Quantification of the Coercivity Factor in Soft Magnetic Materials at Different Frequencies Using Topological Data Analysis

The kinetics of magnetic domain structure in soft magnetic materials is crucial for the understanding of their functional properties, such as coercivity and loss. We have developed a high-speed and real-time magnetic domain measurement system based on the magnetic-optical Kerr effect (MOKE) microscope. High-speed evolution of domain structures of yttrium iron garnet (YIG) thin film under ac magnetic field and its frequency-dependent hysteresis curves were measured by the system. Subsequently, we combined persistent homology (PH) with principal component analysis (PCA), a dimensionality reduction method, to extract topological information on domain structure and analyze the complex magnetization process. We successfully extracted physically meaningful features of the frequency-dependent magnetic domain structures. As a result, using the machine-learning outputted features, the coercivity contributing factors of the magnetization reversal process were visualized onto domain structures. We also found that the occurrence of the coercivity factors increases along excitation magnetic field frequency, indicating the increase of loss. These findings provide new insight into the relationship between coercivity and magnetic domain structure dynamics.


Quantification of the Coercivity Factor in Soft Magnetic
Materials at Different Frequencies Using Topological Data Analysis

I. INTRODUCTION
W ITH the rapid development of electric vehicles around the world, the demand for power electronics technology is increasing.To improve the drive efficiency of motors and inverters of electric vehicles, soft magnetic materials with low-energy loss (iron loss) under ac magnetic field are needed.For example, loss in armature cores of generators is a problem even at relatively low frequencies such as 50-60 Hz [1].
In soft magnetic materials, it is important to elucidate the influence of microstructure on functional properties.In particular, the external magnetic field required for magnetization reversal is called coercivity, a physical quantity that leads to magnetic losses.The coercivity is quite sensitive not only to the structure of the material but also to the magnetization process due to the external magnetic field.it has been suggested that losses originating from magnetic wall motion have different origins depending on the magnetization process [2].Therefore, the complex dynamics of the magnetic domain structure are considered to have a significant effect on the losses.Domain wall (DW) dynamics have been quantitatively studied for simple systems such as magnetic nanowire DW engineering, vortex gyration, and pulse nucleation [3], [4], [5].However, it is not easy to quantitatively explain the causal relationship between DW dynamics and losses in complex magnetic domain structures.In this study, to target high-speed magnetization reversal phenomena (up to 240 Hz), we developed a measurement system to acquire both the microscopic magnetic domain structure and the magnetic hysteresis curve, which expresses the macroscopic function.Specifically, we developed a system that combines a Kerr optical microscope and a high-speed camera.The magneto-optical effect, XMCD, and PEEM in combination with the pump-probe method are often used to observe dynamic magnetic domain structure dynamics [6], [7], [8].However, these methods are not suitable for capturing stochastic and non-reproducible magnetization reversal phenomena because they take time averages of magnetic domains.We focused on real-time observation using a high-speed camera to analyze the randomness of the magnetization process.
Subsequently, we applied topological data analysis (TDA) and machine learning to clarify the mechanism of the domain structure movement.TDA is a topology-based method for quantifying the topological information of a material's structure and has been applied to various fields of materials science, including glass, polymers, and magnetic materials [9], [10], [11].A combination of TDA and material science has succeeded in connecting materials' structure and function, such as static domain structure propagation and energy loss [12], [13], [14].Therefore, it is expected to be able to link complex magnetic domain dynamics with its function (coercivity) by extracting features of magnetic domain dynamics using TDA.

A. Development of High-Speed Measurement System
We have developed a real-time measurement using a high-speed camera to analyze the dynamics of magnetic  domain structures.In previous studies, real-time observation of magnetic domains in the dc to power frequency range (up to 60 Hz) has been performed.However, there are few examples of measurement systems that synchronously acquire the external field and magnetic domains in high-speed domain structure measurement above the range [15], [16].
A schematic and a photo of the developed system are shown in Fig. 1.For the experiment, magnetic domain observations were performed from dc to 240 Hz excitation frequency, and frequency-dependent hysteresis curves were also obtained.The measurement principle is based on the AxioScope Vario (Zeiss), which is capable of measuring the out-of-plane magnetization distribution due to the polar Kerr effect.The camera can acquire images at up to 10 000 frames/s, which enables clear images of magnetic domains.To synchronize the acquisition of the magnetic domain image and the external field, we performed a sampling of the voltage applied from the power supply to the electromagnets.
A DAQ device is used for voltage sampling, and the sampled voltage can be converted to the magnetic field by a calibration line.First, we set the data sampling rate of the camera and DAQ device to the same rate.Second, we used a function generator to send the start trigger toward both simultaneously.This makes it possible to acquire the magnetic domain and magnetic field synchronously.Furthermore, the power supply unit and DAQ device can be operated on a PC.Waveform setting and voltage sampling can be performed all at once using homemade software with an original GUI based on C#.Using this system, it is possible to acquire magnetic domains and external fields at data sampling rates of up to 10 kS/s, making it possible to easily acquire large-scale magnetic domain structure data.

