Introduction

The reciprocal relations of thermodynamics are a fundamental tool to analyze and understand the physics of transport phenomena1. Since the beginning of the 19th century it was clear to Jean Charles Athanase Peltier and later demonstrated by Lord Kelvin, that for a material at a given absolute temperature T a relation exists between the Seebeck coefficient ε (given by the ratio between the measured electromotive force and the applied temperature difference) and the Peltier coefficient ∏ (the ratio between the measured heat current and the applied electric current): ∏ = εT 2,3. This remarkable reciprocity was later found to be part of a wider set of relations, as theoretically demonstrated by Onsager under the assumption of the reversibility of the microscopic physical processes governing macroscopic non-equilibrium thermodynamic effects4.

Reciprocal relations can also be used to analyze transport phenomena which involve not only the electric charge and the heat, but also the spin. Spincaloritronic phenomena5,6,7 can provide additional tools to the field of spintronics, envisioned to be a faster and lower energy consuming alternative to classical electronics8. One of the key building blocks for “spintronic circuits” is the spin battery, a device which can drive a spin current into an external circuit. Spin batteries are fundamental for spintronic devices and may be developed exploiting spincaloritronic effects9. Spincaloritronic devices may also be used in the development of novel thermoelectric heaters/coolers operating at the microscale10,11. The idea of a reciprocity between heat and spin was initially proposed and proven for metals where the spin current is carried by electrons12, but such a reciprocal relation cannot be easily proven in the case of ferrimagnetic insulators where the spin current is carried by thermally excited spin waves13. A typical device, where spincaloritronic effects are found and can be exploited for experiments, is a bilayer made of a ferrimagnetic insulator (e.g. yttrium iron garnet, YIG) and a non magnetic metal with a strong spin-orbit coupling (e.g. platinum, Pt)14. In these devices the spin current is generated longitudinally (along the x axis), normal to the film surface.

In the case of devices which exhibit the spin Peltier effect (SPE) a longitudinal (x axis) heat current is generated, caused by the flow of a transverse (y axis) electric current in the Pt layer13. Conversely, in the case of the spin Seebeck effect (SSE) a transverse (y axis) electric voltage is generated in the Pt layer and caused by the longitudinal temperature gradient parallel to the spin current (x axis)15. Although experimental evidence of both effects has been already obtained16,17, the quantitative demonstration of their reciprocity and the connection with intrinsic properties of the layers, has yet to be proven13,18,19,20. To this end here we provide the first experimental evidence of the reciprocal relation between the thermal and the electric quantities associated to the SPE and the SSE in a bulk YIG/Pt bilayer. The relation for a YIG/Pt bilayer has the following form21,22

$$\frac{-\,{\rm{\Delta }}{T}_{{\rm{SP}},{\rm{x}}}}{{I}_{{\rm{e}},{\rm{y}}}}=\frac{{\rm{\Delta }}{V}_{{\rm{e}},{\rm{y}}}}{{I}_{{\rm{q}},{\rm{x}}}}T$$
(1)

The left hand side of Eq. (1) refers to the SPE: ΔTSP,x is the temperature difference generated across the YIG layer as consequence of the electric current Ie,y flowing in the Pt film. The right hand side of Eq. (1) refers to the SSE: ΔVe,y is the voltage drop across the Pt film caused by the heat current Iq,x flowing across the device and T is the average temperature of the YIG. The experiments are performed measuring the SPE and the SSE on the same YIG/Pt device. The experimental value which represents, within the uncertainty, both the SPE and the SSE response of the specific device is (6.2 ± 0.4) × 10−3 KA−1.

Results

Device geometry and measurement principle

The geometry of the bulk YIG/Pt bilayer is shown in Fig. 1a.

Figure 1
figure 1

(a) Geometry of the YIG/Pt bilayer with tYIG = 0.545 mm, Ly = 4.95 mm, Lz = 3.91 mm and tPt = 5 nm. (b,c) schemes of the device in the SPE and SSE configurations. Heat currents and magnetic moment currents are along x (longitudinal direction), electric effects are along y (transverse direction), the magnetic field and the magnetization are along z. (d,e) sketches of the experimental setup for SPE and SSE, respectively. (f) Experimental result of the SPE heat current, −Iq,SP , as function of the electric current Ie,y at positive magnetic saturation. (g) Experimental result of the SSE voltage ΔVe,y as function of the heat current Iq,x at positive magnetic saturation.

