Geometric symmetry and size-dependent skyrmion phase transitions in magnetic nanostructures

Abstract: We investigate the interplay of geometric symmetry, size, and external magnetic fields in regulating individual skyrmion states within magnetic nanostructures. By analyzing nanodisks, nanosquares, and nanorectangles, we demonstrate that rotational symmetry in nanodisks enables rich topological phase transitions, from ferromagnetic states to skyrmions, skyrmioniums, and multi-states, as their diameter increases. In contrast, square and rectangular structures exhibit suppressed topological complexity due to corner-induced demagnetization effects and reduced symmetries. Under perpendicular magnetic fields, nanodisks show field-driven transitions between skyrmionium and skyrmion states. By leveraging asymmetry, square and rectangular nanostructures stabilize skyrmions over a broader parameter range than nanodisks. These findings highlight geometric symmetry as a critical design parameter for tailoring skyrmion stability and functionality in spintronic applications such as multi-state memory and reconfigurable logic devices.

• The paper demonstrates that nanodisks with rotational symmetry enable a cascade of skyrmion phase transitions as diameter increases from 40 to over 70 nm.
• Micromagnetic simulations using the LLG equation capture the balance among exchange, DMI, Zeeman, and demagnetization energies defining phase boundaries.
• Field-tuned transitions between ferromagnetic, skyrmion, skyrmionium, and multiskyrmion states showcase practical design insights for advanced spintronic devices.

Geometric Symmetry and Size-Dependent Skyrmion Phase Transitions in Magnetic Nanostructures

Introduction

The paper "Geometric symmetry and size-dependent skyrmion phase transitions in magnetic nanostructures" (2605.17939) addresses the crucial role of geometric symmetry, size, and external magnetic fields in determining skyrmion stability and phase transitions within nanostructured magnetic materials. Skyrmions—topologically nontrivial spin textures—are of considerable interest for high-density spintronics due to their stability, scalability, and low power requirements. However, precision control and manipulation at the nanoscale are constrained by geometric and micromagnetic factors, with the interplay of symmetry, shape, and demagnetization effects being insufficiently characterized.

Theoretical Framework and Simulation Methodology

Micromagnetic simulations were performed using the LLG equation, discretized in Mumax3 with sub-nanometer grid sizes to ensure convergence. The Gibbs free energy model incorporated symmetric exchange, interfacial DMI, magnetic anisotropy, Zeeman, and demagnetization energies. Simulations focused on ground-state topology at $T = 4.2~\text{K}$, minimizing thermal activation effects to isolate deterministic phase boundaries.

Parameter sets ($M_s = 914$~kA/m, $A = 11.2$~pJ/m, $D = 4.1$~mJ/m$^2$, $K_u = 6$~MJ/m$^3$, $\alpha = 0.1$) were selected to represent ideal multilayer configurations, recognizing that experimental implementations may reach lower values, yet the qualitative phase behaviors remain robust. Both geometric and magnetic boundary conditions were explicitly implemented, modeling nanodisks, squares, and rectangles of 1~nm thickness, varying lateral dimensions up to 200~nm.

Skyrmion Phase Diversity: Symmetry and Size Effects

Simulation results demonstrate that rotational symmetry in nanodisks enables a cascade of topological phase transitions as diameter increases, enabling states ranging from ferromagnetic to skyrmion, skyrmionium, and multiskyrmion configurations.

Figure 1: Size-dependent evolution of magnetic topological states in nanodisks, squares, and rectangles; disk symmetry enables multistate transitions, squares and rectangles are restricted by demagnetizing fields.

In disks, skyrmions and skyrmioniums interchange between 40–80~nm diameter; multistate regions emerge above 70~nm. Enhanced rotational symmetry ($C_N$) suppresses distortion and demagnetizing perturbations, allowing stable high-order defects. Conversely, squares and rectangles are restricted to single skyrmion states at sharply defined size ranges; corner-induced demagnetization fields disrupt order, limiting complexity.

The aspect ratio (1.5:1 in rectangles) further reinforces shape anisotropy, aligning spins along the long axis and reducing DMI influence. The suppression of complex phases in low-symmetry geometries is quantitatively linked to demagnetization factors and local energy density distributions.

