## Introduction

Magnetic hysteresis is a fundamental function of permanent magnets. The development of permanent magnets, which has a long history since the investigation of KS steel in 19171, can be expressed as the technological effort to enlarge the magnetic hysteresis loop2,3,4,5,6. The magnetic hysteresis of permanent magnets is mainly characterized by the factors of high saturation magnetization, high remanent magnetization, large coercivity, and good squareness. Among them, only the saturation magnetization is an intrinsic property of the material; the others are related to magnetization reversal processes that largely depend on the microstructure of the magnets5,6. These characteristics are the macroscopic consequences of numerous microscopic magnetic domain nucleations and displacements inside magnet bodies. Therefore, magnetic domain observations are critical for permanent magnet studies.

Although magnetic domain observations of permanent magnets have been studied extensively using various techniques, such as magnetooptical Kerr effect (MOKE) microscopy, Lorentz transmission electron microscopy, and hard or soft X-ray magnetic circular dichroism (XMCD) microscopy7,8,9,10,11,12,13,14,15,16,17, all these techniques are limited to observations of magnetic domains on surfaces and thin films. Owing to the presence of a surface demagnetization field and surface defects, the surface magnetic domain structure does not reflect the real magnetic domain structure within a magnet body. Moreover, most conventional magnetic domain observation techniques, such as MOKE microscopy, require mirror-polished surfaces. The polishing process significantly alters intrinsic magnetic properties such as the magnetocrystalline anisotropy of the surface, which prevents researchers from observing the real magnetic domain structure that governs the hysteresis loops of bulk magnets.

Until recently, micromagnetic simulation was the only method utilized to study the magnetic domain structure inside a magnet body. Furthermore, variations in the magnetic domain structure along the magnetic hysteresis have been calculated for various microstructure models of permanent magnets18,19,20,21,22. However, even using modern computation environments, the maximum size of the calculation models has been limited to several hundred cubic nanometers, which is orders of magnitude smaller than the single grain size of a real permanent magnet19,20,21. This limitation is attributed to the need for the cell size of the calculation model to be smaller than the exchange length of the permanent magnet on the order of nanometers, i.e., 4 nm for Nd2Fe14B. A 1/100 or 1/1000 scale model was adopted to emulate the microstructure of a real permanent magnet. Compared with a real permanent magnet, the calculated coercivity of the model tends to be much larger than the experimental coercivity due to the limited model size and the lack of complexity. Recently, a model size of 2 μm with a sufficiently large distribution of grain orientation was reported to reproduce experimental magnetization curves22.

The attempt to experimentally observe the magnetic domain structure inside a magnet body was first demonstrated by magnetic tomography measurements of a soft magnetic Fe-Si rod (8 mm in diameter) with a neutron beam23. Neutrons are suitable for magnetic tomography measurements of bulk magnets because of their sufficiently long penetration depth in magnetic materials. However, neutron magnetic tomography cannot be applied to permanent magnet studies because the 35 μm spatial resolution is considerably larger than the typical magnetic domain size of permanent magnets. Recently, magnetic tomography measurements with soft X-ray24 and hard X-ray25,26,27,28,29 beams have been reported. For these X-ray techniques, a spatial resolution on the order of 100 nm or less is available. The application of soft X-ray magnetic tomography is limited to magnetic thin films because the penetration depth of a soft X-ray beam is on the order of 100 nm24. In contrast, a hard X-ray beam has a moderate penetration depth on the order of micrometers. Furthermore, using a circularly polarized hard X-ray beam, magnetic imaging based on XMCD microscopy is possible for noble metals and rare earth elements16, which are essential elements for hard magnetic materials. Thus, hard X-ray magnetic tomography is considered to be a suitable approach for the three-dimensional (3D) magnetic domain observations of bulk permanent magnets. However, this approach has one technical issue that must be addressed. Most previous reports of hard X-ray magnetic tomography studied the 3D distribution of vector or scalar magnetization components in soft-magnetic Gd-Fe(Co) alloys at zero or a very weak external magnetic field25,26,27. Although the X-ray magnetic tomography measurement of a single grain of a Nd-Fe-B permanent magnet with a diameter of 6 μm was reported, this was performed in a thermally demagnetized state without applying an external magnetic field28. In a more recent study, hard X-ray magnetic tomography measurements with a moderate magnetic field (~0.5 T) were reported29. The magnetic field strength in this study was insufficient to investigate permanent magnets. To perform X-ray tomography measurements of permanent magnets along the magnetic hysteresis curve, a large electro- or superconducting magnet, with a magnetic field of several Tesla, needs to be combined with the X-ray tomography setup; such an experimental system is technically challenging to construct.

