Visualizing the strongly reshaped skyrmion Hall effect in multilayer wire devices

The study investigates how intrinsic topology, geometric confinement, disorder, and skyrmion size shape current-driven skyrmion motion in multilayer wire (racetrack) devices. Prior works established room-temperature skyrmions in chiral multilayers and their manipulation by spin–orbit torques, but practical deployment is hampered by transverse deflection (SkHE), pinning in granular films, edge interactions, and limited understanding at high skyrmion densities. Conflicting experimental reports exist on SkHE magnitude, saturation, and size dependence. The authors aim to quantify ensemble skyrmion dynamics across current and magnetic field, delineate motion regimes (stochastic creep, deterministic creep, plastic flow), measure SkHE behavior including saturation and dependence on edge position and skyrmion size, and reconcile observations with simulations to separate intrinsic from extrinsic (pinning, geometry) contributions.

The introduction surveys: (i) room-temperature skyrmions in heavy metal/ferromagnet multilayers and SOT control; (ii) device concepts for racetrack memory, logic, and neuromorphic computing; (iii) Skyrmion Hall effect arising from Magnus force, beneficial for defect avoidance but potentially detrimental due to large transverse angles (>30° reported); (iv) efforts to reduce or cancel SkHE in ferrimagnetic and synthetic antiferromagnetic systems; (v) extrinsic factors in sputtered films including granularity, defects, and edge interactions with reports of pinning, annihilation, expulsion, or repulsion; and (vi) the need to study dense skyrmion arrays (10–100× higher densities than prior sparse studies). Prior experimental findings on SkHE magnitude, saturation, and size dependence are conflicting, motivating a systematic, statistically significant study.

Experimental: Pt(3)/Co(1.2)/MgO(1.5)]15 multilayers on Si/SiO2 were patterned into 2 µm-wide, 10 µm-long wires. Current pulses (20 ns width, ambipolar, J = ±(1.0–5.8)×10^11 A/m^2) were applied in situ within an MFM under out-of-plane fields µ0H = 75–165 mT. Skyrmions were nucleated via established pulsing under field until stripe domains broke into skyrmions. MFM images (lift 20–30 nm, low-moment tips) were acquired before/after each pulse. An image registration protocol (2D geometric transform in MATLAB) aligned frames; skyrmions were identified and tracked to extract per-pulse displacement vectors, velocity v_s (displacement over effective 20 ns) and deflection angle θ_s relative to current. Over 20,000 motion instances were analyzed. Statistics were compiled across currents and fields, and binned by distance from wire edge (x) and skyrmion diameter d_s (from MFM, 80–200 nm). Motion regimes were identified via proportion moving along current (P_M) and θ_s(v_s) saturation. Fits: v_s(J) with an exponential function and θ_s(v_s) with a sigmoidal function; creep vs plastic flow demarcated at θ_s = 0.5 θ_sat. Micromagnetic simulations: MuMax3 effective medium model of a 2×4 µm^2 wire with parameters rescaled from experiments (A = 5.05 pJ/m, M_s = 0.23 MA/m, K = 0.070 MJ/m^3, D = 0.37 mJ/m^2, α = 0.05). Anti-damping-like SOT with effective spin-Hall angle 0.1; zero temperature; no field-like torque; grain-free. Varied µ0H (84–93 mT) to tune d_s (129–164 nm). Simulated current-driven motion at J = 9.5×10^11 A/m^2 to extract v and θ_s vs d_s. Particle model simulations: Modified Thiele equation for point-like skyrmions with damping α_d = 1.34 and Magnus α_m = 1.0 (intrinsic θ_i = 37°), skyrmion-skyrmion repulsion, and pinning via localized harmonic traps of radius R_pin and strength F_pin = 1.0. Simulations with 1500 particles and 1200 pinning sites under fixed drive in the plastic flow regime. Varied R_pin to emulate effective size changes (larger R_pin ≈ smaller d_s). Measured v_s and θ_s as functions of 1/R_pin ∝ d_s. Additional runs with θ_i = 45° tested robustness.

