Observation of nonlinear thermoelectric effect in moge/y3fe5o12

ABSTRACT Thermoelectric effects refer to the voltage generation from temperature gradients in condensed matter. Although various power generators are made from them, all the known effects,

such as Seebeck effect, require macroscopic temperature gradients; since the sign of the generated voltage is reversed by reversing the temperature gradient, the net voltage disappears when

the temperature distribution fluctuates temporarily or spatially with a macroscopic temperature gradient of zero. It is impossible to utilize such temperature fluctuations in the

conventional thermoelectric effects, a situation which limits their application. Here we report the observation of a second-order nonlinear thermoelectric effect; we develop a method to

measure nonlinear thermoelectricity and observe that a superconducting MoGe film on Y3Fe5O12 generates a voltage proportional to the square of the applied temperature gradient. The nonlinear

thermoelectric generation demonstrated here provides a way for making power generators that produce electric power from temperature fluctuations. SIMILAR CONTENT BEING VIEWED BY OTHERS

BIPOLARITY OF LARGE ANOMALOUS NERNST EFFECT IN WEYL MAGNET-BASED ALLOY FILMS Article 08 January 2024 ROOM-TEMPERATURE NONLINEAR TRANSPORT AND MICROWAVE RECTIFICATION IN ANTIFERROMAGNETIC

MNBI2TE4 FILMS Article Open access 19 December 2024 MAGNETO-OPTICAL DESIGN OF ANOMALOUS NERNST THERMOPILE Article Open access 27 May 2021 INTRODUCTION Typical thermoelectric effects

established so far are Seebeck effect1 and Nernst effect2. Seebeck effect refers to the voltage generation along the temperature gradient, while Nernst effect refers to the voltage

generation normal to the gradient. In both, the amplitude of the generated voltage is proportional to that of the temperature gradient, and disappears in the absence of macroscopic

temperature gradients of a scale larger than the sample size. Here we report the observation of second-order nonlinear thermoelectric voltages, whose amplitude is proportional to the square

of the temperature gradient. To explore the nonlinear thermoelectric effect, a superconductor molybdenum-germanium (MoGe) on a magnet is a useful materials system3. It is a typical system

showing nonlinear resistance that can be controlled by using magnetic fields. In MoGe, a superconducting vortex-liquid state appears4,5 (see the phase diagram in Fig. 1d, determined6 based

on the resistivity measurement), in which a vortex, a string-shaped defect of superconductivity, can easily move by applying a current. The vortex motion driven by a current induces temporal

changes in the local phase of the superconducting wave function around the vortex core and gives rise to the electromotive force due to the phase change. Therefore, the vortex liquid phase

exhibits finite resistance in spite of its superconductivity7 (see Fig. 1c). In MoGe films on a ferrimagnetic insulator Y3Fe5O12 (YIG), the in-plane vortex resistance is externally nonlinear

with respect to the current when a magnetic field is applied in the in-plane direction3. The nonlinear resistance has been attributed to the asymmetry of magnetic fluctuations between the

top and bottom surfaces of the MoGe film (Fig. 1b), which causes different vortex nucleation rates8 (Bean-Livingston barriers) between the surfaces and gives rise to different resistance

depending on the vortex flow directions. The field-induced transition to the vortex liquid phase is so sharp that this nonlinear resistance can be turned on and off by changing the fields,

an advantage which realizes controllable experiments. By using this materials system, as shown in Fig. 1a, b, we investigate a nonlinear thermoelectric effect, where the thermoelectric

voltage due to the vortex Nernst effect9 becomes nonlinear with respect to the temperature difference. RESULTS SAMPLE AND MEASUREMENT SETUP The key for sensitively measuring a thermoelectric

effect is the use of higher-harmonic lock-in methods10,11. In conventional linear thermoelectric measurement, an a.c. current, \(I\sin \omega t\), is applied to a heater attached to an end

of the sample while the other end is connected to a heat bath. Since the Joule heating is proportional to the square of the applied current, the temperature difference ∆_T_ across the sample

is proportional to \({I}^{2}{\sin }^{2}\omega t\propto \cos 2\omega t+{{{{\rm{constant}}}}}\). Therefore, the linear thermoelectric voltage (\(\propto\) ∆_T_) temporally oscillates at 2_ω_,

which can be sensitively detected by a 2_ω_ lock-in method using the a.c. current as a reference. Then, how can we measure the nonlinear thermoelectric voltage proportional to (∆_T_)2

sensitively? One might think that a similar 4_ω_ lock-in method could be used, since the nonlinear thermoelectric voltage \([\propto {\left(\Delta T\right)}^{2}]\propto

