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[Fuyuki Ando](https://orcid.org/0009-0003-7789-8170), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Hiroto Adachi](https://orcid.org/0000-0002-6844-6477), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051)

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[Adiabatic transverse thermoelectric conversion enhanced by heat current manipulation in artificially tilted multilayers](https://mdr.nims.go.jp/datasets/e7348124-b73a-48dc-9e86-771ca6b48295)

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Manuscript_ODTC of ATML_Ando*Contact author: ANDO.Fuyuki@nims.go.jp    Adiabatic transverse thermoelectric conversion enhanced by heat current manipulation in artificially tilted multilayers  Fuyuki Ando,1,* Takamasa Hirai,1 Hiroto Adachi,2 and Ken-ichi Uchida1,3 1Research Center for Magnetic and Spintronic Materials, National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan 2Research Institute for Interdisciplinary Science, Okayama University, 3-1-1 Tsushimanaka, Kita-ku, Okayama, Okayama 700-8530, Japan 3Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan   We phenomenologically formulate and experimentally observe an adiabatic transverse thermoelectric conversion enhanced by a heat current re-orientation in artificially tilted multilayers (ATMLs). By alternately stacking two materials with different thermal conductivities and rotating its multilayered structure with respect to a longitudinal temperature gradient, off-diagonal components in the thermal conductivity tensor are induced. This off-diagonal thermal conduction (ODTC) generates a finite transverse temperature gradient and Seebeck-effect-induced thermopower in the adiabatic condition, which is superposed on the isothermal transverse thermopower driven by the off-diagonal Seebeck effect (ODSE). In this study, we calculate and observe the two-dimensional temperature distribution and the resultant transverse thermoelectric conversion in ATMLs comprising thermoelectric Co2MnGa Heusler alloys and Bi2-aSbaTe3 compounds. By changing the tilt angle from 0° to 90°, the transverse temperature gradient obviously appeared in the middle angles and the transverse thermopower increases up to -121.9 μV/K in Co2MnGa/Bi0.2Sb1.8Te3-based ATML at the tilt angle of 45° whereas the isothermal contribution is estimated to be -82.6 μV/K from the analytical calculation. This hybrid action derived from ODTC results in the significant enhancement of the maximum reduced conversion efficiency from 3.1% to 8.1% in calculation and from 2.3% to 4.4% in experiment.   I. INTRODUCTION A transverse thermoelectric effect in solids, which mutually converts charge and heat currents in the orthogonal directions, has attracted much attention in terms of fundamental physics, materials science, and thermoelectric applications [1–4]. This unique geometry offers a practical advantage, that is, simplification of a thermoelectric device architecture only using a single material, whereas classical thermoelectric devices utilizing the Seebeck effect consist of multiple thermoelectric materials and electrodes connecting in series and forming a π-shaped structure. The most representative transverse thermoelectric effect is the Nernst effect  [1]. Recently, associated with the development of spin caloritronics [4–7] and topological materials science [8–10], the studies on both the ordinary and anomalous Nernst effects have rapidly advanced and achieved the superior transverse thermoelectric performance than ever before owing to the non-trivial band topology [11–20] or extrinsic mechanisms [21–23]. Meanwhile, ODSE as the other principle has been intensively studied for wide variety of materials having microscale anisotropy, i.e., the goniopolar materials [24–28] and (p  n)-type thermoelectric superlattices [29,30], and having macroscale anisotropy, i.e., ATMLs [31–35]. The ODSE materials exhibit higher performances than those by the Nernst effects. Also, the hybridization of the multiple principles in a single material also provides a key for the further advance of materials science for transverse thermoelectrics [36,37]. These recent progresses are expected to develop the next core technologies for energy harvesting and thermal management. In addition to the insight into materials characteristics, the thermal boundary conditions, i.e., isothermal or adiabatic, also have a significant influence on the transverse thermoelectric performance [26]. The standard definition of the figure of merit for transverse thermoelectric conversion 𝑧𝑥𝑦𝑇 is given as [38–41]: *Contact author: ANDO.Fuyuki@nims.go.jp    𝑧𝑥𝑦𝑇 =𝑆𝑥𝑦2𝜌𝑥𝑥𝜅𝑦𝑦𝑇. (1)  where 𝑆𝑥𝑦 ≡ 𝐸𝑥 ∇𝑦𝑇⁄ |∇𝑥𝑇=0 is the isothermal transverse thermopower by the