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

The effective manipulation of magnetization at ultrafast speeds is crucial for numerous technological applications. Spin-orbit torques (SOTs), induced by spin currents and accumulations from spin-orbit interaction (SOI), offer an efficient method for magnetization switching in heavy metal-containing systems at room temperature. SOTs are also active in laser-excited ferromagnetic structures in the terahertz (THz) range, relevant to the frequencies of antiferromagnetic material excitations. The application of current-induced torques for magnetic state manipulation opens possibilities for future technologies, including memristors and neuromorphic computing. However, the dynamic behavior of SOTs at high frequencies remains largely unexplored, both experimentally and theoretically, within a realistic material-specific framework. Existing experimental methods, such as spin-torque ferromagnetic resonance and second-harmonic techniques, provide indirect measurements or are limited to the quasi-static regime. Theoretical calculations based on realistic electronic structures are often restricted to static perturbations, employing phenomenological equations to describe magnetization dynamics. This gap in understanding the time or frequency dependence of SOTs at high frequencies limits our ability to fully utilize their potential for ultrafast applications. SOTs are conventionally decomposed into field-like and damping-like components. While field-like torques induce precessional motion, damping-like torques influence the magnitude and direction of the magnetic moment. Effective manipulation typically requires substantial damping-like torques, although field-like torques can also induce switching with assisting fields. This research aims to address the knowledge gap by investigating SOTs from a microscopic theory of magnetization dynamics in realistic materials, considering collective spin excitations and intrinsic interfacial and bulk spin-orbit-related mechanisms.

Several studies have explored SOTs experimentally by analyzing their angular dependence relative to the magnetization direction and their frequency dependence via spin-torque ferromagnetic resonance. These methods, however, often provide indirect measurements of frequency dependence through rectified voltage or second-harmonic techniques, limited to quasi-static regimes. Theoretically, some work has calculated SOTs using realistic electronic structures, predicting angular dependencies but often neglecting frequency dependence due to limitations to static perturbations. Consequently, the literature lacks a comprehensive understanding of the high-frequency behavior of SOTs.

This work investigates spin-orbit torques using a microscopic theory of magnetization dynamics in realistic materials. This theory incorporates collective spin excitations in the presence of spin-orbit interaction, inherently accounting for intrinsic interfacial and bulk spin-orbit-related mechanisms such as the inverse spin galvanic effect and the spin Hall effect. The study uses a multi-orbital tight-binding Hamiltonian, with parameters obtained from density functional theory (DFT) calculations using the real-space linear-muffin-tin orbitals method within the atomic sphere approximation (RS-LMTO-ASA). This Hamiltonian captures the electronic structure of ferromagnetic/heavy metal bilayers. An effective intra-atomic Coulomb interaction (U) and an on-site spin-orbit interaction (H_SO) are included. The ground-state magnetization is computed self-consistently. The magnetization dynamics induced by an external AC electric field are described using the spin continuity equation, incorporating spin currents, Zeeman torque, and local spin-orbit torque. The spin-orbit torques are obtained within linear response theory, considering both in-phase and out-of-phase contributions to the external electric field. The torques are decomposed into field-like, damping-like, and longitudinal components in the local frame of reference of the atomic magnetic moments. The effective magnetic field, which produces an equivalent perturbation on the magnetic moments, is calculated using linear response theory and expressed in terms of the magnetic-charge current response and magnetic susceptibility. The analysis focuses on Fe/W(110) and Co/Pt(001) bilayers, examining the frequency and angular dependencies of the SOTs and effective magnetic fields.

The study reveals a strong frequency dependence of the spin-orbit torques (SOTs), varying by more than an order of magnitude compared to static values, particularly near the ferromagnetic resonance frequency. A previously unobserved longitudinal component of the torque, capable of altering the magnetization length, is identified. Although its amplitude is an order of magnitude smaller than the transverse components, it exhibits a complex frequency dependence and is enhanced when the magnetization deviates from high-symmetry directions. The angular dependence of the torques is significantly influenced by the excitation frequency, leading to non-trivial variations near resonance. The ratio between damping-like and field-like components of the SOT shows large variations near the resonance, offering a way to tune the torques by adjusting the electric field frequency or pulse characteristics. Surprisingly, the effective magnetic fields, despite the strong frequency dependence of the torques and magnetization, remain almost frequency independent across the investigated range. This indicates the intrinsic timescale for charge-to-spin conversion is much faster than that for spin dynamics. The angular variation of the effective fields, similar to the SOTs, shows higher-order contributions in magnetization direction. The longitudinal component of the effective field is comparable to the transverse components. The study also suggests that the relation between effective field and SOT cannot be generalized by simply inserting frequency-dependent quantities, requiring a consistent computation using the derived equation. The effective fields are shown to be weakly frequency-dependent, suggesting their quasi-static values can be used to interpret terahertz dynamics, except in cases of strong interband transitions in the terahertz range.

The findings of this research challenge conventional approaches to modeling spin-orbit torques, which often treat charge carriers and ferromagnetic units separately. The observed contrasting frequency dependence of torques and effective fields highlights the need for a unified approach considering both charge and spin responses. The identified longitudinal component of the SOT suggests a change in magnetization length related to spin accumulation direction, potentially enhancing AC-unidirectional magnetoresistance (USMR). The ability to manipulate the ratio of damping-like to field-like torque components by varying frequency opens new avenues for controlling magnetization dynamics using pulsed excitations. This complements existing methods focused on current pulse amplitude and length, providing a refined control mechanism for spintronic devices.

This study provides a comprehensive investigation of spin-orbit torques and their associated effective magnetic fields from gigahertz to terahertz frequencies. The discovery of a longitudinal torque component and the contrasting frequency dependence between torques and effective fields offer crucial insights for advancing spintronics. Future research could explore the application of these findings to materials with strong interband transitions in the terahertz range or to antiferromagnetic materials for novel device applications.

The study is primarily theoretical and computational. While the model incorporates realistic material properties, experimental validation is needed to confirm the predicted frequency and angular dependencies of the torques and effective fields across a broader range of materials and conditions. Further investigation into the interplay between the longitudinal torque component and the unidirectional magnetoresistance effect would also be beneficial.