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[Hiroto Adachi](https://orcid.org/0000-0002-6844-6477), [Fuyuki Ando](https://orcid.org/0009-0003-7789-8170), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Rajkumar Modak](https://orcid.org/0000-0001-7939-3289), [Matthew A. Grayson](https://orcid.org/0000-0003-4914-5043), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051)

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[Fundamentals and advances in transverse thermoelectrics](https://mdr.nims.go.jp/datasets/de7a7c7d-16cd-4dff-b0b2-bc798878cd33)

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Fundamentals and advances in transverse thermoelectricsApplied PhysicsExpress      APEX REVIEW • OPEN ACCESSFundamentals and advances in transversethermoelectricsTo cite this article: Hiroto Adachi et al 2025 Appl. Phys. Express 18 090101 View the article online for updates and enhancements.You may also likeSpin caloritronics in metallic superlatticesT Seki, K Uchida and K Takanashi-Thermoelectric conversion of heat fluxes:analytical and experimental approachMounir Amokrane and Bertrand Nogarede-Adaptive genetic algorithm-based designof gamma-graphyne nanoribbonincorporating diamond-shaped segmentwith high thermoelectric conversionefficiencyJingyuan Lu,  , Chunfeng Cui et al.-This content was downloaded from IP address 126.157.117.119 on 02/09/2025 at 14:36https://doi.org/10.35848/1882-0786/adf700/article/10.1088/1361-648X/ad4761/article/10.1088/0964-1726/21/8/085018/article/10.1088/0964-1726/21/8/085018/article/10.1088/1674-1056/acb768/article/10.1088/1674-1056/acb768/article/10.1088/1674-1056/acb768/article/10.1088/1674-1056/acb768/article/10.1088/1674-1056/acb768https://pagead2.googlesyndication.com/pcs/click?xai=AKAOjstLSVWcPfougD-8vVP9RmWF63GkO8ZL1xRkeM9AvSGYjQ9bfFQkyl4QHOeT3HZmlQrk8hktLQ8wH7QMn_rC1EMYa45RVoCDXmKe7MM1ADhP2UWpFQgsFdln7-Yj6snDTnn5SKqM-hKGR2ohdl_JbzGgE9cro36qhFOIQmpKo11nSx9B-B49S2mfcZwVH8WXLVusvB1jd17Gh1OzuVc7EgBnV9ls7zm1XdPAwMv4CN7x54LBtXUC7InHR2Dll1jqvKo7sgv4pIWOg4PjTwtypCuGuLDtw7n7HiaTgDD6wGtNvkWsLne04nIfB7G9spnBb_ErbVNXvRQCyPFtFHOK1njUJMZvqTzlIbZI3ZiF6fu3sbynN8W6&sig=Cg0ArKJSzGXgBia1TPFY&fbs_aeid=%5Bgw_fbsaeid%5D&adurl=https://www.electrochem.org/248/registration%3Futm_source%3DIOP%26utm_medium%3Dbanner%26utm_campaign%3DIOP_248_Early_Reg%26utm_id%3DIOP%2B248%2BEarly%2BRegistrationFundamentals and advances in transverse thermoelectricsHiroto Adachi1* , Fuyuki Ando2 , Takamasa Hirai2 , Rajkumar Modak2,3 , Matthew A. Grayson4 , andKen-ichi Uchida2,3*1Research Institute for Interdisciplinary Science, Okayama University, Okayama 700-8530, Japan2Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, Tsukuba 305-0047, Japan3Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, Kashiwa 277-8561, Japan4Department of Electrical and Computer Engineering, Northwestern University, Evanston, Illinois 60208, United States of America*E-mail: hiroto.adachi@okayama-u.ac.jp; UCHIDA.Kenichi@nims.go.jpReceived June 13, 2025; revised July 28, 2025; accepted July 31, 2025; published online September 2, 2025Transverse thermoelectric effects interconvert charge and heat currents in orthogonal directions due to the breaking of either time-reversalsymmetry or structural symmetry, enabling simple and versatile thermal energy harvesting and solid-state cooling/heating within single materials.In comparison to the complex module structures required for the conventional Seebeck and Peltier effects, the transverse thermoelectric effectsprovide the complete device structures, potentially resolving the fundamental issue of multi-module degradation of thermoelectric conversionperformance. This review article provides an overview of all currently known transverse thermoelectric conversion phenomena and principles, aswell as their characteristics, and reclassifies them in a unified manner. The performance of the transverse thermoelectric generator, refrigerator,and active cooler is formulated, showing that thermal boundary conditions play an essential role in discussion on their behaviors. Examples ofrecent application research and material development in transverse thermoelectrics are also introduced, followed by a discussion of futureprospects. © 2025 The Author(s). Published on behalf of The Japan Society of Applied Physics by IOP Publishing Ltd1. IntroductionThermoelectric conversion, which converts thermal and elec-trical energy in solids, is one of the promising technologies forrealizing a sustainable society.1) The most widely usedthermoelectric generation principle is the Seebeck effect,discovered by T. J. Seebeck in 1821, which can be used togenerate an electric voltage and current in the direction parallelto a temperature gradient ∇T, i.e. the longitudinal direction.The ratio of the generated longitudinal electric field E to theapplied temperature gradient ∇T is called the Seebeckcoefficient S = E/∇T, which represents the thermopower dueto the Seebeck effect. As shown in Fig. 1(a), a thermoelectricconversion module based on the longitudinal Seebeck effecthas a structure in which a large number of pairs of p-type andn-type conductors are connected electrically in series andthermally in parallel. Since the Seebeck coefficient of p-type(n-type) conductors is positive (negative), the thermoelectricvoltage of each conductor is added consecutively, and theoutput of the module is proportional to the number ofconductor pairs. Although each conductor only generates athermoelectric voltage of the order of mV, by integrating them,a large, practical voltage can be obtained. In the reciprocalprocess, when a charge current J is applied to such a module,the longitudinal heat current generated by the Peltier effect canbe used to heat or cool the surface of the module depending onthe J direction [Fig. 1(b)].The thermoelectric conversion performance of materials isoften evaluated using the dimensionless figure of merit zTwith T being the absolute temperature. As discussed inSect. 3 in more detail, zT for the Seebeck effect is propor-tional to the electrical conductivity and the square of theSeebeck coefficient and inversely proportional to the thermalconductivity. For a long time, zT exceeding 1 was anindicator to realize practical applications of the thermoelectricconversion. As a result of active materials research in the 21stcentury, various materials with zT far exceeding 1 have beensynthesized and discovered.1) However, despite the revolu-tionary progress in materials research, the range of applica-tions of thermoelectric power generation and cooling/heatingtechnologies is still limited. Part of the reason for thissituation is the complicated structure of the longitudinalthermoelectric modules based on the Seebeck and Peltiereffects [Figs. 1(a) and 1(b)]. In this module structure, whichhas four junctions per pair of p-type and n-type conductors,the interfacial electrical and thermal resistances increasesignificantly; even if excellent thermoelectric characteristicsare obtained at the material level, the energy conversionefficiency decreases when the multicomponent modules areconstructed.1) There are also problems with the thermal andmechanical durability of the modules and the manufacturingcost due to complexity.To quantify this situation, Fig. 2 compares the ideal reducedconversion efficiency h̄ (red curve), i.e. the efficiency normal-ized by the Carnot efficiency, with the h̄ values observed inactual longitudinal thermoelectric modules (blue plots) as afunction of the device figure of merit ZTave, where ZTave isdefined as the average material zT within the module measure-ment temperature range2) (note that lowercase z and uppercaseZ represent the material and device figures of merit, respec-tively). The h̄ values refer to various thermoelectric modules,including Bi2Te3-based,3–5) PbTe-based,6) GeTe-based,7,8)MgSi2-based,9) Mg3Sb2-based,10,11) Skutterudite-based12) andhalf-Heusler-alloy-based13,14) systems. In all cases, the ob-served efficiencies are much lower than the ideal values.Despite the development of thermoelectric modules with highZTave, the conversion efficiency has not increased drasticallydue to excess resistance, and the larger the ZTave, the greater thedeviation from the ideal value. Following Refs. 8, 14–16, theresistance loss ratio is defined to be the ratio of the total excessresistance, including junctions and electrodes, to the totalresistance of thermoelectric materials themselves. For simpli-city, we assume that the resistance loss ratio is the same forboth electrical and thermal resistances, and plot the ZTaveContent from this work may be used under the terms of the Creative Commons Attribution 4.0 license. Any further distribution of thiswork must maintain attribution to the author(s) and the title of the work, journal citation and DOI.090101-1© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdApplied Physics Express 18, 090101 (2025) APEX REVIEWhttps://doi.org/10.35848/1882-0786/adf700https://crossmark.crossref.org/dialog/?doi=10.35848/1882-0786/adf700&domain=pdf&date_stamp=2025-09-02https://orcid.org/0000-0002-6844-6477https://orcid.org/0000-0002-6844-6477https://orcid.org/0009-0003-7789-8170https://orcid.org/0009-0003-7789-8170https://orcid.org/0000-0002-5577-8018https://orcid.org/0000-0002-5577-8018https://orcid.org/0000-0001-7939-3289https://orcid.org/0000-0001-7939-3289https://orcid.org/0000-0003-4914-5043https://orcid.org/0000-0003-4914-5043https://orcid.org/0000-0001-7680-3051https://orcid.org/0000-0001-7680-3051mailto:hiroto.adachi@okayama-u.ac.jpmailto:UCHIDA.Kenichi@nims.go.jphttps://creativecommons.org/licenses/by/4.0/https://doi.org/10.35848/1882-0786/adf700dependence of h̄ since the resistance loss ratio varies from10%–40% (Fig. 2). These curves agree well with the experi-mental data, indicating that high-performance longitudinalthermoelectric modules exhibit a resistance loss of severaltens of percent.One way to solve these problems is to implementtransverse thermoelectric conversion.17) Using the transversethermoelectric effects, it is possible to generate an electricvoltage and current in a direction perpendicular to ∇T, i.e. thetransverse direction [Figs. 1(c) and 1(d)]. The material figureof merit /z T S Txy yx yy xx2 r k= for transverse thermoelectrics isdefined in a similar way to that for longitudinal thermo-electrics, where the transverse thermopower Syx is defined asthe generated electric field in the y direction normalized byapplied ∇T in the x direction (see Sects. 3 and 4 for details).Here, κxx is the thermal conductivity for the x direction andρyy is the electrical resistivity for the y direction. The voltageand power induced by the transverse thermoelectric effectscan be increased by increasing the length and area of thematerial in the direction perpendicular to ∇T, respectively,and it is not necessary to form a large number of junctionstructures [Fig. 1(c)]. Therefore, the transverse thermoelectricconversion is suitable for reusing thermal energy distributedover a wide area. Furthermore, transverse thermoelectricconversion modules do not require junctions, and there areno problems of interfacial electrical and thermal resistancesand thermal degradation on the hot side [note that twoelectrical contacts for extracting output can be attached to thecold side, as depicted in Fig. 1(c)]. Thus, the efficiencycalculated from the material’s zxyT (red curve in Fig. 2) canbe expected in experimental modules. The transversethermoelectric conversion is also expected to improve thedurability of the modules and reduce manufacturing costs.While transverse thermoelectric conversion has many ad-vantages, there are still various issues that remain to beaddressed, such as the fact that the transverse thermopowerhas not yet reached a practical level and that the importanceof thermal boundary conditions, which are essential fordiscussing the performance of the transverse thermoelectricdevices, has not been fully recognized. Importantly, sincevarious thermoelectric conversion principles are being stu-died independently in different fields, there is currently nounified paradigm for classifying these phenomena.In this article, we review the fundamentals and advances oftransverse thermoelectrics. The article is organized as follows.First, in Sect. 2 we systematically organize the classification ofthe transverse thermoelectric conversion phenomena andsummarize their mechanisms and characteristics. The trans-verse thermoelectric conversion phenomena require symmetrybreaking for charge and heat carriers. Depending on how thissymmetry is broken, the transverse thermoelectric conversionphenomena can be broadly classified into two categories: thosethat occur due to time-reversal symmetry breaking and thosedue to structural symmetry breaking. Here, the structuralsymmetry breaking in this article means that the sample hasa certain structural asymmetry with respect to an axis taken inthe direction of a charge or heat current, at either microscale ormacroscale. In other words, this terminology is defined as thebreaking of the sample’s spatial inversion symmetry about theleft and right sides of that axis. In Sects. 3 and 4 we formulatethe performance of a transverse thermoelectric generator,refrigerator, and active cooler for the cases with time-reversalFig. 1. Schematics of (a) longitudinal thermoelectric generation, (b) longitudinal thermoelectric cooling/heating, (c) transverse thermoelectric generation, and(d) transverse thermoelectric cooling/heating. J (∥y) and ∇T (∥x) denote the charge current and temperature gradient, respectively. Direction of J in (a) and (c)corresponds to the direction of the electric field driven by the thermoelectric effects. In (b) and (d), the direction of J is shown for the cooling operation.090101-2© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWsymmetry breaking and structural symmetry breaking, respec-tively. Although a thermoelectric device also acts as a heatpump, this heating mode is not dealt with in this article due tothe limited space. Unlike the longitudinal thermoelectricconversion based on the Seebeck and Peltier effects, weshow that thermal boundary conditions in the electric-fielddirection play an essential role in the performance of thetransverse thermoelectric conversion. For the transverse ther-moelectric generator, the expressions of the figure of merit andefficiency vary depending on whether the thermal boundaryconditions are isothermal or adiabatic, but the important pointis that they are interchangeable with each other; the transversethermoelectric generation is not inherently more or lessefficient than the longitudinal thermoelectric generation, con-trary to the discussion in Ref. 18 Importantly, depending onthe thermal boundary conditions, a correction term due to thetransverse temperature gradient and Seebeck effect occurs inthe transverse thermopower. The experiments shown in Sect. 5demonstrate the importance of the thermal boundary conditionand the contribution of the correction term. In Sect. 6 wesummarize research activities for developing applications ofthe transverse thermoelectric conversion and introduce newconcepts for further performance improvements. Section 7 isdevoted to the conclusions and prospects. We hope that thisarticle will serve as a cornerstone for basic and appliedresearch on transverse thermoelectrics and provide an oppor-tunity for interdisciplinary fusion of research on varioustransverse thermoelectric conversion phenomena that havebeen studied in different communities.2. Classification of transverse thermoelectric con-version phenomenaThe purpose of this section is to establish a unified classificationfor various transverse thermoelectric conversion phenomena.The transverse thermoelectric conversion phenomena caused bytime-reversal symmetry breaking include magneto-thermoelec-tric and thermo-spin effects, which have been studied mainly inthe field of spin caloritronics.19–21) Here, the magneto-thermo-electric effects refer to the thermoelectric effects depending onmagnetic fields or magnetization, while the thermo-spin effectsrefer to the conversion between a heat current and a spin current,i.e. a flow of spin angular momentum. Some of thesephenomena were discovered in the 21st century, and researchin this regard is still in its infancy. Although the transversethermoelectric conversion phenomena caused by structuralsymmetry breaking have been studied in the field of thermo-electrics for a long time,22,23) they are often called by completelydifferent