Skyrmion pinning energetics in thin film systems

A key issue for skyrmion dynamics and devices are pinning effects present in real systems. While posing a challenge for the realization of conventional skyrmionics devices, exploiting pinning effects can enable non-conventional computing approaches if the details of the pinning in real samples are quantified and understood. We demonstrate that using thermal skyrmion dynamics, we can characterize the pinning of a sample and we ascertain the spatially resolved energy landscape. To understand the mechanism of the pinning, we probe the strong skyrmion size and shape dependence of the pinning. Magnetic microscopy imaging demonstrates that in contrast to findings in previous investigations, for large skyrmions the pinning originates at the skyrmion boundary and not at its core. The boundary pinning is strongly influenced by the very complex pinning energy landscape that goes beyond the conventional effective rigid quasi-particle description. This gives rise to complex skyrmion shape distortions and allows for dynamic switching of pinning sites and flexible tuning of the pinning.


Supplementary Notes
Supplementary Note 1 Supplementary Fig. 1 provides additional information regarding the skyrmion occurrences in the thin film sample. Besides the probability of finding a skyrmion at certain positions, it shows the positions of skyrmions in the beginning and at the end of the observations in Supplementary Fig. 1a. Whereas the newly nucleated skyrmions (blue dots) are evenly spread over the sample, the skyrmions at the end of every video (red dots) are as expected often found at pinning sites. The histograms along the x and y axis indicate furthermore that the skyrmion distribution is uniform along the sample axes throughout the measurement time.
Within the observed area, the positions featuring strong pinning appear evenly spread. The green annotations in Supplementary Fig. 1a exemplarily provide a few distances in micrometers between pinning spots. To study the spatial distribution of pinning sites, we therefore also look at the Fourier transformation of the histogram of skyrmion center positions. The radial dependence of the FFT intensity is depicted in Supplementary Fig. 1b as a function of real space distances. The distances denoted in Supplementary Fig. 1a are indicated by the green solid lines and lay within the FFT region with high intensities. This indicates a range of characteristic distances between positions at which skyrmions appear for a significant amount of time. We therefore conclude the existence of typical lengths under which strong pinning centers occur. In the analysis, we restrict ourselves to the radial dependence of the FFT since the angular distribution is homogeneous meaning that there is no preferred axis in the sample along which pinning sites are observed.
The detailed investigation of the underlying physical reason for this repeated appearance of pinning centers is beyond the scope of this paper. However, since locally varying material parameters result in the occurrence of pinning sites, this effect is potentially featured by a periodic appearance of impurities which can be governed by the physics of the manufacturing procedure of the amorphous multilayer stack. Therefore, the sample growth must be considered when fabricating a stack with specific pinning properties.

Supplementary Note 2
Skyrmions at pinning site 1 can appear at slightly varying positions depending on the skyrmion size. Supplementary Fig. 2a shows skyrmion boundaries for two different occurring skyrmion center of mass positions, which are indicated in Supplementary Fig. 2b. The SB position coincides at one side for both skyrmion sizes indicating a pinned behavior. However, the extension is different on the other side and determined by the size. Therefore, different skyrmion center positions can occur at this particular site (see Supplementary Fig. 2b). The size difference belonging to this variation of the center positions is visible in Supplementary Fig. 2c.

Supplementary Note 3
The small skyrmions appear between the regions of lowered anisotropy introduced in the simulation. Supplementary Fig. 3a shows the probability map of skyrmion center occurrences as presented in Fig.  5. Supplementary Fig. 3c,e show the perpendicular magnetization component of two such simulated skyrmions at different positions exemplarily. As proposed by our SB pinning concept, the area with reduced anisotropy is thereby occupied mainly by the SB depicted in white. Similarly, Supplementary  Fig. 3b,d,f show the center occurrence probability and magnetization components along the z-direction for two simulated skyrmions, respectively. Now, the large skyrmions do not fit between the boxes anymore with their core. Instead, they arrange above or below to maximize the overlap of their SB with the reduced anisotropy region while keeping the same overlap small for their core.
During that process, the skyrmions are also deformed with average eccentricities of 0.05 and 0.11 for the small and the large skyrmion, respectively.