Figure 2: Spatial maps of magnetic energy and demagnetization energy density reveal symmetry-driven energy concentration and corner pinning effects.

Field-Driven Topological Transitions

Phase diagrams as a function of applied perpendicular field $B$ and lateral size $M_s = 914$0 reveal field-tunable transitions, especially pronounced in nanodisks.

Figure 3: Nanodisk topological phase diagram: field and size-dependent transitions between ferromagnetic, skyrmion, skyrmionium, and multiskyrmion states.

Four primary phases are resolved: ferromagnetic for large fields/small diameters, skyrmionium/skyrmion transitions induced by critical fields ($M_s = 914$1–$M_s = 914$2), stable skyrmion domains at moderate sizes and fields, and multistate regimes in larger disks under weak fields. Zeeman energy competes with DMI and exchange; once $M_s = 914$3 exceeds $M_s = 914$4, skyrmionium collapses to skyrmion, signaled by stepwise jumps in topological charge ($M_s = 914$5).

Squares and rectangles remain fundamentally limited: only skyrmion and ferromagnetic states are stabilized, with phase boundaries scaling approximately linearly with $M_s = 914$6 and $M_s = 914$7, reflecting geometric and corner-dominated energy modulation.

Figure 4: Topological structure phase diagram for magnetic squares: field enhances single skyrmion stability range but restricts complex defect formation.

Figure 5: Magnetic rectangles (aspect ratio 1.5:1): asymmetric shape broadens field-window for skyrmion stability but reduces high-order state complexity.

Energetic Analysis

Magnetic energy density calculations quantify symmetry-driven advantages for nanodisks. Disk structures exhibit higher total energy density, sustained by synergy among exchange, DMI, and anisotropy terms, forming closed vortex or skyrmionium states with substantial deviation from the easy axis.

Figure 6: Magnetic energy density versus size: disks maintain high density across wide range, squares and rectangles show lower energy due to demagnetization-induced suppression.

In the presence of a low external field ($M_s = 914$8), disk structures retain elevated energy density relative to squares and rectangles; the Zeeman term intensifies competition but symmetry in disks allows continued stabilization of vortex/skyrmion configurations.

Figure 7: Energy density as function of size under $M_s = 914$9: rotational symmetry sustains disk energy advantage, rectangles/squares benefit in energy efficiency for storage.

Fractional topological charge ($A = 11.2$0) observed in finite-sized magnets is attributed to incomplete spin coverage at boundaries; this effect has implications for quantifying defect robustness in realistic devices.

Implications and Future Directions

The findings provide a rigorous theoretical basis for designing skyrmion-based spintronic devices:

• Nanodisks enable multi-state memory and reconfigurable logic applications due to their support for diverse and switchable topological states; enhanced stability against thermal fluctuations is achieved through symmetry-driven energy barriers.
• Squares and rectangles deliver enhanced robustness and lower energy density, making them suited for binary storage and racetrack memory applications, where topological diversity is less critical but resistance to disturbance is paramount.
• The phase diagrams (size-field tunability) allow precise encoding and real-time switching between topological states, potentially realizing beyond-binary memory architectures and nanoscale field sensors.
• Nanofabrication tolerances (<10~nm) are crucial in device realization, as minor deviations can shift topological boundaries and alter operational stability.

From a theoretical standpoint, integrating finite-temperature effects remains necessary for direct experimental matching, especially in devices operating at room temperature. Further exploration of material parameters, layer engineering techniques, and asymmetric geometries may enable tailored energy landscapes and defect stability.

Conclusion

Geometric symmetry and lateral size are identified as central parameters governing skyrmion phase diversity and stability in magnetic nanostructures. Rotationally symmetric nanodisks support multistate transitions, while squares and rectangles are restricted by demagnetization fields and symmetry breaking. External fields tune transitions and operational windows, particularly broadening skyrmion stability in asymmetric rectangles. These results form a quantitative framework for functional device design in spintronics, demonstrating the importance of symmetry-controlled micromagnetic engineering.