Full size image

The device is composed of a bulk YIG parallelepiped with a thin film of Pt sputter deposited on one side. The temperature gradient ∇xT and the heat current density jq,x are directed along the x axis. The electric voltage ΔVe,y and the electric current density je,y are directed along the y axis. The magnetic moment current density, jM, is along the x direction and transports magnetic moments directed along the z direction. The magnetic field H and the magnetization M of the bulk-YIG are also directed along the z axis (Fig. 1b,c). The measurement setup for both SPE and SSE is shown in the sketches of Fig. 1c,e. The YIG/Pt device is sandwiched between two thermal reservoirs (held at Th and Tc respectively) in order to form a closed thermal circuit. The temperature of the two reservoirs can be externally controlled and the two heat currents between the device and each of the reservoirs are measured simultaneously by sensitive heat flux detectors. The heat flux technique is chosen to avoid measurement uncertainties due to the hardly reproducible thermal contacts23,24,25,26. In order to minimize heat leakages in the thermal circuit, the whole setup is operated in vacuum. Technical, constructional and measurement details are reported in the Methods section.

Spin Peltier effect

In the SPE, an electric current Ie,y flowing in the Pt layer generates a magnetic moment current jM along the x direction as a result of the spin Hall effect27. The adjacent ferrimagnetic bulk YIG acts as a passive component and shows a longitudinal (x axis) heat current associated to the magnetic moment current injection28,29,30. The measured heat current is however also including the Joule heat contribution generated by the electric current flowing in the Pt layer. In order to separate the Joule and spin Peltier contributions, their intrinsic differences have to be exploited: the spin Peltier signal increases linearly with the Ie,y current and changes sign when the magnetization (along the z axis) or the Ie,y are inverted (odd parity), while Joule heating is proportional to \({I}_{{\rm{e}},{\rm{y}}}^{2}\) and does not change sign under an inversion (even parity)13,18.

The thermal problem of the SPE can be represented by the equivalent circuit of Fig. 2 (see also Supplementary material).

Figure 2
figure 2

Equivalent thermal circuit of the SPE measurements of the YIG/Pt bilayer. The YIG layer is represented by the SPE generator, ΔTSP , and by the thermal resistance \({ {\mathcal R} }_{{\rm{YIG}}}\). The Pt layer has a Joule heat current source, Iq,JH, and a thermal resistance \({ {\mathcal R} }_{{\rm{Pt}}}\). The circuit includes two thermal contact resistances \({ {\mathcal R} }_{\mathrm{cont},{\rm{c}}}\) and \({ {\mathcal R} }_{\mathrm{cont},{\rm{h}}}\) taking into account both the thermal resistance of the contacts and the presence of the heat flux sensors. The difference Iq,h − Iq,c = Iq,JH provides the Joule heat. In isothermal conditions, with Th = Tc = T, we have \({\rm{\Delta }}{T}_{{\rm{SP}}}= {\mathcal R} {I}_{{\rm{q}},{\rm{s}}}+{I}_{{\rm{q}},\mathrm{JH}}({ {\mathcal R} }_{{\rm{h}}}-{ {\mathcal R} }_{{\rm{c}}})/2\) where \({ {\mathcal R} }_{{\rm{h}}}={ {\mathcal R} }_{{\rm{Pt}}}/2+{ {\mathcal R} }_{{\rm{YIG}}}+{ {\mathcal R} }_{\mathrm{cont},{\rm{h}}}\), \({ {\mathcal R} }_{{\rm{c}}}={ {\mathcal R} }_{\mathrm{cont},{\rm{c}}}+{ {\mathcal R} }_{{\rm{Pt}}}/2\), Iq,s = (Iq,h + Iq,c)/2 is the half sum and \( {\mathcal R} ={ {\mathcal R} }_{{\rm{h}}}+{ {\mathcal R} }_{{\rm{c}}}\) is the total resistance (see Supplementary material).

