Permanent-magnet-based transverse thermoelectric generator with high fill factor driven by anomalous Nernst effect Open Access

In the urgent issue of sustainable energy solutions, thermoelectric generation technology attracts much attention as a promising avenue for harvesting waste heat and converting it into electricity. The conventional thermoelectric generation is driven by the Seebeck effect, which induces an electric field E in the direction parallel to a temperature gradient ∇T. Thus, the Seebeck effect is classified into a longitudinal thermoelectric effect. Due to the parallel relationship between E and ∇T, a thermoelectric generator (TEG) based on the Seebeck effect needs to consist of many legs of p- and n-type semiconductors, which are arranged in isolation to prevent a short-circuit fault and connected with each other in series on the hot and cool sides by electrode materials to enhance the output power. Such a Π-shaped module configuration imposes two problems that reduce the output power. The first problem of the Π-shaped structure is that the temperature gradient and optimum current density in the thermoelectric materials respectively decrease due to the contact thermal and electrical resistances at the electrode junctions, which especially deteriorate faced on the hot side.^1–3 The second problem is caused by the low fill factor, which is the area density of the active thermoelectric materials within the module. As the fill factor decreases, the power density proportionally decreases, and a part of the input heat flow is wasted as radiation and convection heat losses.^4–6 The fill factor of conventional bulk TEGs based on the Seebeck effect is typically in the range of 30%–60%^5,7,8 and at most ∼70%.⁹ These two problems prevent efficient power generation, resulting in limited applications of the thermoelectric generation technology.

However, conventional transverse TEGs have other problems. The aforementioned second problem regarding the fill factor still remains in conventional transverse TEGs. A scalable Nernst device with the coil geometry has been recently demonstrated with a relatively high fill factor (∼70%) by thinning the insulator layers.¹³ Meanwhile, the fill factor of transverse TEG with thermopile geometry was still 26% because it requires adequate space to form a zigzag circuit by single thermoelectric materials and electrodes.¹⁴ Therefore, an effective way to improve the fill factor in thermopile geometry needs to be established. Furthermore, ANE and the ordinary Nernst effect in many materials work only under external magnetic fields.^10,11,22,23 In the case of ANE, the elongation of magnetic materials along the y-axis increases the ANE-induced thermoelectric voltage, whereas m necessarily directs along the hard axis of shape magnetic anisotropy (z-axis) due to the orthogonal relationship between E_ANE and m. To solve this issue, the use of permanent magnets such as SmCo₅- and Nd₂Fe₁₄B-type magnets has recently been proposed, which enables the magnetic-field-free operation of ANE owing to large coercivity and remanent magnetization.^23–25 However, the use of permanent magnets in thermopile geometry has not been demonstrated so far.

In this study, we present a concept of bulk transverse TEGs based on ANE to realize a high fill factor and magnetic-field-free operation simultaneously. Figures 1(a) and 1(b) show the schematics of the components and module structure for the proposed transverse TEG, respectively. Transverse TEG consists of two different and alternately stacked permanent magnets with positive and negative S_ANE intermediated by thin insulator layers. As a proof-of-concept demonstration, we fabricate permanent-magnet-based transverse TEG comprising many SmCo₅- and Nd₂Fe₁₄B-type permanent magnet slabs having positive and negative S_ANE, respectively.²³ The use of permanent magnets as components in the thermopile structure allows us to easily construct TEG with a fill factor of ∼80%, which is much higher than the fill factors used in the literature.^13,14 All the SmCo₅ and Nd₂Fe₁₄B slabs are confirmed to positively contribute to the total transverse thermoelectric output by the lock-in thermography (LIT) method.^23–28 An ANE-induced thermoelectric voltage is generated without an external magnetic field and its magnitude is ideally equal to the calculated one from S_ANE of SmCo₅ and Nd₂Fe₁₄B. As a result of its structural design, the maximum ANE-induced output power reaches 177 μW at a temperature difference of ΔT = 75 K, which corresponds to an ANE-induced power density of 65 μW/cm². This power density is the largest value among transverse TEGs based on ANE.²⁹ 

The important point of transverse TEG structure in Fig. 1(b) is to utilize not only positive and negative S_ANE values but also the magnetic attractive force due to the remanent magnetization of permanent magnets. The series circuit can be formed by inserting thin insulator layers between the neighboring magnets and electrically connecting the ends of magnets to sum up the transverse thermoelectric voltage in a zigzag manner. The circuit structure drastically increases the fill factor by reducing superfluous spaces for electrical wiring between the magnets. The magnetic attractive forces between permanent magnets allow the construction of a dense stack of themselves and stabilization of the remanent magnetization along the z-axis, owing to the stray field of neighboring magnets.

