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## Creator

[Fuyuki Ando](https://orcid.org/0009-0003-7789-8170), [Yuya Sakuraba](https://orcid.org/0000-0003-4618-9550), [Weinan Zhou](https://orcid.org/0000-0003-2946-9913), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051)

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This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Fuyuki Ando, Yuya Sakuraba, Weinan Zhou, Takamasa Hirai, Ken-ichi Uchida; All electrical detection of thermal Hall angle by on-slab thermocouples. Appl. Phys. Lett. 2 December 2025; 127 (22): 222408 and may be found at https://doi.org/10.1063/5.0302040.[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[All electrical detection of thermal Hall angle by on-slab thermocouples](https://mdr.nims.go.jp/datasets/8cdfbfa5-6316-429b-9d89-c0bc3f7aa1ae)

## Fulltext

Sample HTPD article for RSIAll electrical detection of thermal Hall angle by on-slab thermocouples   Fuyuki Ando,1,a) Yuya Sakuraba,1,a) Weinan Zhou,1 Takamasa Hirai,1 and Ken-ichi Uchida1,2  1National Institute for Materials Science, Tsukuba 305-0047, Japan 2Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan a) Authors to whom correspondence should be addressed.  ANDO.Fuyuki@nims.go.jp and SAKURABA.Yuya@nims.go.jp   ABSTRACT We propose a simple and reliable measurement method for the thermal Hall effect by the use of thermocouples fabricated directly on a target material. The thermal Hall angle has been typically characterized by measuring diagonal and off-diagonal terms of thermal conductivity tensor with manually attached bulk thermometers such as thermocouples and resistance temperature sensors. However, the presence of contact thermal resistance between the material and thermometers and the ambiguity of actual temperature information under a magnetic field hinder the quantitative evaluation of the thermal Hall angle. In this work, to characterize the temperature gradient with low contact thermal resistance, we deposit a thermoelectric NiCu thin film directly on the target materials, pattern it into a crossbar shape, and attach two Au wires at the hot and cold junctions to form a closed electrical circuit, i.e., an on-slab thermocouple. Owing to the ideal fourfold symmetry of the structure, we can determine the thermal Hall angle simply from the ratio of the measured thermoelectric voltages in the longitudinal (input) and transverse (output) directions for the on-slab thermocouples under the out-of-plane magnetic field. The measurements for Co2MnGa Heusler alloy and Ni slabs by the proposed method find that the thermal Hall effect was clearly detected in a wide temperature range from 60 K to 330 K, which is quantitatively consistent with the literature. This approach offers a simple and reliable route to investigate the thermal Hall effect and reveal the wide variety of carrier transport and thermal conversion phenomena.  MANUSCRIPT Thermal Hall effect (THE), i.e., Righi-Leduc effect is the thermal analog of the electrical Hall effect1–20. THE generates a heat current density jq,THE orthogonal to an input heat current density jq,in and magnetic field H with the magnitude H (ordinary THE) or magnetization M with the magnitude M (anomalous THE) as, 𝐣q,THE =𝜅𝑥𝑦𝜅𝑦𝑦𝐣q,in ×𝐇|𝐻|(or 𝐌|𝑀|) (1) mailto:ANDO.Fuyuki@nims.go.jpmailto:SAKURABA.Yuya@nims.go.jp2        = tan 𝜃THE 𝐣q,in ×𝐇|𝐻|(or 𝐌|𝑀|) where κyy and κxy are the diagonal and off-diagonal terms of thermal conductivity tensor, respectively, and tan 𝜃THE = 𝜅𝑥𝑦 𝜅𝑦𝑦⁄  is the thermal Hall angle. Various carriers such as electron, phonon3,5, and magnon4,8,10,16, can induce THE reflecting the scattering mechanism2,7, Berry curvature5,6,9, and topologically protected edge currents15,16 in condensed matters including superconductors and insulators. Especially for Kitaev quantum spin liquids, the half-quantized 𝜅𝑥𝑦 value is crucial evidence for