Machine learning guided processing, microstructure and coercivity mapping in M type strontium hexaferrite

Abstract

M-type hexaferrites are technologically important rare-earth-free permanent magnets in which coercivity (H_c) emerges from a strongly coupled and non-linear interplay between processing conditions, microstructural evolution, and intrinsic magnetic parameters. Here, we develop an experimentally grounded, physics-informed machine-learning framework by studying the multidimensional influence of processing parameters (processing temperature and time), microstructural descriptors (grain size), and intrinsic magnetic properties (saturation magnetization (M_s) and magnetocrystalline anisotropy constant (K₁)) on H_c. Multiple machine-learning models were trained and extreme gradient boosting (XGBoost) yielded the highest predictive accuracy (test and train ). A prediction function using nested for-loops was constructed based on the trained XGBoost model and uniform manifold approximation and projection (UMAP) mapping combined heat map consisting of 5600 points was constructed to analyse experimentally actionable design rules for optimization of H_c as a function of five-dimensional input. High H_c is achieved by maintaining optimal combination of grain size within the single-domain regime (500–700 nm), moderate processing temperatures, limited processing durations to suppress grain coarsening, maximizing K₁ (2.5–3 × 10⁶ erg cm⁻³) and reduced M_s (20–30 emu g⁻¹) to limit demagnetizing effects. These findings suggests that enhancement in H_c requires simultaneous control of processing kinetics, microstructural length scales, and intrinsic magnetic parameters, rather than isolated optimization of any single variable. The extracted trends are consistent with established micromagnetic principles, demonstrating that machine learning trained on experimental data can capture physically meaningful processing-structure-property correlations.

Supplementary Information

The online version contains supplementary material available at 10.1038/s41598-026-55574-x.

Introduction

The accelerating demand for sustainable, high-performance permanent magnets has become a significant challenge in modern materials science, driven by the rapid expansion of electric vehicles, renewable energy technologies, power electronics, and advanced communication systems^1,2. Although, rare-earth-based magnets such as Nd–Fe–B and Sm–Co offers high energy density and strong magnetic performance, their high material cost, supply-chain vulnerability, and environmental impact have intensified global efforts to develop alternative permanent magnet materials that are economically viable and scalable for large-scale production³.

Among the various rare-earth-free magnetic materials, M-type strontium hexaferrites (SrFe₁₂O₁₉) continue to occupy a central position due to their high K₁, high H_c, excellent chemical and thermal stability, corrosion resistance, and low raw material cost. Owing to these advantages, they remain the most widely produced permanent magnets yet paradoxically comprise only 31% of global market share⁴. This discrepancy raises an important question: Are hexaferrites really compatible with rare-earth magnet industry? Many experimental and theoretical studies have manipulated superexchange interactions, defect chemistry and microstructural behavior to tune hexaferrite in order to compete with rare earth magnet industry^5–8.

The most fundamental magnetic characteristic, H_c plays a central role as it directly governs maximum energy product, magnetic hardness, magnetic stability and resistance to demagnetisation. It is governed by a highly coupled interplay between intrinsic magnetic parameters and extrinsic microstructural features. H_c is extraordinarily sensitive to processing-induced microstructural evolution^7–12. Synthesis route, processing temperature and time, thermal profile, atmosphere, and dopant chemistry collectively influence phase formation, grain growth kinetics, defect concentration, and domain structure. Among these variables, grain size plays a particularly important role as it governs the magnetic domain configuration. As grain size decreases, hexaferrites undergo a transition from multi-domain to single-domain structure, thereby suppressing domain wall motion and enhancing H_c. However, further grain refinement beyond the single-domain limit leads to the onset of superparamagnetic behaviour, where thermal fluctuations dominate magnetization and H_c collapses¹¹. Conversely, excessive grain growth during high-temperature or prolonged processing promotes multi-domain formation which significantly reduce H_c despite improvements in crystallinity. This inherent competition between crystallinity, grain growth, and domain structure renders H_c optimization fundamentally non-linear and multi-dimensional. Processing conditions that enhance one aspect of magnetic performance often degrade another, making it difficult to identify optimal synthesis windows through conventional experimental approaches^13,14. As a result, H_c optimization in M-type hexaferrites has traditionally relied on trial-and-error strategies which are time-consuming, system-specific and fails to navigate the coupled effects of multidimensional parameters^6,7,15. Studies including first principal investigation, for instance, impact of ionic substitution (M) on SrFe_12−xM_xO₁₉^12,16–18 and magnetic structure of SrFe₁₂O₁₉^19,20 have provided foundational understanding of magnetic ordering, K₁ and magnetization reversal mechanisms. Yet, establishing a unified framework that simultaneously accounts for processing conditions, microstructural descriptors, intrinsic magnetic parameters, and H_c remains an unresolved challenge.

