The thermoelectric effect that converts heat to electricity in solid-state devices is attracting much attention as a promising candidate for a carbon-free power generation from waste heat1. Thermoelectric generation based on conventional Seebeck effect employs a longitudinal geometry2, in which an electric voltage is generated along a temperature gradient \(\nabla T\) (Fig. 1a). The longitudinal geometry is not favored for applications because it requires a thermopile composed of multiple and alternatively connected thermoelectric materials of different types3, which is complex to cover a large-area heat source. This limitation can be overcome by a thermoelectric device in transverse geometry using the spin Seebeck effect (SSE)4,5,6,7,8 that generates an electric voltage in the direction perpendicular to a temperature gradient9 (Fig. 1b). For practical applications, however, a thermoelectric conversion efficiency of SSE is insufficient and must be substantially enhanced.

Fig. 1: Schematics of SSE and theoretical results.
figure 1

a Process of conventional Seebeck effect where \(\nabla T\) and \({J}_{c}\) denote the temperature gradient and charge current, respectively. b Process of SSE in FM/NM structure where \(\nabla {T}_{z},\,M,\,{J}_{s}\), and \({{{{{\boldsymbol{\sigma }}}}}}\) denote the vertical temperature gradient, magnetization, spin current, and spin polarization vector, respectively. c Model structure where FM consists of two ferromagnetic layers with different exchange interaction \({J}_{{ex}}\). d Theoretical and numerical results of the ratio \({r}_{{{{{{\rm{SSE}}}}}}}\) (defined in the main text) as a function of \({J}_{{ex},2}/{J}_{{ex},1}\).

In normal metal (NM)/ferromagnet (FM) heterostructures (Fig. 1b), a basic building block for SSE, the thermoelectric conversion via longitudinal SSE consists of three separate processes; First, a vertical temperature gradient generates a temperature difference \({T}_{m-e}\) between magnons in FM and electrons in NM (described by the magnon temperature model10,11,12) and a magnon spin current associated with a magnon chemical potential gradient \(\nabla {\mu }_{m}\) in FM (described by the magnon drift-diffusion model13,14). Second, a thermal spin pumping from FM to NM occurs. Third, a spin current generates a transverse voltage via inverse spin Hall effect (ISHE) of NM. These three processes can be independently controlled so that the enhancement of SSE in each process is multiplied. Previous studies have focused mainly on the improvement of the second process by increasing the spin-mixing conductance at the NM/FM interface15,16,17 and the third one by employing NM with large spin Hall angle18,19. In addition, lowering the damping of FM20 for the first process or reducing the thermal conductivity of FM21 enhances the SSE efficiency by several factors.

In this work, we report an efficient way to enhance the SSE signal through the first process by reducing the exchange interaction \({J}_{{ex}}\) at the hotter region of FM. As described below, a reduction of \({J}_{{ex}}\) increases a magnon heat current, which in turn increases \({T}_{m-e}\) and thus the SSE signal. Moreover, as \(\nabla {\mu }_{m}\) contributes to a magnon spin current, the SSE can enhance by increasing \(\nabla {\mu }_{m}\) at a given temperature gradient. Following the Bloch’s \({T}^{3/2}\) law, the most important factor for the magnon chemical potential is \({J}_{{ex}}\) because the magnon density at a given temperature is determined by the Curie temperature. Therefore, a reduction of \({J}_{{ex}}\) at the hotter region of FM increases a magnon spin current, which also enhances the SSE signal.

In what follows, we discuss two different theoretical mechanisms to describe the SSE based on the magnon temperature model10,11,12 and the magnon drift-diffusion model13,14. Both mechanisms may coexist14 but we consider these separately for simplicity. We first describe the magnon temperature model10,11,12 for an insulator/FM1/FM2/NM structure where \({J}_{{ex}}\) differs between FM1 and FM2 (Fig. 1c). From the thermal circuit model equivalent to the layered structure of Fig. 1c and ignoring magnon relaxation, the SSE voltage \({V}_{{{{{{\rm{SSE}}}}}}}\) is found to be proportional to (Supplementary Note 1):