terminologies, which may confuse non-specialists. Forexample, the transverse thermoelectric conversion in a materialhaving charge carriers with opposite signs along different crystalorientations, i.e. axis-dependent conduction polarity (ADCP), iscalled goniopolarity24–26) when it is caused by a single band, but(p× n)-type transverse thermoelectrics27,28) when it is caused bymultiple bands, despite the fact that the details of the bandstructure cannot be determined only by thermoelectric measure-ments. We argue that the leading classification of the transversethermoelectric effects should be based on the symmetry andphenomenology, which can be determined from thermoelectricmeasurements alone. Therefore, we propose a classification ofthe transverse thermoelectric effects comprising three hierarchies:symmetry (hierarchy 1), phenomenology (hierarchy 2) andmechanism (hierarchy 3). Figure 3 shows the currently knowntransverse thermoelectric generation phenomena organized basedon this classification, which is a refinement of the classificationin Ref. 17. In the following subsections, we outline themechanisms and characteristics of each phenomenon.Although we will mainly focus on the thermoelectric generation,the phenomena in Fig. 3 have their Onsager reciprocals, enablingtransverse thermoelectric cooling/heating (Table I). Recently, ahigher-order transverse thermoelectric effect called the transverseThomson effect29) has been observed experimentally, but it is notdiscussed in this article.2.1. Transverse thermoelectric effects with time-re-versal symmetry breakingThe majority of the transverse thermoelectric effects requiretime-reversal symmetry breaking. In other words, the trans-verse thermoelectric effects listed in this subsection occur inconductors under magnetic fields or in magnetic materialswith magnetization. Here, “magnetization” in this articleincludes not only spontaneous magnetization of ferromagnetsand ferrimagnets but also weak magnetization of cantedantiferromagnets.2.1.1. Ordinary Nernst effect. The oldest known trans-verse thermoelectric conversion phenomenon is the Nernsteffect, discovered in 1886 by A. V. Ettingshausen and W.Nernst.30) They found that when ∇T and the magnetic field Hare applied to a conductor in directions orthogonal to eachother, a thermoelectric voltage is generated in the direction ofthe cross product of ∇T and H. The transverse thermopowerdue to the Nernst effect in normal metals is proportional tothe magnitude of the magnetic field H, while that insemimetals often exhibits a nonlinear dependence on H dueto the concerted contributions of electron and hole transport.This phenomenon is often called the Nernst–Ettingshauseneffect, but in this article we simply call it the Nernst effectFig. 2. Calculated and observed reduced conversion efficiency h̄ as afunction of the average figure of merit ZTave. Blue curves are calculated withchanging the resistance loss ratio from 10% to 40%. Blue data points areestimated from the experimental results in Refs. 3–14.090101-3© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWFig. 3. Classification of the transverse thermoelectric effects and schematics of (a) the ordinary Nernst effect (ONE), (b) anomalous Nernst effect (ANE),(c) Seebeck-effect-driven ordinary Hall effect, (d) Seebeck-effect-driven anomalous Hall effect (SAHE), (e) spin-Seebeck-effect-driven (SSE-driven) inversespin Hall effect (ISHE), (f) spin-dependent-Seebeck-effect-driven (SdSE-driven) ISHE, (g) off-diagonal Seebeck effect due to macroscale anisotropic structure,and (h) off-diagonal Seebeck effect due to microscale anisotropic structure. H,M, Js, e−, and h+ denote the magnetic field, magnetization, spatial direction of aspin current, conduction electrons and holes, respectively. In (g), the gray (orange) arrows show the direction of charge (heat) currents in the constituentconductors of the artificially tilted multilayer (ATML). Thermopower tensor is antisymmetric Sxy = − Syx when a magnetic field or magnetism breaks time-reversal symmetry, or symmetric Sxy = Syx when a crystal lattice or layered macrostructure breaks structural symmetry.090101-4© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWand its reciprocal the Ettingshausen effect. Nowadays, thisphenomenon is widely called the ordinary Nernst effect(ONE) to distinguish it from the ANE, described later.The ONE is caused by the Lorentz force acting onconduction electrons and/or holes in a conductor under amagnetic field [Fig. 3(a)]. The ONE-induced transversethermopower in typical metals is very small, but it is knownthat several Dirac/Weyl semimetals and low-dimensionalmaterials exhibit high transverse thermoelectric conversionperformance due to the ONE.31–35) For example, the figure ofmerit of BiSb alloys reaches zxyT> 0.3 at 100–200 K whenan external magnetic field of ∼1 T is applied.31) However,the drawback of the ONE is that an external magnetic fieldmust be applied to generate the transverse thermopower. Toovercome this drawback, it has been demonstrated that thethermoelectric power generation based on the ONE can beachieved without applying an external magnetic field byembedding permanent magnets in a module, where theremanent magnetization of the permanent magnets appliesstray magnetic fields to the adjacent conductors.36) However,there are still issues to be addressed, such as the fact that thefill factor of ONE materials that are responsible for the powergeneration decreases by the amount of the embeddedpermanent magnets.2.1.2. Anomalous Nernst effect. The ANE is a phenom-enon in which a thermoelectric voltage is generated in thedirection of the cross product of ∇T and the spontaneousmagnetization M in a magnetic material [Fig. 3(b)]. Thetransverse thermopower ST in a magnetic material under amagnetic field is phenomenologically described as thesummation of the contributions proportional to H, i.e.ONE, and magnetization M, i.e. ANE37):( )S Q H Q M, 1H MT 0 0m m= +where μ0 is the vacuum permeability and QH (QM) is theproportionality factor of the H-dependent (M-dependent)term. Equation (1) shows that the anomalous Nernst coeffi-cient SANE (=QMμ0Ms with Ms being the saturation magne-tization) can be extracted by extrapolating the H dependenceof ST from the high-field region, in which M is saturated, tozero field. Thus, if M is aligned in one direction, the ANEcan generate a thermoelectric voltage and power evenwithout applying an external magnetic field. Note that themagnitude of the transverse thermopower due to the ANE isdetermined by SANE, not QM since QM is just a phenomen-ological parameter defined for convenience to separate theANE from the ONE. Based on Eq. (1), SANE is oftencompared in terms of M to discuss the scaling behavior.However, since the ANE and related transport properties areestimated for uniformly magnetized materials, they should becompared in terms of Ms, not M. In fact, SANE is notcorrelated with Ms in many magnetic materials; the scalingbehavior does not hold even for simple ferromagnetic metals,such as Fe, Ni and Co.37)The mechanism of the ANE differs from that of the ONE. Ingeneral, the ANE originates from an intrinsic mechanism, i.e. afictitious magnetic field in a momentum space derived fromthe Berry curvature in electronic band structures, and/or anextrinsic mechanism, i.e. skew scattering, side jump andmagnon-electron drag.38–40) Recent experiments and calcula-tions show that the ANE can be enhanced through the intrinsicmechanism by the topology of the electronic structures, and itbecomes one of the hot topics in condensed matter physics.Although SANE is in the order of 0.1 μV/K in conventionalferromagnetic metals (e.g. Fe, Ni and Co), Co-based Heuslercompounds (i.e. Co2MnGa and Co2MnAl1−xSix) exhibit atransverse thermopower of SANE∼ 6 μV/K due to theirtopological electronic structure.41–46) In addition to the Co-based Heusler compounds, large SANE has been observed invarious systems including the binary Fe-based alloys,47–51)Mn-based pnictide (i.e. YbMnBi2),52) Co-based shandite (i.e.Co3Sn2S2),53–55) U-based compound (i.e. UCo0.8Ru0.2Al),56)magnetic multilayer films,57,58) amorphous metals with nanos-cale precipitates59–62) and rare-earth permanent magnets (i.e.SmCo5-type magnets).63,64)To realize practical applications of the ANE, it is necessaryto find and develop magnetic materials with larger SANE.Guidelines for designing SANE can be obtained by separatingit into two components,44,65):( )S , 2xx xy xxANE AHEr a r a= -where ρxx, ρAHE and αxx (αxy) are the longitudinal electricalresistivity, anomalous Hall resistivity and diagonal (off-diagonal) component of the thermoelectric conductivitytensor, respectively. Here, the thermoelectric conductivity,also called the Peltier conductivity, is the tensor that relatesan induced charge current density to an applied temperaturegradient, the unit of which is A/Km. The first term on theright-hand side of Eq. (2) shows the direct transversethermoelectric conversion due to αxy; a recent trend inimproving SANE is to find materials with large αxy causedby the Berry curvature of the electronic bands near the Fermilevel. The second term on the right-hand side of Eq. (2)appears due to the anomalous Hall effect (AHE)66) inducedby the longitudinal carrier flow through the Seebeck effect.However, even though the output of the ANE has improvedin recent years, SANE is still more than an order of magnitudesmaller than the Seebeck coefficients of thermoelectricmaterials in practical use. Therefore, further performanceTable I. List of the transverse thermoelectric generation phenomena and their Onsager reciprocals.Heat-to-charge current conversion Charge-to-heat current conversionOrdinary Nernst effect Ordinary Ettingshausen effectAnomalous Nernst effect Anomalous Ettingshausen effectSeebeck-effect-driven ordinary Hall effect Ordinary-Hall-effect-driven Peltier effectSeebeck-effect-driven anomalous Hall effect Anomalous-Hall-effect-driven Peltier effectSpin-Seebeck-effect-driven inverse spin Hall effect Spin-Hall-effect-driven spin Peltier effectSpin-dependent-Seebeck-effect-driven inverse spin Hall effect Spin-Hall-effect-driven spin-dependent Peltier effectOff-diagonal Seebeck effect Off-diagonal Peltier effect090101-5© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWimprovements through materials research are necessary torealize thermoelectric applications of the ANE.Here, we should mention the non-relativistic counterpart ofthe ANE; the topological Nernst effect [Fig. 3(b)]. In thebreaking of the spin-rotation invariance, the Nernst effectoccurs in the absence of the spin–orbit interaction due to non-coplanar spin structures, e.g. in skyrmions and non-coplanarantiferromagnets.67–70) Although only a limited number ofmaterials exhibit the topological Nernst effect, this phenom-enon also functions as the transverse thermoelectric conver-sion.2.1.3. Seebeck-effect-driven ordinary and anomalousHall effects. As part of efforts to further improve magneto-thermoelectric conversion performance, a transverse thermo-electric conversion mechanism appearing in hybrid materialscomposed of magnetic and thermoelectric materials wasproposed and demonstrated in 2021.71) This mechanism,originally named the Seebeck-driven transverse thermoelec-tric generation, operates when a closed circuit is constructedby connecting the ends of the magnetic and thermoelectricmaterials and is driven by the concerted action of the Seebeckeffect in the thermoelectric material and AHE in the magneticmaterial. In other words, this mechanism corresponds toartificially modulating the second term on the right-hand sideof Eq. (2) by constructing hybrid materials. However, thename “Seebeck-driven transverse thermoelectric generation”could be confused with the transverse thermoelectric conver-sion due to the off-diagonal Seebeck effect described below.In this article, we thus call this mechanism the Seebeck-effect-driven AHE (SAHE).SAHE has a high degree of freedom in material design.The transverse thermopower in the hybrid structure depictedin Fig. 3(d) is expressed as,( ) ( )/S SrS S , 3T ANEAHETE MTE Mrr r= -+-where ρM(TE) and SM(TE) are the longitudinal resistivity andSeebeck coefficient of the magnetic (thermoelectric) material,respectively, and r is the size ratio determined by thedimensions of the magnetic and thermoelectric materials.71)Here, we consider a situation in which magnetic and thermo-electric materials are separated by an insulating layer to avoidthe shunting effect. When the in-plane areas of the magneticand thermoelectric materials are the same, r is determined bythe thickness ratio of the two: r= tTE/tM with tM(TE) being thethickness of the magnetic (thermoelectric) material. Thesecond term on the right-hand side of Eq. (3) represents thecontribution of SAHE. By optimizing the combination andsize ratio of the magnetic and thermoelectric materials, it ispossible to obtain a much larger transverse thermopower thanSANE alone. When a thin film of a magnetic material iscombined with a slab of a thermoelectric material, r becomesvery large, and ST approaching 100 μV/K has beenobserved.71) When a slab of a magnetic material is combinedwith a slab of a thermoelectric material, ST is limited to around10–20 μV/K due to smaller r but such an all-bulk hybridstructure is more suitable for increasing output power.72)Recently, direct-contact SAHE has also been demonstrated,in which a magnetic material is directly deposited on thesurface of a thermoelectric material, rather than forming aclosed circuit.73) Direct-contact SAHE functions with asimpler device structure, but its transverse thermopowerdecreases due to the shunting effect compared to the valuefor the structure depicted in Fig. 3(d). Since there is a vastamount of research data that has been accumulated over manyyears on the AHE and the Seebeck effect, there is still plentyof room for improving the performance of SAHE. However,its versatility is lower than that of the ANE; whereas the ANEworks in both in-plane and perpendicularly magnetized con-figurations, SAHE works only in the perpendicularly magne-tized configuration. Furthermore, even if ST is enhanced, zxyTand the power generation efficiency of SAHE are not alwayshigher than those of the ANE.74)It is worth mentioning that, if a magnetic material showingthe AHE is replaced with a nonmagnetic material showingthe ordinary Hall effect, the Seebeck-effect-driven ordinaryHall effect should occur [Fig. 3(c)]. In principle, thetransverse thermoelectric conversion due to the Seebeck-effect-driven ordinary Hall effect exists in any hybridmaterial comprising two or more materials under a magneticfield. However, at present, there are no reports where itscontribution is explicitly observed.2.1.4. Spin Seebeck and spin-dependent Seebeckeffects. As is clear from the phenomena discussed above,magnetic materials play an important role in the developmentof the transverse thermoelectric conversion, but magnetizationhas been treated as a macroscopic parameter that determinesthe symmetry of the phenomena. Since the beginning of the21st century, there has been a lot of activity in the field ofspintronics, which aims to utilize the spin degree of freedom,an origin of magnetism, from a microscopic perspective. Inthis field, various transport phenomena related to a spin currenthave been discovered and elucidated. As part of this activity,thermoelectrics and thermal transport have been integrated intospintronics, giving rise to the field of spin caloritronics.19–21)Spin caloritronics has rapidly progressed since the discovery ofthe SSE, in which a spin current is generated from a heatcurrent in a magnetic material.75–78)The SSE enables spin-current-driven transverse thermo-electric generation in combination with the spin-to-chargecurrent conversion phenomenon called the ISHE79–81)[Fig. 3(e)]. When ∇T is applied perpendicular to a bilayerstructure consisting of a metal film on a magnetic materialhaving in-plane M, a spin current with the spatial direction Jsis generated near the metal/magnetic-material interface due tothe SSE. This spin current is then converted into a chargecurrent in a direction perpendicular to both ∇T and Mthrough the spin–orbit interaction, i.e. ISHE, in the metalfilm. In spin caloritronics, the SSE in the configuration shownin Fig. 3(e) is called the “longitudinal” SSE because ∇T andJs are parallel to each other. As a result of converting the spincurrent into the charge current in the orthogonal direction bythe