the existence of the chiral Majorana edge mode11,14,20. Meanwhile, the presence of 𝜅𝑥𝑦 has an influence on the thermoelectric and thermo-spin conversion phenomena21–26. For example, the appearance of a transverse temperature gradient by THE induces the parasitic Seebeck (Nernst) thermopower on the intrinsic transverse (longitudinal) thermopower, resulting in the modulation of thermoelectric performances from those purely by the transverse (longitudinal) thermoelectric effect25. Thus, the quantitative evaluation of tan 𝜃THE reveals the wide variety of carrier transport phenomena and develops materials with thermal conversion functionalities. However, the methodology to determine tan 𝜃THE still has room for improvement with respect to its complexity and reproducibility. The typical tan 𝜃THE  analysis relies on the simultaneous measurements of temperature gradients for longitudinal and transverse directions, xT and yT, respectively, under the assumption of a thermally adiabatic condition in the transverse direction (𝐣q,𝑦 = 𝜅𝑦𝑥∇𝑥𝑇 + 𝜅𝑦𝑦∇𝑦𝑇 = 0), where one can estimate tan 𝜃THE = 𝜅𝑥𝑦 𝜅𝑦𝑦⁄ = − 𝜅𝑦𝑥 𝜅𝑦𝑦⁄  as tan 𝜃THE = ∇𝑦𝑇 ∇𝑥𝑇⁄ . (2) A precise temperature measurement to determine xT and yT is required because the  tan 𝜃THE value is usually less than 1%27. However, the contact thermal resistances between target materials and thermometers and the deviation from the thermally adiabatic condition are the major sources of errors in this process. Furthermore, the magneto-thermoelectric or magneto-resistance (-capacitance) effect18 on thermometers need to be taken into account for the accurate estimation of tan 𝜃THE, which makes the analysis procedure complicated. To solve these problems, K. Tomioka, K. Uchida, R. Iguchi et al. and T. Imamura, T. Hirai, K. Oyanagi et al. developed noncontact tan 𝜃THE measurement and analysis methods using periodic-laser-heating-based and magnetic-field-modulation-based lock-in thermography technique under the nearly adiabatic condition17,19. Their work offers a higher reliability of tan 𝜃THE than ever before but is hardly applied below room temperature due to a drastic decrease in infrared radiation intensity. Meanwhile, J. Xu, J. He, J.S. Zhou et al. developed the electrical detection of THE by the Seebeck voltage contrast using two kinds of wires with different Seebeck coefficients S (Cu and CuNi)28. Their study provides an inspiration for the reproducible and versatile method for THE but did not focus on the quantitative evaluation of 3  tan 𝜃THE by eliminating the parasitic thermoelectric voltage, e.g., the Nernst and spin Seebeck effects derived from the target material. From these backgrounds, a thermal Hall measurement method which one can easily use with high reliability and reproducibility needs to be established. Here, we propose a simple method to measure tan 𝜃THE based on the on-slab thermocouples. Figure 1(a) shows schematic and photographs of the detailed measurement configuration. A target slab sample with a square-shape is bridged between the heat source and heat sink, where the thermally adiabatic condition is realized by connecting the corners of the slab to the heat source/sink and applying jq,in in the diagonal direction. A thermoelectric thin film is directly deposited on the slab to minimize the contact thermal resistance with high reproducibility and patterned into a crossbar shape with a typical microfabrication technique. Metallic wires are attached at the ends of the thermoelectric thin film which serves as the hot and cold junctions of the on-slab thermocouple. Owing to the ideal fourfold symmetry of the crossbar structure with the length 𝑙𝑥(= 𝑙𝑦), tan 𝜃THE can be estimated just from the ratio of the simultaneously measured longitudinal and transverse thermoelectric voltages, Vx and Vy, as below, 𝑉𝑦 𝑉𝑥⁄ = ∆𝑆 ∙ ∇𝑦𝑇 ∙ 𝑙𝑦 ∆𝑆 ∙ ∇𝑥𝑇 ∙ 𝑙𝑥⁄  = tan 𝜃THE.                          (3) Here, S is the difference in S between the thermoelectric thin film and metallic wire, for which the magnetic field dependence is generally non-zero but cancels out for the estimation of tan 𝜃THE. Thus, our method does not need to account the magneto-thermoelectric contribution or convert them into the temperature information. This simple and reliable approach will be a guide in various research fields to easily access THE and reveal the unconventional carrier transport and conversion phenomena. The fabrication process of the on-slab thermocouple system in this study is as follows. We used polycrystalline CMG and commercially available Ni slabs with a size of 7.0 × 7.0 × 0.8 mm3 and 7.0 × 7.0 × 0.5 mm3, respectively. The synthesis process and reference tan 𝜃THE value for CMG slabs were reported in the previous studies12,17,29,30. The surface of the CMG and Ni slabs was mechanically polished using sandpapers with the minimum grit of 2000 and abrasive slurry with the alumina particle size of 3, 1, and 0.3 μm for the following deposition of uniform thin films. Figures 1(b) and 1(c) show the ferromagnetic properties of pieces of CMG and Ni measured by a magnetic property measurement system (MPMS; Quantum Design). As shown in Fig. 1(b), M for both samples saturate less than 1 T at 300 K. Then, we define the saturation magnetization Msat by the average M value in the range of 1–2 T of the M-H curve and show the temperature dependence of Msat, where the Msat value is almost constant below 330 K. We used a NiCu film with a composition ratio of Ni:Cu = 45:55 as the on-slab thermocouple material, whose S was measured to be -30.4 μV/K at room temperature by a Seebeck Coefficient / Electric Resistance 4  Measurement System (ZEM3 M8, ADVANCE RIKO, Inc.). To prevent Vx and Vy from the contamination by the Nernst effect or electrical shunting in the metallic slabs, we first covered a central area of the top surface with a 2-μm-thick insulating epoxy photoresist (SU-8 2002, Kayaku Advanced Materials, Inc.31) by a spin coater and cured it at 180℃. The thermal resistance of 2-μm-thick SU-8 layer is roughly estimated to be ~0.3 K/W from the thermal conductivity of 0.3 W/mK31,32 and the dimension of 5 mm × 5 mm × 2 µm, which is several orders of magnitude smaller than those of commonly used metal wires connecting target materials and bulk thermometers (> 100 K/W)27. A crossbar shape with a width of 0.1 mm and a length of 3.8 mm was patterned with a positive resist on the insulating SU-8 layer using the photolithography technique. A 100-nm-thick NiCu thin film was deposited by co-sputtering Ni and Cu at room temperature using dc magnetron sputtering. Finally, a crossbar pattern of NiCu was obtained via a lift-off process. At the ends of the NiCu crossbar, Ta(10 nm)/Au(100 nm) electrodes with a size of 200 × 200 μm2 were deposited to strictly define the distance between the hot and cold junctions being 3.4 mm as shown in Fig. 1(a). After fixing Au blocks with a size of 200 × 200 × 25 μm3 on the Au electrodes by silver epoxy, we attached Au wires with S of 1.96 μV/K at room temperature33 on the Au blocks by a wire bonding method. Because the Ta/Au electrodes and Au blocks are sufficiently thin, the temperature difference appears mainly in the in-plane direction of the NiCu thin film and in the Au wires connecting the NiCu to the isothermal sample holder electrodes. The electrical insulation between the CMG slab and on-slab thermocouples (> 20 MΩ) was confirmed by a multimeter. Thus, both the electrical insulation and low thermal resistance were achieved by the 2-μm-thick SU-8 layer. We observed the in-plane temperature distribution on the CMG and Ni samples under the application of xT using infrared thermography. The samples with the SU-8/NiCu/Au films was placed on a homemade holder with a multifunction probe for a physical property measurement system (PPMS; Quantum Design), bridged between two Cu blocks in the x-direction with a distance of ~8 mm, one of which acted as a heat source by applying a heater power Pin in an embedded chip heater and the other as a heat sink to apply xT. The top surfaces of the samples were covered with black ink having an emissivity over 0.94 and observed by an infrared camera at room temperature and atmospheric pressure in the absence of a magnetic field. Figure 2(a) shows a thermography image for the CMG sample captured after applying Pin of 0.36 W and waiting for 15 min to stabilize xT.  