ratio of the applied temperature gradient in y-axis (yT) and generated electric field in x-axis (Ex) under the temperature gradient in x-axis xT = 0 as shown in Fig. 1(a). Also, 𝜌𝑥𝑥 ≡ 𝐸𝑥 𝑗𝑐,𝑥⁄ |∇𝑥𝑇=0 and 𝜅𝑦𝑦 ≡ −𝑗𝑞,𝑦 ∇𝑦𝑇⁄ |∇𝑥𝑇=0 are respectively the isothermal electrical resistivity and thermal conductivity with jc,x and jq,y being the charge current density in x-axis and heat current density in y-axis. Delves [42,43] and Horst [44,45] pointed out that when the isothermal condition of xT = 0 is replaced by the adiabatic condition of jq,x = 0, 𝑧𝑥𝑦𝑇 changes to an adiabatic figure of merit 𝑧𝑥𝑦∗ 𝑇  due to the appearance of finite xT and modification to the adiabatic transverse thermopower 𝑆𝑥𝑦∗ , electrical resistivity 𝜌𝑥𝑥∗ , and thermal conductivity 𝜅𝑦𝑦∗ . Note that the adiabatic physical quantities are highlighted by the superscript * in this work. xT typically originates from ODTC of the applied jq,y, such as the thermal Hall effect (i.e., Righi-Leduc effect) or anisotropy of thermal conductivity. However, the adiabatic transverse thermoelectric properties modulated by ODTC-induced xT has generally been considered for the cooling applications  [12,42,44,46,47] or treated just as a correction term to explain why magneto-thermoelectric coefficients differ in the same materials, e.g., the difference in thermal boundary conditions due to the aspect ratio [19,26,43,45]. Thus, there has been no attempt to constructively utilize ODTC to improve transverse thermoelectric performance. Materials which actively control the heat current direction have been developed to form unique thermal circuits harvesting waste heat, which are often called thermal metamaterials [48–50]. Figure 1(b) shows a schematic of ATML consisting of two materials with different thermal conductivities 𝜅, where the effective thermal conductivities in parallel and perpendicular to the stacking plane (𝜅∥ and 𝜅⊥) are different from each other. Owing to this structure-induced anisotropic thermal conductivity, the heat current bends from y-axis when yT is applied oblique to the stacking plane. Here, the off-diagonal component of the thermal conductivity tensor 𝜅𝑥𝑦 = −𝑗𝑞,𝑥 ∇𝑦𝑇⁄  in ATMLs is expressed by the analytical matrix calculation in Appendix A with Refs.  [51–53] as   𝜅𝑥𝑦 = (𝜅∥ − 𝜅⊥) sin 𝜃 cos 𝜃, (2)  where θ denotes the tilt angle of the stacking plane to x-axis. The (𝜅∥ − 𝜅⊥) term increases as the difference in 𝜅 between the constituent materials increases. Then, the thermally adiabatic limit and open circuit condition [jq,x = jc,x = 0 in Eq. (A1)] gives a relationship between 𝜅𝑥𝑦 and xT as follows:   ∇𝑥𝑇 = −𝜅𝑥𝑦𝜅𝑥𝑥∇𝑦𝑇. (3)  The previous studies experimentally observed sizable bending angles up to 26° in copper/stainless use steel- FIG 1. Schematic of transverse thermoelectric conversion in (a) isothermal condition and (b) adiabatic condition for an anisotropic material, such as ATML. By rotating the multilayer by θ to the x-axis, ODTC is induced so that the bent heat current direction modulates Ex. (c) Schematic and (d) infrared image of the measurement configuration for the two-dimensional temperature distribution and transverse thermoelectric conversion in CMG/BT-based ATML. *Contact author: ANDO.Fuyuki@nims.go.jp    based ATMLs, which corresponds to ∇𝑥𝑇 ∇𝑦𝑇⁄ ~ −0.5  owing to the large −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  [48,52] . Thus, there is abundant room for utilizing the bending heat current for transverse thermoelectrics by hybridizing transverse and longitudinal thermoelectric effects driven by yT and xT. In this study, we phenomenologically formulate and experimentally demonstrate the enhancement of 𝑆𝑥𝑦∗  and transverse thermoelectric conversion efficiency by combining ODSE induced by yT and the Seebeck effect induced by xT in ATMLs. We synthesized the ATML slabs comprising thermoelectric Co2MnGa Heusler alloys and Bi2-aSbaTe3 compounds with various θ from 0° to 90°. Through the observation of two-dimensional temperature distribution using an infrared camera while applying yT, the θ dependence of ODTC-induced ∇𝑥𝑇 ∇𝑦𝑇⁄  was characterized and confirmed the consistency with the analytically calculated −𝜅𝑥𝑦 𝜅𝑥𝑥⁄ . As the result of the heat current re-orientation, the higher transverse thermopower of -121.9 μV/K than Sxy of -82.6 μV/K obtained by the conventional matrix calculation was observed in Co2MnGa/Bi0.2Sb1.8Te3-based ATML. Owing to the ODTC-induced enhancement of transverse thermopower, the analytically calculated 𝑧𝑥𝑦∗ 𝑇 reaches 0.28 in maximum at room temperature whereas the maximum 𝑧𝑥𝑦𝑇  is calculated to be 0.13, which corresponds to the 165% improvement for the transverse thermoelectric conversion efficiency. We experimentally observed the enhancement of conversion efficiency from 2.3% in the isothermal configuration to 4.4% in the adiabatic one using the same Co2MnGa/Bi0.2Sb1.8Te3-based ATML sample. Thus, the heat current manipulation in the adiabatic condition provides a distinct strategy on the material developments and device designs for transverse thermoelectrics.  