B. Obtaining Magnetic Domain Datasets
Bi-substituted yttrium iron garnet (YIG) single-crystal with a film thickness of 320 µm was employed in this study as a validation sample for the developed system.This material has perpendicular magnetic anisotropy and a large Kerr rotation angle, making it easy to obtain clear magnetic contrast [14].Therefore, we considered it an appropriate material for the verification of the measurement system.The external magnetic field was fixed at an amplitude of 350 Oe, and As a pre-processing step, the acquired images were subjected to background processing using saturated images to remove Illuminance irregularities.Assuming out-of-plane domain structures, a Gaussian filter was used for smoothing and Otsu binarization was used for noise processing to transform the magnetization distribution into two components (up and down).The ac hysteresis curve was created by calculating the sum of the brightness of the magnetic domain images as the magnetization M z and plotting it together with the sampled magnetic field.

C. Feature Extraction Using PH
For the quantification of domain structure, we used persistent homology (PH), a method of TDA.This method quantifies the number, size, spatial distribution, and crowding of the structures in images.This is realized by tracking the evolution of connected pieces and holes by calculating the homology groups of an increasing sequence of complexes (called filtration).The specific process is as follows [17].First, a Manhattan distance is assigned to each pixel based on the boundary between black and white pixels [Fig.2(a)].There are two types of assignments: white-based, which considers the white direction as positive, and black-based, which is the opposite.Next, by changing the threshold of the Manhattan distance, the structures are swelled and contracted [Fig.2(b)].During this process, island-like structures and hole structures emerge or disappear by connecting each other.The threshold at which they emerge is called "birth," and the threshold at which they disappear is called "death."Pairs of (b, d) are called generators, and the number of generators is mapped to a 2-D histogram with b and d as axes, resulting in the persistent diagram (PD) shown in Fig. 2(c).
The PD contains the information of connected components and hole components of the image, which are called Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
0th homology (H 0 ) for the former and 1st homology (H 1 ) for the latter.In the case of 0th homology, as shown in Fig. 2(d), the position of the plot on the PD changes corresponding to the structure of images, allowing the quantification of the magnetic domain structure on the PD.Additionally, since the PD contains information on the morphology of the magnetic domain structure, it is guessed that it also includes information on the magnetic energy of the system.Therefore, it is expected that the PD will be a useful descriptor for analyzing energetics in domain structure.In this analysis, feature extraction was performed using Homcloud a Python API capable of executing PH analysis [17].To extract diverse information due to the significant changes in domain shapes depending on frequency, we extracted both 0th and 1st homology information.By combining this with machine learning, we created features that can explain the dynamics of magnetic domain structures.

D. Dimensionality Reduction Using PCA
To create the topological features with high explainability, we performed dimensionality reduction as a machine learning process.As a dimensionality reduction method, we considered principal component analysis (PCA), which enhances interpretability [18].PCA is a representative example of unsupervised machine learning, projecting high-dimensional vectors (1) Each basis vector of the projection matrix is an eigenvector that encompasses the essential information of the entire data, and the vectors after projection are called principal components.The principal components are conventionally named PC1, PC2, . . ., PCm with smaller numbers indicating that they retain more of the original data's information.Furthermore, PCA is a linear mapping, and the obtained eigenvectors are orthogonal to each other.Therefore, the Euclidean distance between the generated data is preserved.In summary, it is expected that a data space that maximizes variance and effectively utilizes the information generated from the original data space.In this analysis, we reduced the dimensions using PCA and examined the meanings of the features.

