Pinning and hysteresis in the field dependent diameter evolution of skyrmions in Pt/Co/Ir superlattice stacks

We have imaged Néel skyrmion bubbles in perpendicularly magnetised polycrystalline multilayers patterned into 1 µm diameter dots, using scanning transmission x-ray microscopy. The skyrmion bubbles can be nucleated by the application of an external magnetic field and are stable at zero field with a diameter of 260 nm. Applying an out of plane field that opposes the magnetisation of the skyrmion bubble core moment applies pressure to the bubble and gradually compresses it to a diameter of approximately 100 nm. On removing the field the skyrmion bubble returns to its original diameter via a hysteretic pathway where most of the expansion occurs in a single abrupt step. This contradicts analytical models of homogeneous materials in which the skyrmion compression and expansion are reversible. Micromagnetic simulations incorporating disorder can explain this behaviour using an effective thickness modulation between 10 nm grains.

*Correspondence to k.zeissler@leeds.ac.uk EXCHANGE STIFFNESS MEASUREMENTS The exchange stiffness was extracted from the temperature dependence of the saturation magnetisation which can be described by the Bloch law sufficiently far below the Curie temperature. The Bloch law is given by 1,2 ( )/ = 1 − ( where C = 0.0294 in case of the fcc lattice that is the case here, S = 1, kB is the Boltzmann constant, T is the temperature, and a = 0.355 nm is the lattice constant.

DMI MEASUREMENTS
Two methods were used to extract the DMI strength: asymmetric bubble expansion and Brillouin light scattering (BLS). The bubble expansion gives a localized value of D at pinning sites whereas BLS returns the average measured over a larger area and hence is less susceptible to local pinning potentials.
In the asymmetric bubble expansion method, a bubble was nucleated in a single trilayer 3,4 . An outof-plane magnetic field was pulsed and domain wall creep velocity was extracted from the visible bubble expansion. The domain wall velocity was measured for different static in-plane fields which act with or against the DMI field at the domain wall on either side of the bubble, modifying the wall energy (see figure Supplementary 1 (a)). The minima in the velocity versus in-plane field shows the point where the in plane field cancels the effective in plane field induced at the wall by the DMI, HDMI, and can be used to extract the DMI strength D using the formula where MS is the saturation magnetisation and Δ is the domain wall width and is given by √ / , in which A is the exchange stiffness and Keff is the effective perpendicular anisotropy constant. Two trilayer films were measured (Ta(4.8 nm)/Pt(6.2 nm)/Co(0.5 nm)/Ir(1.5 nm) and Ta(3.7nm)/Pt(4.5 nm)/Co(1.0 nm)/Ir(3.0 nm)). D was measured to be 0.6+0.1 mJ/m 2 for the 0.5 nm cobalt layer and 0.3+0.1 mJ/m2 for the 1.0 nm layer. In the Brillouin light scattering method 5,6 , the DMI strength is extracted by measuring an asymmetry in the Stokes and anti-Stokes frequencies of light that has been inelastically scattered from propagating spin waves. In a sample with notable DMI strength, spin waves of a given wavelength propagating in opposite directions have different energies. This behaviour is known as propagation nonreciprocity and occurs when the sample is magnetised inplane and the spin wave vector is perpendicular to the magnetisation, the Damon-Eshbach geometry. The frequency shifts of the inelastically scattered light with respect to the incident laser beam frequency is directly proportional to the DMI strength where kSW is the magnon wavevector, fS is the Stokes frequency, fAS is the anti-Stokes frequency, and γ is the gyromagnetic ratio. DMI was calculated using D=(Δf π MS t)/(2γk) where γ=190 GHz/T is the gyromagnetic ratio, and t is the thickness. The saturation magnetisation MS was measured to be 1.

EFFECTIVE THICKNESS MICROMAGNETIC SIMULATION
Supplementary figure 1 (d) shows the simulated defect free skyrmion diameter as function of magnetization. The N=10 repeat stack is compared to the effective thickness simulation where only one thick magnetic layer is considered. This shows that an effective thickness simulation is a good approximation of the multilayer system.

GRAIN SIZE OF POLYCRYSTALINE Pt/Co/Ir
Bright field transmission electron microscopy images where acquired on a representative trilayer Pt/Co/Ir polycrystalline sample. Ten images where taken at x50k and ten images where taken x80k magnification (see supplementary figure 1 (e) for an image taken at x50k). In total 20 different locations on the sample were investigated. Supplementary figure 1 (f) shows the percentage area of the different grain sizes observed, whilst the inset shows the frequency of the observed grains sizes. Whilst around 2/3 of the sample area is covered by grains less than 7 nm diameter, a second peak in the distribution was observed around 10 nm with 1/3 of the sample area being covered with these larger grains. This bi-modal distribution is not atypical for sputtered films in this thickness range. According to Kim et al. 9 magnetic skyrmions get pinned most strongly by grains of comparable size. In our structures the skyrmion has a diameter of 130 nm-270 nm and hence modelling the disorder with the average large grain size of 10 nm is reasonable for the observed distribution of grain sizes.