Superstrength permanent magnets with iron-based superconductors by data- and researcher-driven process design

The study addresses how to overcome grain boundary limitations in polycrystalline iron-based superconductors (IBSs) to realize next-generation superstrength permanent magnets. IBSs possess high superconducting transition temperatures (Tc ~60 K), minimal electromagnetic anisotropy, and very high upper critical fields (Hc2 > 50 T), making them promising for applications such as MRI, accelerators, and MAGLEV. Despite favorable intrinsic grain boundary transport compared to cuprates (higher critical misorientation angles and robust polycrystalline Jc ~10^4–10^5 A/cm²), polycrystalline IBSs still fall short of single crystals and thin films in current density due to microstructural and compositional sensitivities. The research question is how to design synthesis processes that optimize transport current properties and trapped-field performance in bulk IBSs. The authors propose integrating machine learning (Bayesian optimization) with researcher-guided process design to navigate complex synthesis parameter spaces and improve critical current density and magnet performance.

Prior work has established that IBSs (including 1111, 122, and 11 phases) exhibit higher critical misorientation angles (e.g., 5–9° in Co-doped Ba122) than cuprates (2–3°), supporting better grain boundary transport. Polycrystalline IBS bulks and wires have achieved Jc values significantly exceeding polycrystalline cuprates, though still below single crystals and films. Modeling of polycrystalline behavior and grain boundary properties has progressed, yet the coupling between microstructure, composition, and superconducting transport remains debated. In materials informatics, machine learning has been widely used for Tc modeling and property optimization, but synthesis-focused process informatics lags due to the high dimensionality of process parameters and a lack of databases containing processing conditions. Existing datasets like Materials Project and SuperCon lack detailed processing-property links, and recent data-mining efforts highlight the need for process-centric databases. For IBS processing, two guidelines have been used: (i) high-temperature, long-duration processing (~900 °C) with texturing (wires), and (ii) low-temperature, short-duration processing to create small grains. The present work explores a complementary, data-driven route.

The authors developed a complementary process design combining a data-driven loop (Bayesian optimization using custom software BOXVIA) and a researcher-driven loop. A shared database captured synthesis parameters and measured properties to iteratively guide both loops. Researchers defined the parameter space and initial data, focusing on spark plasma sintering (SPS) parameters for mechanically alloyed (Ba0.6K0.4)Fe2As2 precursor: x = ramp rate (°C/min), y = maximum temperature (°C), and z = dwell time (min). Bayesian optimization suggested synthesis conditions; after each synthesis, critical current density Jc(B, 5 K) was measured and used to update the model. Approximately 40 iterative trials were conducted to optimize Jc at 3 T as the target metric, balancing global exploration and local refinement. Two 30 mm diameter, 6 mm thick bulks were fabricated with the optimized conditions: Bulk1 (data-driven) at (+49.8 °C/min, 556 °C, 32.47 min) and Bulk2 (researcher-driven) at (+50 °C/min, 600 °C, 5 min). Critical current density Jc(B, 5 K) was measured to assess performance. Trapped field experiments used a stacked pair of the K-doped Ba122 bulks subjected to field cooling: cooled to ~5 K in a 7 T external field using a cryocooler, the field was removed at 4.8 T/h at 5 K. Trapped field was measured at two positions: the center of the spacer between the bulks and the center of the stack surface, as a function of temperature (0.5 K/min sweep). Magnetic hysteresis at 5 K was also recorded. Microstructural and compositional characterization employed STEM and atomic-resolution ADF-STEM with elemental mapping (Ba, K, Fe, As) to identify grain morphology, planar defects (stacking faults), and dopant distribution. Numerical comparisons using finite element method (FEM) models are referenced for interpreting magnet performance.

