Introduction

Iron-based high-temperature superconductors (IBSs)1 feature a square iron lattice that serves as a fundamental framework for unconventional superconductivity. These materials are created from a variety of parent compounds, encompassing 1111-, 122-, and 11-phase superconductors and various constituent elements. The excellent properties of IBSs have garnered significant attention across fundamental and applied research fields2, primarily due to their remarkably high superconducting transition temperature (Tc ~ 60 K)3 and due to the recent exploration of their topological characteristics4,5. The variety of potential applications of IBSs further highlight their significance, including the prospect of novel Majorana platforms for topological quantum computation2,6, the development of next-generation superstrength permanent magnets7,8,9 and advancements in superconducting electronics devices10,11.

In magnets, IBSs are better than Nb-based superconductors by functioning at higher temperatures and in stronger magnetic fields. These superconductors are polycrystalline materials that can be fabricated through a more scalable process than cuprate superconductors3. Additionally, IBSs exhibit minimal electromagnetic anisotropy and have an exceptionally elevated upper critical field (Hc2) exceeding 50 T, which is more than double that of Nb-based superconductors12,13,14,15,16. Consequently, IBSs can significantly enhance the performance of particle accelerators, medical magnetic resonance imaging (MRI) scanners, MAGLEV trains, and other devices that rely on conventional Nb-based superconducting magnets3,8.

Due to the enigmas associated with unconventional high-Tc superconductivity, the anisotropic superconducting pairing17,18 and the juxtaposition of high Hc2 and a short coherence length12,13,14,15,16, IBSs are notable owing to their exceptional intrinsic grain boundary transport characteristics10. Katase et al. showed that the critical misorientation angle (θc), which is the angle at which superconducting current initiates decay at grain boundaries, for Co-doped Ba122 (5°–9°)10 is more than twice that in cuprate superconductors (2°–3°). Similarly, elevated θc values have been documented in the 1111 and 11 phases, serving as a shared trait among IBSs19,20,21. In addition, a robust critical current density (Jc ~ 104–5 A/cm2) has been reported in IBSs, exceeding the values observed in polycrystalline cuprate superconductors by several orders of magnitude. Intriguingly, this increased current density has been observed in polycrystalline bulks9,22,23,24,25 and wires3,26,27,28, which consist of many crystal grains with diverse random orientations. Nonetheless, it is essential to acknowledge that this Jc value is notably lower than that of single crystals29,30 and thin films (Jc ~ 106–7 A/cm2)10,11,19, thus not fully reflecting the inherent potential of IBSs. In recent years, significant progress has been made in modeling polycrystalline materials31,32 and evaluating grain boundary characteristics10,19,20,21. However, given the sensitivity of superconducting properties to microstructures and chemical compositions33,34,35,36,37,38, the intricate interplay through which these factors dictate the resultant characteristics remains contentious. Consequently, the establishment of process design principles to enhance transport current properties in IBS materials is of great interest.

In recent years, machine learning, which is a subset of artificial intelligence (AI), has been integrated with materials science to solve a variety of problems, including the modeling of the critical temperature (Tc) of superconductors via database analysis39,40,41,42 and the optimization of material properties43,44,45,46. Nonetheless, the penetration into synthesis is somewhat restricted compared with its application in the construction of comprehensive materials databases. Synthesis processes involve an extensive array of parameters, ranging from factors related to equipment configurations to empirical variables manipulated by researchers. These intricate parameters and variables deviate significantly from well-established databases, such as the Materials Project47 and SuperCon48 databases, which offer valuable repositories of known information. Recent endeavors have resulted in the creation of novel databases through meticulous data mining efforts49,50,51, although databases featuring processing conditions and material properties are lacking. Therefore, examples of process informatics utilizing large databases are limited in the literature. Consequently, process informatics harnessing the potential of expansive databases is relatively unexplored. Despite the promising strides in applying machine learning to materials science, particularly in characterizing material properties, the intricate and multifaceted landscape of synthesis poses distinct challenges that demand further exploration and innovation to fully exploit the potential of process informatics.

