Effects of anisotropy field dispersion and grain boundary on coercivity and squareness ratio for HDDR-processed NdFeB powders Open Access

NdFeB magnetic powders produced by the hydrogenation-decomposition-desorption-recombination (HDDR) process consist of the grains smaller than a single domain particle.^1–5 Therefore, HDDR-processed NdFeB magnets powders are expected to have a high squareness and high coercivity (H_c) for obtaining the high maximum energy product (BH_max) that is required in highly efficient motors of hybrid vehicles or electric vehicles. The squareness ratio was defined as the value of the magnetic field at 90% of the remanent magnetization divided by H_c. However, the squareness ratio is much lower than 1.0, and the H_c of the magnetic powders is lower than a third of the average H_k. As a result, the BH_max of the magnet is lower than that of sintered magnets. Regarding the organization formation process using the HDDR process, a chemical reaction between NdFeB and H₂ occurs, and the NdH₂ phase and Fe₂B phase in the matrix of the α-Fe phase were observed, and after dehydrogenation, fine NdFeB recombines.^4,5 From these processes, we believe that the NdFeB main phase could be covered with a thin Fe-rich layer with low H_k (referred to as a deterioration layer) because there is much α-Fe after hydrogenation. Previous studies have shown that the low H_c is caused by the deterioration layer of a grain surface and the low squareness ratio is caused by anisotropy field (H_k) dispersion.^6,7 The H_k of the deterioration layer was 10% of the main phase of the grain. However, it was unclear how the relation between the magnetic properties of grain boundaries and the H_k dispersion of the grains affect the H_c and the squareness ratio. In this study, we simulated the effects of the H_k dispersion of the grains on the H_c and the squareness ratio in consideration of the magnetic properties of the grain boundaries and the deterioration layer of the grain surface using a micromagnetic simulator.

The H_c and the squareness ratio were calculated from a MH-loop. The time change of magnetization of each grain in the powder was calculated by using an EXAMAG Landau-Lifshitz-Gilbert (LLG) simulator.⁸ The LLG equation shown below is calculated by FEM.

M is the magnetization, t is the time, γ is the gyromagnetic ratio, H_eff is the effective field (which is the sum of the applied, static, anisotropy, and exchange fields), α is the damping constant, and M_s is the saturation magnetization. Figure 1 shows the calculation model of the NdFeB powder. The powder contains 64 grains, and each grain is a cube with a side length of 40 nm including cube cells with a side length of 2 nm. The grain-boundary is 2 nm wide and the deterioration layer of the grain surface is also 2 nm wide. Magnetic characteristics of the main phase of the grains are typical values of NdFeB magnet materials; the saturation magnetization (M_s) was 1.61 T, H_k was 6077 kA/m, and the exchange stiffness constant between the cube cells was 1.0×10^-11 J/m. As for the grain boundary containing soft magnetic inclusion, M_s was 0.805 T, H_k was 1 kA/m, and the exchange stiffness constant was 6.25×10^-12 J/m.⁸ As for the deterioration layer, H_k was 10 % of the main phase and M_s and the exchange stiffness constant were the same as the main phase.⁶ The grain and the grain boundary each have a uniaxial anisotropy, and the easy axis was along the y-direction as shown in Fig. 1. The exchange stiffness constant between different materials was the average value. The applied magnetic field was ±8000 kA/m. The time-step of the LLG calculation was 1.0×10^-12 sec.

The MH-loops were calculated by changing the angle of the easy axis for the applied magnetic field. The angle was changed from 0 to 90° in 10° increments. Figure 2 shows the angular dependence of the H_c for the nonmagnetic and soft magnetic grain boundaries (a) without and (b) with the deterioration layer. The H_c was calculated by using the projection component of the magnetization on the easy axis. As shown in Fig. 2 (a), the H_c for the soft magnetic grain boundary is clearly lower than that for the nonmagnetic grain boundary. This is because the magnetization of the soft magnetic grain boundary with a low H_k is easy to reverse, and the exchange field between the grain and the grain boundary is added to the applied magnetic field. When some magnetizations of the grains began to reverse, other magnetizations promptly reversed due to magnetic domain wall motion. A low H_c was obtained around 45° for the nonmagnetic and soft magnetic grain boundaries. The ratios of the H_c at 0° and 45° for the nonmagnetic and soft magnetic grain boundaries are 0.63 and 0.66, respectively. These values are higher than the Stoner-Wohlfarth model, and the graphs are left-right asymmetry at 45° due to the static field.

When the grains were overlaid by the deterioration layers (Fig. 2 (b)), the H_c was angle independent until 45° regardless of whether the grain boundary had magnetic material. For angles over 45°, the H_c increased like the Stoner-Wohlfarth model. Thus, the magnetizations reversal process was incoherent rotation, which was dominated by the deterioration layer. First, the magnetization of the deterioration layer reversed, and then, the magnetization of the main-phase reversed due to the exchange interaction between the deterioration layer and the main phase. The difference in H_cs between both grain boundaries was insignificant and the H_cs were close to the experimental value.

The model and magnetic characteristics, except for the H_ks of the main phase and the deterioration layer, are the same as those of the previous section. The average H_k of the main phase (<H_k>) was 6077 kA/m, and the coefficient variation of H_k (σH_k/<H_k>; σH_k: standard deviation) was changed from 5 to 30% in 5% increments. The <H_k> was assumed as the typical value because the experimental value had yet to be measured. The H_k of the deterioration layer was 10% of the main phase. The σH_k/<H_k> dependences of H_c and the squareness ratio are summarized in Fig. 3. The H_c strongly depends on the σH_k/<H_k> for the powder with the soft magnetic grain boundary and without the deterioration layer (blue line). The H_cs of the other conditions are independent of the σH_k/<H_k>. The H_c is close to the experimental value for the powder with the deterioration layer and the powder with the soft magnetic grain boundary, without the deterioration layer, and a σH_k/<H_k> value of 30%.

The squareness ratio of the soft magnetic grain boundary is 1.0 up to a σH_k/<H_k> value of 25%, because the exchange interaction between the grain and the grain boundary layer causes the magnetic domain wall motion and the magnetization to instantly reverse at H_c. On the other hand, the squareness ratio for the nonmagnetic grain boundary decreases as the σH_k/<H_k> increases. This is because the nonmagnetic grain boundary layer contributes to the pinning of the magnetization reversal, and the magnetization reverses in order from the main-phase of the grain with low H_k. The squareness ratio is close to the experimental value for the powder with the nonmagnetic grain boundary and with a σH_k/<H_k> value from 25 to 30%.

Figure 4 shows the comparison of demagnetization loops between the nonmagnetic grain boundary and the experiment. The powder had the deterioration layer and a σH_k/<H_k> value of 30%. It was clear that the loop for the powder with the nonmagnetic grain boundary was close to the experimental loop, with the squareness ratio being especially close to the experimental value. However, the H_c was slightly larger than the experimental value, and the value of <H_k> + 3σH_k was too large compared to the H_k of the single crystal.⁹ We think c-axis dispersion of the grains should be also considered in the simulation.

When the grain boundary was non-magnetic and the σH_k/<H_k> of the grain was from 25 to 30%, the H_c and squareness ratio were close to the experimental values. In the future, the effects of the H_k and the c-axis dispersions of the grains will be investigated. Moreover, because some magnetic area in the grain boundary was experimentally observed, the effects of the magnetic grain boundary area on the magnetic characteristics will be investigated.

We thank Dr. Mishima of Aichi Steel Corporation for the data and his advice on the experimental bonded magnet.