Spin-orbit torques and their associated effective fields from gigahertz to terahertz

Ultrafast manipulation of magnetisation is central to envisioned spintronic technologies, relying on torques generated either by spin-polarized currents (spin-transfer torque) or by spin currents and accumulations induced by spin–orbit interaction (spin-orbit torques, SOTs). SOTs enable efficient switching in systems with heavy metals or topological insulators and are relevant from GHz to THz frequencies, including for antiferromagnets. Despite their importance for ultrafast applications, the frequency dependence of SOTs has remained largely unexplored in realistic, material-specific treatments; experiments often infer it indirectly and theory has mostly focused on static perturbations using phenomenological dynamics. The research question addressed here is how SOTs and their associated effective magnetic fields depend on frequency and magnetisation direction in realistic ferromagnet/heavy-metal bilayers, and whether additional torque components arise that can affect the magnetisation magnitude. The purpose is to provide a microscopic, dynamical understanding of SOTs from GHz to THz and to clarify how effective fields relate to torques across this range, informing control strategies for ultrafast spintronic devices.

Experimental studies typically quantify SOTs via angular dependence with respect to magnetisation and probe frequency dependence indirectly using spin-torque ferromagnetic resonance, or assess quasi-static effective fields via second-harmonic techniques. First-principles and realistic electronic-structure calculations have predicted SOT angular dependences but largely in the static limit, leaving frequency dependence to phenomenological magnetisation dynamics models. Conventionally, SOTs are decomposed into field-like (odd in magnetisation) and damping-like (even) components, linked to precession and alignment/antialignment with spin accumulation; specific combinations can deterministically switch magnetisation without external fields. Standard theoretical approaches often separate charge carriers and the ferromagnetic unit with sd-like coupling and use static spin continuity, or model torque as M × exchange-correlation field; these approximations neglect full time-dependent, coupled charge–spin dynamics. This work addresses these gaps with a unified microscopic framework incorporating both magnetic and charge responses and spin–orbit coupling, enabling direct computation of frequency- and angle-dependent SOTs and effective fields.

The study uses a microscopic atomistic theory based on the spin continuity equation for local spin moments Mi(t), including spin currents to neighbors, Zeeman torque from an external magnetic field, and local spin–orbit torque from atomic SOI. Bilayer systems with a ferromagnet (FM) on a heavy metal (HM) are considered, focusing on C2v-symmetric Fe/W(110) (in-plane easy axis along x), with Co/Pt(001) results provided in Supplementary Information. An external spatially uniform AC electric field E(t) = E0 cos(ωt) is applied along x, generating spin currents along ẑ and a spin accumulation δs ∝ ẑ × E(t) (polarized along ŷ) at the FM/HM interface; the electric field also perturbs orbital angular momentum in the FM. The magnetisation is reoriented within selected planes by an external magnetic field to probe angular dependence. Within linear response theory (Kubo formalism) and random phase approximation, the time-dependent SOT is computed via the perturbing vector potential A(t) ∝ −sin(ωt). The torque is decomposed into in-phase and out-of-phase components relative to the driving field: τSOT(t) = E0[τin(ω) cos(ωt) − τout(ω) sin(ωt)], with amplitude τSOT(ω) and phase φ(ω). In the local frame, τSOT(ω) = τFL(ω) m × δs + τDL(ω) m × (m × δs) + τ||(ω) m||, identifying field-like, damping-like (antidamping-like), and longitudinal components. Effective magnetic fields are obtained by relating the magnetisation response to both the charge-current vector potential and to a fictitious magnetic field via susceptibilities, yielding Beff(ω) from δM(ω) = χMA(ω)A(ω) and δM(ω) = χMB(ω)B(ω), leading to Beff that reproduces the same δM under E(t); Beff is similarly decomposed into field-like, damping-like, and longitudinal projections and into in-phase/out-of-phase parts. Electronic structure is modeled by a multi-orbital tight-binding Hamiltonian with nine orbitals per site (s,p,d) and two spin channels for one FM layer atop four HM layers. Parameters are derived from DFT RS-LMTO-ASA calculations to obtain hopping and on-site terms. An effective intra-atomic Coulomb interaction U acts on d orbitals at mean-field level, and an on-site spin–orbit term HSO = −∑ λ L·S (λ from RS-LMTO-ASA) is included for d orbitals. Ground-state magnetisation is computed self-consistently by adjusting d-level centers to match DFT occupations at fixed Fermi energy; a constant energy-level broadening of 68 meV is used. Magnetic and torque response functions are calculated within RPA. Frequency sweeps cover static to THz (including ferromagnetic resonance), and angular sweeps rotate magnetisation in zx, zy, and xy planes. Ratios of damping-like to field-like components are evaluated for in-phase, out-of-phase, and amplitude responses.