A. Data Acquisition Result
The frequency dependence of the magnetic domain images and the magnetic hysteresis curves obtained using the developed system are shown (Fig. 3).The hysteresis curves are extracted for ten continuous loops at each frequency.Focusing on the hysteresis curves, they have closed shapes at 5 and 60 Hz. while at 240 Hz, the hysteresis curve is open, and an increase in coercivity was confirmed.Such behavior of the magnetic hysteresis curve expanding under high-frequency magnetic fields has been confirmed in many soft magnetic materials and also in iron garnet materials, including YIG [19], [20].Additionally, focusing on the lower left of the hysteresis curve, as indicated by the bidirectional arrow, dispersion occurs in the position of the nucleation jump with increasing frequency.This suggests that the path of the magnetization process differs with each loop, and dispersion in the energy loss associated with DW movement occurs.
Focusing on the magnetic domain structure near coercivity (Fig. 3), it can be seen that the domain morphology changes in the order of maze → maze + stripe → stripe + bubble as the frequency increases.This trend has also been confirmed in iron garnet materials like YIG [21].Therefore, we succeed in obtaining a reliable dataset.The system developed in this study enabled capturing the random magnetization process at both macroscopic hysteresis and microscale levels.Additionally, PDs for the 0th and 1st homology groups were created for each magnetic domain image.Looking at the PDs, the distribution of generators changes depending on frequency, confirming that features of different magnetic domain structures were extracted well.Also, even at the same frequency, it was confirmed that the shape of the PDs changes with each loop, suggesting that PH can extract information on the randomness of domain structures dependent on frequency.In previous studies, the magnetization process was altered by changing the position of defects, and different magnetization processes and energy losses were quantitatively analyzed using feature extraction with PH [12].
Therefore, in this analysis, it is expected that PD can extract information on different magnetization processes and energy losses, namely, coercivity information in the hysteresis curve for each loop.

B. Feature Engineering Using PH and PCA
Subsequently, we created low-dimensional features representing the magnetic domain structure using PH and PCA.First, 6000 magnetic domain structures (2000 for each frequency) were converted to PDs.During the conversion, we created PDs based on white and black for both the 0th and 1st homology groups, resulting in four PDs for each magnetic domain image.Next, PDs were transformed into feature vectors whose elements are the value of the generators blurred by the Gaussian kernel [17] and combined to create an 8320-D vector.By stacking these vectors for all images, we created a feature matrix of size 6000 × 8320.
Since the features extracted by PH are high-dimensional, we aimed to design features with high interpretability by reducing their dimensions, focusing on the interpretation of the phenomenon.The results of dimensionality reduction are shown in Fig. 4(a).Data points are continuously distributed from saturation to coercivity along the path of magnetization reversal indicated by the black arrow.This trend of continuous data distribution aligns with the previously reported feature extraction results of the quasi-static process in Permalloy and YIG, indicating that feature extraction using PH demonstrates high robustness across various soft magnetic material datasets [13], [14].Additionally, the cumulative contribution rate of PC1 and PC2 was 78.5%, indicating a successful reduction.Therefore, it is suggested that the features created by PH and PCA are robust descriptors for the fast magnetization reversal phenomenon.Focusing on the coercivity region indicated by the red frame, it can be seen that the coercivity points correspond to the line of PC2 = 0, and the absolute value of PC1 increases with frequency.Thus, it is suggested that PC1 is a useful feature representing coercivity and frequency.Furthermore, at 240 Hz, the variance of the distribution is larger and sparser compared to other frequencies, suggesting that randomness in the magnetization process can be extracted using the feature space.
From these observations, it is considered that the extracted features can transform the complex magnetization reversal dynamics into an interpretable form.Moreover, the extracted eigenvector w 1 can be mapped and visualized on the PD of both 0th and 1st homology [Fig.4(b)].The dark red areas on the reconstructed PDs indicate components that contribute to the increase in PC1.It is also considered that the contributing factors to coercivity have been extracted.

C. Visualization of Coercivity Factors
In PH, it is possible to visualize where the generator points on the PDs correspond to in the original structure.(called inverse analysis).Therefore, by conducting inverse analysis using reconstructed PDs [Fig.4(b)], we can visualize the contributing factors of coercivity onto domain structures as marked by red dots [Fig.5(a)].
Looking at the visualization results, at 5 Hz, points were placed in areas corresponding to topological defects such as branches and ends of the magnetic domain structure.Topological defects correspond to areas where the periodic structure of magnetic domains and ends is disturbed, resulting in locally high-energy sites [22].Therefore, the points placed in the visualization results are considered to be areas where the system becomes energetically unstable due to the complexity of the magnetic domains toward coercivity.Similarly, at 60 Hz, it was found that the points are concentrated in areas where the DWs are slightly curved.This can be interpreted as areas surrounded by DWs that become high-energy.Additionally, at 240 Hz, points were placed around bubble domains (enclosed by green frame), indicating that areas where magnetic domains are divided were extracted.The same visualization was performed for the 1st homology group, and it was confirmed that points Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
were placed in similar trends.From these results, it can be considered that the visualized points are sites that inhibit magnetization reversal as contributing factors of coercivity.In this way, we were able to visualize factors contributing to frequency-dependent coercivity in a data-driven manner from magnetic domain structures.
Subsequently, to statistically analyze the occurrence of coercivity factors, the number of visualized factors was counted for all 6000 images, and the counted number for each image is shown as histograms [Fig.5(b)].The number of those factors is counted as the total number of 0th and 1st homology groups.It should be noted that in the vicinity of saturation, magnetic domains are not formed, and the region where the number of coercivity factors is zero is excluded.First, peaks in the histograms were confirmed for all frequencies.The region where the peaks occur (gray area) corresponds to points near coercivity.As shown in the figure, the peak shift in the direction of increasing coercivity factors was confirmed at 240 Hz.In addition, the distribution spreads with frequency in the direction of increasing coercivity factor.From these observations, it is considered that the average occurrence of coercivity factors increases with frequency, contributing to the increase in coercivity.Furthermore, since fast magnetization reversal is considered a non-equilibrium process, it is suggested that the spread of the distribution detects the instability of the system in the high-frequency region.
In this analysis, we were able to demonstrate a new method of coercivity analysis that captures the increase in coercivity from the microstructure of the magnetic domain structure.

