Impact of intergranular phase variations on the anomalous Nernst effect in Nd-Fe-B permanent magnets

INTRODUCTION

Thermoelectric technology enables the conversion of heat into electricity, and vice versa, offering a promising alternative to meet the growing demand for sustainable energy. Conventional thermoelectric generators (TEGs) operate based on the Seebeck effect, a longitudinal thermoelectric phenomenon where the induced electric field aligns parallel to the temperature gradient. As a result, optimizing thermoelectric output in Seebeck-based TEGs necessitates multiple legs of p- and n-type semiconductors connected in series, typically arranged in a Π-shaped configuration^[1,2]. However, constructing this Π-shaped configuration involves numerous electrode junctions, leading to potential drawbacks such as a low fill factor^[3,4], high power loss^[5], and reduced device durability^[6].

An alternative approach is to develop TEGs based on the transverse thermoelectric effect, where the induced electric field is perpendicular to the temperature gradient. This reduces the number of required junctions, simplifying the TEG design and addressing the limitations of conventional longitudinal TEGs. An important transverse thermoelectric effect for power generation is the Nernst effect. The Nernst effect generates a charge current perpendicular to both the temperature gradient and either an applied magnetic field (H) or the material’s magnetization (M), referring to as the ordinary Nernst effect (ONE) when driven by H, or as the anomalous Nernst effect (ANE) when driven by the M of the material. A major limitation of both ONE and ANE in many materials is the need to apply a continuous H^[7-21], which complicates their application in TEGs. To achieve zero-field operation in transverse Nernst-based TEGs, it is essential to achieve high thermoelectric conversion performance of ANE in magnetic materials with high coercivity (H_c) and remanent magnetization (M_r), such as permanent magnets, and integrate them into TEG devices. Another viable approach is to utilize ONE and ANE with other transverse thermoelectric effects, such as the off-diagonal Seebeck/Peltier effect^[22,23].

Among the currently available permanent magnets, Nd-Fe-B and Sm-Co-based magnets exhibit high H_c and M_r at room temperature^[24-26]. Miura et al.^[27] observed a significant positive anomalous Nernst coefficient (S_ANE) in the SmCo₅-based sintered magnets (+3.5 × 10^-6 VK^-1) and negative S_ANE (-8.7 × 10^-7 VK^-1) in the Nd₂Fe₁₄B-based sintered magnets. By combining these two permanent magnets, Ando et al.^[1] developed a transverse TEG device with a high fill factor that achieved an ANE-driven power generation (power density) of 177 μW (65 μWcm^-2) at a temperature difference of 75 K, using a 273 K heat sink, the record-high value among transverse TEGs utilizing ANE. However, this value is still a few orders of magnitude lower than that of conventional Seebeck-based TEGs. Therefore, enhancing the S_ANE value of the permanent magnets is crucial and requires fundamental research that focuses not only on exploring new materials but also on advancing our understanding of microstructural factors.

Many efforts to enhance S_ANE in magnetic materials are aimed at optimizing the Berry curvature contribution in electronic band structures^[20,28,29]. In contrast, recently, Gautam et al.^[30] demonstrated a new direction for improving S_ANE from the viewpoint of microstructure engineering. The formation of nonmagnetic copper nanoclusters in an amorphous ferromagnetic Fe-based matrix was shown to enhance both electrical conductivity (σ_xx) and thermal conductivity (κ) of the alloys, with an optimal nanocluster size increasing the S_ANE value by 70%. This raises the question of how microstructure engineering, traditionally employed to optimize coercivity and remanence in the permanent magnets^[31-35], influences their σ_xx, κ, and S_ANE.

This study aims to investigate how variations in grain size and intergranular phase (IGP) influence σ_xx, κ, and S_ANE in the Nd-Fe-B permanent magnets. These magnets were fabricated from rapidly solidified Nd-Fe-B ribbon powders with an initial nano-sized grain structure^[36,37]. During the different processing stages of hot-pressing, hot-deformation, and grain boundary diffusion process (GBDP), significant microstructural changes occur^[38-41], which will be systematically analyzed. The findings are expected to shed light on the relationship between microstructural evolution and transport properties in the Nd-Fe-B magnets, contributing to the design and optimization of permanent magnet materials for transverse thermoelectric applications.

