Research ArticlePHYSICS

The spin Nernst effect in tungsten

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Science Advances  03 Nov 2017:
Vol. 3, no. 11, e1701503
DOI: 10.1126/sciadv.1701503

Abstract

The spin Hall effect allows the generation of spin current when charge current is passed along materials with large spin-orbit coupling. It has been recently predicted that heat current in a nonmagnetic metal can be converted into spin current via a process referred to as the spin Nernst effect. We report the observation of the spin Nernst effect in W. In W/CoFeB/MgO heterostructures, we find changes in the longitudinal and transverse voltages with magnetic field when temperature gradient is applied across the film. The field dependence of the voltage resembles that of the spin Hall magnetoresistance. A comparison of the temperature gradient–induced voltage and the spin Hall magnetoresistance allows direct estimation of the spin Nernst angle. We find the spin Nernst angle of W to be similar in magnitude but opposite in sign to its spin Hall angle. Under an open-circuit condition, this sign difference results in the spin current generation larger than otherwise. These results highlight the distinct characteristics of the spin Nernst and spin Hall effects, providing pathways to explore materials with unique band structures that may generate large spin current with high efficiency.

INTRODUCTION

The giant spin Hall effect (SHE) (1) in heavy metals (HMs) with large spin-orbit coupling has attracted great interest owing to its potential use as a spin current source to manipulate magnetization of magnetic layers (24). Recently, it has been reported (5, 6) that the spin Hall conductivity of 5d transition metals depends on the number of 5d electrons, indicating that the observed SHE is due to the topology and filling of the characteristic bands at the Fermi surface (7, 8). Spin current in solids can be produced not only by charge current but also by heat current (9). Understanding the coupling between spin current and heat current is the central subject of spin caloritronics (10). It is now well understood that a temperature gradient applied across a magnetic material, typically a magnetic insulator, results in spin accumulation that can be used to generate spin current in neighboring nonmagnetic materials via the spin Seebeck effect (1117).

It has been predicted theoretically (1823) that in nonmagnetic materials with strong spin-orbit coupling, the heat current can be converted into spin current. The effect, often referred to as the spin Nernst effect, generates spin current that scales with the energy derivative of the spin Hall conductivity. Here, we show direct probe of the spin Nernst effect in amorphous-like W, which has the largest spin Hall angle among the 5d transition metals (6, 24, 25). When an in-plane temperature gradient is applied across W/CoFeB/MgO heterostructures, we observe longitudinal and transverse voltages that vary with the magnetic field similar to those of the spin Hall magnetoresistance (SMR) (2631). The W layer thickness dependence of the longitudinal voltage is compared to that of the SMR to estimate the size and sign of the spin Nernst angle. We find that the spin Nernst angle of W is slightly smaller (~70%) than its spin Hall angle and the two angles have opposite signs.

RESULTS

The film structure used is sub.|dN HM|1 FM|2 MgO|1 Ta (thickness in nanometers), where HM is Ta or W and the ferromagnetic metal (FM) is Co20Fe60B20 (hereafter referred to as CoFeB). We first study the electrical transport properties of the films. The inverse of the device longitudinal resistance (1/RXX) multiplied by a geometrical factor (L/w), the sheet conductance GXX = L/(wRXX), is plotted as a function of the HM layer thickness (dN) in Fig. 1 (A and B) for the Ta and W underlayer films, respectively (see inset of Fig. 1A for the definitions of L and w as well as the coordinate system). We fit the data with a linear function to estimate the resistivity (ρN) of the HM layer. The fitting results are shown by the blue solid lines: We obtain ρN of ~183 and ~130 μΩ·cm for Ta and W, respectively. Note that W undergoes a structural phase transition (6, 24, 32) when its thickness is larger than ~6 nm, as indicated by the change in GXX at this thickness. The SMR, RSMR = ΔRXX/RXXZ, is plotted as a function of the HM layer thickness in Fig. 1 (C and D). We define ΔRXX = RXXYRXXZ as the resistance difference when the magnetization of the CoFeB layer is pointing along the y direction (RXXY) and z direction (RXXZ). The thickness dependence of RSMR is consistent with previous reports (6, 31).