ISHE, the SSE can be the mechanism of the “transverse”thermoelectric conversion. We also note that, instead of theISHE, the inverse Rashba–Edelstein effect at the junctioninterface of different materials can be used to induce the spin-current-driven transverse thermoelectric conversion.82,83)The energy carrier of the SSE is the collective motion ofthermally excited localized magnetic moments, i.e. spin wavesor magnons.76,78,84–89) Thus, it can operate even if themagnetic layer is an insulator, enabling the thermoelectricconversion using an insulator, which is not be possible with090101-6© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWother phenomena.90,91) At present, the thermopower generatedby the SSE is on the same order as the anomalous Nernstcoefficient, which is insufficient for thermoelectric applica-tions. However, physics and material science researches areunderway to improve the efficiency of the heat-to-spin currentconversion in magnetic materials and spin-to-charge currentconversion in metals, as well as to scale up materials fromsimple bilayer films to multilayers92) and bulk composites.93)The spin currents are carried by elementary particles orquasiparticles having spin angular momentum. Historically,conduction-electron spin currents and magnon spin currentshave been actively studied in spintronics and spin calori-tronics. As mentioned above, the carriers of the spin currentsgenerated by the SSE are magnons. In magnetic conductors,the spin currents can also be generated via conduction-electron transport under ∇T, which is called the spin-dependent Seebeck effect (SdSE). This was directly observedby Slachter et al. in 2010 using a non-local method.94) As thename suggests, this phenomenon originates in the spindependence of the Seebeck coefficient in a magnetic con-ductor. It is also possible to convert the spin currentgenerated by SdSE into a transverse voltage using theISHE95); the SdSE can also be the driving principle for thetransverse thermoelectric conversion [Fig. 3(f)]. However, itis difficult to distinguish the transverse thermoelectric powergenerated by this mechanism from the magnon-driven SSEand the interface effect on the ANE.2.2. Transverse thermoelectric effects with structuralsymmetry breakingIn the transverse thermoelectric conversion, there are alsophenomena that do not depend on magnetic fields ormagnetization. These phenomena are driven by the aniso-tropy of the structure and/or electronic transport properties.As mentioned above, various names are used for thetransverse thermoelectric conversion phenomena due tostructural symmetry breaking. In this article, we classifythem into the phenomena due to macroscale and microscaleanisotropy structures.2.2.1. Off-diagonal Seebeck effect due to macroscaleanisotropic structure. In an artificial multilayer structureconstructed by alternately stacking two conductors and cuttingthe stack at a certain angle, anisotropic transport propertiesoccur even if the constituent conductors exhibit isotropicelectron/hole transport. When the off-diagonal componentsof the thermopower tensor become finite due to this aniso-tropy, the artificially tilted multilayer (ATML) functions as atransverse thermoelectric conversion element22,23,96–109)[Fig. 3(g)]. Since this transverse thermoelectric conversionoriginates from the Seebeck coefficient, it is called the off-diagonal Seebeck effect, which is clearly different from theNernst effects that appear under time-reversal symmetrybreaking. As described later, the phenomenological formula-tion differs in part between the transverse thermoelectricconversion principles in systems with time-reversal symmetrybreaking and structural symmetry breaking.The transverse thermoelectric conversion based on theoff-diagonal Seebeck effect has been studied for a longtime,22,23) and it is possible to design a transverse thermo-power and figure of merit by selecting appropriate consti-tuent materials and optimizing their tilt angle and thicknessratio. While the figure of merit for the off-diagonal Seebeckeffect in the ATML is improved by a high transversethermopower, high effective electrical conductivity, andlow effective thermal conductivity for the hybrid material,the characteristics required of each constituent materialdiffer from those of conventional thermoelectric materials.In addition to the large difference in the Seebeck coefficientbetween the two materials, the contrast in the electrical andthermal conductivities between them is important, whichinduces nonuniform charge and heat current distributions[Fig. 3(g)]. In other words, even if the difference in theSeebeck coefficient is large, if the electrical and thermalconductivities of the two materials are the same, thetransverse thermoelectric conversion does not occur. Thetransverse thermopower takes its maximum value when thetilt angle is 45°, but the other parameters exhibit differentbehaviors. The smaller the tilt angle, the higher the effectiveelectrical conductivity in the direction of the charge currentand the lower the thermal conductivity in the direction ofthe heat current in the ATML. Based on these situations, themaximum value of the figure of merit for the off-diagonalSeebeck effect in the ATML is obtained at an angle smallerthan 45°.107,108) However, if the tilt angle is too small, it isdifficult to process and synthesize the ATML in practice andits thermoelectric performance deteriorates due to theinfluence of boundary conditions.98,99) Thus, the contrastin the electrical and thermal conductivities should not be toolarge. Many studies on ATMLs have been conducted usingsintered bulk materials, and zxyT values exceeding 0.2 havebeen achieved at around room temperature.108) The trans-verse thermoelectric conversion performance of the ATMLcan be calculated analytically, but in many systems, theexperimentally obtained zxyT is much smaller than the idealvalue due to the interfacial electrical and thermal resistancesbetween the two materials. Despite this situation, inSmCo5/Bi0.2Sb1.8Te3 (BST) ATML developed by Andoet al. extremely small interfacial electrical and thermalresistances were achieved, and zxyT close to the ideal valuewas obtained experimentally, highlighting the importance ofinterface engineering in the ATML.108) The transversethermoelectric conversion performance of SmCo5/BSTATML has been further improved by utilizing the aniso-tropic electrical and thermal conductivities of the SmCo5magnets and hybridizing their ANE; zxyT now reaches 0.3 ataround room temperature.109)2.2.2. Off-diagonal Seebeck effect due to microscaleanisotropic structure. Even without constructing ATMLs,the transverse thermoelectric conversion can be achieved insingle crystals showing anisotropic electronic conductionproperties. In the single crystals of several metals, such asSb, Gd, Ho, Er and Bi, the Seebeck coefficient is of the samesign but anisotropic.110–112) These materials having “uni-polar” anisotropy make it possible to construct a transversethermoelectric conversion element simply by cutting them sothat the off-diagonal components of the thermopower tensorare finite. It is difficult to construct high-output powergeneration or cooling/heating devices from such single-crystal metals because of their small transverse thermopowercaused by the unipolar anisotropy in the Seebeck coefficient.In fact, due to their unipolar nature, the temperature gradientcan never be purely orthogonal to the electric field.113)Nevertheless, the off-diagonal Seebeck effect in single-090101-7© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWcrystalline metals is being considered for use in heat fluxsensors because of their low thermal resistance.114)It is also possible for single crystals to exhibit ADCP,though early experimental reports115–118) did not recognizetheir utility as transverse thermoelectrics until it was pointedout.27) In recent years, even more single-crystalline materialswith ADCP have been discovered.24–26,119–122) Unlike theaforementioned unipolar anisotropy, these materials exhibit“ambipolar anisotropy”, in which the sign of the Seebeckcoefficient reverses depending on the crystal orientation.Therefore, by cutting the material along the direction betweenthe axis showing the large positive Seebeck coefficient andthe axis showing the large negative Seebeck coefficient andby applying a temperature gradient to allow the conductionelectrons and holes to diffuse in the transverse direction,excellent transverse thermoelectric performance can be ob-tained through the off-diagonal Seebeck effect [see Fig. 3(h)].For example, it has been reported that zxyT of single-crystal-line Re4Si7 reaches 0.7 at 980 K.25) This value is outstandingcompared to other principles, but there are issues such as thelimited choice of materials, the need for large single crystals,and the fact that the large figure of merit is obtained only athigh temperatures.The transverse thermoelectric conversion arising fromADCP is referred to by various names. When ambipolaranisotropy arises from a single metallic band with an openFermi surface and alternating positive and negative Gaussiancurvature, this metallic band is called a goniopolar band, andthese materials are referred to as goniopolar materials.24)When ambipolar anisotropy arises in semiconductors orsemimetals whose separate n-type and p-type bands resultin n-type conduction dominating one axis and p-type con-duction dominating orthogonally, these materials are called(p× n)-type transverse thermoelectrics,27,28) (see Hierarchy 3in Fig. 3). However, as mentioned at the beginning of Sect. 2the details of the band structures cannot be determined fromthermoelectric measurements alone, and the terminology ofthe thermoelectric effects should be determined based ontheir symmetry and phenomenology. Thus, we have mergedthe effects mentioned in this subsection into “off-diagonalSeebeck effect due to microscale anisotropic structure” (seeHierarchy 2 in Fig. 3).3. Formulation of transverse thermoelectric conver-sion with time-reversal symmetry breakingIn this section, we present the formulation of the efficiency ofboth longitudinal and transverse thermoelectric devices. Thissubject has a long history as can be seen in manytextbooks123–126) and publications.91,127–129) However, lessknown is the fact that an apparent expression for the efficiencyvaries with the choice of independent variables appearing inthe transport equations, as well as with the symmetry of thetransport coefficient matrices. For the longitudinal device,choosing the charge current density and the temperaturegradient (j,∇T) as independent variables leads to the conven-tional expression for the efficiency.123,124) An alternativechoice of the electric field and the temperature gradient(E,∇T) as independent variables yields another expressionfor the efficiency.130) Of course, these two efficiency expres-sions are physically the same and equivalent, but expressed byseemingly different parameters.On the other hand, in the case of the transverse device, thethermal boundary condition in the transverse direction, ormore precisely in the direction of the electric field, inevitablyspecifies the allowed choice of independent variables. Forexample, under the adiabatic condition, the charge and heatcurrent densities (j, q) are the appropriate independent vari-ables, while under the isothermal condition the charge currentdensity and temperature gradient (j,∇T) are the appropriateindependent variables. As expected, these two thermalboundary conditions yield two different expressions for theefficiency.131,132) In the presence of the Nernst effect but theabsence of the Seebeck effect, these two efficiency expres-sions are the same and equivalent, similar to the case of thelongitudinal device. However, in the simultaneous presenceof the Nernst and Seebeck effects, these two expressions areno longer the same quantity, representing two physicallydifferent situations. Moreover, inclusion of the thermal Halleffect (THE) or the off-diagonal component of the thermalconductivity tensor further modifies the efficiency expression(note that THE is often called the Righi–Leduc effect133)).Below, following the line of argument of Ref. 134, we presentan as simple as possible formulation of the performance oflongitudinal and transverse thermoelectric devices. In Sect. 3.1we present two formulations for the performance of longitudinalthermoelectric generators that are expressed by two differentfigures of merit. Next, in Sect. 3.2 we discuss the performanceformulation for transverse thermoelectric generators, and showhow the efficiency expression differs depending on the iso-thermal or adiabatic boundary condition in the transversedirection. Then, in Sect. 3.3 we discuss the performance oftransverse refrigerators and active coolers.135) In Sect. 4 wediscuss the case for the off-diagonal Seebeck effect.24,26,28)In this section, we discuss transverse devices with time-reversal symmetry breaking, while in the next section we discusstransverse devices with structural symmetry breaking. Animportant distinction between time-reversal symmetry breakingand structural symmetry breaking is that the off-diagonalcomponents of the thermopower, electronic thermal conductivityand electrical resistivity tensors [see Eqs. (25)–(27) below]are antisymmetric (Sxy=− Syx, κxy=− κyx, ρxy=− ρyx) for theformer and symmetric (Sxy= Syx, κxy= κyx, ρxy= ρyx) for thelatter.136) It is a useful distinction for clarifying how fundamen-tally different these phenomena are from a tensor representa-tional standpoint.3.1. Revising performance of a longitudinal thermo-electric generatorBefore formulating the efficiency of a transverse thermo-electric generator, we revisit the efficiency of a longitudinalthermoelectric generator. The system is shown in Fig. 4(a),where p-type and n-type thermoelectric materials are ther-mally connected in parallel and electrically connected inseries. This device operates under a temperature bias betweena hot reservoir Th and a cold reservoir Tc.First, we present the conventional formulation of theefficiency123,124) using the charge current density and thetemperature gradient (j,∇T) as independent variables. Next,we present an alternative formulation of the efficiency usingthe electric field and the temperature gradient (E,∇T) asindependent variables.130)3.1.1. Conventional expression. We begin with thefollowing transport equations for the heat current density q090101-8© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWand electric field E137):( )qq TSjjTT, 4xyxyxyk= -n n⎜ ⎟ ⎜ ⎟⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠( )EEjjSTT, 5xyxyxyr= +n n⎜ ⎟ ⎜ ⎟⎜ ⎟⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠where Sν, κν and ρν are respectively the Seebeck coefficient,thermal conductivity, and electrical resistivity of each ther-moelectric element with the carrier index ν= p, n. Althoughwe assume an isotropic material in the above transportequations, this is merely for the sake of simplicity. In thecase of an anisotropic material, we can start from thetransport equations represented with tensor coefficients(see, e.g. Eq. (26.12) of Ref. 137). Then, after multiplyingthe x component of Eq. (4) with the cross-section of eachelement A= lylz, as well as integrating the x component ofEq. (5) along the x axis, we obtain:( )Q TSJ K T , 6x x= + D( )V RJ S T , 7xD = - + Dwhere Qx= Aqx, ΔV=− Exlx, Jx= Ajx, ΔT= Th− Tc,S= Sp− Sn, R= (ρp+ ρn)lx/A and K= (κp+ κn)A/lx. Theabove linear-response relation can be expressed by using amatrix as,( )QVJT, 8x xD=D⎛⎝⎞⎠⎛⎝⎞⎠where the matrix  is defined by,( ) TS KR S.=-The determinant of  is given by ( ) zTdet 1kr= + ,where κ= κp+ κn, ρ= ρp+ ρn and the quantity,( )zTTS, 92kr=is the figure of merit of the longitudinal device. Note that,because of the conditions R> 0 and K> 0, which are set bythe second law of thermodynamics,137) the sign of det isalways positive. This means that the linear-response relation[Eq. (8)] is physically stable, so that there is no constraint onthe value of zT except that it should have a positive sign.Therefore, the figure of merit for the longitudinal device isunbounded, namely,( )zT0 . 