EXPERIMENTAL

RESULTS AND DISCUSSION

Figure 1A presents the magnetization curves of the HP and HD samples, with the y-axis representing M and the x-axis corresponding to H. The y-intercept of the graph corresponds to M_r, which represents the sample M at zero H. The saturation magnetization (M_s) of the samples was measured under the maximum H. The HP sample displays a low remanence to saturation magnetization ratio (M_r/M_s) of 0.63 and a slightly rounded demagnetization curve in the second quadrant which is a typical feature for the isotropic permanent magnets. In contrast, the HD sample shows an increase in the M_r/M_s ratio to 0.94, indicating a large degree of texture in the studied magnet^[49]. Furthermore, the demagnetization curves become more square-shaped, reflecting a substantial enhancement in the crystallographic texture of the Nd₂Fe₁₄B grains after hot-deformation. Consequently, the µ₀M_r of the magnet improves significantly, increasing from 0.72 T in the HP state to 1.30 T in the HD state.

Figure 1. (A) Magnetization curves of HP and HD magnets, showing the transition from isotropic to anisotropic magnetic properties; (B) Magnetization curves of HD magnets after Dy-Nd-Cu, Nd-Cu, and Pr-Cu GBDP, illustrating retained anisotropy with varying coercivity enhancements. HP: Hot-pressed; HD: hot-deformed; GBDP: grain boundary diffusion process.

Figure 1B presents the magnetization curves for the Dy-Nd-Cu, Nd-Cu, and Pr-Cu GBDP magnets, with the HD magnet included for comparison. These magnets retain the anisotropic loop shape, though their M is reduced due to the dilution of Nd₂Fe₁₄B phase after the diffusion process. µ₀M_r values for the Dy-Nd-Cu, Nd-Cu, and Pr-Cu GBDP magnets are 1.01, 0.99, and 1.06 T, respectively. In contrast, their μ₀H_c increases significantly, rising from 1.00 T in the initial HD magnet to 2.28 T, 1.62 T, and 1.85 T after the Dy-Nd-Cu, Nd-Cu, and Pr-Cu diffusion process, respectively. The corresponding values of µ₀M_r, µ₀M_s, M_r/M_s, μ₀H_c, and maximum energy product (BH)_max for each magnet are given in Supplementary Table 1 of the supplementary information.

Supplementary Figure 1 presents the XRD patterns of the studied magnets. In the HP magnets, the (410), (214), and (330) reflections dominate, indicating a random grain orientation. After HD, however, the (006), (105), and (004) reflections become prominent, signifying strong c-axis crystallographic alignment^[50,51]. This high degree of alignment is preserved after GBDP, as evident in the XRD of Dy-Nd-Cu, Nd-Cu, and Pr-Cu magnets and in agreement with M_r/M_s ratio data [Supplementary Table 1].

To investigate the microstructural changes following processing, microstructural analyses were conducted on the studied magnets. Due to differences in grain size, the HP magnet was examined using TEM [Figure 2A], while the HD and GBDP HD magnets were observed using SEM [Figure 2B-E]. A backscattered electron scanning electron microscopy (BSE-SEM) image of the HP magnet is provided in Supplementary Figure 2 to illustrate the difficulty of observing its fine-grained structure using SEM. The HP magnet [Figure 2A] exhibits fine, equiaxed grains with sizes less than 100 nm. These isotropic grains contribute to the low M_r/M_s ratio observed in Figure 1A. After HD [Figure 2B], the Nd₂Fe₁₄B grains evolve into well-aligned, platelet-like grains with sizes exceeding 200 nm in lateral direction. Note that the brightly imaged regions in BSE-SEM images indicate the presence of a Nd-rich phase existing in the grain boundary region of the HD sample. This microstructural transformation explains the increase in the M_r/M_s ratio after HD [Figure 1A], which shifts the magnet’s characteristics from isotropic to anisotropic. The observed anisotropic grains after HD is consistent with the previous reports^[38]. However, unlike in^[38], platelet-shaped grains were not observed in the HP magnet in this study, likely due to the short duration (less than five minutes) of the hot pressing process.