• Strong frequency dependence of SOTs: Near ferromagnetic resonance, all components of the torque exhibit large magnitude changes (exceeding an order of magnitude relative to static values) and sign reversals, with pronounced angular dependence due to magnetisation-direction-dependent resonance frequencies.
• Discovery of a longitudinal torque component τ||(ω): Even when the magnetisation aligns with the spin accumulation M || δs, a nonvanishing longitudinal component appears in the local-frame projection. It can modify the magnetisation magnitude; its amplitude is about one order of magnitude smaller than the transverse SOTs. Resonant features in τ|| originate from coupling to transverse excitations via spin–orbit interaction.
• Effective magnetic fields are nearly frequency independent: Despite strong dynamical changes in torques and magnetisation, the associated effective fields Beff(ω) show weak dependence on frequency from the static limit up to THz, including at resonance. This arises from cancellation of the interaction-renormalization factor (1 + χ(ω)U)−1 between magnetic-charge-current and magnetic susceptibilities. Angular variations in Beff projections are present, with higher-order magnetisation-direction terms notably affecting the field-like component.
• Tunability via frequency and phase: Ratios TDL/TFL for in-phase and out-of-phase components become large where TFL crosses zero; amplitude ratios indicate damping-like torques dominate at low frequencies. This suggests control of torque composition by frequency choice or pulse timing (to select in-phase vs out-of-phase response).
• Material-specific example (Fe/W(110)): For M along z, a resonance around ~1.5 THz is observed; large enhancements of torque components occur for rotations in zx and zy planes. Longitudinal effective fields are comparable in magnitude to transverse ones, yet yield weaker longitudinal SOTs due to smaller longitudinal magnetic responses (excitation energies in the eV range).
• Implications for transport effects: The longitudinal SOT component, tied to spin accumulation relative to M, relates to unidirectional spin Hall magnetoresistance (USMR). Enhanced AC-USMR is predicted when M is slightly tilted from δs and driven near resonance.

The results close a key knowledge gap by providing a microscopic, frequency-resolved description of SOTs and their effective fields in realistic bilayers. They show that while torques vary strongly with frequency, the effective fields producing equivalent magnetisation dynamics remain largely constant across GHz–THz, indicating that charge-to-spin conversion occurs on timescales much faster than spin dynamics. Consequently, the common static relation between torque and effective field cannot be naively generalized to finite frequencies; Beff must be computed consistently from coupled charge–spin response functions, and the appropriate dynamical torque in atomistic spin dynamics is m(t) × Beff(t). Experimentally, quasi-static effective fields measured by second-harmonic techniques can be used to interpret THz dynamics in typical systems. However, in materials with electronic structures supporting strong interband transitions in the THz range, Beff(ω) may acquire significant frequency dependence. The pronounced frequency and phase sensitivities of SOTs suggest practical control strategies: tailoring frequency and timing (e.g., two-pulse protocols) to preferentially enhance in-phase or out-of-phase torque components, optimizing switching or precessional control without large static assisting fields. The identification of a longitudinal torque component provides a route to manipulate magnetisation magnitude and to enhance AC-USMR signals by small angular offsets near resonance, offering new read/write avenues in spintronic devices.

This work presents a unified microscopic framework that captures the full frequency and angular dependence of spin–orbit torques and their associated effective magnetic fields in ferromagnet/heavy-metal bilayers. It reveals (i) substantial dynamical modifications of SOTs around ferromagnetic resonance, (ii) a finite longitudinal torque capable of altering magnetisation length, and (iii) an almost frequency-independent effective field across GHz–THz in typical systems. These insights enable frequency- and phase-based control of torque composition and magnetisation dynamics, suggesting protocols using tailored pulses to optimize switching and detection, and indicating opportunities to enhance AC-USMR. Future research directions include extending calculations to materials with strong THz interband transitions to map regimes where Beff(ω) gains structure, exploring antiferromagnetic systems and intrinsically 2D magnets for ultrafast control, and integrating temperature and disorder effects to guide device-level implementations.

Findings are based on linear response (small-amplitude excitations) within a tight-binding framework parametrized from DFT and mean-field treatment of Coulomb interactions, with a constant broadening. Results are primarily demonstrated for a monolayer Fe/W(110) bilayer (with Co/Pt(001) in Supplementary), so quantitative conclusions are material- and geometry-specific. The near frequency-independence of effective fields may not hold in systems with strong THz interband transitions or where charge–spin conversion timescales approach spin-dynamic timescales. Effects of temperature, disorder, and multilayer thickness variations are not explicitly treated.