10< ¥Since the linear-response relations have been set up, we arenow ready to calculate the efficiency of the Seebeckgenerator [Fig. 4(a)]. In the following, in line with theargument of Ref. 134, we consider only a term linear inΔT. This means that the Joule heating RJx2, which otherwiseenters the efficiency evaluation through the thermal powerinput, can be safely neglected in our calculation. The Carnotefficiency is given by,( )TT, 11Ch =Dwhere T= (Th+ Tc)/2, and the reduced efficiency / Ch h h=relative to the Carnot efficiency is defined by,¯ ( )TT, 12outhh =Dwhere h is the thermal power input from the hot reservoirand R Jxout L2= is the electrical power output dissipated bythe load resistance RL.In order to evaluate the right-hand side of Eq. (12), weneed to know the charge current Jx determined by,( )JVR, 13xemftot=where Vemf is the electromotive force and Rtot= R+ RL is thetotal resistance of the circuit. Using Vemf= SΔT andQxh = , the efficiency can be expressed as a function ofu= RL/R as {( ) }/u u zT u1 12 1h = + + + - , where zT isdefined in Eq. (9). The maximum reduced efficiency h isobtained for the stationary value u zT1= +* , whichyields,¯ ( )zTzT1 11 1. 14h =+ -+ +3.1.2. Alternative expression. In the preceding subsec-tion, we have formulated the efficiency of the longitudinalthermoelectric device by choosing (j,∇T) as independentvariables. Below, we discuss an alternative formulation of theefficiency by choosing (E,∇T) as independent variables.Fig. 4. Schematics of the longitudinal thermoelectric device [(a)] con-sidered in Sect. 3.1 and transverse thermoelectric device [(b)] considered inSect. 3.2. In (a), p-type and n-type thermoelectric materials are thermallyconnected in parallel and electrically connected in series, with both cross-sections being the same as A = lylz and the device is sandwiched between ahot reservoir Th and a cold reservoir Tc. In (b), a transverse thermoelectricmaterial is sandwiched between Th and Tc. lx, ly and lz denote the length ofthe material along the x, y and z directions, respectively.090101-9© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWWe first invert Eqs. (6) and (7) as follows:( )QTSRV K T , 15x = - D + ¢D( )JRVSRT1, 16x = - D + Dwhere /K K TS R2¢ = + . This defines the thermal conduc-tivity at zero electric field:( )TS. 172k kr¢ = +The above linear-response relation can be represented as,( )QJVT, 18xx= DD̂⎛⎝⎞⎠⎛⎝⎞⎠where the matrix  is defined by,// /TS R KR S R1.=- ¢-̂ ⎜ ⎟⎛⎝⎞⎠The determinant of  is calculated to be det =( )( )/R K z T1¢ - ¢ , where,( )z TTS, 192k r¢ =¢is a new figure of merit defined by using k¢ 130) Note that, forEq. (18) to be physically stable, det should be positive,from which we obtain a condition z T1 0- ¢ > . This meansthat the figure of merit is bounded as,( )z T0 1. 20¢ <We next calculate the reduced efficiency for the presentchoice of independent variables. Using Qxh = ,Vemf= SΔT, and Jx= Vemf/Rtot with Rtot= RL+ R, thereduced efficiency [Eq. (12)] is calculated to be h ={( ) ( )}/u u z T u1 12 1+ ¢ - + - , where u= RL/R. The max-imum reduced efficiency h is obtained for the stationaryvalue u z T1= - ¢* , which yields,¯ ( )z Tz T1 11 1. 21h =- - ¢+ - ¢3.1.3. Summary of longitudinal device. To summarizethis subsection, we have formulated the efficiency of alongitudinal thermoelectric generator based on the Seebeckeffect in two different ways. In the former we have calculatedthe maximum efficiency [Eq. (14)] by choosing (j,∇T) asindependent variables, whereas in the latter we have calcu-lated the maximum efficiency Eq. (21) by choosing (E,∇T)as independent variables.130) We note that, in deriving thetwo different efficiency expressions, we assume no physicaldifference in the condition of the thermoelectric device; thedifference in the efficiency expressions stems from thedifference in the choice of the independent variables. Inother words, the two seemingly different efficiency expres-sions for the longitudinal device correspond to two differentrepresentations of the same quantity in two different para-meters zT and z T¢ . Indeed, from Eqs. (9), (17) and (19), wefind that zT and z T¢ are related by,( )z TzTzT1. 22¢ =+Then, substituting Eq. (22) into the efficiency expression ofEq. (21), we come to the efficiency expression of Eq. (14). Inthis sense, we note that these two expressions are equivalent,and achieving z T 1¢ = limit is as difficult as achievingzT=∞ limit.3.2. Performance of transverse thermoelectric gen-eratorIn this subsection, we formulate the efficiency of a transversethermoelectric generator, as shown in Fig. 4(b), where atransverse thermoelectric material showing ONE/ANE issandwiched between Th and Tc. Below, it is shown that athermal boundary condition in the direction of the electricfield [along the y axis in Fig. 4(b)] plays a crucial role in theformulation.In the literature,91,125,129,131,132,134) the efficiency of thetransverse device has been discussed by assuming that onlythe Nernst/Ettingshausen effects are present in the device,while the Seebeck/Peltier effects are absent. In this subsec-tion, we discuss a more realistic situation where both theNernst/Ettingshausen and Seebeck/Peltier effects are present.Below, we show that the efficiency expression is substan-tially modified by the simultaneous presence of the Nernst/Ettingshausen and Seebeck/Peltier effects. Moreover, wediscuss the influence of THE, which also affects theefficiency expression.We use the following transport Eqs.137):( )qq TSjjTT, 23xyxyxyk= -^ ^⎜ ⎟ ⎜ ⎟⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠( ) EEjjSTT, 24xyxyxyr= +⎜ ⎟ ⎜ ⎟⎜ ⎟⎛⎝⎞⎠⎛⎝⎞⎠⎛⎝⎞⎠where the transport coefficient matrices are given by,( )SS SS S, 25xx xyyx yy= ⎜ ⎟⎛⎝⎞⎠( ) , 26xx xyyx yykk kk k= ⎛⎝⎞⎠( ). 27xx xyyx yyrr rr r=^ ⎛⎝⎞⎠Note that the resistivity in the above equation is measuredunder a zero temperature gradient, which coincides with theconventional definition of electrical resistance.138) Under theopen-circuit condition along the x axis ( jx= 0), equationsnecessary for the efficiency calculation are given by,( )q T T TS j , 28x xx x xy y xy yk k= -  -  +( )q TS j T T , 29y yy y yy y yx xk k= -  - ( )E j S T S T. 30y yy y yy y yx xr= +  + 3.2.1. Adiabatic condition in transverse direction. Wefirst discuss the adiabatic boundary condition in the direction090101-10© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWof the electric field [along the y axis in Fig. 4(b)] since this isthe “natural” case for the transverse device according toRef. 134. In this subsection, we are concerned with the casewhere the transport coefficient matrices in Eqs. (25)–(27)have antisymmetric off-diagonal components, i.e. Syx=−Sxy,κyx=−κxy and ρyx=−ρxy. This symmetry applies to ONE andANE devices.Now, using the adiabatic condition along the y axis (qy= 0),the above equations can be solved for ∇xT and Ey, yielding,( )( )( ) ( )TTSj q1, 31xxyxxyxxxaa ak k = -( )( )( )( )E jSq . 32y yy yyxxxxaaark= -In these equations, we defined the following new coefficientsrenormalized by the Seebeck effect and THE:( )( )( )( )( )( )( )S S SS S SrrS S r,,1 ,1 ,1 ,xy xy yy xyx yx yy yxx xx x y yxyy yy x y yxyy yy x y yxaaaaak kk k= - Q= - Q= - Q Q= - Q Q= - Q Qwhere Qx= κxy/κyy, Qy= κyx/κyy and ryx= κyy/κxx. Note thatfor an isotropic material (κxx= κyy), we have Q≡Qx=−Qydue to the symmetry κyx=− κxy. From this and Syx=− Sxy,we obtain,( )( ) ( )S S S S , 33xy yx xy yya a= - = - Q( ) ( )( ) ( ) 1 , 34xx yy xxa a 2k k k= = + Q( )( )S S 1 . 35yy yya 2= + QNote also that, due to the variable change from ∇xT to qx, theadiabatic resistivity,( )( )( )( )( ) ( )( )( )( )( )( )TS TS STS TS, 36yy yyyyyyxy yxxxyyyyyyyxxxaa 2aa aaa 2aa 2ar rk krk k= + += + -appears in Eq. (32), where we used the symmetry Syx=− Sxyand κyx=− κxy to move to the second line. Now, multiplyingEq. (31) with A= lylz and integrating Eq. (32) along the yaxis, we obtain,( )( )( )( ) ( )/TS TK l lJKQ1, 37xyx yy xaa aD = - +( )( )( )( )( ) /V R JSK l lQ , 38yyxx yxaaaD = - +where ( ) ( ) /K A lxx xa ak= , ( )( ) ( ) /R l l lyy y x za ar= .The linear-response relation described above can beexpressed as,( )TVJQ, 39yxDD= ⎜ ⎟⎛⎝⎞⎠⎛⎝⎞⎠where the matrix  is defined by,( )( )( )( )( ) ( )( )( )( )//TSK l l KRSK l l1. 40yxx yyxx yaa aaaa=-̂⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟Note that in moving from Eqs. (37) and (38) to Eqs. (39)and (40), we used the symmetry Syx=− Sxy and κyx=− κxy.The determinant of  is calculated as det =̂( )( )( ) ( ) ( )/R K z T1 xya a a+ , where the quantity,( )( )( )( ) ( )z TTS, 41xyyxxx yyaa 2a ak r=is the adiabatic figure of merit for the transverse device. Notethat det  is positive for positive values of ρ(a). This meansthat, under the condition ρ(a)> 0, the linear-response relation[Eq. (39)] is physically stable, so that there is no constraint on( )z Txya except that it should have a positive sign. Therefore,( )z Txya is unbounded, namely,( )( )z T0 . 42xya < ¥The remaining steps to calculate the reduced efficiency[Eq. (12)] of the transverse device in the adiabatic limit arebasically the same as in the previous section. The thermal powerinput from the hot reservoir is given by Qxh = and thecurrent flowing through the circuit is given by Jy= Vemf/Rtot,where Rtot= RL+R(a) with RL being the load resistance.Using ( ( ) )( ) ( )/ /V S l l Kxy y xemfa ah= -  , the reduced efficiencyis calculated as {( ) ( )}( )/u u z T u1 1xy2 a 1h = + + + - , whereu=RL/Rtot. Then, the maximum reduced efficiency h isobtained for the stationary value ( )u z T1 xya= +* , whichyields,¯ ( )( )( )z Tz T1 11 1. 43xyxyaah =+ -+ +3.2.2. Isothermal condition in transverse direction. Next,we discuss the isothermal boundary condition in thedirection of the electric field. As in the previous subsec-tion, we are concerned with the case where the transportcoefficient matrices in Eqs. (25)–(27) have antisymmetricoff-diagonal components, i.e. Syx =− Sxy, κyx =− κxy andρyx =− ρxy, whose symmetry applies to the ONE/ANEdevices. Assuming the vanishing temperature gradientalong the y axis in Eqs. (28), (29) and (30), we obtain,( )q T TS j , 44x xx x xy yk= -  +( )E j S T. 45y yy y yx xr= + Note that, in contrast to the adiabatic case, there is nocontribution from THE. Now, multiplying Eq. (44) withA= lylz and integrating Eq. (45) along the y axis, we obtain,090101-11© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEW( ) ( )/Q TS l l J K T , 46x xy y x y= + D( ) ( )/V RJ S l l T , 47y yx y xD = - + Dwhere K= κxxA/lx and R= ρyyly/(lxlz). The above linear-response relation can be represented as,( )QVJT, 48x yD=D⎛⎝⎞⎠⎛⎝⎞⎠where the matrix  is defined by,( )( )( )//TS l l KR S l l. 49yx y xyx y x=--̂ ⎜ ⎟⎛⎝⎞⎠Note that in moving from Eqs. (46) and (47) to Eqs. (48) and(49), we used the symmetry Syx=− Sxy. The determinant of is calculated to be,( )( )KR z Tdet 1 ,xyi= -̂where( )( )z TTS, 50xyyxyy xxi2r k=is the isothermal figure of merit for the transverse device.Note that, for Eq. (48) to be physically stable, det shouldbe positive. Then, we have a condition ( )z T1 0xyi- > , whichmeans that ( )z Txyi is bounded, namely,( )( )z T0 1. 51xyi <Finally, we calculate the reduced efficiency h [Eq. (12)].Using Vemf= Sxy(ly/lx)ΔT and Rtot= R+ RL in Eq. (13), hcan be calculated as {( ) ( )}( )/u u z T u1 1xy2 i 1h = + - + - ,where u= RL/R. The maximum efficiency is obtained forthe stationary value ( )u z T1 xyi= -* , which yields,¯ ( )( )( )z Tz T1 11 1. 52xyxyiih =- -+ -3.2.3. Figure of merit defined by Delves. In this part,we comment on the figure of merit defined by Delves.132)Here, Delves referred to those as “A authors” who start fromour Eqs. (44) and (45), and denoted the corresponding figureof merit as z TE* with an asterisk. When we compare ourEqs. (44) and (45) with Eqs. (4) and (5) of Ref. 132, we findthat Delves’s 11k* corresponds to our κxx in Eq. (44). Note that11k* coincides with the usual definition of the thermalconductivity under zero charge current.Delves132) also referred to those as “B authors” who startby inverting Eqs. (28) and (30) in the following form:q TTSE ,x xx xxyykr= -   -j EST1,yyyyyxyyxr r= - where( )TS, 53xx xxyxyy2k kr = -is the thermal conductivity under zero electric field, which isdenoted by Delves as κ11 without an asterisk.132) Note thatTHE is discarded here, and the symmetry Syx=− Sxy is used.Multiplying these two equations with their cross-sections lylzand lxlz, we obtain,( )( )/QTS l lRV K T , 54xxy y x= - D + D( )( )/JRVS l lRT1, 55yyx y x= - D + Dwhere /K l l lxx y z xk =  and R= ρyylxlz/ly. Then, usingVemf=− Sxy(ly/lx)ΔT and Jy= Vemf/Rtot, where Rtot= RL+R(a), the reduced efficiency [Eq. (12)] is calculated to be{( ) }/u u z T u1 1xy2 1h = + ¢ + + - , where u= R/RL, and thecorresponding figure of merit is given by,( )z TS, 56xyyxyy xx2r k =and from Eqs. (50) and (53), this quantity satisfies,( )( )( )z Tz Tz T1. 57xyxyxyii =-Note that z Txy is denoted by Delves as zET.132) Themaximum efficiency is obtained for s z T1 xy= + * , whichyields,¯ ( )z Tz T1 11 1. 58xyxyh =+  -+  +From the above argument, we find the connection betweenz TE* and zET defined by Delves,132) and ( )z Txyi [Eq. (50)] andz Txy [Eq. (56)] defined in this study. In short, the relations aregiven by,( )( )z T z T , 59E xyi=*( )z T z T. 60E xy= To check this correspondence, we substitute the aboverelations into Eq. (57), then we find,( )z Tz Tz Tz Tz Tz T1,1, 61EEEEEE=-=+***which is exactly the same as Eq. (4) [the second one among twoEq. (4)’s] of Ref. 132. Therefore, z TE* defined by Delves132)coincides with the isothermal figure of merit. Moreover, in thelimit of neglecting the Seebeck effect and THE, zET coincideswith the adiabatic figure of merit, ( )zxya [Eq. (41)]. In this sense,two formulations of the figure of merit in Refs. 131 and 132 arethe same and equivalent. Therefore, there is no conflict betweenthem, and the corresponding description about their conflict17,26)should be corrected accordingly.090101-12© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEW3.2.4. Summary of transverse device. To summarizethis subsection, we have formulated the maximum efficiencyof a transverse thermoelectric generator for two thermalboundary conditions in the transverse direction: the adiabaticboundary condition [Eq. (43)], and isothermal boundarycondition [Eq. (52)]. We note here that, for the Seebeck-effect-driven anomalous/ordinary Hall effects and SSE/SdSE-driven ISHEs, the efficiency expression is different from whatwe have discussed in the previous subsections, because theseeffects involve composite/hybrid materials.74,91) For example,the efficiency of the SSE device depends on the thickness ratioof the two layers of a magnetic insulator and the attached metal.Let us discuss the result obtained in Sect. 3.2. In the absenceof the Seebeck effect and THE (Syy= 0 and Q= 0) whichassumed as customary in the literature,91,125,129,131,132,134) fromEqs. (36), (41) and (50), we obtain a simple relation similar toEq. (22):( )( )( )( )z Tz Tz T1. 62xyxyxyiaa=+Substituting Eq. (62) into Eq. (43), we obtain Eq. (52). Thatis, these two expressions for the efficiency coincide with eachother. If we invert Eq. (62), we have,( )( )( )( )z Tz Tz T1, 63xyxyxyaii=-from which we see that achieving the ( )z T 1xyi = limit isequivalent to achieving the ( )z Txya = ¥ limit. The equivalenceof the energy conversion efficiency in the adiabatic andisothermal conditions can be visually confirmed in Figs. 5(a)and 5(b), where zT for the Seebeck effect is set to be zerobecause of Syy= 0.Next, we consider the case in the presence of the Seebeckeffect while in the absence of THE, i.e. Syy ≠ 0 andQ= 0. Inthis case, from Eqs. (36), (41) and (50), we have thefollowing relation:( )( )( )( )z Tz Tz T zT1, 64xyxyxyaii=- +where, in comparison to Eq. (63), the denominator has anadditional contribution coming from the Seebeck effect,/zT S Tyy yy yy2 k r= . This equation is solved for ( )z Txyi , giving,( )( )( )( )( )z Tz T zTz T11. 65xyxyxyiaa=++Substituting the above equation into Eq. (51), we obtain aconstraint more severe than Eq. (42):( )( )z TzT01, 66xya <which means that the smaller the Seebeck effect, the largerthe upper limit of the adiabatic figure of merit for thetransverse device. This behavior can be visually found inFigs. 