Thermoelectric signature of individual skyrmions

We experimentally study the thermoelectrical properties of individual sykrmions in Pt/Co/Ru multilayers. Skyrmions are nucleated by current pulse injection in micropatterned Pt/Co/Ru stripes. In-situ magnetic force microscopy (MFM) is used to characterize the number, size, and spatial distribution of the skyrmions. A transverse thermal gradient from a microheater allows to generate an anomalous Nernst effect (ANE) signal proportional to the out-of-plane magnetization of the wire, which is detected electrically. To derive the thermoelectrical signature of individual skyrmions, the stray field of the MFM tip is used to delete individual skyrmions one by one while the respective thermoelectrical signals are compared. The observed signature is attributed to the ANE of the skyrmions spin structure. We demonstrate the thermoelectrical detection and counting of individual skyrmions and study the field dependence of their ANE signature. Possible topological contributions to the experimentally observed thermoelectrical signature are discussed. Such thermoelectrical characterization allows for non-invasive detection and counting of skyrmions and enables fundamental studies of topological thermoelectric effects

where N ANE is the ANE coefficient per magnetic moment, μ 0 is the vacuum permeability, l is the wire length, ∇T x ̅̅̅̅̅ is the averaged ∇T x and M z is the averaged zcomponent of the magnetization between the contacts. The condition for saturation is denoted by M z = M s , the saturation magnetization.
The ANE in the lithographically patterned devices was first characterized by ANE hysteresis measurements. In Figure 1b  To extract the temperature distribution and ∇T x ̅̅̅̅̅ finite element modelling (detailed in Supporting Information) of the device and experimental conditions was implemented 20-21 . The resulting temperature distribution for a microheater current I heater of 4.3 mA is shown in Figure 1f. The average thermal gradients along the x,y,z axes in the device are found to be ∇T x ̅̅̅̅̅ = 4.2 K/µm, ∇T y ̅̅̅̅̅ = 2.9 mK/µm and ∇T z ̅̅̅̅̅ = 0.2 mK/µm, respectively.
Given the order of magnitude difference between ∇T x ̅̅̅̅̅ on one hand, and ∇T y ̅̅̅̅̅ and ∇T z ̅̅̅̅̅ on the other hand, we neglect the latter two contributions to the ANE in further analysis.
In Figure 1g, we show the simulated surface temperature profile across the microstripe along the dashed line in Figure 1f. Inside the microstripe (inset), the temperature T stripe increases by about 8.6 K. Using the microstripe as a resistive thermometer, the averaged T stripe can be experimentally determined as function of I heater (see the Supporting Information). As shown in Figure 1h (black circles), T stripe increases quadratically with I heater as expected for Joule heating and the experimentally determined temperature increase of 8.6 K for the maximum I heater = 4.3 mA agrees well with the simulations.
Similar ANE hysteresis loops as in Figure 1b were measured for a range of input heat currents I heater = 0.2 -4.3 mA. In Figure 1h  To controllably define the number of skyrmions between the contacts for the ANE device, we used a single skyrmion annihilation procedure using an MFM tip 25 . Here, the confined stray magnetic field gradient around the tip apex allows the precise local annihilation of single skyrmions. In Figure 3a, we show an MFM image after nucleation of five skyrmions in an applied magnetic field of +34.84 mT. For a given external applied magnetic field, i.e. +11.3 mT, the local total field (applied field + stray field of the tip) under the MFM tip is always below the skyrmion annihilation field (between 50 mT and 60 mT). As such this allows us to image the skyrmions without significant tipsample interaction. In contrast, when the applied field is exceeds +24 mT, the total local field under the MFM tip is greater than the annihilation threshold. The assistance of such a bias field leads to a precision methodology to delete individual skyrmions at  Figure 3a-c, the number of skyrmions is first reduced from five (a) to three (b) and finally to one (c). In Figure   3d-f, the number of skyrmions is first reduced from four (d) to two (e) and finally to one

COMSOL simulation parameters
Thermal modeling of the microheater and surrounding device topology was performed using FEM modelling with COMSOL. The Joule heating module was implemented to estimate the temperature rise and thermal gradient within the ANE device. The geometry and dimensions of the modeled sample match that of the device.