First, we investigated the thermoelectric and magnetic properties of the SmCo₅- and Nd₂Fe₁₄B-type permanent magnet slabs. We used SmCo₅ and Nd₂Fe₁₄B circular disks with a thickness of 0.5 mm and diameter of 20 mm, having the easy axis of magnetic anisotropy along the thickness direction, which were commercially available from Magfine Corporation. The surface of Nd₂Fe₁₄B disks was coated by a Ni–Cu–Ni plating layer with a thickness of ∼15 µm to prevent oxidization.³⁰ To quantitatively estimate S_ANE of these magnets without applying an external magnetic field, we used the LIT method. The LIT method enables measurement of the anomalous Ettingshausen effect (AEE), the reciprocal effect of ANE, and the estimation of S_ANE through the Onsager reciprocal relation: Π_AEE = S_ANET with Π_AEE being the anomalous Ettingshausen coefficient. The experimental procedure to estimate Π_AEE is the same as that used in previous reports.^23–28  Figure 2(a) shows the measurement results of S_ANE and Π_AEE at 300 K in the absence of external magnetic fields. The S_ANE values for SmCo₅ and Nd₂Fe₁₄B in the remanent states were estimated to be +3.5 × 10⁻⁶ and −8.7 × 10⁻⁷ V/K, respectively, the magnitude and sign of which are consistent with those reported in Ref. 23. Figures 2(b) and 2(c), respectively, show the electrical conductivity σ measured by a four-probe method and the thermal conductivity κ estimated from thermal diffusivity measurements using a laser flash analyzer and differential scanning calorimetry. According to the obtained S_ANE, σ, and κ values, the transverse thermoelectric figure of merit z_ANET (=$S ANE 2 σT/κ$⁠) for SmCo₅ and Nd₂Fe₁₄B was estimated to be 4.2 × 10⁻⁴ and 2.5 × 10⁻⁵ at 300 K, respectively. Meanwhile, we measured the magnetization M curves for these magnets while applying an external magnetic field H along the thickness direction to confirm the remanent magnetization. Figure 2(d) shows the results of the M–H curves obtained at room temperature. Both the SmCo₅ and Nd₂Fe₁₄B slabs show a remanent magnetization comparable to the saturation magnetization and a large coercivity, enabling the magnetic-field-free operation of ANE.

The next step is to fabricate high-density TEG using pairs of SmCo₅ and Nd₂Fe₁₄B slabs. Figure 2(e) shows the photograph of our transverse TEG consisting of 12 alternately stacked SmCo₅/Nd₂Fe₁₄B pairs. We inserted ∼0.05-mm-thick paper towel layers soaked with heat-resistant glue (Aron Alpha Tough-power) between the slabs to avoid reduction of the ANE voltage due to shunting effects. After the glue cured, we cut the stacked disks into a rectangular shape with a size of 8.2 mm (||∇T) × 16.5 mm (||E) × 16.5 mm (|| the stacking direction) so that the Nd₂Fe₁₄B surfaces were exposed on the planes attaching to electrodes and heat source/sink by removal of the Ni–Cu–Ni plating layers. The ends of neighboring SmCo₅ and Nd₂Fe₁₄B slabs were manually and carefully connected by attaching indium to form a zigzag series circuit without short-circuit fault. Then, the stacked SmCo₅ and Nd₂Fe₁₄B slabs were magnetized by applying a pulse magnetic field of +8 T in the stacking direction. The adhesion of slabs is reinforced by their magnetic attractive force. In this structure, the ANE voltage in all the slabs positively contributes to the total voltage owing to the positive (negative) S_ANE of SmCo₅ (Nd₂Fe₁₄B) when the ∇T and m directions in all the slabs are the same. The inset of Fig. 2(e) shows a top view of TEG, which confirms that the slabs are stacked with high density and separated by thin insulator layers.