5(c), 5(e) and 5(g). In contrast, as shown in Figs. 5(d),5(f) and 5(h), the relation between ( )z Txyi and h is independentof zT because the Seebeck effect does not occur in the ydirection under the isothermal condition.Finally, in the generic situation of the simultaneouspresence of the Seebeck effect and THE, i.e. Syy ≠ 0 and Q≠ 0, a similar analysis is much more involved and the resultsare not summarized in the form presented above.3.3. Performance of transverse thermoelectric refrig-eratorIn this subsection, we consider the performance of atransverse thermoelectric refrigerator/heat pump. Here, asmentioned in the Introduction, we focus only on the coolingmode and discuss the maximum attainable temperaturedifference of the device as a measure of its performance.Note that the thermoelectric refrigerator discussed here isdifferent from compressor-based refrigerators and magneto-caloric refrigerators.139) In the previous two sections, wediscarded the Joule heating effect since we used an approx-imation that is valid up to linear order in the temperature bias.By contrast, in order to investigate the maximum temperaturedifference, we need to take the Joule heating into account,and determine the temperature distribution in the refrigerator.In discussing the maximum attainable temperature differ-ence of a refrigerator, we need to evaluate the rate of the heatremoval c from the cold reservoir. Then, when we useFig. 4(b) for the present problem, first we need to replace theload resistance with a current source for applying a chargecurrent density j to the device. Furthermore, it is convenientto interchange the role of the hot and cold reservoirs asTh↔ Tc125) merely due to the fact that this choice simplifiesthe calculation by allowing us to calculate c at the originx = 0. Therefore, in this subsection, we assume that the left-Fig. 5. Relations between ( )zxyi , ( )zxya and h for various values of zT. ( )zxyi ( ( )zxya )and zT denote the transverse figure of merit in the isothermal (adiabatic)condition and the longitudinal figure of merit for the Seebeck effect,respectively.090101-13© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWhand side (right-hand side) of the device in Fig. 4(b) is incontact with Tc (Th).We first need to obtain the differential equation thatdetermines temperature distribution in the device. The rateof heat evolution is given by −∇ · qE, where qE= q+ fj isthe energy current density with f being the electrochemicalpotential. Then, assuming the steady-state condition, thetemperature distribution in the device is determined by,140)· · · ( )q q E j 0. 67E- = - + =Below, since we assume the boundary condition jx= 0, theJoule heating term is given by E · j= Eyjy in Eq. (67).3.3.1. Isothermal condition in transverse direction. Wefirst discuss the isothermal boundary condition in the direc-tion of the electric field [along the y axis in Fig. 4(b)],because the result for this case is known in theliterature.141,142) Applying the isothermal condition ∇yT= 0to Eqs. (28)–(30), we have,( )q T TS j , 68x xx x xy yk= -  +( )q T TS j , 69y yx x yy yk= -  +( )E j S T. 70y yy y yx xr= + Then, substituting these Eqs. into Eq. (67), we obtain,( )T TS j S j T j2 0, 71xx x yx x y yx y x yy y2 2k r +  +  + =where we used the symmetry Sxy=− Syx, as well as theequations ∇y∇xT=∇x∇yT= 0 and ∇ · j=∇yjy= 0 underjx= 0.In order to solve Eq. (71), we follow an approximatemethod introduced in Refs. 125, 141. We first replace jy withits spatial average jy sá ñ , substitute it into Eq. (71), and obtain,( )T a T b 0, 72x x2 +  + =where /a S j2 yx y xxs k= á ñ and /b jyy y xxs2r k= á ñ . We next seeka solution to the above equation in the form of a power seriesexpansion as,( ) ( )T x T c xcx2, 73c 12 2= + +where c1 and c2 are coefficients to be determined. Then,substituting Eq. (73) into Eq. (72) and setting x = 0, weobtain ac1+ c2+ b= 0. In addition, using the boundarycondition T(x= lx)= Th, we have c l l Txcx1 222+ = D , whereΔT= Th− Tc as before. From these two equations, bydiscarding terms higher in a2, ab, the two coefficients c1 andc2 are approximately obtained as c1=ΔT/lx+ (a/2)ΔT+(b/2)lx and c2=− aΔT/lx− b, and the temperature distribu-tion is given by,( ) ( ) ( )T x TxlT ba Tlx x l12. 74x xxc= + D - +D-⎜ ⎟⎡⎣⎢⎛⎝⎞⎠⎤⎦⎥⎡⎣⎢⎤⎦⎥Given the temperature distribution above, we are now in aposition to calculate the rate of the heat removal from thecold reservoir ( )Aq x 0xc = = , where A= lylz. SubstitutingEq. (74) into Eq. (68) and multiplying A, we obtain c =( ) ( )/ /K T R J T S l l J2 y yx y x y2h- D - - , where J l l jy x z y s= á ñ ,and K and R are defined below Eq. (47). We completethe square with respect to Jy in the above equation as( )( ) ( )( )/ / /K T R J J T S R l l2 2y y yx y xc2h2 2 2= - D - - + * , where( )/ /J T S l l Ry yx y xh= -* . Then, we see that the maximumvalue ( ( ) ) ( )/ /T S l l R K T2yx y xc h2 2 2= - D is obtained forJ Jy y= *. Finally, the maximum temperature differenceTmaxD is obtained for zero heat load c at the cold reservoir,yielding,( )( )T z T12, 75xymaxih2D =where ( )zxyi is defined by Eq. (50). Note that, as emphasized inRef. 125, Eq. (75) is the same as the result obtained by theexact treatment.142) Note also that the isothermal figure ofmerit ( )zxyi is bounded as expressed by Eq. (51).3.3.2. Adiabatic condition in transverse direction. Wenext consider the adiabatic condition in the direction of theelectric field [along the y axis in Fig. 4(b)]. Taking intoaccount the adiabatic condition qy= 0, we start fromEqs. (31) and (32), and rewrite them as follows:( )( ) ( )q T TS j , 76x xx x xy ya ak= -  +( )( ) ( )E j S T , 77y yy y yx xb ar= + where a new quantity ( )yybr is defined by,( )( )( )( )TS, 78yy yyyyyyba 2ar rk= +and ( )xxak , ( )yyak , ( )Syya , ( )Sxya and ( )Syxa are defined below Eq. (32).Note that ( )yybr in Eq. (77) is different from the adiabaticresistivity defined by Eq. (36), and its appearance in thepresent situation is already commented on in Table 7.7.1 ofRef. 125. Note also that to obtain the above equation, we usedthe symmetry ( ) ( )S Sxy yxa a= - . Then, after substituting Eqs. (76)and (77) into Eq. (67), we obtain the same equation as Eq. (72)but with the coefficients a and b being replaced by,( )( )aS j2,yx yxxasak=á ñ( )( )bj.yy yxxbs2ark=á ñSince the temperature distribution is obtained in the same way,we now proceed in exactly the same manner as in the precedingsubsection, and obtain the rate of the heat removal fromthe cold reservoir as ( )( ) ( )/K T R J2 yca b 2= - D - -( )( ) /T S l l Jyx y x yha , where ( )( ) ( ) /K l l lxx y z xa ak= , ( )( ) ( ) /R l l lyy y x zb br=and J l l jy x z y s= á ñ . This is transformed as ( )Kca= -( )( ) ( )( )( ) ( ) ( )/ / /T R J J T S R l l2 2y y yx y xb 2h2 a 2 b 2D - - +* , where( )( ) ( )/ /J T S l l Ry yx y xha b= -* . From this result, it can be seen thatc is maximized for J Jy y= *, yielding the following maximumtemperature difference:( )( )T z T12, 79xymaxbh2D =090101-14© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWwhere( )( )( )( ) ( )z TTS, 80xyyxxx yyba 2a bk r=is the corresponding figure of merit for the transverserefrigerator in the adiabatic operation. Note the differencebetween ( )zxyb [Eq. (80)] and ( )zxya [Eq. (41)], where ( )zxyb isbounded. Indeed, in the absence of THE, we have,( )( )zzT011, 81xyb <+due to Eq. (51).Finally, we note the practical aspect of the transversethermoelectric cooler. In this subsection, we have assumed arectangular-shaped device [Fig. 4(b)] to formulate the max-imum temperature difference and figure of merit for thetransverse thermoelectric refrigerator. However, the actualcooling performance depends on the geometric shape of therefrigerator due to the competition between the transversethermoelectric conversion and Joule heating, as well as heatrelease conditions at the hot side. In a similar manner to theconventional Peltier refrigerator that achieves a large tem-perature difference with multistage cascading,143) the tem-perature difference in the transverse thermoelectric refrig-erator can be increased by constructing a so-called infinite-stage device.144,145) Also in this device, the absence ofinterfacial thermal resistance at stacking boundaries is asignificant advantage of the transverse thermoelectric con-version.3.4. Performance of transverse active coolerBefore ending this section, we would like to discuss theactive cooling.135) As discussed above, the active coolingsystem is designed to dissipate heat from the hot part of anobject, so that the heat drained from the hot part is maximizedwhile ΔT is minimized. The key is that, since the heat isdrained from the hot part, the Fourier heat conduction andPeltier heat act in a constructive way. Then, the quantity thatcharacterizes the performance is the effective thermal con-ductivity κeff that drains the heat from the hot part. While thisidea is developed for the longitudinal device, below we applythe same argument to the transverse device with time-reversalsymmetry breaking.In Sect. 3.3, when discussing the transverse refrigerator,we interchanged the role of hot and cold reservoirs asTh↔ Tc. Because the following calculation applies to theactive cooling device, here we need to undo this interchangeand assume that the left-hand side (right-hand side) of thedevice is in contact with Th (Tc) as in Fig. 4(b).First, we consider the case under the isothermal conditionin the transverse direction. Starting from Eq. (71) andperforming the same calculation as in Sect. 3.3.1, we obtainthe temperature distribution:( ) ( ) ( )T x TxlT ba Tlx x l12, 82x xxh= - D - -D-⎜ ⎟⎡⎣⎢⎛⎝⎞⎠⎤⎦⎥⎡⎣⎢⎤⎦⎥where /a S j2 yx y xxs k= á ñ and /b jyy y xxs2r k= á ñ . Note that thisexpression is also obtained by interchanging Th and Tc inEq. (74). Then, substituting this temperature distribution intoEq. (68), we obtain the rate of heat removal from the hotreservoir ( )l l q x 0y z xh = = as ( )/K T R J2 yh2= D - -( )/T S l l Jyx y x yc , where J l l jy x z y s= á ñ , and K and R are definedbelow Eq. (47). Completing the square with respect toJy in the above equation, we obtain K Th = D -( )( ) ( )( )/ / /R J J T S R l l2 2y y yx y x2c2 2 2- +* , where Jy =*( )/ /T S l l R2yx y xc2- . Then, by setting J Jy y= *, the maximumheat removal from the hot reservoir h* is obtained as( )( )/ /K T T S R l l2yx y xh c2 2 2= D +* . Now we introduce theheat current density ( )/q l ly zh h= * * , and rewrite the aboveequation as (( )) ( ) ( )/ /q T l T S l2xx x yx yy xh c2 2k r= D +* . Then,defining /T T lx x = D~, which is positive by definition, theabove equation can be represented as,( )q T , 83xx xheffk= ~*where the effective thermal conductivity is defined by,( )( )TTPF2, 84xx xxeffic2k k= +Dand the isothermal power factor for the transverse device isgiven by,( )( ) SPF . 85yxyyi2r=Next, we consider the case under the adiabatic condition inthe transverse direction. In this case, repeating the samediscussion as in Sect. 3.3.2, we only need to make thefollowing substitutions to the isothermal result:( ),yy yybr r( ),xx xxak k( )S S ,yx yxawhere ( )yybr is defined by Eq. (78) and other quantities aredefined below Eq. (32). Note that ( )yybr explicitly depends ontemperature. Since this is the temperature felt by the drivingcharge current parallel to the y axis, we approximateT= (Th+ Tc)/2. Then, repeating the same calculation asabove, we obtain Eq. (83), in which the effective thermalconductivity is given by,( )( )TTPF2, 86xx xxeffac2k k= +Dand the adiabatic power factor for the transverse device isgiven by,( )( )( )( )SPF . 87yxyyaa 2br=4. Formulation of transverse thermoelectric conver-sion with structural symmetry breakingSo far, we have considered situations where the transportcoefficient matrices in Eqs. (25)–(27) have antisymmetric off-diagonal components. For the ONE/ANE devices, thiscondition is satisfied. However, the off-diagonal Seebeckeffect is of geometric origin. Therefore, the off-diagonalcomponents of the transport coefficient matrices are sym-metric, satisfying Syx= Sxy, κyx= κxy and ρyx= ρxy. In this090101-15© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWsection, we discuss the efficiency of these thermoelectricgenerators in the first two subsections, and then formulate theperformance of these thermoelectric refrigerators and activecoolers in the last two subsections. This treatment goes intogreater depth than prior review articles, which consider amore introductory perspective to transverse thermoelectricsbased on structural symmetry breaking.146)4.1. Transverse thermoelectric generator in adiabaticconditionUnder the adiabatic boundary condition for the presentsystem, instead of Eq. (36), we have,( )( )( )( )( )( )TS TS, 88yy yyyyyyyxxxaa 2aa 2ar rk k= + +where we used the symmetry Syx= Sxy and κyx= κxy. Then,instead of Eqs. (39) and (40), we have,( )( )( )( )( ) ( )( )( )( ) //S TK l l KRSK l l1, 89yxx yyxx yaa aaaa¢ =--+⎛⎝⎜⎜⎜⎜⎞⎠⎟⎟⎟⎟where the determinant is also modified as det ¢ =̂( )( )( ) ( ) ( )/R K z T1 xya a a- . Note that the adiabatic figure of merit( )zxya in the above equation is defined by Eq. (41), but theadiabatic resistivity therein is redefined by Eq. (88).Consistent with this change, instead of Eq. (42), the adiabaticfigure of merit for the present system is bounded, i.e.( )( )z T0 1. 90xya <Then, instead of Eq. (43), the maximum efficiency of thepresent system is given by,¯ ( )( )( )z Tz T1 11 1, 91xyxyaah =- -+ -under the adiabatic boundary condition.4.2. Transverse thermoelectric generator under iso-thermal conditionUnder the isothermal boundary condition for the presentsystem, instead of Eq. (49), we have,( )( )( ) //S T l l KR S l l, 92yx y xyx y x¢ =- ⎜ ⎟⎛⎝⎞⎠where we used Syx= Sxy. Then, the determinant also changesas ( )( ) KR z Tdet 1 xyi¢ = + , which means that, instead ofEq. (51), the isothermal figure of merit for the present systemis unbounded, i.e.( )( )z T0 . 93xyi < ¥In line with this change, the maximum efficiency of thepresent system is given by,¯ ( )( )( )z Tz T1 11 1, 94xyxyiih =+ -+ +under the isothermal boundary condition.4.3. Transverse thermoelectric refrigeratorHere, we briefly comment on the performance of an off-diagonal Peltier refrigerator. Let us first discuss the iso-thermal boundary condition in the direction of charge current[along the y axis in Fig. 4(b)]. In deriving Eq. (72) for thetransverse thermoelectric refrigerators, we calculate the ex-pression of the coefficient a by using the symmetrySxy=− Syx as,aS j S jS j2,xy yxxyx yxxyx yxxk kk=- +=then we perform replacement j jy y s á ñ . By contrast, in thepresent case, we have the symmetry Sxy= Syx so that thecoefficient a vanishes, i.e. a = 0 for the off-diagonal Peltierdevice under the isothermal condition, while the othercoefficient becomes /b jyy y xx2r k= . Then, performing thesame calculation as in Sect. 3.3, the maximum attainabletemperature difference is obtained as,( )( )T z T12, 95xymaxic2D =where the appearance of the cold-side temperature Tc issimilar to the result for longitudinal Peltier devices.124) Notethe difference from Eq. (75). Note also that ( )zxyi for the off-diagonal Peltier device [Eq. (95)] is unbounded, as shown inEq. (93).We next discuss the off-diagonal Peltier refrigerator underthe adiabatic condition. Repeating the same calculation as inSect. 3.3, we find a = 0 and ( ) ( )/b jyy y xxb 2 ar k= , where ( )yybr isdefined by Eq. (78). Then, the maximum temperaturedifference is calculated to be,( )( )T z T12, 96xymaxbc2D =where ( )zxyb is defined by Eq. (80).4.4. Transverse active coolerFinally, we discuss the performance of the active coolerbased on the off-diagonal Peltier effect. Let us first considerthe case of isothermal condition in the transverse direction. Inthis case, as in the previous section, we need to determine thetemperature distribution by solving Eq. (72). Then, we obtainthe temperature distribution expressed by Eq. (82), wherea = 0 and /b jyy y xx2r k= should be noted. Then, substitutingthis temperature distribution into Eq. (68), we obtain the rateof heat removal from the hot reservoir ( )l l q x 0y z xh = = as( ) ( )/ /K T R J T S l l J2 y yx y x yh2h= D - + , where Jy= lxlzjy.Completing the square with respect to Jy in the aboveequation, we obtain ( )( )/K T R J J2 y yh2= D - - + *( )( )/ /T S R l l2yx y xh2 2 2, where ( )/ /J T S l l R2y yx y xh= -* . Then,by setting J Jy y= *, the maximum heat removal from thehot reservoir h* is obtained as K Th = D +*( )( )/ /T S R l l2yx y xh2 2 2. Now we define ( )/q l ly zh h= * * , andrewrite the above equation as ( )/q T lxx xh k= D +*( )/T S l2yx yy xh2 2 r . Then, with the definition of /T T lx x = D~,the above equation can be transformed to Eq. (83), in whichthe effective thermal conductivity for the present case isgiven by,090101-16© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEW( )( )TTPF2, 97xx xxeffih2k k= +Dand the isothermal power factor for the transverse device isgiven by,( )( ) SPF . 