Fig. 1 Longitudinal resistance and SMR of HM|CoFeB|MgO heterostructures.

(A and B) Sheet conductance GXX = L/(wRXX) versus HM layer thickness dN for the Ta (A) and W (B) underlayer films. The solid lines show a linear fit to the data in an appropriate range of dN. Schematic of the measurement setup is illustrated in the inset of (A). The inset of (B) is the expanded y-axis plot of the main panel. (C and D) SMR (RSMR = ΔRXX/RXXZ) plotted against dN for the Ta (C) and W (D) underlayer films. The red solid lines are fit to the data using Eq. 1. Parameters used in the fitting are summarized in Table 1.

The transverse resistance of the films is shown in Fig. 2 (see inset of Fig. 1A for details of the measurement setup). The inset of Fig. 2A shows the transverse resistance (RXY) versus the out-of-plane field HZ for a Ta underlayer film. We define 2ΔRXY = RXYZRXYZ , that is, the anomalous Hall resistance, as the difference in RXY when the magnetization is pointing along +z and −z. In Fig. 2 (A and B), ΔRXY is plotted as a function of HM layer thickness. |ΔRXY| decreases with increasing dN largely due to current shunting into the HM layer. To estimate the anomalous Hall angle, ΔRXY is divided by RXXZ, multiplied by a geometric factor (L/w), and divided by a constant (xF) that accounts for the current shunting effect into the HM layer: Embedded Image, where tF and ρF are the thickness and resistivity of the FM layer, respectively. The HM layer thickness dependence of the normalized anomalous Hall coefficient RAHE/xF = (ΔRXYL)/(RXXZwxF) is plotted in Fig. 2 (C and D) for the Ta and W underlayer films, respectively. We find that the normalized anomalous Hall coefficient shows a significant HM layer thickness dependence, particularly for the W underlayer films.

Fig. 2 HM layer thickness dependence of the anomalous Hall resistance.

(A and B) HM layer thickness (dN) dependence of the anomalous Hall resistance (ΔRXY) for the Ta (A) and W (B) underlayer films. The inset of (A) shows RXY versus HZ for sub.|~1.1 Ta|1 CoFeB|2 MgO|1 Ta (thickness in nanometers). The definition of ΔRXY is schematically illustrated. (C and D) Normalized anomalous Hall coefficient RAHE/xF = (ΔRXYL)/(wRXXZxF) plotted against dN for the Ta (C) and W (D) underlayer films. The solid lines show the calculated RAHE/xF using Eq. 2 with three different values of Im[GMIX]. Parameters used in the calculations are summarized in Table 1, except for Im[GMIX], which is noted in the legend.

We next show the thermoelectric properties of the films. Figure 3A shows a sketch of the setup to study the Seebeck coefficient of the films. A heater is placed near one side of the substrate to create a temperature gradient across the substrate. The difference in the temperature between the hot (TH) and cold (TL) sides of the substrate (ΔT = THTL), across a distance D, is measured using an infrared camera. The longitudinal (Seebeck) voltage VXX = V(x1) − V(x2) is measured between two points of the device separated by a distance L = x2x1 < D. The temperature of position x1 is higher than that of x2 (see Fig. 3A). The ΔT dependence of VXX is shown in Fig. 3 (B and C) for the Ta and W underlayer films, respectively. The data are fitted with a linear function to extract the Seebeck coefficient S ~ −(VXX/L)/(ΔT/D) (33) from the slope, which is plotted as a function of dN in Fig. 3 (D and E). S approaches ~−4 μV/K when the HM layer thickness is thin for both film structures, which we consider provides information of the Seebeck coefficient of CoFeB (we assume that the MgO and the oxidized Ta capping layers have negligible contribution to VXX). In contrast, the thick limit of dN gives the Seebeck coefficient of the HM layer: We estimate S of ~−2 and ~−12 μV/K for Ta and W, respectively.

