Injection locking in DC-driven spintronic vortex oscillators via surface acoustic wave modulation

Spintronic technology offers promising solutions to the critical challenges posed by rapid advances in artificial intelligence, particularly in enhancing scalability and reducing power consumption.^1–3 While spintronics has already seen industrial implementation, a key focus of ongoing research is the development of more efficient methods for manipulating and electrically reading magnetization dynamics.

The spin-transfer torque oscillator (STO) is a fundamental component in spintronic circuits. It is an intrinsically non-linear device with a three-layered structure in which a non-magnetic spacer separates two ferromagnetic layers. STOs offer notable frequency tunability through the use of electrical currents, magnetic fields, ambient noise, or interactions with other STOs.^4–10 

STOs employing magnetic vortices have been demonstrated to generate signals in the sub-GHz range with high output powers^8,11–13 and narrow frequency linewidths.^14–16 This frequency range is typically associated with the vortex’s gyrotropic mode, corresponding to a rotation of the core centered around its equilibrium position. The gyrotropic mode can be excited by dc currents above a characteristic threshold, determined by the geometry and material parameters, via spin-transfer torque (STT).^17–19 Efficiently manipulating the gyrotropic dynamics of the magnetic vortex in STOs is of great interest due to the prominent applications it has to offer, such as non-volatile data storage^20,21 and spin wave generation.^22–24 Generating non-linear effects in magnetic vortices, such as high frequencies and vortex core (VC) polarity reversal, typically requires high input currents. This results in increased Joule heating and ohmic losses that degrade energy efficiency. In contrast, voltage-driven magnetization excitation techniques can be more effective because they use electric fields rather than currents, thereby minimizing Joule heating and improving energy efficiency.^25,26 One efficient way to excite magnetization dynamics using voltage gates is through hybrid magneto-electric methods.²⁵ This approach typically combines piezoelectric and magnetoelastic materials. When a voltage is applied, it induces mechanical deformation in the piezoelectric material, which interacts with the magnetoelastic material to generate magnetization dynamics through the Villari effect.^27–38 An interdigital transducer (IDT) on a piezoelectric substrate allows precise control of the mechanical deformations. The effective excitation of the gyrotropic mode of magnetic vortices by spin acoustic waves (SAWs) has been demonstrated both theoretically³⁷ and experimentally.³⁹ 

To implement large networks of STOs, it is essential to synchronize their frequencies globally since variations between devices can cause frequency discrepancies. This work theoretically demonstrates that SAWs can control the frequency of the DC current-driven gyrotropic mode in vortex STOs. Figure 1 illustrates the multi-layered system considered. The three-layer column stack comprises a polarizer, a spacer, and a free cobalt–iron–boron (CoFeB) layer in a vortex configuration. The disks are 125 nm in diameter with a 5 nm thick free layer. DC current is introduced into the vortex-based STO by means of metallic electrodes contacting the upper and lower ferromagnetic layers. These electrodes, which are only a few nanometers thick, are significantly smaller than the SAW wavelength, which is in the micrometer range. This size difference ensures the minimal disruption of SAW propagation, effectively preserving the strain effects on the magnetic vortex. Incorporating these electrodes improves electrical contact, which is critical for efficient SAW generation and effective vortex-SAW coupling. Our predictions, based on the simplified Thiele approach and validated by micromagnetic simulations, suggest that this method can achieve injection locking. This work provides an avenue for the efficient and low-power synchronization of large STO networks.

Here, we considered only the elastic tensor component $ϵ X X$ (longitudinal strain). The transverse strain components, $ϵ Y Y$ and $ϵ Z Z$⁠, were neglected for simplicity. This approximation assumes that the longitudinal strain plays the dominant role in the magnetoelastic interaction under the conditions considered. Parameter $B 1$ represents the first-order magnetoelastic coupling constant. In this manuscript, we define $ϵ X X(t)≡ A S A Wcos( ω S A Wt)$⁠, where $A S A W$ and $ω S A W$ are the amplitude and frequency of the SAW, respectively. This strain component acts similar to a unidirectional anisotropy along the x-direction.³⁹ This term can have two different effects on magnetic vortices. First, it can deform the core profile of a static vortex centered on the disk. Second, for vortices with displaced cores, it can drive gyrotropic motion. To displace the vortex core and break the spatial symmetry, thereby enhancing the magnetoelastic coupling,³⁷ we apply an in-plane external magnetic field at an angle of 45 $°$ to the x axis (see Fig. 1).

To achieve injection locking using the Thiele formalism, we numerically integrated the model outlined in Eq. (1), incorporating the STT term from Eq. (2), the alternating magnetoelastic coupling from Eq. (3), and the second-order expansion of the dyadic dissipation from Eq. (4). This model also includes the established exchange and magnetostatic interactions, an external magnetic field, and the Oersted field. The analysis was performed with a DC current of 3 mA, a SAW amplitude of $A S A W=60× 10 − 5$⁠, and a magnetic field of 5 mT applied at a $θ B= 45 °$ angle to the $x ^$ direction (for more details about the parameters, see Table I). Figure 2(a) shows a 2D plot of the power spectral density (PSD), illustrating the variation in the vortex STO’s gyrotropic frequency as a function of the SAW’s swept frequency. Here, $Δω=ω− ω g$ represents the deviation of the oscillator frequency $ω$ from its natural gyrotropic mode frequency $ω g$⁠, and $ΔΩ= ω S A W− ω g$ represents the deviation of the SAW frequency $ω S A W$ from $ω g$⁠. The plot highlights the locking bandwidth $ΔS$⁠, where frequency locking is achieved, and we also observe phase locking within this range. Figure 2(b) shows the locking bandwidth $ΔS$ as a function of the SAW amplitude, revealing a linear increase in $ΔS$ with the SAW amplitude within the studied range. These results confirm that nonlinear effects on vortex dynamics, such as vortex core deformation, enable injection locking in a current-driven vortex via SAW. Furthermore, the bandwidth of the injection lock can be tuned by adjusting the SAW amplitude. Importantly, numerical simulations show that without these nonlinear effects, the locking bandwidth $Δ$S approaches zero, underscoring the critical role of vortex deformation in facilitating injection locking in this system. These are the main findings of this manuscript.