{I}_{{{{{\rm{H}}}}}}^{4}\), where _I_H refers to the current applied to the heater and \(\Delta T\propto {I}_{{{{{\rm{H}}}}}}^{2}\). However, we found that the 4_ω_ lock-in method does not

work properly, because the 4_ω_ harmonic component is contaminated by the multiplication of linear thermoelectric responses (\(\propto {I}_{{{{{\rm{H}}}}}}^{2}\)); the order of the lock-in

detection should be carefully chosen for reliable measurement (see Methods for details). Here we have developed a second-order lock-in method to measure the nonlinear thermoelectric

voltages. First, two heaters are attached to both ends of the sample (Fig. 2d). Then, a.c. currents with the same frequency \(\omega /2\pi\) are applied to the heaters, superimposed with

d.c. currents. Owing to the d.c. currents, the heat generated at each heater contains a 1_ω_-oscillating component as an a.c.-d.c. cross term as well as a 2_ω_-oscillating component. The

values of the heater currents are tuned so that the same amount of heat is generated from the two heaters (see Methods for details). When the phase difference \(\phi\) between these heater

currents is zero, an identical amount of heat is generated at the two heaters all the time, and the temperature difference across the sample is kept zero. This condition can be confirmed by

the disappearance of the thermoelectric voltage at 1_ω_ and 2_ω_. After establishing the condition, the relative phase _ϕ_ between the heater currents is displaced from zero. When _ϕ_ = _π_,

the two heaters are heated alternately at 1_ω_ (the 2_ω_-oscillating temperature difference remains zero since 2_ϕ_ = 2_π_), and the temperature difference oscillating at 1_ω_ finally

appears across the sample without 2_ω_ components. The second-order nonlinear thermoelectric effect converts this 1_ω_ temperature difference into a 2_ω_-oscillating voltage, which can be

detected in terms of the 2_ω_ lock-in method by using the heater current as a reference. The expected _ϕ_ dependence of this 2_ω_ lock-in voltage V2_ω_ is (see Fig. 2a, b, and Supplementary

Note 1) $${V}_{2\omega }\propto \,\cos \phi {\sin }^{2}\left(\frac{\phi }{2}\right).$$ (1) This characteristic dependence provides a touchtone for discriminating nonlinear thermoelectric

effects. Figure 2c shows a schematic illustration of the sample system used in the present study. The sample is an amorphous MoGe film with the thickness 150 nm, sputtered on YIG. YIG

exhibits large magnetic susceptibility, resulting in a difference in the electromagnetic fluctuation amplitude between the top and bottom surfaces of the MoGe3. To apply ∆_T_ along the _y_

axis, the sample was sandwiched by two heaters (Fig. 2d). Following the above procedure, we measured the second harmonic lock-in voltage V2_ω_ along the \(x\) axis between the electrodes

(Fig. 2c, d) using the current \({I}_{{{{{\rm{H}}}}}1}\sin (\omega t+\phi )\) applied to the Heater-1 as a reference. We also applied the current \({I}_{{{{{\rm{H}}}}}2}\sin \omega t\) to

the Heater-2. Here \(\phi\) is the phase difference between the two heater currents. We define the normalized second harmonic voltage as \(\Delta {V}_{2\omega }={V}_{2\omega }\left(\phi

\right)-{V}_{2\omega }\left(\phi+\pi \right)\). OBSERVATION OF NONLINEAR THERMOELECTRIC EFFECT In Fig. 2e, we show the second harmonic voltage \(\Delta {V}_{2\omega }\) measured for the

YIG/MoGe at \(4.43\) K and \(\phi=\pi\) (blue solid curve), a condition where the \(1\omega\)-oscillating temperature difference across the sample is maximized. Importantly, a clear \(\Delta

{V}_{2\omega }\) signal appears at \(3\) T \( < {|B|} \, < \, 4\) T. This field range coincides with the appearance condition for the superconducting vortex-liquid phase in the MoGe

at the present temperature (see the resistivity \(\rho\) in Fig. 2e), signaling that the second-harmonic nonlinear signal \(\Delta {V}_{2\omega }\) appears only when vortices are mobile.