Fig. 3 Seebeck coefficient of HM|CoFeB|MgO heterostructures.

(A) Schematic illustration of the measurement setup for temperature gradient–induced longitudinal voltage. The bright square represents part of the substrate, and the dark region indicates the area where the device is located. D = 0.7 cm; L = 0.6 cm; w = 50 μm. (B and C) Longitudinal (Seebeck) voltage VXX measured as a function of the temperature difference ΔT for sub.|~7.0 Ta|1 CoFeB|2 MgO|1 Ta (B) and sub.|~5.6 W|1 CoFeB|2 MgO|1 Ta (C). The horizontal and vertical error bars represent the uncertainty of the temperature gradient and the variation of the voltage under a fixed temperature gradient, respectively. (D and E) Seebeck coefficient S ~ −(VXX/L)/(ΔT/D) plotted against dN for the Ta (D) and W (E) underlayer films. The error bars denote the variation of S due to the uncertainty of the temperature gradient. The horizontal dashed lines are guides to the eye, which provide an estimate of the Seebeck coefficient of the HM and FM layers.

The off-diagonal component of the thermoelectric properties is summarized in Fig. 4. The experimental setup to study the temperature gradient–induced transverse voltage is depicted in Fig. 4A. A typical hysteresis loop obtained by measuring the HZ dependence of the temperature gradient–induced transverse voltage VXY = VXY(y2) − VXY(y1) (see Fig. 4A for the definitions of y1 and y2) is shown in Fig. 4B. Similar to the anomalous Hall resistance, we define 2ΔVXY, that is, the anomalous Nernst voltage, as the difference in VXY when the magnetization is pointing along +z and −z. Figure 4C shows the ΔT dependence of ΔVXY for a W underlayer film. Within the applied temperature gradient, the response is linear. We thus fit a linear function to obtain the anomalous Nernst coefficient SANE = (ΔVXY/L)/(ΔT/D) from the slope (here, L = y2y1).

Fig. 4 HM layer thickness dependence of the anomalous Nernst effect.

(A) Schematic illustration of the measurement setup for temperature gradient–induced transverse voltage. The bright square represents part of the substrate, and the dark region indicates the area where the device is located. D = 0.7 cm; L = 0.6 cm; w = 50 μm. (B) Transverse voltage VXY versus HZ of sub.|~3.4 W|1 CoFeB|2 MgO|1 Ta when a temperature difference of ΔT ~ 2.5 K is applied. The definition of ΔVXY is schematically drawn by the blue arrow. (C) ΔT dependence of the anomalous Nernst voltage ΔVXY for the same sample described in (B). The horizontal and vertical error bars represent the uncertainty of the temperature gradient and the variation of the voltage under a fixed temperature gradient, respectively. The red solid line shows a linear fit to the data. (D and E) Anomalous Nernst coefficient SANE = (ΔVXY/L)/(ΔT/D) plotted against dN for the Ta (D) and W (E) underlayer films. (F and G) dN dependence of the normalized anomalous Nernst coefficient SANE/xF = (ΔVXYD)/(LΔTxF) for the Ta (F) and W (G) underlayer films. The error bars in (D) to (G) denote the variation of quantities due to the uncertainty of the temperature gradient. The solid lines in (F) and (G) show the calculated SANE/xF using Eq. 6 with three different values of θSN. Parameters used in the calculations are summarized in Table 1.

