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[Fuyuki Ando](https://orcid.org/0009-0003-7789-8170), [Yebin Lee](https://orcid.org/0000-0002-0737-1635), [Takamasa Hirai](https://orcid.org/0000-0002-5577-8018), [Ken-ichi Uchida](https://orcid.org/0000-0001-7680-3051)

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[Phenomenological formulation of hybrid transverse magnetothermoelectric conversion in artificially tilted multilayers](https://mdr.nims.go.jp/datasets/ee40e2e6-1b66-4249-8716-75eac2938aed)

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Manuscript_ODTC of ATML_Ando*Contact author: ANDO.Fuyuki@nims.go.jp    Phenomenological formulation of hybrid transverse magneto-thermoelectric conversion in artificially tilted multilayers  Fuyuki Ando,1,* Yebin Lee,1 Takamasa Hirai,1 and Ken-ichi Uchida1,2 1National Institute for Materials Science, 1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan 2Department of Advanced Materials Science, Graduate School of Frontier Sciences, The University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, Chiba 277-8561, Japan   Hybridizing multiple transverse thermoelectric effects in a single material synergistically enhances the dimensionless figure of merit for transverse thermoelectric conversion owing to the square-law characteristics of the superimposed transverse thermopowers. Here, we phenomenologically formulate the appearance of transverse magneto-thermoelectric phenomena, i.e., the anomalous Nernst effect and Seebeck-driven anomalous Hall effect, in artificially tilted multilayers consisting of magnetic and thermoelectric materials, where the off-diagonal Seebeck effect originally exists due to the anisotropic composite structure. Because the contributions of the anomalous Nernst, Seebeck-driven anomalous Hall, and off-diagonal Seebeck effects to transverse thermopower have different artificial structure dependences with each other, the total thermoelectric performance maximizes by optimizing the structure to balance these effects. We confirm the appearance of these thermoelectric effects by performing a finite element analysis of transverse thermoelectric fields in artificially tilted multilayers consisting of ferromagnetic Co2MnGa Heusler alloy and thermoelectric semiconductor Bi2Te3. This work will be a guideline to design the optimum structural parameters for superior transverse thermoelectric performance through the hybridization of spin- and magnetism-related thermoelectric phenomena in artificially tilted multilayers.   I. INTRODUCTION Transverse thermoelectric effects, where charge and heat currents are converted in the orthogonal directions in solids, has recently attracted attention in the fields of solid-state physics, materials science, and thermal management applications [1–4]. This unique conversion geometry enables a simplification of thermoelectric device architecture only using a single material and attaching electrodes at the ends, which is a practical advantage against the conventional Seebeck-effect-based devices consisting of many pairs of p- and n-type thermoelectric elements [5–8]. Owing to this junctionless structure, transverse thermoelectric devices have potential to exhibit ideal thermoelectric performance expected from the constituent material. Multiple principles of the transverse thermoelectric conversion have been investigated so far. The most representative ones are the ordinary Nernst effect in nonmagnetic conductors [1] and anomalous Nernst effect (ANE) in magnetic materials. Especially, associated with the development of spin caloritronics [3,9–11] and topological materials science [12–14], both theoretical and experimental studies on ANE have rapidly progressed realizing superior transverse thermoelectric