Figure 2. (A) BF-TEM image of the hot-pressed (HP) magnet; along with BSE-SEM images of (B) the hot-deformed (HD) magnet; (C) Dy-Nd-Cu grain boundary diffusion processed (GBDP); (D) Nd-Cu GBDP; and (E) Pr-Cu GBDP HD magnets. BSE-SEM: Backscattered electron scanning electron microscopy; BF-TEM: bright-field transmission electron microscopy.

Figure 2C-E illustrates the microstructures of the RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP magnets. These images reveal a significant change in the thickness of RE-rich IGP following RE-Cu GBDP, evident from the increased areal fraction of the bright phase in BSE-SEM images, from 6% in the HD magnet [Figure 2B] to 20%, 19%, and 28% after Dy-Nd-Cu, Nd-Cu, and Pr-Cu GBDP, respectively [Figure 2C-E]. The formation of a thick RE-rich IGP, which magnetically isolates Nd₂Fe₁₄B grains, is known to enhance coercivity in the GBDP magnets^[39-41]. In addition, grain misorientations and the increased volume fraction of the IGP observed in the RE-Cu GBDP magnets likely account for the reduced M_r in these magnets, as presented in Figure 1B.

We investigated the microstructure and the distribution of constituent and diffused elements in HD and RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP magnets using high-angle annular dark field (HAADF)-scanning transmission electron microscopy (STEM) and STEM-energy dispersive X-ray spectroscopy (EDS) techniques [Figure 3A-D]. Our observations revealed an increase in thickness and segregation of RE-Cu elements within the IGP of the GBDP magnets. Additionally, we identified the formation of (Nd,Dy)₂Fe₁₄B and (Nd,Pr)₂Fe₁₄B phase on the outer surfaces of Nd₂Fe₁₄B grains in the Dy-Nd-Cu and Pr-Cu GBDP magnets, respectively [Figure 3B and D]. Since Dy₂Fe₁₄B and Pr₂Fe₁₄B exhibit larger magnetic anisotropy fields than Nd₂Fe₁₄B at room temperature^[52], the formation of Dy-rich and Pr-rich shell regions in the Dy-Nd-Cu and Pr-Cu GBDP magnets contributes to a higher coercivity compared to the Nd-Cu GBDP magnet.

Figure 3. STEM-HAADF images and STEM-EDS elemental maps showing the distribution of constituent and diffused elements for (A) the HD magnet; (B) the Dy-Nd-Cu GBDP HD magnet; (C) the Nd-Cu GBDP HD magnet; and (D) the Pr-Cu GBDP HD magnet. HD: Hot-deformed; GBDP: grain boundary diffusion process; STEM-HAADF: high-angle annular dark field scanning transmission electron microscopy; STEM-EDS: STEM-energy dispersive X-ray spectroscopy.

Figure 4A summarizes the σ_xx of the studied magnets, showing a slight increase after HD and subsequent GBDP. No notable differences in σ_xx were observed among the RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP magnets. Figure 4B presents the κ and the lattice contribution to thermal conductivity (κ_lat) of the studied magnets. The κ_lat is extracted by subtracting the electronic contribution (κ_e) from the total κ_lat: = κ - κ_e. The κ_e is estimated using the Wiedemann-Franz Law κ_e = Lσ_xxT, where L represents the Lorenz number (2.44 × 10^-8 WΩK^-2) and T is the absolute temperature^[30]. Figure 4B shows that the κ_lat exhibits a slight increase after HD but rises significantly following GBDP. This contributes to the increase in κ in the HD and GBDP magnets compared to the HP magnet. The κ_e values of the studied magnets are presented in Supplementary Figure 3; κ_e shows only a slight increase after HD and GBDP.

Figure 4. (A) Electrical conductivity (σ_xx) and (B) total thermal conductivity (κ) and lattice thermal conductivity (κ_lat) of the HP, HD, and RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP HD magnets. HP: Hot-pressed; HD: hot-deformed; GBDP: grain boundary diffusion process.

