviations (σ) for each target variable. Selection criteria were established to target the sparsely populated tri-target property region identified during literature analysis, specifically requiring a freezing temperature (T_f) below 298 K, a permittivity (ε(@T_f)) exceeding 3300, and a stability index over 0.08. To prioritize experimental success and ensure high predictive reliability, we retained only candidates whose relative predictive uncertainty remained below 30% for all three target properties (σ/μ ≤ 0.3 for T_f, ε(@T_f), and stability). This constraint was applied based on the rationale that high variance indicates a lack of sufficient training data in that chemical region, thereby reducing the probability of accurate experimental reproduction.

Experimental procedures

Synthesis of (Sr_0.48Na_0.26Bi_0.26)(Ti_1-xSn_x)O₃ (0 ≤ x ≤ 0.02) and BaTiO₃ (BTO) ceramic samples was conducted via the solid-state reaction method. High-purity precursor powders, including SrCO₃ (≥99.9%, Sigma-Aldrich), Na₂CO₃ (>99%, Daejung), Bi₂O₃ (nanopowder, 99.9%, Thermo Scientific), TiO₂ (nanopowder, ≥99.5%, Sigma-Aldrich), SnO₂ (Daejung), and BaCO₃ (≥99%, Daejung), served as the starting materials. These precursors were weighed according to the target stoichiometry and ball-milled in an ethanol medium for 72 h to ensure thorough mixing. Subsequent to drying, calcination was performed in air for 4 h to initiate phase formation, with SNBTS and BTO processed at 900 °C and 1100 °C, respectively. The resulting powders were subjected to a secondary ball-milling process for 24 h to refine particle size and enhance chemical homogeneity. Green bodies were then formed by pressing the powders into disks with a diameter of 10 mm and a thickness of 1 mm under uniaxial pressure. Final densification was achieved through sintering in air; SNBTS pellets were held at 1200 °C for 2 h, while BTO pellets were sintered at 1300 °C for 4 h.

The final dimensions of the sintered specimens were approximately 8.2 mm in diameter and 0.7 mm in thickness. Specimens were subjected to mechanical polishing, including a final diamond polishing step, ensuring a smooth surface for FE-SEM observations and ferroelectric measurements. Surface preparation for electrical characterization involved the application of high-temperature sintering silver paste (Winner Technology, Gyeonggi-do, Korea) to both faces of the sintered pellets, followed by firing at 710 °C to form conductive electrodes. Temperature-dependent dielectric permittivity and loss tangents were recorded using an LCR meter (Agilent 4263 A) across the specified frequency range. Microstructural observations and surface morphology analysis of the specimens were carried out using field-emission scanning electron microscopy (FE-SEM, Merlin, Zeiss). Crystal structures were analyzed using X-ray diffraction (XRD, D8-Advance, Bruker). The diffraction patterns were recorded in the 2θ range of 20°–80° with a step interval of 0.01° and a scan speed of 2°/min. Polarization-electric field (P–E) hysteresis loops were measured using a ferroelectric test system (Radiant Technologies, Germany) at a frequency of 20 Hz with a sine wave profile. A high-voltage amplifier was utilized to increase the applied voltage up to a maximum of 4200 V during the measurements.

For Rietveld refinement, sintered pellets were crushed into powders and measured using the same Bruker D8-Advance diffractometer with Cu Kα radiation. Full-range powder XRD patterns were collected over a 2θ range of 20°–130° with a step size of 0.01° and a counting time of 0.5 s per point. During measurement, the powder sample was rotated at 15 rpm to improve powder averaging. Rietveld refinement was performed using TOPAS with a cubic perovskite average-structure model based on the SrTiO₃ reference structure. The lattice parameter, scale factor, background, and peak-profile parameters were refined, while the atomic coordinates were fixed to the ideal pseudocubic crystallographic positions. The refined lattice parameters, unit-cell volumes, and reliability factors are summarized in Supplementary Table 5.

Bipolar strain–electric field (S–E) curves were measured at room temperature using an aixACCT ferroelectric/piezoelectric measurement system equipped with a high-voltage amplifier and a SIOS laser displacement sensor. Ag-electroded ceramic pellets were driven up to ±60 kV cm^-1 at 10 Hz using a linear triangular waveform. The longitudinal displacement ΔD was measured by the laser displacement sensor and converted to strain using S = ΔD/t, where t is the measured sample thickness. The effective large-signal coefficient d₃₃^* was calculated from the finite-field strain response as S_max/E_max.