In this study, we focus on an optimization process for enhancing the critical current properties of polycrystalline (Ba,K)Fe2As2, also known as K-doped Ba122. To achieve this enhancement, we have integrated Bayesian optimization43,52, a machine-learning-based approach, which empowers data-driven experiments even when operating with limited prior data. Bayesian optimization is a method for effectively exploring the maximum and minimum values of a given unknown function, denoted as f(x), that has several key attributes. This method operates without constraints concerning the nature of the unknown function f(x). Moreover, this technique adeptly avoids the pitfall of converging toward local maxima or minima, and it demonstrates proficiency across diverse materials science applications53. Our adopted methodology is based on a collaborative framework. Initially, we design a researcher-driven process that ensures the simultaneous accumulation of data to promote a data-driven process. This collaborative relationship is reliant on the continuous reference to the furnished dataset. Ultimately, these two processes yield an optimal parameter set. This parameter set then facilitates the fabrication of permanent magnet prototypes under various conditions. We subject these prototypes to experiments to measure their magnetic properties. Subsequently, we draw comparisons between these experimental findings and numerical models based on the finite element method (FEM).

Results and discussion

Process design

We designed a process by combining collaborative researcher-driven efforts with data-driven approaches. Figure 1 illustrates the framework in which machine learning and researchers could share and reference the same experimental data while independently designing the process. Researchers provided a search space for process parameters and initial data based on human experience and theoretical strategies. These data were input into a machine learning algorithm to predict the synthesis conditions that yield superior properties. Researchers could synthesize samples according to the proposed conditions, perform measurements, and update the database. In machine learning, this “data-driven loop” was repeated to perform a balanced global search and local fine-tuning using the customized software BOXVIA54. Simultaneously, the researchers provided a general framework and experimental data for the machine learning and design processes (“researcher-driven loop”). Researchers analyzed the shared database, including data obtained from the data-driven loop; then, they planned the next process and synthesized samples. The connection of these two independent loops allowed for the easy removal of biases and the widening of their perspective. These loops helped expanding machine learning data, including almost optimized, good-quality data, which could contribute to increasingly efficient process design.

Fig. 1: Conceptual schematic of the complementary data- and researcher-driven process designs.
figure 1

The “data-driven” loop, which provides the synthesis conditions for Bulk1, is driven by machine learning using the tailored software BOXVIA [54]. The “researcher-driven” loop, which provides the synthesis conditions for Bulk2, is based on human experience and theory-based strategies. A shared database is continually updated by both loops, contributing to increasingly efficient process design.

In the first phase, researchers methodically sorted through quantifiable process parameters, identifying those with considerable influence on the resultant properties. The researchers then focused on three pivotal process parameters denoted as x (ramping rate), y (maximum temperature), and z (dwell time). These parameters govern the spark plasma sintering process, which could be applied to the mechanically alloyed (Ba0.6K0.4)Fe2As2 precursor powder obtained through a high-energy milling process55 (comprehensive details are provided in the Methods-Sample Preparation section). The three-dimensional process search space was meticulously defined, serving as the backdrop for preliminary optimization efforts undertaken by the researchers themselves. These initial findings supported the data-driven process design, which was an aspect described in the Methods-Bayesian Optimization and Software section. Notably, these results led to the development of BOXVIA, a dedicated Bayesian optimization software54. The sample synthesis was guided by Bayesian recommendations for the process parameters. Following each synthesis, the critical current density (Jc) is ascertained. This cyclical process was carried out until the zenith was reached, at which point the optimal parameters were identified. Given the pivotal role of the magnetic field dependence of Jc(B) in shaping the trapped field performance of superconducting permanent magnets, our optimization metric was dependent on the Jc(3 T) magnetic field strength. This parameter served as the basis of our optimization endeavors, steering the entire optimization process.

