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## Creator

[Atsushi Takahagi](https://orcid.org/0000-0003-0362-2772), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Abdulkareem Alasli](https://orcid.org/0000-0002-1681-0492), [Sang J. Park](https://orcid.org/0000-0003-1684-4876), [Hosei Nagano](https://orcid.org/0000-0003-4926-2768), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051)

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This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: https://doi.org/10.1038/s41567-025-02936-3[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

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[Observation of the transverse Thomson effect](https://mdr.nims.go.jp/datasets/3c7843eb-4c95-438f-b1c5-a8cb8d0946a0)

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Microsoft Word - Takahagi_revised-manuscript_MDR1  Observation of the transverse Thomson effect Atsushi Takahagi1,*, Takamasa Hirai2, Abdulkareem Alasli1, Sang Jun Park2, Hosei Nagano1, and Ken-ichi Uchida2,3,*  Affiliations 1 Department of Mechanical Systems Engineering, Nagoya University, Nagoya, Japan. 2 National Institute for Materials Science, Tsukuba, Japan. 3 Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa, Japan. *e-mail: takahagi.atsushi.u3@s.mail.nagoya-u.ac.jp; UCHIDA.Kenichi@nims.go.jp   The thermoelectric Thomson effect, predicted in the 1850s by William Thomson, produces volumetric heating/cooling in a conductor due to the concerted action of the Seebeck and Peltier effects. Recently, transverse thermoelectrics studies on the Nernst and Ettingshausen effects have progressed rapidly to enable versatile thermal management technologies and to explore topological transport properties. However, a transverse Thomson effect, arising from the concerted action of the Nernst and Ettingshausen effects, has not yet been observed. Here, we report the observation of the transverse Thomson effect in a conductor. We observed volumetric heating/cooling in a semimetallic Bi88Sb12 alloy induced by a charge current, temperature gradient, and magnetic field applied orthogonally to each other using thermoelectric imaging techniques. We found that the heating/cooling can be switched by the field direction. Our experiments and analyses reveal the essential difference between the conventional and transverse Thomson effects; the former depends sorely on the temperature derivative of the Seebeck coefficient, while the latter depends not only on the temperature derivative of the Nernst coefficient but also on its magnitude. The observation of the transverse Thomson effect 2  fills a missing piece in the history of thermoelectrics and provides a new principle for active thermal management technologies.  The Thomson effect generates volumetric heating/cooling when a charge current and temperature gradient T are applied in the same direction within a conductor1,2 (Fig. 1c). The heat production rate per unit volume due to the Thomson effect is given by  𝑞 𝜏 𝐣 ∙ ∇𝑇 1   where TE and jc are the Thomson coefficient and charge current density, respectively. This effect originates from the longitudinal thermoelectric effects that interconvert charge and heat currents in parallel directions, that is, the Seebeck3 and Peltier4 effects (Fig. 1a,b). In the presence of T, the Peltier heat current generated by jc varies with position when the longitudinal thermoelectric coefficient of a conductor has finite temperature T dependence. Consequently, Thomson heat release or absorption occurs due to the finite divergence of the heat current in the bulk of the conductor. Through the Onsager reciprocal relation between the Seebeck and Peltier effects5,6, TE is determined by the T derivative of the Seebeck coefficient SSE as follows:   𝜏 𝑇𝑑𝑆𝑑𝑇2   This is known as the first Thomson (or Kelvin) relation1,2. While the Seebeck coefficient is regarded as a constant parameter in linear-response thermoelectrics, its T dependence is essential for the Thomson effect. Therefore, the Thomson effect is classified as a higher-order longitudinal thermoelectric phenomenon. In addition to the fundamental Seebeck, Peltier, and Thomson effects, the longitudinal magneto-thermoelectric effects, which depend on the magnitude and direction of a magnetic field, were also observed and systemized (Fig. 1d-f)7,8. 