II. METHODS To obtain a superior ODSE-induced transverse thermopower, Co2MnGa (CMG), Bi0.2Sb1.8Te3 (BST), and Bi2Te3 (BT) were selected as the constituent materials for ATMLs. The synthesis processes and measurement methods for thermoelectric transport properties for each sintered body are detailly described in Ref. [37]. Table I shows the summary of the measurement results: the Seebeck coefficient SSE, electrical resistivity ρ, and thermal conductivity κ for CMG, BST, and BT. It is found that the transport properties, especially κ, are greatly different between CMG and BST (BT), which is favorable to obtain both large ODSE and ODTC in ATMLs.  The CMG/BST- and CMG/BT-based ATML slabs with various θ values were prepared by the spark plasma sintering method. The sintered CMG with a diameter of 20 mm was sliced into many disks with a thickness of 1 mm using a diamond wire saw. The CMG disks and BST (BT) po wders were alternately filled into a graphite die and sintered at 450℃ with a uniaxial pressure of 30 MPa and soaking time of 60 min under high vacuum. The average thickness of BST (BT) was 0.7 mm, which corresponds to the thickness ratio of CMG t = dCMG/dCMG+dBST(BT) = 0.59 with d being the thickness of each layer. The sintered CMG/BST and CMG/BT multilayers were cut into rectangular slabs with a size of 10  8  1 mm3 and θ = 0, 15, 30, 45, 60, 75, and 90°. The two-dimensional temperature distribution and transverse thermoelectric conversion for CMG/BST- and CMG/BT-based ATMLs were measured using an infrared camera and homemade thermoelectric property measurement setup. Figure 1(c) shows a schematic of the measurement configuration, where the ATML sample was bridged between two anodized Al blocks in the y-direction with a distance of ~6 mm, one of which acted as a heat source by applying a charge current to connected chip heaters and the other as a heat sink to generate yT. Here, a region of interest in the ATML sample floated in the air keeping close to the adiabatic condition in x-axis, defined as the adiabatic configuration. Meanwhile, by attaching a high thermal conductivity layer to the bottom surface of ATML, the ODTC-induced xT can be canceled due to the thermal shunting effect in the parallel thermal circuit [54]. In this work, a 0.5-mm-thick thermally oxidized Si substrate was attached to the ATML sample by a diamond thermal grease, defined as the isothermal configuration. To measure a transverse thermoelectric voltage Vx, two Al-1%Si wires were directly attached on the 10  8 mm2 top surface with a distance of ~6.5 mm in the x-direction by using a wire bonder. To apply a load current Iload, TABLE I. Thermoelectric transport properties for CMG, BST, and BT from Ref. [37].   CMG BST BT SSE (10-6 V/K) -32.1 170.6 -110.3 ρ (10-6 Ωm) 1.25 7.81 5.46 κ (W/mK) 18.7 1.2 1.5  *Contact author: ANDO.Fuyuki@nims.go.jp    the 8  1 mm2 side surfaces were covered with a silver paste connected to copper wires. Figure 1(d) shows an infrared image of CMG/BT-based ATML with θ = 45° placed on the sample holder. The central part of 10  8 mm2 surface was covered with an insulating black ink having a high emissivity over 0.94. The observation area for thermography using an infrared camera is defined as a 3.6  3.6 mm2 square centered on the midpoint of the two Al-1%Si electrodes.   III. RESULTS A. Off-diagonal thermal conduction  We begin with the analytical calculation of the thermal conductivity tensor for CMG/BST- and CMG/BT-based ATMLs following Appendix A and Refs. [51–53]. The measured κ in Table I and t are substituted into Eqs. (A1) and (A2) to calculate 𝜅𝑖𝑗 . According to the equations, t and θ are variables for 𝜅𝑖𝑗  as well as κ of CMG, BST, and BT. Figures  FIG 2. (a)-(c) Contour plots of the calculated (a) 𝜅𝑥𝑥, (b) 𝜅𝑥𝑦, and (c) −𝜅𝑥𝑦 𝜅𝑥𝑥⁄ . White dotted lines indicate t = 0.59. (d) Thermal images of CMG/BST-based ATMLs at t = 0.59 and various θ values under the application of yT. (e)-(f) The x- and y-axes line profiles of the average temperature signals in the y- and x-directions for the adiabatic and isothermal configurations. (g) The plot of effective xT and yT for CMG/BST-based ATML at θ = 45° in the adiabatic and isothermal configurations under various temperature differences between the heat source and sink. (h) The θ dependence of ∇𝑥𝑇 ∇𝑦𝑇⁄  for CMG/BST-based ATMLs in the adiabatic and isothermal configurations with the calculated adiabatic (−𝜅𝑥𝑦 𝜅𝑥𝑥⁄  at t = 0.59) and isothermal limits. *Contact author: ANDO.Fuyuki@nims.go.jp    2(a)−(c) shows the contour plots of 𝜅𝑥𝑥 , 𝜅𝑥𝑦 , and −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  as functions of t and θ for CMG/BST-based ATML. Figure 2(a) shows that 𝜅𝑥𝑥 monotonically increases as t (θ) increases (decreases) due to the enhanced contribution of CMG with high κ. Meanwhile, 𝜅𝑥𝑦  in Fig. 2(b) shows a different trend maximizing at 45° due to the sin 𝜃 cos𝜃 contribution.  