Visualization of magnetic dipolar interaction based on scanning transmission X-ray microscopy

Using scanning transmission X-ray microscopy (STXM), in this report we visualized the magnetic dipolar interactions in nanocrystalline Nd-Fe-B magnets and imaged their magnetization distributions at various applied fields. We calculated the magnetic dipolar interaction by analyzing the interaction between the magnetization at each point and those at the other points on the STXM image.


Introduction
Nd-Fe-B magnets have recently attracted much attention as they have become increasingly used [1] and intensively examined [2,3,4,5,6]. In nanocrystalline Nd-Fe-B magnets, whose particle size is less than the single-domain size, magnetic dipolar interactions become essential in magnetization reversal processes. However, the significance of dipolar energy in these magnets is not yet understood, because a quantitatively measuring dipolar energy in magnetic materials is difficult. Therefore, we consider that it is important to visualize the magnetic dipolar interaction for revealing the nature of the magnetization reversal processes, and demonstrate a visualization of the magnetic dipolar interaction in permanent magnets using scanning transmission X-ray microscopy (STXM).
In STXM, a Fresnel zone plate (FZP) focuses the X-rays to a diffraction-limited focus, and images are acquired by raster-scanning the sample through the focus while measuring the transmitted intensity with a relatively large X-ray detector. In our experiment, we used a FZP with an outermost zone width of 25 nm and a focus size (or spatial Resolution) of about 30 nm. Additionally, we used a combination of a central stop and an order-sorting aperture to block zero-order light not diffracted by the zone plate and higher-order light. The samples, mounted on TEM grids, are first held in the focal position of the beam. The sample is then scanned with respect to the focused soft X-ray beam using an interferometer-controlled piezo stage. Transmitted light is measured using a photomultiplier with a scintillator. Essentially, STXM is a microscope that takes advantage of the X-ray magnetic circular dichroism (XMCD) effect. Because the XMCD effect is proportional to the magnetization of a specific element and is sensitive to the projection of the magnetization along the direction of photon propagation, the STXM image of the magnetic contrast shows the real space distribution of element-specific magnetization with high spatial resolution. Therefore, STXM is a powerful tool to investigate the magnetic microstructure of Nd-Fe-B magnets [7].
In this paper, we analyzed the STXM image of the magnetization distribution with a special attention to the direction perpendicular to the surface. We calculated the potential energy of the magnetic dipolar interaction between the magnetization at each point and those at the other points on the STXM image in order to obtain the distribution of the dipolar energy. Finally, we visualized the magnetic dipolar interaction in the nanocrystalline Nd-Fe-B magnets at remanent states after applying 0, 1, 2, 4, 6, and 10 kOe.

Formalism
To visualize the magnetic dipolar interaction energies, we must calculate them from the magnetization distribution, which can be obtained from the STXM image [8]. The potential energy of the magnetic dipolar interaction U is given by In this study, we assume the sample is thin enough to calculate the magnetization as M = (0, 0, m(x, y)). By substituting M = (0, 0, m(x, y)) into eq.(1), we obtain where r = (x, y, z) is the coordinate of the observation point, r = (x , y , z ) is the coordinate of the source point, 2a is the width of the sample in the x direction, 2b is the width of the sample in the y direction and 2c is the thickness of the sample. The relation between the magnetic dipolar interaction energy E d and the magnetic field H is given by where H z is the z component of H. On the other hand, the relation between U and H is given by H = −∇U . Therefore, we can obtain E d as the function of U as follows: From eq.(2), the term of U in eq.(4) is given by where r xy = √ (x − x ) 2 + (y − y ) 2 is the length between the observation point and the source point. By using eq.(4) and eq.(5), we can visualize the distribution of the dipolar energy as shown in Fig.1(b) from the STXM image of the magnetization distribution as shown in Fig.1(a).

Results
By using the method shown in Fig.1, we visualize the magnetic dipolar interaction in the nanocrystalline Nd-Fe-B magnet from the STXM image. We can obtain the two-dimensional magnetic dipolar interaction energy E dxy , which also represents the density of the magnetic dipolar interaction energy E d at (x, y), by substituting eq.(5) into eq.(4)