98yxyyi2r=We also discuss the case under the adiabatic condition inthe transverse direction. As in Sect. 3.4, we only need tomake the following substitutions to the isothermal case:( ),yy yybr r( ),xx xxak k( )S S ,yx yxawhere ( )yybr is defined by Eq. (78) and other quantities aredefined below Eq. (32). Note that ( )yybr explicitly depends ontemperature, which is approximated T= (Th+ Tc)/2 as be-fore. Then, repeating the same calculation as in the iso-thermal case, we obtain Eq. (83), where the effective thermalconductivity is given by,( )( )TTPF2, 99xx xxeffah2k k= +Dand the adiabatic power factor for the transverse device isgiven by,( )( )( )( )SPF . 100yxyyaa 2br=5. Measurement of transverse thermopowerThe transverse thermoelectric effects can be investigated bymeasuring an electric field or voltage in a sample in thetransverse direction (y direction) while applying a uniformtemperature gradient in the longitudinal direction (x direc-tion). In the measurement of the transverse thermoelectriceffects due to structural symmetry breaking, which isindependent of a magnetic field H and/or magnetization M,the positions of electrodes for measuring the voltage must becarefully set up because asymmetry in the electrode positionscause the longitudinal thermopower to be superimposed onthe transverse thermopower. It is easy to separate thetransverse thermoelectric effects due to time-reversal sym-metry breaking, which show an odd dependence on H and/orM in the z direction, from the longitudinal thermopower.Here, the pure transverse thermopower can be extracted bymeasuring the H dependence of the transverse voltage V andcalculating (V(+H)− V(−H))/2 with V(+H) and V(−H)respectively being V obtained when positive and negativemagnetic fields are applied. In the case of the longitudinalthermoelectric effects, where the temperature gradient andelectric field are parallel, the voltage per temperature differ-ence and the electric field per temperature gradient are thesame value. In contrast, in the case of the transversethermoelectric effects, these are different values dependingon the aspect ratio of the sample. Therefore, to discuss theperformance of the transverse thermoelectric effects, theelectric field per temperature gradient, i.e. the transversethermopower, must be compared.The following focuses on the measurement methods for theheat-to-charge current conversion phenomena (mainly ANE).On the other hand, methods for clarifying the charge-to-heatcurrent conversion phenomena have also been established.147)To measure the phenomena that output heat currents, includingthe anisotropic magneto-Peltier, anomalous Ettingshausen, andspin Peltier effects, various temperature detection techniques,such as the direct contact with thermocouple wires,148)integration of on-chip thermocouples,149) lock-in thermographymethod150–154) and lock-in thermoreflectance method,155) areemployed. In all of these methods, lock-in detection to separatethermoelectric responses from Joule heating and magnetic fielddependence measurements to distinguish different magneto-thermoelectric and thermo-spin effects are indispensable.5.1. Measurement configuration for Nernst effect inthin filmsIn the fields of spin caloritronics and topological materialsscience, the ANE has been widely measured not only in bulkmaterials but also in thin-film devices. In contrast to simplemeasurements for bulk materials, there are two experimentalconfigurations for thin films because of their huge differencebetween the in-plane and out-of-plane dimensions, and theircharacteristics need to be understood.37,57) One configurationis the perpendicularly magnetized (PM) configuration, whereH (∇T) is applied along the out-of-plane (in-plane) direction[Fig. 6(a)]. The PM configuration is widely used for thequantitative estimation of SANE since the magnitude of ∇T canbe precisely measured by various experimental techniquesincluding thermometers attached to hot and cold sides of asubstrate or heat baths,57) on-chip thermometers grown on thesubstrate,42) and an infrared camera.44,45,47,48,58) Thus, byestimating the ANE-induced electric field through the extrac-tion of the zero-field intercept of the H dependence of thetransverse electric field Ey and by normalizing it by ∇T, onecan determine SANE. However, the PM configuration oftenrequires the application of large H to align the magnetizationof films along the out-of-plane direction to overcome thestrong demagnetization field. The other configuration is the in-plane magnetized (IM) configuration, where H (∇T) is appliedalong the in-plane (out-of-plane) direction [Fig. 6(b)]. The IMconfiguration is suitable for thermal energy harvesting and heatflux sensing, discussed later, as ANE-based thermoelectricgeneration works simply by forming films onto heat sources.However, the IM configuration is not suitable for estimatingSANE quantitatively because the temperature difference be-tween the top and bottom of thin films is hard to quantify (notethat it is possible to quantify SANE even in the IM configurationby measuring the out-of-plane thermal conductivity of thinfilms using the time-domain thermoreflectance method156)).Comparing the transverse thermopower in the PM and IMconfigurations is also used to separate the contribution of theANE from that of the SSE-driven ISHE, which appears only inthe IM configuration due to the symmetry of the ISHE.157,158)Note that the situations discussed in this subsection areapplicable to the ONE in thin films.Figures 6(c) and 6(d) show an example of the experimentalresults of the ANE measured in the PM and IM090101-17© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWconfigurations, respectively, for a 50 nm thick Co2MnGa thinfilm, epitaxially grown on a single-crystalline MgO (001)substrate.37) In both the configurations, the observed trans-verse thermoelectric signals showed an odd dependence on Hand its magnitude saturated in the high-H region, where themagnetization of the Co2MnGa film saturated. This is atypical behavior of the ANE. In the IM configuration, a largeANE-induced thermopower was observed to appear even atzero field because the strong demagnetization field in the out-of-plane direction stabilized the remanent magnetization inthe in-plane direction [Fig. 6(d)], a situation preferable forpractical applications. In contrast, a tiny ANE-inducedthermopower was observed at zero field in the PM config-uration [Fig. 6(c)]. To generate the ANE-induced thermo-power in the absence of an external magnetic field in the PMconfiguration, it is necessary to use a magnetic thin filmhaving large perpendicular magnetic anisotropy.18,55)5.2. Effect of thermal boundary conditionsAs is clear from the formulation in Sects. 3 and 4, it isimportant to take the thermal boundary conditions in theelectric field direction into account in the measurements ofthe transverse thermoelectric effects. This is because underadiabatic conditions, the transverse thermopower is modu-lated by the concerted action of the Seebeck effect and THE(off-diagonal thermal conductivity) for the transverse thermo-electric effects due to time-reversal symmetry breaking(structural symmetry breaking). It has been experimentallyshown that this correction term can improve the performanceof the transverse thermoelectric conversion when the sign ofSxy and the correction term are the same.26,159)In the following, we show the results of ANE measure-ments for SmCo5-type bulk permanent magnets to confirmthat the effect of the thermal boundary conditions is too largeto be ignored even in ferromagnetic metals, of which THE istypically small. In SmCo5, Sxy and the correction term shouldhave the same sign.63) It is difficult to achieve an idealadiabatic or isothermal condition in an experiment, but whichcondition is closer depends on the aspect ratio of thematerial.26,29) When the length of the sample along thetemperature gradient is much shorter than the width alongthe electric field, the thermal boundary condition along theelectric-field direction can be regarded as almost isothermal[Fig. 7(a)]. When the length of the sample along thetemperature gradient is long, it is closer to adiabatic[Fig. 7(b)]. In most experiments, the sample dimension alongthe transverse electric-field direction (y direction) is smallerthan that along the longitudinal temperature gradient ∇xTdirection (x direction), resembling the adiabatic thermalboundary condition shown in Fig. 7(b). In this configuration,a transverse temperature gradient along the y direction ∇yT isfinite due to the contribution from THE. Consequently,according to Eq. (33), the observed transverse thermopower( )/E Ty x-* is the combination of the isothermal transversethermopower ( )∣/E Ty x T 0y-  = and the longitudinal thermo-power Syy multiplied by the ratio of ∇yT to ∇xT:( ) ( )∣( )ETETSTT. 101yxyx Tyyyx0y-=-+ =*⎜ ⎟⎛⎝⎞⎠Therefore, unless the sample dimension along the transverseelectric field is long enough to realize ∇yT∼ 0, the observedthermopower is either underestimated or overestimated,depending on the sign of the second term in the right-handside of Eq. (101); the choice of sample dimensions plays aFig. 6. (a),(b) Schematics of the PM and IM configurations for the measurements of the ANE. (c) Magnetic field H dependence of the transversethermopower Ey/∇xT for the Co2MnGa film at room temperature, measured in the PM configuration. Ey, ∇xT and μ0 denote the transverse electric field,longitudinal temperature gradient and vacuum permeability, respectively. (d) H dependence of the transverse voltage Vy for the Co2MnGa film, measured in theIM configuration. This result was obtained when a heater power of 160 mW was applied to the top heat bath. Exact composition of this film isCo53.0Mn23.8Ga23.2. Data in (c) and (d) are reconstructed from Fig. 3 of Ref 37)090101-18© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWcrucial role in transverse thermopower measurements. Toaccurately determine Sxy from direct thermopower measure-ments, the sample dimension along the y direction should belarger than that along the x direction.We now experimentally demonstrate the influence ofsample dimensions on the transverse thermopower using abulk polycrystalline SmCo5 slab. The SmCo5 block, com-mercially available from Magfine Corporation, Japan, wascut into two rectangular slabs with dimensions along the x, yand z directions, respectively, being 10.0, 7.7 and 0.8 mmand 10.0, 2.1 and 0.8 mm using a diamond wire saw, whereonly the length along the y direction, d, is significantlydifferent. The slabs have an easy axis of magnetization alongthe z direction. The measurement configuration is illustratedin Fig. 7(c). The two SmCo5 slabs were bridged between twoheat baths separated by 8.0 mm and firmly attached to theheat baths. To apply a temperature difference between theheat baths, chip heaters were attached to the hot side. Thesample holder was fixed at the center of a superconductingmagnet, where a magnetic field ranging from μ0H= 5 T to−5 T was applied along the z direction. The H dependence ofthe transverse voltage Vy along the y direction was recordedby attaching Cu wires at the ends of the slabs at variousvalues of the temperature difference. To accurately measurethe temperature gradient on the SmCo5 slabs, the top surfaceof the slabs was coated with insulating black ink with anemissivity of > 0.94, and steady-state temperature imageswere captured using an infrared camera.Figure 7(d) shows the H dependence of the transverseelectric field Vy/d for the SmCo5 slabs, recorded under alongitudinal temperature gradient of ∣∇xT∣∼ 1× 103 K/m. Inboth the slabs, the transverse electric field changed its signwith large hysteresis with respect to the H reversal. The Hdependence of Vy/d corresponds to the magnetization curveof the SmCo5 permanent magnet, indicating the appearanceof the ANE. To determine the magnitude of the M-dependenttransverse thermopower for the SmCo5 slabs, the M-odd-dependent component of Vy/d under the saturation state,Vodd/d, was extracted for each ∣∇xT∣ value by extrapolatingthe signal in the high-H region to zero field. Here, the finiteH-linear component was observed only for the slab withd= 2.1 mm, suggesting the presence of the Seebeck voltagebetween the SmCo5/Cu-wire contacts induced by the or-dinary (H-dependent) THE in SmCo5 [Fig. 7(d)]. Althoughthe SmCo5 slabs were cut from the same ingot, they exhibiteddifferent Vodd/d values [Fig. 7(e)]. The M-dependent trans-verse thermopower for the SmCo5 slab with d= 2.1 mm(d= 7.7 mm) was found to be 5.1 μV/K (4.1 μV/K); thedifference in the transverse thermopower is much larger thanthe measurement errors. Recall the fact that the slab withd= 2.1 mm (d= 7.7 mm) is closer to the adiabatic (iso-thermal) condition. Therefore, the transverse thermopowerunder the adiabatic condition is expected to be overestimatedcompared to that under the isothermal condition because ofthe same sign of Sxy and the correction term induced by theSeebeck effect and anomalous (M-dependent) THE. Thisexpectation is consistent with the observed difference in theM-dependent transverse thermopower. Here, we note that thecorrection term discussed theoretically in Sect. 3 is deter-mined by the absolute Seebeck coefficient of a magneticmaterial (i.e. SmCo5), while the observed difference in thetransverse thermopower in Fig. 7 depends on the relativeSeebeck coefficient between SmCo5 and Cu. However, sincethe Seebeck coefficient of Cu (<2 μV/K) is much smallerthan that of SmCo5 (−19 μV/K 63)) at room temperature, themagnitude and sign of the correction term in the present setupis determined mainly by the transport properties of SmCo5.The experimental results shown here offer guidelines forFig. 7. (a),(b) Schematics of the measurement setups for the transverse thermoelectric generation in the nearly isothermal and adiabatic conditions. (c)Schematic of the measurement setup for the transverse thermopower in SmCo5-type permanent magnet slabs with different values of the width d. (d) Hdependence of the transverse electric field Vy/d for the SmCo5 slabs with d = 7.7 mm (red and orange) and 2.1 mm (black and gray) at ∣∇xT∣ ∼ 1 × 103 K/m.H-independent offset of the transverse electric field is subtracted from the raw data. Red and black (orange and gray) curves show the H dependence of Vy/dafter (before) subtracting the H-linear component estimated by linear fitting of the signals in the high-H region. (e) ∣∇xT∣ dependence of the M-odd-dependentcomponent of the transverse electric field Vodd/d, which is estimated by extrapolating the Vy/d signal in the high-H region to zero field.090101-19© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWdetermining the appropriate sample aspect ratio and measure-ment setup for transverse thermopower measurements, mini-mizing parasitic signals from the Seebeck effect and THE.We also note that considering typical sample dimensions, theestimation of SANE from the measurement of the anomalousEttingshausen effect and the Onsager reciprocal relationreduces the influence of the correction term.29)A similar effect occurs not only in the Nernst effects butalso in the transverse thermoelectric conversion with struc-tural symmetry breaking. In the latter case, THE does notoccur due to the absence of a magnetic field or magnetization,but the transverse thermopower can be modulated by the off-diagonal thermal conductivity derived from anisotropictransport properties. To accurately determine each transportcoefficient, it is necessary to quantify or suppress thecontribution coming from the correction term. However,since the performance of the transverse thermoelectric con-version can be improved by superimposing the correction-term contribution, its utilization is important in terms ofapplications.26,159)It should also be noted that the underestimation or over-estimation of the transverse thermopower due to the thermalboundary conditions does not occur in thin films. Thetemperature of thin films is strongly thermalized to that ofsubstrates, which have a large heat capacity. Thus, even ifTHE and structure-induced off-diagonal thermal conductivityare finite, the transverse thermopower is always obtained inthe isothermal condition.6. Application of transverse thermoelectric conver-sionAlthough the transverse thermoelectric conversion is not yetin practical use, experimental verification is being activelyconducted to realize next-generation heat flux sensing andenergy harvesting technologies by utilizing the variouscharacteristics introduced in Sect. 2 In this section, weshow examples of experiments aimed towards applicationsof the transverse thermoelectric conversion, followed bypresenting new concepts and future prospects for furtherperformance improvements.6.1. Heat