3  Transverse thermoelectric effects that interconvert charge and heat currents in perpendicular directions have attracted attention for their relevance to fundamental physics and thermal management applications9-16. The Nernst17 (Ettingshausen18) effect is a representative transverse thermoelectric effect that generates a charge (heat) current in the direction of the cross-product of an applied heat (charge) current and magnetic field in a conductor (Fig. 1g,h). Although the performance of longitudinal thermoelectric modules reduces due to the presence of multiple junctions in complex Π-shaped thermopile structures, transverse thermoelectric modules based on the Nernst and Ettingshausen effects can be constructed without such junctions, preserving the energy conversion efficiency of the material19-21. Owing to these advantages, materials with high transverse thermoelectric conversion performances are actively being investigated for practical applications. As a part of these efforts, a giant Nernst effect was discovered in Dirac and Weyl semimetals11,14.Most of the fundamental thermoelectric effects were discovered in the 19th century1-4,17,18 and have been systematically studied for a long time. However, higher-order transverse thermoelectric effects in conductors remain unobserved and the transverse Thomson effect (TTE) is one of such phenomena. Phenomenologically, TTE is expected to occur when a charge current, temperature gradient, and magnetic field are applied orthogonally to each other in a conductor. In the presence of transverse T, the transverse heat current generated by the Ettingshausen effect varies with position when the transverse thermoelectric coefficient of the conductor has finite T dependence, producing volumetric heat release or absorption (Fig. 1i). However, the experimental verification of TTE has not progressed due to challenges in distinguishing it from other thermal effects. Here, we report the observation of TTE in a conductor. We achieved this by developing a thermoelectric imaging technique that isolates TTE signals from temperature modulations caused by the Peltier and Ettingshausen effects, as detailed below. Despite their phenomenological similarity, TTE fundamentally differs from the longitudinal Thomson effect owing to its transverse nature. The heat production rate per unit volume due to TTE is expressed as  4  𝑞 ≡ 𝜏𝐇|𝐇|𝐣 ∙ ∇𝑇 3  𝜏 𝑇𝑑𝑆𝑑𝑇2𝑆 4   where SNE is the Nernst coefficient and the Onsager reciprocal relation between the Nernst and Ettingshausen effects is assumed (Methods)22. Importantly, TTE is determined not only by the T derivative of the Nernst coefficient (first term on the right-hand side of equation (4)) but also by its magnitude (second term on the right-hand side of equation (4)), while longitudinal TE is determined sorely by the T derivative of the Seebeck coefficient (equation (2)); the first Thomson relation does not apply to TTE. When heat release or absorption owing to TTE is measured, TTE becomes the experimentally observable quantity rather than T(dSNE/dT).  A material with large TTE, that is, both a large magnitude and T dependence of SNE, should be selected to observe TTE. Here, we focus on the semimetallic Bi88Sb12 alloy, which exhibits a strong magnetic-field-dependent ordinary Nernst effect around room temperature23. A polycrystalline Bi88Sb12 alloy was synthesized using the spark plasma sintering method under the same conditions as those described in refs. 16,24 and was observed to have isotropic transport properties (Supplementary Note 1). Figure 2a presents the T dependence of SNE of our Bi88Sb12 slab measured by the method in refs. 25,26. To obtain the pure SNE values free from parasitic thermoelectric effects, we paid close attention to the thermal boundary conditions during the measurements (Methods). We confirmed that the Bi88Sb12 slab exhibits a large magnitude and finite T dependence of SNE around room temperature. Figure 2b shows the H dependence of TTE and T(dSNE/dT) for Bi88Sb12 at 320 K, calculated using the measured SNE values. The sign of T(dSNE/dT) is positive for all H values, whereas that of TTE switches from positive to negative as H increases because of the competition between the first and second terms on the right-hand side of equation (4). Figure 2c illustrates the TTE measurement system. To clarify the temperature distribution induced by TTE, we adopted a thermoelectric imaging technique based on lock-in thermography8,27-29. A rectangular Bi88Sb12 slab sample was fixed to two plates with temperature control modules 5  (Methods). Uniform T was applied to the sample along the y direction while maintaining the average temperature at 320 K, where the average temperature and the temperature difference between the sample edges T were monitored through steady-state temperature images with an infrared camera. A square-wave-modulated AC charge current Jc with amplitude Jc, frequency f = 5 Hz, and zero offset and an out-of-plane H with magnitude H were applied along the x and z directions, respectively. When Jc and H are applied to the Bi88Sb12 sample, the Ettingshausen effect generates a heat current Jq in the y direction, causing a temperature change at the y-axis sample edges (Fig. 1h). When T, Jc, and H are co-applied, TTE heats or cools the entire sample. The resulting temperature modulation was measured using the same infrared camera. The first harmonic component of the temperature modulation was extracted from the thermal images using the Fourier analysis and converted to lock-in amplitude A and phase  images to selectively detect the thermal response proportional to Jc, that is, the temperature modulation due to thermoelectric effects. Here, A reflects the magnitude of the temperature