The study of magnetism in curved geometries is a burgeoning field in modern magnetism research. Curvature significantly impacts the statics and dynamics of magnetic textures, potentially leading to unconventional spin textures with unique physical properties stemming from the interplay of geometry, topology, and chirality. Curvature introduces two additional energies: geometrically induced anisotropy energy and effective Dzyaloshinskii-Moriya interaction (DMI) energy. These energies drive novel phenomena in three-dimensional curved magnetic structures, including magnetochiral effects, topologically induced magnetic patterns, and the Spin-Cherenkov effect of spin waves. Magnetic skyrmions, topologically protected spin textures with twisted magnetic moment structures, are promising data carriers in spintronics. Their stability is influenced by DMI and anisotropy energy. The interaction between skyrmions and curvature is thus a crucial area of research. Previous studies suggested skyrmion stabilization by curvature in spherical shells or when the radius is comparable to the size of a curvilinear defect. More recent work demonstrated skyrmion movement in magnetic nanotubes under high current density without annihilation. This current work builds upon these findings by introducing curvature into chiral magnetic materials to investigate magnetization evolution in three-dimensional nanostructures, employing micromagnetic simulations to characterize skyrmion nucleation, annihilation, and structure as a function of nanostructure size. The simulated three-dimensional structures are experimentally feasible, having been fabricated using various chemical and physical methods.

The literature review extensively cites previous research on magnetism in curved geometries, highlighting the impact of curvature on magnetic textures and the emergence of unconventional spin textures. It summarizes the influence of curvature-induced energies (geometrically induced anisotropy and effective DMI) on the observed phenomena. The review also covers existing literature on magnetic skyrmions, their topological protection, applications in spintronics, and the influence of DMI and anisotropy on their stability. Previous studies on skyrmion stabilization by curvature in specific geometries (spherical shells, curvilinear defects) are discussed, along with the most recent research on skyrmion dynamics in nanotubes. The review establishes a strong foundation for the current research by contextualizing the study within the existing body of knowledge on skyrmions and curved geometries.

The researchers utilized micromagnetic simulations to model the magnetization dynamics in chiral nanostructures. The Landau-Lifshitz-Gilbert (LLG) equation, which describes the magnetization dynamics, was solved using the MuMax3 code. The effective magnetic field in the LLG equation incorporates exchange, DMI, magnetostatic, and Zeeman energy terms. The simulations used material parameters corresponding to FeGe (saturation magnetization Ms, exchange constant A, DMI constant D, and Gilbert damping constant α). Nanotubes were discretized into small cells (2 × 2 × 2 nm³). The topological number (Q), a measure of the magnetization configuration, was calculated using different formulations depending on the geometry (2D plane for thin films and a modified approach for curved surfaces, considering the curvature). To visualize the magnetic states, 3D representations along with mz-images in 2D planes were used; data from circular layers with a fixed radius was selected for the 2D representations. The x-axis and arc length (sarc) formed the coordinates for these 2D plots.

The study revealed key findings on skyrmion behavior in both nanotubes and curved thin films. **Nanotubes:** * Skyrmion formation in nanotubes occurs through the breaking of helical stripes on the nanotube's sides. This process does not involve an abrupt change in the topological number, unlike in thin films. * Skyrmions exist over a wide range of magnetic fields; thinner nanotubes exhibit a larger field range for skyrmion stability. * The skyrmion configuration in nanotubes differs from thin films, showing an elliptical shape with increasing size and deformation from the outer to inner circular layers. The skyrmion number is determined by the periodicity of helical stripes which is related to the nanotube thickness and radius. * The nucleation field decreases, while the annihilation field increases with decreasing nanotube thickness. This leads to a larger field range for skyrmion existence in thinner nanotubes. Larger nanotube radius generally means smaller nucleation and larger annihilation fields. **Curved Thin Films:** * Three distinct hysteresis processes were observed depending on the film's curvature radius (R): * **R > 600 nm:** Behavior similar to 2D thin films, involving maze domains transitioning to skyrmions. * **150 nm < R < 600 nm:** No skyrmion formation, with a gradual transition from uniformly magnetized states to stripe domains. * **R < 150 nm:** Similar to nanotubes, skyrmions form through helical stripe breaking. * The susceptibility at H = 0 increases rapidly with decreasing R for R < 150 nm. * The nucleation field generally increases with increasing R (R > 600 nm), while the annihilation field remains relatively unchanged. The nucleation field is smaller for R < 150 nm, reaching a minimum around R ≈ 70 nm. * The skyrmion size in curved films is dependent on the curvature radius (R), with smaller radii resulting in smaller skyrmions. For larger R the skyrmion is nearly circular but for smaller R the shape becomes more elliptical.

The findings highlight the significant role of curvature in shaping skyrmion behavior in three-dimensional chiral nanostructures. The gradual skyrmion formation in nanotubes without abrupt topological number changes, contrasted with thin film behavior, points to distinct nucleation mechanisms governed by the curved geometry. The three different hysteresis processes observed in curved thin films demonstrate the strong influence of curvature on the magnetization evolution, transitioning from 2D-like behavior at large radii to nanotube-like behavior at small radii. The size and shape of skyrmions are strongly affected by the curvature, demonstrating tunability. These results suggest that curvature is a viable method to tune and control skyrmion stability. The observed size dependence opens avenues for engineering skyrmion properties by controlling nanostructure dimensions. This work provides valuable insights into the interplay of curvature, chirality, and skyrmion stability, potentially guiding the design of novel spintronic devices.

This research investigated skyrmion behavior in three-dimensional chiral nanostructures, revealing distinct nucleation mechanisms and stability characteristics in nanotubes and curved thin films. The findings demonstrate the significant impact of curvature on skyrmion formation, stability, and size. The observation of three different hysteresis processes highlights the tunability of skyrmion properties through curvature control. This work suggests that manipulating curvature offers a promising pathway for stabilizing skyrmions at lower magnetic fields, potentially paving the way for room-temperature skyrmion applications. Future research could explore the optimization of material parameters and curvature to achieve zero-field room temperature skyrmion stability.

The study primarily relies on micromagnetic simulations, which have inherent limitations in precisely representing real material properties and effects. While the chosen material parameters correspond to FeGe, variations in real material properties might influence the observed skyrmion behavior. The simulations consider specific geometrical parameters, and a more extensive parameter space exploration could offer a more complete understanding of skyrmion behavior under diverse conditions. The model might not capture all possible complex interactions in real three-dimensional nanostructures.