Fabrication of prototype permanent magnets

Two relatively large disk-shaped bulk samples of K-doped Ba122 (Ba0.6K0.4Fe2As2) were synthesized for trapped field measurements using the optimized processing conditions that were evaluated in the aforementioned process design. Each sample had a diameter of 30 mm and a thickness of 6 mm. The Bulk1 and Bulk2 samples were fabricated utilizing specific process parameters derived from the data-driven and researcher-driven approaches, respectively. For Bulk1, the parameters (x, y, z) were set to +49.8 °C/min, 556 °C, and 32.47 min, while for Bulk2, the parameters were set to +50 °C/min, 600 °C, and 5 min. These meticulously chosen parameters ensured the controlled synthesis of the samples and the consistency of the experimental results.

Critical current properties

Figure 2 presents an overview of the dependence of Jc on the magnetic field at 5 K for both the data-driven and researcher-driven samples. In both methodologies, the optimization efforts yielded enhancements in Jc, although the trends exhibited some variations. Under the researcher-driven approach, the magnetic field dependence of Jc demonstrated a steep increase towards 0 T, resulting in a significant increase in the maximum Jc value. Conversely, the data-driven approach displayed a gradual magnetic field dependence, with improvements in Jc across various magnetic field strengths, but particularly at 3 T. After conducting approximately 40 trials, the optimized process conditions obtained through the researcher-driven and data-driven approaches (Bulk2 and Bulk1, respectively) resulted in distinct outcomes. Specifically, the researcher-driven approach (Bulk2) yielded the highest Jc value, reaching 1.2 ×105 A/cm2 under zero magnetic field conditions. Conversely, the data-driven approach (Bulk1) achieved the highest Jc value under the designated field conditions (3 T). Notably, the achieved Jc values, which exceeded 1 × 105 A/cm2 under low magnetic field conditions, were some of the best for randomly oriented polycrystalline IBS bulk materials9,22,23,24,25.

Fig. 2: Magnetic field dependence of the critical current density at a temperature of 5 K, Jc (B, 5 K).
figure 2

The data are presented for samples that were prepared through both data-driven (Bulk1) and researcher-driven (Bulk2) approaches.

The trends observed in this study could be interpreted as follows. When exploring simple conditions, such as Jc in the absence of an external magnetic field, researchers determined it was straightforward to formulate hypotheses and devise optimal processes. Researchers could refer to measurement results effectively fed back to process design, such as reducing the impurity peak intensity in X-ray diffraction patterns and improving uniformity of grain size from electron microscopy images (see Nanostructure and chemical composition). Under these conditions, researchers could efficiently find a coarse optimum. However, utilizing machine learning for optimization could be applicable to properties under specific conditions (as demonstrated in this study for a 3 T magnetic field). In these cases, a multitude of factors, such as multiband effects, electromagnetic anisotropy, and flux pinning, undergo intricate interactions, making a gradual and methodical trial-and-error approach feasible without the interference of preconceived notions. In a broad sense, Bayesian optimization offered a good blend of global exploration and precise local optimization53,56. In this study, researchers mainly undertook the definition of the parameter space and initial coarse optimization. Consequently, emphasis was placed on local optimization, yielding potential candidates proximate to the optimal values. The researcher-driven approach optimized the maximum sintering temperature x in 50 °C increments so that the optimized x was 600 °C (the parameter set was (x, y, z) = (+50 °C/min, 600 °C, 5 min) and was used for the synthesis of Bulk2). In contrast, Bayesian optimization was carried out in 1 °C increments; thus, the precision was relatively fine. By employing the machine learning approach, the finest parameter set refinement through local fine-tuning was achieved, (x, y, z) = (+49.8 °C/min, 556 °C, 32.47 min), which was used for the synthesis of Bulk1.

Nanostructure and chemical composition

Figure 3 shows the nanostructures and compositional analyses of Bulk1 (data-driven) and Bulk2 (researcher-driven). In Fig. 3a, the microstructure of Bulk2 revealed a closely interconnected arrangement characterized by fine-grained crystals (measuring several tens of nm). This characterization was achieved by researchers employing a short sintering duration at 600 °C. Conversely, Bulk1, resulting from the Bayesian optimization procedure (entailing extended sintering at a lower temperature), exhibited a trend of segregating into minute grains of several tens of nanometers. In addition, there were large grains ranging from 100 to 300 nm in size, as shown in Fig. 3b. The scanning transmission electron microscopy (STEM) images (Fig. 3c, d) show the introduction of planar defects within the grains, which occurred at intervals approximately twice the superconducting coherence length of ~5 nm. The dimensions of these planar defects were approximately 10 nm, with an average spacing of a few nanometers, which was indicative of their nature as stacking faults. Further confirmation was obtained through the atomic resolution ADF–STEM image (Fig. 3e) coupled with elemental mapping (visible in Fig. 3f–i), which verified the uniform substitution of K into the Ba site across both samples.

