Angular dependence of coercivity in isotropically aligned Nd-Fe-B sintered magnets

Nd-Fe-B sintered magnets are widely used in various applications such as traction motors for hybrid electric vehicles, electric power steering motors for automobiles, and compression motors for air conditioners and refrigerators. Recently, the focus for improving the magnetic properties has been on increasing the coercivity to obtain excellent heat durability and demagnetizing field durability for these applications. Furthermore, the need to reduce the use of heavy rare-earth metals such as Dy or Tb is stronger than before because of limited resources and production of these materials. However, the best method to improve the coercivity of these materials is unclear. Therefore, it is necessary to elucidate the coercivity mechanism to determine how to improve this property. The coercivity resulting from coherent rotation of magnetization is expected to be proportional to the anisotropy field of the material expressed by the following equation:¹ 

where H_cJ is the coercivity of the magnet, H_A is the anisotropy field, K is the anisotropy energy and Js is the saturation magnetization. However, the observed coercivity is always much lower compared with the anisotropic field. This phenomenon is called Browns paradox.¹ 

It has been considered that Nd₂Fe₁₄B grains are surrounded by a Nd-rich phase,^2,3 which has been treated as a nonmagnetic phase, and that all grains are magnetically isolated from other grains by this grain boundary phase. It is also considered that there are no pinning sites inside the Nd₂Fe₁₄B grain. Based on this postulation and evidence, the coercivity mechanism has been discussed. Until now, two main coercivity mechanisms have been proposed and discussed. One is that the nucleation of magnetization reversal generates somewhere in the low anisotropy area at the grain boundary, and the reversal of the magnetization is driven by the coherent rotation of magnetization.^4–6 The other is that the nucleation of magnetization reversal happens by activation volume resulting from the thermal fluctuation in the weak anisotropy area near the grain boundary, and the nucleation of reverse magnetization grows to the magnetization reversal domain then the reversed domain spreads to the entire grain.^7–10 However, the physical reality of the activation volume is not clear. It seems that the process to generate nucleation of reverse magnetization by thermal activation volume has caused ambiguity in the coercivity mechanism. The angular dependence of the coercivity has been used the most to evaluate which mechanism agrees with the experimental results. However, it is difficult to evaluate which model agrees well with the experimental results.

In our previous papers, we reported the relationship between the alignments of magnets and their coercivities.^11–14 Through these experimental data, it was found that the coercivity decreased as the alignment improved, and decreased to 70% of the coercivity of an isotropically aligned magnet in perfectly aligned magnets. These results were difficult to explain by the coherent rotation model, but could be qualitatively explained by magnetic domain wall motion. We also evaluated the alignment distribution function using electron back-scattered diffraction (EBSD) and calculated the alignment dependence of the coercivity decrease ratio (β), which is defined in Equation (2):

where H_cJ is the coercivity of the aligned magnets and H_cJisotropy is the coercivity of the isotropically aligned magnet.

From EBSD, the alignment distribution functions (P(θ)) of these magnets were close to the Gaussian distribution. Using these alignment distributions, the alignment (α=Br/Js), in which Br is the residual magnetization and Js is the saturation magnetization, is determined by Equation (3).

The coercivity was also calculated by Equations (4) and (5) using the postulation that every grain independently follows Kondorskii law (1/cos θ).¹⁵ 

where the area from 0 to θ₁ represents the magnetization reversal area.

Through these calculations, it was found that the observed residual magnetization and α agreed well with the calculation results, but the observed β did not agree with our calculations. The entire curve of the alignment dependence of the coercivity decrease ratio is difficult to explain by Eqs. (4) and (5). This result is caused by the calculated β, which includes the postulation that every grain independently followed the magnetic domain wall motion.¹³ However, when we applied the observed β to Equation (5), the calculated θ₁ obtained from Eq. (5) was different from the expected magnetization reversal area (θ₁) that was calculated from Eq. (4), as shown in Figure 1.^14,16 These experimental results showed that our postulation that every grain reverses independently and follows the 1/cos θ law is not true. This discrepancy between the experimental data and calculation results could be explained by considering that the domain wall was pinned at the grain boundary of the tilted grains and that several grains reversed at the same time in one domain wall jump.