Figure 2f shows the \(\phi\) dependence of \({V}_{2\omega }\) measured with the fixed magnetic field \(B=-3.75\) T at \(T=4.34\) K. With the change in \(\phi\), \({V}_{2\omega }\) exhibits

complicated oscillation. The observed \(\phi\) dependence is in perfect agreement with the characteristic behavior, predicted for the nonlinear thermoelectric effect as Eq. (1) (see Fig. 2b,

f). By tuning the phase \(\phi\) to \(\phi=2\pi\), at which the temperature difference generated by the heaters is absent, the value of \({V}_{2\omega }\) decreases to \(\sim 0\), showing

that the observed \({V}_{2\omega }\) signal is caused by the temperature difference generated in the sample, not by the simple heating of the sample. The absence of the \({V}_{2\omega }\)

signal in a MoGe/SiO2 sample also shows that the influence of the sample heating is negligibly small (see Supplementary Note 3 for details). All these are evidence that the observed

\({V}_{2\omega }\) signal is the second-order nonlinear thermoelectric voltage caused by vortex motion. Note that the \({V}_{2\omega }\) picks up the second-order response with respect to

the temperature difference (see Supplementary Note 4 for details). The \({V}_{2\omega }\) signal disappears (gray dots in Fig. 2f) when the system temperature is higher than the transition

temperature of the superconductivity, \({T}_{{{{{\rm{c}}}}}}\), supporting the interpretation. We also confirmed that the spatial temperature inhomogeneity is small along the sample length

(see Supplementary Note 6 and 7 for details), and confirmed that the \({V}_{2\omega }\) signal appears at different values of \(\omega /2{{{{\rm{\pi }}}}}\) (see Supplementary Note 9 for

details). In Fig. 3a, we show the heater-current amplitude \({I}_{{{{{\rm{H}}}}}}\) dependence of \({V}_{2\omega }\) for the YIG/MoGe. By increasing \({I}_{{{{{\rm{H}}}}}}\) from \(0.35\) mA

to \(3.25\) mA, the \({V}_{2\omega }\) signal proportional to \(\cos \phi {\sin }^{2}\left(\phi /2\right)\) becomes more distinct with the peak amplitude \({A}_{2\omega }\). In Fig. 3b, we

plot the \({A}_{1\omega }\) dependence of \({A}_{2\omega }\), where \({A}_{1\omega }\) refers to the peak amplitude of the \(1\omega\) linear thermoelectric voltage, measured simultaneously

by using the lock-in method. The linear thermoelectric voltage is proportional to the temperature gradient applied to the sample. The data in Fig. 3b shows that \({A}_{2\omega }\) is

proportional to \({A}_{1\omega }^{2}\), demonstrating that the observed second harmonic voltage is proportional to the square of the applied temperature gradient. Figure 3c shows the

temperature \(T\) dependence of \(\Delta {V}_{2\omega }\) for the YIG/MoGe at \(\phi=\pi\). With the increase in \(T\) from \(3.54\) K, the \(\Delta {V}_{2\omega }\) peak shifts to lower

fields (blue solid curves in Fig. 3c). The field dependence of the resistivity \(\rho\) of the sample at each \(T\) is shown as the red solid curves in Fig. 3c, which shows that the \(\Delta