Temperature change generated by the microheater
The microstripe and the microheater could be used as temperature sensors exploiting their resistance change, induced by the Joule heating of the microheater. To this end, their temperature coefficients of resistance are calibrated. The ANE device used for the actual ANE-measurements does not allow 4-wire measurement of the microheater.
Therefore, in the same fabrication process as for the ANE device a separate device with the same geometry was fabricated but with additional contacts. During the resistance measurements, both devices, were put in a temperature-controlled air bath unit, where temperatures could be stabilized in the range from 288 K to 314 K. The resistance for both the microstripe and the microheater shows a linear dependency in the temperature range calibrated (Fig. S1a), as expected for metals. The resulting temperature coefficients of resistance α, were determined as 8.7 10 −4 for the microstripe and the microheater 1.5 10 −3 , respectively.
The known temperature coefficient allows to backward calculate the temperature from a given resistance value.
By this means, the temperature increase in the microheater and microstripe were experimentally determined for different microheater currents, as shown in Fig

Skyrmion nucleation experiment and field dependency of single skyrmion
As described in the main text, three independent skyrmion nucleation and deletion sequences were performed using the same microstripe. In Fig. S2a-c

Fit for the calculated stray field of single skyrmions
The local magnetic configuration in the sample was probed using an MFM in 'lift mode' at a distance of z = 220 nm (lift height + cantilever oscillation amplitude) to the sample surface, mapping the cantilever phase shift Δφ.
To determine the area of the skyrmions from the phase shift images, we fitted the measured Δφ from the cross-section (along two axis, x and y) of every skyrmion data with a Gaussian function as: This fitting model was verified by employing the same fitting procedure to a calculated MFM phase signal obtained from a simulated skyrmion. To obtain the simulated MFM image we followed the Tip Transfer Function (TTF) approach [6,7]. The stray field distribution of a typical skyrmion at the MFM measurement height was calculated from its magnetization distribution as resulting from micromagnetic simulations according to [8].
The radius of the simulated skyrmion was in the same range as the experimentally   Fig. 4a) shows a very good agreement with the experimental data. Therefore, we conclude, that the simulations strongly overestimate the spatial variations in thermal gradient. One possible reason for these deviations are edge effects resulting form the unrealistic idealized square cross section of the microstripe in the simulation..
Nevertheless, we used the COMSOL results of the thermal gradient to estimate the upper limits of the uncertainty of the simulated data, complemented with possible error contributions from the calculated skyrmion area: For each skyrmion an effective local thermal gradient was obtained by averaging over its area using the COMSOL simulations with an assumed error of 10% due to position and shape inaccuracies.
Additionally, an error is allowed in the calculation of the skyrmion area. To this end, we assumed an error in the relationship between the σ of the fitted Gaussian and the effective skyrmion radius (r sky ≈ √2σ) of more than 12% (r sky ≈ √2 ± 0.25σ). In Fig.   S6b we show how the effective skyrmion radius is thereby affected. The ANE voltage for every single skyrmion in this specific approach can be calculated using the following equation: V ANE = −N ANE μ 0 M z ∇T x;local A tot /w where ∇T x;local represents the local average thermal gradient and w = 2µm the width of the microstripe. The other parameters were previously obtained and mentioned in the manuscript. In Fig. S5c we show the comparison between the measured ANE voltage from the sequence shown in Fig. S2a depicted as the red dots and the simulated ANE voltage. Taking in to account the above discussed error margins leads to a range of possible simulated ANE values that is shown here as the shaded region.
We attribute the fact that the shaded region it mostly found at higher negative ANE values than the experimental data to an overestimation of the thermal gradient in the COMSOL simulations.