$${V}_{{{{{{\rm{SSE}}}}}}}\propto {T}_{m-e}=\frac{{Q}_{1}^{{{{{{\rm{FM}}}}}}}{R}_{1}^{{{{{{\rm{FM}}}}}}}+{Q}_{2}^{{{{{{\rm{FM}}}}}}}{R}_{2}^{{{{{{\rm{FM}}}}}}}}{{R}_{1}^{{{{{{\rm{FM}}}}}}}+{R}_{2}^{{{{{{\rm{FM}}}}}}}+{R}_{{{{{{\rm{F}}}}}}2{{{{{\rm{|F}}}}}}1}^{{{{{{\rm{int}}}}}}}+{R}_{{{{{{\rm{F}}}}}}1{{{{{\rm{|N}}}}}}}^{{{{{{\rm{int}}}}}}}}{R}_{{{{{{\rm{F}}}}}}1{{{{{\rm{|N}}}}}}}^{{{{{{\rm{int}}}}}}}$$

where \({Q}_{i}^{{{{{{\rm{FM}}}}}}}\) is the magnon heat current of FMi (i = 1, 2), \({R}_{i}^{{{{{{\rm{FM}}}}}}}\) is the magnon heat resistance of FMi, and \({R}_{{{{{{\rm{F}}}}}}2{{{{{\rm{|F}}}}}}1}^{{{{{{\rm{int}}}}}}}\) (\({R}_{{{{{{\rm{F}}}}}}1{{{{{\rm{|N}}}}}}}^{{{{{{\rm{int}}}}}}}\)) is the interface magnon heat resistance at the FM2/FM1 (FM1/NM) interface. Assuming no loss of magnon heat current at the FM2|FM1 interface (i.e., \({R}_{{{{{{\rm{F}}}}}}2{{{{{\rm{|F}}}}}}1}^{{{{{{\rm{int}}}}}}}\to 0\)) and \({R}_{1}^{{{{{{\rm{FM}}}}}}}+{R}_{2}^{{{{{{\rm{FM}}}}}}}\,\gg \,{R}_{{{{{{\rm{F}}}}}}1{{{{{\rm{|N}}}}}}}^{{{{{{\rm{int}}}}}}}\), Eq. (1) is simplified as

$${V}_{{{{{{\rm{SSE}}}}}}}\propto \frac{{Q}_{1}^{{{{{{\rm{FM}}}}}}}{R}_{1}^{{{{{{\rm{FM}}}}}}} \,+{Q}_{2}^{{{{{\rm{FM}}}}}}{R}_{2}^{{{{{{\rm{FM}}}}}}}}{{R}_{1}^{{{{{{\rm{FM}}}}}}}+{R}_{2}^{{{{{\rm{FM}}}}}}}{R}_{{{{{{\rm{F}}}}}}1{{{{{\rm{|N}}}}}}}^{{{{{{\rm{int}}}}}}}.$$

Using \({Q}_{i}^{{{{{{\rm{FM}}}}}}}={\kappa }_{i}\nabla T\) and \({R}_{i}^{{{{{{\rm{FM}}}}}}}={d}_{{Fi}}/({\kappa }_{i}A)\) with the magnonic heat conductivity \(\kappa\), the temperature gradient \(\nabla T\), the FM thickness \({d}_{F}\), and the area \(A\) of the structure, \({V}_{{{{{{\rm{SSE}}}}}}}\) is found to be proportional to \({\kappa }_{1}{\kappa }_{2}\left({d}_{F1}+{d}_{F2}\right)/({\kappa }_{1}{d}_{F2}+{\kappa }_{2}{d}_{F1})\). Given \({\kappa }_{i}\propto {\left({J}_{{ex},i}\right)}^{-1/2}\) [14] (Supplementary Table 1), the ratio \({r}_{{{{{{\rm{SSE}}}}}}1}\) of \({V}_{{{{{{\rm{SSE}}}}}}}\) with \({J}_{{ex},1}\,\ne \,{J}_{{ex},2}\) to \({V}_{{{{{{\rm{SSE}}}}}}}\) with \({J}_{{ex},1}={J}_{{ex},2}\) for the magnon temperature model is given as


Equation (3) shows that \({r}_{{SSE}1}\) is larger than 1 for \({J}_{{ex},2}\, < \,{J}_{{ex},1}\) (Fig. 1d). It is because a reduced \({J}_{{ex}}\) at the hotter region of FM (i.e., FM2) decreases \({R}_{2}^{{{{{{\rm{FM}}}}}}}\), which in turn increases \({V}_{{{{{{\rm{SSE}}}}}}}\).