{V}_{2\omega }\) peaks appear concomitant with the sudden change in \(\rho\) signaling the appearance of the vortex liquid state at each temperature. The results support our interpretation

that the second harmonic voltage is due to the vortex motion in the vortex liquid phase in the MoGe/YIG. PHENOMENOLOGICAL MODEL FOR NONLINEAR VORTEX NERNST EFFECT We formulated the nonlinear

vortex Nernst effect in the MoGe/YIG in terms of a phenomenological model3. In the model, we assume that the vortex flow \({{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}(+\Delta T)\) in the \(-{{y}}\)

direction [\({{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}\left(-\Delta T\right)\) in the \(+{{{y}}}\) direction] due to the temperature difference \(\Delta T\) \([-\Delta T]\) is governed by the

nucleation process12,13 of vortex strings at the interfaces of the MoGe (Fig. 4a). The vortex nucleation rate \(\propto {e}^{-\Delta F/{k}_{{{{{\rm{B}}}}}}T}\) is determined by the

nucleation energy barrier \(\Delta F\) of a vortex at the interface14, where \({k}_{{{{{\rm{B}}}}}}\) is the Boltzmann constant. The energy barrier \({\Delta F}_{{{{{\rm{YIG}}}}}}\) at the

MoGe/YIG interface is less than that at the opposite interface, \({\Delta F}_{{{{{\rm{vac}}}}}}\), since the greater magnetic susceptibility in YIG reduces the magnetostatic energy of the

stray field created in the YIG by a nucleated vortex3 (see Supplementary Note 2). The asymmetry of the energy barrier, \({\Delta F}_{{{{{\rm{YIG}}}}}} \, < \, {\Delta

F}_{{{{{\rm{vac}}}}}}\), induces a difference in the vortex nucleation rate between the interfaces under \(\Delta T\), and leads to an asymmetric thermal vortex flow

\(|{{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}(+\Delta T)|\ \ne \ |{{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}\left(-\Delta T\right)|\) (Fig. 4a). The asymmetric vortex flow generates an electric field

according to \({{{{\bf{E}}}}}\propto {{{{\bf{B}}}}}\times {{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}\), giving rise to the vortex Nernst effect proportional to \({\left(\Delta T\right)}^{2}\): a

nonlinear vortex Nernst effect. We calculated the electric field due to the nonlinear vortex Nernst effect as (see Methods and Supplementary Note 2 for details)

$${E}_{{{{{\rm{N}}}}}2}\propto \frac{{S}_{\phi }}{\eta d}\left(\frac{\Delta {F}_{{{{{\rm{YIG}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}{e}^{-\frac{\Delta

{F}_{{{{{\rm{YIG}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}}-\frac{\Delta {F}_{{{{{\rm{vac}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}{e}^{-\frac{\Delta

{F}_{{{{{\rm{vac}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}}\right){\left(\Delta T\right)}^{2},$$ (2) where \({S}_{\phi }\), \(\eta\), and \(d\) are the entropy carried by a vortex, the vortex viscosity,

and the thickness of the MoGe film, respectively. In Fig. 4c, we show the \(B\) dependence of \({E}_{{{{{\rm{N}}}}}2}\) calculated from Eq. (2), which reproduces the experimental data (see

Fig. 4b). The measurement method presented here will allow the exploration of nonlinear thermoelectric effects in various candidate materials, such as bilayer conductors that exhibit

nonreciprocal resistance15 and polar ferromagnetic conductors16. The nonlinear thermoelectric effect can be used for making temperature-fluctuation sensors and voltage generators using

out-of-equilibrium temperature fluctuations. The effect also bridges a gap between thermoelectricity and nonlinear physics in a solid. It will certainly be important to find matter that

exhibits the effect at room temperature. METHODS SAMPLE PREPARATION A Y3Fe5O12 (YIG) film with the thickness \(\sim\)1 μm was grown on a Gd3Ga5O12 (GGG) (111) substrate with the thickness

520 μm by liquid phase epitaxy. An amorphous MoGe film with the thickness 150 nm was sputtered on the YIG by RF sputtering in the Argon atmosphere of 2.8 × 10−1 Pa. During the sputtering,

the substrate was rotated at 3000 rpm and kept at room temperature by water cooling17. The length and the width of the MoGe/YIG sample are 8.0 mm and 1.0 mm, respectively. MEASUREMENT SETUP

