Hysteretic excitation in non-collinear antiferromagnetic spin-torque oscillators: A terminal velocity motion perspective

We present a novel theoretical framework for non-collinear anti-ferromagnetic spin torque oscillators by unifying spin dynamics under the Poisson bracket formalism. By shifting from traditional torque-based descriptions to a continuous operational symmetry perspective, we develop two complementary viewpoints: a vector perspective that identifies infinite degenerate Rigid Body Precessional (RBP) states—arising from the fact that exchange energy is solely a function of the Rigid Body Rotational Transformation (RBRT) generator (i.e., total magnetic momentum)—and a particle perspective that decomposes the dynamics into Center-of-Mass (CM) translation and Relative Motion (RM) oscillation. Utilizing time-dependent RBRT and rigid body uniform translational transformation techniques, we analytically resolve the rapid (∼10 ps) transient evolution into a stable RBP state driven by spin–orbit torques (SOTs) and damping. We further demonstrate that the out-of-plane anisotropy component of the uniaxial crystalline anisotropy lifts the exchange degeneracy, triggering a long-term (∼1 ns) oscillatory decay toward a finalized steady state s, characterized by uniform spin z-components and a 120° inter-spin locking angle. The dynamics of this state are accurately governed by our proposed terminal velocity motion model for CM variables [H.-H. Chen et al., arXiv:2305.11020 (2023)], in which exchange coupling is effectively transformed into kinetic energy with a light effective mass, leading to a transient excitation time of approximately 10 ps—nearly two orders of magnitude shorter than that of typical ferromagnetic systems (∼1–2 ns). This model precisely predicts SOT-driven transients, hysteretic excitation process, and the dynamic phase diagram across the full current range. Finally, we account for the theoretical mismatch in the sub-critical current regime by identifying a “rigid-body breaking” effect: a surge in effective friction caused by the self-resonance of RM variables induced by the translation of the CM variable, as mediated by the in-plane anisotropy component of the uniaxial crystalline anisotropy.