Magnetic droplet soliton pairs

The study addresses whether the reference layer (RL) in spin-torque nano-oscillators with perpendicular magnetic anisotropy can itself host and dynamically sustain a magnetic droplet soliton alongside the well-studied free-layer (FL) droplet. Solitons, including magnetic droplets, are particle-like solutions of nonlinear wave equations and have been widely explored theoretically, numerically, and experimentally, with most magnetic droplet work focusing on the FL in STNOs. However, at high currents the RL can become dynamically unstable (e.g., back-hopping and RL modes have been reported in MTJs), suggesting that RL dynamics may be significant. This work tests the hypothesis that the RL is not merely a static polarizer at high current densities, but can nucleate a droplet driven by spin-transfer torque and coexist with an FL droplet. Demonstrating and characterizing such droplet pairs is important for understanding coupled nonlinear soliton dynamics and for potential applications leveraging their tunable, broadband microwave signatures.

Prior work established magnetic droplets in STNOs and related nanostructures, often treating the RL as a fixed polarizer while focusing on FL solitons. Droplets have also been realized with pure spin currents (spin Hall effect) and explored in simulations via VCMA. Extensive theory describes dissipative droplets, their nucleation boundaries, drift instabilities, and interactions, including multi-contact configurations. In parallel, device physics has documented RL instabilities such as back-hopping and RL eigenmodes at high bias in MTJs, indicating that RL dynamics can be excited by STT. Zhang–Li torque effects due to lateral current components can modify droplet size and stability, particularly with polarity-dependent inward/outward pressure on droplet perimeters. These bodies of work motivate exploring RL participation in droplet formation and coupled FL–RL soliton dynamics in all-perpendicular STNOs.

Devices: Nanocontact STNOs with strong PMA in both layers. Multilayer stack on oxidized Si: seed Ta(4)/Cu(14)/Ta(4)/Pd(2); all-perpendicular pseudo-spin valve [Co(0.35)/Pd(0.7)]×5/Co(0.35)/Cu(5)/[Co(0.22)/Ni(0.68)]×4/Co(0.22); cap Cu(2)/Pd(2). Mesas (8 µm × 16 µm) patterned by optical lithography and dry etching; insulated by 30-nm SiO2 (CVD). Nanocontacts (50–150 nm diameter; device used: 60 nm) fabricated by EBL and RIE through SiO2; top metallization 500 nm Cu/100 nm Au by lift-off. Measurements: Magnetic field applied normal to film plane; custom 40-GHz probe station allowed control of field strength/polarity/direction. Device contacted with GSG probe via 40-GHz bias-Tee. DC current from Keithley 6221; dc voltage measured by Keithley 2182. Negative current defined as electron flow from FL to RL. Microwave signal emitted upon auto-oscillation was coupled via bias-Tee, amplified with a low-noise amplifier, and recorded by spectrum analyzer (R&S FSU 20 Hz–67 GHz), 5 MHz resolution bandwidth. For resistance-based phase identification, absolute Rdc and differential resistance dV/dI were recorded; constant device resistance and parabolic Joule-heating background were subtracted in plots. Micromagnetic simulations: GPU-based MuMax3. Simulation grid 512×512×3 cells; cell size 3.90625×3.90625×3.90625 nm³. Region1 (bottom) modeled RL ([Co/Pd]); Region2 (top) FL ([Co/Ni]); middle layer Cu spacer. Thicknesses: RL 7.8125 nm, FL 3.90625 nm. Drive current region: cylindrical nanocontact of 80-nm diameter; Oersted field included. Zhang–Li torque omitted (simulated current path without lateral component). Interlayer exchange coupling set to 0. Magnetic parameters: Ku (FL)=340 kJ/m³, Ku (RL)=375 kJ/m³ (from out-of-plane FMR); Ms (FL)=716.2 kA/m, Ms (RL)=730 kA/m; γ/2π=28 GHz/T; Aex=10 pJ/m; damping α=0.03; current polarization P=0.4; spin-torque asymmetry parameter Λ=1.3. To emulate back-hopping and mutual STT, at each time step the real-time magnetization of one layer served as the polarization for the other. Absorbing boundary conditions via smoothly increasing edge damping. Applied field and initial magnetization set to 89.7° relative to plane to avoid singularities and mimic setup uncertainty. Magnetization components averaged over 128×128 cells (≈500×500 nm²) around the NC (NC region ≈2% of sampled area); data sampled every 6 ps. Simulations primarily at T=0 K; an additional finite-temperature run at T=300 K was performed for a periodic DP case to assess thermal effects.

