Eigenmodes of synthetic antiferromagnetic skyrmions

Technical Summary: Eigenmodes of Synthetic Antiferromagnetic Skyrmions

Problem Statement
Magnetic skyrmions are topologically non-trivial spin textures with potential for spintronic and magnonic applications, such as racetrack memory and logic gates. However, ferromagnetic (FM) skyrmions face limitations, including large minimum sizes due to stray fields and the skyrmion Hall effect, which causes deflection from current-driven trajectories. Synthetic antiferromagnets (SAFs), consisting of antiferromagnetically coupled ferromagnetic layers, offer a solution by reducing stray fields and canceling the skyrmion Hall effect. While the dynamic modes of single FM skyrmions and isolated SAF skyrmions in disk geometries have been studied, a systematic understanding of the collective eigenmodes in confined SAF skyrmion chains—specifically the interplay between geometric confinement, intralayer skyrmion-skyrmion coupling, and interlayer antiferromagnetic coupling—remains incomplete.

Methodology
The authors investigate the excitation modes of confined SAF skyrmions using micromagnetic simulations. The study employs two primary computational approaches:

1. Eigenvalue Method: Solving the linearized Landau-Lifshitz-Gilbert (LLG) equation without damping to obtain eigenvalues and eigenvectors. This method is numerically efficient and guarantees the identification of all eigenmodes regardless of excitation.
2. Ringdown Method: Simulating the full system dynamics with a small-amplitude external excitation (sinc or Gaussian pulses) and performing time-domain Fourier analysis to obtain power spectral density (PSD).

The simulations utilize the magnum.np solver with cubic discretization cells of $1 \times 1 \times 1 \text{ nm}^3$. Material parameters include exchange stiffness $A = 15 \text{ pJ/m}$, perpendicular anisotropy $K_u = 1 \text{ MJ/m}^3$, interfacial DMI $D = 3 \text{ mJ/m}^2$, and RKKY coupling $\sigma = -3 \times 10^{-4} \text{ J/m}^2$. The study covers various geometries: single skyrmions in square and rectangular confinement, unequal layer thicknesses, and chains containing 2, 5, and 25 skyrmions. Visualization of modes involves phase-amplitude mapping of the $m_z$ component and tracking skyrmion core trajectories via the first moment of topological charge density.

Key Contributions and Results

• Single Skyrmion in FM vs. SAF:

  • In a single FM layer, the authors recover the standard hierarchy: a low-frequency gyrotropic mode (CW or CCW depending on core polarity), a breathing mode, and a higher-frequency rotational mode.
  • In a SAF square geometry, the antiferromagnetic coupling creates a frustrated system where the preferred rotation directions of the two layers (due to opposite core polarities) compete. This results in two nearly degenerate gyrotropic modes where both layers rotate in the same sense, but with different amplitudes.
  • Modifying the geometry from square to rectangular lifts the degeneracy. Instead of circular gyrotropic motion, the system exhibits translational modes. In these modes, the two layers rotate in opposite senses (CW in one, CCW in the other), causing the x-components of their motion to cancel and resulting in a net linear translation of the SAF skyrmion core along the long axis of the rectangle.

• Effect of Layer Asymmetry:

  • In SAFs with unequal layer thicknesses (e.g., 2 nm top, 1 nm bottom), the near-degeneracy of the gyrotropic modes is lifted significantly. The mode frequency splits based on which rotation sense is preferred by the thicker layer, demonstrating that geometric asymmetry can tune the mode frequencies.

• Skyrmion Chains (Collective Modes):

  • For chains of 2 and 5 skyrmions, the authors identify collective translational and breathing modes with standing-wave-like spatial profiles.
  • Translational Modes: In chains, these modes exhibit in-phase or out-of-phase motion between neighboring skyrmions. The authors find that a standing-wave model ($A_m(n) = a \sin(k_m n)$) successfully describes horizontal translational modes but fails for vertical translational modes, which can exhibit uniform motion across the chain.
  • Breathing Modes: The SAF geometry supports both in-phase (top and bottom layers breathing together) and out-of-phase (layers breathing oppositely) breathing modes. The out-of-phase modes are unique to the SAF bilayer structure and have no direct analogue in single-layer FM systems.

• Signal Propagation:

  • In a chain of 25 SAF skyrmions, the authors demonstrate signal propagation along the chain using a Gaussian pulse excitation. The signal propagates with a velocity of approximately 300 m/s, which is about 15% faster than in comparable ferromagnetic chains (approx. 255 m/s). The propagation involves the excitation of the top-bottom in-phase breathing band.

Significance
The paper provides a systematic description of the collective dynamics of SAF skyrmions, highlighting how geometric confinement and interlayer coupling fundamentally alter the excitation spectrum compared to ferromagnetic systems. Key findings include the emergence of translational modes in rectangular geometries due to frustrated gyrotropic dynamics and the existence of distinct out-of-phase breathing modes. The demonstration of signal propagation with velocities comparable to or exceeding those in FM chains suggests that SAF skyrmion chains are viable candidates for information transport in magnonic devices, offering the additional benefits of reduced stray fields and the suppression of the skyrmion Hall effect. The work establishes SAF skyrmion chains as a controllable system for studying collective spin dynamics relevant to microwave, magnonic, and neuromorphic functionalities.