To validate the predictions of the simplified Thiele model, we performed micromagnetic simulations using the full LLG equation. Unlike the Thiele approach which simplifies the system, micromagnetic simulations take into account all degrees of freedom, leading to variations in quantitative results especially in non-linear regimes. All parameters were consistent with those in the Thiele model (see Table I), except for the applied current $J$⁠, which was set to 7 mA. The simulations were performed with the Mumax3 software package⁴⁶ on the Aithericon platform.⁴⁷ 

Figure 3 shows the results obtained from the micromagnetic simulations demonstrating the injection locking behavior. Panels (a) and (b) show that the locking behavior observed in these simulations is consistent with the predictions of the Thiele model. A fixed angle of $θ B= 45 °$ and $A S A W=10× 10 − 5$ was used for these simulations. In addition, we investigated how the locking bandwidth depends on the amplitude of the magnetic field. Figure 3(c) shows that the locking bandwidth increases linearly with the magnetic field amplitude, with simulations performed at a fixed angle of $θ B= 45 °$ and $A S A W=25× 10 − 5$⁠. This increase in bandwidth is attributed to the field-induced deformation of the magnetic vortex, which enhances the coupling with the SAW. As reported by Koujok et al.,³⁷ the magnetic field breaks the symmetry of the vortex along the direction of the applied field, hence facilitating this coupling. To further validate these findings, we examined the dependence of the locking bandwidth on the applied field angle $θ B$⁠. Figure 3(d) shows that the maximum bandwidth occurs at an angle of $θ B= 45 °$⁠. In these simulations, we considered a fixed field $B e x t=5$ mT and $A S A W=10× 10 − 5$⁠.

To assess the impact of SAWs on the enhancement of signal properties under typical thermal conditions, we investigated their effect on the emitted signal of the vortex STO at a temperature of 300 K. Figure 4 shows the fast Fourier transform (FFT) of the temporal magnetization components $M x+i M y$⁠. In the absence of SAWs [Fig. 4(a), blue peak], the vortex STO’s signal has a relatively low amplitude at the gyrotropic frequency. When SAWs with $A S A W=1× 10 − 5$ are applied, a significant increase in the signal amplitude is observed [Fig. 4(b), red peak]. This result is consistent with the locking behavior.^48,49 Hence, injection locking via SAW enables efficient low-power linewidth narrowing, which is especially beneficial for applications requiring precise signal filtering or maintaining signal integrity over long distances.

In this study, we investigated the injection locking of DC current-driven vortex excitations by SAWs. Using the simplified Thiele formalism, we elucidated how vortex deformation induces non-linear behavior and identified the resulting locking bandwidth and its dependence on SAW amplitude. These results were validated by micromagnetic simulations using the full LLG equation, which confirmed the predicted behavior. Our simulations also revealed a linear relationship between the locking bandwidth and the applied magnetic field. In addition, analysis of the angle of the magnetic field further supported our understanding of the vortex deformation effects. At room temperature, we demonstrated that SAWs can significantly increase the oscillator signal amplitude while narrowing its bandwidth. These findings are critical for advancing experimental investigations and developing efficient low-power synchronization mechanisms for large-scale STO-based networks.

This work has been supported by the European union via the European Research Council within the Starting Grant No. 101042439 “CoSpiN” and by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—TRR 173-268565370” (Project B01). D.R.R. and G.F. were supported by Project No. 101070287—SWAN-on-chip—HORIZON-CL4-2021-DIGITAL-EMERGING-01, the project PRIN 2020LWPKH7 “The Italian factory of micromagnetic modelling and spintronics” and the project PRIN20222N9A73 “SKYrmion-based magnetic tunnel junction to design a temperature SENSor-SkySens,” funded by the Italian Ministry of University and Research (MUR) and by the PETASPIN Association (www.petaspin.com). D.R. also acknowledges funding from the project PE0000021, “Network 4 Energy Sustainable Transition—NEST,” funded by the European Union—NextGenerationEU, under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.3—Call for Tender No. 1561 dated 11.10.2022 of the Italian MUR (CUP C93C22005230007) and support of the project D.M. 10/08/2021 n. 1062 (PON Ricerca e Innovazione), funded by the Italian MUR. A.R. acknowledges financial support from CIP2022036.

The authors have no conflicts to disclose.

R. Moukhader: Formal analysis (equal); Investigation (equal); Software (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). D. R. Rodrigues: Formal analysis (equal); Funding acquisition (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal). A. Riveros: Formal analysis (equal); Investigation (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). A. Koujok: Software (equal); Validation (equal); Visualization (equal); Writing – review & editing (equal). G. Finocchio: Funding acquisition (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). P. Pirro: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Supervision (equal); Validation (equal); Writing – review & editing (equal). A. Hamadeh: Conceptualization (equal); Funding acquisition (equal); Investigation (equal); Methodology (equal); Software (equal); Supervision (equal); Validation (equal); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.