In this study, we performed hard X-ray magnetic tomography measurements in an advanced high-coercivity Nd-Fe-B sintered magnet along the magnetic hysteresis curve using separately equipped electro- and superconducting magnets. By varying the magnetic field from +5.5 T to −5.5 T, we successfully visualized the actual magnetic domain dynamics of a Nd-Fe-B-based permanent magnet along its magnetic hysteresis, that is, the nucleation of reversed domains followed by the expansion of domains with domain wall displacements. Scanning electron microtomography was employed for the same observing volume to elucidate the correlation between the 3D magnetic domain structure and the microstructure of the Nd-Fe-B magnet. Consequently, some unexpected behaviors of the local magnetization reversals were observed.

## Materials and methods

### Nd-Fe-B magnet

Nd-Fe-B magnets originally invented in 198230,31 have been widely used as the highest-performance permanent magnets5,6. Recently, Nd-Fe-B magnets have gained further significance because they are indispensable for high-energy-efficiency motors and generators32. The advanced Nd-Fe-B magnet used in this study consists of unprecedentedly fine Nd2Fe14B grains with very high coercivity compared with conventional Nd-Fe-B sintered magnets33,34,35 (see Supplementary S1 and Supplementary Fig. S1). The fine-grained Nd-Fe-B sintered magnet was prepared using a pressless sintering process (PLP) with He-jet milled Ga-doped Nd-Fe-B powder36,37. The average diameter of the Nd-Fe-B alloy powder was 1.2 μm, and the composition was Nd23.2Pr7.39Co0.89Al0.21Cu0.04Ga0.29B0.86Febal in wt. %. The grain boundary modification process using a Tb70Cu30 eutectic alloy further improved the coercivity38. After the grain boundary diffusion process at 860 °C for 10 h and subsequent annealing at 460 °C for 1.5 h, the coercivity reached as large as 2.7 T at ambient temperature.

### Micropillar of the Nd-Fe-B magnet

The micropillar of the Nd-Fe-B magnet was fabricated using a focused ion beam (FIB) technique with Ga ions. The acceleration voltage was varied from 30 kV for rough shaping up to a 30 μm target width to 5 kV for a final target width of 18 μm to minimize the FIB damage on the magnetic properties of the micropillar sample. After the FIB process, a 10-nm-thick Ta film was deposited on the pillar surface as an oxidation protection layer.

### X-ray scalar magnetic tomography measurements combined with strong magnetic fields

X-ray scalar magnetic tomography measurements were performed at BL39XU of SPring-8 using a previously reported setup26. Circularly polarized monochromatic hard X-ray radiation was generated using a Si 111 double-crystal monochromator and a 0.45-mm-thick diamond phase retarder. The X-ray beam was then microfocused to a spot with dimensions of 100 (horizontal) × 130 (vertical) nm2 in full width at half maximum (FWHM) at the sample position. The X-ray energy was tuned to 6.726 keV, where the X-ray magnetic circular dichroism signal takes a maximum at the Nd L2 edge. A micropillar PLP magnet sample was mounted on high-precision stages with x-z translations and rotation θ about the vertical (z) direction. Polarization-averaged X-ray absorption (XAS) projection images, μ(X, Z, θ) = (μ+ + μ)/2, were taken by two-dimensionally scanning the sample in the x-z plane, which is perpendicular to the incident X-ray beam with sampling steps of 100 nm in both the x and z directions. Here, $$\mu ^ + = {\int} {\mu _i^ + \left( {x\cos \theta - y\sin \theta ,x\sin \theta + y\cos \theta } \right)dy}$$ $$\left[ {\mu ^ - = {\int} {\mu _i^ - \left( {x\cos \theta - y\sin \theta ,x\sin \theta + y\cos \theta } \right)dy} } \right]$$ represents the X-ray absorption for the right (left) circular polarization integrated along the X-ray beam direction (y). The magnetic projection images, $${\Delta}\mu \left( {x,z,{\uptheta}} \right) = \mu ^ + - \mu ^ -$$, were recorded simultaneously by monitoring the XMCD signals using a lock-in amplifier locked to an X-ray helicity modulation frequency of 37 Hz. The XAS and XMCD projection images were collected at sample rotation angles of −90° ≤ θ ≤ 85° with a step of 5° (see Supplementary S3 and Supplementary Fig. S3a). The projection image area was 32 µm (x) × 10 µm (z) to cover the region of interest with a height of 10 µm and 15 µm below the tip of the pillar, as indicated in yellow in Fig. 1a. The acquisition time for obtaining the full tomographic data was 11 h.

