Hysteretic Excitation in Non-collinear Antiferromagnetic Spin-Torque Oscillators: A Terminal Velocity Motion Perspective

TL;DR

This work develops a Poisson-bracket–based framework for non-collinear AFM spin-torque oscillators, revealing an infinite degeneracy of rigid-body precession states and a TVM description that reduces complex multi-sublattice dynamics to center-of-mass and relative-motion coordinates. The theory shows fast SOT-driven transients collapse onto M-aligned RBUT/RBP states, while a slow out-of-plane anisotropy–driven RM oscillation decays toward a TVM steady state with a light exchange–origin inertia $m_{ ext{eff}}=(3A_{ ext{ex}})^{-1}$ and terminal velocity $rac{dΦ_c}{dt}=a_{J0}/α$. An out-of-plane anisotropy lifts degeneracy, yielding slow RM dynamics around the RBUT states and causing self-resonant RM bursts that explain rigidity breakdown at sub-critical currents; this mechanism is supported by macrospin and TVM simulations. The TVM framework accurately predicts threshold currents, hysteretic excitation, and the driven frequency across the full current range and provides a scalable coarse-graining path to massive lattices, enabling predictive design of NC-AFM STOs for sub-THz and neuromorphic applications.

Abstract

We present a theoretical framework for non-collinear antiferromagnetic spin torque oscillators (NC-AFM STO) by unifying spin dynamics under the Poisson Bracket formalism. Shifting from traditional torque-based descriptions to an operational symmetry perspective, we develop two complementary viewpoints: a vector perspective identifying infinite degenerate Rigid Body Precession (RBP) states where exchange energy depends solely on the total magnetic momentum, and a particle perspective decomposing dynamics into Center-of-Mass (CM) translation and Relative Motion (RM) oscillation. Using time-dependent rotational and translational transformation techniques, we analytically resolve the rapid (~10 ps) transient evolution into a stable RBP state driven by SOT and damping. We demonstrate that the out-of-plane anisotropy (OPA) lifts the exchange degeneracy, triggering a long-term (~1 ns) oscillatory decay toward a steady state characterized by uniform spin z-components and a 120-degree inter-spin locking angle. This state is accurately governed by our Terminal Velocity Motion (TVM) model [arXiv:2305.14013], where exchange coupling transforms into kinetic energy with a light effective mass. The model precisely predicts SOT-driven transients, hysteretic excitation, and the dynamic phase diagram. Finally, we account for the sub-critical current regime mismatch by identifying a 'Rigid-Body Breaking' effect: a surge in effective friction caused by the self-resonance of RM variables induced by CM translation, mediated by the in-plane anisotropy (IPA).

Figures (21)

