Generative learning facilitated discovery of high-entropy ceramic dielectrics for capacitive energy storage

Dielectric capacitors are essential for power electronics due to their high power density, but their energy density (Ue) lags far behind electrochemical storage, limiting miniaturization and integration. Achieving high U requires simultaneously maximizing polarization (Pm), minimizing residual polarization (Pr), and increasing breakdown strength (Eb). Conventional ferroelectrics (e.g., BaTiO3, BiFeO3, PZT) offer large Pm but suffer from large Pr and low Eb. High-entropy design, which increases configurational entropy by multi-element substitution, can create locally diverse polarization configurations (smaller, weakly coupled PNRs), reduce hysteresis (lower Pr), and enhance Eb via lattice distortion, grain refinement, and amorphous-phase formation. However, the vast multicomponent compositional space makes empirical discovery inefficient, and machine learning efforts are hampered by small datasets. The study aims to integrate phase-field simulations with a generative learning framework to efficiently discover high-entropy Bi(Mg0.5Ti0.5)O3-based dielectrics with superior energy storage, overcoming data scarcity and the combinatorial explosion in design space.

Prior work has shown that high-entropy ceramics can enhance dielectric energy storage by inducing local disorder and polymorphic distortions that reduce hysteresis and increase breakdown strength. Reports include high-entropy enhancements in Bi2Ti2O7-based films with distorted nanocrystalline/amorphous phases and (K,Na)NbO3-based ceramics achieving higher Eb and delayed polarization saturation. More broadly, high-entropy oxides exhibit tunable structural/electrochemical properties. Data-driven materials design via machine learning has accelerated discovery across materials, yet faces challenges under data scarcity, leading to overfitting and poor generalization. Generative models, which learn data distributions to produce new, similar samples, offer a path to augment limited materials data. Phase-field methods have been employed to study relaxor formation and energy storage behavior, and recent studies engineered relaxors by entropy to boost performance. These strands motivate combining physics-based simulation with generative/inverse design to navigate high-dimensional spaces efficiently.

The study follows a theory-to-experiment workflow: (1) Phase-field simulations; (2) Generative learning and inverse design; (3) Directed synthesis and characterization. - Phase-field simulations: Time-dependent Ginzburg–Landau (TDGL) equations model polarization dynamics with total free energy including Landau bulk, gradient, elastic, and electrostatic contributions. Increasing configurational entropy Sconfig is represented via enhanced dipole disorder and local stochastic strain (e_ij^s = Q_ijkl P_k P_l + c x, with c following a Gaussian distribution reflecting doping/entropy levels). Simulations on BiFeO3-, BaTiO3-, and PbTiO3-based systems compute local polarization distributions, P–E loops, and dischargeable energy density U as Sconfig varies (0.69R to 1.64R). - Generative learning framework: Starting from 77 experimental compositions of Bi1−a−b−c La_a Sr_b Ca_c (Mg0.5Ti0.5)1−d−e−f Mn_d Zr_e Hf_f O3 (BMT-based), an encoder–decoder neural network separately encodes A-site and B-site compositions into latent variables z and reconstructs compositions (loss combines MMD between z and prior with binary cross-entropy). Latent spaces are visualized via PCA. A classifier (ANN with k-fold CV) is trained to distinguish high-Ue samples, enabling sampling of candidate high-performance compositions. A Gaussian Mixture Model (GMM) models latent density; Metropolis–Hastings MCMC samples z to generate novel candidates similar to high-performance regions. A regression ensemble integrating ANN and LightGBM, with physical/atomic descriptors (normalized), predicts Ue and its uncertainty; hyperparameters are tuned via random search and Bayesian Optimization. A ranking metric λ = α·rank(Upredict) + β·rank(Uuncertainty) prioritizes candidates balancing high predicted performance and controlled uncertainty (active-learning inspired). This pipeline generated 2144 candidate high-performance compositions (Ue>65 J cm⁻³), from which five top-ranked were selected for experiments. - Experimental synthesis and characterization: Five BMT-based high-entropy films (C-1…C-5) were fabricated by chemical solution deposition on Pt substrates. Precursors (Bi, La, Sr, Ti, Mg, Mn, Zr sources) were dissolved and stabilized; films were spin-coated (4500 rpm, 30 s), pyrolyzed at 200/300/450 °C (5 min each), and rapid thermally annealed at 640 °C for 2 min, yielding ~160 nm films. Structural analysis used GI-XRD; microstructure by FE-SEM and HR-TEM; domains by PFM; dielectric properties by impedance analyzer (1 kHz–1 MHz); ferroelectric P–E loops and energy storage by a ferroelectric workstation (1 kHz). Breakdown strength Eb was analyzed via Weibull statistics; leakage currents were measured versus field. Performance was further evaluated across temperature (20–150 °C), frequency, and cycling (up to 10^5 cycles).