In recent years, machine learning (ML) has emerged as a powerful tool for materials discovery and optimization, offering the ability to uncover hidden correlations within high-dimensional datasets. Recently, Bhandari et al.²¹ employed ML models for H_c prediction of rare-earth-based permanent magnets. Kovacs et al.²² used ML models to optimize the chemical composition and the microstructure of the rare-earth magnet. In this study, we have used experimental-data-driven ML to extract hidden correlations from high-dimensional datasets. The present dataset reflects realistic experimental conditions, synthesis variability and microstructural evolution encountered in laboratory and industrial settings unlike computational datasets derived from idealized first-principles calculations. Multiple ML models including linear parametric, gradient-based regression, tree-based ensemble, kernel-based probabilistic model and artificial neural networks (ANN) were trained and XGBoost model yielded the highest predictive accuracy (). The trained XGBoost model was used to construct a predictive function and nested for loops were employed over practical range of each five input variable (processing temperature, processing time, K₁, M_s and grain size). This approach generated a total of 5600 input combinations for which corresponding H_c was predicted using the trained model. The UMAP mapping combined heat map enabled the mapping of multivariate parameter space and identification of optimal regions associated with high predicted H_c. Figure 1 depicts detailed methodology used for mapping of processing-microstructure-property relationships in M-type strontium hexaferrite using machine learning models. To our knowledge, this is the first study to quantitatively extract a processing-microstructure-H_c design map for M-type hexaferrites directly from experimental literature data. This study also provides pathway for the rational design of rare-earth-free permanent magnets.

Fig. 1.

A schematic illustration of physics informed machine learning approach using different ML models (linear regression, polynomial regression, eXtreme Gradient Boosting (XGB), stochastic gradient descent (SGD) regression, random forest, artificial neural network (ANN) and Gaussian Process Regression (GPR)) to predict the coercivity of the M-type hexaferrite using multiple input (processing time, processing temperature, grain size, saturation magnetisation and magnetocrystalline anisotropy). The heat map and UMAP for processing-microstructure-property relationships in strontium M-type hexaferrite is also portrayed.

Experimental dataset

The input experimental database of M-type hexaferrite materials was assembled from literature^{4,7,8,15,23–52}. The compiled dataset consist of processing temperature, processing time, K₁, M_s and grain size as the input descriptors. Among the various synthesis variables reported for M-type hexaferrites, processing temperature and processing time were selected as the primary processing descriptors. These parameters play a decisive role in phase formation, densification, defect annealing and grain growth, and are consistently reported across a wide range of synthesis routes, including conventional ceramic processing, sol–gel, co-precipitation, and related chemical approaches. In hexaferrite, H_c primarily depends on grain size due to its well-established influence on magnetic domain structure and therefore, grain size was chosen as the principal microstructural descriptor^8,28,29,53. In addition to microstructure, intrinsic magnetic properties, specifically M_s and K₁ were included. The dependency of H_c on these parameters is explained in supplementary file Sect. 1 and Fig. S1. The H_c was selected as the target property owing to its central importance in determining magnetic hardness and technological applicability of permanent magnets. To ensure physical consistency, data corresponding to incomplete phase formation, poorly crystallized samples, or superparamagnetic behaviour were excluded.