Advanced structural analysis

Local ferroelectric behavior and polarization switching characteristics were investigated using piezoresponse force microscopy (PFM; NX-10, Park Systems). Domain images were acquired over a 1 × 1 μm² area in PFM mode with an AC drive voltage of 2 V and a scan rate of 0.3 Hz. To correlate the microscopic switching with macroscopic remanent polarization (P_r), a DC bias of 10 V was applied. The evolution of the polar state was recorded sequentially: in the pristine state, during the application of the DC bias, and immediately following its removal. Post-acquisition data analysis, including phase and amplitude mapping, was performed using the XEI software.

Atomic-scale structural investigations were conducted using a Cs-corrected and monochromated transmission electron microscope (TEM/STEM; Themis Z, FEI) operated at an accelerating voltage of 200 kV. TEM specimens were prepared via focused ion beam (FIB; Nova Nanolab, FEI) milling to ensure electron transparency. High-angle annular dark-field (HAADF) STEM images were recorded at a high magnification of 7.6 Mx. The precise atomic positions of the A-site and B-site cations were determined through two-dimensional Gaussian fitting using a customized Python-based script. This process enabled the construction of polarization and amplitude maps by calculating the displacement of the A-site cations relative to the center of the surrounding B-site lattice, providing projected maps of local cation off-centering relative to the surrounding B-site lattice.

Lattice vibrations and structural symmetry were probed via Raman spectroscopy (LabRAM HR Evolution, Horiba) using a 532 nm excitation laser. Calibration was performed using a standard silicon wafer prior to measurement. Raman spectra were collected in the range of 0–1000 cm^–1, with each spectrum resulting from the average of five individual scans for a maximized signal-to-noise ratio. Processing of the raw spectral data utilized an asymmetric least squares (ALS) algorithm (λ = 120,000, p = 0.0001) for precise baseline subtraction. Peak identification was executed through a segmented detection strategy guided by the second derivative of the spectrum, with prominence and distance constraints tailored to different frequency zones. Final spectral deconvolution was performed by fitting the corrected data to a sum of pseudo-Voigt profiles using the lmfit library, achieving a high goodness-of-fit (R² ≈ 0.998) and enabling the quantitative analysis of individual phonon modes.

Author contributions

K.S. conceived the research, performed data curation, conducted the modeling, and performed most experiments. Y.K. helped with TEM measurements, and J.K. assisted in model optimization. B.J. and H.C. assisted with P–E measurements and dielectric temperature (ε–T) characterization. N.K. and J.R. assisted with strain–electric field (S–E) curve measurements and related electromechanical characterization. H.W.J. supervised the project and provided guidance. The manuscript was prepared by K.S. with contributions from Y.K., J.K., and H.W.J.

Peer review

Peer review information

: Nature Communications thanks Xudong Qi and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

The authors disclose support for the research of this work from the National Research Council of Science & Technology (NST) grant funded by the Korea government (MSIT) (No. GTL25021-230), the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (MSIT), South Korea (RS-2024-00421181), and the InnoCORE program of the Ministry of Science and ICT (1.250021.01), and the Nano & Material Technology Development Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science and ICT (RS-2024-00405016).

Data availability

All data supporting the findings of this study are available within the article and its Supplementary Information files. The datasets generated and analyzed during this study, including the curated database derived from literature mining (1,202 entries), the master list of elemental properties used for physics-guided descriptor engineering, the high-throughput virtual screening (HTS) top-ranked results, and the 74-entry SNBT/NBT-ST-family benchmark dataset, have been deposited in the Zenodo repository under the accession code 10.5281/zenodo.20039899. Source data for the figures presented in this paper are available from the corresponding author upon reasonable request.

Code availability

The computational framework and custom codes developed for this study, including the Python-based pipeline for the deep ensemble model and high-throughput virtual screening, have been permanently archived and are publicly available in the Zenodo repository under the accession code 10.5281/zenodo.19028580.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74376-3.