modulation, which is defined in the range of A  0, and  reflects the sign of the temperature modulation and its time delay due to the thermal diffusion. The influence of the Peltier effect at the x-axis edges was minimized by capturing the center of the x-axis long sample (note that the Peltier effect generates Jq in the x direction). According to equation (3), the sign of the TTE-induced temperature change should reverse by reversing H; thus, we obtain the H-odd-dependent thermoelectric signals through Aodd = |A(+H)exp[i(+H)] A(H)exp[i(H)]|/2 and odd = arg[A(+H)exp[i(+H)] A(H)exp[i(H)]] from thermal images captured under positive and negative magnetic fields to eliminate field-independent (H-even-dependent) signals, such as the conventional Peltier and Thomson (magneto-Peltier and magneto-Thomson) effects.  Figure 2d shows the images of steady-state T, Aodd, and odd for the Bi88Sb12 sample at Jc = 1 A, T = 40 K, and 0H = 200 mT, where0 is the vacuum permeability. The steady-state T image confirms that applied T along the y direction is uniform and T along the x direction caused by the Peltier and thermal Hall effects is negligibly small (|xT| < 0.1 K/mm). We also confirmed that T along the z direction is almost zero by a simulation (Supplementary Note 2). The Aodd image shows that a temperature change occurs around the edges of the sample. The odd values of this thermal 6  response differ by 180 between the right and left edges. The temperature change on the right (left) edge has almost the same phase as (opposite phase to) the input current and represents increasing (decreasing) T, indicating that the temperature change signals are due to the Ettingshausen effect. Under these conditions, the volumetric thermal response caused by TTE is expected to appear throughout the sample. However, the Ettingshausen signal at the sample edges obscures the TTE signals (Fig. 2d). To separate TTE from the Ettingshausen signal, we measured only the Ettingshausen effect in the absence of T (at T = 0 K), while maintaining the average sample temperature at 320 K (Fig. 2e). The TTE signal can be extracted from the difference between the results of Fig. 2d and 2e, that is, by calculating Adiff = |Aodd(T)exp[iodd(T)Aodd(T = 0 K)exp[iodd(T = 0 K) and diff = arg[Aodd(T)exp[iodd(T)Aodd(T = 0 K)exp[iodd(T = 0 K) to eliminate the Ettingshausen background except for its T dependence. Figure 2f shows the steady-state temperature difference Tdiff = T(T)  T(T = 0 K), Adiff, and diff images at Jc = 1 A, T = 40 K, and 0H = 200 mT. Except for the regions near the sample edges, the sample exhibits clear volumetric temperature change, where Adiff is almost uniform and diff is constant around 90. This diff value is the same as the phase due to Joule heating30, which generates a single volumetric heat source; therefore, the observed temperature change originates from volumetric heat release. Near the sample edges, the volumetric signal is obscured by a residual Ettingshausen signal due to its T dependence. Nevertheless, this volumetric temperature change signal is well separated from the residual Ettingshausen signal around the center of the sample because the heat diffusion length of Bi88Sb12 at 5 Hz is ~ 0.4 mm, much smaller than the sample width (Methods).  To confirm the fundamental properties of the observed volumetric temperature change, we measured the Jc and T dependences of Adiff and diff systematically. Figure 3a presents the Adiff and diff images for the Bi88Sb12 sample for various values of Jc at T = 40 K and 0H = 200 mT. As shown in the y-axis profiles of Adiff and diff in Fig. 3b, the thermal signal reaches its maximum value at the sample center. Hereafter, we discuss the detailed behavior using the averaged values of Adiff and diff taken within a 1 mm width at the center, unaffected by the edges. We found that Adiff increased in proportion to Jc and diff remained unchanged with respect to Jc (Fig. 3c). Figure 3d-f shows the T 7  dependence of Adiff and diff at Jc = 1 A and 0H = 200 mT, measured while holding the average sample temperature at 320 K. Adiff is also proportional to T and diff is independent of T. These features of the volumetric thermal response satisfy equation (3) with respect to Jc and T. Significantly, the volumetric temperature change disappears when H || Jc (Extended Data Fig. 1). This behavior is consistent with the features of TTE.  Figure 4a,b presents the Adiff and diff images and their line profiles along the y direction for the same Bi88Sb12 sample at various H values at Jc = 1 A and T = 40 K. We found that as H increased, Adiff peaked at 200 mT, decreased to 400 mT, and increased again for 0H > 400 mT (upper graph of Fig. 4c). diff shifted from 90 to 90 at around 400 mT, indicating a reversal in the thermal response from heating to cooling (lower graph of Fig. 4c). This thermal response reversal occurs near the H value where TTE changes sign and cannot be explained by the behavior of T(dSNE/dT), which remains positive at all H values (Fig. 2b).  