• Figure 1: (Color online) Schematic illustration of the NC-AFM spin-torque nano-oscillator (STNO) Zhao2021. (a) The bilayer setup comprises a heavy metal (HM) base and a NC-AFM free layer. An in-plane charge current $J$ along the $x$-direction generates a vertical spin current via the spin Hall effect, with the spin polarization oriented along the $z$-axis. (b) Configuration of the three-sublattice spins $\mathbf{m}_{1}, \mathbf{m}_{2},$ and $\mathbf{m}_3$. Notably, the three uniaxial crystalline anisotropic symmetric axes for the three sublattices' spins respectively are all defined to lie on the $(x, y)$ plane.
• Figure 2: (Color online) Schematics of a continuous transformative viewpoint of the Hamiltonian formulation (left panel) and its generalization to any other physical quantities (right panel). The black and purple curves denote the transformed trajectories driven by the generators $H$ and $G$, respectively. The green circles mark the state points on the two curves, which are designated by the parameters time $t$ and $a$, respectively. The bold red arrows indicate the infinitesimal displacements taken at the moment $t_0$ or at $a_0$ on the trajectories, respectively. Their respective projections $(d\phi_{i},dp_{i})$, shown by the thin red arrows, are $((\partial H/\partial p_{i})_{0}dt,(-\partial H/\partial\phi_{i})_{0}dt)$ and $((\partial G/\partial p_{i})_{0}da,(-\partial G/\partial\phi_{i})_{0}da)$, respectively.
• Figure 3: (Color online) Two types of perspectives in spin dynamics are presented by the Poisson brackets based continuous transformation (Panel(a)). The first (left panel) is the vector (geometry) perspective, where $\mathbf{M}$ is the total magnetic moment of the three exchange-coupled spins $\mathbf{m}_{i}$ ($i=1,2,3$) and $\mathbf{e}_{\mathrm{n}z}$ is its unit vector. The second (right panel) is the classical particle perspective, with the $x$ axis indicating the arrangement direction of the three non-linear strings coupled particles (all having the same elastic coefficient $A_{\mathrm{ex}}$) and the phase plane $\phi_{\mathrm{n}}-p_{\mathrm{n}}$ (normal to the $x$ axis) being used to designate the states of the particles. Panels (b) and (c) schematically present two examples of degenerate RBP (RBUT) states under the two perspectives, specifically for the absence of anisotropy ($k=0$). Both states share the same constant non-zero total moment $M_{\mathrm{n}z}$ (where $\dot{P}_{\mathrm{n}c}=0$ and $\dot{\Phi}_{\mathrm{n}c}$ is a non-zero constant), and $M_{\mathrm{n}x(y)}= 0$. However, they feature different spin configurations: State I in Panel (b) has identical $z$ components of the spins and equal phase angle differences ($\Phi_{-\mathrm{n}12}=\Phi_{-\mathrm{n}23}=2\pi/3$). State II in Panel (c) has non-identical $z$ components and non-equal phase angle differences ($\Phi_{-\mathrm{n}12}\neq\Phi_{-\mathrm{n}23}$). The terms RBP and RBUT signify a rigid body motion (precession/translation) where the angles between spins are static ($\dot{\theta}_{12(23)}=0$), and the RM variables are static ($\dot{\Phi}_{-\mathrm{n}12(23)}=0$ and $\dot{P}_{-\mathrm{n}12(23)}=0$). Panels (d) and (e) display two instances of $M_{\mathrm{n}z}$-degenerate RBP states where the total moment is tilted, resulting in $M_{x(y)}\neq0$. The difference between them lies in the spin trajectories: Panel (d) shows a state where at least one of the spins' trajectories enclosesthe$z$axis. This motion is a mixed state, comprising the RBUT and the RM's EO around this RBUT state, characterized by dynamic RM variables ($\dot{\Phi}_{-12(23)}\neq0$ and $\dot{P}_{-12(23)}\neq0$); while, Panel (e) shows a state where no spin's trajectory encloses the $z$ axis. This is a pure RM's EO around one of the stationary RBUT states, characterized by a static CM momentum ($\dot{P}_{\mathrm{c}}=0$) and zero time-averaged angular velocity ($\langle\dot{\Phi}_{\mathrm{c}}\rangle_{T}=0$). Notably, the red double arrow in the right sub-panel of (e) signifies a tiny oscillatory amplitude of $\Phi_\mathrm{c}$ around some point, which results in $\langle\dot{\Phi}_\mathrm{c}\rangle_{T} = 0$. Schematics (f) and (g) present the phase portraits in the CM and RM phase spaces for the two cases introduced in Panels (d) and (e), respectively. In the CM space, the red horizontal line indicates the CM's uniform translation, while the red double arrows signify its oscillation around an equilibrium point along the $\Phi_{\mathrm{c}}$ axis. In the RM space, the light transparent yellow area marks the set of RBUT states (or RBP states with $M_{x(y)}= 0$) that are degenerate to $M_{z}$ ($P_{\mathrm{c}}$). Furthermore, the green circle enclosing one of these RBUT states (State I or II) indicates the trajectory of the RM's elastic oscillation (EO).
• Figure 4: (Color online) RBP state relaxation to stability via SOT and damping cooperation. Macrospin simulated snapshots show the RBP state relaxation to stability, driven by the cooperative action of the SOT and damping. Panel (a) Initial State: The system starts with the total magnetization direction $\mathbf{e}_{\mathrm{n}z}$misaligned with the spin polarization vector $\mathbf{p}$ ($\mathbf{e}_{\text{n}z} \neq \mathbf{p}$), also characterized as an RBUT state with an EO. Panel (b) Final State: The system evolves into one of the stable $M_{z}$-degenerate RBP (RBUT) states with the total momentum's direction aligning with the polarization ($\mathbf{e}_{\text{n}z} = \mathbf{p}$). Parameters: $a_{J0} = 0.01$ ($J =1.8962\times 10^{-9}\text{A/cm}^2$), $\alpha=0.01$, and $k = 0$. More details about such a relaxation process are presented in the supplementary animation in the attachment (see 'SOT-Damp eliminated RM.mp4').
• Figure 5: (Color online) Snapshots of three selected degenerate SOT-driven RBP (RBUT) states. These macrospin simulation snapshots show three degenerate RBP (RBUT) states for $a_{J0}=0.01$ ($J=1.8962\times 10^{-9}\mathrm{A}/\mathrm{cm}^2$), $\alpha=0.01$, and $k=0$. The spin-polarization vector $\mathbf{p}$ points toward the $z$ axis, and the colors on the arrows indicate the $m_{z}$ values of the three spins. Panels (a)-(c) illustrate the three distinct states: (a) the state where $m_{zi}=-1/3$ ($\Phi_{-12(23)}=2\pi/3$); (b) the state with $m_{z1}=-1$ and $m_{z2}=-m_{z3}$ ($\Phi_{-23} = \pi$); and (c) the state with $m_{z1}\neq m_{z2}\neq m_{z3}$. More details about the SOT-driven RBP (RBUT) states are presented in the supplementary animation in the attachment (see 'degenerate state 120 degree.mp4', 'degenerate state 180 degree .mp4', and 'degenerate state other.mp4').