The observed increase in σ_xx and κ following HD and GBDP can be attributed to the grain growth^[53,54]. However, the substantial rise in κ_lat after GBDP suggests the involvement of an additional mechanism. To investigate this, microstructural analyses comparing the IGP of HP, HD and GBDP magnets-represented by the Dy-Nd-Cu GBDP magnet-were conducted, as illustrated in Figure 5A-C. The HP magnet exhibits a thin amorphous IGP [Figure 5A], which is retained in the HD magnet [Figure 5B]. However, this phase transforms into a thick crystalline IGP after GBDP [Figure 5C and D]. This crystallization of the IGP after GBDP is likely a key factor driving the pronounced increase in κ_lat, as crystalline IGPs typically exhibit higher phonon mean free paths and reduced phonon scattering compared to their amorphous counterparts, thereby improving κ^[30,55,56]. Additionally, the prolonged heat treatment during GBDP may also contribute to the increase in κ_lat by reducing point defect scattering. The reduced content of ferromagnetic elements (Fe + Co) in the IGP after GBDP contributes to the enhanced coercivity [Figure 1B] by reducing the M of the IGP^[39-41,57]. However, the impact of this compositional change on σ_xx and κ still requires further investigation. The mechanism of IGP thickening and crystallization can be described as follows. During GBDP, the diffusion source, consisting of eutectic alloys, melts and infiltrates the magnet through the grain boundaries. This process increases the thickness and volume fraction of the RE-rich phase in the IGP and modifies its composition. The thickening of the IGP during GBDP, combined with prolonged annealing, likely explains the observed crystallization.

Figure 5. High resolution HAADF-STEM images and superimposed STEM-EDS maps of Nd and Fe obtained across the intergranular phase (IGP) region in (A) HP, (B) HD, and (C and D) Dy-Nd-Cu GBDP HD magnets, along with the corresponding concentration depth profiles of Nd, Fe, Co, Dy, and Cu. HP: Hot-pressed; HD: hot-deformed; GBDP: grain boundary diffusion process; STEM-HAADF: high-angle annular dark field scanning transmission electron microscopy; STEM-EDS: scanning transmission electron microscopy - energy dispersive X-ray spectroscopy.

Here we show the transverse thermoelectric conversion properties of the studied magnets. Figure 6A presents the A_odd and ϕ_odd images of the studied magnets at f = 1.0 Hz and J_c = 1.0 A, measured in the M_r state under zero H. Uniform current-induced temperature modulation is clearly observed across the entire surface of the magnet slabs. To quantitatively estimate the anomalous Ettingshausen coefficient Π_AEE (=S_ANET), the A_odd per unit charge current density j_c, i.e., A_odd/j_c, was measured at different f ranging from 1.0 Hz to 10.0 Hz, as shown in Figure 6B. The A_odd values at each f were obtained by averaging the A_odd values over the marked rectangular area (1.2 × 4.5 mm²) in Figure 6A. A clear decrease in A_odd/j_c with increasing f was observed, which is well replicated by considering thermal diffusion in the sample using the one-dimensional heat diffusion equation in the frequency domain (solid lines in Figure 6B)^[27]. Finally, the steady-state value of A_odd/j_c, corresponding to f → 0 Hz (A_odd,_0Hz/j_c), was calculated from fitting the curve in Figure 6B. The signal with ϕ_odd approximately 180º indicates that the bottom surface of the sample (-y direction) is being heated, as illustrated in Figure 6C.

Figure 6. (A) A_odd and ϕ_odd images at J_c = 1.0 A, and f = 1 Hz for HP, HD, and RE-Cu (RE = Dy-Nd, Nd, Pr) GBDP HD magnets; (B) Frequency dependence of the A_odd/j_c (C) ϕ_odd (D) Π_AEE and S_ANE, (E) z_ANET for same studied magnets. HP: Hot-pressed; HD: hot-deformed; GBDP: grain boundary diffusion process; S_ANE: anomalous nernst coefficient; AEE: anomalous Ettingshausen effect.