The HM layer thickness dependence of anomalous Nernst coefficient SANE is plotted for the Ta and W underlayer films in Fig. 4 (D and E, respectively). |SANE| decreases with increasing dN for the Ta underlayer films, whereas it shows a peak at around dN ~ 3 nm for the W underlayer films. Similar to the anomalous Hall resistance, the presence of the HM layer can shunt the Hall voltage. To account for this effect, SANE is divided by xF. The normalized anomalous Nernst coefficient SANE/xF is plotted as a function of dN in Fig. 4 (F and G). We find a larger variation of SANE/xF with dN for the W underlayer films than that for the Ta underlayer films.

Recent studies have shown that the spin current generated within the HM layer modifies the anomalous Hall resistance via a nonzero imaginary part of the spin-mixing conductance at the HM/FM interface (26, 27, 34). The large variation of the normalized anomalous Nernst coefficient with dN for the W underlayer films indicates that a temperature gradient can cause the spin current generation in the W layer that results in the modification of the off-diagonal component. To evaluate the temperature gradient–induced spin current generation (due to the spin Nernst effect) in a more explicit way, we studied the external field dependence of the Seebeck voltage in analogy to the SMR. The experimental setup is the same with that of Fig. 3A: Here, a large external magnetic field is applied during the measurements.

In Fig. 5 (A and B), we show the longitudinal (Seebeck) voltage VXX = V(x1) − V(x2) of Ta and W underlayer films, respectively, plotted as a function of the external field directed along the y axis (HY). The temperature difference ΔT across the substrate is ~3.5 K. For the W underlayer films (Fig. 5B), we find a peak-like structure around zero field (signals are shifted vertically for clarity so that the large field limit of VXX is equal to zero). The peak found in the VXX versus HY plot decays to zero when |HY| ~ |HK|, where HK is the effective anisotropy field required to force the magnetization to point along the film plane (see fig. S1 for the magnetic properties of the heterostructures). The peak amplitude ΔVXX = VXXYVXXZ defined schematically in Fig. 5B is equivalent to the difference in VXX when the magnetization is pointing along the y axis (VXXY) and the z axis (VXXZ). This definition is in accordance to that of SMR. We have also studied VXX as a function of HX and HZ (the results are shown in fig. S2). In contrast to VXX versus HY, we find no clear feature in the HX and HZ dependence of VXX. These results suggest that the thermal analog of the anisotropic magnetoresistance (AMR) is small in CoFeB (35). Note that the AMR of the CoFeB layer here is ~0.1% (31), much smaller than that of the Ni-based soft magnetic materials (36). The small temperature gradient–induced AMR-like voltage (VXX versus HX; see fig. S2) found here also indicates that the possible contribution from the combination of AMR and interfacial spin-orbit coupling (37, 38) on ΔVXX may be small. We also find little evidence of proximity-induced magnetism (3941) in W and Ta, which may influence the temperature gradient–induced voltage via AMR in the HM layer.

Fig. 5 Signatures of the spin Nernst magnetoresistance.

(A and B) Longitudinal (Seebeck) voltage VXX versus HY for sub.|~3.1 Ta|1 CoFeB|2 MgO|1 Ta (A) and sub.|~3.4 W|1 CoFeB|2 MgO|1 Ta (B) when a temperature difference ΔT of ~3.5 K is applied. The definition of ΔVXX is schematically drawn. (C) dN dependence of spin Nernst coefficient SSNE = (ΔVXX/L)/(ΔT/D) (solid circles) and the scaled SMR SRSMR (open circles) for the W underlayer films. The solid lines show the calculated SSNE using Eq. 5 with three different values of θSN. Parameters used in the calculations are summarized in Table 1. (D) dN dependence of θSNSH obtained from SSNE/(SRSMR) and the relation described in Eq. 7. The error bars in (C) and (D) denote the variation of quantities due to the uncertainty of the temperature gradient.