FOR NONLINEAR VORTEX NERNST EFFECT The measurement was performed by a standard lock-in technique10,11 using Physical Property Measurement System (Quantum Design, Inc.). When a temperature

difference, ∆_T_, is applied to a material, the generated thermoelectric voltage _V_ can be expanded as \(V={a}_{1{{{{\rm{st}}}}}}\Delta T+ {a}_{2{{{{\rm{nd}}}}}}{\left(\Delta

T\right)}^{2}+\cdots\), where \({a}_{n{{{{\rm{-th}}}}}}\ (n={{{\mathrm{1,\ 2}}}}, \cdots )\) is the \(n\)-th order thermoelectric coefficient. When the input temperature difference

oscillates at the frequency \(\omega /2\pi\), the _n_-th harmonic voltage detected with a lock-in amplifier is given by \({V}_{n{\omega }}\left(t\right)=\frac{1}{{t}_{0}}{\int

}_{t-{t}_{0}}^{t} \sin \left(n{\omega }{t}^{{\prime} }+ {\theta }_{0}\right)V\left(t^{\prime} \right){dt}^{\prime},\) where \({t}_{0}\) is a sufficiently large averaging time and \({\theta

}_{0}\) is a set phase for the amplifier. In conventional linear thermoelectric measurement, a temperature difference is generated via the Joule heating induced by the a.c. current

\(\propto\) sin _ωt_ applied to a heater attached to an end of the sample, \(\Delta T\propto {\sin }^{2}\omega t\), and the \(2\omega\) lock-in voltage is measured. Similarly, one might

think that a nonlinear thermoelectric effect \(\propto {\left(\Delta T\right)}^{2}\) can be measured via the 4_ω_ lock-in measurement. However, the 4_ω_ lock-in signal may include not only

the second harmonic response but also the third harmonic one which may appear even without inversion symmetry breaking. To address this issue, the order of the lock-in detection should be

lowered, which is achieved if the input temperature difference oscillates in a first-harmonic manner, \(\Delta T\propto \sin \omega t\). To create \(\Delta T\propto \sin \omega t\) and

detect the second-order nonlinear thermoelectric voltage selectively, two heaters (Heater-1 and Heater-2) were attached to the MoGe and GGG surfaces, respectively (Fig. 2d). The currents

\({I}_{{{{{\rm{H}}}}}1}\sin (\omega t+\phi )+{I}_{{{{{\rm{DC}}}}}1}\) and \({I}_{{{{{\rm{H}}}}}2}\sin \omega t+{I}_{{{{{\rm{DC}}}}}2}\) were applied to the Heater-1 and the Heater-2,

respectively. Due to the Joule heating, ∆_T_ is generated in the MoGe film along the thickness direction (_y_ direction in Fig. 2c, d): $$\Delta T= \, {R}_{1}{\left[{I}_{{{{{\rm{H}}}}}1}\sin

(\omega t+\phi )+{I}_{{{{{\rm{DC}}}}}1}\right]}^{2}-{{R}_{2}\left({I}_{{{{{\rm{H}}}}}2}\sin \omega t+{I}_{{{{{\rm{DC}}}}}2}\right)}^{2}\\= \,

{\widetilde{I}}_{{{{{\rm{DC}}}}}1}^{2}-{\widetilde{I}}_{{{{{\rm{DC}}}}}2}^{2}+2{\widetilde{I}}_{{{{{\rm{DC}}}}}1}{\widetilde{I}}_{{{{{\rm{H}}}}}1}\sin (\omega t+\phi

)-2{\widetilde{I}}_{{{{{\rm{DC}}}}}2}{\widetilde{I}}_{{{{{\rm{H}}}}}2}\sin \omega t \\ + {\widetilde{I}}_{{{{{\rm{H}}}}}1}^{2}{\sin }^{2}\left(\omega t+ \phi

\right)-{\widetilde{I}}_{{{{{\rm{H}}}}}2}^{2}{\sin }^{2}\omega t.$$ (3) Owing to the a.c. and d.c. currents, the resultant temperature difference contains the 1\(\omega\)-oscillating

components as the a.c.-d.c. cross terms. Here, \({R}_{1}\) (\({R}_{2}\)) represents the prefactor relating the Joule heating of the Heater-1 (Heater-2) to \(\Delta T\), and