- Dual-loop optimization successfully identified distinct optimal SPS conditions for different targets: the researcher-driven process maximized zero-field Jc, while Bayesian optimization maximized Jc at 3 T. - Process parameters: • Bulk1 (data-driven): x = +49.8 °C/min, y = 556 °C, z = 32.47 min. • Bulk2 (researcher-driven): x = +50 °C/min, y = 600 °C, z = 5 min. - Critical current density at 5 K: • Bulk2 reached the highest zero-field Jc = 1.2 × 10^5 A/cm^2. • Bulk1 achieved the highest Jc at the target field of 3 T; both bulks exceeded 1 × 10^5 A/cm^2 under low fields, among the best for randomly oriented polycrystalline IBS bulks. - Trapped field performance: • Field-cooled trapped field of 2.83 T at ~5 K measured at the center of a stacked bulk pair (30 mm diameter, 6 mm thickness each) after 7 T magnetization; this is ~2.7× the prior IBS bulk trapped-field record (1.03 T by Weiss et al.). • The trapped field substantially exceeds fields generated by IBS test coils of similar scale (e.g., ~0.27–0.31 T at 4.2 K for Ba122 coils reported by Pyon et al.). - Field stability: Demonstrated magnetic field stability exceeding 0.1 ppm/h for a practical 1.5 T permanent magnet (from the IBS bulk), relevant for MRI requirements. - Microstructure: • Bulk2 (researcher-driven, 600 °C, short dwell): closely connected fine grains (tens of nm). • Bulk1 (data-driven, ≤600 °C, long dwell): bimodal grain size with minute grains (tens of nm) and larger grains (100–300 nm), with dense intragranular planar defects (stacking faults) of ~10 nm size spaced by a few nm to ~10 nm (about twice the coherence length ~5 nm). • Elemental mapping confirms uniform K substitution on Ba sites in both samples. • The data-driven process reveals a third processing pathway: extended low-temperature sintering producing bimodal grain sizes and dense intragranular defects that act as strong flux pinning centers; grain boundary spacing distributions become bipolarized, aiding current pathways. - Scaling implication: Given relatively small bulk size and flat J(B) in IBSs, larger bulks are expected to trap stronger fields; operation at 10–20 K (near liquid H2) is promising for practical systems.

The complementary data-driven and researcher-driven loops mitigated bias and leveraged both human intuition and algorithmic exploration. Researcher-driven optimization efficiently identified coarse optima for simple targets (e.g., maximizing zero-field Jc via higher maximum temperature and short dwell to produce fine, connected grains). Bayesian optimization excelled at optimizing under specific operating conditions (e.g., Jc at 3 T), navigating complex trade-offs among multiband effects, anisotropy, and flux pinning. The data-driven pathway uncovered a microstructural regime with bimodal grain sizes and dense intragranular stacking faults whose spacing is comparable to a few multiples of the coherence length, providing effective vortex pinning. Conceptually, a bimodal grain network increases coordination number in the grain boundary graph, facilitating more percolative pathways across low-angle boundaries and enhancing current transport. These microstructural features underpin the record trapped field (2.83 T at ~5 K) and the excellent field stability (>0.1 ppm/h at 1.5 T), demonstrating the feasibility of IBS bulk permanent magnets meeting or approaching MRI stability requirements. The results suggest that further optimization and scaling (larger bulks, engineered defect landscapes) could substantially increase trapped fields due to the relatively flat J(B) of IBSs.

This work introduces a dual-loop process informatics framework—integrating Bayesian optimization (BOXVIA) with researcher-guided design—to optimize spark plasma sintering parameters of K-doped Ba122 bulks. The approach delivered: (i) top-tier polycrystalline Jc (>10^5 A/cm^2 at low fields), (ii) record trapped field of 2.83 T at ~5 K (2.7× prior IBS bulk record), and (iii) field stability better than 0.1 ppm/h at 1.5 T. Nanostructural analysis revealed a previously underexplored regime characterized by bimodal grain sizes and dense intragranular stacking faults that serve as effective pinning centers and improve current percolation. These findings establish a roadmap for engineering microstructures in IBS bulks to realize superstrength permanent magnets suitable for applications like MRI, particularly at 10–20 K with compact cryocoolers or liquid hydrogen cooling. Future research should (a) elucidate mechanisms controlling bimodal grain formation and its quantitative impact on Jc and trapped field, (b) expand the process parameter space and database (including atmosphere, pressure, precursor chemistry) for richer machine learning, (c) scale bulk dimensions and stack architectures, and (d) integrate modeling (FEM) with micromechanical and pinning landscape design for predictive optimization.

- Mechanistic understanding of how nonuniform (bimodal) grain size distributions form during extended low-temperature SPS and how they quantitatively impact Jc remains incomplete. - Optimization explored a limited SPS parameter set (ramp rate, max temperature, dwell time); other factors (pressure, atmosphere, precursor stoichiometry, milling conditions) were not systematically varied. - Dataset size was modest (~40 trials), which may limit generalizability and the robustness of the surrogate model. - Demonstrations used relatively small bulks (30 mm diameter, 6 mm thickness) and low operating temperature (~5 K) for the record trapped field; scalability to larger sizes and higher temperatures needs validation. - Full author affiliation mapping and some experimental details (e.g., exact FEM model parameters) were not provided in the excerpt, constraining comprehensive interpretation.