Weakly pinned skyrmion liquid in a magnetic heterostructure

Magnetic skyrmions are topologically distinct particles whose thermally activated motion could be used to implement probabilistic computing paradigms. While solid-liquid phase transitions in skyrmion lattices have been demonstrated, the behavior of a skyrmion liquid and the effects of pinning are largely unknown. Here we demonstrate the formation of a weakly pinned skyrmion liquid in a magnetic heterostructure. By inserting a Ru wedge layer at the ferromagnet/heavy metal interface we evaluate the dependence of skyrmion dynamics on the skyrmion size and density. Our experiments demonstrate that the diffusion of skyrmions is largest in dense liquids with small skyrmions. The thermal motion of skyrmions at room temperature easily overcomes the narrow distribution of pinning site energies in the granular film structure, satisfying a key requirement of probabilistic device architectures. Micromagnetic simulations support the findings and also reveal the existence of a thermally activated high-frequency collective oscillation.

Figure 1

(a) Schematic of the Ta/Pt/CoFeB/Ru wedge/Pt multilayer sample. (b) Perpendicular MOKE measurement of the sample under out-of-plane applied fields at the same position (V) on the wedge as the following images. The MOKE signal is averaged over the $88\times 67$ $\mu {\mathrm{m}}^2$ image. (c)–(j) Polar MOKE microscopy images showing magnetic contrast at one position on the wedge as a function of applied magnetic field, with (c) 0.0 mT, (d) 0.05 mT, (e) 0.10 mT, (f) 0.15 mT, (g) 0.21 mT, (h) 0.26 mT, (i) 0.31 mT, and (j) 0.36 mT. The scale bar in (c) is 20 $\mu \mathrm{m}$ in length. All images are at the same scale.

Figure 2

MOKE microscopy images of the skyrmion state at 0.26 mT applied out-of-plane field for different thicknesses of the Ru wedge. The positions are referred to in the text using Roman numerals. For increasing Ru thickness the skyrmion densities in skyrmions/$\mu {\mathrm{m}}^2$ are (a) 0.07 (I), (b) 0.11 (II), (c) 0.15 (III), (d) 0.13 (IV), (e) 0.17 (V), and (f) 0.20 (VI). The scale bar indicates 10 $\mu \mathrm{m}$.

Figure 3

(a) Zero-magnetic field MOKE microscopy image showing a stripe domain pattern at position (II). The scale bar indicates 20 $\mu \mathrm{m}$. (b) At the same position (II) as (a) a skyrmion array with a density of 0.11 skyrmions/$\mu {\mathrm{m}}^2$ forms in 0.26 mT applied field. (c) and (d) are as (a) and (b) but at position (V) where the skyrmion array has a density of 0.17 skyrmions/$\mu {\mathrm{m}}^2$. The insets in (a)–(d) show the static structure functions of the microscopy images. (e) Radial pair distribution function for the skyrmion arrays at 0.26 mT at position (II) and (f) position (V). (g) Skyrmion-skyrmion pair interaction derived from the skyrmion arrays at 0.26 mT at position (II) and (h) position (V).

Figure 4

(a) Radially averaged static structure functions at position (II) for different applied magnetic fields, corresponding to the images shown in Fig. 1. (b) Correlation length as a function of applied magnetic field at the six sample positions with different Ru thickness. (c) Normalized intermediate scattering function as a function of wave vector, shown for different lag times, $\mathrm{\Delta }t$ at position (II). (d) Normalized intermediate scattering function against lag time at 0.26 mT applied field for the six sample positions. (e) Decay time of the intermediate scattering function peak at the six sample positions as a function of applied magnetic field. (f) Decay time of the intermediate scattering function peak as a function of skyrmion radius. (g) Decay time of the intermediate scattering function peak as a function of skyrmion density. (h) Dependence of the normalized intermediate scattering time on a wait time for positions (II) and (V).

Figure 5

(a)–(e) Images of a few skyrmions at position (V) at different times. (f) Extracted pinning sites from averaged video stills, a whiter color within the red circle indicates greater occupation of a site. (g) Histogram of skyrmion occupation time calculated using $8\times 8$ pixel areas. (h) Two-site correlation function calculated using the pinning probability (left-hand axis) plotted with the radial pair distribution function (right-hand axis). See text for details.

Figure 6

(a) Snapshot of part of a micromagnetic simulation at 160 mT applied field. Inset shows the static structure function. (b) Intermediate scattering function as a function of momentum for different delay times. (c) Self-intermediate scattering function as a function of delay time $k$ increases in the direction of the arrow from $2.7\times {10}^6$ to $5.0\times {10}^7{\mathrm{m}}^{-1}$ in steps of $2.4\times {10}^6{\mathrm{m}}^{-1}$. (d) Mean square displacement of the skyrmions $\langle \mathrm{\Delta }{r}^2\rangle$ as a function of lag time $\mathrm{\Delta }t$. The inset shows $\langle \mathrm{\Delta }{r}^2\rangle /\mathrm{\Delta }t$, which is fitted to $D_0[1+4/ln\left(\mathrm{\Delta }t\right)]$ where $D_0$ is the diffusion coefficient. (e) Histogram of skyrmion occupation time calculated using $8\times 8$ pixel areas. (f) Two-site correlation function calculated using the pinning probability (left-hand axis) plotted with the radial pair distribution function (right-hand axis).

Figure 7

(a) The sixfold bond-orientational order parameter calculated for simulations with (black) and without (red) pinning as a function of temperature. Magnetization snapshots from simulations as a function of temperature with grains at (b) 10 K, (c) 50 K, (d) 100 K, (e) 200 K and without grains at (f) 10 K, (g) 50 K, (h) 100 K, (i) 200 K. The insets show the static structure function.

Figure 8

(a) The time dependence of $M_z$ spatially averaged across the simulation. The inset shows the FFT of the data (black) with a Savitsky-Golay average (red). (b) The time dependence of $M_x$ (black) and $M_y$ (red) spatially averaged across the simulation. The inset shows the FFT of the $M_x$ data (black) with a Savitsky-Golay average (red). (c) The detrended intermediate scattering function at four different particular wave vectors at 300 K. The curves are offset for clarity. (d) The detrended intermediate scattering function as a function of temperature at a wave vector of $4\times {10}^6{\mathrm{m}}^{-1}$.