performance due to non-trivial band topology [15–25] or extrinsic charge-spin scattering mechanisms [26–28]. Recently, Zhou et al. demonstrated a unique way to boost the transverse thermopower by a hybrid action of the Seebeck and anomalous Hall effects in thermoelectric semiconductor/ferromagnetic metal heterostructures, which is called the Seebeck-driven magneto-thermoelectric conversion or Seebeck-driven anomalous Hall effect (SAHE) (note that we adopt the latter terminology in this paper) [29–32]. Meanwhile, as the transverse thermoelectric effect unrelated to magnetism, the off-diagonal Seebeck effect (ODSE) has been intensively studied for wide variety of materials including goniopolar materials [33–37], p  n-type superlattices [38,39], and artificially tilted multilayers [40–45]. These ODSE materials tend to exhibit higher transverse thermoelectric performance than those by spin- or magnetism-related principles. Hybridization of the multiple transverse thermoelectric effects in a single material paves a new way to realize an unprecedentedly high thermoelectric performance [46–48]. Uchida et al. first demonstrated the hybrid transverse magneto-thermoelectric conversion in Bi88Sb12/Bi0.2Sb1.8Te3-based artificially tilted multilayers, where the magneto-Seebeck and ordinary Nernst effects mainly in the Bi88Sb12 layers are superimposed on ODSE by the application of an external magnetic field. As a result, the total transverse thermoelectric cooling performance was largely modulated from -10% to +15% [46]. ANE was also superimposed by embedding magnetic layers in artificially tilted multilayers [47], which was achieved *Contact author: ANDO.Fuyuki@nims.go.jp    even without an external magnetic field by the use of permanent magnet materials [48]. The figure of merit zxyT for this hybrid transverse thermoelectric conversion is expressed as  𝑧𝑥𝑦𝑇 =𝑆𝑥𝑦2𝜌𝑥𝑥 𝜅𝑦𝑦𝑇 =[𝑆ODSE + 𝑆𝑥𝑦(𝑀 or 𝐻)]2𝜌𝑥𝑥𝜅𝑦𝑦𝑇 =𝑆ODSE2 + 𝑆𝑥𝑦(𝑀 or 𝐻)2 + 2𝑆ODSE ∙ 𝑆𝑥𝑦(𝑀 or 𝐻)𝜌𝑥𝑥𝜅𝑦𝑦𝑇.  (1) Here, the total transverse thermopower Sxy is the sum of the transverse thermopower due to ODSE (SODSE) and that dependent on the magnetization M or magnetic field H [Sxy(M or H)]. ρxx is the electrical resistivity along the direction of a charge current, κyy is the thermal conductivity along the direction of a heat current, and T is the absolute temperature. Because zxyT is proportional to the square of Sxy, Sxy(M or H) produces the 2SODSESxy(M or H) term in addition to Sxy(M or H)2, which offers the greater improvement of zxyT than that solely by Sxy(M or H)2. However, there remains a problem that the Sxy(M or H) value is unclear in artificially tilted multilayers, which differs from the transverse magneto-thermopower of magnetic layers, such as the anomalous Nernst coefficient SANE, due to the inhomogeneous temperature distribution and electrical shunting with the adjacent layers [47]. Furthermore, Sxy(M or H) has a structural parameter dependence different from that of SODSE. Thus, we need a guideline for the optimum composite structure to balance SODSE and Sxy(M or H) and maximize transverse thermoelectric performance. In this study, we phenomenologically developed the formulation of the hybrid transverse magneto-thermopower Sxy [=SODSE+Sxy(M or H)] for artificially tilted multilayers consisting of magnetic and thermoelectric materials as shown in Fig. 1. Electric fields E induced by ANE and SAHE in magnetic layers are formulated for multilayers with a tilt angle θ = 0°. The coordinate rotation in x-y plane yields the ODSE-induced E and resultant Sxy. To confirm the appearance of these thermoelectric effects, we perform a thermoelectric coupling analysis using a finite element method (FEM) for the artificially tilted multilayers consisting of a ferromagnetic Co2MnGa Heusler alloy and thermoelectric Bi2Te3 compound. The analysis shows that the Sxy(M or H) term has a unique structural parameter dependence which modifies the optimum composite structure and a sizable impact on Sxy and zxyT.   