\({\widetilde{I}}_{{{{{\rm{H}}}}}1,{{{{\rm{DC}}}}}1}\) (\({\widetilde{I}}_{{{{{\rm{H}}}}}2,{{{{\rm{DC}}}}}2}\)) is defined as

\({{\widetilde{I}}^{2}}_{{{{{\rm{H}}}}}1,{{{{\rm{DC}}}}}1}={R}_{1}{I}_{{{{{\rm{H}}}}}1,{{{{\rm{DC}}}}}1}^{2}\)

(\({{\widetilde{I}}^{2}}_{{{{{\rm{H}}}}}2,{{{{\rm{DC}}}}}2}={R}_{2}{I}_{{{{{\rm{H}}}}}2,{{{{\rm{DC}}}}}2}^{2}\)). Before the measurement of the nonlinear vortex Nernst voltage, the values of

\({I}_{{{{{\rm{DC}}}}}1}\), \({I}_{{{{{\rm{DC}}}}}2}\), \({I}_{{{{{\rm{H}}}}}1}\), and \({I}_{{{{{\rm{H}}}}}2}\) were tuned so that the same amount of heat is generated from the two heaters

when \(\phi=0\), which is realized under \({\widetilde{I}}_{{{{{\rm{DC}}}}}1}={\widetilde{I}}_{{{{{\rm{DC}}}}}2}={\widetilde{I}}_{{{{{\rm{DC}}}}}}\) and

\({\widetilde{I}}_{{{{{\rm{H}}}}}1}={\widetilde{I}}_{{{{{\rm{H}}}}}2}={\widetilde{I}}_{{{{{\rm{H}}}}}}\) (see Supplementary Note 1 for the detailed procedure). When the relative phase

\(\phi\) between the heater currents is changed from \(0\) to \(\pi\), the two heaters are heated alternately at \(1\omega\), and the temperature difference oscillating at \(1\omega\) is

produced across the sample without \(2\omega\) components: \(\Delta T\propto \sin \omega t\). The validity of this method is confirmed by measuring the generated temperature gradient with

film-thermometers attached to the sample or by measuring the Nernst effect for a reference Permalloy sample (see Supplementary Notes 7 and 8 for details). The nonlinear vortex Nernst effect

converts this \(1\omega\) temperature gradient into a \(2\omega\)-oscillating transverse voltage \(\propto {\left(\nabla T\right)}^{2}\), and the 2\(\omega\) lock-in voltage \({V}_{2\omega

}^{2{{{{\rm{nd}}}}}}\) should exhibit a characteristic \(\phi\) dependence: $${V}_{2\omega }^{2{{{{\rm{nd}}}}}}\propto {a}_{2{{{{\rm{nd}}}}}}\cos \phi {\sin }^{2}\left(\frac{\phi

}{2}\right),$$ (4) which is different from the \(2\omega\) “linear” vortex Nernst voltage \({V}_{2\omega }^{1{{{{\rm{st}}}}}}\) arising from the fifth and sixth terms in Eq. (3):

$${V}_{2\omega }^{1{{{{\rm{st}}}}}}\propto {a}_{1{{{{\rm{st}}}}}}{\sin }^{2}\phi .$$ (5) This shows that the nonlinear thermoelectric voltage can be distinguished from the conventional

linear thermoelectric voltage in terms of the \(\phi\) dependence of \({V}_{2\omega }\). In the experiments, the second harmonic lock-in voltage \({V}_{2\omega }\) in the MoGe in the \(x\)

direction is measured with a lock-in amplifier (LI5640, NF Corporation) by using the Heater-1 current \({I}_{{{{{\rm{H}}}}}1}\sin \left(\omega t+\phi \right)\) as a reference. All the

\({\Delta V}_{2\omega }\)-\(B\) and \({V}_{2\omega }\)-\(\phi\) data were anti-symmetrized with respect to the magnetic field \(B\). PHENOMENOLOGICAL MODEL FOR NONLINEAR VORTEX NERNST EFFECT