As the compiled dataset originates from independent experimental studies, unavoidable variation arise from difference in synthesis protocols, varied dopants, measurement techniques and microstructural characterization methods. Rather than eliminating this variability, it is intentionally retained, as it reflects realistic experimental uncertainty and processing diversity encountered in laboratory and industrial practice. In order to avoid error, a detailed fundamental study of input database was performed. Figure 2a illustrates the bivariate dependence of H_c on grain size and K₁. As observed H_c first increases with grain size and attains a maximum value at approximately 700 nm, suggesting formation of a single domain structure which enhances resistance to magnetisation reversal. Beyond 700 nm, H_c decreases due to the transition from single domain to multi domain structure. In multi-domain region, H_c decreases as domain wall motion becomes dominant¹¹. Additionally, increased H_c generally results from high K₁ values. The optimum H_c value governs from combinational effect of K₁ and grain size. It should be noted that minor secondary maxima visible in the Fig. 2a are due to the discrete experimental data points. Further, the variation of H_c on processing temperature and grain size is shown in Fig. 2b. In this we have focused the grain size up to 800 nm to show the effect in single domain limit. It can be observed that at low temperature as well as high temperature formation of single domain structure is energetically not favoured. However, the formation of hexaferrite with maximal values of H_c is most favourable upto 1100 °C^7,13.

Fig. 2.

(a) Variation of magnetocrystalline anisotropy constant & grain size (nm) with coercivity (kOe) and (b) processing temperature (°C) & grain size (nm) with coercivity (kOe).

Partial dependence plots of H_c on different inputs (grain size, processing temperature, processing time, K₁ and M_s) is shown in Fig. 3. A strong positive variation of H_c on grain size is depicted, showing dominant dependence of H_c  on grain size in single domain region for M-type hexaferrite. In contrast, H_c shows a negative variation with processing temperature and processing time due to grain coarsening and grain growth under elevated thermal treatment. H_c shows a positive contribution from K₁ and an inverse relationship with M_s, consistent with ^7,11.

Fig. 3.

Partial dependence plots illustrating partial dependence of coercivity on key parameters including grain size, processing temperature, processing time, magnetocrystalline anisotropy constant, saturation magnetisation.

Prior to analysis, all descriptors were converted to consistent units and standardized to remove scale bias among variables. Notably, these plots are learned from experimentally collected data, showing that the ML model successfully captures fundamental structure-processing-property relationships in M-type hexaferrites. These plots shows is dominantly dependent on grain size and provides a pathway for optimization through controlled processing conditions and microstructural engineering.

Machine learning framework

The objective of the ML framework employed in this work is to establish quantitative relationships between H_c and experimentally accessible processing parameters, microstructural descriptors and intrinsic magnetic properties of M-type hexaferrites.

Classical machine learning models

Machine learning (ML) models are computational algorithm trained to establish relationships between a space of independent variables and a space of dependent variable . The complete dataset is represented as a collection of input-output pairs, . The inputs are often concatenated and written as where N denotes the total number of datapoints and each input is a D-dimensional vector.

Let f:X→ Y that maps each input to an output . The objective of ML model is to learn this mapping from the by identifying the underlying pattern between input and output such that

1

where . This is done via splitting of dataset into two different subsets in the ratio of A : (100-A) with A as training dataset on which different M-L models learn the pattern from the data and 100-A is known as test dataset on which M-L models test the dataset to check accuracy of the M-L model. The dataset was split into 80:20 (train: test dataset ratio) for the model deployment.

The relationship was modelled using four different classes of classical ML approaches,

• (i)Linear parametric models (Linear and Polynomial Regression),
• (ii)Gradient-based regression (Stochastic Gradient Descent (SGD) Regression),
• (iii)Tree-based ensemble model (Random Forest and XGBoost Regression).
• (iv)Kernel-based probabilistic model (Gaussian Process Regression).

Artificial neural network

Artificial neural network (ANN) was used to understand the highly complex pattern associated with predicting H_c of the hexaferrite. A non-linear leaky ReLu activation function defined as

2

was applied between the input and hidden layers to enhance learning and training stability often observed in small-scale neural networks. The model was trained using Adam optimisation algorithm (a gradient-based method) implemented using the Tensorflow platform employing Keras interface.