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In adiabatic conditions the SPE corresponds to the direct measurement of ΔTSP , the temperature difference generated between the two faces of the bulk YIG sample. Previous experimental works have succeeded to extract ΔTSP out of the Joule heat component by using an AC technique13,18. Here, to test the reciprocal relation in a stationary state, we employ a DC technique in which we set isothermal conditions at the thermal baths, Th = Tc = T, and measure simultaneously the two heat currents: Iq,c and Iq,h. The difference of the heat flux signals Iq,h − Iq,c = Iq,JH provides the Joule heat only (see Fig. 3a), while the half sum Iq,s = (Iq,h + Iq,c)/2 contains the SPE signal (see Fig. 3b,c).

Figure 3
figure 3

Heat current signals measured on the YIG/Pt bilayer during the SPE experiment. (a) Joule heat signal given by the difference Iq,JH = Iq,h − Iq,c. (b) SPE by magnetic field inversion. Half sum signal Iq,s = (Iq,h + Iq,c)/2 caused by a rectangular waveform of the magnetic field |μ0Hs| = 50 mT, for two steady values of electric current (±40 mA, orange/purple). (c) SPE by electric current inversion. Iq,s caused by a rectangular waveform of the electric current |Ie,y| = 40 mA, for two steady values of magnetic field (±50 mT, orange/purple). In both cases the spin Peltier signal Iq,SP is obtained as half of the variation at the inversion Iq,SP = ΔIq,s/2. The variation ΔIq,s is taken at the inversion instant and is computed from the extrapolation of the linear fit taken a few seconds after the inversion. This method permits to avoid the contributions of spurious induced voltage spikes after the inversions. The values reported in Fig. 1f) are the result of the average of 10 inversions.

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We first detect the SPE generated by setting a constant electric current (i.e. Ie,y = 40 mA, see Fig. 3b, orange points) and periodically inverting the magnetic field. The half sum signal Iq,s has a change of ±2Iq,SP at each inversion. Equivalently, when the sign of the electric current in the Pt film is changed (i.e. Ie,y = −40 mA, Fig. 3b, purple points), a sign inversion of the change \(\mp \)2Iq,SP occurs. The field inversion allows to detect the small contribution of the spin Peltier heat current (a few μW) superimposed to the Joule heating background (a few mW).

As a second step we measure the SPE signal Iq,s at the constant magnetic field μ0Hs = +50 mT while the applied current Ie,y = ±40 mA is periodically inverted, Fig. 3c, orange points. Conversely when we apply μ0Hs = −50 mT and the applied current Ie,y = ±40 mA is inverted, we obtain the curve of Fig. 3c, purple points. The current inversion method provides the same results, within the uncertainty, as the magnetic field inversion one, provided one takes into account the presence of small spurious offset signals as discussed in the Methods section. This second method allows to detect the SPE signal as function of the applied magnetic field. Therefore it permits the determination of the hysteresis loop of YIG31 (more details about this experiment are reported in Supplementary materials).

At the saturating magnetic field μ0Hs = +50 mT, by applying different values of Ie,y and by deriving the corresponding values of −Iq,SP as shown in Fig. 3b,c, we are able to obtain the linear relation between the SPE heat current and the electric current data of Fig. 1f. By a linear fit we find

$$\frac{-\,{I}_{{\rm{q}},{\rm{SP}}}}{{I}_{{\rm{e}},{\rm{y}}}}=\mathrm{(5.1}\pm \mathrm{0.3)}\times {10}^{-5}\,{{\rm{WA}}}^{-1}$$
(2)

