Observation of transverse spin Nernst magnetoresistance induced by thermal spin current in ferromagnet/non-magnet bilayers

Electric generation of spin current via spin Hall effect is of great interest as it allows an efficient manipulation of magnetization in spintronic devices. Theoretically, pure spin current can be also created by a temperature gradient, which is known as spin Nernst effect. Here, we report spin Nernst effect-induced transverse magnetoresistance in ferromagnet/non-magnetic heavy metal bilayers. We observe that the magnitude of transverse magnetoresistance in the bilayers is significantly modified by heavy metal and its thickness. This strong dependence of transverse magnetoresistance on heavy metal evidences the generation of thermally induced pure spin current in heavy metal. Our analysis shows that spin Nernst angles of W and Pt have the opposite sign to their spin Hall angles. Moreover, our estimate implies that the magnitude of spin Nernst angle would be comparable to that of spin Hall angle, suggesting an efficient generation of spin current by the spin Nernst effect.

, where 0 is the centre location of the laser beam, x is the distance from the centre, and σ is the standard deviation. The power intensity of the laser decays exponentially from the beam centre in z-direction along the thickness of the samples. The Gaussian laser beam has full width half maximum of ~3σ, equivalent to its experimental diameter of 5µm. The finite element modeling in COMSOL software has been performed using the material parameters obtained under the same experimental conditions as the thermoelectric measurement. The absorption coefficients of W, Pt and CoFeB are 8.66×10 5 cm -1 , 7.78×10 5 cm -1 and 8.9×10 5 cm -1 , respectively and the total reflectance of CoFeB(2 nm), W(3 nm)/CoFeB(2 nm), and Pt(3 nm)/CoFeB(2 nm) are measured as 0.119, 0.244, and 0.255, respectively. The temperature profiles were calculated utilizing the above parameters for the simulation area of 10 µm × 1 mm, of which size is equivalent to the experimented structure.
The absorbed electromagnetic wave in a thin film deposited on the SiO 2 /Si substrate is evaluated by the absorption coefficients that is the characteristic feature of particular thickness of respective materials. In an ultra-thin film structure, absorption process could be affected by the multiple reflection and related interference effects, which is considered by the laser is moved away from the centre of the structure, net ∆ 7 appears. Supplementary Figure 4 shows the lateral temperature differences along the x-direction (∆ 7 ) which were calculated by integrating temperature over the locally-excited area from −2 ;< to +2 ;< in the y-direction, where the thermal spin current is mostly converted into a transverse voltage.
Here, ;< is the standard deviation of the temperature distribution. As a result, similar lateral temperature difference between left and right edge ∆ 7 for both W/CoFeB and Pt/CoFeB samples is obtained to be ~24 K while the ∆ 7 is ~17 K for CoFeB. When a laser locates at the edge of the sample, the effective length for temperature gradient and thermal voltage generation, A and B were defined by 2 ;< in x-direction and 4 ;< in y-direction, respectively.
Here we show that the COMSOL simulation result is reasonable by performing an additional measurement. First of all, we note that a typical RT curve measurement does not allow us to estimate the temperature gradient caused by the laser illumination by the following reason. The laser illumination thermally excites a local area where sample temperature is efficiently increased; however, the local heating makes it difficult to measure the sample temperature by measuring the resistance because the resistance increase of the illuminated area (~a few micron range) does not markedly contribute to the total resistance of the sample (~mm size).
In order to resolve the size difference issue between the sample and the laser spot, we fabricated a sample with a narrow wire structure as illustrated in Supplementary Fig. 5a. Note that in this new sample, the resistance is dominated by the narrowest wire region of which width is even smaller than the laser spot size. We first measured the resistance of the sample as a function of the temperature in physical property measurement system (PPMS, Quantum design), showing a linear relation between the resistance and the sample temperature ( Supplementary Fig. 5b). Then, we measured the variation of the resistance with a laser illumination. When the laser (55 mW) is on the wire, the resistance is increased from 1202 Ω to 1232 Ω as shown in Supplementary Fig. 5c, corresponding to the increase in temperature from 296 K to 324 K. On the other hand, when the laser is moved away from the wire of ~10 µm, the resistance is 1213 Ω, which corresponds to the temperature of 307 K. As a result, ∆ 7 in the sample of 10 µm width is estimated to be 17 K, which is comparable to that obtained by simulation. Because of the non-identical sample structure, this experiment cannot determine the temperature profile in the real sample of Fig. 3; however, this confirms that the laser illumination can induce a temperature difference in the sample with a similar order of magnitude as the calculated value.