We next describe the magnon drift-diffusion model13,14 for the same structure of Fig. 1c. Ignoring \({T}_{m-e}\) and solving magnon diffusion equations in FMs coupled with spin diffusion equation in NM gives \({V}_{{{{{{\rm{SSE}}}}}}}\) as (Supplementary Note 1).

$${V}_{{{{{{\rm{SSE}}}}}}}={\theta }_{{SH}}\nabla T\frac{2e{l}_{N}d\left[\left({d}_{F1}^{2}{l}_{F2}^{2}{L}_{1}+{d}_{F2}^{2}{l}_{F1}^{2}{L}_{2}\right){\sigma }_{F1}+2{d}_{F1}{d}_{F2}{l}_{F1}^{2}{L}_{1}{\sigma }_{F2}\right]}{{{\hslash }}{d}_{N}{l}_{F1}^{2}{l}_{F2}^{2}\sigma {\sigma }_{F1}}{{{{{\rm{csch}}}}}}\frac{{d}_{N}}{{l}_{N}}{{{\sinh }}}^{2}\frac{{d}_{N}}{2{l}_{N}},$$

where \({\theta }_{{SH}}\) is the spin Hall angle of NM, \({l}_{N}\) is the spin diffusion length of NM, \({l}_{{Fi}}\) is the magnon diffusion length of FMi, \(\sigma\) is the charge conductivity of NM, \({\sigma }_{{Fi}}\) is the magnon spin conductivity of FMi, \({L}_{i}\) is the spin Seebeck coefficient of FMi, \({d}_{N}\) is the NM thickness, and \(d\) is the electrode distance to measure \({V}_{{{{{{\rm{SSE}}}}}}}\). We obtain Eq. (4) with assumptions of continuous magnon chemical potential at the FM2/FM1 interface and \({d}_{F1(2)}\,\ll \,{l}_{F1(2)}\) to simplify \({V}_{{{{{{\rm{SSE}}}}}}}\) (Supplementary Note 1). Given \({\sigma }_{F}\propto L\propto {J}_{{ex}}^{-1/2}\) and \({l}_{F}\propto {J}_{{ex}}^{1/2}\) [14] (Supplementary Table 1), we obtain the ratio \({r}_{{{{{{\rm{SSE}}}}}}2}\) of \({V}_{{{{{{\rm{SSE}}}}}}}\) with \({J}_{{ex},1}\,\ne \,{J}_{{ex},2}\) to \({V}_{{{{{{\rm{SSE}}}}}}}\) with \({J}_{{ex},1}=\,{J}_{{ex},2}\) for the magnon drift-diffusion model as


Similar to \({r}_{{{{{{\rm{SSE}}}}}}1}\) [Eq. (3)], \({r}_{{{{{{\rm{SSE}}}}}}2}\) is larger than 1 for \({J}_{{ex}2}\, < \,{J}_{{ex}1}\) (Fig. 1d). It is because a reduced \({J}_{{ex}}\) at the hotter end of FM increases a gradient of magnon chemical potential \(\nabla {\mu }_{m}\), which in turn increases \({V}_{{{{{{\rm{SSE}}}}}}}\).

Since the above analytic theories are obtained with several crude approximations, we also carry out numerical simulations based on the stochastic Landau-Lifshitz-Gilbert equation for thermal spin pumping (Supplementary Note 2), following the procedure of Ref. 22. Numerical results (symbols in Fig. 1d) show a qualitatively similar trend with the analytic theories. All these results support our argument that a reduced \({J}_{{ex}}\) at the hotter region of FM (= FM2) enhances \({V}_{{{{{{\rm{SSE}}}}}}}\).

To experimentally demonstrate the enhancement of \({V}_{{{{{{\rm{SSE}}}}}}}\) by reducing \({J}_{{ex}}\) at the hotter region, we employ a W (4 nm)/CoFeB (2 nm)/AlOx (2 nm) wire device, in which a ZrO2 gate oxide and a Ru gate electrode are incorporated (Fig. 2a; see details in the Methods section). In this structure, we reduce \({J}_{{ex}}\) at the hotter region of CoFeB by utilizing gate voltage (VG)-induced oxygen migration, which is known to efficiently modulate magnetic properties of FM through redox reactions23,24. VG-induced oxygen migration allows us to control the oxidation state (i.e., magnetic properties) of FM in single sample and thus to avoid possible ambiguities caused by sample-to-sample variations.