We performed a numerical simulation of the thermoelectric transport for the Bi88Sb12 sample using the finite difference method to clarify the origin of the H dependence of the volumetric temperature change in detail (Methods). The simulated y-axis profiles of Adiff and diff in Fig. 4d show the quantitatively estimated thermal responses of TTE based on equation (3) by substituting the measured SNE values and thermophysical properties of Bi88Sb12 in the transport model (Extended Data Fig. 2). The behaviors of the experimentally observed and calculated Adiff and diff values were similar (Fig. 4b,d). The magnitude of the simulated Adiff was larger than that of the experimental one because we assumed that all heat due to TTE contributed to the change in the sample temperature by neglecting the heat loss to the sample holder in the simulation. The calculated H dependence of the temperature modulation induced by TTE in Fig. 4e indicates that Adiff has two peaks at 200 and 700 mT and diff changes from 90 to 90 around 400 mT, well consistent with the experimental results in Fig. 4c. Note that, although TTE is almost zero around 400 mT (Fig. 2b), the calculated diff is sensitive to whether the sign of TTE is positive or negative. This causes a subtle difference in the H dependence of diff between the experimental and analytical results. This result implies that although the H values at which the sign reversal occurs are slightly different between the observed lock-in 8  thermography signals and TTE estimated from measured SNE, its difference can be quantitatively explained by the simulation considering the T or position dependence of TTE due to applied T. Therefore, all the observed behaviors of the volumetric thermal change relative to Jc, T, and H align with equation (3), providing evidence of TTE observation. We revealed that the observable physical quantity of TTE is TTE, not T(dSNE/dT), which is a key distinction from the longitudinal Thomson effect. To further investigate the behaviors of TTE, we divided the Adiff and diff signals into T(dSNE/dT) and 2SNE components based on the simulation. Figure 5a,b shows the calculation results for the T(dSNE/dT) component. diff remains 90 except for the H region with tiny temperature dependence of SNE, indicating that this component mainly induces the volumetric heat release due to positive T(dSNE/dT) of Bi88Sb12 (Fig. 2b). Conversely, the 2SNE component induces the volumetric heat absorption, where diff is always 90 (Fig. 5c,d). The magnitude of the thermal response due to 2SNE increases with increasing H, consistent with the H dependence of |SNE| in Fig. 2a. Thus, at low (high) H, the T(dSNE/dT) (2SNE) component dominates the total TTE signal in Bi88Sb12 (Extended Data Fig. 3a), leading to the sign reversal of the volumetric temperature change. Furthermore, the simulated y-axis profiles of diff in Fig. 5a show that the T(dSNE/dT) contribution induces heat release (absorption) and weakens (enhances) dominant cooling due to the 2SNE contribution in the right (left) half of the sample when 0H > 600 mT. This behavior explains why the magnitude of the TTE signal in the left half of the sample is larger than that in the right half for 0H > 600 mT in Fig. 4b and 4d. The observation of TTE fills a missing piece in thermoelectrics, providing new insights into condensed matter physics and thermal management applications. As demonstrated in this study, the sign of the TTE-induced temperature change is controlled by the direction of the magnetic field and the magnitude of the temperature change is determined not only by the temperature derivative of SNE but also by the magnitude of SNE. This feature distinguishes TTE from the longitudinal Thomson effect determined by the first Thomson relation. In our Bi88Sb12, the signs of the T(dSNE/dT) and 2SNE components are opposite around room temperature, resulting in the sign reversal of the volumetric temperature change in its field dependence and the compensation of the thermoelectric performance 9  of TTE. This compensation leads to TTE the magnitude of which is only ~15% of TE for Bi88Sb12 (Extended Data Fig. 3). In other words, in materials where both components in TTE have the same sign, the performance of TTE can be further improved. Owing to the presence of the 2SNE component, materials showing the large Nernst effect, such as WTe214,31, NbSb213, NbP32,33, and MnBi2Te434-37, can be good candidates for the emergence of large TTE. To improve the T(dSNE/dT) component, phase change materials may be useful38,39. According to refs. 11,13,14,32, Cd3As2, WTe2, NbSb2, and NbP are expected to have the T(dSNE/dT) and 2SNE components with the same sign at cryogenic temperatures. Materials with giant TTE may boost cooling operation of transverse thermoelectric devices, in a similar manner to the recent demonstration that the longitudinal Thomson effect improves the performance of the Peltier cooling39. The observation of TTE in nonmagnetic materials suggests the existence of anomalous TTE in magnetic materials, which depends on the direction of spontaneous magnetization. Similar to TTE in nonmagnetic materials, anomalous TTE should become apparent in materials where both the temperature derivative and magnitude of the anomalous Nernst coefficient are large. Based on the analogy with the ordinary and anomalous Nernst effects, carrier mobility and Berry curvature should be important for the appearance of ordinary and anomalous TTEs, respectively. Observing anomalous TTE remains a task for future work; however, this phenomenon can be used to drive TTE-based thermal devices without applying an external magnetic field. The experimental methods established in this study will be useful for observing such unobserved higher-order thermoelectric phenomena.   