In Fig. 5C, we plot SSNE = (ΔVXX/L)/(ΔT/D), which we refer to as the spin Nernst coefficient, as a function of the W layer thickness. |SSNE| takes a maximum at dN ~ 3 to 4 nm, similar to that of the SMR shown in Fig. 1D. These results indicate that the interfacial magnetoresistance caused by the Rashba interaction, which takes a maximum at an HM layer thickness close to one lattice constant (42), is not the main source of the voltage (SSNE) found here [see figs. S3 and S4 for discussions on the effects of the FM layer (CoFeB) and an unintended out-of-plane temperature gradient (15, 4345) on the voltage measurements].

To account for these results, a drift-diffusion model is extended to describe spin transport in a bilayer system. The HM layer thickness dependence of the SMR and the anomalous Hall coefficient are described by the following equations (26, 31)Embedded Image(1)Embedded Image(2)where θSH and λN are the spin Hall angle and the spin diffusion length of the HM layer, respectively; θAH is the anomalous Hall angle of the FM layer; and gS = 2ρNλNGMIX, where GMIX is the spin-mixing conductance of the HM/FM interface. Here, for simplicity, we have neglected the contribution of longitudinal spin current absorption on the SMR (31).

Furthermore, we assume that a temperature gradient (T) applied across a sample can generate spin current Q (flow of spin angular momentum carried by electrons) via the spin Nernst effect in a similar way an electric field E (or current) generates spin current through the SHE, that isEmbedded Image(3)where indices k and j denote the spin and flow direction of the spin current, respectively; ek is a unit vector; is the reduced Planck constant; e is the electron’s charge; and SN and θSN are the Seebeck coefficient and the spin Nernst angle of the HM layer, respectively. For simplicity, we do not consider the spin Hall and spin Nernst effects of the FM layer because θSH of FM has been reported to be small compared to that of the HM layers (4648). However, in the FM layer, the anomalous Hall and anomalous Nernst effects generate a transverse charge current JT when E and T are applied. The transverse charge current (opposite to the electron flow) isEmbedded Image(4)where Embedded Image is a unit vector representing the magnetization direction of the FM layer, and SF and θAN are the Seebeck coefficient and the anomalous Nernst angle of the FM layer, respectively.

We assume that a temperature gradient Embedded Image is applied under an open-circuit condition. The change in the longitudinal voltage Embedded Image when the magnetization of the FM layer is pointing along the y axis (VXXY) and z axis (VXXZ), ΔVXX = VXXYVXXZ, is expressed asEmbedded Image(5)

Similarly, the difference in the transverse voltage Embedded Image when the magnetization reverses its direction from +z to −z, 2ΔVXY = VXYZVXYZ, readsEmbedded Image(6)

Equations 5 and 6 represent the dN dependence of the spin Nernst and anomalous Nernst coefficients, respectively. The Seebeck coefficient of the HM/FM bilayer, defined as S = xFSF + (1 − xF)SN, is obtained experimentally using the relation S ~ −(VXX/L)/(ΔT/D), and the results are shown in Fig. 3 (D and E). We note that when θSN = 0, SSNE = SRSMR: The functional form of SSNE and RSMR is the same.

The first term (θSHS) in the curly bracket of Eq. 5 appears because of the open-circuit condition. The electrons initially move from the hot to cold side when a temperature gradient is applied (the Seebeck coefficients of the FM and HM layers are all negative). Once the electrons reach the edge of the patterned structure, an internal electric field EINT develops because of charge accumulation at the edges. The direction of EINT is such that it cancels the electron flow driven by the temperature gradient, resulting in a net zero current. However, spin current can be generated via the SHE when a nonzero EINT exists, thus contributing to the SMR. The second term (θSNSN) in the curly bracket of Eq. 5 corresponds to the contribution to the SMR that results from a direct conversion of heat current to spin current. Similar classification also applies to the terms in the curly brackets of Eq. 6.