We used a separate setup of an electromagnet and a superconducting magnet elsewhere in the facility to apply a strong magnetic field to the sample. The sample holder was detached from the X-ray tomography setup and transferred to an external electromagnet (maximum field of 2.8 T) or superconducting magnet (8 T) when applying a magnetic field. The sample holder was reattached to the X-ray apparatus after the application of the magnetic field. X-ray tomography measurements were performed for the magnetic remanent state of the sample at zero field. This procedure was repeated while varying the magnetic field from 5.5 T (positive saturation) to −5.5 T (negative saturation). Thus, all the magnetic tomography data in this study are for the remanent state along the demagnetizing process. Reproducibility in the sample position was within ±3 µm using a kinematic sample holder mount (see Supplementary S2 and Supplementary Fig. S2a), allowing for smooth sample reattachment and reliable data acquisition. High stability of the experimental environment was ensured during the measurements (see Supplementary S2 and Supplementary Fig. S2b).

### Reconstruction of three-dimensional magnetic images

The standard algorithm of the algebraic reconstruction technique (ART) was applied to 36 projection images collected at angles from –90° to +85° with a step of 5° for reconstructing three-dimensional (3D) XAS images. For the reconstruction of 3D magnetic images, a modified ART algorithm was applied to the measured XMCD projections. Among the components of the magnetization vector, $${{{\boldsymbol{m}}}}(x,y,z) = \left( {m_x,m_y,\;m_z} \right)$$, where my is the only component parallel to the easy magnetization axis and is assumed to have a finite value. The mx and mz components are zero because of the strong uniaxial anisotropy of the hard magnet considered in this study. Therefore, we modified the ART algorithm to consider the cosθ factor added in the standard Radon transformation that appears in this restricted case of XMCD projections as $${\Delta}\mu \left( {x,z,{\uptheta}} \right) \propto {\int} {m_y\cos \theta dy}$$. The spatial resolution of the reconstructed 3D magnetic image was estimated to be 360 nm26. The tomographic reconstructions were performed using house-made macros running on Igor Pro 8 (https://www.wavemetrics.com/products/igorpro). ImageJ (https://imagej.nih.gov/ij/) software was used to render the 3D representations and generate movies.

### Scanning electron microtomography

Microtomography was performed after the magnetic tomography measurement. Serial backscattered electron SEM images of the (x-y) plane were obtained by slicing the micropillar with a 50 nm step using an FIB with Ga ions.

### 3D fast Fourier transformation

Fast Fourier transformation (FFT) was applied to the reconstructed 3D magnetic image data to obtain the 3D Fourier transform spectrum of a magnetic image. The central part of the reconstructed magnetic image that corresponds to the sample area was extracted with a size of 17.8 × 17.8 × 9.8 µm3 and processed by FFT without a window function. The resulting resolution in reciprocal space was Δqx = Δqy = 0.056 µm–1 and Δqz = 0.10 µm–1.