As a result, −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  in Fig. 2(c) shows a unique θ dependence which minimizes at 64° and relatively moderate t dependence. In the case of CMG/BT-based ATML, −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  minimizes at 62°, suggesting that the optimum t and θ to maximize ODTC depend on the balance of κ for constituent materials. To experimentally confirm the heat current manipulation by ODTC, we observed and analyzed the two-dimensional temperature distribution for our ATMLs in the adiabatic and isothermal configurations. Figure 2(d) shows thermal images in the area indicated by Fig. 1(d) under the application of yT for CMG/BST-based ATMLs in the adiabatic configuration. Here, the input heater power was tuned so that the temperature difference between the heat source and sink stabilized at 10.0 ± 0.1 K. All the thermal images represent layered patterns with different angles corresponding to θ, which reflects the difference in κ between CMG and BST. The temperature drastically changes in the BST layers due to the lower κ. To quantitatively discuss ODTC-induced xT and yT in our ATMLs, we plot the x- and y-axes position dependence of the temperature signals averaged in the y- and x- directions for the adiabatic and isothermal configurations as shown in Fig. 2(e)−(f). The coordinate origin is defined at the lower left corner of the thermal images. In the adiabatic configuration, the negative slope in the x-axis plot is clearly observed only in the middle θ values, whereas those are almost zero at θ = 0° and 90°. This characteristic is consistent with 𝜅𝑥𝑦 ∝sin 𝜃 cos 𝜃  in Fig. 2(b), directly suggesting the appearance of ODTC-induced xT. The y-axis plot shows the decrease in the positive slope as θ increases, which can result from the increase in 𝜅𝑦𝑦  of the ATML samples according to the matrix calculation in Appendix A. Let us see the result in the isothermal configuration in Fig. 2(f). Because of the formation of a parallel thermal circuit with the Si substrate, yT is smaller than that in Fig. 2(e) even when the temperature difference between the heat source and sink is the same. Importantly, the negative slope in the x-axis plot is relatively suppressed although not perfectly canceled, suggesting the thermal shunting with the attached Si substrate. The reason for the nonlinear slopes in the x- and y- axis plots in all the θ values is unclear but probably related to the thermal shunting with the side electrodes and heat source/sink through the Si substrate.  We compare the experimentally measured ∇𝑥𝑇 ∇𝑦𝑇⁄  in the adiabatic and isothermal configurations associated with the analytically calculated −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  for CMG/BST-based ATMLs. Figure 2(g) shows the plot of effective xT and yT for CMG/BST-based ATML with θ = 45° in the adiabatic and isothermal configurations under the various temperature differences between the heat source and sink, where the ∇𝑥𝑇 and ∇𝑦𝑇 values are determined by the slope of linear fit in Fig. 2(e)−(f). Also, the ∇𝑥𝑇 ∇𝑦𝑇⁄  values are evaluated by the slope of linear fit in Fig. 2(g). The θ dependence of ∇𝑥𝑇 ∇𝑦𝑇⁄  for CMG/BST-based ATMLs as well as the calculated −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  at t = 0.59 are depicted in Fig. 2(h). According to Eq. (3), in the ideal adiabatic and isothermal condition, ∇𝑥𝑇 ∇𝑦𝑇⁄  is equal to −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  and zero, respectively. The measured ∇𝑥𝑇 ∇𝑦𝑇⁄  in the adiabatic configuration is obviously closer to −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  than that in the isothermal configuration, suggesting the induction and suppression of ODTC in the adiabatic and isothermal configurations, respectively. The  ∇𝑥𝑇 ∇𝑦𝑇⁄  values quantitatively agree with −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  especially at low θ region and reach the minimum value of -0.40 at θ = 60° in the adiabatic configuration owing to the large difference in κ in Table I, which is expected to have a significant impact on transverse thermoelectric performance. The reason for the deviation at higher θ is mentioned in Discussion section. Hereby, we demonstrated the manipulation of the heat current direction and appearance of xT through ODTC for ATMLs in the adiabatic condition.  