Predictive performance

The predictive performance of the different ML models was evaluated using common statistical parameter, R squared given as

3

where MSE is the mean square error and is given as

4

and variance is represented as

5

Alternatively, R squared can also be expressed as

6

where RSS represents the residual sum of squares which is given as

7

and TSS represents the total sum of squares which is given as

8

The loss function is represented using the sum of MSE and first or second order norms of the target parameters. This results in model convergence and reduces overfitting in parameterized ML models²¹.

As the input values of different variables have several order magnitude difference, therefore data was normalised using scikit-learn StandardScaler. It transforms each feature to dimensionless variable consisting of unit variance and zero mean, , where μ is the mean and σ is the standard deviation of feature x. In this study, we use scikit-learn library for the Python programming language to study predictive data analysis.

Table 1 compares the score of the different ML models used in this study. As observed linear model possess least value of suggesting a complex non-linear relationship between dependent and independent variables²¹. Figure 4 compares predicted and experimental H_c for the test dataset.

Table 1.

Comparison of score of the different machine learning models.

Model
	Train	Test
Linear Regression	0.662	0.663
Polynomial Regression	0.626	0.712
SGD Regression	0.644	0.714
ANN (Adam)	0.893	0.901
Random Forest	–	0.806
Gaussian Process Regression	–	0.793
XGBoost	0.970	0.953

Fig. 4.

(a–g) Comparison of predicted coercivity and experimental coercivity using different machine learning models.

The predictive performance of the different ML models indicates that H_c response is only partially captured by linear, polynomial and SGD regression. The ANN architecture using four layers (dense layers = 40, activation function = Leaky_ReLU, negative slope = 0.01, dense layers = 20, activation function = ReLU, dense layers = 10, activation function = linear, and output layer = 1, and epochs = 1000, batch size = 10) achieved high predictive performance. To prevent the case of overfitting and enhance generalization, the EarlyStopping callback from tensorflow.keras.callbacks library was used during training. The training process was terminated at epoch 120 when no further enhancement in the validation loss was seen and the model attained a training loss of 0.0018. As ANN training is stochastic in nature, slight difference in performance metrics was observed across different runs. This behaviour is due to the random weight initialization, data shuffling, and mini-batch ordering. While train-test splits reduces variability but it cannot be completely eliminated and therefore, the performance of ANN was evaluated based on repeated runs.

The hyperparameters in random forest model were selected as n_estimators = 500, max_depth = 6, min_samples_split = 4, min_samples_leaf = 2 and random state = 42. The Gaussian process regression model utilized a composite kernel that combines a radial basis function and a white noise term. In random forest and gaussian process regression model, leaveoneout cross validation was used using LeaveOneOut from sklearn.model_selection library to evaluate performance of the model. In this every data point in dataset was selected sequentially as test sample, whereas remaining N-1 samples were employed for training. The procedure was repeated for all N samples, so that each observation is used exactly once for validation. The model performance was assessed via averaging the prediction error over all leaveoneout cross validation iterations. For XGBoost, the hyperparameters were selected as learning rate = 0.03 and number of boosting iterations = 100. As observed XGBoost yielded highest predictive accuracy (test and train ) and a mean absolute error of 0.100.

Processing-microstructure-property mapping

A central outcome of this study is the establishment of a quantitative processing-microstructure-property mapping which maximize H_c by optimising processing parameters and microstructure in M-type hexaferrites. As the XGBoost regression model yielded the highest , it was subsequently used to construct a prediction function to estimate the H_c as a function of five dimensional input. To systematically explore the input data, nested for loops were applied over a practical range of each five inputs (grain size was varied from in steps of 100 nm, processing time was varied from 3–6 h with steps of 1 h, processing temperature was varied from 800 to 1200 °C with increment of 100 °C, M_s was varied from with step size of and K₁ was varied from 0.5 to 2.5 erg cm⁻³ in steps of ). The combination of these nested loops resulted in a unique input parameter sets. For every input parameter set, the XGBoost based prediction function was used to predict the H_c value resulting in a dataset consisting of 5600 input-output combinations. It is interesting to note that despite being generated from XGBoost based modelling, the 5600 input-output combination mimics the real experimental behaviour as prediction function is learnt from experimental data.