The Π_AEE and S_ANE values were calculated using $$ \prod_{\mathrm{AEE}}=S_{\mathrm{ANE}} T=\frac{\kappa\left|\Delta T_{\mathrm{AEE}}\right|}{j_{\mathrm{c}} L} $$, where L is the sample thickness and ∆T_AEE represents the temperature difference between the top and bottom surfaces of the sample induced by AEE. This temperature difference is determined as ∆T_AEE = 2A_odd,_0Hz (M_s/M_r)^[45]. An M_s/M_r correction is applied to address the incomplete saturation of the studied magnets’ M during the AEE measurements. To ensure accurate determination of the M_s/M_r values for the LIT sample slabs, their magnetic hysteresis loops were measured using a pulse B-H tracer [Supplementary Figure 4], and the results are summarized in Supplementary Table 2. The estimated values of Π_AEE and S_ANE are summarized in Figure 6D. The Π_AEE (S_ANE) increases from -7.9 × 10^-5 V (-2.6 × 10^-7 VK^-1) in the HP state to -1.3 × 10^-4 V (-4.4 × 10^-7 VK^-1) after HD. Further GBDP using Dy-Nd-Cu results in additional increments, reaching to -1.5 × 10^-4 V (-5.0 × 10^-7 VK^-1). It is worth noting that, due to the crystallographic texture formed during hot-deformation, the transverse thermopower at the M_r state (zero H) in HD and RE-Cu (RE = Dy-Nd, Nd, and Pr) GBDP Nd-Fe-B magnets is nearly identical (85%-86%) to that of S_ANE.

The transverse thermoelectric conversion performance of ANE was evaluated using the dimensionless figure of merit, z_ANET, expressed as $$ \frac{S_{\mathrm{ANE}}^{2} \sigma_{x x} T}{\kappa} $$^[45]. As shown in Figure 6E, z_ANET increases significantly after HD, primarily due to the substantial rise in S_ANE. However, while GBDP further enhances S_ANE, the concurrent increase in κ partially counterbalanced this improvement, resulting in slight increment in z_ANET for the Dy-Nd-Cu and Pr-Cu GBDP magnets. The highest z_ANET value achieved in this study is 8.1 × 10^-6, observed in the Nd-Cu GBDP magnet.

S _ANE can be decomposed into two components: S_ANE = ρ_xxα_xy - ρ_AHEα_xx ≡ S_I + S_II^[6,28]. Here, ρ_xx (= 1/σ_xx) is the longitudinal electrical resistivity, α_xy is the transverse thermoelectric conductivity, ρ_AHE is the anomalous Hall resistivity, and α_xx is the longitudinal thermoelectric conductivity. The S_I component reflects the direct conversion of a temperature gradient into a transverse electric field through α_xy. In contrast, the S_II component represents the transverse electric field due to the Seebeck-effect-induced carrier flow bent by anomalous Hall effect (AHE). S_II can be rewritten as $$ S_{\mathrm{II}}=-\frac{\rho_{\mathrm{AHE}}}{\rho_{x x}} \alpha_{x x} \rho_{x x}=-\tan \theta_{\mathrm{AHE}} S_{x x} $$, where θ_AHE is the anomalous Hall angle.

The S_xx values of the studied magnets are shown in Figure 7A. All the magnets exhibit negative S_xx values, with the absolute values decreasing after HD and subsequent GBDP. Figure 7B shows the ρ_yx as a function of the H for the studied magnets. It is worth mentioning that the ρ_yx curve follows the magnetization curves of the magnets and the slope of ρ_yx as a function of external magnetic field becomes negligible after saturation of magnetization [Figure 7B]. This suggests that the ordinary Hall contribution to the ρ_yx is negligible compared to anomalous Hall contribution in the present study^[58]. Additionally, the graph exhibits H-odd symmetry characteristic, as it is symmetric with respect to the origin. This indicates that the magneto-Seebeck effect also contributes negligibly. The ρ_AHE is determined by extrapolating ρ_yx at μ₀H = 14 T to the zero field. Supplementary Figure 5 presents the ρ_AHE values and the anomalous Hall angle (tanθ_AHE) of the studied magnets. The absolute values of ρ_AHE and tanθ_AHE are gradually decreased after GBDP. Notably, although ρ_AHE slightly decreases after HD, tanθ_AHE increases. Further studies are required to clarify the underlying mechanism of these changes. Figure 7C presents the contributions of S_I and S_II to the total S_ANE of the studied magnets. All the Nd-Fe-B magnets show comparable magnitudes for S_I and S_II, but with opposite signs. The S_I values for the studied magnets remain relatively constant after HD and GBDP, with slight reductions observed in the Nd-Cu and Pr-Cu GBDP magnets. In contrast, the S_II values exhibit a clear and significant decrease as the magnets undergo HD and GBDP, which correlates with the substantial changes in S_xx and ρ_yx [Figure 7A and B] observed following these treatments. This result suggests that microstructural changes following HD and GBDP of rapidly solidified Nd-Fe-B magnets have a more pronounced effect on S_II than on S_I. The reduction of the opposing S_II was observed alongside grain growth and crystallographic texture formation resulting from HD, as well as the transformation of the amorphous IGP into a crystalline phase after GBDP. This reduction leads to an increase in the total S_ANE. The α_xy, which is often used to evaluate the thermoelectric performance of ANE, is shown in Figure 7D. It can be seen that the absolute value of α_xy only experiences a slight increase after HD and a slight decrease after GBDP, indicating that it remains relatively unchanged throughout these processes.