10   Fig. 1: Thermoelectric effects. a, Schematic of the Seebeck effect. When a temperature gradient T is applied to a conductor, an electric voltage V is generated along the T direction. b, Schematic of the Peltier effect. When a charge current Jc is applied to a conductor, a heat current Jq is generated along the Jc direction and induces a heat release or absorption𝑄 at the edges of the conductor, that is, the interfaces with different materials. c, Schematic of the Thomson effect. When Jc and T are applied to a conductor in parallel directions, volumetric 𝑄 is generated throughout the conductor. d-f, Schematics of the magneto-Seebeck, magneto-Peltier, and magneto-Thomson effects, which are categorized as longitudinal magneto-thermoelectric effects. The longitudinal thermoelectric effects in a-c are known to be dependent on the magnitude and direction of a magnetic field H. g, Schematic of the Nernst effect. When T is applied to a conductor, V is generated in the direction of the cross product of T and H. h, Schematic of the Ettingshausen effect. When Jc is applied to a conductor, Jq is generated in the direction of the cross product of Jc and H and induces 𝑄 at the side edges of the conductor. i, Schematic of the transverse Thomson effect. When Jc, T, and H are applied to a conductor in perpendicular directions, volumetric 𝑄 is generated throughout the conductor. The blue spheres in the schematics represent electron transport. In semiconductors and semimetals, hole transport also contributes to these thermoelectric effects in the same manner.  11   Fig. 2: Transverse thermoelectric conversion properties and lock-in thermography measurements. a, T dependence of the Nernst coefficient SNE of Bi88Sb12 for various values of the magnetic field magnitude H. b, H dependence of the transverse Thomson coefficient TTE (equation (4)) and its component T(dSNE/dT) at 320 K calculated using the data in a. c, Schematic of the set-up for the lock-in thermography measurement of TTE. To excite TTE, Jc (with magnitude Jc), T (with the temperature difference between the sample edges T), and H were applied along the x, y, and z directions, respectively. d, Steady-state temperature T, H-odd-dependent lock-in amplitude Aodd, and phase odd images for Bi88Sb12 at Jc = 1 A, T = 40 K, and 0H = 200 mT. e, Steady-state T, Aodd, and odd images at Jc = 1 A, T = 0 K, and 0H = 200 mT. f, Steady-state temperature Tdiff, amplitude Adiff, and phase diff images obtained by calculating the difference between the results of d and e.  12   Fig. 3: Charge current and temperature gradient dependences of volumetric temperature change. a, Adiff and diff images for the Bi88Sb12 sample for various values of Jc at T = 40 K and 0H = 200 mT. b, y-axis profiles of Adiff and diff for various values of Jc. c, Jc dependence of Adiff and diff at T = 40 K and 0H = 200 mT. d, Adiff and diff images for various values of T at Jc = 1 A and 0H = 200 mT. e, y-axis profiles of Adiff and diff for various values of T. f, T dependence of Adiff and diff at Jc = 1 A and 0H = 200 mT. The y-axis profiles in b and e are obtained by averaging the raw profiles in the entire regions of the images in a and d. The data in c and f are obtained by averaging the Adiff and diff values between the dotted lines in b and e, respectively. The error bars represent the standard deviation of the data.  13   Fig. 4: Magnetic field dependence of volumetric temperature change. a, Adiff and diff images for the Bi88Sb12 sample for various H values at Jc = 1 A and T = 40 K. b, Observed y-axis profiles of Adiff and diff for various H values. c, Observed H dependence of Adiff and diff at Jc = 1 A and T = 40 K. d, Simulated y-axis profiles of Adiff and diff for various H values at Jc = 1 A and T = 40 K, obtained by substituting the experimentally determined thermoelectric and thermophysical properties of Bi88Sb12 into the finite difference model. e, Simulated H dependence Adiff and diff at Jc = 1 A and T = 40 K. The y-axis profiles in b are obtained by averaging the raw profiles in the entire regions of the images in a. The data in c and e are obtained by averaging the Adiff and diff values between the dotted lines in b and d, respectively. The error bars represent the standard deviation of the data. 14   Fig. 5: Verification of components of transverse