Skyrmions have clear advantages for future spintronic devices over domain walls in terms of information carrying and storage, however there are still many open questions left and inherent challenges in the transition to applications. Here we study the effect of an additional interfacial spin-transfer torque (ISTT) [1], as well as the well-established spin-orbit torque (SOT), on skyrmion collections [2] – group of skyrmions dense enough that they are not isolated from one another – in magnetic multilayers, by comparing modelling using Boris [3], with experimental results.<br/>We show that skyrmion collective motion has additional effects not reproduced when compared to modelled isolated skyrmions [1]. In particular, as the skyrmion collection density increases the effect of local disorder is reduced, leading to reduced threshold current densities, and a sharper dependence of skyrmion Hall Angle (SkHA) on velocity in the depinning regime, in agreement with experimental results. When including both the ISTT and SOT, the small SkHA observed in experiment are in good agreement, also explaining the observed diameter-independent SkHA [4], as seen in Figure 1. Contrary to the model which considers the SOT as the only driving torque, and which predicts a monotonically increasing SkHA with decreasing diameter, inclusion of ISTT results in SkHA closely aligned to experimental results, particularly when the effect of local disorder and skyrmion-skyrmion interactions are taken into account.<br/>Our results shed further light on collective current-driven skyrmion motion in magnetic multilayers and highlight the significance of the ISTT in understanding current-driven skyrmion motion in magnetic multilayers.

![](https://assets.underline.io/uploads/markdown_image/1/image/219d9cf02b4f240924efc702e30f317a.jpg)

Figure 1. Collective SkHA dependence on skyrmion diameter for (a) N = 50 Sk/µm2 and (b) N = 100 Sk/µm2. (c) Experimentally obtained SkHA from skyrmion collections with a range of diameters with an average current density of 2×1011 A/m2. The overlaid dashed and solid lines show the ideal isolated skyrmion case – independent of current density – for the SOT and ISTT+SOT, respectively. The solid blue line in panel (b) represents a Savitzky-Golay fit to the simulated skyrmion motion, for a current density of Jc = 2×1011 A/m2.



Collective Skyrmion Motion Under the Influence of an Additional Interfacial Spin-Transfer Torque

An efficient method for computing thermodynamic equilibrium states at the micromagnetic length scale is introduced [1], based on the Markov chain Monte Carlo method. Trial moves include not only rotations of spins, but also a change in their magnetization length. The method is parameterized using the longitudinal susceptibility, reproduces the same Maxwell-Boltzmann distribution as the stochastic Landau-Lifshitz-Bloch equation, and is applicable both below and above the Curie temperature. The algorithm has been implemented in Boris [2], is fully parallel [3], can be executed on graphical processing units, and efficiently includes the long range dipolar interaction. Applications to finite-temperature hysteresis loop modelling, chiral magnetic thin films, granular magnetic media, and artificial spin ices are discussed. A typical application is exemplified in relation to Figure 1, where finite-temperature hysteresis loops in a 2 nm thin Co film with interfacial Dzyaloshinsky-Moriya interaction are computed. Simulation of such hysteresis loops in chiral films is problematic with quasi-zero temperature methods such as steepest descent energy minimizer, since the nucleation and annihilation of skyrmions and labyrinth domain structure is principally a thermally activated process.<br/><br/>Furthermore, the method is also extended to two-sublattice systems, allowing modelling of ferrimagnetic and antiferromagnetic materials at the micromagnetic length scale. Using a multi-layered computation, which includes both ferromagnetic and antiferromagnetic layers, exchange bias may be simulated in bilayers, as well as more complex multi-layered structures including synthetic antiferromagnetic stacks. Moreover we show the method may also be applied to study of thermal and athermal training effects in exchange bias.

![](https://assets.underline.io/uploads/markdown_image/1/image/c87224dc80ce91d19d61166abaca5e0c.jpg)

Figure 1 - Hysteresis loops in a 2 nm thin Co film on Pt, using a uniform 300 K temperature (thin dashed line), and a Gaussian temperature profile varying from 300 K at extremities to 400 K at the centre (thick dashed line). The insets show the perpendicular magnetization component on the increasing field sweep, with blue denoting magnetization into the plane, and red out of the plane.

Parallel Micromagnetic Monte Carlo Method for Computation of Thermodynamic Equilibrium States in One and Two-Sublattice Systems

**Session chair:** Olivier Boulle, SPINTEC


HOE - Magnetic Skyrmions

technical paper

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This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

**Session chair:** Yasushi Kanai, Niigata Institute of Technology


Callum MacKinnon

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Collective Skyrmion Motion Under the Influence of an Additional Interfacial Spin-Transfer Torque

Parallel Micromagnetic Monte Carlo Method for Computation of Thermodynamic Equilibrium States in One and Two-Sublattice Systems

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