Fig. 3: Microstructures and nanostructures of the samples.
figure 3

Subfigures (a) and (b) represent the microstructures of researcher-driven Bulk2 and data-driven Bulk1, respectively. Subfigures (a) through (d) show STEM images, while subfigures (e) through (i) show an atomic resolution image (e) alongside corresponding elemental mapping images for Ba (f), K (g), Fe (h), and As (i).

To date, two distinctive process guidelines have emerged for the production of IBSs with enhanced critical current properties. (i) One approach involves the cultivation of large grains, accomplished through extended processing durations at elevated temperatures of ~900 °C, coupled with c-axis texturing achieved by employing cold-working techniques in wires26,27,28. (ii) Conversely, an alternative strategy focuses on the generation of small grains achieved by adopting short processing durations at low temperatures9,22,23,33,36,55. Nonetheless, our data-driven process introduced a novel “third” pathway characterized by a bimodal grain size profile, with small grains and dense intragranular defects. This novel approach was realized through extended processing durations executed at remarkably low temperatures, specifically below 600 °C. Notably, considering that the average spacing exceeded the superconducting coherence length of 2ξc (~2.4 nm)12 by several multiples, the intragranular defects effectively contributed to the flux pinning of vortices. Despite these advancements, certain aspects, such as the mechanism governing the formation of nonuniform grain size distributions and the resultant effects on the critical current, remain unclear at this point. By conceptualizing the grain boundary network as a graph, the bimodal grain size distribution led to a greater number of adjacent grains compared to that in scenarios featuring uniform grain sizes. This increase in coordination number could facilitate the creation of optimal pathways for superconducting currents, including interfaces such as low-angle grain boundaries.

Permanent magnet properties

Trapped field and temperature dependence

Figure 4 shows the temperature dependence of the trapped field in the stacked pair of K-doped Ba122 bulk materials. The bulk pair underwent a field-cooling process to reach a temperature of approximately 5 K, which was achieved using a cryocooler while subjected to an external field of 7 T. After the removal of the external field, which was performed at a sweep rate of 4.8 T/h at 5 K, the trapped field within the bulk pair was measured at two distinct positions: at the center of the spacer (referred to as the “center”), positioned between the two bulks, and at the center of the stack surface (referred to as the “surface”). These measurements were taken as a function of temperature using a sweep rate of 0.5 K/min. After the field-cooling process, the maximum trapped field recorded was 2.83 T at the center of the bulk pair. This measurement was approximately 2.7 times larger than the previous trapped field record achieved using iron-based superconducting magnets, which was 1.03 T, as reported by Weiss et al.9. Moreover, the measured trapped field surpassed the generated field strength of IBS test coil magnets by one order of magnitude. Notably, Pyon et al. crafted small coils with an outer diameter of 46 mm and a height of 28 mm utilizing Ba122 round wires. In their study, the researchers reported measurements of 0.31 T and 0.27 T at 4.2 K for Na-doped and K-doped Ba122, respectively57. Similarly, Wang et al. manufactured a single pancake K-doped Ba122 coil with an outer diameter of 34.8 mm and a height of 4.62 mm58.

Fig. 4: Variations in the trapped magnetic field with temperature, as measured experimentally at both the center and the surface of the bulk pair.
figure 4

The inset within the figure shows a photograph of the bulk pair. For reference, data from Weiss et al.9 are presented.