Thomson coefficient. a, Simulated y-axis profiles of Adiff and diff due to the T(dSNE/dT) component for the Bi88Sb12 sample for various values of H at Jc = 1 A and T = 40 K. The data were calculated using the thermoelectric and thermophysical properties of Bi88Sb12 and the thermoelectric transport model based on the finite-difference method. b, Simulated H dependence Adiff and diff due to the T(dSNE/dT) component at Jc = 1 A and T = 40 K. c, Simulated y-axis profiles of Adiff and diff due to the 2SNE component for various H values at Jc = 1 A and T = 40 K. d, Simulated H dependence Adiff and diff due to the 2SNE component at Jc = 1 A and T = 40 K. The data in b and d are obtained by averaging the Adiff and diff values between the dotted lines in a and c, respectively. The error bars represent the standard deviation of the data. 15  Methods Measurement procedures We prepared a Bi88Sb12 sample with lengths of 13.0, 3.1, and 0.3 mm along the x, y, and z directions, respectively, to measure TTE by the lock-in thermography method. The side edges of the Bi88Sb12 sample were fixed to two plates using thermal conductive silicone adhesive, where the contact area between the sample and plate was 13.0 × 0.5 mm2 (Fig. 2c). The plates were made of anodized aluminum and were electrically insulated from the sample. A ceramic heater and Peltier module were attached to each plate to control the magnitude of the applied temperature gradient and the average temperature of the sample. Cu wires were electrically connected to the 3.1 × 0.3 mm2 surfaces of the Bi88Sb12 slab using indium. A magnetic field was applied to the sample in the z direction by an electromagnet, where a maximum magnetic field is 700 mT and its nonuniformity is less than 2% within the sample size. An insulating black ink with an infrared emissivity > 0.94 was coated on the sample surface to increase the emissivity and ensure its uniformity. We performed lock-in thermography measurements after the temperature gradient and average temperature of the sample reached a steady state. Note that the slight fluctuation of the average sample temperature does not affect the lock-in thermal images because A and  reflect only periodic temperature change in linear response to the applied AC charge current. The measurement time for the A and  images for each condition was 10 min.  Formulation of Nernst and Ettingshausen effects From Ohm’s law, Fourier’s law, and Onsager reciprocal relations, the linear response relations of the charge current density jc and heat current density jq can be written as22  𝐣𝐣𝜎 𝜎𝑆𝑇𝜎𝑆𝑇 𝜅𝑇𝐄∇𝑇/𝑇 5   where E,  S, and  denote the electric field, electrical conductivity, Seebeck coefficient, and thermal 16  conductivity tensors, respectively. The matrix formula commonly used in the field of thermoelectrics is obtained by replacing jc and E in equation (5) as follows:  𝐄𝐣𝜌 𝑆𝑆𝑇 𝜅′𝐣∇𝑇6   where (= 1) is the electrical resistivity and ’ S2T. When a magnetic field is applied in the z direction, equation (6) in the x-y plane is described as  ⎝⎛𝐸𝐸𝑗 ,𝑗 , ⎠⎞⎝⎜⎛𝜌 𝜌𝜌 𝜌𝑆 𝑆𝑆 𝑆𝑆 𝑇 𝑆 𝑇𝑆 𝑇 𝑆 𝑇𝜅 𝜅𝜅 𝜅 ⎠⎟⎞⎝⎛𝑗 ,𝑗 ,∇ 𝑇∇ 𝑇⎠⎞ 7   in an isotropic material, where xx = yy,Sxx = Syy, ’xx = ’yy,xy = yx,Sxy = Syx, and ’xy = ’yx.  First, we consider the Nernst-effect-induced electric field Ey when a temperature gradient xT and magnetic field Hz are applied in the x and z directions, respectively. In the Nernst measurement, the x and y directions are typically under open-circuit conditions (jc,x = jc,y = 0). Here, assuming the isothermal condition in the y direction (yT = 0), the measured thermopower is obtained from equation (7) as  𝐸 ,∇ 𝑇𝑆 8   where the subscript i denotes the isothermal condition. Thus, the pure Nernst coefficient can be obtained from Nernst measurements under the isothermal condition in the transverse direction. In contrast, assuming the adiabatic condition in the y direction (jq,y = 0), the thermopower is calculated as  17  𝐸 ,∇ 𝑇𝑆 𝜃 𝑆 ≡ 𝑆∗ 9   where the subscript a denotes the adiabatic condition and th (’xy/’xx) is the thermal Hall angle. Therefore, the observed thermopower includes not only the contribution from the Nernst effect but also that from the Seebeck and thermal Hall effects, making it difficult to extract the pure Sxy contribution.   Next, we consider the temperature gradient yT induced by the Ettingshausen effect when a charge current density jc,x and magnetic field Hz are applied in the x and z directions, respectively. In the Ettingshausen measurement, the y direction is typically under an open-circuit condition (jc,y = 0). Here, assuming the isothermal condition in the x direction (xT = 0), the measured charge-to-heat current conversion ratio is obtained from equation (7) as  𝜅 ∇ 𝑇𝑗 ,𝑆 𝑇 10   Thus, the pure Nernst coefficient can be obtained from Ettingshausen measurement under the isothermal condition in the longitudinal direction. In contrast, assuming the adiabatic condition in the x direction (jq,x = 0, i.e., xT  0), the charge-to-heat current conversion ratio is calculated as  𝜅 ∇ 𝑇𝑗 ,𝑆 𝜃 𝑆 𝑇 𝑆∗ 𝑇 11   where ’xy << ’xx is assumed (note that th of BiSb alloys was reported to be a few %40). Similar to the Nernst measurements, under the adiabatic condition, the observed temperature change signal includes not only the contribution from the Ettingshausen effect but also that from the Peltier and thermal Hall effects considering the Onsager reciprocal relations.   