We consider a superconducting vortex system in a MoGe film attached to YIG. In the vortex liquid phase, vortex strings can move freely in the MoGe under a driving force. For simplicity, we

assume that the vortex flow is determined by the nucleation process of vortex strings at the MoGe surfaces3. The nucleation rate of a vortex is given by \(P\propto {e}^{-\Delta

F/{k}_{{{{{\rm{B}}}}}}T}\), where \(\Delta F\), \({k}_{{{{{\rm{B}}}}}}\), and \(T\) are the nucleation energy barrier of a vortex at the interface14, the Boltzmann constant, and temperature.

Since \(\Delta F\) is affected by the magnetic susceptibility in the vicinity of the MoGe surfaces and YIG exhibits the larger susceptibility than that of vacuum, the nucleation energy

barrier \({\Delta F}_{{{{{\rm{YIG}}}}}}\) at the MoGe/YIG interface is less than that at the opposite MoGe/vacuum interface, \({\Delta F}_{{{{{\rm{vac}}}}}}\), which results in the

difference in \(P\) between the top and bottom MoGe surfaces. In the presence of the temperature difference \(\Delta T\) (see Fig. 4a), vortex strings feel a thermal force,

\({{{{{\bf{f}}}}}}_{{{{{\rm{th}}}}}}={S}_{\phi }\Delta T/d(-{{{{{\bf{e}}}}}}_{y})\), where \({S}_{\phi }\), \(d\), and \((-{{{{{\bf{e}}}}}}_{y})\) are the entropy carried by a vortex, the

thickness of the MoGe film, and a unit vector along the \(y\) axis, respectively. This thermal force accelerates a vortex string to a speed \({{{{\bf{v}}}}}\), and the vortex string feels a

frictional force, \({{{{{\bf{f}}}}}}_{{{{{\rm{f}}}}}}=-\eta {{{{\bf{v}}}}}{{{{\boldsymbol{,}}}}}\) where \(\eta\) is the viscosity. In a steady state, the thermal force is in balance with

the frictional force, \({{{{{\bf{f}}}}}}_{{{{{\rm{th}}}}}}+{{{{{\bf{f}}}}}}_{{{{{\rm{f}}}}}}=0\), which leads to the terminal velocity of the vortex string

\({{{{{\bf{v}}}}}}_{{{{{\rm{v}}}}}}={S}_{\phi }\Delta T/\eta d(-{{{{{\bf{e}}}}}}_{y})\). When the temperature at the MoGe surface in contact with vacuum (YIG) is \(T+\Delta T/2\) (\(T-\Delta

T/2\)), the vortex flow \({{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}(\Delta T)\) in the \(-y\) direction becomes finite. The amplitude \({J}_{{{{{\rm{v}}}}}}\left(\Delta T\right)\) of

\({{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}(\Delta T)\) is described as $${J}_{{{{{\rm{v}}}}}}\left(\Delta T\right)\propto \left[{P}_{{{{{\rm{vac}}}}}}\left(T+\frac{\Delta

T}{2}\right)+{P}_{{{{{\rm{YIG}}}}}}\left(T-\frac{\Delta T}{2}\right)\right]\frac{{S}_{\phi }}{\eta }\frac{\Delta T}{d}.$$ (6) Here, \({P}_{{{{{\rm{vac}}}}}}(T)\propto {e}^{-{\Delta

F}_{{{{{\rm{vac}}}}}}/{k}_{{{{{\rm{B}}}}}}T}\) [\({P}_{{{{{\rm{YIG}}}}}}(T)\propto {e}^{-{\Delta F}_{{{{{\rm{YIG}}}}}}/{k}_{{{{{\rm{B}}}}}}T}\)] is the vortex nucleation rate at the

MoGe/vacuum [MoGe/YIG] interface. The vortex flow generates an electric field according to \({{{{\bf{E}}}}}\propto {{{{\bf{B}}}}}\times {{{{{\bf{J}}}}}}_{{{{{\rm{v}}}}}}\), and we calculate

the electric field due to the nonlinear vortex Nernst effect \(\propto {\left(\Delta T\right)}^{2}\) as $${E}_{{{{{\rm{N}}}}}2}\propto {\left.\frac{{\left(\Delta

