Field-free spin-orbit torque perpendicular magnetization switching in ultrathin nanostructures

Magnetization switching through spin-orbit torque (SOT) is of great recent interest for SOT-MRAM because it promises faster write speed, lower write energy, and higher endurance than spin-transfer torque MRAM. A typical SOT-MRAM cell integrates an MTJ on a heavy-metal layer. An in-plane charge current in the heavy metal generates a perpendicular spin current via the spin Hall effect, which then exerts SOT on the magnetic free layer. While SOT has switched both perpendicular and in-plane magnetization, perpendicular switching is preferred for speed and scalability. However, perpendicular SOT switching typically requires an additional in-plane bias magnetic field, which is undesirable due to reduced thermal stability and potential cell crosstalk. Prior approaches to eliminate the external magnetic field include structural asymmetry to create a unidirectional effective field, built-in in-plane bias fields using in-plane magnetized layers, antiferromagnetic heavy metals, ferroelectric layers to leverage polarization or strain, dual heavy-metal layers to generate competing spin currents, adding spin-transfer torque via out-of-plane current, and engineering the ferromagnet geometry. Recent micromagnetic simulations suggested field-free SOT perpendicular switching by jointly controlling lateral size, interfacial DMI, and current density within an intermediate range, without altering standard SOT-MRAM architecture. In this work, we employ both micromagnetic and atomistic spin dynamics (ASD) simulations to study nucleation and growth during the two-step switching. The ASD model resolves atomistic spin moments on the actual lattice, enabling accurate description of early-stage nucleation where embryos are too small for micromagnetics. We show that lateral confinement, interfacial DMI, and SOT synergistically enable deterministic nucleation and growth, yielding field-free perpendicular switching when nanomagnet size, DMI strength, and current are all in intermediate ranges. We further train a decision-tree machine learning model on ~1000 benchmarked micromagnetic datasets to rapidly identify successful switching regimes in the ternary space of size, DMI, and current density.

The paper reviews multiple strategies to achieve field-free SOT-induced perpendicular switching: (1) structural asymmetry such as thickness gradients in the magnetic, oxide, or heavy-metal layers to generate a unidirectional effective perpendicular field; (2) built-in in-plane bias fields via an in-plane magnetized reference layer in the MTJ or antiferromagnetic heavy metals with in-plane magnetized sublattices; (3) integrating ferroelectrics to exploit electric-field-controlled polarization or piezoelectric strain; (4) adding a second heavy-metal layer to provide competing spin currents; (5) applying out-of-plane current to add spin-transfer torque; and (6) geometrical engineering of the ferromagnetic layer. Recent simulations indicated that by appropriately choosing lateral nanomagnet size, interfacial DMI at the ferromagnet/heavy-metal interface, and applied current density, one can achieve deterministic, magnetic-field-free switching without altering the conventional SOT-MRAM stack. This work builds on and extends those findings by providing both micromagnetic and atomistic insights and by delivering a data-driven predictive model.