II. METHODS Figure 1 shows a multilayer system for the tensor calculation of Sxy. Magnetic and thermoelectric  FIG 1. Schematic of transverse magneto-thermoelectric conversion in artificially tilted multilayers consisting of magnetic and thermoelectric materials. S’ANE and S’SAHE are defined as E’x and E’y divided by ’yT and ’xT, respectively. S’ANE refers to the transverse thermopower induced by ANE in the magnetic layers with M along the z’-direction, whereas S’SAHE by ANE and SAHE through the contribution from the neighboring thermoelectric layers. The tensor calculation of Sxy by all the ANE, SAHE, and ODSE contributions in the artificially tilted multilayers is performed by rotating the coordinate with θ from the x’y’z’ to xyz Cartesian coordinate. *Contact author: ANDO.Fuyuki@nims.go.jp    materials are alternately stacked in the y’-direction of x’y’z’ Cartesian coordinate system. The thickness ratio of the magnetic layers relative to the total thickness t = dM/(dM+dTE) is fixed, where dM and dTE are the thicknesses of the magnetic and thermoelectric layers, respectively. When M is oriented along the z’-direction and a temperature gradient T is applied in the x’-y’ plane, E is induced in the cross-product (T×M) direction by ANE and SAHE. S’ANE and S’SAHE are respectively defined as the generation of E’x and E’y induced by ’yT and ’xT. Table I shows thermoelectric, electrical, and thermal transport phenomena for constituent materials which are dealt in or applicable to our framework. In this work, we simplify the model as much as possible focusing on the essential phenomena to discuss the importance of the hybrid strategy. For the diagonal terms in thermoelectric tensor Sij, we assume an isotropic Seebeck coefficient as Sxx = Syy (= S), which can be extended to the isotropic magneto-Seebeck effect. Note that, however, the anisotropic (magneto-)Seebeck effect is out of our framework. For the off-diagonal terms, we account ANE induced by M as Sxy = -Syx (= SANE) at mz = +1, excluding the ordinary Nernst effect induced by H and SODSE (Sxy = Syx) of constituents induced by the microscopic structural symmetry breaking [4]. Also, we assume isotropic longitudinal electrical and thermal conductivities as σxx = σyy (= σ) and κxx = κyy (= κ), which include magneto-resistance and magneto-thermal resistance effect as far as they are isotropic. For the off-diagonal terms, only AHE induced by M is accountable, whereas the ordinary and symmetric Hall effects [49], ordinary and anomalous Righi-Leduc effects [50,51], and off-diagonal thermal conduction by structural symmetry breaking [52] are not. Please note that the following discussion is valid when the effects in the “Phenomena” column in Table I are accounted. We list the main physical quantities used in this study in Table II. First, we formulate S’ANE in a multilayer system. The effective temperature gradient applied to the magnetic layers in the y’-direction is described as tκTET/[tκTE+(1-t)κM] with κM and κTE respectively being thermal conductivities of the magnetic and thermoelectric materials. The generated E’x for the magnetic layers with 𝑆ANEM  is reduced by the factor tσM/[tσM+(1−t)σTE] due to electrical shunting with the neighboring thermoelectric layers, where σM and σTE are electrical conductivities of the magnetic and thermoelectric materials, respectively. Consequently, the transverse magneto-thermopower S’ANE normalized by ’yT is expressed as  𝑆′ANE(𝑚𝑧)  =𝑡𝜅TE𝑡𝜅TE + (1 − 𝑡)𝜅M𝑡𝜎M𝑡𝜎M + (1 − 𝑡)𝜎TE 𝑆ANEM ∙ 𝑚𝑧.  (2) Here, mz is the z’ component of the unit magnetization vector (−1  mz  +1). Next, we formulate S’SAHE as well as S’ANE. When ’xT is applied, a charge current is induced by the difference of the Seebeck coefficients between the magnetic and thermoelectric materials (𝑆TE − 𝑆M) . The shunting charge current in magnetic layers is in TABLE I. Thermoelectric, electrical, and thermal transport phenomena for constituents applicable to this study.   Symmetry Phenomena Not applicable Sii Sxx = Syy (= S) Magneto-Seebeck effect Anisotropic magneto-Seebeck effect Sij Sxy = -Syx (= SANE) Anomalous Nernst effect Ordinary Nernst effect Off-diagonal Seebeck effect σii (σ) σxx = σyy (= σ) Magneto-resistance Anisotropic magneto-resistance σij σxy = -σyx (= σAHE) Anomalous Hall effect Ordinary and symmetric Hall effect κii κxx = κyy (= κ) Magneto-thermal resistance Anisotropic magneto-thermal resistance κij κxy = -κyx (= 0) / Ordinary and anomalous Righi-Leduc effects Off-diagonal thermal conduction  *Contact author: ANDO.Fuyuki@nims.go.jp    turn converted into transverse E’y by AHE [31]. Then, 𝑆ANEM  is modified into  𝑆ANEM −(1 − 𝑡)(𝑆TE − 𝑆M)𝜎TEtan𝜃AHEM𝑡𝜎M + (1 − 𝑡)𝜎TE , (3)  Because E’y appears only in the magnetic layers as shown in Fig. 1, S’SAHE for the whole multilayer system is expressed as  𝑆′SAHE(𝑚𝑧) = −𝑡 [𝑆ANEM −(1 − 𝑡)(𝑆TE − 𝑆M)𝜎TEtan𝜃AHEM𝑡𝜎M + (1 − 𝑡)𝜎TE ] ∙ 𝑚𝑧 .  (4)  Utilizing S’ANE and S’SAHE in a multilayer system, we obtain Sxy for an artificially tilted multilayer system with a finite θ value. The thermoelectric tensor in the x’y’z’ coordinate is given by  𝑆′𝑖𝑗 = (𝑆′𝑥𝑥 𝑆′ANE 0𝑆′SAHE 𝑆′𝑦𝑦 00 0 𝑆′𝑧𝑧). (5)  Here, S’xx and S’yy are described as follows:  𝑆′𝑥𝑥 =𝑡𝜎M𝑆M + (1 − 𝑡)𝜎TE𝑆TE𝑡𝜎M + (1 − 𝑡)𝜎TE, (6) 𝑆′𝑦𝑦 =𝑡𝜅TE𝑆M + (1 − 𝑡)𝜅M𝑆TE𝑡𝜅TE + (1 − 𝑡)𝜅M . (7)  To transform Eq. (5) for the case of an artificially tilted multilayer, we introduce a rotation of θ around the z’-axis. This modifies the original x’y’z’ Cartesian coordinate with x = x’cosθ+y’sinθ and y = −x’sinθ+y’cosθ by the Jacobian matrix of the coordinate transformation: 𝐽 = (cos 𝜃 sin 𝜃 0−sin 𝜃 cos 𝜃 00 0 1). (8)  As a result, Sxy component in the thermoelectric tensor Sij for artificially tilted multilayers is expressed as   𝑆𝑥𝑦 = (𝑆′𝑥𝑥 − 𝑆′𝑦𝑦) sin 𝜃 cos 𝜃                                 +𝑆′ANE(𝑚𝑧) cos2 𝜃 − 𝑆′SAHE(𝑚𝑧) sin2 𝜃. (9) TABLE II. Summary of the main physical quantities.  Structural parameter θ [°] Tilt angle t [a.u.] Thickness ratio Constituent material S [V/K] Seebeck coefficient σ [S/m] Electrical conductivity κ [W/mK] Thermal conductivity SANE [V/K] Anomalous Nernst coefficient tanθAHE [a.u.] Anomalous Hall angle (= -σAHE/σ) mz [a.u.] Unit magnetization vector in z(z’)-direction Multilayer system (θ = 0°, x’y’z’ coordinate) S’xx [V/K] Longitudinal thermopower E’x induced by ’xT S’yy [V/K] Longitudinal thermopower E’y induced by ’yT S’ANE [V/K] Transverse thermopower E’x induced by ’yT S’SAHE [V/K] Transverse thermopower E’y induced by ’xT Artificially tilted multilayer system (θ ≠ 0°, xyz coordinate) SODSE [V/K] Transverse thermopower by ODSE Sxy [V/K] Hybrid transverse thermopower by ODSE, ANE, and SAHE ρxx [Ωm]  Electrical resistivity κyy [W/mK] Thermal conductivity zxyT [a.u.] Transverse thermoelectric figure of merit  *Contact author: ANDO.Fuyuki@nims.go.jp    The first term (S’xx−S’yy)sinθcosθ corresponds to SODSE as the Goldsmid proposed [41] and the second term S’ANE(mz)cos2θ−S’SAHE(mz)sin2θ to the magnetization-dependent term Sxy(M or H). For a fixed t value, the derivative of Sxy by θ is expressed as  𝑑𝑆𝑥𝑦𝑑𝜃= (𝑆′𝑥𝑥 − 𝑆′𝑦𝑦) cos 2𝜃                                    −[𝑆′ANE(𝑚𝑧) + 𝑆′SAHE(𝑚𝑧)] sin 2𝜃. (10)  Then, the stationary θ value at dSxy/dθ = 0 satisfies  tan 2𝜃 =𝑆′𝑥𝑥 − 𝑆′𝑦𝑦𝑆′ANE(𝑚𝑧) + 𝑆′SAHE(𝑚𝑧) , (11)  which provides