B. Adiabatic transverse thermopower We phenomenologically formulate the adiabatic transverse thermopower 𝑆𝑥𝑦∗  by ODSE and the ODTC-induced Seebeck effect for anisotropic materials such as ATMLs. The two-dimensional thermoelectric tensor in the x-y plane for the heat current q and electric field E is introduced as follows [42–45]:  ( 𝑗𝑞,𝑥𝑗𝑞,𝑦𝐸𝑥𝐸𝑦 ) =(  𝑆𝑥𝑥𝑇 𝑆𝑥𝑦𝑇𝑆𝑦𝑥𝑇 𝑆𝑦𝑦𝑇−𝜅𝑥𝑥 −𝜅𝑥𝑦−𝜅𝑦𝑥 −𝜅𝑦𝑦𝜌𝑥𝑥 𝜌𝑥𝑦𝜌𝑦𝑥 𝜌𝑦𝑦𝑆𝑥𝑥 𝑆𝑥𝑦𝑆𝑦𝑥 𝑆𝑦𝑦 )  ( 𝑗𝑐,𝑥𝑗𝑐,𝑦∇𝑥𝑇∇𝑦𝑇) .  (4)   Here, we consider generation of Ex by the application of yT under an open-circuit condition in the y-*Contact author: ANDO.Fuyuki@nims.go.jp    direction (jc,y = 0). Then, the relevant linear-response equations can be written as   𝑗𝑞,𝑥 = 𝑆𝑥𝑥𝑇𝑗𝑐,𝑥 − 𝜅𝑥𝑥∇𝑥𝑇 − 𝜅𝑥𝑦∇𝑦𝑇, (5) 𝑗𝑞,𝑦 = 𝑆𝑦𝑥𝑇𝑗𝑐,𝑥 − 𝜅𝑦𝑥∇𝑥𝑇 − 𝜅𝑦𝑦∇𝑦𝑇, (6) 𝐸𝑥 = 𝜌𝑥𝑥𝑗𝑐,𝑥 + 𝑆𝑥𝑥∇𝑥𝑇 + 𝑆𝑥𝑦∇𝑦𝑇, (7) 𝐸𝑦 = 𝜌𝑦𝑥𝑗𝑐,𝑥 + 𝑆𝑦𝑥∇𝑥𝑇 + 𝑆𝑦𝑦∇𝑦𝑇. (8)  Equations (5)−(8) in the adiabatic condition for the x-direction (jq,x = 0) can be solved for yT and Ex as  ∇𝑦𝑇 =𝑆𝑦𝑥∗𝜅𝑦𝑦∗𝑇𝑗𝑐,𝑥 −1𝜅𝑦𝑦∗𝑗𝑞,𝑦 , (9) 𝐸𝑥 = 𝜌𝑥𝑥∗ 𝑗𝑐,𝑥 −𝑆𝑥𝑦∗𝜅𝑦𝑦∗𝑗𝑞,𝑦 , (10)  where the adiabatic transverse thermopower 𝑆𝑥𝑦∗  and 𝑆𝑦𝑥∗  are described by using the isothermal transverse thermopower 𝑆𝑥𝑦 and 𝑆𝑦𝑥 as follows:  𝑆𝑥𝑦∗ = 𝑆𝑥𝑦 −𝜅𝑥𝑦𝜅𝑥𝑥𝑆𝑥𝑥 , (11) 𝑆𝑦𝑥∗ = 𝑆𝑦𝑥 −𝜅𝑦𝑥𝜅𝑥𝑥𝑆𝑥𝑥 , (12)  where the first term represents the intrinsic ODSE and the second term the ODTC-induced Seebeck effect. In the case of ATMLs, both the Seebeck and thermal conductivity tensors are symmetric (see Appendix A), i.e., 𝑆𝑥𝑦 = 𝑆𝑦𝑥 , 𝜅𝑥𝑦 = 𝜅𝑦𝑥 , and hence 𝑆𝑥𝑦∗ = 𝑆𝑦𝑥∗ . Note that those for the Nernst and thermal Hall effects are antisymmetric due to Onsager’s reciprocal relations, i.e., 𝑆𝑥𝑦 = −𝑆𝑦𝑥 , 𝜅𝑥𝑦 = −𝜅𝑦𝑥 , and hence 𝑆𝑥𝑦∗ = −𝑆𝑦𝑥∗ . Firstly, we calculated and measured 𝑆𝑥𝑥 to confirm the anisotropic thermoelectric property in ATMLs as the origin of 𝑆𝑥𝑦. Fig. 3(a) shows the contour plots of 𝑆𝑥𝑥 for CMG/BST-based ATML as functions of t and θ calculated from Eq. (A11). As t decreases (increases) and θ increases (decreases), 𝑆SE of BST (CMG) layers dominantly contributes to 𝑆𝑥𝑥  for CMG/BST-based ATML. Fig. 3(b) shows the θ dependence of 𝑆𝑥𝑥 at t = 0.59, where 𝑆𝑥𝑥  was measured using the Seebeck-coefficient/electric-resistance measurement system (ZEM-3, ADVANCE RIKO Inc.). The measured 𝑆𝑥𝑥 value changes depending on θ from negative values for   FIG 3. (a) Contour plot of the calculated 𝑆𝑥𝑥 for CMG/BST-based ATML. (b) The θ dependence of the measured and calculated 𝑆𝑥𝑥  at t = 0.59 with the tiny error bar. (c)-(d) Contour plots of the calculated (c) 𝑆𝑥𝑦 and (d) 𝑆𝑥𝑦∗ . (e) The θ dependence of the measured 𝐸𝑥 ∇𝑦𝑇⁄  in the adiabatic and isothermal configurations and calculated 𝑆𝑥𝑦 and 𝑆𝑥𝑦∗  for both CMG/BT- and CMG/BST-based ATMLs. The inset is the ∇𝑦𝑇 dependence of 𝐸𝑥 for CMG/BST-based ATML at θ = 45°. *Contact author: ANDO.Fuyuki@nims.go.jp    θ ≤ 15° due to the electrical shunting effect in CMG layers to large positive value for θ > 15°, reproducing the analytically calculated values. Thus, we can expect the experimental observation of 𝑆𝑥𝑦 by ODSE. Then, we analytically calculate 𝑆𝑥𝑦 and 𝑆𝑥𝑦∗ . Figure 3(c) shows the contour plot of 𝑆𝑥𝑦 as functions of t and θ for CMG/BST-based ATML, calculated based on Eq. (11) and the thermoelectric transport tensors for ATMLs in Appendix A.  The  𝑆𝑥𝑦 value minimizes at 45° in a similar manner to 𝜅𝑥𝑦  in Fig. 2(b). On the other hand, because 𝑆𝑥𝑥  and −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  have the different t and θ dependences as shown in Figs. 3(c) and 2(c), 𝑆𝑥𝑦∗  exhibits a distinct behavior with 𝑆𝑥𝑦 [Fig. 3(d)]. Owing to the same sign of 𝑆𝑥𝑦  and −𝜅𝑥𝑦𝑆𝑥𝑥 𝜅𝑥𝑥⁄  for θ > 15°, the peak 𝑆𝑥𝑦∗  is greatly higher than the peak 𝑆𝑥𝑦  in this study: 𝑆𝑥𝑦∗  reaches -167.7 μV/K at t = 0.47 and θ = 64°, whereas 𝑆𝑥𝑦 -82.7 μV/K at t = 0.61 and θ = 45°. Thus, we naturally expect the improvement of the transverse thermopower through the contribution of ODTC.  Now, we are in the position to experimentally characterize the transverse thermopower 𝐸𝑥 ∇𝑦𝑇⁄  in the adiabatic and isothermal configurations comparing with 𝑆𝑥𝑦 and 𝑆𝑥𝑦∗ . Figure 3(e) shows the measurement results for CMG/BST- and CMG/BT-based ATMLs. The 𝐸𝑥 ∇𝑦𝑇⁄  values are determined by the slope of linear fit as shown in the inset of Fig. 3(e). Reflecting the opposite sign of SSE for BST and BT, CMG/BST- and CMG/BT-based ATMLs also show the opposite sign of 𝐸𝑥 ∇𝑦𝑇⁄  (see Eqs. (A6), (A7), and (A10) of Appendix A for details). For CMG/BT-based ATML, 𝑆𝑥𝑦∗  is larger than 𝑆𝑥𝑦 in the overall θ region because 𝑆𝑥𝑥  is negative regardless of θ. Importantly, the experimentally measured 𝐸𝑥 ∇𝑦𝑇⁄  in the adiabatic configuration is also larger than that in the isothermal configuration regardless of θ, completely matching the analytical calculation. Meanwhile, for CMG/BST-based