Given the multivariate nature (five input variables) of the dataset (5600 unique input-output pairs), a nonlinear dimensionality reduction technique i.e. UMAP mapping and heat map was used to visualize the five dimensional space in a two-dimensional embedding which conserved both global and local structural relationships⁵⁴. The main aim of this data-extensive grid-based evaluation is the mapping of multivariate parameter space and identification of optimal regions associated with high predicted H_c.

By integrating ML predictions with physically interpretable trends, this mapping provides practical guidance for tailoring magnetic performance in rare-earth-free permanent magnets. Figure 5a-c illustrate the pairwise mean predicted heatmap for H_c between processing temperature & processing time, grain size & processing temperature and M_s & K₁. Figure 5a shows variation of H_c with processing temperature (800–1200 °C) and processing time (3–6 h). High H_c is generally observed at low processing temperature and a pronounced reduction is observed at 1200 ℃ particularly for high processing time. Figure 5b shows variation of H_c with grain size (400–1000 nm) processing temperature (800–1200 ℃). It can be clearly observed that high H_c is observed at low processing temperature (800–900 ℃) and intermediate grain size (500–700 nm). Figure 5c shows variation of H_c with K₁ and M_s (20–55 emu g⁻¹). High H_c is observed at relatively high K₁ (> 1.5 erg cm⁻³) particularly at low M_s (< 25 emu g⁻¹). Overall, Fig. 5a–c concludes that optimal values of H_c results from low processing temperature (< 1100 ℃), low processing time (< 5 h), intermediate grain size (500–700 nm), high K₁ (> 1.5 erg cm⁻³) and low values of M_s (< 25 emu g⁻¹). These combinational results are consistent with previous experimental studies.

Fig. 5.

Mean predicted heatmap of (a) processing time vs. processing temperature, (b) processing temperature vs. grain size & (c) saturation magnetisation vs. magnetocrystalline anisotropy constant, (d) UMAP decode map of grain size (e) UMAP region mapping of five-dimensional input space (processing time, processing temperature, grain size, magnetocrystalline anisotropy constant & saturation magnetisation) with predicted output coercivity and UMAP decode map of (f) saturation magnetisation and (g) processing time.

The UMAP mapping shown in Fig. 5d–g effectively optimised the combinational set of five dimensional input parameters for high H_c values and elucidate the underlying processing-microstructure-property relationships in M-type hexaferrite. Each point on UMAP plot (Fig. 5e) represents a unique input combination and spatial clustering shows similarity in the input feature space. The colour of each point represents the magnitude of H_c and top 5% H_c region in Fig. 5d–g is marked in red boundary. This red boundary region indicates that optimal H_c is not randomly distributed but corresponds to specific cluster in feature space i.e. associated with well-defined input parameter combination. The UMAP decode map is shown in Fig. 5d,f and g and to understand the influence of interdependent variables, specific regions in the UMAP decode map is highlighted using green, black and blue boundary. The green boundary region represents localized zones associated with high H_c values as it coincides with high H_c values in Fig. 5e and also partially overlap top 5% H_c region. As observed green boundary regions in Fig. 5d, f and g is associated with intermediate grain size in the range 500–700 nm (Fig. 5d), low M_s in the range 20–30 emu g⁻¹ (Fig. 5f) and low mean processing temperature in the range 800–900 ℃ (Fig. 5g) showing that these are the dominant factors which yielded highest H_c value (Fig. 5e). Further, it can be observed that black boundary region associated with grain size in the range 900–1000 nm (Fig. 5d) and M_s in the range 50–55 emu g⁻¹ (Fig. 5f) in the same cluster coincides with low H_c values (Fig. 5e). It shows that differences in microstructure and magnetic response effects the H_c response.