The model calculations are compared to the experimental results presented in Figs. 1 (C and D), 2 (C and D), 4 (F and G), and 5C to find a parameter set that best describes the results. The calculation results are shown by the solid lines in each figure, and the parameters extracted (θSH, θAH, λN, θSN, θAN, Re[GMIX], and Im[GMIX]) are summarized in Table 1 (see Materials and Methods for details of the fitting process). The spin Hall angles (θSH) estimated for the Ta and W underlayers are consistent with those from previous reports (2, 6, 24, 25, 31). These results show that the model can account for all results shown in Figs. 1 to 5 using a single set of parameters listed in Table 1. Note that the spin-mixing conductance obtained from the fitting is mostly consistent with that from previous reports (see Materials and Methods for details).

Table 1 Parameters used to describe the experimental results.

Resistivity (ρN), Seebeck coefficient (SN), spin diffusion length (λN), spin Hall angle (θSH), and spin Nernst angle (θSN) of the HM layer and resistivity (ρF), Seebeck coefficient (SF), anomalous Hall angle (θAH), and anomalous Nernst angle (θAN) of the FM layer in the HM/FM/MgO heterostructure. Re[GMIX] and Im[GMIX] represent the real and imaginary parts of the spin-mixing conductance GMIX at the HM/FM interface. N/A, not applicable. Ω, ohm; μΩ, microohm.

View this table:

To illustrate the effect of the spin Nernst effect on the transport properties more clearly, the spin Nernst and anomalous Nernst coefficients are numerically calculated using Eqs. 5 and 6 with three different spin Nernst angles, θSN = −θSH, θSN = 0, and θSN = θSH. The open circles in Fig. 5C represent the scaled SMR (SRSMR) calculated using the results of Figs. 1D and 3E. As described above, SRSMR lies on the θSN = 0 line. This demonstrates that the internal electric field EINT partly contributes to the spin current generation. In contrast, the spin Nernst coefficient SSNE (solid circles) lies closer to the θSN = −θSH line. When the signs of θSN and θSH are opposite, contribution from the heat current–induced spin current adds constructively to the EINT-induced spin current. Note that for the Ta underlayer films, the expected spin Nernst coefficient using Eq. 5 and the parameters defined in Table 1 (with θSN = −θSH) is ~0.01 (μV/K). This is smaller than the experimental resolution, and we consider that this is the reason why we find no characteristic feature in the voltage measurements (Fig. 5A).

Furthermore, we show that the spin Nernst angle θSN can be extracted experimentally, without relying on model parameters such as the spin mixing conductance. From Eqs. 1 and 5, we obtainEmbedded Image(7)

In Fig. 5D, we plot θSNSH obtained by using the results of Figs. 1D, 3E, and 5C (and Eq. 7). The plot shows that the signs of the spin Nernst and spin Hall angles are opposite and the magnitude of the former is somewhat smaller than that of the latter. [Meyer et al. (49) have studied the spin Nernst effect in Pt/YIG and found that the signs of two angles are also opposite for Pt; however, the spin Nernst angle of Pt was reported to be larger than its spin Hall angle].

From numerical calculations, we find that θSNSH is not susceptible to the values of the spin-mixing conductance and the degree of longitudinal spin absorption (that is, the spin polarization of the FM layer), which influences the absolute values of RSMR and SSNE (31). The calculations also show that θSNSH is not significantly influenced by contribution(s) from the anomalous Hall/anomalous Nernst effects and the spin Hall/spin Nernst effects, if any, of the FM layer as long as the HM layer thickness dN is larger than λN (details will be reported elsewhere). When dN is smaller than λN, these effects can influence the value of θSNSH: The slight increase in θSNSH at small dN found in Fig. 5D may be due to this contribution. We thus consider that the large dN limit of θSNSH provides a better estimate, from which we find θSNSH ~ −0.7.