• Experimental demonstration that both FL and RL can host magnetic droplet solitons that coexist as a droplet pair (DP) in all-perpendicular STNOs.
• Identification of phases via Rdc, dV/dI, and PSD (frequency 0–5 GHz): parallel (P), anti-parallel (AP), single FL droplet (D), AP single droplet (AP-D), and droplet pairs in P (DP) and AP (AP-DP) configurations.
• Stable single FL droplet nucleation at μ0H=+0.2 T with negative current: D forms at I=−4 mA (j≈−1.4×10^8 A/cm²), seen as a clear Rdc step and weak, narrow microwave signal; remains highly stable over ~10 mA current range with minimal noise.
• With increasing |I|, transition to DP marked by dramatic onset of low-frequency broadband microwave noise and a decrease in Rdc that saturates around ~17 mΩ (about halfway between P and D), indicating reduced stability with annihilation/re-nucleation and drift.
• At μ0H=−0.2 T (AP background): AP-D nucleates at I=+6.8 mA (j≈+2.4×10^8 A/cm²) with small ΔRdc≈6 mΩ (vs ≈40 mΩ for P-state D), indicating smaller AP-droplet size; AP-D becomes unstable around I≈+10 mA (j≈3.5×10^8 A/cm²). AP-DP emerges at I≈+13 mA (j≈4.6×10^8 A/cm²) with high noise. RL reverses under sufficient negative current, switching AP→P and reestablishing a stable D.
• Zhang–Li torque from lateral current components yields polarity-dependent inward/outward pressure, breaking P/AP symmetry: AP-droplets are smaller; PSD power and shape differ between P-DP and AP-DP; P-DP exhibits roughly twice the maximum integrated power compared to AP-DP, with low-frequency peaks (0–1 GHz) shifting to higher frequencies with increasing field.
• Phase diagrams (current–field) from dV/dI and integrated PSD map broad stability of single droplets (linear STT-like nucleation boundaries) and regions of DP; at |μ0H|≳0.46 T, increased noise and smeared boundaries complicate distinguishing D-drift from DP.
• Distinguishing features in field sweeps: DP noise occupies lower frequencies with a two-peak-like PSD, whereas D-drift noise diminishes toward 0 GHz; in DP regime Rdc often has positive slope; D is flat; drift shows descending Rdc.
• Micromagnetic simulations corroborate RL droplets and classify DP dynamics into quasi-periodic, periodic, and chaotic regimes controlled by field and current. • Ordinary FL droplet at μ0H=0.5 T, I=−4 mA: uniform precession at ~14 GHz (near Zeeman limit), slowed by interlayer interaction; no drift; tiny RL response at same frequency. • Quasi-periodic DP at μ0H=0.5 T, I=−6 mA: alternating expansion/contraction between FL and RL droplets; time scales: fast precession ~14 GHz; small wiggles period ~0.45 ns; large Mz peaks ~every 2 ns; occasional disruptions every 10–15 ns. • Periodic DP at μ0H=0.5 T, I=−12 mA: both droplets present continuously; mutual gyration ~2.4 GHz around near-center points; mutual precession frequencies nearly equal and reduced to ~12.7 GHz (sub-Zeeman), highlighting strong coupling. • Chaotic DP at μ0H=0.1 T, I=−6 mA: intermittent presence/absence of droplets, lack of correlation between FL and RL Mz; in-plane uniform precession only when RL droplet absent.
• Finite-temperature (300 K) simulation of periodic DP preserves overall behavior and 12.7 GHz precession but introduces random fluctuations, intermittent RL droplet disappearance, and enhanced low-frequency noise—consistent with experimental broadband noise.
• Experimentally, the largest broadband noise signals appear at highest currents, likely dominated by droplet drift; a fully periodic DP state was not unambiguously observed in experiments.

The results directly demonstrate that the reference layer in all-perpendicular STNOs can host a spin-torque-driven magnetic droplet that coexists and interacts strongly with the conventional free-layer droplet. This addresses the core question of RL involvement and shows that RL cannot be treated as a static polarizer at high current densities. The coexistence leads to rich nonlinear dynamics manifesting as strong, broadband microwave noise and distinct resistance signatures. The polarity-dependent Zhang–Li torque breaks P/AP symmetry, explaining observed differences in droplet sizes, transition thresholds, PSD power, and spectral shapes. Micromagnetic simulations reproduce RL droplet formation and reveal that the pair dynamics can be quasi-periodic, periodic, or chaotic depending on field and current, with characteristic precession and modulation time scales that align with the experimentally observed low-frequency noise envelopes (peaks around ~0.15 GHz and broad falloff). Simulations further explain sub-Zeeman precession and mutual frequency locking in periodic DP due to strong interlayer coupling through STT. Including finite temperature induces stochastic fluctuations that enhance drift and low-frequency noise, matching experiment. Overall, the findings broaden the STNO phase space to include coupled soliton pairs and offer insights into nonlinear interaction mechanisms relevant across soliton-bearing systems.

This work establishes that both the free and reference layers in all-perpendicular STNOs can support coexisting magnetic droplet solitons, expanding the known current–field phase diagram to include droplet pair states. Experimentally, single FL droplets are highly stable with minimal noise, while droplet pairs exhibit strong, broadband microwave noise and reduced resistance—signatures of intermittent dynamics and drift. Micromagnetic simulations corroborate RL droplet formation and classify pair dynamics into periodic, quasi-periodic, and chaotic regimes, with coupling effects producing sub-Zeeman precession and near-equal frequencies across layers. These results underscore the need to account for RL dynamics in spin-torque-driven solitonic phenomena and suggest practical applications such as tunable incoherent microwave sources (radio lighting). Future research directions include: systematic exploration of temperature effects; incorporating Zhang–Li torque and spin-diffusion in simulations; varying interlayer exchange coupling via spacer thickness to tune interaction strength; direct time-resolved imaging of droplet pair dynamics; and engineering device geometries to stabilize periodic DP regimes for coherent multi-soliton functionalities.

• Experimental frequency readout of droplet perimeter precession via GMR is suppressed due to all-perpendicular geometry; phase identification relies on Rdc, dV/dI, and broadband PSD, which can blur distinctions between D-drift and DP at high fields/currents.
• Some regions (|μ0H|≳0.46 T, high |I|) show smeared transitions and overlapping noise signatures, complicating unambiguous phase assignment.
• Micromagnetic model simplifications: zero interlayer exchange coupling; omission of Zhang–Li torque; simplified current path; zero-temperature assumption for most runs; MuMax default of fixed polarizer overcome by a custom mutual-polarization scheme but still an approximation. Thermal and spin-transport effects at high currents are not fully captured.
• Simulated NC diameter (80 nm) differs from measured device (60 nm), limiting quantitative correspondence.
• A periodic DP regime evident in simulations was not clearly observed experimentally, possibly due to thermal/drift effects and device-specific parameters.