On the other hand, blue boundary region in Fig. 5d, f and g coincides with low H_c values (Fig. 5e). As observed this region corresponds increased grain size in the range 800–1000 nm (Fig. 5d), high M_s in the range 45–55 emu g⁻¹ (Fig. 5f) and high processing temperature in the range 1150–1200 ℃ (Fig. 5g). The separation of this region from high H_c region suggest it represent different input feature space.

Overall, these green, black and blue boundary region provides an interpretation of UMAP embedding and reveals that top 5% H_c region is governed by intermediate grain size, low M_s and low mean processing temperature. On the other hand, low H_c region is associated with excessive grain growth and high M_s. The high processing temperature results in excessive grain growth and multi-domain behaviour resulting in low H_c values.

Overall, the processing–microstructure–property design map established in this work provides a predictive and physically interpretable framework for tailoring synthesis and processing conditions to achieve high-H_c, rare-earth-free permanent magnets. By transforming experimentally reported data into a quantitative design tool, the present approach bridges empirical magnet processing and data-driven materials engineering, offering a transferable strategy for optimization of functional magnetic materials.

Conclusions

In this work, an experimentally grounded, physics-informed machine-learning framework has been developed to quantitatively understand strongly coupled relationship between processing conditions, microstructural evolution and intrinsic magnetic parameters for optimising H_c in M-type hexaferrites. By learning directly from experimentally reported data, ML models incorporated experimental conditions, synthesis variability, and microstructural evolution that govern real materials performance. In order to avoid error, a detailed fundamental study of input database was performed using partial dependence analysis and bivariate dependence plot which identifies H_c is dominantly governed by grain size and reveals a well-defined optimal regime associated with single-domain microstructures.

Multiple ML models including linear parametric, gradient-based regression, tree-based ensemble, kernel-based probabilistic model and ANN were compared. Among the evaluated ML models, ensemble-based XGBoost achieved the highest predictive accuracy, with approaching 0.953, highlighting the strongly non-linear and coupled nature of H_c evolution in hexaferrite system. The trained XGBoost model was used to construct a predictive function and nested for loops were employed over practical range of each five input variable (processing temperature, processing time, K₁, M_s and grain size) generating a total of 5600 input-output combinations. The UMAP mapping combined heat map enabled the mapping of multivariate parameter space and identify experimentally actionable design rules for optimizing H_c in strontium M-type hexaferrites. High H_c is achieved by maintaining grain sizes within the single-domain regime (500–700 nm), employing moderate processing temperatures with limited processing durations to suppress grain coarsening, maximizing K₁ and minimising M_s to limit demagnetizing effects. Further, the dominance of each variable over other is also observed. These findings suggests that enhancement in H_c requires simultaneous control of processing kinetics, microstructural length scales and intrinsic magnetic parameters, rather than isolated optimization of any single variable.

Beyond M-type hexaferrites, the proposed framework can be readily extensible to other magnetic materials, including ferrites, heusler alloys, and functional magnetic oxides, where magnetic performance similarly arises from coupled processing and microstructural effects. More broadly, this work demonstrates that interpretable machine learning trained on experimental literature data can serve as a practical design tool for development of high-performance magnetic materials.

Supplementary Information

Below is the link to the electronic supplementary material.

Author contributions

Harshit Nashier: Conceptualization, Investigation, Methodology, Data curation, Software, Visualization, Formal Analysis, Writing—Original draft preparation, Anuja Dhingra: Conceptualization, Investigation, Methodology, Data curation, Software, Visualization, Formal Analysis, Writing—Original draft preparation, Rajesh Kumar: Conceptualization, Formal Analysis, Visualization, Validation, Writing—Reviewing and Editing, O. P. Thakur. : Conceptualization, Supervision, Funding support, Writing- Reviewing and Editing, Raghvendra Pandey: Conceptualization, Supervision, Validation, Visualization, Writing—Reviewing and Editing.

Data availability

The literature-derived dataset and trained model parameters will be made available upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.