Thermoelectric microscopy of magnetic skyrmions

The magnetic skyrmion is a nanoscale topological object characterized by the winding of magnetic moments, appearing in magnetic materials with broken inversion symmetry. Because of its low current threshold for driving the skyrmion motion, they have been intensely studied toward novel storage applications by using electron-beam, X-ray, and visible light microscopies. Here, we demonstrate another imaging method for skyrmions by using spin-caloritronic phenomena, that is, the spin Seebeck and anomalous Nernst effects, as a probe of magnetic texture. We scanned a focused heating spot on a Hall-cross shaped MgO/CoFeB/Ta/W multilayer film and mapped the magnitude as well as the direction of the resultant thermoelectric current due to the spin-caloritronic phenomena. Our experimental and calculation reveal that the characteristic patterns in the thermoelectric signal distribution reflect the skyrmions’ magnetic texture. The thermoelectric microscopy will be a complementary and useful imaging technique for the development of skyrmion devices owing to the unique symmetry of the spin-caloritronic phenomena.

Circumferential profile of thermoelectric image obtained by two-dimensional detection. Figure S3 shows the circumferential profiles of the magnetic circular objects in the amplitude image measured at the field label H in Fig. 3b. We calculated the profiles for three objects labelled α, β, and γ (Fig.   S3b). The averaged profiles are calculated using the values in the annulus with the radius of the small (large) ring being about 230 nm (360 nm). As we discussed in the Results section of the main text (Simulation of thermoelectric images due to skyrmions), the Néel skyrmions are expected to show the uniform amplitude profile along the circumference while the trivial bubbles are expected to show the nonuniform or asymmetric profile (see Fig. 4f). Figure S3c shows, however, that the profiles do not have sufficient S/N ratio for experimental distinction.

Skyrmion diameter dependence of the thermoelectric images.
The diameter of the skyrmions is estimated by comparing the experimental and calculated results. Figure   S4a shows the skyrmion diameter (d) dependence of the simulated thermoelectric images. By comparing the diameter of the ring-like pattern in the amplitude image at H of Fig. 3b and the calculated amplitude images, the skyrmions' diameter is found to be around 1000 nm. Interestingly, Fig. S4a indicates that although the magnitude decreases as d decreases (see Fig. S4c), the spatial variation of the j c direction around the skyrmion's centre remains even when the optical laser spot size is comparable to d (Fig. S4b). This is due to the combination of steeply-changing magnetic structure of the skyrmions and spin-caloritronic phenomena.
The sign of the thermoelectric signals is reversed between one and the other halves of each skymion. Thus, even if the heating diameter is comparable to the skyrmions, the shift of the heating centre from the skyrmion's centre ensures finite thermoelectric signals. We note that apart from the skyrmions' core, the spatial distribution is simply broadened following to the optical limitation.

Extraction of thermoelectric image due to spin caloritronic phenomena.
The thermoelectric images shown in the main text were obtained after subtracting periodic environmental noise and background offset from the raw thermoelectric images. Firstly, the periodic noise with high spatial frequency (the wavy pattern in Fig. S6) was eliminated using a high-pass filter in the following procedure.
The raw image was converted into a Fourier transformation (FT) image, the points with high wave vectors and large intensity were subtracted. Then, the FT image was converted back into the thermoelectric image.
Secondly, the uniform background offset was subtracted, which is due to the electronic circuit, such as the characteristic offset of the amplifier and the thermoelectric current due to the environmental temperature difference inside the circuit. The offset value was calculated as the averaged value over the regions A-D shown in the leftmost images of Fig. S6, where A-D locate outside the Hall cross and no spin-caloritronic signals are expected. In the thermoelectric images after the offset subtraction, negative/positive signals can be found on the top/bottom of the horizontal section of the Hall cross. These signals might be due to the magnetic moments lying in-plane at the Hall cross edges and/or the asymmetric in-plane temperature gradient induced by edge illumination 1 ; while at inside, the in-plane temperature gradient is radial and thus the signal is induced only when the out-of-plane magnetic moment m z distributes asymmetrically in the heating spot, at the edges, the in-plane temperature gradient is formed only toward the other edge and thus it simply reflects . In addition, similar situation can happen for edges where the electrical sensitivity changes, i.e. around the boundary of the centre part of the Hall cross. We note that these parasitic signals do not affect the signals inside the cross area of the Hall cross, which we discussed in Fig. 3b.

Power dependence of thermoelectric image.
To check that the magnetic texture in our sample is not changed by the laser heating, we measured the laser power P in dependence of the thermoelectric images. Figure S7 shows that the distribution of the thermoelectric signals does not change when P in < 1.14 mW, confirming that the experimental results shown in the main text, measured at P in = 0.84 mW, are not affected by laser light effects 2 . Note that small change in the patchy pattern can be found at 1.28 and 1.43 mW irradiation. Figure S7. Irradiation power P in dependence of thermoelectric images.

Estimation of the magnitude of the anomalous Nernst and spin Seebeck effects.
In order to estimate the magnitude of the ANE and SSE contributions, we compared the magnitude of the transverse thermopower in the MgO/CoFeB/Ta/W multilayer film with the in-plane (IP) and out-of-plane (OP) magnetization 3,4 . For the IP (OP) magnetized system, both the ANE and SSE appear (only the ANE appears). The sample for this measurement was made on the thermally oxidized Si substrate, where the width w is 2.0 mm and the length is 6.0 mm. The MgO/CoFeB/Ta/W film was deposited on the whole surface of the substrate and annealed in the same manner as the device used for the thermoelectric imaging. The sample was loaded on a heat bath and cramped by a heater block, which is thermally isolated from the environment so that the entire applied heater power P heater flows to the heat bath through the sample 5 . The length of the heater block ' l is 5.0 mm and the thermoelectric voltage V induced within ' l was measured using a voltmeter. With applying P heater = 150 mW to the heater block and the magnetic field to the sample, we measured the thermoelectric voltage along the length direction. The recorded voltage V is converted into the transverse thermopower S using the relation by estimating the temperature gradient T applied to the sample. For the IP magnetized system, T is simply determined by the thermal conductivity  of the sample layer: For the OP magnetized system, T is approximately determined by subs and thickness t subs of the substrate because the thermal resistance is governed by the substrate: Figure S8 shows the magnetic field magnitude H dependence of S for the IP and OP magnetized samples. As The obtained values are S ANE = 0.22 V/K and S SSE = 0.14 V/K, of which the sign and magnitude are consistent with previous reports 6,7 . For the above calculation, we assumed the thermal conductivities used for the temperature profile calculation in Fig. 4a.