The alignment dependence of θ₁, which was calculated using Equation (4), and θ₁, which was calculated using Equation (5) from the observed βs, are shown in Figure 1.¹⁶ From this figure, the observed θ₁s agree with that of the low alignment magnets in the curve of θ₁. When we used alignment distributions (P(θ)), the angular dependence of the coercivity, which is shown in Figure 2, qualitatively explained the experimental results.¹⁶ 

The difference between the observed θ₁ and the calculation results becomes smaller as the alignment nears the isotropical alignment and converges at 45° in isotropically aligned magnets. Therefore, the observed angular dependence of coercivity of an isotropically aligned magnet is expected to agree well with the calculation results. It is important to check the angular dependence of coercivity of isotropically aligned magnets to validate our proposed coercivity mechanism.

An isotropically aligned magnet with a composition of Nd₁₅Co_1.0B₆Fe_bal., which was provided from Shin-Etsu Chemical Co., Ltd (Tokyo, Japan) was used to determine the angular dependence of coercivity. The isotropically aligned magnet was cut and ground to a diameter of 4mm and a length of 4mm for measurement of the magnetic properties. The angular dependence of the coercivity of the isotropically aligned magnet was measured by a vibrating sample magnetometer (VSM) with a high-temperature superconducting coil (VSM-5HSC by Toei Industry Co. Ltd.) with a 5 Tesla magnetic field. Fig. 3 shows a schematic diagram of the angular dependence coercivity for isotropically aligned magnets, which shows the applied magnetic field angle ϕ to the magnetization direction. Before measurement by VSM, the isotropically aligned magnet was magnetized by an 8 Tesla magnetic field to achieve saturation of magnetization.

Figure 4 shows the demagnetization curve of the isotropically aligned magnet. The coercivity monotonically decreased as the angle (θ) between the magnetization direction and the magnetic field increased. Figure 5 shows the experimental data of the angular dependence of coercivity and the calculation results obtained by Equation (5), which was discussed in detail in our previous paper.^14,16

In this equation, Q(ω, τ) is the alignment distribution of Nd₂Fe₁₄B grains and is constant for the isotropically oriented magnet. ω₁ is the magnetization reverse angle calculated using Equation (5).

The angular dependence of the coercive force was calculated using ω₁.^14,16

In this equation, H_cJ(ϕ) is the coercive force measured in a direction parallel to the reverse magnetic field that is applied at an angle ϕ to the easy magnetization direction. The experimental data agreed well with the calculation results until ϕ =60°. This result showed that the angular dependence of the coercivity was well explained by the 1/cos θ law, which showed that every grain independently reverses or the unit of the reverse grains is so small and the angular dependence of the coercivity could be evaluated by the statistical method using Equations (5) and (6).

In the calculation of the coercivity of isotropically aligned magnets, the coercivity monotonically decreased to $1/2$ at 90°. At 90°, the coercivity changed discontinuously to 0, as shown in Fig. 5 At ϕ >60°, a difference between the experimental data and the calculation results was clear. This discrepancy might result from degradation of the surface layer by tooling or low coercivity grains present in the magnets.

This result, whereby the observed angular dependence of the coercivity agreed well with the calculation results that every grain independently follows the 1/cos θ law, reinforces our hypothesis about the magnetization reversal process and the coercivity mechanism of Nd-Fe-B sintered magnets; in aligned magnets, the magnetic domain wall is pinned at the grain boundary of the tilted grain and, when the magnetic domain wall is de-pinned from this pinning site, the magnetic domain wall jumps through several grains. In aligned magnets, the crust of the grains simultaneously reversed because of the pinning and jumping of the magnetic domain wall. It is evident that the observed expanded magnetization reversal area in the aligned magnets reflects the average size of the crust of the grains and the strength of the pinning force of these materials.

This work was partially supported by Japan Science and Technology Agency (JST), Collaborative Research Based on Industrial Demand (No. 20110111). We also appreciate Takehisa Minowa and Hajime Nakamura of Shin-Etsu Chemicals Inc. for providing isotropically oriented magnets.