The trapped field of a superconducting bulk magnet depends on its size and current density, as in the case of a coil magnet. Considering the relatively small size of the prototype magnet (3 cm in diameter) and the rather flat Jc(B) dependence of IBSs, a strong magnetic field could be expected in a larger sized magnet32. Since the 1 T magnetic field of the previous record was held at approximately 18 K (i.e., nearly three times higher than 5 K), strong magnet applications at 10–20 K can be expected, where it could be operated at a high thermal stability using a compact cryocooler. This temperature range is close to the boiling point of liquid hydrogen (~20 K), indicating that liquid hydrogen cooling could be considered for practical iron-based superconducting magnets.

Hysteresis loop

Figure 5 shows the magnetic hysteresis loop acquired at 5 K while sweeping the external magnetic field at a rate of 4.8 T/h. The sweep followed a sequence of 0 T → 7 T → − 7 T → 7 T after zero-field cooling of the bulk pair to 5 K. An external magnetic flux of approximately 2.5 T led to the magnetic flux reaching the center of the bulk pair. At 7 T, the hysteresis loop exhibited a pronounced increase as a result of robust flux pinning and a highly irreversible magnetic field. A numerical model was employed, which displayed excellent agreement with the experimental results, albeit with a marginally larger simulated hysteresis loop.

Fig. 5: Experimental and simulated magnetic hysteresis loops at a temperature of 5 K obtained through measurements at the center of the bulk pair.
figure 5

The hysteresis loop was obtained by zero-field cooling the sample to 5 K, followed by progressively increasing the external magnetic field from 0 to 7 T, subsequently decreasing it from 7 to –7 T and ultimately increasing it again from −7 to 7 T (as indicated by the arrows).

Magnetization process and flux creep state

Figure 6 shows the temporal evolution of the magnetic flux density during the magnetization and subsequent flux creep of the bulk pair at 5 K. The measured flux density decreased smoothly during the magnetization process, and the sample magnetization was completed without flux jumps. The sweep rate for demagnetizing the superconducting magnet from 7 to 0 T was 4.8 T/h. This sweep rate was faster than the typical sweep rates used during the magnetization of similar polycrystalline-type bulk magnets, MgB259,60,61,62 (e.g., 0.6 T/h, Badica et al.60), suggesting that relatively fast magnetization could be employed in Ba122 at low temperatures such as 5 K, potentially because of its high thermomagnetic stability. The magnetic flux density in the flux creep state, denoted as BT(t), could be reasonably approximated by the following equation:

$${B}_{{\rm{T}}}(t)/{B}_{{\rm{T}}}(0)=1.046-4.80\times {10}^{-3}\,\mathrm{ln}(t).$$
(1)
Fig. 6: Magnetization process and subsequent flux creep state of the bulk pair at a temperature of 5 K.
figure 6

The bulk pair was cooled to 5 K in the presence of an external field of 7 T. The external field (dashed line) was then gradually reduced from 7 T to 0 T at a sweep rate of 4.8 T/h. Following the removal of the external field (from t = ~5250 s), the magnetic field decreased gradually due to the thermal activation of the trapped quantized magnetic flux, indicating the flux creep state.

The flux creep rate demonstrated here exhibited comparable behavior to those of previously reported Ba122 bulk magnets (1.013–3.51 × 10−3 ln(t) at 5 K9) and MgB2 bulk magnets (1.018–2.84 × 10−3 ln(t) at 20 K59). The logarithmic time dependence observed in the decay of the flux density implied that thermal fluctuations of the quantized magnetic flux accounted for flux creep. This flux creep corresponded to a process in which the quantized magnetic flux was released from its confinement at the flux pinning potential U due to the influence of thermal energy kBT; following an exponential rate: exp(–U/kBT).

Numerical modeling

Our findings demonstrated remarkable agreement between the experimental and modeling results for magnetization and subsequent flux creep. This agreement strongly implied that the chosen model parameter for the flux creep exponent of the EJ power law representing the nonlinear resistivity of the superconductor (n = 35) characterized these bulk samples effectively. This value was either comparable to or even lower than those reported in tape-based studies63,64. Moreover, the favorable alignment of critical current density (Jc) values between the large and small bulk specimens (which were assumed in the superconducting properties of the model) signified the successful translation of the synthesis process to a large scale. This success was attributed to the inherent characteristics of polycrystalline materials. Furthermore, this alignment suggested that the supercurrent, Jc(B), flowed uniformly throughout the bulk material.