The area without black ink coating apparently shows lower temperature because of the lower infrared radiation intensity. If the SU-8 layer has a large thermal resistance yielding any temperature difference between the CMG top surface and on-slab thermocouples, the top-view temperature distribution shows a discontinuity between the CMG and SU-8 layer top surfaces. However, the temperature distribution covered with the black ink continuously changes at the edges of the insulating SU-8 layer, which validates a negligibly small contact thermal resistance between the CMG sample and on-slab thermocouples. To characterize xT and yT on the NiCu crossbar, we obtain line profiles for 3.4 × 0.5 mm2 [red rectangular area in Fig. 2(a)] and 5  0.5 × 3.4 mm2 [blue rectangular area in Fig. 2(a)], respectively. Figure 2(b) shows the longitudinal temperature distribution in the red and blue rectangular areas under various Pin for the CMG sample, where an almost linear slope ensures the homogeneous xT in the region of interest. The difference in xT between the red and blue rectangular areas are 9.5 for CMG and 13.5% for Ni, which deviations are inevitable in this diagonal configuration due to the nonuniform heat currents. In this work, to take this nonlinearity of xT into account, tan 𝜃THE is defined as 1.095 × 𝑉𝑦 𝑉𝑥⁄  for CMG and 1.135 × 𝑉𝑦 𝑉𝑥⁄  for Ni. Meanwhile, Fig. 2(c) shows the transverse temperature distribution in the blue rectangular area under various P for the CMG sample. Under Pin of 0.36 W, yT is estimated to be 0.004 ± 0.007 K/mm by the slope of linear fit while xT being -1.256 ± 0.004 K/mm. Thus, under no magnetic field, the applied temperature gradient is solely along x-direction (yT = 0).  We measured the magnetic field dependence of Vx and Vy under various Pin. In the PPMS chamber with a purged state (vacuum level is about 10 Torr of He gas), Vx and Vy were measured using dc nanovoltmeters (Keithly 2182) after reaching each magnetic field and waiting 20 s to stabilize at the thermal equilibrium state. The PPMS cryostat enables a stable temperature and magnetic field control suitable for the sensitive THE measurement27. Figure 3(a) shows the result of Vx for the CMG sample at 300 K. The Vx values are hardly changed by the application of the magnetic field, which is one of the characteristics of the NiCu-based alloy and preferable for the thermal Hall measurement. Figure 3(b) shows the average Vx (Vx,ave) values for various Pin in Fig. 3(a) as a function of Tx obtained from results in Fig. 2(b). The thermopower of the NiCu/Au-based on-slab thermocouple is characterized to be -29.7 μV/K from the slope of linear fit, which agrees with S between the NiCu thin film and Au wire (= SNiCu-SAu = -32.4 μV/K). Thus, Vx is purely attributed to the thermoelectric voltage of the on-slab thermocouple. Meanwhile, Fig. 3(c) shows the magnetic field dependence of Vy for the CMG sample at 300 K. The Vy values reverse depending on the sign of the applied magnetic field and saturate less than 1 T, which obviously reflects the magnetization process of the CMG sample in Fig. 1(b). Thus, the observed Vy predominantly originates from the anomalous THE in the CMG sample. In this study, the saturated Vy (Vy,sat) is defined as the average Vy above 1 T to estimate the anomalous thermal Hall angle tan 𝜃ATHE with the