DISCUSSION

The anomalous Nernst and anomalous Hall angles (5054) of CoFeB also have opposite signs (see Table 1), which results in a larger anomalous Nernst effect than otherwise. Theoretically, the signs of the Nernst and Hall angles do not necessarily have to match (33), since the Nernst angle (θAN and θSN) is defined by the energy derivative of the corresponding Hall conductivity near the Fermi energy, which can be positive or negative regardless of the sign of the Hall angle (θAH and θSH). Thus, the sign and the magnitude of the Nernst angle can be very different from the Hall angle. The recently reported spin Hall tunneling spectroscopy (55) and/or the temperature gradient–induced magnetization measurements (56) may provide access to information on the energy-level dependence of the Hall conductivity and can be used to verify the relationship between the Hall and Nernst angles.

We briefly discuss contributions from other effects that may influence the signal due to the spin Nernst effect (see table S1 for more details). It has been reported that an unintended out-of-plane temperature gradient may develop during the application of an in-plane temperature gradient (15, 4345). Under this circumstance, the anomalous Nernst effect of the FM layer can contaminate the signals observed in the voltage measurements. We observe this longitudinal voltage (VXX) in film structures without the HM (W) layer and thicker FM (CoFeB) layer under the application of HY. However, the HY dependence of VXX is distinct: The values of VXX when the magnetization points along +y and −y are different for the anomalous Nernst voltage caused by the unintended out-of-plane temperature gradient (fig. S3, L to N), whereas the values lie at the same level for the spin Nernst coefficient induced by the in-plane temperature gradient (Fig. 5B). For similar reasons, the combined effect of the spin Seebeck effect within the FM layer and the inverse SHE of the HM layer under an out-of-plane temperature gradient can be excluded. The size of the unintended out-of-plane temperature gradient scales with the thickness of the CoFeB layer, and it is smaller than the detection limit for the 1-nm-thick CoFeB layer used here (see fig. S3, I to N). We have also confirmed that the spin Nernst coefficient SSNE is negligible for heterostructures without the W layer (for example, in sub.|1 CoFeB|2 MgO|1 Ta) (see fig. S3, K to N).

The results presented here not only provide insights into the thermoelectric generation of spin current in HMs with strong spin-orbit coupling but also have important implications on expanding the search of materials that can generate spin current. The spin Nernst effect may be able to generate spin current from materials that are not possible with the SHE, for example, in systems where the density of states at the Fermi level is zero. The two-dimensional chalcogenides and the Weyl semimetals, in which the Fermi level coincides with the Dirac point, are of particular interest. The spin Nernst effect may thus broaden material research on the spin current generation beyond the current reach of the SHE.

MATERIALS AND METHODS

Sample preparation and measurements

All films were deposited using magnetron sputtering on nondoped silicon substrates coated with ~100-nm-thick thermal oxides (SiOx). Films were post-annealed at ~300°C for 1 hour before the device patterning processes. Optical lithography and Ar ion etching were used to pattern the films into wires and Hall bars. Contact pads made of 5 Ta|100 Au (in nanometers) were formed by a liftoff process.

All measurements were performed at room temperature. A temperature gradient across the substrate was applied by placing a ceramic heater on one side of the substrate and a heat-absorbing Cu block on the other side. The substrate was fixed to the heater/Cu block using a thermally conducting double-sided tape made of Al. The temperature profile of the system was studied using an infrared camera with Si substrates coated with a blackbody matt (the surface emissivity is calibrated). The camera was used to ensure that the temperature gradient across the substrate is uniform. Because of the necessity of this coating, the temperature profile of the device under investigation cannot be monitored in real time: Once the sample is coated with the blackbody matt, it is difficult to perform the voltage measurements. Since the temperature gradient across the substrate largely depends on the contact between the substrate and the heater/Cu block, we checked its variation by placing the substrate to the setup multiple times and monitored the temperature profile using the infrared camera. The variation of the temperature gradient was ~±10% of the average value. The horizontal error bars in Figs. 3 (B and C) and 4C reflect this variation. The vertical error bars in the same figures represent the distribution of the voltage when measurements are repeated multiple times under the same contact between the substrate and the heater/Cu block. The vertical error bars are smaller than the symbols, suggesting that the measurements are stable and the temperature gradient do not evolve once the substrate is fixed. Thus, the dominant source of the measurement error originates from the uncertainty in the actual value of the temperature gradient across the substrate: The error bars in Figs. 3 (D and E), 4 (D to G), and 5 (C and D) reflect this uncertainty.