Figure 7 shows a 3D visualization of the distributions of the (a) current density and (b) trapped magnetic flux density within the bulks obtained from the field-cooling magnetization (FCM) model when the magnetizing field initially reached 0 T (t = 5250 s). The assembled disks exhibited collective bulk-like behavior, where the distributions of current density and magnetic flux density aligned with the characteristic trapped field patterns observed in magnetized bulk superconductors61,62,65. A high magnetic flux density was observed in the central region, accompanied by a related decrease in current density. Conversely, this trend was reversed toward the periphery of the bulk due to the inherent nature of the critical current density, Jc(B). In addition, a slight asymmetry between Bulk1 and Bulk2 was observed. The current density at the center was greater for Bulk1 than for Bulk2, where the local (trapped) magnetic field was the highest. However, at the periphery, Bulk2 had a higher current density than Bulk1, where the local (trapped) magnetic field was the lowest. This result arose due to the differences in Jc(B) between the two bulks: there was a crossover at approximately 0.5 T between the relative magnitudes of Jc(B) (Fig. 1). Notably, a peak trapped field of 3.17 T was calculated at the center of Bulk1.

Fig. 7: Numerical FEM simulation results.
figure 7

3D representation of the simulated distributions for (a) current density and (b) trapped magnetic flux density within the bulk materials (top bulk = Bulk1, bottom bulk = Bulk2). These simulations were generated using the FCM model at the moment when the magnetizing field initially reached 0 T at t = 5250 s.

Temporal uniformity of the permanent magnet

Following the FCM process at 10 K, the temperature was subsequently reduced to 5 K to stabilize the trapped flux. The temporal evolution of the trapped field was investigated over a duration of approximately 300,000 s, as shown in Fig. 8. Remarkably, the trapped magnetic field, which was 2.0 T at the center and 1.5 T at the surface, exhibited near-constant temporal behavior, displaying negligible decay even after three days. The time-dependent variations (t in seconds) of the trapped magnetic field, denoted as BT, both at the center and at the surface, could be approximated by the following equations:

$${B}_{{\rm{T}}}({\rm{tesla}})=2.069-2.35\times {10}^{-9}\,\mathrm{ln}(t),$$
(2)
$${B}_{{\rm{T}}}({\rm{tesla}})=1.510-2.23\times {10}^{-9}\,\mathrm{ln}(t),$$
(3)
Fig. 8: Temporal evolution of the trapped magnetic field.
figure 8

Panels (a) and (b) represent the flux density measured at the center (a) and the surface (b) of the bulk pair. The bulk pair was magnetized at a temperature of 10 K and subsequently cooled to 5 K to establish a state of frozen flux. To provide context, the decay curves at rates of –1 and –10 ppm/h are included for reference.

Exhibiting a remarkable level of magnetic field stability, the observed behavior surpassed the decay rate benchmark of –0.1 ppm/h. This value is considered crucial for medical MRI scanners and imperative for achieving exceptionally accurate cross-sectional images. This exceptional stability can be attributed to the robust pinning of magnetic flux quanta, likely arising from their interactions with nanoscale defects within the material.

In summary, we successfully demonstrated an iron-based superconducting permanent magnet with an unparalleled level of performance. This achievement stemmed from a meticulously devised synthesis process that seamlessly integrated researcher-driven and data-driven approaches. This amalgamation served to optimize the intricate nanostructure inherent in polycrystalline materials. The fundamental conclusions of this study were summarized as follows:

  • The achievement of an iron-based superconducting permanent magnet with a practical magnetic field strength was demonstrated successfully. This strength notably surpassed the prior record by a factor of 2.7 (compared to 1.03 T), which was accompanied by an excellent level of temporal magnet stability. A decay rate of less than 0.1 ppm/h met the stringent specification mandated for medical MRI scanners while maintaining a magnetic field strength of 1.5 T.