Fig. 2: Gate voltage control of thermoelectric voltage.
figure 2

a Schematic illustration of a gate voltage (VG) modulation of transverse thermoelectric voltage (VTE) in a W/CoFeB/AlOx device with a ZrO2 gate oxide and a Ru gate electrode. b Normalized RH versus temperature in a W (4 nm)/CoFeB (1 nm)/AlOx (2 nm) structure for VG = +13 V (blue) and VG = −13 V (red). c VTE in a W (4 nm)/CoFeB (2 nm)/AlOx (2 nm) structure as a function of magnetic field (Bx) and (d), magnetic field angle (\({\varphi }_{{{{{{\rm{B}}}}}}}\)) at \({V}_{G}\) = +13 V (blue), −13 V (red), and 0 V (black). e Laser power dependence of \({\triangle V}_{{TE}}\). f VG effect on VTE in a Ti (3 nm)/CoFeB (2 nm)/AlOx (2 nm) structure for \({V}_{G}\) = +13 V (blue), −13 V (red), and 0 V (black). Here, all measurements were done at a laser power of 30 mW except for the experiment of ΔVTE versus laser power shown in (e).

To confirm that VG modulates \({J}_{{ex}}\) of CoFeB, we measure temperature-dependent anomalous Hall resistance (RH) of a W (4 nm)/CoFeB (1 nm)/AlOx (2 nm) sample with VG = ±13 V (equivalent to 3.25 MV/cm). Here VG is applied for 5 min at 100 °C, and the measurement is carried out with the gate floating. This is possible due to the non-volatile nature of the gate effect in our sample, which persists even after VG is turned off. As CoFeB (1 nm) has the in-plane magnetization, we apply an out-of-plane field of 9 T to measure RH. This field is much larger than the demagnetization field (~ 1 T) so that VG-induced change in the magnetic anisotropy does not affect RH versus temperature. Figure 2b shows that the normalized RH gradually decreases with temperature, and in particular the sample with VG = −13 V has a stronger temperature dependence. Since the magnetic moment of the CoFeB does not change at the measurement temperature up to 380 K (Supplementary Note 3), this result shows that the Curie temperature and \({J}_{{ex}}\) of CoFeB decrease with a negative voltage. Note that in our sign convention, oxygen ions are drifted toward (away from) the CoFeB/AlOx interface for a negative (positive) VG. Therefore, the result of Fig. 2b indicates that the oxidation reduces \({J}_{{ex}}\) of CoFeB near the CoFeB/AlOx interface.

We next measure a thermoelectric voltage (VTE) at various VG. In the sample, a vertical temperature gradient (\(\nabla {T}_{z}\)) is generated by irradiating a focused laser (laser power = 30 mW) with a wavelength of 660 nm, and then VTE is measured while sweeping a magnetic field (Bx) in the direction transverse with the voltage probe or rotating a magnetic field of 100 mT in the x-y plane (azimuthal angle \({\varphi }_{B}\)). Figure 2c shows the results; for the case of without applying VG (black symbols), VTE is positive for a positive Bx (i.e., M//+x where M is the CoFeB magnetization), and changes its sign when M is reversed. Defining ΔVTE = [VTE (M//+x) – VTE (M//−x)]/2, we find that ΔVTE is 0.7 μV for the case without VG but is largely modulated by applying VG; ΔVTE is 5.5 μV for VG = +13 V, which is almost eight times greater than that without VG. Furthermore, ΔVTE even changes the sign by a negative VG; ΔVTE = −4.3 μV for VG = −13 V.

The VG-induced modulation of ΔVTE is further confirmed by the angle-dependent VTE measurement that exhibits \({{\cos }}{\varphi }_{B}\)–dependence (Fig. 2d). This angular dependence is consistent with the symmetry of spin thermoelectrics5 where VTE maximizes when M is aligned in the x-direction perpendicular to both \(\nabla {T}_{z}\) and the voltage probes. We also measure ΔVTE as a function of laser power to confirm that ΔVTE originates from \(\nabla {T}_{z}\). Figure 2e shows that ΔVTE increases linearly with the laser power at both VG polarities, demonstrating that the VG-induced ΔVTE enhances by increasing \(\nabla {T}_{z}\) in the sample.