the appropriate branch to identify the maximum Sxy value. Furthermore, according to the following equation,  𝑑𝑆𝑥𝑦𝑑𝜃|𝜃=45°= −[𝑆′ANE(𝑚𝑧) + 𝑆′SAHE(𝑚𝑧)], (12)  one can find the optimum θ value to maximize Sxy shifts above (below) 45° when S’ANE(mz)+S’SAHE(mz) is negative (positive). Then, let us introduce physical criteria to enhance and constructively hybridize ODSE, ANE, and SAHE. For the first term [SODSE = (S’xx−S’yy)sinθcosθ], S’xx−S’yy is obtained by the transformation of Eqs. (6)–(7) as follows,  𝑆′𝑥𝑥 − 𝑆′𝑦𝑦 = [𝑡𝜎M𝑡𝜎M + (1 − 𝑡)𝜎TE−𝑡𝜅TE𝑡𝜅TE + (1 − 𝑡)𝜅M] 𝑆M + [(1 − 𝑡)𝜎TE𝑡𝜎M + (1 − 𝑡)𝜎TE −(1 − 𝑡)𝜅M𝑡𝜅TE + (1 − 𝑡)𝜅M] 𝑆TE. (13) When inequal relationships of σM > σTE and κM > κTE are assumed, the opposite sign of SM and STE is necessary to constructively superpose the first and second terms in Eq. (13) with the same sign. Meanwhile, one can achieve a constructive hybridization of ANE and SAHE for the higher value of second term [S’ANE(mz)cos2θ-S’SAHE(mz)sin2θ] by satisfying the following sign criterion as,  𝑆ANEM ∙ (𝑆TE − 𝑆M) ∙ tan𝜃AHEM < 0. (14)  Because the sign of second term depends on that of mz, one can constructively superpose the first (ODSE) and second (ANE and SAHE) terms in Eq. (9) by optimizing the magnetization direction. Thus, to obtain the higher Sxy, one needs to select the constituent materials simultaneously taking Eq. (13) and the inequality (14) into account. To discuss the quantitative impact of the superposition of the second (ANE and SAHE) term on the transverse thermoelectric performance, we analytically calculate Sxy and zxyT for the artificially tilted multilayers based on the ferromagnetic Co2MnGa Heusler alloy with large SANE and tanθAHE  [18,19] and thermoelectric Bi2Te3 compound with larger S than Co2MnGa. Table III summarizes the transport properties for Co2MnGa and Bi2Te3 [31,47,53]. This material combination satisfies the sign criterion (14), ensuring the constructive hybridization of ANE and SAHE. We also performed an FEM simulation by COMSOL Multiphysics software to confirm the appearance of ANE and SAHE in artificially tilted multilayers. 2-dimensional Co2MnGa/Bi2Te3-based artificially tilted multilayers with an area of 10.0 × 8.0 mm2 and θ of 0, 15, 30, 45, 60, 75, and 90° were constructed. The thicknesses of the Co2MnGa and Bi2Te3 layers are respectively set to be 1.0 mm and 0.7 mm, where t is 0.59, which is comparable to the experimentally demonstrated value in ref. [47]. Temperatures on the top and bottom sides are set to be 303.15 K and 293.15 K, respectively. The interfacial electrical and thermal resistance between Co2MnGa and Bi2Te3 layers are set to be zero, limiting our framework into an ideal SAHE-enhanced regime.   TABLE III. Transport properties for Co2MnGa and Bi2Te3 [31,46].   SM, STE [10−6 V/K] σM, σTE [105 S/m] κM, κTE [W/mK] 𝑆ANEM [10−6 V/K] tan𝜃AHEM  [a.u.] Co2MnGa −32.1 7.99 18.7 6.9 0.089 Bi2Te3 −110.3 1.83 1.5 n.a. n.a.  *Contact author: ANDO.Fuyuki@nims.go.jp    III. RESULTS A. Tensor calculation  Figure 2(a)–(b) shows the tensor calculation results of S’ANE(mz = +1)cos2θ and −S’SAHE(mz = +1)sin2θ for the Co2MnGa/Bi2Te3-based artificially tilted multilayer.  S’ANE(mz = +1)cos2θ increases with the increase of t and decrease of θ due to the cos2θ contribution and reaches 6.9  10-6 V/K at θ = 0° and t = 1.0, that is, 𝑆ANEM  of plain Co2MnGa. Meanwhile, −S’SAHE(mz = +1)sin2θ increases with increasing both t and θ due to the sin2θ contribution, which interestingly exhibits relatively larger values in the wide range of t and θ than S’ANE(mz = +1)cos2θ. This characteristic is attributed not only to the absence of inhomogeneous temperature distribution and electrical shunting but also to the constructive hybridization of ANE and SAHE according to the inequality (14). Figure 2(c) shows the calculated SODSE for the Co2MnGa/Bi2Te3-based artificially tilted multilayer. Regardless of the mz value, the finite SODSE is induced due to the anisotropy of the Seebeck coefficient (S’xx−S’yy) and maximized at θ = 45° because of the sinθcosθ contribution.   