  • By synergizing expert researcher knowledge with the capabilities of machine learning, a successful process design was executed. This process was initiated with a relatively limited dataset. Through Bayesian optimization, the optimal process parameters were predicted, particularly for the critical current performance under complex conditions.

  • Noteworthy differences appeared within the nanostructures of the finest samples generated through these researcher-driven and data-driven approaches. The researcher-driven technique yielded a tightly packed grain boundary network structure, maintaining an approximate spacing of 25 nm. Conversely, the data-driven process led to a distinctive bimodal grain boundary arrangement, with spacing dimensions of approximately 25 mm and a broad range of 50–200 nm. Furthermore, this approach introduced densely packed intragranular defects with spacings on the order of several nanometers.

  • Numerical FEM simulations showed excellent agreement with the experimental results, suggesting the existence of a uniform supercurrent distribution Jc(B) within the bulk material. This attribute, which is characteristic of polycrystalline superconducting materials, represented a significant advantage.

  • The magnet structure consisted of a randomly oriented polycrystalline material, which is notably the most robust among unconventional high-Tc superconductors. The production of this superconductor could be efficiently scaled through conventional ceramic processes, aided by the straightforward predictability and optimization achievable using numerical models. Consequently, these iron-based superconductors hold substantial promise as pioneering, next-generation, and high-strength magnets suitable for practical applications. This potential extends to challenging environments, such as liquid hydrogen cooling (~20 K).

Methods

Sample preparation

K-doped Ba122 bulk samples were synthesized using mechanically alloyed precursors, which were generated through a high-energy milling process22 using a planetary ball–mill apparatus (Fritsch, P-7) and subsequently consolidated using spark plasma sintering55. To optimize the critical current density, small disk-shaped bulk samples were meticulously prepared under diverse process conditions. Each disk had dimensions of 10 mm in diameter and 1.3 mm in thickness. The precursor powders were formulated by weighing elemental metals within a controlled environment, specifically a glove box, maintained under a highly purified Ar atmosphere. The molar ratio of Ba:K:Fe:As was 0.6:0.4:2:2, indicating a 40% K doping concentration at the Ba sites. The mechanical alloying of K-Ba122 powder was achieved through a planetary ball–mill setup. For the spark plasma sintering (SPS) procedure, the precursor powder was compacted into a graphite mold with an inner diameter of 10 mm. This assembly was subsequently positioned within the SPS apparatus. The temperature was increased by following a programmed trajectory, gradually reaching the desired temperature (“y” °C) at a controlled rate of “x” °C/min. Throughout this process, a constant uniaxial pressure of 50 MPa was applied using an SPS apparatus (LABOX-315R, Sinter Land, Japan). Once the maximum temperature was sustained for a duration of “z” min, the pressure was gradually reduced, allowing the samples to cool to room temperature.

Microstructural and nanostructural analyses

Microstructural investigations were conducted using a STEM instrument (ARM-200F, JEOL, Japan) operating at an acceleration voltage of 200 kV. This analysis was augmented with energy-dispersive X-ray spectroscopy (JED-2300, DrySD100GV, JEOL, Japan) for comprehensive chemical composition analysis. In the STEM framework, the electron beam convergence angles and detection angles were carefully set. Specifically, a low-angle annular dark field (LAADF) STEM configuration was established with convergence angles ranging from 40 to 160 mrad, while high-angle annular dark field (HAADF) STEM was configured with detection angles ranging from 90 to 370 mrad.

Magnetization analyses

Magnetization analyses were carried out by applying magnetic fields reaching 7 T using a SQUID VSM magnetometer (MPMS3, Quantum Design, Germany) for small sample cuts (~0.5 × 1.2 × 2.8 mm3). The temperature Tc, which signified the superconducting transition, was defined as the point at which the transition reached 90%. The extended Bean model was used to calculate the critical current density Jc from the magnetic hysteresis loop.