flux sensingA heat flux sensor is a device that enables simultaneousdetection of the magnitude and direction of a heat flow,which can be an essential component for smart thermalmanagement systems. However, practical applications of heatflux sensors have been limited due to several problems. Thecommercially available sensors are based on the Seebeckeffect and consist of a serially connected 3D array of twodifferent thermoelectric materials. Therefore, the heat fluxsensors based on the Seebeck effect require a durablesubstrate or thick plate to provide mechanical stability.Hence, the conventional sensors are mainly applicable toflat surfaces and their mechanical flexibility is limited. Due tothe presence of thick substrates, these sensors have largethermal resistance that changes the heat flow distribution tobe detected. Although the sensitivity of the heat flux sensorsbased on the Seebeck effect is proportional to the number ofthe thermoelectric material junctions and to the sensor size,the complex structure and low mechanical durability make itdifficult to construct highly sensitive sensors. As an alter-native to the sensors having these drawbacks, thin-film-basedheat flux sensors driven by the transverse thermoelectriceffects, e.g. the SSE-driven ISHE160) and ANE,37,48,161,162)were proposed and demonstrated. Due to its simpler structureand improved transverse thermopower, ANE-based heat fluxsensors have recently become mainstream. In the ANE-basedsensor, a lateral thermopile structure consisting of twodifferent alternately arranged and serially connected wireswith different SANE values is used. The simple thermopilestructure and the symmetry of the ANE make the sensorsthin, reducing their thermal resistance. If magnetic materialsshowing large ANE can be formed on thin flexible sheets,flexible heat flux sensors can be constructed.48,161) Thesefeatures of the ANE-based heat flux sensors are advantageousover the conventional heat flux sensors based on the Seebeckeffect.The sensitivity of the heat flux sensor is determined by theoutput thermoelectric voltage over a heat flux density q.Thus, the open-circuit voltage is more important than theoutput power. To increase the open-circuit voltage due toANE VANE, one often uses an ANE-based heat flux sensorconsisting of a thermopile structure in which one end of amagnetic wire is electrically connected to the opposite end ofan adjacent magnetic wire [Fig. 8(a)]. In this zigzag-shapedthermopile structure, when the magnetization (heat flux) isalong the width (thickness) direction of the wires, i.e. in theIM configuration, the ANE voltage is generated along thelength of the wires l and the total output voltage isproportional to the number of the connected magnetic wiresn. This structure is thus more advantageous for increasing theopen-circuit voltage than plain blocks or sheets. The sensi-tivity of the ANE-based heat flux sensor VANE/q is thusproportional to (SANEln)/κ with κ being the thermal con-ductivity of the magnetic material in the out-of-plane direc-tion.Here, we show a proof-of-concept demonstration of aflexible heat flux sensor based on the ANE. We deposited the100 nm thick amorphous Sm20Co80 film on a 50 μm thickflexible polyethylene naphthalate (PEN) sheet, where amor-phous Sm20Co80 exhibits substantial SANE, coercivity andremanent magnetization and can be formed on anysubstrate.161) We formed 50 Sm20Co80 wires with a widthof 50 μm and a length of 10 mm arranged in a parallelconfiguration at an interval of 150 μm. The wires were thenconnected in series using Au wires to make the total devicearea of 10× 10 mm2, where q was estimated through thenormalization by this area. The total length of the Sm20Co80wires for this configuration reaches 500 mm. We measuredthe voltage V between the ends of the thermopile structure byapplying an in-plane external magnetic field along the widthdirection of the wires and an out-of-plane heat current. Thedetails of the experimental procedures are shown in Ref. 161As shown in Fig. 8(b), clear ANE signals with hysteresisbehaviors were observed in the H dependence of V and theirmagnitude increases with increasing q across the device. Thesaturated V values for μ0H> 0.1 T indicate the negligiblysmall ONE contribution, enabling the simple extraction of theANE contribution. Figure 8(c) shows the ANE voltage,represented by Vodd, as a function of q,161) where Vodd at0.2 T was extracted by the field-odd component of V at∣μ0H∣= 0.2 T and Vodd at zero field was extracted from thezero-field V values for the positive-to-negative and negative-090101-20© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWto-positive sweep of H. The sensitivity of the ANE-basedheat flux sensor was estimated to be VANE/q∼Vodd/q= 1.12 ± 0.02× 10−7 VW−1m2 and 0.89 ± 0.02×10−7 VW−1m2 in the saturation magnetization (0.2 T) andremanent magnetization (0.0 T) states, respectively.Importantly, the use of magnetic thin films with finitecoercivity and remanent magnetization makes it possible tosense a heat flux using the ANE even in the absence of anexternal magnetic field. Therefore, the design and control ofmagnetic anisotropy are essential for the application of theANE. Although the sensitivity of the sensor used in thisdemonstration is lower than that of commercial sensors basedon the Seebeck effect, the ANE has strong advantages inflexibility and low thermal resistance, extending applicationsof heat flux sensors. To further improve the performance ofthe ANE-based heat flux sensors, extensive effort has beendevoted to developing new materials with larger SANE andlower κ as well as to optimizing the device structure. Sinceamorphous materials can be fabricated on any surface andhave lower κ than crystalline materials, they are promisingcandidates for the development of low-cost, high-perfor-mance heat flux sensors with large areas.59–62,161)In addition to the ANE, the off-diagonal Seebeck effect inanisotropic single-crystal metals114) and conductive layeredoxides formed on off-cut substrates with tilted crystalplanes163–167) are also expected to be used as a thermalsensor. These artificially tilted conductive oxides are alsobeing developed for other applications including light sensingand THz emission168) via the transverse thermoelectricconversion.6.2. Thermal energy harvestingTransverse thermoelectric devices are useful for harvestingand utilizing widely distributed thermal energy. By installingsimple block- or sheet-shaped transverse thermoelectricdevices on a heat source surface, one can extract a chargecurrent and voltage along the surface direction from atemperature gradient perpendicular to the surface. Kiriharaet al. pointed out that the SSE-driven ISHE is suitable for thethermal energy harvesting from curved or complex-shapedheat source surfaces and demonstrated that SSE devices canbe fabricated using simple coating90) or spray160) processes.Similar functionality can be obtained using the ONE (ANE)in the in-plane magnetic field (magnetized) configuration,enabling the fabrication of flexible transverse thermoelectricdevices based on thin films deposited on flexiblesubstrates161) or amorphous metal ribbons.59–62) However,challenges include the difficulty of extracting a large chargecurrent and electrical power from thin films and the fact thatthe thermopower generated by the ONE, ANE and SSE-driven ISHE has not yet reached sufficient levels for practicalapplications.Experimental demonstrations of thermal energy harvestingusing the off-diagonal Seebeck effect, which exhibits hightransverse thermoelectric conversion performance, have alsobeen reported. Figure 9 compares the maximum powerdensity normalized by the square of the temperature gradient(∝ the power factor) for thermoelectric conversion modulesdriven by various principles.4,9,10,99–101,108,167,169–172) Themaximum power density of the ATML-based modules drivenby the off-diagonal Seebeck effect stands out, being 2–3orders of magnitude larger than that of the modules driven bythe ONE and ANE. Recently developed SmCo5/BST ATMLexhibits excellent transverse thermoelectric conversion per-formance, with the maximum power density of the modulescomposed of 14 ATML elements reaching 56.7 mW/cm2 at atemperature difference of 152 K.108) SmCo5/BST ATML hashigher thermal conductivity than longitudinal thermoelectricmaterials, making it difficult to maintain a large temperaturedifference in steady-state conditions; the efficiency of theATML-based module falls short of that of longitudinalmodules driven by the Seebeck effect. However, the max-imum power density normalized by the square of thetemperature gradient in ATMLs exceeds that of the long-itudinal modules based on the Seebeck effect (Fig. 9).Therefore, ATML-based transverse thermoelectric modulesare suitable for thermal energy harvesting because they cangenerate large power even in environments where only smalltemperature differences can be generated. In addition, intransverse thermoelectric modules with thermopile structures,by alternately changing the height of each element, a built-inFig. 8. (a) Schematic of the ANE-based heat flux sensor. Sensor comprisesthe amorphous Sm20Co80 wires and Au wires arranged alternately andconnected in series and formed on a flexible polyethylene naphthalate (PEN)sheet. q denotes the heat flux density with the magnitude q. (b) H dependenceof V for the Sm20Co80/Au sensor for various values of q. (c) q dependence ofVodd at ∣μ0H∣ = 0.2 and 0.0 T. Data in (b) and (c) are reconstructed from Fig.6 of Ref. 161.090101-21© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWheat sink can be incorporated to promote heat release to theatmosphere.108) This built-in heat sink structure cannot beachieved in longitudinal modules where rigid plates arepresent on the top and bottom surfaces. Even withoutsandwiching the thermoelectric module between hot andcold heat baths, the built-in heat sink structure enablessubstantially large power generation simply by placing iton a heat source due to the improved heat release to theatmosphere. In particular, SmCo5/BST ATML has remanentmagnetization and functions as a permanent magnet; when itcan be attached to a heat source through a magnetic attractiveforce, the contact thermal resistance between the module andheat source decreases even when thermal interface materialsare inserted between them, enabling the effective generationof a temperature difference from the heat source. As shown inRef. 108, when the thermopile module comprisingSmCo5/BST ATML elements is placed on a hot plate, theelectrical power generated solely by heat release to the airreaches a value sufficient to drive small sensors.6.3. Hybrid transverse magneto-thermoelectric con-versionIn this subsection, we introduce one of the promising strategiesfor improving transverse thermoelectric conversion performance:hybrid transverse magneto-thermoelectric conversion.106,107,109)As classified in Fig. 3, there are various driving principles forthe transverse thermoelectric conversion, but the phenomenawith time-reversal symmetry breaking and those with structuralsymmetry breaking have been studied independently indifferent fields. The interdisciplinary fusion of these phe-nomena not only leads to the creation of new research fieldsbut also significantly contributes to improving the transversethermoelectric conversion performance. Here, imagine asituation where multiple transverse thermoelectric conversionprinciples manifest in a single material, i.e. either a homo-geneous or composite/hybrid material. As an example, if thetransverse thermopower due to the off-diagonal Seebeck effectSOD and that due to ONE SONE are simultaneously manifested,the total transverse thermopower is simply the sum of thesetwo contributions: SOD+ SONE. Since ∣SOD∣> ∣SONE∣ in manycases, the contribution of the ONE to the total transversethermopower is small. In this situation, the figure of merit isproportional to the square of the total transverse thermopower,and described as( )( )z T S SS S S S2 . 102zy OD ONE2OD2ONE2OD ONEµ += + +The third term on the right-hand side of Eq. (102) isimportant. Even if SONE is small, the modulation of thefigure of merit due to the ONE increases significantly due tothe synergistic effect with SOD. In this example, by increasingthe magnetic-field/magnetization-independent SOD contribu-tion, the influence of magnetic fields or magnetization on thefigure of merit can be enhanced. If three or more transversethermoelectric conversion phenomena simultaneously occurin a single material, more diverse synergistic effects can beobtained.The effectiveness of the hybrid transverse magneto-ther-moelectric conversion has been experimentally demonstratedin several systems. In Ref. 106, ATML composed ofBi88Sb12, which exhibits large magneto-thermoelectric ef-fects, and BST, which exhibits large Seebeck and Peltiereffects, was used to enhance the transverse thermoelectricconversion performance by applying a magnetic field. Theobserved performance enhancement in this system originatesfrom the hybrid thermoelectric conversion involving threeeffects: the off-diagonal Seebeck (Peltier), magneto-Seebeck(Peltier) and ordinary Nernst (Ettingshausen) effects[Fig. 10(a)]. Figure 10(b) shows the charge current Jdependence of the surface temperature averaged over theBi88Sb12/BST junction in ATML under the steady state,where the contribution of Joule heating results in a parabolicbehavior and that of the thermoelectric effects results in theshift of the parabolic curves. The systematic experimentalresults reported in Ref. 106 indicate that a temperaturegradient is generated in a direction perpendicular to theapplied charge current and the steady-state cooling perfor-mance due to the transverse thermoelectric conversion isdramatically modulated by the reversal of the magnetic field.In the experimental setup depicted in Fig. 10(b), when apositive magnetic field is applied, the direction of the net heatcurrent generated by the off-diagonal Peltier effect is thesame as that by the ordinary Ettingshausen effect, improvingthe transverse thermoelectric conversion performance. Thefigure of merit at room temperature for Bi88Sb12/BST ATMLis 0.20 (0.14) when a positive (negative) magnetic field isapplied, and this field-dependent modulation is much largerthan the figure of merit for the ONE alone in the Bi88Sb12alloy. This is attributed to the contribution of the cross termin Eq. (102). Furthermore, in this study, it has beendemonstrated that by replacing BST in ATML with thepermanent magnet Nd2Fe14B, the magneto-thermoelectriceffects can be superimposed without applying an externalFig. 