## Results and discussion

### Magnetic tomography experiment

The magnet sample was microfabricated into a rectangular pillar (18 μm in width and 40 μm in length), as shown in Fig. 1a. The c-axis of each Nd2Fe14B grain, which is the easy magnetization axis, was aligned along the short axis of the pillar, and the magnetic field was applied in this direction. This direction is referred to as the easy axis in this study. Before the magnetic tomography measurement, the magnetization reversal behavior of a larger pillar sample of 1.1 mm in width and 3 mm in length (easy axis along the short axis) was investigated using the first-order reversal curve (FORC) measurement34,39,40,41,42. Figure 1b shows the FORCs measured using a vibrating sample magnetometer (VSM). The FORC measurement was performed at an elevated temperature of 120 °C to saturate the sample under a maximum field of 2.6 T generated from the electromagnet of the VSM system. The FORCs represent the magnetization curves with the magnetic field H starting from Hr on the major magnetic hysteresis curve. The major magnetic hysteresis loop is filled with FORCs by varying Hr from zero to negative saturation. Then, the FORC diagram shown in Fig. 1c was obtained by plotting the second-order differentiated magnetization with respect to H and Hr. A perpendicularly prolonged pattern is observed on the downward diagonal line of H = | Hr | . This is a typical pattern for a narrow coercivity distribution with a large interaction field41,42, which corresponds to the short-axis demagnetization field μ0NxMs ≈ 0.5 T, where μ0, Nx, and Ms represent the permeability in vacuum, demagnetization factor along the short-axis, and saturation magnetization, respectively. Furthermore, the tail in the high-field region along the line of H = | Hr | was found, indicating the existence of unsaturated grains in the high-field region, which is considerably larger than the coercivity41.

The magnetic tomography measurement was performed using a focused hard X-ray beam at an X-ray energy of the Nd L2 edge at BL39XU of SPring-8. The observed volume is a 10 μm height region located 15 μm from the tip of the micropillar, as shown in Fig. 1a. Thus, the observed volume is estimated to contain ~4000 grains. This large number of grains within the observed volume is sufficient to reproduce the real magnetic domain behavior in bulk Nd-Fe-B sintered magnets.

The micropillar sample was set into an external electro- or superconducting magnet for applying a magnetic field. The sample was then mounted on a magnetic tomography measurement setup. Next, the measurement was performed for the magnetic remanent state of the sample, and this process was repeated by changing the external magnetic field. After finishing all magnetic tomography measurements, microstructure tomography was carried out by sequentially recording the scanning electron microscopy (SEM) of backscattered electron (BSE) images while slicing the sample using a Ga-ion FIB36.

Figure 1d shows the integrated XMCD signal for the observed volume as a function of the external magnetic field H, which corresponds to the averaged magnetization in the volume. The magnetic hysteresis curve of the millimeter-sized sample is also plotted as the black solid line, which was obtained by applying a maximum external magnetic field of 14 T using a VSM with a superconducting magnet. XMCD integration was performed for the volume excluding the outermost surface of 1 μm thickness because the outermost surface grains of the micropillar lose their hard magnetic properties attributed to FIB damage. Therefore, the XMCD curve of the micropillar agrees with the magnetic hysteresis curve of the millimeter-sized sample, indicating that the magnetic domain structure of the micropillar observed by XMCD is unaffected by the surface and well represents that of the bulk magnet.

The reconstructed X-ray absorption (XAS) image of the cross-sectional plane of the micropillar is shown in Fig. 1e. The white spots in the XAS image are nonmagnetic Nd-rich phases, such as metallic Nd, fcc-NdO, and Nd2O336, because the XAS image corresponds to the Nd density distribution. The corresponding plane of the BSE-SEM image can be precisely identified by utilizing these Nd-rich regions of the XAS image as 3D position markers (see Supplementary Video 1), as shown in Fig. 1f. In addition to these white spots, many wavy obscured patterns were observed in the XAS image. Some of them correspond to the Nd-rich regions seen in the BSE-SEM image in Fig. 1f; however, some uncorrelated patterns remain due to the artifacts in tomographic reconstruction from the limited number of projections and the weak transmitted X-ray intensity passing through the relatively thick sample (25 µm in a diagonal direction).