All the above results are consistent with spin thermoelectric voltages induced by \(\nabla {T}_{z}\), but VG-induced sign change of ΔVTE demands a further investigation. This sign change is caused by the fact that CoFeB is a metallic FM and thus not only SSE of W/CoFeB but also anomalous Nernst effect (ANE) of CoFeB itself contributes to ΔVTE. For a W/CoFeB bilayer where W has a negative spin Hall angle25, the ANE of CoFeB has the opposite sign to the SSE of W/CoFeB26. Then, an important question is whether the ANE or the SSE changes with VG. To address this question, we examine VG-induced ΔVTE in a Ti (3 nm)/CoFeB (2 nm)/AlOx (2 nm) sample. It is expected that the SSE contribution to ΔVTE is negligible in this sample because of a negligibly small spin Hall angle of Ti27. Figure 2f shows that the angle-dependent VTE’s of Ti/CoFeB sample at different VG’s are almost identical regardless of VG. Since VTE of Ti/CoFeB sample is predominantly determined by the ANE of CoFeB, this result shows that the effect of VG on ANE is negligible. Therefore, the VG-induced ΔVTE in the W/CoFeB sample is almost entirely governed by the VG-induced modulation of SSE and a negative VG increases the magnitude of the SSE since the sign of SSE is negative for the W/CoFeB sample. This VG-induced sign change of ΔVTE in the W/CoFeB sample evidences that the VG-induced modulation of SSE signal is sufficiently large to exceed the ANE signal that is almost independent of VG. We also investigate samples with other NM electrodes of Pt or Ta (Supplementary Note 4). Both samples show that the magnitude of the SSE increases at negative VG, consistent with the results of the W/CoFeB sample. Furthermore, the VG modulation efficiency is closely related to the magnitude and sign of the spin Hall angle of the NM layer.

The above results unambiguously show that the gate voltage modifies the SSE. We attribute the main mechanism of the gate effect to the oxygen ion migration and associated oxidation of CoFeB because it exhibits non-volatile nature and has a threshold voltage. Figure 3a shows that ΔVTE does not change significantly for |VG| < 5 V. This threshold behavior may be related to the energy barrier for oxygen migration23,24. On the other hand, when VG exceeds the threshold, ΔVTE increases with VG. This result demonstrates that ΔVTE can be continuously controlled by VG in a reversible manner.

Fig. 3: Effect of CoFeB oxidation on VTE in W/CoFeB/AlOx structure.
figure 3

a \({\triangle V}_{{TE}}\) as a function of \({V}_{G}\) in W (4 nm)/CoFeB (2 nm)/AlOx (2 nm) (laser power = 30 mW). b VTE as a function of magnetic field (Bx) in W (4 nm)/CoFeB (2 nm)/AlOx (1.5 nm) with varying plasma oxidation time from 0s to 150s. c ΔVTE as a function of plasma oxidation time tox with VG = 0 V (black), VG = +12 V (blue) and −12 V (red). d, e VTE as a function of magnetic field (Bx) in a W (4 nm)/CoFeB (2 nm)/Ti (1 nm)/AlOx (2 nm) structure (d) and a W (4 nm)/Ti (1 nm)/CoFeB (2 nm)/AlOx (2 nm) structure (e) with VG = +12 V (blue) and −12 V (red) (laser power = 30 mW).

To independently confirm that the observed gate effect originates from the oxidation of CoFeB, we directly oxidize the CoFeB layer by plasma oxidation. We fabricate W (4 nm)/CoFeB (2 nm)/AlOx (1.5 nm) samples where the oxidation state of CoFeB is modulated by varying the oxidation time (tox), which ranges from 0 to 150 s. Figure 3b shows that VTE strongly depends on tox; ΔVTE is positive for tox = 0 s, gradually decreases with tox, and becomes negative when tox exceeds 75 s. It indicates that the magnitude of SSE increases with tox, which is the same trend as applying a negative VG (Fig. 2). This result can be understood by the fact that a negative VG makes oxygen ions to migrate from oxide to CoFeB, leading to the oxidation of CoFeB. Note that the oxidation-induced modulation of ΔVTE is also seen in samples with thicker CoFeB (Supplementary Note 5).