Whether isothermal or adiabatic conditions should be applied depends on the aspect ratio of the sample in the x-y plane41. In a rectangular sample, when the x-axis length is much larger than the 18  y-axis length, the sample is nearly isothermal in the x direction and adiabatic in the y direction. As described above, the Nernst and Ettingshausen measurements are affected by the thermal boundary conditions along the electric field, that is, the y and x directions, respectively. Consequently, the Ettingshausen measurements are better suited for estimating Sxy than the Nernst measurements for the aspect ratio of our sample. Thus, the T dependence of the Nernst coefficient in Fig. 2a was estimated through the Ettingshausen measurements using the lock-in thermography method, in which the influence of the Peltier and thermal Hall effects was suppressed.   Formulation of longitudinal Thomson effect We consider the heat production rate per unit volume 𝑞 generated when a charge current density jc,x and temperature gradient xT are applied in the x direction. 𝑞 comprises the Joule heating and the divergence of the heat current and is obtained using equation (7) as follows:  𝑞 𝐸 𝑗 , div 𝑗 ,                                                                            𝜌 𝑗 , 𝑆 ∇ 𝑇 𝑗 , ∇ 𝑆 𝑇𝑗 , 𝜅 ∇ 𝑇          𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑆𝑑𝑆 𝑇𝑑𝑇𝑗 , ∇ 𝑇            𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑆 𝑆 𝑇𝑑𝑆𝑑𝑇𝑗 , ∇ 𝑇𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑇𝑑𝑆𝑑𝑇𝑗 , ∇ 𝑇                            12   The last term of the last line in equation (12) is the heat generated by the concerted action of jc,x and xT, that is, the longitudinal Thomson effect (equations (1) and (2)). The last term of the fourth line in equation (12) shows that the Sxx components contributing to the longitudinal Thomson effect generated by the Joule heating and heat-current divergence terms have different signs and eliminate each other. Hence, the observable physical quantity of the longitudinal Thomson effect is determined only by the temperature derivative of the Seebeck effect.  Formulation of transverse Thomson effect We consider the heat production rate per unit volume 𝑞 generated when a charge current density jc,x, 19  temperature gradient yT, and magnetic field Hz are applied in the x, y, and z directions, respectively. The y direction is assumed to be under the open-circuit condition (jc,y = 0). We solve 𝑞  for two thermal boundary conditions in the x direction. First, 𝑞  for the isothermal condition in the x direction (xT = 0), is obtained using equation (7) as follows:   𝑞i 𝐸 𝑗 , div 𝑗 ,                                                                                𝜌 𝑗 , 𝑆 ∇ 𝑇 𝑗 , ∇ 𝑆 𝑇𝑗 , 𝜅 ∇ 𝑇         𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑆𝑑𝑆 𝑇𝑑𝑇𝑗 , ∇ 𝑇             𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑆 𝑆 𝑇𝑑𝑆𝑑𝑇𝑗 , ∇ 𝑇𝜌 𝑗 , ∇ 𝜅 ∇ 𝑇 𝑇𝑑𝑆𝑑𝑇2𝑆 𝑗 , ∇ 𝑇          13   When the magnetic field is reversed, the first and second terms of the last line in equation (13) do not change their sign and only the last term reverses its sign. This is because the first and second terms are determined only by the diagonal components of the transport tensors, whereas the last term is determined by the off-diagonal component. The last term of the last line indicates the volumetric heat release or absorption due to TTE, which is generated by the concerted action of jc,x and yT. The proportionality factor of the H-odd-dependent term is defined as TTE to be   𝜏 ≡ 𝑇𝑑𝑆𝑑𝑇2𝑆 14   Therefore, experimentally measured TTE includes not only the temperature derivative of the Nernst coefficient but also its magnitude. This feature differs from the longitudinal Thomson coefficient, which is determined only by the temperature derivative of the Seebeck coefficient (equation (2)). The reason for this difference is that the Sxy components of TTE generated by the Joule heating and heat-current divergence terms have the same sign and enhance each other (see the last term of the fourth line in equation (13)). Note that the formulation of TTE for anisotropic materials is shown in Supplementary Note 3. 20   Next, we derive 𝑞  for the adiabatic condition in the x direction (jq,x = 0, i.e., xT  0). From equation (7), xT is generated by the heat current due to the Peltier and thermal Hall effects:  ∇ 𝑇𝑆 𝑇𝑗 , 𝜅 ∇ 𝑇𝜅15   𝑞  with finite xT in equation (15) can be obtained as  𝑞 𝐸 𝑗 , div 𝑗 ,                                                                                                                𝜌 𝑗 , 𝑆 ∇ 𝑇 𝑆 ∇ 𝑇 𝑗 , ∇ 𝑆 𝑇𝑗 , 𝜅 ∇ 𝑇 𝜅 ∇ 𝑇𝜌 𝑗 , 1𝑆 𝑇𝜌 𝜅∇𝜅 𝜅𝜅∇ 𝑇                                                𝑆 𝜃 𝑆𝑑𝑑𝑇𝑆 𝜃 𝑆 𝑇 𝑗 , ∇ 𝑇𝜌 𝑗 , 1𝑆 𝑇𝜌 𝜅∇𝜅 𝜅𝜅∇ 𝑇                                           𝑇𝑑 𝑆 𝜃 𝑆𝑑𝑇2 𝑆 𝜃 𝑆 𝑗 , ∇ 𝑇     16   The last term of the last line in equation (16), due to TTE caused by jc,x and yT, includes the contribution of the Seebeck and thermal Hall effects and differs from that in the isothermal condition. Therefore, the transverse Thomson coefficient *TTE in the adiabatic condition is  𝜏∗ 𝑇𝑑𝑆∗𝑑𝑇2𝑆∗ 17   indicating that *TTE is determined by S*xy and affected by the thermal boundary conditions.  