RKKY-like interactions between two magnetic skyrmions

Technical Summary: RKKY-like Interactions between Two Magnetic Skyrmions

Problem Statement
While magnetic skyrmions are recognized as promising candidates for spintronic and neuromorphic computing applications, the fundamental nature and microscopic origins of skyrmion-skyrmion interactions remain poorly understood. It is established that isolated skyrmions in vertically magnetized chiral films typically repel one another. However, recent observations indicate that attractive interactions can emerge under specific conditions, such as tilted external fields, thermal frustration, or varying sample thickness. Despite these observations, a clear theoretical framework explaining the conditions for attraction versus repulsion, and the underlying physics governing these interactions, has been lacking.

Methodology
The authors investigate skyrmion-skyrmion interactions in chiral magnetic films using a combination of numerical simulations and analytical modeling.

• Model: The system is modeled as a chiral magnetic film of thickness $d$ in the $xy$-plane. The total magnetic energy includes exchange, Dzyaloshinskii–Moriya interaction (DMI), magnetic anisotropy (controlled by either an external field $H$ or crystalline anisotropy $K$), dipolar energy, and Zeeman energy.
• Dynamics: Magnetization dynamics are governed by the Landau–Lifshitz–Gilbert (LLG) equation.
• Simulation: The authors use the Mumax3 package to numerically solve the LLG equation on films of size $600 \text{ nm} \times 600 \text{ nm} \times 0.5 \text{ nm}$ with periodic boundary conditions. They fix one skyrmion at the origin and vary the position of a second skyrmion to map the potential energy surface $E(x, y)$.
• Analytical Approach: To understand the origin of observed phenomena, the authors derive asymptotic solutions for the stable spin texture equations (Eqs. 2 and 3) for in-plane skyrmions, analyzing the behavior of the spin angles $\Theta$ and $\Phi$ far from the skyrmion core.

Key Contributions and Results

1. Discovery of RKKY-like Oscillatory Interactions:
   The study reveals that interactions between skyrmions in in-plane magnetized chiral films (induced by tilted fields or anisotropy) are intrinsically Ruderman-Kittel-Kasuya-Yosida (RKKY)-type. Unlike the monotonic repulsion seen in perpendicular films, these interactions oscillate between attraction and repulsion with a well-defined period ($\lambda \approx 90\text{--}91 \text{ nm}$).

   • Anisotropy: The interaction is highly anisotropic. Along the axis perpendicular to the skyrmion's center spin direction, the energy decreases monotonically (repulsion). Along the axis parallel to the center spin, the energy profile exhibits multiple minima and maxima, indicating alternating attractive and repulsive forces.

2. Mechanism: The "Wavy Tail" Structure:
   The authors identify the microscopic origin of this oscillation as an intrinsic, periodic "wavy tail" in the spin texture of in-plane skyrmions.

   • Analytical Proof: Asymptotic analysis of the spin texture equations shows that the azimuthal angle $\Phi$ of the magnetization varies linearly with distance, creating a periodic structure with a period $\lambda = 4\pi A / D$. This period depends only on the exchange stiffness ($A$) and DMI strength ($D$), making it robust against variations in field strength or anisotropy constants.
   • Interaction Logic:
     • Attraction: When two skyrmions are separated by an integer multiple of the period ($n\lambda$) along the oscillatory axis, their wavy tails are in phase. They can share the same tail structure, reducing the total formation energy, analogous to covalent bonding in molecules.
     • Repulsion: When separated by half-integer multiples ($(n+1/2)\lambda$), the tails are out of phase. Connecting them requires significant deformation of the spin structure, increasing the total energy and resulting in a repulsive force.

3. Universality and Phase Diagrams:
   The RKKY-like behavior is shown to be universal for all tilted skyrmions, regardless of whether the tilt is caused by an external magnetic field or crystalline magnetic anisotropy.

   • Phase Diagrams: The authors construct phase diagrams in the parameter space of anisotropy strength and tilt angle. They identify a critical tilt angle ($\theta > 23^\circ$) required for the emergence of the RKKY-like interaction. Below this angle, interactions remain purely repulsive or skyrmions become unstable. The results hold for both asymmetric (field-controlled) and symmetric (crystalline) anisotropy, as well as for both bulk and interfacial DMI.

4. Skyrmion Molecules:
   The study demonstrates that two skyrmions bound by this attractive interaction can form a "bi-skyrmion molecule." Simulations show that when a spin-orbit torque is applied to only one skyrmion in the pair, the entire molecule moves as a single rigid object, maintaining the bond length, provided the separation corresponds to an energy minimum.

Significance
The paper claims to provide a new physical principle for understanding and engineering skyrmion interactions. By revealing that skyrmion-skyrmion interactions are not merely repulsive but can be tuned to be oscillatory and attractive via the intrinsic wavy tail structure, the findings offer a mechanism for:

• Designing "skyrmion molecules" and complex skyrmion superstructures.
• Controlling the collective behavior of multiple skyrmions in racetrack and logic devices.
• Introducing an additional degree of freedom (separation distance relative to the oscillation period) for manipulating skyrmion dynamics in spintronics and neuromorphic computing applications.

The authors emphasize that these findings are based on fundamental physics derived from the spin texture equations and confirmed by numerical simulations, establishing a universal framework for tilted skyrmion interactions.