To characterize the fill factor, we observed the infrared image of the top surface of SmCo₅/Nd₂Fe₁₄B-based TEG [Fig. 2(f)]. The fill factor can be estimated from the contrast of thermally emitted infrared intensity due to the difference in infrared emissivity between the metallic surfaces with low emissivity and paper towels with high emissivity. Here, the line profiles are taken along the z-axis as shown in Fig. 2(g), where the infrared intensity at each z-position is averaged along the y-axis. The contrast of infrared intensity was clearly observed between the slabs and their spaces, and the relative area of the slabs was estimated to be 81%, when the threshold infrared intensity value is set to be 95 [Fig. 2(g)].

where j_c and j_q,AEE denote the charge current density and AEE-induced heat flow density, respectivily. As a result of J_q,AEE generation, the temperature on the surfaces of these magnetic materials is modulated. By extracting the transient first-harmonic temperature modulation signals of the thermal images and transforming them into lock-in phase φ and amplitude A by Fourier analysis, the pure contribution of thermoelectric effects, such as AEE and the Peltier effect, can be visualized without contamination by Joule heating.^26,27 In magnetized permanent magnets, the AEE-induced temperature modulation free from the Peltier effect can be measured in the absence of an external magnetic field in areas far from the sample edges.²³ The spatial distribution of modulation allows us to confirm that all the SmCo₅ and Nd₂Fe₁₄B slabs correctly operate without short-circuit faults. Figures 3(b) and 3(c) show examples of the φ and A images at f = 1.0 Hz and J_c = 1 A, respectively. The thermoelectric signals on the top surface of SmCo₅/Nd₂Fe₁₄B-based TEG were clearly observed with high area density. Importantly, the SmCo₅ and Nd₂Fe₁₄B slabs show spatially homogeneous A signals, which is consistent with the feature of AEE with m along the z-axis [Fig. 3(c)]. This uniformity confirms that J_c flows in a zigzag way [indicated by black arrows in Fig. 3(b)] without short-circuit fault. Because of the opposite direction of J_c and opposite sign of Π_AEE between SmCo₅ and Nd₂Fe₁₄B, the resultant temperature modulation due to AEE shows the same sign in all the slabs, which was confirmed by the almost identical φ values observed on the whole area of the top surface [Fig. 3(b)]. As shown in Fig. 3(c), the SmCo₅ slabs show larger A signals than Nd₂Fe₁₄B slabs due to larger Π_AEE [Fig. 2(a)]. Figures 3(d) and 3(e) show the line profiles of φ and A taken along the z-axis, where the signals are averaged along the y-axis, within the area indicated by a 4.5 mm square in Figs. 3(b) and 3(c), respectively. The Nd₂Fe₁₄B slabs show a large φ delay and rapid drop in A as f increases, which can be explained by the fact that Nd₂Fe₁₄B has lower κ than SmCo₅ [Fig. 2(c)].²⁷ Thus, LIT measurements reveal that all SmCo₅ and Nd₂Fe₁₄B slabs positively contribute to transverse thermoelectric conversion without any short-circuit fault, while the superfluous space for electrical wiring between neighboring slabs is significantly reduced.

We are in a position to demonstrate magnetic-field-free transverse thermoelectric power generation in SmCo₅/Nd₂Fe₁₄B-based TEG. Figure 4(a) shows a schematic and photograph of the setup for four-terminal measurements of thermoelectric power, where TEG was sandwiched by a heater and heat sink. When TEG is used, the current source will be replaced to a load resistor so that one can obtain the output power without an external power supply. The heat sink’s temperature was controlled by flowing coolant at 273 K. The actual ΔT inside TEG was estimated by measuring the thermal image of its side surface coated by black ink. The hot and cool side temperatures (T_h and T_c) are defined as the highest and lowest temperatures of the thermal image in TEG along the heat flow direction.