FIG 2. Contour plots of (a) S’ANE(mz = +1)cos2θ, (b) −S’SAHE(mz = +1)sin2θ, (c) SODSE, and (d) Sxy for the Co2MnGa/Bi2Te3-based artificially tilted multilayer. (e) A schematic of the hybridization of ODSE, ANE and SAHE (f) The θ dependence of SODSE, S’ANE(mz = +1)cos2θ, −S’SAHE(mz = +1)sin2θ, and Sxy at t = 0.59. (g) Maximum values of S’ANE(mz = +1)cos2θ at θ = 0°, −S’SAHE(mz = +1)sin2θ at θ = 90°, SODSE at θ = 45°, and Sxy at θ = 47° with 𝑆ANEM  of the plain Co2MnGa. *Contact author: ANDO.Fuyuki@nims.go.jp     Figure 2(d)-(e) shows the total Sxy with mz = +1 according to Eq. (9) and a schematic of the hybridization of ODSE, ANE and SAHE in the Co2MnGa/Bi2Te3-based artificially tilted multilayer. The absolute value of Sxy increases because both S’ANE(mz = +1)cos2θ and −S’SAHE(mz = +1)sin2θ are positively superimposed on SODSE [Fig. 2(d)]. The generation of E by ODSE, ANE, and SAHE is described in Fig. 2(e) according to the material hierarchy. The ODSE-induced E is strongly localized in the Bi2Te3 layers with large STE. Meanwhile, the ANE- and SAHE-induced E appear in the Co2MnGa layers, which are oriented in the x-direction and perpendicular to the multilayer, respectively. To visualize the quantitative impact of the hybridization concept, we plot the θ dependence of SODSE, S’ANE(mz = +1)cos2θ, −S’SAHE(mz = +1)sin2θ, and Sxy at t = 0.59 in Fig. 2(f). It is found that −S’SAHE(mz = +1)sin2θ modifies Sxy more than S’ANE(mz = +1)cos2θ, resulting in the shift of optimum θ to a higher angle than 45°. Here, we compare the maximum values of S’ANE(mz = +1)cos2θ at θ = 0°, −S’SAHE(mz = +1)sin2θ at θ = 90°, SODSE at θ = 45°, and Sxy at θ = 47° with 𝑆ANEM  of the plain Co2MnGa. Although S’ANE and −S’SAHE are solely lower than 𝑆ANEM  of 6.9  10-6 V/K due to the reduced volumetric ratio of Co2MnGa, Sxy reaches 32.4  10-6 V/K owing to the constructive hybridization with SODSE. Importantly, the optimum θ and t are different between Sxy and SODSE, which respectively maximizes to be 32.7  10-6 V/K at θ = 47° and t = 0.67 and 29.7  10-6 V/K at θ = 45° and t = 0.63. According to Appendix, the modification of θ and t also changes ρxx and κyy from 2.28  10-6 Ωm and 7.95 W/mK at θ = 45° and t = 0.63 to 2.19  10-6 Ωm and 8.78 W/mK at θ = 47° and t = 0.67. Considering these θ and t dependences of ρxx and κyy, we estimate that zxyT based on the maximum Sxy is 14% higher than that based on the maximum SODSE owing to the square-law characteristics [Eq. (1)]. Thus, these results provide two important ideas to employ the strategy of hybrid transverse magneto-thermoelectric conversion: a requirement to rearrange the artificial structure to maximize Sxy and a potentially large impact on the transverse thermoelectric performance toward thermoelectric applications.  B. Finite element calculation Figure 3 (a)–(b) shows the results of the thermoelectric coupling analysis for temperature T and electric potential V distributions in the Co2MnGa/Bi2Te3-based artificially tilted multilayer with θ = 45° and mz = 0. The application of T in the y-direction [Fig. 3(a)] induces the oblique V gradient [Fig. 3(b)], that is, the generation of E for both the x- and y-directions (Ex = -dV/dx and Ey = -dV/dy), which respectively imply ODSE and the conventional Seebeck effect.  