Supplementary Note 2. Magnetic field dependence of thermoelectric signal
To verify the effect of magnetic field on the spin Nernst magnetoresistance (SNMR), we repeated the measurement for the W(3 nm)/CoFeB(2 nm) sample with different magnetic fields of 30 mT, 60 mT, and 100 mT. Supplementary Figure 6 demonstrates that the thermoelectric signals are almost independent of the magnetic field when it is large enough to saturate the magnetization to field direction.

Supplementary Note 3. Thermoelectric signals depending on laser position along the yaxis
We perform the measurement of the thermoelectric Hall voltage (the same measurement as Fig.   2) with varying position along the y-axis (Supplementary Fig. 7a). We find that the transverse thermoelectric signals are almost identical for the measurements with different y-positions ( Supplementary Fig. 7b-d), indicating that there is no considerable effect of the conventional Seebeck effect in our measurement configuration. This is attributed to the local excitation by the laser heating (diameter ~5 µm) in the elongated sample structure: 10 µm × 1 mm in which the ∆ D between the two ends of the sample is not significantly generated by the laser illumination. This confirms the SNMR signal in our sample is mostly dominated by the ∆ 7 .

Supplementary Note 4. Planar Hall effect for W/CoFeB structure
Supplementary Figure 8 shows the planar Hall effect (PHE) in CoFeB (2 nm) and W (3 nm)/CoFeB (2 nm) samples which are the same samples for the transverse spin Nernst magnetoresistance (SNMR) measurement shown in Fig. 2. The W/CoFeB sample shows a larger PHE signal than the CoFeB sample by a factor of ~20. In order to study the origin of the enhancement in the PHE, we investigated the angular dependence of the magnetoresistance in the W/CoFeB sample. We measured the longitudinal (R xx ) and transverse (R xy ) resistance in W(t W )/CoFeB(2 nm) samples with t W =2, 4, and 5 nm while rotating the sample on three major planes of the x-y, y-z, and z-x planes under a magnetic field of 9 T. The angle of each plane is denoted as , , and , respectively, as indicated in Supplementary Fig. 9. The ΔR xx represents the spin Hall magnetoresistance (SMR), anisotropic magnetoresistance (AMR), and a sum of the SMR and AMR for the -scan, -scan, and -scan, respectively. Supplementary   Figures 9-11 show that the SMR is much more dominant than the AMR in the W/CoFeB sample.
Moreover, the ΔR xy with is attributed to the PHE which is the same amount as the ΔR xx , demonstrating the enhancement in the PHE is mostly contributed by the transverse SMR.
Supplementary Figure 12 shows the W thickness dependence of the SMR, which is consistent with that of the SNMR.

Supplementary Note 5. Decomposition of thermoelectric Hall signals in HM/CoFeB
We studied the dependence of the HM thickness on the thermoelectric Hall signal in W(t W )/CoFeB and Pt(t Pt )/CoFeB structures. Supplementary Figure 13 shows the angular dependence of the thermoelectric Hall voltages for each sample at various laser locations. As thermoelectric Hall voltages consist of and 2 components (Vθ, V 2θ ), which are related to ∆ 6 and ∆ 7 , respectively, we decomposed them by fitting the measured data with Equation (S2).
= O cos + &O sin 2 + , … (S2) where C is a d.c. offset of the thermoelectric Hall voltage which depends on the sample or measurement configuration. The offset is simply subtracted when the SNMR is analyzed in Fig.   2 and Supplementary Fig. 13 because it does not have an angular dependence that is a characteristic feature of the SNMR. The extracted Vθ as a function of laser position from the centre of the sample in W/CoFeB and Pt/CoFeB are shown in Supplementary Fig. 14, while the V 2θ components are presented in Fig. 3a,b. The Vθ in all samples decreases as the laser position moves away from the centre of the sample which is explained by the reduction of the heating area on the structure, or smaller