3D magnetic images are reconstructed from the measured XMCD projection images by a scalar magnetic tomographic reconstruction, which represents the distribution of a magnetic vector component parallel to the easy axis. Figure 2a, b show the cutaway 3D BSE-SEM and magnetic images observed from the same region, respectively. A 1 μm thick surface rim of the sample was excluded. The gray regions in the magnetic image are the binarized nonmagnetic Nd-rich regions from the BSE-SEM image. In the 3D magnetic image, the domain structure is observed clearly as red and blue regions for the positive and negative magnetizations along the easy axis, although randomly distributed white ghost patterns are seen. Figure 2c shows the cutaway 3D magnetic images at different external fields μ0H along with the magnetic hysteresis curve starting from positive saturation to negative saturation; the details are presented in Supplementary Fig. S4. The reversed domains appear from the edges at μ0H = −1 T (Fig. 2c, (ii)) because of the damaged soft magnetic surface layer. The areal fraction of the reversed domains increases gradually with decreasing external magnetic field, but they stay located at the edges until μ0H = −2.2 T (Fig. 2c, (iv)). Then, the reversed domains largely expand at μ0H = −2.4 T (Fig. 2c, (v)) and approach negative saturation. The existence of unreversed grains can be confirmed near negative saturation (Fig. 2c, (vii), and (viii)), which is expected from the FORC diagram measurement (Fig. 1c).

### Magnetic domain structures of thermally demagnetized and coercivity states

Figure 3a, b show the cross-sectional 3D magnetic images for the thermally demagnetized and coercivity states (μ0H = −2.7 T), respectively. The red, yellow, and blue frame colors correspond to the (x-y), (y-z), and (x-z) cross-sectional planes, respectively, as schematically shown in Fig. 3c. Although both the thermally demagnetized and coercivity states are macroscopically demagnetized, their microscopic domain structures are considerably different. Stripe domains are clearly found along the easy axis with a width of 2 4 grains for the thermally demagnetized state. The domain walls run straight, irrespective of the grain boundary. On the other hand, large and irregular domains are found in the coercivity state, and their domain walls are pinned at the grain boundary.

In general, these domain structures, i.e., straight stripe domains in the thermally demagnetized state and large domains in the coercivity state, reflect the intergrain exchange and magnetostatic interactions existing in the magnet. Micromagnetics simulation based on the Landau-Lifschitz-Gilbert equation was performed to calculate the domain structure of the thermally demagnetized state with and without the intergrain exchange interaction, as shown in Supplementary Fig. S5. As a result, both magnetic domain structures are similar, indicating that the magnetostatic interaction is the major role for the straight stripe domains in the thermally demagnetized state. On the other hand, the large domains observed in the coercivity state suggest the existence of nonnegligible intergrain exchange interactions even in the advanced coercivity Nd-Fe-B sintered magnet15.

Fourier transformation analysis was performed to examine the characteristic features of the 3D magnetic domain structure. Figure 4a, b compare the fast Fourier transform (FFT) spectra of the 3D magnetic images of the thermally demagnetized and coercivity states (μ0H = −2.7 T), respectively, where q = (qx, qy, qz) represents a wavenumber vector in reciprocal space. These FFT spectra correspond to spin-polarized small-angle neutron scattering (SP-SANS) spectra43. The SP-SANS spectra include both the magnetic and structural information, whereas the FFT spectra in Fig. 4, calculated from the 3D magnetic domain images, comprise magnetic information only. The thermally demagnetized state (Fig. 4a) exhibits the 3D FFT spectra of a thin doughnut-like pattern that lies in the (qx-qz) plane, with the main Fourier components confined to the region of |qy | < 0.2 µm−1. In the inset of Fig. 4a shown with an expanded color scale, the doughnut shape with a radius of |q | ~0.25 µm−1 is more visible. These spectra reflect the stripe domain structure extending along the y-axis (easy axis), as indicated in Fig. 3a. Prominent peak structures at q = (±0.22 μm−1, 0, 0) in the (qx-qy) plane and at q = (0, 0, ±0.22 μm−1) in the (qy-qz) planes correspond to the magnetic structure with a mean half-period of 2.3 μm, which is the magnetic domain width. The coercivity state (Fig. 4b) shows the 3D FFT spectra with a strong component concentrated at the origin in the region of |qx | , |qy | , |qz | < 0.15 µm−1. The spectrum with strong low-frequency components indicates the formation of a large and less periodic magnetic domain structure of the coercivity state, as shown in Fig. 3b. However, Fig. 4b shows that there are side-band peaks at q = (±0.22 μm−1, 0, 0) and (0, 0, ±0.22 μm−1), which are similar to those in the demagnetized state. This fact indicates that the magnetic domain structure of the coercivity state is weakly modulated with the same periodicity as in the thermally demagnetized state. The above analysis suggests that the reciprocal vector component of the magnetic structure qx-z = (qx, 0, qz) with $$\left| {q_{x - z}} \right| = \sqrt {q_x^2 + q_z^2} = 0.22\;\mu {{{\mathrm{m}}}}^{ - 1}$$ is determined as the characteristic length of 2.3 μm of this magnetic domain irrespective of the magnetization states.

