Nucleation and stability of skyrmions in three-dimensional chiral nanostructures

Curvature in magnetic systems substantially modifies magnetic energy landscapes via geometrically induced anisotropy and effective Dzyaloshinskii–Moriya interaction (DMI), enabling unconventional spin textures and magnetochiral effects in curved films, wires, and nanotubes. Magnetic skyrmions—topologically protected spin textures—are promising information carriers, and their stability is influenced by DMI and anisotropy. Prior studies suggest curvature can stabilize skyrmions in special geometries (e.g., spherical shells) and enable robust dynamics in nanotubes. This work investigates how curvature affects nucleation, stability, structure, and field-driven evolution of skyrmions in three-dimensional chiral nanostructures (nanotubes and circularly curved films) using micromagnetic simulations, aiming to clarify nucleation mechanisms, stability ranges, geometry dependence, and potential curvature-enabled control of skyrmions.

Theoretical and experimental efforts have established that curvature introduces effective anisotropy and DMI in magnetic systems, leading to novel static and dynamic phenomena (e.g., magnetochiral effects, topologically induced patterns, spin-Cherenkov effect). Skyrmions have been observed in non-centrosymmetric materials and multilayers, with stability governed by DMI and anisotropy. Curvature has been predicted and shown to stabilize skyrmions in spherical shells and near curvilinear defects, while magnetic nanotubes can host and transport skyrmions without annihilation under large currents. Recent multilayer experiments demonstrated room-temperature skyrmions at zero magnetic field with tunable size and density by nanostructure design, motivating the exploration of curvature as an additional stabilization and control parameter.

Magnetization dynamics were simulated by solving the Landau–Lifshitz–Gilbert (LLG) equation with effective field derived from exchange, DMI, magnetostatic, and Zeeman energies, implemented in MuMax3. Nanotubes and curved thin films were discretized with a cell size of 2 × 2 × 2 nm³. Material parameters correspond to FeGe: saturation magnetization Ms = 3.84 × 10^5 A/m, exchange constant A = 8.78 × 10^-12 J/m, interfacial DMI constant D = 1.58 × 10^-3 J/m², and Gilbert damping α = 0.2. Field-driven hysteresis processes were computed under out-of-plane magnetic fields (Hz). To quantify topology, the skyrmion topological number Q was computed. In a 2D plane, Q = (1/4π) ∬ (∂m/∂x × ∂m/∂y) · m dx dy. For curved geometries, magnetization on circular layers of fixed radius ri was mapped to an x–sarc plane (sarc is arc length from y=0), and Q was evaluated as Q = (1/4π) ∬ (∂m/∂x × ∂m/∂sarc) · m dx ds, with ds = √(dy² + dz²). Visualization included 3D vector plots and mz maps in the x–sarc plane for selected circular layers to reveal stripe and skyrmion patterns. Size and shape of skyrmions were characterized by fitting elliptical isolines (mz=0) to extract semimajor (a) and semiminor (b) axes across layers and fields. Geometry parameters explored included nanotube length L = 400 nm with varied outer radius R and inner radius r (thickness R−r), and curved films derived from a flat film (L=400 nm, W=250 nm, thickness=10 nm) bent to circular arcs with radius R spanning 40 nm to infinity.