negligible contribution of the ordinary THE. Figure 3(d) shows a plot of Vy,sat as a function of the average Vx (Vx,ave) to directly estimate tan 𝜃ATHE from the slope of the linear fit with the standard deviation. As a result, we determine the tan 𝜃ATHE values of the polycrystalline CMG to be 2.45 ± 0.02% at 300 K, which is comparable to the literature values (2.65 and 2.94%)12,17, ensuring the reliability of the proposed method. The lower tan 𝜃ATHE value than that measured in Ref. 17 is attributed to the lower sintering temperature (850℃) of the CMG polycrystal resulting in the relatively disordered L21-type structure. 6  Now we are in a position to characterize the thermal Hall effect in the wide temperature range. Figure 4(a) shows the magnetic field dependence of tan 𝜃THE at 60, 180, and 300 K for the CMG sample. In this temperature range, tan 𝜃THE clearly reflects the magnetization curves of CMG saturating less than 1 T, which confirms the validity to use Vy,sat for the determination of tan 𝜃ATHE. Figure 4(b) shows tan 𝜃ATHE at temperatures ranging from 60 K to 330 K for the CMG and Ni samples, where tan 𝜃ATHE of -0.53 ± 0.02% at 300 K for the Ni sample agrees with those in the literature (-0.45 and -0.51%)17,34 as well as the CMG sample. The tan 𝜃ATHE values for the CMG sample show the increase trend as temperature increases with the small error bars differing from the magnetization behavior in Fig. 1(c). To explain this temperature dependence of the anomalous THE, L. Xu, X. Li, L. Ding et al. confirmed that the anomalous transverse version of the Wiedemann-Franz law held in the CMG system12, which is a linear relationship between the anomalous Hall effect and anomalous THE: 𝜅𝑥𝑦𝐴  is estimated as 𝐿𝑥𝑦A 𝜎𝑥𝑦𝐴 𝑇 with 𝐿𝑥𝑦A  being the anomalous transverse Lorenz number independent of temperature and 𝜎𝑥𝑦𝐴  being the anomalous Hall conductivity35,36. Because 𝜎𝑥𝑦𝐴  and 𝜅𝑦𝑦 of the CMG system show the moderate temperature dependence from 60 K to 330 K, tan 𝜃ATHE (= 𝜅𝑥𝑦𝐴 𝜅𝑦𝑦⁄ ) monotonically increases with increasing temperature12. To confirm the robustness of our method in the wide temperature range, we performed a thermal cycle test using the Ni sample in the order 330, 60, 330, 60, and 330 K. Figure 4(c) shows the magnetic field dependence of tan 𝜃THE measured at 330 K for three times, where the similar thermal Hall signals are obtained even after the Ni sample experienced the temperature sweep from 60 K to 330 K. As a result, the estimated tan 𝜃ATHE values for 60 and 330 K are reproduced under this thermal cycle condition [Fig. 4(d)], which ensures that the on-slab thermocouple system is mechanically stable against the temperature. Finally, we note the future perspective and limitation of our proposed method. In this study, we chose the polycrystalline CMG slab with the large tan 𝜃ATHE to demonstrate the validity of our method. However, the characterization of tan 𝜃THE lower than 1% for other materials might be problematic especially at cryogenic temperature. In Fig. 4(a), the signal-to-noise ratio of tan 𝜃THE decreases as temperature decreases due to not only the decrease in tan 𝜃THE of the CMG sample but also the decrease in S of the NiCu thin film, which is a typical behavior of the Seebeck effect in metals. Thus, to study the cryogenic phenomena such as the phonon Hall and quantum thermal Hall effects, the development of on-slab resistance temperature sensors instead of thermocouples is required. For the requirement of the deposition and microfabrication of uniform thin films, our proposed method is limited to the materials whose surface can be processed into the mirror finish by mechanical or chemical polishing. In conclusion, we propose a simple and reliable measurement method for tan 𝜃THE based on the on-slab thermocouples. The on-slab thermocouples minimize the contact thermal resistance between target materials and thermocouples