The thermal resistance \( {\mathcal R} \) of the whole stack consisting of sensors, sample and additional thermal contacts (e.g. thermal paste or thermally conducting layers) is experimentally measured by setting a heat current value and measuring the temperatures of the two thermal reservoirs by two thermocouples. The result is \( {\mathcal R} =\mathrm{(119}\pm \mathrm{2)}\) KW−1. With \({\rm{\Delta }}{T}_{{\rm{SP}}}= {\mathcal R} {I}_{{\rm{q}},\mathrm{SP}}\), the measured spin Peltier coefficient is

$$\frac{-\,{\rm{\Delta }}{T}_{{\rm{SP}}}}{{I}_{{\rm{e}},{\rm{y}}}}=\mathrm{(6.1}\pm \mathrm{0.4)}\times {10}^{-3}\,{{\rm{KA}}}^{-1}$$
(3)

and since the current density flowing in the Pt film is je,y = Ie,y/(tPtLz) with Lz = 3.9 mm and tPt = 5 nm we are finally able to obtain the intrinsic SPE coefficient:

$$\frac{-\,{\rm{\Delta }}{T}_{{\rm{SP}}}}{{j}_{{\rm{e}},{\rm{y}}}}=\mathrm{(1.19}\pm \mathrm{0.08)}\times {10}^{-13}\,{{\rm{Km}}}^{2}{{\rm{A}}}^{-1}.$$
(4)

Spin Seebeck effect

In the SSE experiment the bulk YIG is the active layer which generates a magnetic moment current when subjected to a temperature gradient, while the Pt is the passive layer in which the injected magnetic moment current is converted into a transverse electric potential. The SSE signal measured across the Pt film ΔVe,y changes sign when the magnetization of the YIG is inverted in sign (odd parity).

The measurement of the SSE is performed by setting a value of the heat current Iq,x traversing the YIG/Pt device and measuring the consequent voltage ΔVe,y found on the Pt layer when the YIG layer is at magnetic saturation. The set of values obtained ΔVe,y versus the heat current Iq,x is shown in Fig. 1f and a linear fit gives

$$\frac{{\rm{\Delta }}{V}_{{\rm{e}}}}{{I}_{q,x}}=\mathrm{(2.1}\pm \mathrm{0.1)}\times {10}^{-5}\,{{\rm{VW}}}^{-1}$$
(5)

A geometry independent (intrinsic) spin Seebeck coefficient can be defined as ∇yVe/jq,x with ∇yVe = ΔVe/Le,y and jq,x = Iq,x/Aq with Aq = Le,y × Lz. With Le,y = 4.17 mm, the dimension of the Pt electrode used to detect the voltage drop, and Lz = 3.91 mm we have

$$\frac{{\nabla }_{{\rm{y}}}{V}_{{\rm{e}}}}{{j}_{q,x}}=\mathrm{(8.2}\pm \mathrm{0.3)}\times {10}^{-8}\,{{\rm{VmW}}}^{-1}$$
(6)

Finally the spin Seebeck coefficient SSSE = ∇yVe/∇xT, given by the ratio between the transverse gradient of the electric potential ∇yVe in Pt and the longitudinal gradient of the temperature ∇xT in YIG (as it is often defined in literature) can be obtained

$${S}_{{\rm{SSE}}}=-\,\frac{{\nabla }_{{\rm{y}}}{V}_{{\rm{e}}}}{{j}_{q,x}}{\kappa }_{{\rm{YIG}}}=-\,\mathrm{(5.4}\pm \mathrm{0.2)}\times {10}^{-7}\,{{\rm{VK}}}^{-1}\,$$
(7)

using the bulk value of the thermal conductivity of YIG: κYIG = 6.63 Wm−1 K−1 32. This result concurs with another experimental bulk YIG \(|{S}_{{\rm{SSE}}}|\simeq 4\times {10}^{-7}\) VK−1 33.