### Local magnetic domain behaviors

Some interesting local magnetic domain behaviors were observed from the careful examination of the 3D magnetic domain structure. One is the intrinsic nucleation and annihilation of magnetic domains inside the permanent magnet. In conventional 2D magnetic domain observations, similar nucleation and annihilation of magnetic domains have already been observed15. However, these observations could be the tips of large domains existing underneath the surface, and there is no method to distinguish them from the 2D magnetic domain observations using MOKE microscopy. Figure 5a, b show the real nucleation and annihilation of magnetic domains experimentally observed for the first time. The surrounding areas were dimmed to highlight the grains of interest. At μ0H = −2.2 T (Fig. 5a), where the shoulder of the magnetic hysteresis curve appears, a reversed grain exists in isolation from the other reversed domains. This grain has a prolate shape, and the long axis is tilted from the easy axis on the (y-z) plane. The short axis of the grain is considered to be parallel to the c-axis because Nd2Fe14B grains preferentially grow perpendicular to the c-axis44. This finding indicates that the grain with a tilted c-axis would be the nucleation site, as discussed by the micromagnetic simulations22. At μ0H = −3.5 T (Fig. 5b), which is close to the negative saturation of the magnetic hysteresis curve, an unreversed row of three grains is observed, which also exists in isolation from the other unreversed domains. This is the intrinsic annihilation site of the unreversed domain.

We also observed unexpected local magnetic domain behaviors. Figure 5c shows a reversed magnetic grain chain found at μ0H = −2.4 T. This reversed grain chain bends on the (y-z) plane. Surprisingly, this grain chain on the (x-z) plane runs perpendicular to the easy axis, whereas the magnetization of each grain aligns along the easy axis. This magnetization state is energetically unfavorable because of the strong magnetostatic interaction field along the grain chain. This finding suggests the existence of a locally strong anisotropic intergrain exchange interaction perpendicular to the c-axis to compensate for the magnetostatic interaction field. Such an anisotropic feature of the grain boundary phases was reported in Nd-Fe-B sintered magnets33,45.

Figure 5d shows the existence of a grain in which the magnetization reverses back against the magnetic field. The upper cross-sectional 3D magnetic images at μ0H = −2.2 T show partially reversed grains. The reversed domain seems to come from the outer upper region of the observation area because the left and upper sides of the (y-z) and (x-z) planes, respectively, are the top end of the observation region. This reversed domain largely shrinks on the lower cross-sectional 3D magnetic images at μ0H = −2.4 T near the coercivity. Near the coercivity, the reversed magnetic domains largely expanded, as shown in Fig. 2 and Supplementary Fig. S4, with decreasing magnetic field, which caused a significant change in the local magnetostatic interaction field. Thus, it is plausible that this unnatural magnetic domain behavior is a consequence of the reformation of the magnetic domain structure attributed to the large change in the local magnetostatic interaction field.

### Summary

We demonstrated the first 3D observation of magnetic domain behaviors of a bulk permanent magnet using hard X-ray MCD tomography with high field electro- and superconducting magnets. The 3D magnetic domain structure was precisely correlated with the 3D microstructure tomography image obtained from the same volume. The 3D magnetic domain structure dynamics were successfully visualized along the magnetic hysteresis curve. This technique opens up a new stage of permanent magnet study. The characteristic lengths of the thermally demagnetized and coercivity states were evaluated from the FFT analysis, and some unexpected local magnetic domain behaviors were observed. More detailed characteristic features of magnetic domain dynamics can be revealed from the 3D magnetic domain observation method with more statistical analysis, including machine learning technologies. Enhancing the signal intensity and throughput in magnetic tomography measurements are the next challenges for this purpose and for time-resolved 3D microscopy of magnetization dynamics in permanent magnets in the future.