The magnetic-field-free operation of ANE-induced power generation was demonstrated by measuring the thermoelectric voltage V by applying a load current I_load to TEG. Figure 4(c) shows the I_load dependence of the output power P at various values of ΔT, where P is the product of V and I_load. The high P, 177 µW at maximum when ΔT = 75 K and I_load = 41 mA, was obtained owing to both the large V_oc and small internal resistance of module R_module, which is the slope of the I_load–V curve. The R_module of TEG was estimated to be 9.9 mΩ at ΔT = 21 K, whereas the value calculated from the σ values and dimensions of SmCo₅ and Nd₂Fe₁₄B is 9.1 mΩ; the deviation of R_module from the calculated value due to the electrodes and junctions is only +8%. The inset of Fig. 4(c) shows the ΔT dependences of the maximum output power P_max and power density per unit area ω_max, which is proportional to the fill factor. Owing to the high P_max of 177 μW and high fill factor of ∼80%, the resultant ω_max reaches 65 μW/cm² at ΔT = 75 K, which is the highest value among transverse TEGs based on ANE.²⁹ 

Finally, we discuss future developments and applications of permanent-magnet-based transverse TEG. Since the fill factor of TEG is determined by the thickness ratio between magnetic materials and insulator layers, we can further increase the fill factor and ω_max by using thinner insulator layers. Although the demonstrated P_max of 177 μW can be used, for example, as stand-alone power supply for wireless sensor network systems, further improvement of z_ANET in permanent magnets is required for widespread thermoelectric applications. This is because commercial TEGs based on the Seebeck effect exhibit 70–80 mW/cm² at temperature differences of 60–70 K around room temperature,^32,33 which are still three orders of magnitude larger than that of the presented transverse TEG based on ANE. Decreasing κ and increasing S_ANE in permanent magnets are the significant tasks, especially for those with negative S_ANE, since the small negative S_ANE of Nd₂Fe₁₄B is a bottleneck in the output of our TEG. Additionally, the thermal durability of TEG is limited by that of the remanent magnetization in Nd₂Fe₁₄B-type permanent magnets, which gradually decreases beyond ∼400 K.³⁴ Meanwhile, since the remanent magnetization of SmCo₅- and Nd₂Fe₁₄B-type permanent magnets around room temperature is at the highest level, our TEG enables energy harvesting from everywhere permanent magnets are used.

In conclusion, we constructed ANE-based transverse TEG with high fill factor and demonstrated its magnetic-field-free operation. As a proof-of-concept demonstration, we used SmCo₅- and Nd₂Fe₁₄B-type permanent magnets with large coercivity, large remanent magnetization, and S_ANE of opposite sign. The pairs of SmCo₅ and Nd₂Fe₁₄B slabs were integrated into high-density TEG, whose fill factor surprisingly reaches ∼80%. Using the LIT method, all the SmCo₅ and Nd₂Fe₁₄B slabs were confirmed to positively contribute to transverse thermoelectric conversion without any short-circuit fault. An ANE-induced output power of 177 μW was generated at a temperature difference of 75 K without external magnetic field, which corresponds to the largest power density of 65 µW/cm² among TEGs based on ANE. This study manifests the importance of developing permanent magnet materials with large positive and negative S_ANE values together with large remanent magnetization. The device architecture proposed here will serve as a design guideline for high-performance transverse thermoelectric power generation.

The authors thank H. Sepehri-Amin and Y. Sakuraba for valuable discussions and K. Suzuki and M. Isomura for technical supports. This work was supported by ERATO “Magnetic Thermal Management Materials” (No. JPMJER2201) from JST, Japan, and NEC Corporation.

The authors have no conflicts to disclose.

Fuyuki Ando: Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Validation (equal); Writing – original draft (lead); Writing – review & editing (equal). Takamasa Hirai: Data curation (supporting); Investigation (supporting); Writing – review & editing (supporting). Ken-ichi Uchida: Conceptualization (lead); Funding acquisition (lead); Investigation (equal); Methodology (equal); Project administration (lead); Supervision (lead); Validation (equal); Writing – review & editing (lead).

The data that support the findings of this study are openly available in Zenodo at https://doi.org/10.5281/zenodo.10005678.³⁵