Discussion

The microscopic and physical origin of the spin Peltier and of the spin Seebeck effects has been investigated in detail26,28,29,30,34,35,36,37,38,39,40,41,42,43. These two effects are the results of two independent physical mechanisms. In YIG the presence of a spin current or, more generally, of a magnetic moment current, carried by thermally excited spin waves, is accompanied by a heat current. In Pt the longitudinal spin polarized current is associated with a transverse electric effect due to the inverse spin Hall effect, described by the spin Hall angle θSH. At the interface between YIG and Pt the spin current is partially injected from one layer into the other44,45. By adopting the thermodynamic description of Johnson and Silsbee46 further developed in refs21,22,28,29,30, the thermomagnetic effects in YIG are described by means of the thermomagnetic power coefficient εYIG, that has an analogous role to the thermoelectric power coefficient ε of thermoelectrics. At the interface, the passage of the magnetic moment current, due to diffusion, is mainly determined by the magnetic moment conductances per unit surface area, vM, of the two layers. Basing on these ideas, it has been possible to work out the reciprocal relation relating the spin Seebeck and spin Peltier effects with the intrinsic and extrinsic parameters of the bilayer. The expression is21,22

$$\frac{-\,{\rm{\Delta }}{T}_{{\rm{SP}}}}{{j}_{{\rm{e}}}}\frac{1}{T}=\frac{{\nabla }_{{\rm{y}}}{V}_{{\rm{e}}}}{{j}_{q,x}}{t}_{{\rm{Pt}}}={\theta }_{{\rm{SH}}}{\mu }_{0}(\frac{{\mu }_{{\rm{B}}}}{e})\frac{1}{{v}_{{\rm{p}}}}\frac{{\varepsilon }_{{\rm{YIG}}}{\sigma }_{{\rm{YIG}}}}{{\kappa }_{{\rm{YIG}}}}$$
(8)

Eq. (8) contains two equal signs. The equal sign at the left is between the SPE and SSE measured quantities and we find the temperature difference between the two faces of YIG, ΔTSP, of the SPE generated by the electric current density je,y and the transverse gradient of the electric potential in Pt, ∇yVe, of the SSE, generated by the heat current density, jq,x, in YIG. The equal sign at the right refers to the relation of both SPE and SSE to intrinsic coefficients. In addition to the expected parameters: θSH, the spin Hall angle of Pt and εYIG, the thermomagnetic power coefficient of YIG, we have: σYIG, the magnetic moment conductivity of YIG, vp, the magnetic moment conductance per unit surface area of the YIG/Pt interface and κYIG, the thermal conductivity of the YIG. μ0 is the magnetic constant, μB is the Bohr magneton and e is the elementary charge. vp depends on the intrinsic conductances, vM of both YIG and Pt and on the ratio t/lM between the thickness t and the magnetic moment diffusion length lM, for each layer. The expression of vp is derived in ref.22 and reported in the Supplementary material.

By just taking the first equal sign of Eq. (8) and integrating over the size of the bilayer device we have

$$\frac{-\,{\rm{\Delta }}{T}_{{\rm{SP}}}}{{I}_{e,y}/({L}_{{\rm{z}}}\cdot {t}_{{\rm{Pt}}})}\frac{1}{T}=\frac{{\rm{\Delta }}{V}_{e,y}/{L}_{q,y}}{{I}_{q,x}/({L}_{q,y}\cdot {L}_{{\rm{z}}})}{t}_{{\rm{Pt}}}$$
(9)

At the left hand side (Lz ⋅ tPt) is the area where the electric current flows. At the right hand side (Lq,y ⋅ Lz) is the area of the thermal contact and corresponds to the region where the spin Seebeck effect rises. The previous relation can be simplified, leading to Eq. (1) and permitting a direct test of the reciprocity by using the experimental values. By taking the average temperature T = (298 ± 2) K, the spin Seebeck experiment gives

$$\frac{{\rm{\Delta }}{V}_{e,y}}{{I}_{q,x}}T=\mathrm{(6.3}\pm \mathrm{0.3)}\times {10}^{-3}\,{{\rm{KA}}}^{-1}\,$$
(10)

which is in excellent agreement with the spin Peltier value of Eq. (3) and verifies experimentally the reciprocal relation, Eq. (1), between spin Seebeck and spin Peltier effects.