Figure 4(a) shows the mz dependence of the V distributions under a temperature difference of 10 K for the Co2MnGa/Bi2Te3-based artificially tilted multilayers with θ = 0, 15, 30, 45, 60, 75, and 90°. It is obvious that ODSE and the Seebeck effect predominantly determine the V distribution due to the larger thermopower than magnetization dependent components. ODSE-induced Ex appears at θ values other than 0° and 90°, which is consistent with the sinθcosθ term in Eq. (9). For lower θ values than 45°, the V distributions between mz = ±1 show negligibly small difference. On the other hand, for higher θ values than 45°, the V distribution is modulated by the sign of mz. Especially at θ = 90°, the contour plots of V for mz = ±1 show the upward- and downward-sloping, which means the opposite sign of Ex depending on the magnetization direction.   To clarify Ex, we plot the x-direction line profile of V(y = 0), which is the average V in the range of −0.1 mm  y  +0.1 mm in Fig. 4(b). For the middle θ values exhibiting ODSE, V(y = 0) shows a step-like behavior due to the inhomogeneous T and thermoelectric property; the Bi2Te3 regions exhibit the negative and steep slope due to the negative xT (T component in the x-direction) and larger S, whereas the Co2MnGa regions the positive and gentle slope due to the positive xT and smaller S. Importantly, only the  slope of the Co2MnGa regions varies depending on the sign of mz, which  suggests the additive contributions by ANE and SAHE and results in the modulation of Ex. Here, the mz-induced modulation of Ex becomes larger as θ increases, which is consistent with the larger contribution by −S’SAHE(mz = +1)sin2θ than that by S’ANE(mz = +1)cos2θ shown in Fig. 2.   FIG 3. Contour plots of (a) T and (b) V for the Co2MnGa/Bi2Te3-based artificially tilted multilayer with θ = 45° and t = 0.59. *Contact author: ANDO.Fuyuki@nims.go.jp    Let us compare the Sxy(M/H) contribution obtained from the tensor calculation and FEM simulation. Figure 5 shows the θ dependence of Sxy(mz = +1)−Sxy(mz = -1), which corresponds to 2[S’ANE(mz = +1)cos2θ−S’SAHE(mz = +1)sin2θ] in Eqs. (2)-(9) and the difference of V(y = 0) between x = ±5 mm divided by the applied temperature difference of 10 K for the FEM simulation in Fig. 4(b). These results are quantitatively consistent with each other, showing the increasing trend with increasing θ. Thus, we confirm the appearance and spatial distribution of the transverse thermoelectric effect solved in Eqs. (2)-(9).   IV. DISCUSSION Finally, we discuss the future perspective to design artificially tilted multilayers using Eqs. (2)-(9) for experiments. As discussed above, the most important message is that, to maximize Sxy, one needs to combine magnetic and thermoelectric materials following the inequalities (10)–(11) and rearrange t and θ values from those optimized only for ODSE. As the exploration of magnetic materials having the larger SANE and tanθAHE than Co2MnGa progresses, the rearrangement proposed in this work will have a significant impact on the transverse thermoelectric performance. Note that although the combination of Co2MnGa and n-type Bi2Te3 provides the constructive hybridization of ANE and SAHE, in the sense of SODSE as Eq. (11), one needs to combine materials with the largely different SM and STE. Thus, the simultaneous tuning of S, SANE, and tanθAHE is crucial to constructively utilize all the thermoelectric effects. As  demonstrated in Figs. 3 and 4, both the ANE and SAHE contributions occur even in the mm-scale thickness and sample size. Thus, we can adopt typical sintering methods, such as spark plasma sintering, to experimentally create the artificially tilted multilayers for the hybrid transverse magneto-thermoelectric conversion. On the other hand, there are three possible sources which may cause a deviation of Sxy from the calculated one. First, this work neglects the magneto-Seebeck, ordinary Nernst, and Hall effects by the application of  H to simplify the discussion. However, the stray field due to magnetic layers will induce these H-dependent magneto-thermoelectric effects and slightly modulate    FIG 4. (a) Contour plots of V and (b) line profiles of V(y = 0) in the x-direction for the Co2MnGa/Bi2Te3-based artificially tilted multilayers with various θ values from 0° to 90° and mz = ±1.  *Contact author: ANDO.Fuyuki@nims.go.jp    Sxy [46,54]. Second, the thermal boundary condition in the x-direction can influence Sxy. When a target material is thermally adiabatic in the E direction, Sxy includes a parasitic Seebeck contribution by the transverse heat current generation, i.e., the off-diagonal thermal conduction due to an axis-dependent conduction polarity [35,52] and Righi-Leduc effect in magnetic layers [51,55,56]. Third, the interfacial electrical resistance or the formation of interfacial diffusion layers between magnetic and thermoelectric layers can modulate the SAHE contribution because the Seebeck-effect-induced electric field in magnetic layers depends on the interface condition. In fact, while the ANE contribution was clearly measured for Co2MnGa/Bi2Te3- and Co2MnGa/Bi0.2Sb1.8Te3- based artificially tilted multilayers, the SAHE contribution was hardly observed [47]. Please note that our framework is limited in an ideally high SAHE regime. On the other hand, if one inputs tan𝜃AHE = 0 in Eq. (4), it transforms into no SAHE limit.   V. CONCLUSIONS We phenomenologically formulated the equations of hybrid transverse magneto-thermopower Sxy for artificially tilted multilayers consisting of magnetic and thermoelectric materials for the optimum structural design. It is found that the S’SAHE(mz)sin2θ term shows larger values in the wide range of t and θ than the S’ANE(mz)cos2θ term due to the constructive contribution by SAHE in addition to ANE. These magnetization-dependent contributions modify the optimum t and θ values to maximize Sxy. In case of the Co2MnGa/Bi2Te3-based artificially tilted multilayer, the hybridization of ANE and SAHE is expected to increase zxyT up to 14% owing to the square-law characteristics of Sxy. We performed an FEM thermoelectric coupling analysis for the Co2MnGa/Bi2Te3-based artificially tilted multilayer, confirming the appearance of the transverse thermoelectric effects solved by phenomenological formulation. This work will be a guideline to design artificially tilted multilayers with high transverse thermoelectric performance through the hybridization of magneto-thermoelectric phenomena.  ACKNOWLEDGMENTS  The authors thank Y. Yamashita and T. Yagi for valuable discussions. 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 Resonac Holdings Corporation.  APPENDIX: CALCURATION OF TRANSPORT TENSORS The electrical resistivity and thermal conductivity tensors ( 𝜌𝑖𝑗  and 𝜅𝑖𝑗 ) are formulated for artificially tilted multilayers based on the Goldsmid’s method [41]. In the x’y’z’ Cartesian coordinate, the electrical resistivities and thermal conductivities in x’- and y’-directions are analytically calculated as follows:  𝜌′𝑥𝑥 =1𝑡𝜎M + (1 − 𝑡)𝜎TE, (A1) 𝜌′𝑦𝑦=𝑡𝜎TE + (1 − 𝑡)𝜎M𝜎M ∙ 𝜎TE , (A2) 𝜅′𝑥𝑥 = 𝑡𝜅M + (1 − 𝑡)𝜅TE, (A3) 𝜅′𝑦𝑦 =𝜅M ∙ 𝜅TE𝑡𝜅TE + (1 − 𝑡)𝜅M . (A4)  Following the same coordinate transformation by a rotation matrix as Eq. (8), 𝑆𝑖𝑗  and 𝜌𝑖𝑗  in the xyz Cartesian coordinate are obtained. The components relevant to this work are shown as  𝜌𝑥𝑥 = 𝜌′𝑥𝑥 cos2 𝜃 + 𝜌′𝑦𝑦 sin2 𝜃, (A5) 𝜅𝑦𝑦 = 𝜅′𝑥𝑥 sin2 𝜃 + 𝜅′𝑦𝑦 cos2 𝜃. (A6)      FIG 5. The θ dependence of transverse magneto-thermopower component Sxy(mz = +1)−Sxy(mz = -1) obtained from the tensor calculation and FEM simulation. *Contact author: ANDO.Fuyuki@nims.go.jp     [1] A. Von Ettingshausen and W. 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