ATML, 𝑆𝑥𝑦∗  is larger than 𝑆𝑥𝑦 for θ > 15° due to the sign change of 𝑆𝑥𝑥  [(Fig. 3(b)]. The experimentally measured 𝐸𝑥 ∇𝑦𝑇⁄  in the adiabatic and isothermal configurations for CMG/BST-based ATML completely reproduces this large and small relationship between 𝑆𝑥𝑦∗  and 𝑆𝑥𝑦 , where 𝐸𝑥 ∇𝑦𝑇⁄  in the adiabatic configuration is larger only for θ > 15°. These results are the direct evidence of the contribution of ODTC to transverse thermoelectric conversion. The 𝐸𝑥 ∇𝑦𝑇⁄  values quantitatively agree with 𝑆𝑥𝑦∗  for θ ≤ 45° but get closer to 𝑆𝑥𝑦  with increasing θ, whose trend is consistent with that of the measured ∇𝑥𝑇 ∇𝑦𝑇⁄  in Fig. 2(h). Although the conventional matrix calculation suggests the optimum 𝑆𝑥𝑦  of -82.6 μV/K at θ = 45° for CMG/BST-based ATML, 𝐸𝑥 ∇𝑦𝑇⁄  measured in the adiabatic condition reaches the much higher value of -121.9 ± 22.2 μV/K, which leads to a significant enhancement of the transverse thermoelectric performance.  C. Transverse thermoelectric performance We phenomenologically formulate the adiabatic transport properties relevant to 𝑧𝑥𝑦∗ 𝑇  for anisotropic materials such as ATMLs. By solving Eqs. (5)−(8), 𝜌𝑥𝑥∗  and 𝜅𝑦𝑦∗  defined by Eqs. (9) and (10) can be expressed as  𝜌𝑥𝑥∗ = 𝜌𝑥𝑥 + (𝑆𝑥𝑦∗ 𝑆𝑦𝑥∗𝜅𝑦𝑦∗+𝑆𝑥𝑥2𝜅𝑥𝑥)𝑇, (13)   FIG 4. (a)-(b) Contour plots of the calculated (a) 𝜌𝑥𝑥 and (b) 𝜌𝑥𝑥∗  for CMG/BST-based ATML. (c) The θ dependence of the measured and calculated 𝜌𝑥𝑥  at t = 0.59 including tiny error bars. (d)-(e) Contour plots of the calculated (d) 𝜅𝑦𝑦, and (e) 𝜅𝑦𝑦∗ . *Contact author: ANDO.Fuyuki@nims.go.jp    𝜅𝑦𝑦∗ = 𝜅𝑦𝑦 −𝜅𝑥𝑦𝜅𝑦𝑥𝜅𝑥𝑥. (14)  Figures 4(a)−(b) show the analytically calculated 𝜌𝑥𝑥 and 𝜌𝑥𝑥∗  as functions of t and θ for CMG/BST-based ATML, indicating the obviously different behaviors between the isothermal and adiabatic conditions as well as 𝑆𝑥𝑦  and 𝑆𝑥𝑦∗ . As shown in Fig. 4(c), we experimentally measured the θ dependence of 𝜌𝑥𝑥 for CMG/BST-based ATML at t = 0.59 using ZEM-3. 𝜌𝑥𝑥 increases as θ increases in correspondence with the analytical calculation. Because the voltage was measured while applying a pulse current, the measured 𝜌𝑥𝑥 values show a relatively close trend to that in the isothermal condition. As well as 𝜌𝑥𝑥 and 𝜌𝑥𝑥∗ , 𝜅𝑦𝑦 and 𝜅𝑦𝑦∗  show the distinct behaviors as functions of t and θ [Figs. 4(d)−(e)]. Due to the symmetric relations of 𝑆𝑥𝑦 = 𝑆𝑦𝑥  and 𝜅𝑥𝑦 = 𝜅𝑦𝑥 , 𝜌𝑥𝑥∗  ( 𝜅𝑦𝑦∗ ) necessarily exhibits a higher (lower) value than 𝜌𝑥𝑥  (𝜅𝑦𝑦). As a result, 𝑧𝑥𝑦∗ 𝑇 is expressed by the replacement of Eq. (1) by Eqs. (10), (12), and (13):  𝑧𝑥𝑦∗ 𝑇 =𝑆𝑥𝑦∗ 2𝜌𝑥𝑥∗ 𝜅𝑦𝑦∗𝑇. (15)  Note that by substituting the relations of 𝑆𝑥𝑦 = −𝑆𝑦𝑥 , 𝜅𝑥𝑦 = −𝜅𝑦𝑥 due to Onsager’s reciprocal theory to Eqs. (13)−(14), the adiabatic transport properties for the Nernst effects can be obtained, where 𝜅𝑦𝑦∗  necessarily exhibits a higher value than 𝜅𝑦𝑦 contrary to the case of ATMLs.  Figure 5(a)−(b) shows analytically calculated 𝑧𝑥𝑦𝑇 and 𝑧𝑥𝑦∗ 𝑇  values for CMG/BST-based ATML. In general, 𝑆𝑥𝑦 (∝ sin𝜃 cos 𝜃) maximizes at θ = 45° and both 𝜌𝑥𝑥  and 𝜅𝑦𝑦  monotonically decrease as θ decreases [Figs. 3(a), 4(a), and 4(c)]. Thus, the conventional calculation for ATMLs in the isothermal limit necessarily suggests the optimum θ lower than 45° to maximize 𝑧𝑥𝑦𝑇  [33–35,37]. In fact, Fig. 5(a) shows that 𝑧𝑥𝑦𝑇  for CMG/BST-based ATML maximizes to be 0.13 at t = 0.56 and θ = 30°. On the other hand, the best θ to maximize 𝑆𝑥𝑦∗  drastically changes from 45° due to the −𝜅𝑥𝑦𝑆𝑥𝑥 𝜅𝑥𝑥⁄  term and positions at 64° for CMG/BST-based ATML [Fig 3(c)]. Figure 5(b) shows that, through balancing between high 𝑆𝑥𝑦∗  and low 𝜌𝑥𝑥∗  and 𝜅𝑦𝑦∗ , 𝑧𝑥𝑦∗ 𝑇 reaches 0.28 at t = 0.44 and θ = 47°, which provides totally different design parameters from the conventional analytical calculation.  Let us compare the transverse thermoelectric performance between the isothermal and adiabatic conditions. Note that the figure of merit cannot be directly used for the fair comparison because the domains of definition are different as follows:  0 < 𝑧𝑥𝑦𝑇 < ∞, (16) 0 < 𝑧𝑥𝑦∗ 𝑇 < 1. (17)  The presence of ODTC and the Seebeck effect makes it more difficult to get the relation between 𝑧𝑥𝑦𝑇 and  FIG 5. (a)-(b) Contour plots of the calcurated (a) 𝑧𝑥𝑦𝑇 and (b) 𝑧𝑥𝑦∗ 𝑇 for CMG/BST-based ATML. (c) The Iload dependence of |Vx| and P at yT = 0.48 K/mm (0.51) in the adiabatic (isothermal) configuration for CMG/BST-based ATML at t = 0.59 and θ = 45°. (d) The θ dependence of the maximum reduced efficiency in the adiabatic and isothermal configurations together with the calculated 𝜂𝑥𝑦 and 𝜂𝑥𝑦∗  for CMG/BST-based ATML at t = 0.59. *Contact author: ANDO.Fuyuki@nims.go.jp    𝑧𝑥𝑦∗ 𝑇  whereas  𝑧𝑥𝑦∗ 𝑇 = 𝑧𝑥𝑦𝑇 (1 + 𝑧𝑥𝑦𝑇)⁄  was obtained under  𝑆𝑥𝑥 = 𝜅𝑥𝑦 = 0. Then, we introduce the maximum reduced efficiency for transverse thermoelectric conversion in the isothermal