- Device/geometry: Modeled standard perpendicular SOT-MRAM cell comprising an MgO/Co20Fe60B20 (1.1 nm) bilayer nanodisk on a Pt underlayer; typical disk diameters 40–63 nm (detailed results shown for 40 and 50 nm). Interfacial DMI at the CoFeB/Pt interface is included (measured D ~0.45 mJ/m² for CoFeB/Pt). For ASD studies of atomistic kinetics, a MgO/Co/Pt stack with a 50 nm Co disk is used to demonstrate generality. - Micromagnetic simulations: Performed mainly with the commercial μ-Pro package (CPU/MPI) and validated with MuMax3 (GPU). The CoFeB disk is discretized with 1×1×0.55 nm³ cells (thickness 1.1 nm). Magnetization dynamics follows the LLG equation augmented with SOT effective field from the spin Hall effect in Pt. Effective fields include exchange, interfacial DMI, anisotropy (interfacial plus shape/PMA), magnetostatic (stray) field, and thermal noise (for selected cases). Boundary conditions and FFT-accelerated magnetostatics were used. Representative parameters for Co20Fe60B20 (1.1 nm): Ms=1.21×10^6 A/m, α=0.027, K_inter=1.3 mJ/m², γ≈2.265×10^5 Hz/(A/m), A=1.9×10^−11 J/m, θ_SH=0.0812; D≈0.45 mJ/m² unless varied. For Co (1.1 nm) used in comparative studies: Ms=1.40×10^6 A/m, K_inter≈1.6 mJ/m², A=2.75×10^−11 J/m, D=2.05 mJ/m², α=0.31, γ≈2.21×10^5 Hz/(A/m). - Spin-orbit torque: Implemented as a Slonczewski-like damping-like SOT with effective field proportional to the applied in-plane charge current density Jc in Pt and spin Hall angle θ_SH. The z-component of the SOT torque scales with mx mz (sign set by current polarity and spin polarization +x). Spatial asymmetry arises from edge magnetization tilting due to DMI under lateral confinement, yielding quadrant-dependent signs of τ_SOT and H_SOT that determine nucleation sites. - Simulation protocols: Time-invariant Jc to study >90° switching and equilibrated states; pulsed Jc to realize full 180° switching via two-step process (drive to mz<0 then relax after pulse). Explored parameter sweeps over disk diameter (40–63 nm), interfacial D (0–>3 mJ/m²), and Jc (~10^13 A/m² range). Output metrics include average mz(t), domain wall trajectories, total out-of-plane torque maps, and energy density changes Δf_tot(t). - Atomistic spin dynamics (ASD): Implemented with in-house AtomMag (validated against Fidimag). A 2D hexagonal lattice of 36,096 Co atoms (lattice parameter ~0.25 nm; monolayer ~4 Å thick) approximates a 50 nm disk. The atomistic LLG includes exchange (nearest-neighbor Heisenberg), interfacial DMI (D_ij = D0 ẑ×r_ij), uniaxial anisotropy, dipolar interactions, and SOT (field-like and damping-like terms; σ along +x). Continuum parameters mapped to atomistic counterparts; anisotropy Ku adjusted to compensate for enhanced demagnetization in monolayer approximation, yielding similar edge tilt as micromagnetics. Time integration with 1 fs step. Thermal stochastic field omitted for nucleation-kinetics clarity. - Data analysis: Nucleation/growth quantified by reversed-domain volume fraction (micromagnetics) and percentage of atoms with Sz<0 (ASD). Growth kinetics fitted to Kolmogorov–Avrami equation Vr=V0[1−exp−(t/τ)^n], extracting τ and n. Domain wall curvature and torque asymmetries analyzed at representative times. - Machine learning: 1069 micromagnetic data groups (input Xi: disk diameter, D, Jc; output Y: equilibrated <mz>) split into 989 training and 80 test sets. Evaluated multiple scikit-learn regressors; decision-tree regression achieved best performance (R^2≈0.90, MSE≈0.01). Model used to rapidly predict switching diagrams (Jc vs D at fixed size; or across sizes), reproducing micromagnetic trends with orders-of-magnitude speedup (<2 minutes on a laptop vs ~1000 CPU-hours for full sweeps).

- Deterministic field-free >90° switching: For a 40 nm diameter, 1.1 nm CoFeB disk on Pt with D≈0.45 mJ/m², a time-invariant current density Jc in an intermediate window induces deterministic flipping of most local moments (average mz equilibrates to a negative value). Example: Jc=1.4×10^13 A/m² drives <mz> to ~−0.17 under steady drive, enabling two-step full reversal via pulsed current. - Full 180° switching with unipolar or bipolar pulses: Applying a pulse of the same magnitude Jc (e.g., 1.4×10^13 A/m²) and turning it off when mz crosses its critical state enables relaxation to the opposite perpendicular state. Demonstrations include pulse durations ~3.5 ns and ~0.3 ns. Because symmetry breaking is not required, repeated switching can be achieved with either unipolar or bipolar pulses. Forward (up→down) and backward (down→up) switching exhibit identical Δf_tot(t) profiles, indicating symmetric energy pathways. - Nucleation site and growth asymmetry: With initial +z magnetization and inward edge tilt (due to positive D), nucleation occurs in the bottom-left (third) quadrant where both τ_SOT,z and H_SOT,z are negative under a positive current along −y. Growth is favored in second and third quadrants where τ_tot,z remains negative ahead of the moving wall, driving unidirectional expansion. Backward switching nucleates in the fourth quadrant with signs reversed but analogous dynamics. - Domain wall torque and curvature: Spatial maps show total out-of-plane torque τ_tot negative in front of the wall for most of the process, ensuring deterministic wall advance. Endpoints of the wall experience larger |τ_tot| than the center, yielding a positive curvature at equilibrium. - Size and current windows: Successful >90° switching requires intermediate disk diameters and currents. For D=0.45 mJ/m², the window spans ~40–63 nm diameter with corresponding intermediate Jc ranges; too small disks have enhanced PMA and reduced edge tilt mip, raising nucleation thresholds; below a tilt threshold (~7.22×10^−2 in normalized edge in-plane magnetization), nucleation is suppressed. - DMI and current windows: For a 40 nm disk, >90° switching occurs for D ~0.2–3.2 mJ/m² at intermediate Jc. Increasing D from small values reduces Jc due to larger edge tilt magnitude mip and tilted-volume fraction Vt, lowering nucleation barriers. At larger D (≈1.2–3.2 mJ/m²), Jc increases despite larger mip and Vt due to domain-wall structural instability and reduced mobility (Walker-like breakdown behavior). For D≥~3.4 mJ/m², the system relaxes to Néel stripe domains after current off, yielding mz≈0 instead of a single-domain switched state. - Kinetics follow Avrami scaling: Micromagnetic reversed-domain volume fraction and ASD atomic-flip fraction fit Kolmogorov–Avrami kinetics with effective growth dimension n≈2, confirming two-dimensional domain growth. Characteristic saturation time τ decreases with increasing Jc (e.g., micromagnetics Co: τ from ~0.24 to ~0.12 ns as Jc increases 1.1→1.5×10^13 A/m²). - Atomistic insight into early nucleation: ASD on a 50 nm Co disk (36,096 atoms) shows spatially varying nucleation speeds across clusters P1–P3 tied to local effective SOT fields and initial tilt, with the fastest flipping where the downward-pointing SOT field is strongest. Central atoms in faster clusters traverse polar angles beyond 90° earlier (e.g., θ ~86°, 92°, 93° sequences), corroborating deterministic embryo formation. - Machine learning acceleration: A decision-tree regressor trained on 989 datasets predicts equilibrated <mz> from (size, D, Jc) with R^2≈0.90 and MSE≈0.01. Predicted switching diagrams closely match micromagnetic results (except rare outliers) while computing in under 2 minutes on a laptop, versus thousands of core-hours for full simulation sweeps.