IV. CONCLUSION
We have developed a measurement system for magnetic domain structure dynamics and succeeded in observing the randomness of the magnetization process.Through TDA, we were able to create features that describe the magnetization reversal dynamics at high frequencies.The occurrence of coercivity factors visualized by PC1 becomes more prominent at higher frequencies, and we were able to discuss the loss increase in the fast magnetization reversal process and the system stability from the point of view of microscale structure formation.Also, PH can be applied to not only binary images but also grayscale in-plane domain structure images [13], so this analysis process can be extended to more practical materials such as electrical steel.For instance, by comparing the locations of coercivity factors with the distribution of crystal orientation and microstructure measured by electron backscatter diffraction (EBSD), it might be feasible to establish new guidelines for designing materials with low iron loss.The findings of our study offer valuable insights for linking the material's structure and its properties achieving low-loss soft magnetic materials.

Fig. 1 .
Fig. 1.(a) Configuration of the developed measurement system.The sampling is triggered by simultaneously sending a start signal toward the DAQ device and camera.(b) Example photograph of microscope and highspeed camera.

Fig. 2 .
Fig. 2. Schematic explanation of PH.(a) Add Manhattan distance to each pixel.(b) Change threshold of distance and island or hole structures appear and vanish (c) PD corresponding to the filtration process.(d) Relation of PD and binary structure images.sinusoidal magnetic fields of three different frequencies (5, 60, and 240 Hz) were applied.The data sampling rate was 2 kS/s for 5 Hz data and 10 kS/s for 60 and 240 Hz data.The external magnetic field and magnetic domain images were acquired at 2000 sampling points for each frequency.The images were recorded as 16-bit information with a field of view size of 350 × 350 µm (image size: 512 × 512 pixels).As a pre-processing step, the acquired images were subjected to background processing using saturated images to remove Illuminance irregularities.Assuming out-of-plane domain structures, a Gaussian filter was used for smoothing and Otsu binarization was used for noise processing to transform the magnetization distribution into two components (up and down).The ac hysteresis curve was created by calculating the sum of the brightness of the magnetic domain images as the magnetization M z and plotting it together with the sampled magnetic field.

Fig. 3 .
Fig. 3. Frequency-dependent hysteresis curve, domain structure, and PD of YIG film.The magnetic domain images on the hysteresis curves are the ones in the demagnetization process.Black and white square images are the saturated domain structures.

Fig. 4 .
Fig. 4. (a) Dimension reduction result by PCA.The explanation ratio for PC1 and PC2 is 51.1% and 27.4%, respectively.In the coercivity region, PC1 increases with frequency, indicating that PC1 expresses frequency and coercivity.(b) Reconstructed PD by PC1's eigenvector.Dark red areas indicate the increasing factor of PC1.

Fig. 5 .
Fig. 5. (a) Red dots in the domain structure are the contributing factors of PC1, namely the coercivity factor.(b) Histogram of the amount of coercivity factors (red dots) for each image.The peak shift indicates that the average amount of coercivity factor occurrence increases with frequency, indicating the increase of coercivity.
Color versions of one or more figures in this article are available at https://doi.org/10.1109/TMAG.2024.3408002.Digital Object Identifier 10.1109/TMAG.2024.3408002 © 2024 The Authors.This work is licensed under a Creative Commons Attribution 4.0 License.
For more information, see https://creativecommons.org/licenses/by/4.0/Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.