and adiabatic limits (𝜂𝑥𝑦 and 𝜂𝑥𝑦∗ ), which enables the fair comparison because the thermoelectric conversion efficiency is expressed by 𝜂Carnot𝜂𝑥𝑦  and 𝜂Carnot𝜂𝑥𝑦∗  with the Carnot efficiency 𝜂Carnot  for both the isothermal and adiabatic cases:  𝜂𝑥𝑦 =√1 + 𝑧𝑥𝑦𝑇 − 1√1 + 𝑧𝑥𝑦𝑇 + 1, (18) 𝜂𝑥𝑦∗ =1 −√1 − 𝑧𝑥𝑦∗ 𝑇1 +√1 − 𝑧𝑥𝑦∗ 𝑇. (19)  By respectively substituting 𝑧𝑥𝑦𝑇 = 0.13 and 𝑧𝑥𝑦∗ 𝑇 =0.28  into Eqs. (18) and (19), 𝜂𝑥𝑦  = 3.1% in the isothermal limit and 𝜂𝑥𝑦∗  = 8.1% in the adiabatic limit are obtained. Thus, the change rate in the conversion efficiency is estimated to be 165%, which provides a great impact on the transverse thermoelectric performance. To experimentally confirm the enhancement of conversion efficiency in the adiabatic condition, we performed the thermoelectric power generation measurements for CMG/BST-based ATML in the adiabatic and isothermal configurations. As shown in Fig. 5(c), the load current Iload dependence of |Vx| was measured under the application of ∇𝑦𝑇 to characterize the output power P (= Iload  |Vx|). Each |Vx| value was obtained after inputting Iload and waiting for 10 seconds to stabilize at the thermal equilibrium state. The |Vx| value almost linearly decreases as Iload increases due to the internal resistance of CMG/BST-based ATML, resulting in the parabolic-shape P curves. Interestingly, despite the applied ∇𝑦𝑇  is smaller in the adiabatic configuration in Fig. 5(c), the obtained P is larger owing to the larger open-circuit voltage (Vx at Iload = 0) by the constructive contribution from ODTC. Then, we experimentally determine the maximum reduced efficiency as follows: 𝜂𝑥𝑦 =𝑇ave∆𝑦𝑇∙𝑃𝐴𝑗𝑞,𝑦. (20) Here, 𝑇ave  is the average temperature, ∆𝑇𝑦  the temperature difference between the edges attached to the heat source and sink, and A is the cross-sectional area to input 𝑗𝑞,𝑦  for CMG/BST-based ATMLs, respectively. The 𝑗𝑞,𝑦  values in the adiabatic and isothermal configurations are estimated from 𝜅𝑦𝑦∗ ∇𝑦𝑇 and 𝜅𝑦𝑦∇𝑦𝑇 as shown in Figs. 4(d)−(e), respectively. Figure 5(d) shows the θ dependence of the measured 𝜂𝑥𝑦 and 𝜂𝑥𝑦∗  together with their analytical calculations for CMG/BST-based ATML at t = 0.59. We find that the measured 𝜂𝑥𝑦∗  is significantly larger than the measured 𝜂𝑥𝑦 for θ > 15° as predicted by the analytical calculations. Thus, the difference in 𝜂𝑥𝑦  and 𝜂𝑥𝑦∗  is due purely to the contribution from the ODTC-induced Seebeck effect. The measured 𝜂𝑥𝑦∗  value reaches 4.4% in maximum at θ = 45° whereas 𝜂𝑥𝑦  remains 2.3%, corresponding to 89% enhancement of the transverse thermoelectric conversion efficiency in the adiabatic configuration. The above calculations and experiments claim that the distinct materials design and performance potential are obtained depending on whether the thermal boundary condition is isothermal or adiabatic.  IV. DISCUSSION We discuss what factors influence the thermal boundary condition toward the precise estimation of  ∇𝑥𝑇 ∇𝑦𝑇⁄  and the resultant transverse thermoelectric performance. Obviously, the heat dissipation at the side surfaces through the convection and radiation enforces the imperfect adiabatic condition even in our measurement setup. Scudder  [26] demonstrated that the x : y aspect ratio of the target material also changes the boundary condition through the thermal short-circuit by heat spreaders, i.e., heat source and sink. The shorter length in the y-direction might be one of the reasons why the measured 𝐸𝑥 ∇𝑦𝑇⁄  for ATMLs in the previous reports were comparable to or less than 𝑆𝑥𝑦  [33–35,37]. Meanwhile, this work finds that, even if the measurement setup and geometry of the samples were unchanged, the measured ∇𝑥𝑇 ∇𝑦𝑇⁄ , 𝐸𝑥 ∇𝑦𝑇⁄ , and 𝜂𝑥𝑦  gradually deviates from the calculated  FIG 6. Schematic of a lateral thermopile module utilizing the ODTC-induced transverse thermopower. *Contact author: ANDO.Fuyuki@nims.go.jp    −𝜅𝑥𝑦 𝜅𝑥𝑥⁄ , 𝑆𝑥𝑦∗ , and 𝜂𝑥𝑦∗  with increasing θ [Figs. 2(h), 3(e), and 5(d)], meaning that the region of interest changes from the adiabatic to isothermal condition. In light of this observation, the erosion of isothermal boundaries from the heat spreaders (∇𝑥𝑇 = 0) has a variable influence in this study, i.e., as 𝜅𝑦𝑦 increases by increasing θ [Fig. 4(d)], the heat spreaders more and more imposes the isothermal confinement on the positions to measure ∇𝑥𝑇 ∇𝑦𝑇⁄ , 𝐸𝑥 ∇𝑦𝑇⁄ , and 𝜂𝑥𝑦. In other words, the decrease in 𝜅𝑦𝑦 or the optimization of x : y aspect ratio will result in the enhancement of the experimental 𝜂𝑥𝑦 of 4.4% ideally to 𝜂𝑥𝑦∗  of 7.6% in the adiabatic limit. Finally, we present how to utilize the ODTC-induced performance enhancement in transverse thermoelectric devices. Figure 6 shows a schematic of a lateral thermopile structure as an example. The important point is that two kinds of