To prove that the gate effect is equivalent to the CoFeB oxidation, we also investigate the effect of gate voltage (VG = ±12 V) on ΔVTE in the samples with different tox’s (Fig. 3c). When applying VG = −12 V (+12 V), ΔVTE becomes more negative (positive), i.e., SSE increases (decreases) for all samples regardless of tox. These results show that the gate effect is equivalent to the CoFeB oxidation. We also check whether the CoFeB oxidation near the CoFeB/AlOx interface (i.e., the hotter region) or the CoFeB/W interface (i.e., the colder region) is important for the SSE change. To check this, the interfaces are intentionally modified by introducing a Ti (1 nm) insertion layer. Figure 3d,e show that the VG effect disappears when the Ti layer is inserted at the CoFeB/AlOx interface whereas an apparent VG effect is still present when the Ti layer is inserted at the W/CoFeB interface. This result confirms that the oxidation state of CoFeB at the CoFeB/AlOx interface, i.e., the hotter region, is crucial for the SSE. Note that oxidation of the CoFeB layer can modulate the shunting effect and consequently alter the SSE and ANE effects. However, the change in shunting estimated from the resistance change by VG is only 3.2% (Supplementary Note 6), which is not sufficient to account for the magnitude and even sign modulation of the VTE shown in Fig. 2c.

In this work, we demonstrate that the SSE signal enhances by gate voltage that modulates the oxidation state of FM. Our theory describes the enhancement of SSE with a reduced exchange interaction at the hotter region of FM.

Possible consequences of the oxidation other than a reduced exchange may include an increased damping, a formation of an antiferromagnetic phase such as CoO and FeOx at the interface28, a reduced saturation magnetization, or a change of magnetic anisotropy. Therefore, we also check whether these changes can explain our experimental observation. Numerical simulations based on the stochastic LLG equation (Supplementary Note 2) show that the increased damping at the hotter region enhances the SSE signal by <50%, which is much weaker than the effect of reduced exchange. Simulations also show that a formation of an antiferromagnetic phase, a reduced saturation magnetization, or a change of magnetic anisotropy of oxidized FM lattices is unable to describe a largely enhanced SSE due to the oxidation. On the other hand, it should be noted that the reduced electronic and phononic heat conductivity in oxidized FM lattices can induce a larger thermal gradient compared to non-oxidized ferromagnet layers, since insulators typically have lower thermal conductivity than metals. Although the net SSE enhancement induced by oxidation can include a contribution from an increased thermal gradient in general, our analysis suggests that the exchange modulation is dominant for our experiment (Supplementary Note 7). Therefore, we conclude that the reduced exchange interaction at the hotter region is a main cause of the observed efficient VG-induced enhancement of SSE.

Although the magnitude of thermoelectric voltages in our device is still far from practical applications for energy harvesters, our work demonstrating a proof-of-concept for enhancing the SSE through the oxygen manipulation in NM/FM/oxide multilayers could pave an efficient way for further development for the SSE-based thermoelectric devices. As we explained in the beginning of this letter, the thermoelectric conversion via SSE consists of three separate processes and the enhancement of SSE in each process can be multiplied. Therefore, our approach, a reduced exchange interaction at the hotter region, can be combined with FM/NM interface engineering and strong spin-orbit-coupled NM to further enhance the SSE signal. Moreover, our approach is far more general than we demonstrate here. The reduced exchange at the hotter region can be realized in magnetic insulators that are widely used for the SSE studies2,15,16,17,18,19,20,21,29,30. It is not limited to single layer FM but is also applicable to multiple layers consisting of two or more FMs having different exchange interaction, yielding a wide variation of material combinations. Moreover, the enhanced SSE by the exchange engineering can be combined with thermoelectric effect induced by couplings among magnon, electron, and phonon systems to further improve the thermoelectric signal31,32,33,34,35,36. Optimization of such multilayers based on the concept we report here paves a way to realize practical applications based on the SSE and broadens the scope of material engineering for the SSE-based thermoelectric devices.


Sample preparation

Samples of W(or Ti)/CoFeB/AlOx structure were grown on thermally oxidized Si substrates by magnetron sputtering with a base pressure of 4.0 × 10−6 Pa. The metallic layers were deposited at a working Ar pressure of 0.4 Pa and a power of 30 W, and the AlOx layer was formed by plasma oxidation of an Al layer with an O2 pressure of 4 Pa and a power of 30 W. A bar-shaped device of 15 μm × 1000 μm was patterned by using photolithography and ion milling process. A gate oxide of ZrO2 (40 nm) was grown at 125 °C by plasma-enhanced atomic layer deposition using a TEMAZ [Tetrakis (ethylmethylamido) zirconium] precursor and O2. The gate electrode (15 μm × 20 μm) of Ru (20 nm) was defined at the centre of the bar-shaped device.