To determine the trapped field characteristics, measurements were performed on a pair of prototype disk-shaped bulk samples across temperature, external magnetic field, and time domains. The two large disk-shaped bulk samples (with a diameter of 30 mm and thickness of 6 mm) were arranged in a vertical stack configuration and separated by an intermediary copper spacer measuring 1.2 mm in thickness. Magnetization was achieved through FCM under a magnetic field of 7 T. Following this step, the stacked assemblies were gradually cooled to 5 K using a Gifford–McMahon (GM) cryocooler (CRTHE05-CSFM, Iwatani Gas). The external magnetic field was then removed at a rate of approximately 4.8 T/h. Trapped magnetic fields were assessed at the surface and at the center of the stacked bulk pair using transverse cryogenic Hall sensors (HGCT-3020, Lake Shore Cryotronics, USA).

Bayesian optimization and software

To optimize the critical current density (Jc), a correlation with the experimental parameters was established such that Jc = f(x, y, z), where f is a black-box function. This process operated without defining a specific equation for f(x, y, z) if the continuity of f(x, y, z) and x, y, and z was assumed. Within the Bayesian optimization algorithm, this function f was characterized using the preliminary dataset and Gaussian process regression52. As a result, the critical current density Jc was described in terms of a Gaussian distribution:

$$J(x,y,z)\approx N(\mu (x,y,z),{\sigma }^{2}(x,y,z)),$$
(4)

where μ(x, y, z) and σ(x, y, z) are the mean and standard deviation, respectively. The subsequent experimental conditions were ascertained through the evaluation of an acquisition function, computed using μ(x, y, z) and σ(x, y, z). Specifically, the expected improvement acquisition function53 was employed, incorporating the likelihood and the extent of enhancement in Jc. The optimization of the experimental conditions was achieved using iterative processes encompassing Gaussian process regression and acquisition function computation, which were performed concurrently with the Jc measurements. Assuming small sample-to-sample variation, one measurement dataset was used for each experimental condition. The preliminary experimental data obtained by the researchers and based on their preliminary global optimization were employed as the initial dataset for Bayesian optimization, which subsequently generated the forthcoming experimental conditions.

Numerical modeling

The numerical modeling framework was based on the 2D axisymmetric H-formulation66, implemented in the commercial software package COMSOL Multiphysics using the AC/DC module “Magnetic Field Formulation” module’s interface. This framework was used successfully to model the magnetization of iron-based bulk superconductors63. The geometry was identical to the experimental assembly, with magnetic field probes integrated into the model at the corresponding locations of the Hall sensors (as outlined in the Magnetization analyses section). The nonlinear resistivity, ρ(J), of the superconducting material was described using the EJ power law. Here, E was proportional to Jn, where n is the flux creep exponent. This “n” value has a direct correlation with bulk magnetization relaxation–i.e., flux creep67,68 (discussed in the Magnetization process and flux creep state section). The magnetic field dependence of the critical current density, Jc(B), was input into the model for Bulk1 and Bulk2 via direct interpolation69 of the experimental data presented in Fig. 1. The magnetizing field was applied and subsequently removed by setting appropriate magnetic field boundary conditions at the outer boundaries of the model.

The simulation of the magnetic hysteresis loop (as outlined in the Hysteresis loop section) was carried out by modifying the boundary conditions corresponding to zero-field-cooling (ZFC) magnetization. In this case, the magnetizing field, Bz, was increased from 0 T to 7 T, then decreased to −7 T, and finally increased to 7 T. A ramp rate of ±4.8 T/h was assumed, mirroring the experimental procedure.

The simulation of FCM followed the same process as described previously70. First, an outer magnetic field boundary condition was set such that Bz(t = 0 s) = 7 T (with the same initial conditions being set for the entire model). The magnetic field was subsequently decreased at a rate of −4.8 T/h, mirroring the experimental procedure, until it reached 0 T at t = 5250 s. The model was run (with a zero external magnetizing field) for an additional 100 min (5250 s ≤ t ≤ 11,250 s) to simulate the effect of flux creep.

Finally, isothermal conditions were assumed due to the slow ramp rate of the external magnetizing field. Thus, the superconductor was assumed to remain at a temperature of 5 K.