T\right)}^{2}}{2}\frac{{\partial }^{2}}{\partial \Delta {T}^{2}}{J}_{{{{{\rm{v}}}}}}\left(\Delta T\right)\right|}_{\Delta T\to 0}.$$ (7) By substituting Eq. (6) into Eq. (7),

\({E}_{{{{{\rm{N}}}}}2}\) is expressed as (see Supplementary Note 2 for details) $${E}_{{{{{\rm{N}}}}}2}\propto \frac{{S}_{\phi }}{\eta d}\left(\frac{\Delta

{F}_{{{{{\rm{YIG}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}{e}^{-\frac{\Delta {F}_{{{{{\rm{YIG}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}}-\frac{\Delta {F}_{{{{{\rm{vac}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}{e}^{-\frac{\Delta

{F}_{{{{{\rm{vac}}}}}}}{{k}_{{{{{\rm{B}}}}}}T}}\right){\left(\Delta T\right)}^{2}.$$ (8) DATA AVAILABILITY The data that support the findings of this study are available from the

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Rectification Project” (No. JPMJER1402) from JST, Japan, CREST (Nos. JPMJCR20C1 and JPMJCR20T2) from JST, Japan, Grant-in-Aid for Scientific Research (S) (No. JP19H05600) from JSPS KAKENHI,

Japan, Grant-in-Aid for Scientific Research (B) (No. JP20H02599) from JSPS KAKENHI, Japan, Grant-in-Aid for Transformative Research Areas (No. JP22H05114) from JSPS KAKENHI, Japan,

Grant-in-Aid for JSPS Fellows (No. JP20J21622) from JSPS KAKENHI, Japan, Murata Science Foundation, and Sumitomo Chemical. H.A. was supported by GP-Spin at Tohoku University. AUTHOR

INFORMATION AUTHORS AND AFFILIATIONS * Department of Applied Physics, The University of Tokyo, Tokyo, 113-8656, Japan Hiroki Arisawa, Yuto Fujimoto, Takashi Kikkawa & Eiji Saitoh * RIKEN

Center for Emergent Matter Science, Wako, 351-0198, Japan Hiroki Arisawa & Eiji Saitoh * Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan Hiroki Arisawa *

Institute for AI and Beyond, The University of Tokyo, Tokyo, 113-8656, Japan Eiji Saitoh * WPI, Advanced Institute for Materials Research, Tohoku University, Sendai, 980-8577, Japan Eiji

Saitoh Authors * Hiroki Arisawa View author publications You can also search for this author inPubMed Google Scholar * Yuto Fujimoto View author publications You can also search for this

author inPubMed Google Scholar * Takashi Kikkawa View author publications You can also search for this author inPubMed Google Scholar * Eiji Saitoh View author publications You can also

search for this author inPubMed Google Scholar CONTRIBUTIONS H.A. and Y.F. contributed equally to this work. H.A. and Y.F. prepared the samples and carried out the experiments with help from

T.K. H.A. analyzed the data with help from Y.F. and T.K. H.A. and Y.F. formulated the theoretical model. E.S. planned and supervised the research. H.A., Y.F., and E.S. prepared the

manuscript. All the authors discussed the results and commented on the manuscript. CORRESPONDING AUTHOR Correspondence to Eiji Saitoh. ETHICS DECLARATIONS COMPETING INTERESTS The authors

declare no competing interests. PEER REVIEW PEER REVIEW INFORMATION _Nature Communications_ thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer

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http://creativecommons.org/licenses/by/4.0/. Reprints and permissions ABOUT THIS ARTICLE CITE THIS ARTICLE Arisawa, H., Fujimoto, Y., Kikkawa, T. _et al._ Observation of nonlinear

thermoelectric effect in MoGe/Y3Fe5O12. _Nat Commun_ 15, 6912 (2024). https://doi.org/10.1038/s41467-024-50115-4 Download citation * Received: 29 March 2024 * Accepted: 01 July 2024 *

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