- Simulations: Increasing Sconfig breaks large ferroelectric domains into smaller, disordered polar nanoregions, shifting P–E behavior from ferroelectric to relaxor-like. Across BFO-, BTO-, and PTO-based models, normalized dischargeable energy density U increases monotonically as Sconfig rises from 0.69R to 1.64R. - Data-driven design: From 77 initial BMT-based datasets, the generative framework produced 2144 high-performance candidates (Ue>65 J cm⁻³). Five top-ranked compositions (C-1…C-5) were selected for directed experiments. - Structure/microstructure: GI-XRD shows C-1 and C-3 are perovskite; C-2, C-4, C-5 exhibit some pyrochlore phase. Peak broadening and intensity reduction indicate grain refinement and increased amorphous content. HR-TEM quantifies local amorphous fraction rising from ~10% (BMT) to ~45% (C-3). - Dielectric behavior: Dielectric constants at 1 kHz decrease from ~310 (BMT) to ~190 (C-3), with improved frequency stability (~10% variation for C-n vs ~30% for BMT). Loss tangents are suppressed; a relaxation peak appears near 100 kHz. - Breakdown and leakage: Weibull Eb (kV cm⁻¹): BMT 1173; C-1 4357; C-2 4232; C-3 5104; C-4 3734; C-5 2987. Weibull modulus β: 7.76 (BMT); 10.19, 10.2, 11.8, 10.51, 16.97 (C-1…C-5). Leakage at 1000 kV cm⁻¹: C-3 4.01×10⁻¹⁰ A cm⁻² vs BMT 3.97×10⁻⁸ A cm⁻² (~100× reduction). - Polarization: At respective Eb, BMT Pm=88 µC cm⁻², Pr=40 µC cm⁻² (Pm−Pr=48); C-1…C-5 Pm−Pr = 76, 72, 96, 45, 97 µC cm⁻². For C-3, Pm=119 µC cm⁻², Pr=23 µC cm⁻². - Energy storage: Pure BMT Ue≈18 J cm⁻³. High-entropy films all improved; best C-3 (Bi0.87La0.08Sr0.05Ti0.41Mg0.39Mn0.15Zr0.05O3) achieved Ue=156 J cm⁻³ at Eb=5104 kV cm⁻¹ (≈8× BMT) with efficiency η≈75% under comparable fields. Directed compositions generally exhibit Eb>4000 kV cm⁻¹, enabling high Ue. - Stability: C-n films show robust cycling up to 1×10⁵ cycles at 2000 kV cm⁻¹. Example: C-3 maintains Ue=33.36 J cm⁻³ and η=77% after 10⁵ cycles at 2000 kV cm⁻¹. Temperature stability (20–150 °C) and frequency stability are superior to BMT; BMT punctures at 2000 kV cm⁻¹ upon heating, whereas C-3 varies modestly (Ue 33.36→29.36 J cm⁻³; η 77%→70%).

The study demonstrates that high configurational entropy modulates local dipole landscapes—fragmenting domains into polar nanoregions and introducing controlled disorder—which jointly reduces hysteresis (lower Pr) and raises breakdown strength (higher Eb). Phase-field simulations predicted the monotonic improvement of dischargeable energy with increasing Sconfig, and experiments validated this trend in BMT-based high-entropy films. The best-performing composition, identified via a generative learning and ranking-based inverse design, achieved a dramatic increase in Ue (156 J cm⁻³) by simultaneously achieving large Pm at high Eb while suppressing Pr. Microstructural evidence (grain refinement, increased amorphous fraction) supports the mechanistic link to improved Eb and lower leakage. The generative framework effectively addressed data scarcity by learning a latent compositional representation, sampling high-performance regions, and integrating prediction uncertainty to prioritize experiments. This approach significantly reduced experimental trial cycles (only five targeted syntheses) and offers a scalable path to navigate vast multicomponent design spaces for dielectric energy storage materials and beyond.

By integrating phase-field simulations with a generative learning-driven inverse design, the work rapidly discovered high-entropy BMT-based dielectrics with superior energy storage. The framework generated and screened thousands of candidates from sparse data, with five targeted experiments yielding across-the-board performance gains and a standout composition (Bi0.87La0.08Sr0.05Ti0.41Mg0.39Mn0.15Zr0.05O3) delivering Ue=156 J cm⁻³ at Eb=5104 kV cm⁻¹—over eightfold higher than pristine BMT. The results confirm that entropy-engineered dipole disorder synergistically tunes Pm, Pr, and Eb, enabling high Ue with good efficiency and robust cycling, temperature, and frequency stability. The methodology provides a generalizable, efficient route for designing multicomponent high-entropy functional materials. Future work could expand the compositional palette and anion/cation sublattices, incorporate additional physics-informed descriptors, refine uncertainty quantification, and couple active learning with automated synthesis for closed-loop discovery.

- Data scarcity: The initial dataset comprised only 77 compositions, which can limit global generalization of the models and raise prediction uncertainty, especially at higher entropy. - Model uncertainty and bias: Although uncertainty-aware ranking was used, prediction errors remain possible; uncertainty increased with compositional complexity. - Phase purity: Some directed compositions (C-2, C-4, C-5) exhibited pyrochlore phases, which may affect performance and complicate structure–property attribution. - Scope of validation: Experiments were limited to thin films (~160 nm) processed by a specific CSD/annealing protocol; scalability to other thicknesses, substrates, or processing routes requires further study. - Property envelope: While high Eb and Ue were demonstrated, long-term reliability beyond 10^5 cycles and under harsher environmental conditions (e.g., humidity, higher temperatures) was not assessed.