The study addresses the key challenge of achieving perpendicular SOT switching without an external magnetic field and without modifying the standard SOT-MRAM stack. The mechanism relies on the synergy among lateral confinement (which enhances PMA and sets edge tilt volume), interfacial DMI (which imparts a chiral in-plane edge magnetization and sets nucleation quadrant), and current-induced SOT (which provides a quadrant-selective out-of-plane torque). This combination creates a deterministic nucleation site and a unidirectional domain wall drive, enabling >90° switching with steady current and full 180° reversal with pulses. The approach sidesteps symmetry-breaking requirements, allowing both unipolar and bipolar pulsing schemes, which simplifies write circuitry and reduces cross-talk risks. The parameter-space mapping shows that successful switching requires intermediate windows in size, DMI strength, and current. Too small disks or too weak DMI limit edge tilt and increase nucleation barriers, while too strong DMI destabilizes walls, reduces mobility (akin to Walker breakdown), and can trap the system in stripe-domain states. The atomistic simulations provide micro-to-atomic scale consistency with micromagnetics, validate 2D growth kinetics, and reveal local embryo dynamics underpinning deterministic nucleation. The machine learning model offers a practical tool to rapidly identify viable design points across multi-parameter spaces, significantly accelerating device design and optimization.

The work computationally demonstrates magnetic-field-free SOT-mediated perpendicular magnetization switching in ultrathin nanostructures without altering standard SOT-MRAM architecture. Full 180° switching is realized via a deterministic two-step route: a current-driven >90° switching followed by relaxation to the reversed single-domain state upon pulse removal. Deterministic nucleation and directed domain-wall propagation emerge from the synergy of lateral confinement, interfacial DMI, and SOT. The switching occurs only within intermediate ranges of nanomagnet size, DMI strength, and current density. Kinetics conform to Avrami 2D growth, and ASD provides atomistic insight into early nucleation. A decision-tree ML model trained on ~1000 micromagnetic datasets predicts switching outcomes with ~90% accuracy and orders-of-magnitude speedup, enabling rapid exploration of device design spaces. Future directions include experimental validation across material stacks and sizes, inclusion of variability and temperature/noise effects in both micromagnetic and ASD models, extending ML predictors to incorporate additional parameters (e.g., interfacial anisotropy, thickness, spin Hall angle), and optimizing current pulse shapes for energy-delay trade-offs.

- The results are obtained from simulations; experimental confirmation in identical stacks and dimensions is not provided in this work. - ASD uses a 2D monolayer approximation with adjusted anisotropy to compensate enhanced demagnetization, which may deviate from true 1.1 nm films; thermal stochastic fields were omitted in ASD nucleation studies to clarify kinetics. - Material parameters (DMI, anisotropy, θ_SH) are taken from literature and may vary with processing; edge roughness, defects, and variability are not explicitly modeled. - The successful switching region requires intermediate ranges of DMI, size, and current; outside these windows switching fails or results in stripe-domain metastability, potentially limiting design margins. - Machine learning predictions are limited by the training dataset coverage and model type; rare outliers indicate possible regime-specific inaccuracies.