Once the reciprocity is verified, we can take the second equal sign of Eq. (8) and obtain an experimental value that can be compared with the theoretical coefficients. By labeling the value taken from equation (8) as KYIG/Pt we find from the experiments

$${K}_{\mathrm{YIG}/\mathrm{Pt}}=\mathrm{(4.0}\pm \mathrm{0.3)}\times {10}^{-16}\,{{\rm{m}}}^{2}{{\rm{A}}}^{-1}\,$$
(11)

We now compare the obtained value with the coefficients known from the literature. For what concerns the spin Hall angle we choose the absolute value of 0.1, similar to what was found experimentally47,48 on samples of YIG with Pt layers of comparable thickness49. In this work we use the spin Hall angle θSH for the current of magnetic moments which has opposite sign with respect to the one for spin currents reported in the literature (i.e. we have negative θSH for Pt and positive for W and Ta)21,22. By using μ0σYIG = vYIGlYIG and employing the values of the coefficients determined in previous works21 (\({\varepsilon }_{{\rm{YIG}}}\simeq -\,{10}^{-2}\) TK−1 and θSH = −0.1) we can quantify the only missing parameter, vp/vYIG, as \({v}_{{\rm{p}}}/{v}_{{\rm{YIG}}}\simeq 9\). This value is compatible with the transmission of the magnetic moment current between the two layers as determined by the intrinsic diffusion lengths lM, by the thicknesses t and by the intrinsic conductances vM of YIG and Pt. By using the expression for vp22 with \({l}_{{\rm{Pt}}}\simeq 7.3\) nm and \({l}_{{\rm{YIG}}} \sim 0.4\,\mu {\rm{m}}\) and with \({v}_{{\rm{YIG}}} \sim {v}_{{\rm{Pt}}}\) we obtain vp/vYIG = 8 which is reasonably close to the measured value.

We are therefore able to evaluate the cooling potentiality of the spin Peltier effect. If we consider the YIG/Pt device able to operate in adiabatic conditions (Iq,c = 0), the temperature change across the device ΔT will be

$${\rm{\Delta }}T={\rm{\Delta }}{T}_{{\rm{SP}}}-{ {\mathcal R} }_{{\rm{YIG}}}{I}_{q,\mathrm{JH}}$$
(12)

where we have assumed \({ {\mathcal R} }_{{\rm{Pt}}}\ll { {\mathcal R} }_{{\rm{YIG}}}\) and \( {\mathcal R} \simeq { {\mathcal R} }_{{\rm{YIG}}}\). By taking the specific device studied in this paper, the electric current which is maximizing ΔT is Ie,y = −12.4 μA giving a maximum temperature change of ΔT = 3.8 × 10−8 K. This value appears so small to discourage any attempt to employ the SPE in practice. However the verification of the validity of the thermodynamic theory for the SPE and SSE (Eq. (8)) offers, in future perspective, the possibility to design and optimize spin Peltier coolers and spin Seebeck generators going beyond the specific bilayer sample used in this experimental study. Work is in progress along this line, however two main preliminary comments are already possible at this stage. The first is to identify the YIG and Pt thicknesses that would optimize the effects. The answer comes from the fact that both materials are active over a thickness of the order of the diffusion length lM. Therefore promising devices would have \({t}_{{\rm{YIG}}} \sim {l}_{{\rm{YIG}}}\) and \({t}_{{\rm{Pt}}} \sim {l}_{{\rm{Pt}}}\). The second is that, as for thermoelectrics, the thermomagnetic YIG is characterized by a figure of merit \({\zeta }_{{\rm{T}}}={\varepsilon }_{{\rm{YIG}}}^{2}{\sigma }_{{\rm{YIG}}}T/{\kappa }_{{\rm{YIG}}}\), which is indeed very small at room temperature, \({\zeta }_{{\rm{T}}}\simeq 4\times {10}^{-3}\) 22. However it is expected that the ζT parameter could improve in the temperature range between 50 and 100 K where experiments50 have reported a much larger SSE (almost a factor 5) than the room temperature value. Finally it is worth to mention that both improvements could further benefit by cascading several devices in thermal series51. For example, with \({\zeta }_{{\rm{T}}} \sim 1\) and using a multilayer with an appropriate compensation of the Joule heat of each layer by using variable cross sections, one could obtain up to an effective ΔT = 20 K for a device containing ~103 junctions. Work along this line has already progressed and future improvements are expected19.