Nanotubes: • Field-driven evolution for R=40 nm, r=20 nm shows three regimes: twisted helical stripe state (Q≈0.16 at Hz=0), skyrmion state (after stripes break at left/right sides around Hz≈0.25 T, Q≈−7.52 indicating ~8 skyrmions), and ferromagnetic state (skyrmion annihilation in two steps: Q jumps to ~−4 at Hz≈1.01 T as four skyrmions near boundaries annihilate; remaining four annihilate at Hz≈1.14 T). • Skyrmions nucleate via gradual breaking of helical stripes at left/right sides; unlike thin films, Q increases smoothly (no abrupt jump) during nucleation. The helical stripe periodicity—and thus skyrmion number—depends on nanotube thickness and radius: smaller R and thinner walls yield higher stripe density and more skyrmions; larger R and thicker walls yield fewer. • Hysteresis (R=80 nm, r=60 nm): from saturated −z state, small stripes nucleate near Hz≈−0.21 T, evolve into twisted helical stripes, then break at Hz≈0.16 T to form skyrmions (nucleation field Hn). Skyrmions persist from ~0.15 T to ~1.25–1.26 T, then annihilate (Han) to ferromagnetic state. • Size dependence (Fig. 2c): decreasing wall thickness lowers Hn and raises Han, widening the stability field range. Larger R tends to reduce Hn and increase Han. Examples: R=40 nm, thickness=30 nm: Hn=0.31 T, Han=1.05 T; R=80 nm, thickness=10 nm: Hn=0.10 T, Han=1.31 T. • Skyrmion structure in nanotubes is a circular truncated cone; cross-sections on circular layers are elliptical and size increases toward the inner layer. For R=80 nm, r=40 nm at Hz≈0.37 T: outer layer (ri=79 nm): a=14.7 nm, b=13.7 nm; middle (ri=61 nm): a=17.4 nm, b=16.1 nm; inner (ri=41 nm): a=24.8 nm, b=19.9 nm. Field dependence shows anisotropy of a and b decreases with increasing field; e.g., for ri=41 nm at H=0.4 T, a−b=4.9 nm, while for ri=79 nm at H=0.9 T, a−b≈0.2 nm. Curved thin films: • Hysteresis and susceptibility: decreasing bend radius R increases remanence and zero-field susceptibility χ(0); χ(0) rises rapidly for R<150 nm (e.g., R=70 nm gives χ(0)=10.4). • Three evolution types depending on R: – R>600 nm (including flat film, R=∞): Maze domains at low fields; distorted nanostripe shrinks; sudden Q jump at H≈0.62 T marks formation of skyrmions (three skyrmions observed), which persist to H≈1.41–1.42 T before annihilation. – 150 nm<R<600 nm (e.g., R=150 nm): Stripes form and transform without skyrmion formation; Q varies but reflects stripe ends rather than true skyrmions; at Hz≈0.56 T stripes vanish to ferromagnetic state. – R<150 nm (e.g., R=60 nm): Behavior akin to nanotubes; stripes twist and break at the left/right sides into skyrmions around Hz≈0.17 T; skyrmions stable up to ≈1.29–1.30 T, then annihilate. • Mechanism: Curvature modifies shape anisotropy and stripe morphology—large R yields maze domains; intermediate R straightens stripes (parallel to x), preventing skyrmion formation; small R induces twisted stripes that break into skyrmions. • Nucleation and annihilation fields vs R (Fig. 9): For R>600 nm, Hn slightly increases with R, Han nearly constant. For R<150 nm, Hn is reduced relative to large-R films; minimum Hn≈0.1 T occurs near R≈70 nm. • Skyrmion size vs R (Fig. 10): Larger R yields nearly circular skyrmions (a≈b); smaller R produces more elliptical and overall smaller skyrmions; both a and b decrease with increasing field.

The simulations show that curvature profoundly influences both the nucleation pathway and stability of skyrmions. In nanotubes and small-radius curved films, stripes twist along the curved surface and break at lateral regions, producing skyrmions without abrupt topological jumps, unlike flat films where skyrmions nucleate via sudden Q changes from stripe breakup. Geometric parameters—tube thickness and radius, film bending radius—control the stripe periodicity, skyrmion number, and the field window of stability. Thinner nanotubes and smaller bending radii reduce nucleation fields and increase annihilation fields, widening operational stability ranges. These curvature-induced effects suggest practical routes to engineer skyrmion stability and size/shape via three-dimensional geometries, potentially lowering the required external fields and enabling robust skyrmion states suitable for spintronic applications. The findings align with prior observations that curvature introduces effective anisotropy and DMI, offering a geometric handle to tailor skyrmion energetics.

This work reveals that in chiral magnetic nanotubes and circularly curved thin films, skyrmions nucleate from the breaking of helical stripes at lateral regions. In nanotubes, nucleation proceeds without abrupt changes in topological number, and thinner tubes exhibit lower nucleation fields, higher annihilation fields, and broader stability ranges. Skyrmions in nanotubes form circular truncated-cone structures with elliptical cross-sections whose size increases toward the inner surface. In curved films, three distinct hysteresis behaviors emerge: flat/large-R films behave like 2D thin films; intermediate R prevents skyrmion formation; small R promotes skyrmion nucleation via twisted-stripe breakup. Curvature thus provides a means to control skyrmion nucleation, stability, and morphology, potentially reducing the external field needed to stabilize skyrmions and enabling curvature-based design of skyrmion devices. Future work should explore parameter regimes and materials to realize room-temperature, zero-field-stable skyrmions controllable by curvature.

The study is based on micromagnetic simulations with specific material parameters corresponding to FeGe and does not include experimental validation. Thermal fluctuations and temperature effects are not discussed, and only selected geometries (sizes, radii, thicknesses) were explored. The topological analysis uses layer-wise mapping in curved geometries, which may approximate the full three-dimensional topology. These factors may limit direct generalization to other materials or geometries without further investigation.