In this study, we used a rectangular sample with a longer x-axis length than a y-axis length for measuring TTE to reduce xT occurrence. Thus, we have discussed TTE assuming the isothermal condition. We obtained similar TTE signals for a sample with a different aspect ratio because the isothermal condition was maintained (Supplementary Note 4).  21  Numerical simulation for TTE The TTE experiment was simulated using a numerical model for the Bi88Sb12 sample based on the finite-difference method. The simulation was performed for the y direction using the following unsteady one-dimensional heat equation:  𝜌 𝐶𝜕𝑇𝜕𝑡𝜅𝜕 𝑇𝜕𝑦𝑞 18   where d and C are the density and specific heat, respectively. The heat source 𝑞 in equation (18) includes the contributions from the Ettingshausen effect at the edges of the sample and TTE occurring throughout the sample as follows:  ⎩⎪⎪⎨⎪⎪⎧ 𝑞 𝑦 0 mm𝑆 𝑇𝑗 ,∆𝑦        𝑞 𝑦 3.1 mm𝑆 𝑇𝑗 ,∆𝑦       due to the Ettingshausen effect𝑞 𝑦 𝑇𝑑𝑆𝑑𝑇2𝑆 𝑗 , ∇ 𝑇 due to TTE19   where y is the length of one node and the sample is divided into more than 100 nodes in the y direction. The equations above were calculated under the adiabatic condition: jq,y(y = 0 mm) = jq,y(y = 3.1 mm) = 0. The y-axis profiles of Aodd and odd were derived using Fourier analysis from the temperature response of each node after the calculations converged when a square-wave-modulated AC charge current jc,x was applied. The jc,x value was assumed to be constant at each node because  of Bi88Sb12 changes only ±3% with respect to T for all H and its H dependence has no effect on TTE (Extended Data Fig. 2). We performed this calculation with and without applying yT. The Adiff and diff values were obtained by determining the difference between the results. We used the measured SNE (Fig. 2a) and thermophysical properties (d = 9260 kg m-3, C = 135 J kg-1 K-1, and  = 3.9 W m-1 K-1 presented in Extended Data Fig. 2) of Bi88Sb12 for the simulation. As C and  do not depend on T and  varies only by ±5% under H, the C and  values were averaged and regarded as the constant 22  parameters.   Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request.  23  References 1 Thomson, W. On a mechanical theory of thermo-electric currents. Proc. R. Soc. Edinburgh 3, 91-98 (1851). 2 Thomson, W. On the dynamical theory of heat. Trans. R. Soc. Edinburgh 21, 123-171 (1857). 3 Seebeck, T. J. Magnetische Polarisation der Metalle und Erze durch Temperatur-Differenz. Abh. Preuss. Akad. Wiss. 265-373 (1822). 4 Peltier, J. C. A. Nouvelles expériences sur la caloricité des courants électrique. 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Ando for valuable discussions and M. Isomura and K. Suzuki for technical supports. This work was supported by ERATO “Magnetic Thermal Management Materials” (JPMJER2201) from JST, Japan; Grant-in-Aid for Scientific Research (B) (19H02585) and Grant-in-Aid for Scientific Research (S) (22H04965) from JSPS, Japan; and NEC Corporation. A.T. was supported by Grant-in-Aid for JSPS Fellows (23KJ1122) from JSPS, Japan.   Author contributions K.U. planned the study; K.U. and H.N. supervised the study; A.T. and K.U. designed the experiments, prepared the samples, verified the formulations for transverse thermoelectric effects, developed the explanation of the experiments, and prepared the manuscript; A.T. performed the experiments for TTE with support from T.H. and performed the numerical simulation; A.A., S.J.P., and A.T. measured the thermophysical properties of the sample. All the authors discussed the results and commented on the manuscript.  Competing interests The authors declare no competing interests.  27   Extended Data Fig. 1: Magnetic field direction dependence. Adiff and diff images for the Bi88Sb12 sample at Jc = 1 A, T = 40 K, and 0H = 200 mT for H || Jc.     Extended Data Fig. 2: Physical properties of Bi88Sb12. a, T dependence of the thermal conductivity  of Bi88Sb12 for various H values.  was measured based on the method of ref. 42 using an electromagnet and a sample holder with T control modules. b, T dependence of the specific heat C of Bi88Sb12 at zero field, measured using a differential scanning calorimeter. c, T dependence of the electrical conductivity  of Bi88Sb12 for various H values, measured using the four-probe method.   28   Extended Data Fig. 3: Transverse vs longitudinal Thomson coefficients of Bi88Sb12. a, H dependence of TTE (equation (4)) and its component T(dSNE/dT) and 2SNE at 320 K. b, H dependence of the longitudinal Thomson coefficient TE (equation (2)) at 300 K in ref. 8.