In summary, we investigated experimentally both the SPE and the SSE in a bulk YIG/Pt device. The thermal observables of these experiments are investigated by means of heat current measurements. This introduces a novel technique for the SPE characterization of a given sample in the DC regime. The experimental results of the SPE and SSE are used to verify for the first time the reciprocal relation between the two. The relation between both effects with the intrinsic and extrinsic parameters of YIG and Pt bilayer offers the possibilities for a more in-depth investigation of the applicability of spincaloritronic devices.

Methods

Device and experimental setup

The YIG/Pt device employed is made of a bulk yttrium iron garnet (YIG) single crystal prepared by crystal growth in high-temperature solutions applying the slow cooling method52. Single crystals which nucleate spontaneously at the crucible bottom and grow to several centimetres sizes have been separated from the solution by pouring out the residual liquid. Afterwards, one crystal was cut parallel to one of its facets to prepare slices and parallelepipeds of the following dimensions: Ly = 4.95 mm, Lz = 3.91 mm, tYIG = 0.545 mm. Carefully grinding and polishing result in a sample with optical smooth surfaces (Rq = 0.4 nm obtained by AFM). After polishing both sides, a thin film of Pt (tPt about 5 nm) was sputtered on the top of one of the Ly × Lz surfaces at University of Loughborough (United Kingdom). Subsequently two 100 nm thick gold electrode strips for electric contacts were deposited. The inner distance between the electrodes is equal to Le,y = 4.17 mm. The Au contacts on the Pt layer are electrically connected to 40 μm diameter platinum leads by silver paste. The SSE voltage is measured by means of a Keithley 2182 nanovoltmeter while the SPE electric current is generated by a Keithley 2601 source meter.

In order to avoid heat leakages all measurements are performed under vacuum (1.6 × 10−4 mbar) by means of a turbomolecular pump. The heat current sensors are miniaturized Peltier cells (5 × 5 mm and 1.9 mm of thickness, RMT Ltd model 1MD04-031-08TEG), calibrated according to the procedure described in ref.24. The characteristic Iq = −SpVp of both heat sensors are described by Sp = 0.97 ± 0.01 V/W, where Iq is the heat current traversing the sensor and Vp is the voltage measured by a nanovoltmeter. The thermal reservoirs are two brass parallelepipeds (1 × 1 × 2 cm) to which one face of each heat sensor is glued with silver paste. The other face of each sensor is clamping the YIG/Pt device. We use an aluminum nitride slab (3 × 3 × 4 mm) with a large nominal thermal conductivity (140–180 Wm−1 K−1) as geometrical adapter to thermally connect the Pt side of the device with the corresponding heat sensor. Thin layers of silicon based thermal grease are used to ensure uniform thermal conductivity through the sections.

Offset subtractions during SPE and SSE measurements

In the case of the electric current inversion method, the spin Peltier heat signals is affected by a spurious heat current offset, Iq,off, (a few percent of the total) that is not present in the magnetic field inversion method and should be therefore subtracted. The reason for this spurious heat current offset is that a conventional Peltier effect arises in presence of different metal contacts in the measurement setup (i.e. the contact between Pt and Au layers and the contacts with the electrical leads). These contacts would give a transverse (along y) heat flux, however the presence of an even very small transverse heat leakage may contribute to a small longitudinal contribution. This spurious conventional Peltier effect Iq,off presents, with respect to the current inversion, the same odd parity as the spin Peltier signal. When using the odd parity of the SPE under magnetic field inversion the spurious effect is cancelled. Conversely, when determining the SPE through the electric current inversion ±Ie,y method, it is summed up. The offset is subtracted by using one measurement with magnetic field inversion method at saturation as a reference. Also the spin Seebeck experiment is affected by a spurious voltage measured at Pt which is the reciprocal of the one found in the spin Peltier with current inversion. A very small transverse leaking heat flux may gives rise to electric effects in the nV range caused by the ordinary Seebeck effect due to the electric contacts between different metals. Again, this offset voltage is eliminated by using one magnetic field inversion point at saturation as a reference.