and simplify the tan 𝜃THE analysis process just from the ratio of the longitudinal and transverse thermoelectric voltages without considering 7  the magneto-thermoelectric effect or converting them into temperatures and thermal conductivities. As the proof-of-concept demonstration, we prepared a NiCu/Au-based on-slab thermocouples on the CMG sample and performed the tan 𝜃THE measurement in the wide temperature range. The tan 𝜃ATHE values show the increase trend following the anomalous transverse Wiedemann-Franz law of CMG with small error bars and quantitatively agree with the literature values measured around room temperature. This simple and reliable method will be helpful to easily measure tan 𝜃THE and reveal the rich variety of carrier transport and conversion phenomena.  The authors thank K. Suzuki, M. Isomura, N. Kojima, and S. Kuramochi for technical support. This work was supported by ERATO “Magnetic Thermal Management Materials” (Grant No. JPMJER2201) from JST, Japan, and Grant-in-Aid for Scientific Research (KAKENHI; Grant No. 24K17610) from JSPS, Japan.  AUTHOR DECLARATIONS Conflict of Interest The authors have no conflicts to disclose. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.   8  REFERENCES 1 A.W. Smith, and A.W. Smith, Phys. Rev. 5, 35 (1915). 2 Y. Zhang, N.P. Ong, Z.A. Xu, K. Krishana, R. Gagnon, and L. Taillefer, Phys. Rev. Lett. 84(10), 2219 (2000). 3 C. Strohm, G.L.J.A. Rikken, and P. Wyder, Phys. Rev. Lett. 95(15), 155901 (2005). 4 Y. Onose, T. Ideue, H. Katsura, Y. Shiomi, N. Nagaosa, and Y. Tokura, Science 329(5989), 297–299 (2010). 5 T. Qin, J. Zhou, and J. Shi, Phys. Rev. B 86(10), 104305 (2012). 6 L. Zhang, New J. Phys. 18(10), 103039 (2016). 7 K. Sugii, M. Shimozawa, D. Watanabe, Y. Suzuki, M. Halim, M. Kimata, Y. Matsumoto, S. 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Cusack, and P. Kendall, Proc. Phys. Soc. 72, 898 (1958). 34 Y. Onose, Y. Shiomi, and Y. Tokura, Phys. Rev. Lett. 100, 016601 (2008). 35 X. Li, L. Xu, L. Ding, J. Wang, M. Shen, X. Lu, Z. Zhu, and K. Behnia, Phys. Rev. Lett. 119, 056601 (2017). 36 L. Xu, X. Li, X. Lu, C. Collignon, H. Fu, J. Koo, B. Fauqué, B. Yan, Z. Zhu, and K. Behnia, Sci. Adv. 6, eaaz3522 (2020).     10  FIGURES     FIG. 1. (a) Schematic and photographs of thermal Hall measurement configuration using the NiCu/Au-based on-slab thermocouples. The photographs show the Co2MnGa (CMG) Heusler alloy slab sample and the measurement setup on a homemade sample holder. (b) Magnetic field H dependence of the magnetization M and (c) temperature T dependence of the saturation magnetization Msat for CMG and Ni.   11     FIG. 2. (a) A thermography image for the CMG sample captured under a heater power Pin of 0.36 W at room temperature. (b) Longitudinal T distributions in the red and blue rectangular areas and (c) transverse T distributions in the blue rectangular area in (a) under various Pin.   12     FIG. 3. (a) H dependence of the longitudinal thermoelectric voltage Vx under various Pin at 300 K for the CMG sample. (b) The average Vx (Vx,ave) as a function of longitudinal temperature difference Tx for various Pin. The calculation curve represents the difference in the Seebeck coefficients between the NiCu thin film and Au wire. (c) H dependence of the transverse thermoelectric voltage Vy under various Pin at 300 K for the CMG sample. (d) The saturation Vy (Vy,sat) as a function of Vx,ave to estimate the anomalous thermal Hall angle tan 𝜃ATHE.   13    FIG. 4. (a) H dependence of tan 𝜃THE at various T under Pin of 0.36 W for the CMG sample. (b) T dependence of tan 𝜃ATHE from 60 K to 330 K for the CMG and Ni samples with the reference values12,17,34. (c) H dependence of tan 𝜃THE at 330 K under Pin of 0.36 W for the Ni sample before and after the thermal cycle from 60 K to 330 K. (d) Variation of  tan 𝜃ATHE at 60 and 330 K during the thermal cycle for the Ni sample.