- Motion regimes: With increasing J, fraction moving along current P_M increases exponentially, reaching ~90% at J ≈ 5.8×10^11 A/m^2, indicating deterministic motion. Creep and plastic flow regimes are distinguished via θ_s(v_s): θ_s grows sigmoidal with v_s and saturates, marking plastic flow onset. - Velocity: Average v_s increases exponentially with J across fields, reaching ~8 m/s at J ≈ 5.8×10^11 A/m^2, with individual skyrmions exceeding 24 m/s. The exponential v_s(J) trend is field-independent, though higher fields increase threshold current and reduce viable current window due to annihilation and stronger pinning. - Skyrmion Hall effect: θ_s increases with v_s and saturates at θ_sat ≈ 22° around v_s ≈ 12 m/s. This saturation and θ_s(v_s) profile are consistent over µ0H = 75–165 mT. The saturated SkHE is notably smaller than typical 40–70° in ferromagnetic multilayers and comparable to ferrimagnetic values (25–35°). - Edge effects (plastic flow): v_s(x) remains high and nearly uniform (~8–10 m/s) across the 2 µm width, with a modest ~20% decrease near the left edge (deflection direction), indicating weakly inelastic edge interactions. θ_s(x) varies parabolically: ~-5° at the left edge, ~+25° at the center (≈ θ_sat), and ~+10° near the right edge. Trends are robust to reversing current direction and to higher field (105 mT), ruling out Oersted fields and highlighting deterministic extrinsic shaping by edge and neighbor interactions. - Size effects (plastic flow): v_s is largely independent of d_s across 80–200 nm (variation ≈10% for fixed J). θ_s increases weakly with d_s across all plastic-flow currents: from ~5–10° for d_s ≤ 100 nm to ~20° for d_s ≈ 200 nm. This contradicts rigid-skyrmion Thiele predictions (θ_s ∝ 1/d_s) and prior creep-regime observations, but is robust across datasets and devices. - Simulations: Grain-free micromagnetics reproduce intrinsic behavior where v increases then saturates with d_s ≥ 150 nm and θ_s decreases from ~40° to ~20° as d_s increases—consistent with rigid-skyrmion theory—thus not explaining the experimental θ_s(d_s) trend in plastic flow. Particle model with pinning shows θ_s decreases strongly for smaller effective d_s (larger R_pin), dropping well below intrinsic θ_i (37–45°), qualitatively matching experiments; v_s decreases with increased pinning (smaller d_s). Results suggest pinning and disorder reshape the effective SkHE, overpowering intrinsic size trends in plastic flow.

The findings demonstrate that in dense skyrmion arrays within multilayer wires, extrinsic factors—disorder-induced pinning, geometric confinement by edges, and inter-skyrmion interactions—significantly reshape the observable SkHE while leaving longitudinal mobility largely intact in plastic flow. The measured θ_sat ≈ 22° indicates an intrinsically moderate Hall response in these stacks or an effective reduction due to disorder, comparable to compensated systems, alleviating concerns about excessive transverse drift in device operation. The edge produces weakly inelastic interactions that deflect skyrmions without annihilation or strong pinning, enabling stable guidance across the width. The weak positive θ_s(d_s) dependence, opposite to rigid-skyrmion predictions, is reconciled by particle-model simulations showing that pinning increasingly suppresses transverse response of smaller skyrmions, reducing their effective Hall angle below the intrinsic value. Meanwhile, v_s insensitivity to size likely reflects internal texture (bubble-like) effects captured by micromagnetics, decoupling longitudinal mobility from pinning-driven θ_s reshaping. Together, these results delineate a practical operating window—plastic flow at moderate currents—where high-throughput, largely linear transport is achievable, while acknowledging and leveraging deterministic extrinsic influences.

The study provides a comprehensive, statistically robust mapping of skyrmion ensemble dynamics in Pt/Co/MgO wire devices. It identifies transitions from stochastic to deterministic creep and to plastic flow, quantifies a reproducible SkHE saturation at ~22°, and reveals that while skyrmion speed is robust to edge position and size, the SkHE is strongly modulated by both. Contradicting rigid, defect-free theoretical expectations, the SkHE increases weakly with skyrmion size in plastic flow. Particle-model simulations attribute this to the interplay of intrinsic Hall response and size-dependent pinning effectiveness, while micromagnetics capture intrinsic trends but not the extrinsic reshaping. Implications include: (i) conventional ferromagnets can exhibit favorable, modest SkHE for devices; (ii) edges can be engineered to guide motion without deleterious annihilation; (iii) material granularity and inter-skyrmion interactions can be used to tailor size-dependent dynamics. Future work should incorporate thermal effects, field-like torques, grain structures, and high-density interactions in simulations, and explore device geometries leveraging controlled edge and pinning landscapes for bespoke Hall responses and neuromorphic functionalities.

- The plastic-flow current window is narrow (J ≈ 5.5–5.8×10^11 A/m^2), limiting analysis of J-dependent trends in θ_s(d_s). - Micromagnetic simulations were grain-free, at zero temperature, and neglected field-like torque, thus not capturing disorder-induced reshaping of SkHE observed experimentally. - Effective medium modeling of the multilayer and finite system size may omit multilayer-specific interactions or thermal activation effects. - Particle model treats skyrmions as point particles and cannot capture internal texture effects (e.g., bubble-like modes) that influence v_s and damping in smaller skyrmions. - Increased field reduces viable current range due to annihilation and stronger pinning, constraining exploration at higher µ0H.