transverse thermoelectric materials with the same sign of −𝜅𝑥𝑦 𝜅𝑥𝑥⁄  and the opposite sign of 𝑆𝑥𝑦∗ , such as our CMG/BT- and CMG/BST-based ATML slabs, are used. By alternately stacking the two materials intermediated by insulator layers and electrically connected side-by-side, a thermally parallel and electrically series circuit is formed. The side surfaces for electrodes need to be thermally isolated from heat spreaders as much as possible. Owing to the unidirectional transverse heat current for each element, the net transverse temperature gradient ∇𝑥𝑇  will originate without canceling out in the entire module. As a result, the transverse thermopower will be enhanced compared with 𝑆𝑥𝑦 [Fig. 3(e)] and positively contribute to the output power.   V. CONCLUSIONS We phenomenologically formulated and experimentally observed an adiabatic transverse thermoelectric performance enhanced by ODTC raising examples of CMG/BT- and CMG/BST-based ATMLs. ODTC induced by an anisotropic thermal conductivity generates a finite transverse temperature gradient and Seebeck-effect-induced thermopower in the adiabatic condition, which is superposed on the isothermal transverse thermopower driven by ODSE. From the two-dimensional temperature distributions on the ATML surfaces in the adiabatic condition, the generation of sizable transverse temperature gradient was observed, which is quantitatively consistent with the calculated ODTC ratio. The resultant adiabatic transverse thermopower in ATMLs was clearly larger than the calculated and measured isothermal ones. By utilizing this ODTC-induced thermopower, we greatly improved the transverse thermoelectric conversion efficiency up to 89% from the conventional isothermal configuration. This work provides a new design for the ODSE materials and devices distinct from the conventional matrix calculations.   ACKNOWLEDGMENTS The authors thank H. Sepehri-Amin and A. Takahagi for valuable discussions and K. Suzuki and M. Isomura for technical supports. This work was supported by ERATO “Magnetic Thermal Management Materials” (No. JPMJER2201) from JST, Grants-in-Aid for Scientific Research KAKENHI (No. 24K17610) from JSPS, and NEC Corporation.  APPENDIX A: CALCURATION OF THERMOELECTRIC TRANSPORT TENSORS FOR ATMLS First, the thermal conductivity tensor for ATML comprising two materials A and B with different thermal conductivities ( 𝜅A  and 𝜅B ) is introduced following Refs.  [51–53]. An A/B-based multilayer exhibits the anisotropic thermal conductivities between the directions parallel and perpendicular to the stacking plane as follows:  𝜅∥ = 𝑡𝜅A + (1 − 𝑡)𝜅B, (A1) 𝜅⊥ =𝜅A ∙ 𝜅B𝑡𝜅B + (1 − 𝑡)𝜅A. (A2)  When we define x- or z- and y-axes as the parallel and perpendicular directions respectively, the thermal conductivity tensor is expressed using Eqs. (A1) and (A2) as  𝜅𝑖𝑗 = (𝜅∥ 0 00 𝜅⊥ 00 0 𝜅∥). (A3)  The off-diagonal components are zero in the original description. To transform this plain multilayer into ATML, we introduce a rotation around z-axis by an angle θ which modifies the original axes to 𝑥′ =𝑥 cos 𝜃 + 𝑦 sin 𝜃 and 𝑦′ = −𝑥 sin𝜃 + 𝑦 cos𝜃 by the Jacobian matrix of the coordinate transformation:  𝐽 = (cos𝜃 sin 𝜃 0−sin 𝜃 cos𝜃 00 0 1). (A4)  *Contact author: ANDO.Fuyuki@nims.go.jp    The thermal conductivity tensor is modified through this rotation for the ATML structure, resulting in the finite off-diagonal components as   𝜅𝑖𝑗 =𝐽𝜅𝑖′𝑗′𝐽𝑇det(𝐽)= (𝜅∥ cos2 𝜃 + 𝜅⊥ sin2 𝜃 (𝜅∥ − 𝜅⊥) sin𝜃 cos𝜃 0(𝜅∥ − 𝜅⊥) sin 𝜃 cos 𝜃 𝜅∥ sin2 𝜃 + 𝜅⊥ cos2 𝜃 00 0 𝜅∥),  (A5)  where JT denotes the transpose of J, and det(J) the determinant.  The Seebeck and electrical resistivity tensors (𝑆𝑖𝑗  and 𝜌𝑖𝑗) are formulated for A/B-based ATML based on the Goldsmid’s method [32]. The Seebeck coefficients and electrical resistivities in the directions parallel and perpendicular to the stacking plane are analytically calculated using thermoelectric transport parameters (𝑆SE,A, 𝑆SE,B, 𝜌A and 𝜌B) as follows:  𝑆SE,∥ =𝑡𝜌B𝑆SE,A + (1 − 𝑡)𝜌A𝑆SE,B𝑡𝜌B + (1 − 𝑡)𝜌A, (A6) 𝑆SE,⊥ =𝑡𝜅B𝑆SE,A + (1 − 𝑡)𝜅A𝑆SE,B𝑡𝜅B + (1 − 𝑡)𝜅A, (A7) 𝜌∥ =𝜌A ∙ 𝜌B𝑡𝜌B + (1 − 𝑡)𝜌A, (A8) 𝜌⊥ = 𝑡𝜌A + (1 − 𝑡)𝜌B. (A9)  Following the completely same coordinate transformation by a rotation matrix as Eqs. (A3)−(A5), 𝑆𝑖𝑗  and 𝜌𝑖𝑗 are obtained. The components relevant to this work are shown as  𝑆𝑥𝑦 = (𝑆SE,∥ − 𝑆SE,⊥) sin𝜃 cos𝜃, (A10) 𝑆𝑥𝑥 = 𝑆SE,∥ cos2 𝜃 + 𝑆SE,⊥ sin2 𝜃, (A11) 𝜌𝑥𝑥 = 𝜌∥ cos2 𝜃 + 𝜌⊥ sin2 𝜃. (A12)     [1] A. Von Ettingshausen and W. Nernst, Ueber das Auftreten electromotorischer Kräfte in Metallplatten, welche von einem Wärmestrome durchflossen werden und sich im magnetischen Felde befinden, Ann. Phys. 265, 343 (1886). [2] M. H. Norwood, Theory of Nernst Generators and Refrigerators, J. Appl. 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