Figure 7. (A) Seebeck coefficient (S_xx); (B) µ₀H dependence of Hall resistivity (ρ_yx); (C) contributions of S_I and S_II to the S_ANE; and (D) transverse thermoelectric conductivity (α_xy) for the studied magnets. S_ANE: Anomalous nernst coefficient.

Finally, we discuss potential strategies to further enhance the S_ANE and z_ANET of Nd-Fe-B permanent magnets. Based on this study and previous work on commercial sintered magnets^[27], we observe that the contributions from S_I and S_II in Nd-Fe-B magnets tend to have opposite signs, leading to a small S_ANE due to the destructive summation of the two components. The formation of crystalline IGP in Nd-Fe-B magnets could potentially increase S_ANE by reducing the S_II magnitude or changing its sign. However, this increase in S_ANE is somewhat offset by the rise in κ resulting in a smaller enhancement of z_ANET. To further enhance both S_ANE and z_ANET in Nd-Fe-B magnets, it is also necessary to increase the S_I component, without significantly increasing κ. This can be achieved by enhancing α_xy, which is fundamentally tied to the Berry curvature at the Fermi level^[59,60]. Tuning the Berry curvature near the Fermi level or equivalently, tuning the Fermi level position, has been shown to effectively enhance the transverse thermoelectric coefficient α_xy, as demonstrated in ref.^[28,61]. On the other hand, a systematic investigation of microstructural features beyond the crystallinity of the IGP, including grain size, shape, IGP thickness, and IGP continuity is still needed to understand how they impact the S_I and S_II components. Although the S_ANE of the magnets studied in this research is still lower than that of commercial sintered magnets reported in previous work^[27], as shown in Supplementary Figure 6, this study has demonstrated that optimizing the microstructure can effectively increase both S_ANE and z_ANET in Nd-Fe-B magnets, paving the way for further advancements in magnet optimization for TEG applications.

CONCLUSIONS

This study addresses a gap in the understanding of how microstructural features can influence the σ_xx, κ, and S_ANE in the Nd-Fe-B permanent magnets fabricated from rapidly solidified ribbon powders, offering insights into improving S_ANE. The findings reveal that S_ANE increases by 68%, from -2.6 × 10^-7 VK^-1 in the HP state to -4.4 × 10^-7 VK^-1 after HD in which grain growth and crystallographic texture are realized without changing the composition of the magnets. S_ANE further increases to -5.0 × 10^-7 VK^-1 after grain boundary structure and composition change from thin amorphous phase to thick crystalline phase by grain boundary diffusion of Dy-Nd-Cu alloy. The increase in S_ANE is primarily due to the reduction of the opposing S_II component following HD and GBDP. Owing to the crystallographic texture formation after HD, almost the same transverse thermopower as S_ANE is obtained in HD and RE-Cu (RE = Dy-Nd, Nd, and Pr) GBDP Nd-Fe-B magnets at the M_r state (zero H). However, the enhancement in S_ANE following GBDP is somewhat counterbalanced by the increase in κ, resulting in a smaller improvement in z_ANET. These findings demonstrate that microstructural optimization can effectively enhance the S_ANE in ultra-fine grained Nd-Fe-B magnets, providing a promising avenue for advancing materials in applications of transverse thermoelectrics.