9. Comparison of the maximum power density per heat transfer areamaxw normalized by the square of the applied temperature gradient ∇T2 forvarious longitudinal and transverse thermoelectric modules. Modules corre-sponding to the green triangle, yellow triangle and blue diamond data pointsare driven by the Seebeck effect,4,9,10,169) ONE/ANE170–172) and off-diagonalSeebeck effect.99–101,108,167) Red stars show the data for the off-diagonalSeebeck effect in SmCo5/BST ATML, which exhibits the current record-high/ Tmax2w  value.108) Inset shows the same data on a logarithmic scale.090101-22© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWmagnetic field due to the stray magnetic field from theremanent magnetization of Nd2Fe14B. In Ref. 107, thecandidate phenomena for the hybrid transverse magneto-thermoelectric conversion have been expanded by inducingthe ANE in ATML comprising the topological ferromagnetCo2MnGa. In Co2MnGa-based ATMLs combined with p-type BST and with n-type Bi2Te3, a magnetization-direction-dependent modulation of the figure of merit significantlylarger than that of the ANE alone was observed due to thehybridization of the off-diagonal Seebeck effect and theANE. In Ref. 109, the hybridization of the off-diagonalSeebeck effect and the ANE has been realized in SmCo5/BSTATML in the absence of an external magnetic field byutilizing the large in-plane remanent magnetization ofSmCo5. Note that in these ATMLs, the transverse thermo-power generated by the ONE (ANE), SONE (SANE), iseffectively reduced because of the shunting effect in adjacentconductors. Nevertheless, the gain from the cross term inEq. (102) is greater than the reduction in SONE (SANE).Examples of the hybrid transverse magneto-thermoelectricconversion are still limited. In principle, the Seebeck-effect-driven ordinary/anomalous Hall effects and SSE/SdSE-drivenISHEs, not used in the above examples, are also integrable intothe hybrid transverse magneto-thermoelectric conversion. In fact,although the thermoelectric output is small, the hybrid transversethermoelectric generation based on the combination of the ANEand SSE-driven ISHE has been demonstrated.91,93,158)Originally, spin-caloritronic phenomena such as the ANE andSSE have been extensively studied from the viewpoint offundamental physics, but their low thermoelectric conversionperformance has limited their impact on applications. Theconcept of the hybrid transverse magneto-thermoelectric con-version provides one of the solutions for practical applications ofthese spin-caloritronic phenomena.6.4. Thermoelectric permanent magnetThe transverse thermoelectric conversion in permanent mag-nets is suitable for power generation and cooling applicationssince they enable the construction of large-area devices andtheir mass production. Their roles discussed so far aresummarized as follows:• Manifestation of the giant anomalous Nernst andEttingshausen effects,63,64)• Zero-magnetic-field operation of the anomalous Nernstand Ettingshausen effects,63,64,109,171)• Zero-magnetic-field operation of the magnetic-field-in-duced magneto-thermoelectric effects, i.e. the ordinaryNernst and Ettingshausen effects and the magneto-Seebeck and Peltier effects, utilizing stray magneticfields from remanent magnetization,36,106)• Reduction of contact thermal resistance between thermo-electric devices and heat sources through a magneticattractive force.108)Therefore, the creation of “thermoelectric permanent mag-nets” that possess high transverse thermoelectric conversionperformance with permanent magnet features is important.However, the above roles are based on the macroscopicproperties of permanent magnets [Fig. 11(a)]. By reexa-mining permanent magnets from a microscopic perspectiveand engineering not only their magnetic properties but alsotheir thermoelectric conversion characteristics, new prospectsfor the transverse thermoelectric conversion become ap-parent.Permanent magnets are not homogeneous materials butrather composite materials or engineered ensembles of inter-faces with micrometer- to sub-micrometer-scale structures[Fig. 11(b)]. The goal of conventional permanent magnetdevelopment has been to achieve large maximum energyproduct ( )BH max, and the optimization of microstructures hassignificantly improved the coercive force, remanent magneti-zation and their temperature-dependent characteristics.173) Thedesign guidelines for the thermoelectric permanent magnetsare completely different from those for conventional magnets;it is necessary to design and control microstructures to improvethe transverse thermoelectric conversion characteristics. Forexample, by incorporating thermoelectric semiconductors intothe microstructure of permanent magnets, the appearance ofthe Seebeck-effect-driven ordinary/anomalous Hall effects inthe magnets is expected. Introducing spin–orbit interactionnear the interface between ferromagnetic phases and nonmag-netic grain-boundary phases may result in the superposition ofthe transverse thermopower due to SSE/SdSE-driven ISHEs[see the right-hand-side magnified illustration in Fig. 11(b)].Furthermore, by appropriately designing the grain size anddistribution, it may be possible to simultaneously reduce thephonon thermal conductivity46,174) [see the left-hand-sideillustration in Fig. 11(b)]. In other words, by optimallydesigning the microstructures of the thermoelectric permanentmagnets, it is possible to independently engineer the transversethermopower, electrical conductivity and thermal conductivity,thereby significantly improving the figure of merit and energyconversion efficiency. To achieve this, it will be necessary togo beyond mere material exploration, and instead focus on theFig. 10. (a) Schematic of the hybrid transverse magneto-thermoelectric cooling based on the off-diagonal Peltier, magneto-Peltier and ordinaryEttingshausen effects in ATML. (b) Charge current J dependence of ΔTave at μ0H = ± 0.8 T for Bi88Sb12/BST ATML. ΔTave denotes the steady-statetemperature change from room temperature averaged over one Bi88Sb12/BST pair.090101-23© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWdesign of hierarchical structures and the control of phaseboundaries within permanent magnets, with the aid of micro-structure analysis techniques and precise transport propertymeasurements. Microstructure engineering of magnetic mate-rials aimed at realizing high-performance transverse thermo-electric conversion has already begun.46,59,61,62,175)7. Conclusion and prospectsIn this review article, we have introduced various phenomenaand principles that realize the transverse thermoelectricconversion, reclassified them into eight categories based onmodern knowledge (Fig. 3), and summarized their character-istics. We have formulated the figure of merit and efficiencyfor the transverse thermoelectric generator, the maximumattainable temperature difference for the thermoelectricrefrigerator, and the effective thermal conductivity for thetransverse active cooler. Importantly, the thermal boundaryconditions in the electric-field direction play an importantrole in the precise discussion and evaluation of the transversethermoelectric conversion performance. Each phenomenonexhibits different thermoelectric conversion characteristicsand the research stage is also completely different. Atpresent, excellent zxyT values have been obtained throughthe ONE in some semimetals at low temperatures and the off-diagonal Seebeck effect in ATMLs and goniopolar materials.However, there are also issues such as limitations on materialselection and the need to apply a magnetic field for transversethermoelectric conversion operations. While many newphysical phenomena and principles have been discovered inthe field of spin caloritronics, the efficiency of the transversethermoelectric generation using magnetic materials and thespin degree of freedom is currently very low, and it remainsat the basic research stage. Nevertheless, the benefits of thetransverse thermoelectric conversion, which enables theconstruction of thermoelectric conversion devices withoutjunctions, are immeasurably significant, and it is expectedthat basic and applied studies will accelerate further. Varioustransverse thermoelectric phenomena that have been investi-gated independently in different fields have been integrated inrecent years, and methods for further improving the perfor-mance have been proposed and demonstrated. If the trans-verse thermoelectric conversion that has high versatility andefficiency is achieved through material exploration, materialdevelopment, microstructure engineering, optimal design ofhybrid materials, and modularization technologies, one canovercome the technical issues faced by existing thermo-electric conversion modules.AcknowledgmentsThe authors thank Y. Goto, J. P. Heremans, K. Hirata,M. Hirschberger, Y. Lee, Y. Mizuguchi, S. Mori, M. Murata,H. Nagano, Y. Nakanishi, K. Oyanagi, S. J. Park,Y. Sakuraba, Y. Sato, H. Sepehri-Amin, A. Takahagi,T. Yagi and W. Zhou for valuable discussions and manycollaborators and group members for preliminary reviews ofthe manuscript. 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Salvador et al., Phys. Chem. Chem. Phys. 16, 12510 (2014).170) M. Murata, K. Nagase, K. Aoyama, A. Yamamoto, and Y. Sakuraba,iScience 24, 101967 (2021).171) F. Ando, T. Hirai, and K. Uchida, APL Energy 2, 016103 (2024).172) M. Chen, J. Wang, K. Liu, W. Fan, Y. Sun, C. Felser, T. Zhu, and C. Fu,Adv. Energy Mater. 14, 2400411 (2024).173) K. Hono and H. Sepehri-Amin, Scr. Mater. 67, 530 (2012).174) M. Nomura, J. Shiomi, T. Shiga, and R. Anufriev, Jpn. J. Appl. Phys. 57,080101 (2018).175) Z. H. Kautsar, B. Madavali, T. Hirai, K. Uchida, and H. Sepehri-Amin,Energy Mater. 5, 500129 (2025).Hiroto Adachi received the B.Sc., M.Sc. and D.Sc. degrees from Kyoto University, Kyoto,Japan, in 1998, 2000 and 2003, respectively. Hewas a Postdoctoral Research Assistant atOkayama University; at Swiss Federal Instituteof Technology (ETH) Zurich; at Institute forMaterials Research, Tohoku University; and atAdvanced Science Research Center, JapanAtomic Energy Agency. From 2014–2015, hewas an Associate Chief Scientist at AdvancedScience Research Center, Japan Atomic EnergyAgency. Since 2016, he has been an AssociateProfessor at Research Institute for Interdisciplinary Science, OkayamaUniversity. His research is focused on spin transport, spin caloritronics andsuperconductivity.Fuyuki Ando received his doctoral degree fromGraduate School of Science, Kyoto University,Japan, in 2020 based on his work on the electric-field effect on magnetic thin films. From 2020–2023, he was employed by PanasonicCorporation, where he was involved in thedevelopment of thermoelectric materials andmodules. Since 2023, he has worked in NationalInstitute for Materials Science (NIMS), Japan, asa NIMS Special Researcher. His research in-terest includes spintronics, magneto-thermo-electrics and unconventional superconductingproperties.090101-26© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEWhttps://doi.org/10.1038/s41467-024-53723-2https://doi.org/10.1038/s41467-024-53723-2https://doi.org/10.1039/D4EE04845Hhttps://doi.org/10.1002/andp.202500127https://doi.org/10.1016/0022-3697(59)90225-2https://doi.org/10.1103/PhysRev.137.A1139https://doi.org/10.1016/0022-3697(65)90112-5https://doi.org/10.1016/0022-3697(65)90112-5https://doi.org/10.1557/adv.2019.150https://doi.org/10.1016/j.sna.2023.114729https://doi.org/10.1016/j.sna.2023.114729https://doi.org/10.1557/PROC-793-S6.1https://doi.org/10.1103/PhysRevB.71.113201https://doi.org/10.1103/PhysRevLett.104.176601https://doi.org/10.1103/PhysRevLett.108.056604https://doi.org/10.1103/PhysRevLett.108.056604https://doi.org/10.1002/adma.202308151https://doi.org/10.1021/acs.chemmater.3c02970https://doi.org/10.1038/s41467-024-48217-0https://doi.org/10.1038/s41467-024-48217-0https://doi.org/10.1103/PRXEnergy.3.043007https://doi.org/10.1063/1.1722971https://doi.org/10.1063/1.1735380https://doi.org/10.1088/0508-3443/13/12/303https://doi.org/10.1063/1.1729172https://doi.org/10.1088/0508-3443/15/1/121https://doi.org/10.1103/PhysRevLett.100.016601https://doi.org/10.1115/1.3636573https://doi.org/10.1115/1.3636573https://doi.org/10.1103/PhysRevApplied.11.054008https://doi.org/10.1103/PhysRevApplied.11.054008https://doi.org/10.1117/12.2229028https://doi.org/10.1016/j.pmatsci.2017.10.005https://doi.org/10.1063/1.1721551https://doi.org/10.1063/1.1931133https://doi.org/10.1063/1.1702670https://doi.org/10.1063/1.1702670https://doi.org/10.1016/0365-1789(63)90005-5https://doi.org/10.1103/PhysRevApplied.15.014011https://doi.org/10.7566/JPSJ.90.122001https://doi.org/10.1103/PhysRevB.96.184422https://doi.org/10.1103/PhysRevB.96.184422https://doi.org/10.1103/PhysRevLett.113.027601https://doi.org/10.1038/srep28233https://doi.org/10.1038/ncomms13754https://doi.org/10.1038/ncomms13754https://doi.org/10.1063/1.5022759https://doi.org/10.1063/1.5022759https://doi.org/10.1038/s41586-018-0143-xhttps://doi.org/10.1103/PhysRevB.101.020415https://doi.org/10.1103/PhysRevB.101.020415https://doi.org/10.1103/PhysRevApplied.21.024039https://doi.org/10.1103/PhysRevLett.110.067207https://doi.org/10.1103/PhysRevB.88.214403https://doi.org/10.1103/gqvy-p5p1https://doi.org/10.1103/gqvy-p5p1https://doi.org/10.1038/srep23114https://doi.org/10.1080/14686996.2022.2138538https://doi.org/10.1002/adma.202303416https://doi.org/10.1016/0921-4534(94)91308-0https://doi.org/10.1016/0921-4534(94)91308-0https://doi.org/10.1115/1.2751141https://doi.org/10.1155/2008/329601https://doi.org/10.1063/1.3194796https://doi.org/10.1063/1.3194796https://doi.org/10.1007/s11664-013-2959-3https://doi.org/10.1002/adma.202305622https://doi.org/10.1039/C4CP01582Ghttps://doi.org/10.1016/j.isci.2020.101967https://doi.org/10.1063/5.0180506https://doi.org/10.1002/aenm.202400411https://doi.org/10.1016/j.scriptamat.2012.06.038https://doi.org/10.7567/JJAP.57.080101https://doi.org/10.7567/JJAP.57.080101https://doi.org/10.20517/energymater.2025.26Takamasa Hirai received the B.E., M.E. and Ph.D. degrees from The University of Tokyo,Japan, in 2015, 2017 and 2020, respectively.Here, he was involved in electric-field control ofmagnetism in ultrathin films. In 2020, he joinedSpin Caloritronics Group in NIMS and startedresearch on spin caloritronics and thermal en-gineering. He is currently a Senior Researcher inNIMS. His research interests are in engineeringelectrical, thermal and thermoelectric transportproperties across interfaces, thin films and hy-brid material systems.Rajkumar Modak is a Project AssistantProfessor at Graduate School of FrontierSciences, The University of Tokyo, Japan. He isan experimental researcher in materials scienceand applied physics, with expertize in mag-netism, spintronics, spin caloritronics and ther-moelectrics. He earned his Ph.D. in Physicsfrom Indian Institute of Technology Guwahati,India, in 2019, where he investigated the multi-functional properties of Ni-Mn-based Heusleralloy films. Prior to his current role, he was aPostdoctoral Researcher (2019–2021), a JSPSFellow (2021–2023), and a NIMS Special Researcher (2023–2024) at NIMS,Japan, where he worked on developing materials and methods to acceleratebasic science and application studies on spin caloritronics.Matthew A. Grayson is a Professor atNorthwestern University, USA. He receivedhis Ph.D. from Princeton University, USA, in1998 and conducted postdoctoral research atUniversity of Maryland, USA, and later atTechnical University of Munich, Germany, as aHumboldt Fellow. In 2007, he joinedNorthwestern University, where his researchspans from novel semiconductor characteriza-tion methods and quantum Hall effect to trans-verse thermoelectrics and electrical impedancetomography.Ken-ichi Uchida received his doctoral degreefrom Tohoku University, Japan, in 2012. Afterworking as an Assistant/Associate Professor(2012–2016) at Tohoku University and as aGroup Leader at NIMS (2016–2023), he hasbeen a Distinguished Group Leader at ResearchCenter for Magnetic and Spintronic Materials,NIMS, since 2023 and a Professor at GraduateSchool of Frontier Science, The University ofTokyo, Japan, since 2024. He has been leadingthe JST ERATO project since 2022 and workingmainly on spin(calori)tronics, thermoelectricsand thermal energy engineering.090101-27© 2025 The Author(s). Published on behalf ofThe Japan Society of Applied Physics by IOP Publishing LtdAppl. Phys. Express 18, 090101 (2025) APEX REVIEW 1. Introduction 2. Classification of transverse thermoelectric conversion phenomena 2.1. Transverse thermoelectric effects with time-reversal symmetry breaking 2.1.1. Ordinary Nernst effect 2.1.2. Anomalous Nernst effect 2.1.3. Seebeck-effect-driven ordinary and anomalous Hall effects 2.1.4. Spin Seebeck and spin-dependent Seebeck effects 2.2. Transverse thermoelectric effects with structural symmetry breaking 2.2.1. Off-diagonal Seebeck effect due to macroscale anisotropic structure 2.2.2. Off-diagonal Seebeck effect due to microscale anisotropic structure 3. Formulation of transverse thermoelectric conversion with time-reversal symmetry breaking 3.1. Revising performance of a longitudinal thermoelectric generator 3.1.1. Conventional expression 3.1.2. Alternative expression 3.1.3. Summary of longitudinal device 3.2. Performance of transverse thermoelectric generator 3.2.1. Adiabatic condition in transverse direction 3.2.2. Isothermal condition in transverse direction 3.2.3. Figure of merit defined by Delves 3.2.4. Summary of transverse device 3.3. Performance of transverse thermoelectric refrigerator 3.3.1. Isothermal condition in transverse direction 3.3.2. Adiabatic condition in transverse direction 3.4. Performance of transverse active cooler 4. Formulation of transverse thermoelectric conversion with structural symmetry breaking 4.1. Transverse thermoelectric generator in adiabatic condition 4.2. Transverse thermoelectric generator under isothermal condition 4.3. Transverse thermoelectric refrigerator 4.4. Transverse active cooler 5. Measurement of transverse thermopower 5.1. Measurement configuration for Nernst effect in thin films 5.2. Effect of thermal boundary conditions 6. Application of transverse thermoelectric conversion 6.1. Heat flux sensing 6.2. Thermal energy harvesting 6.3. Hybrid transverse magneto-thermoelectric conversion 6.4. Thermoelectric permanent magnet 7. Conclusion and prospects Acknowledgments A9