Thermoelectric measurement

Thermoelectric voltage was measured by illuminating a focused laser with a spot size of 5 μm and a wavelength of 660 nm that generated a vertical temperature gradient. During the measurement, the laser spot was positioned at the centre of the devices by monitoring the reflectance of the laser using a photodiode sensor. Prior to the measurement, a gate voltage was applied to the Ru gate electrode for 5 min at 100 °C with a ground connected to the W/CoFeB electrode. Thermoelectric measurements were carried out with the gate floating at room temperature, and each measurement was repeated more than 3 times.

Numerical simulation

We perform atomistic model simulations with stochastic Landau-Lifshitz-Gilbert (LLG) equation for one-dimensional system consisting of ten layers along the z-axis. The system is allowed to have inhomogeneous exchange (including antiferromagnetic exchange) and damping. The stochastic LLG equation for the unit magnetization \({\hat{{{{{{\bf{m}}}}}}}}_{i}\) of the ith layer is \(\frac{d{\hat{{{{{{\bf{m}}}}}}}}_{i}}{{dt}}=-\gamma {\hat{{{{{{\bf{m}}}}}}}}_{i}\times \left({{{{{{\bf{H}}}}}}}_{{{{{{\rm{eff}}}}}},i}{{{{{\boldsymbol{+}}}}}}{{{{{{\bf{H}}}}}}}_{{{{{{\rm{th}}}}}},i}\right)+{\alpha }_{i}{\hat{{{{{{\bf{m}}}}}}}}_{i}\times \frac{d{\hat{{{{{{\bf{m}}}}}}}}_{i}}{{dt}}\), where \(\gamma\) is the gyromagnetic ratio and \({\alpha }_{i}\) is the damping parameter of the ith layer. The effective field \({{{{{{\bf{H}}}}}}}_{{{{{{\rm{eff}}}}}},i}\left({{{{{\boldsymbol{=}}}}}}\frac{2{A}_{{ex},i}}{{M}_{s}}\frac{{\partial }^{2}{\hat{{{{{{\bf{m}}}}}}}}_{i}}{\partial {z}^{2}}{{{{{\boldsymbol{-}}}}}}\frac{2{K}_{h}}{{M}_{s}}{m}_{z}\hat{{{{{{\bf{z}}}}}}}{{{{{\boldsymbol{+}}}}}}{H}_{{{{{{\rm{ext}}}}}}}\hat{{{{{{\bf{x}}}}}}}\right)\) consists of the exchange field, easy-plane anisotropy field, and external magnetic field \({H}_{{{{{{\rm{ext}}}}}}}\), where \({A}_{{ex},i}\) is the exchange stiffness of the ith layer, \({M}_{s}\) is the saturation magnetization, \({K}_{h}\) is the easy-plane anisotropy energy. The thermal fluctuation field \({{{{{{\bf{H}}}}}}}_{{{{{{\rm{th}}}}}},i}\) obeys \(\langle {{{{{{\bf{H}}}}}}}_{{{{{{\rm{th}}}}}},i}(t)\rangle=0\) and \(\langle {{{{{{\bf{H}}}}}}}_{{{{{{\rm{th}}}}}},i}(t){{{{{{\bf{H}}}}}}}_{{{{{{\rm{th}}}}}},j}({t{{\hbox{'}}}})\rangle=\frac{2{k}_{B}{T}_{i}{\alpha }_{i}}{\gamma V{M}_{s}}{\delta }_{{ij}}\delta \left(t-{t}^{{\prime} }\right)\), where \({k}_{B}\) is the Boltzmann constant, \({T}_{i}\) is temperature of the ith layer and \(V\) is the volume.

Following Ref. 22, we calculate the SSE signal from \({j}_{{Tgrad}}^{s}-{j}_{{Tconst}}^{s}\) where \({j}_{{Tgrad}}^{s}\) is the spin pumping current proportional to the time average \(\langle {[{\hat{{{{{{\bf{m}}}}}}}}_{N}\times d{\hat{{{{{{\bf{m}}}}}}}}_{N}{{{{{\boldsymbol{/}}}}}}{dt}]}_{x}\rangle\) with \(N=10\) (i.e., the coldest layer) in the presence of temperature gradient whereas \({j}_{{Tconst}}^{s}\) is the spin pumping current at a constant temperature.