Fitting procedure

Experimental results were fitted using Eqs. 1, 2, 5, and 6. Before carrying out the fitting, we determined the following parameters from the experimental results. The resistivity (ρN) of the HM layers was obtained by the dN dependence of GXX shown in Fig. 1 (A and B). For the resistivity (ρF) of the FM (CoFeB) layer, we used a value from our previous study (57). The Seebeck coefficients of the HM layer (SN) and the FM layer (SF) were estimated from the results presented in Fig. 3 (D and E). We also measured the Seebeck coefficient of the FM layer (SF) independently using a film stack that did not include any HM layer (results are shown in fig. S4I). We found that SF estimated from film stacks with and without the HM layer were similar. The anomalous Hall angle (θAH) and the anomalous Nernst angle (θAN) of the FM (CoFeB) layer can be estimated by the zero HM thickness limit of the normalized RAHE (Fig. 2, C and D) and the normalized SANE (Fig. 4, F and G), respectively.

We first fit RSMR (Fig. 1, C and D) and RAHE (Fig. 2, C and D) using Eqs. 1 and 2 to determine θSH, λN, Re[GMIX], and Im[GMIX]. Note that in many previous studies, a transparent interface [Re[GMIX] ≫ Im[GMIX] and Re[GMIX] ≫ 1/(2ρNλN)] has been assumed to estimate the lower bound of θSH. In such a case, GMIX drops off from Eq. 1 and simplifies the fitting. Here, we used Re[GMIX] and Im[GMIX] as the fitting parameters to account for the dN dependence of RSMR and RAHE. For both underlayer films, we found that Im[GMIX] has to be negative and larger in magnitude than Re[GMIX]. This characteristic GMIX is in agreement with the current induced torque found in similar heterostructures (5860) according to the relation of GMIX and the torque (61). For the Ta underlayer films, the change in RAHE with dN is larger than what is expected from Eq. 2. We infer that there are other effects that are not captured by Eq. 2 (62, 63). With the parameters described in Table 1 (unless noted otherwise), SSNE (Fig. 5C) and SANE (Fig. 4, F and G) were calculated using Eqs. 5 and 6, respectively, with θSN denoted in each figure legend.

SUPPLEMENTARY MATERIALS

Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/3/11/e1701503/DC1

Additional experimental results

Discussion related to other effects that may influence the voltage measurements

fig. S1. Magnetic properties of HM/CoFeB/MgO heterostructures.

fig. S2. Spin Nernst magnetoresistance of Ta and W underlayer films.

fig. S3. Thermoelectric properties of CoFeB thin films without the HM layer.

fig. S4. Comparison of parameters with and without the HM layer.

table S1. Influence of other phenomena on the temperature gradient–induced voltage measurements.

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REFERENCES AND NOTES

Acknowledgments: We thank S. Bosu and S.S.-L. Zhang for fruitful discussions. Funding: This work was partly supported by Japan Society for the Promotion of Science (JSPS) Grants-in-Aid (15H05702 and 26709045), Casio Foundation, Ministry of Education, Culture, Sports, Science and Technology (MEXT) R&D Next-Generation Information Technology, and the Spintronics Research Network of Japan. Y.-C.L. is an International Research Fellow of the JSPS. Author contributions: M.H. and Y.S. planned the study. P.S. and Y.-C.L. carried out microfabrication. P.S., Y.-C.L., and Y.S. measured the samples and analyzed the results with help from S.M. and M.H. S.T. developed the drift-diffusion model. All authors discussed the data and commented on the manuscript. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in the paper are present in the paper and/or the Supplementary Materials. Additional data related to this paper may be requested from the authors.
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